« back to all changes in this revision
Viewing changes to main.aux
- browse files at revision 230
- compare with another revision
- download diff
- download tarball
- view history from revision 230
- Committer: gerald.mwangi at gmx
- Date: 2018-03-08 13:59:24 UTC
- Revision ID: gerald.mwangi@gmx.de-20180308135924-2wdxf3pthzibt3s1
work on modern noether3
added
removed
main.aux
956
956
\newlabel{eq:scalarProdFunctionspace}{{3.4}{67}{Noether's First Theorem: A Modern Version}{equation.3.0.4}{}}
957
957
\newlabel{eq:scalarProdFunctionspace@cref}{{[equation][4][3]3.4}{67}}
958
958
\@writefile{toc}{\contentsline {section}{\numberline {3.1}The action of $\mathbb {G}$ on Functionals}{67}{section.3.1}}
959
\newlabel{eq:lieGroupActionEuclidSpace}{{3.5}{67}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.5}{}}
959
\newlabel{eq:lieGroupActionEuclidSpace}{{3.5}{67}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.5}{}}
960
960
\newlabel{eq:lieGroupActionEuclidSpace@cref}{{[equation][5][3]3.5}{67}}
961
\newlabel{eq:GOmegaElement}{{3.8}{68}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.8}{}}
961
\newlabel{eq:GOmegaElement}{{3.8}{68}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.8}{}}
962
962
\newlabel{eq:GOmegaElement@cref}{{[equation][8][3]3.8}{68}}
963
\newlabel{eq:phiGOmega}{{3.9}{68}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.9}{}}
963
\newlabel{eq:phiGOmega}{{3.9}{68}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.9}{}}
964
964
\newlabel{eq:phiGOmega@cref}{{[equation][9][3]3.9}{68}}
965
\newlabel{eq:Xbasis}{{3.12}{69}{Dual Basis}{equation.3.1.12}{}}
966
\newlabel{eq:Xbasis@cref}{{[equation][12][3]3.12}{69}}
967
\newlabel{eq:XStarBasis}{{3.13}{69}{Dual Basis}{equation.3.1.13}{}}
968
\newlabel{eq:XStarBasis@cref}{{[equation][13][3]3.13}{69}}
969
\newlabel{eq:dualBasis}{{3.14}{69}{Dual Basis}{equation.3.1.14}{}}
970
\newlabel{eq:dualBasis@cref}{{[equation][14][3]3.14}{69}}
971
\newlabel{eq:dualBasisProof1}{{3.16}{69}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.16}{}}
972
\newlabel{eq:dualBasisProof1@cref}{{[equation][16][3]3.16}{69}}
973
\newlabel{eq:dualBasisProof2}{{3.18}{69}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.18}{}}
974
\newlabel{eq:dualBasisProof2@cref}{{[equation][18][3]3.18}{69}}
975
\newlabel{eq:derivativeE}{{3.22}{70}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.22}{}}
976
\newlabel{eq:derivativeE@cref}{{[equation][22][3]3.22}{70}}
977
\newlabel{eq:flowSubdiffF}{{3.23}{70}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.23}{}}
978
\newlabel{eq:flowSubdiffF@cref}{{[equation][23][3]3.23}{70}}
979
\newlabel{eq:subdiffHi}{{3.25}{71}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.25}{}}
980
\newlabel{eq:subdiffHi@cref}{{[equation][25][3]3.25}{71}}
981
\newlabel{eq:flowSubdiffH}{{3.28}{71}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.28}{}}
982
\newlabel{eq:flowSubdiffH@cref}{{[equation][28][3]3.28}{71}}
983
\newlabel{eq:flowSubdiffE}{{3.30}{71}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.30}{}}
984
\newlabel{eq:flowSubdiffE@cref}{{[equation][30][3]3.30}{71}}
985
\newlabel{eq:lieAlgebraPhiGeneral}{{3.33}{72}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.33}{}}
986
\newlabel{eq:lieAlgebraPhiGeneral@cref}{{[equation][33][3]3.33}{72}}
987
\newlabel{eq:lieAlgebraPhi}{{3.36}{72}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.36}{}}
988
\newlabel{eq:lieAlgebraPhi@cref}{{[equation][36][3]3.36}{72}}
989
\newlabel{eq:flowSubdiffFphi}{{3.40}{73}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.40}{}}
990
\newlabel{eq:flowSubdiffFphi@cref}{{[equation][40][3]3.40}{73}}
991
\newlabel{eq:flowSubdiffHphi}{{3.41}{73}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.41}{}}
992
\newlabel{eq:flowSubdiffHphi@cref}{{[equation][41][3]3.41}{73}}
993
\newlabel{eq:flowSubdiffEPhi}{{3.43}{73}{The action of $\mathbb {G}$ on Functionals}{equation.3.1.43}{}}
994
\newlabel{eq:flowSubdiffEPhi@cref}{{[equation][43][3]3.43}{73}}
995
\@writefile{toc}{\contentsline {section}{\numberline {3.2}Embedding Geometrical Constraints into Prior Energies}{73}{section.3.2}}
996
\newlabel{eq:priorMinimmizers}{{3.46}{73}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.46}{}}
997
\newlabel{eq:priorMinimmizers@cref}{{[equation][46][3]3.46}{73}}
998
\newlabel{eq:priorMinimizersG}{{3.47}{73}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.47}{}}
999
\newlabel{eq:priorMinimizersG@cref}{{[equation][47][3]3.47}{73}}
965
\newlabel{eq:GPhi1dim}{{3.10}{68}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.10}{}}
966
\newlabel{eq:GPhi1dim@cref}{{[equation][10][3]3.10}{68}}
967
\newlabel{eq:GPhiAlgebra}{{3.11}{68}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.11}{}}
968
\newlabel{eq:GPhiAlgebra@cref}{{[equation][11][3]3.11}{68}}
969
\newlabel{eq:GOmegaAlgebra}{{3.14}{69}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.14}{}}
970
\newlabel{eq:GOmegaAlgebra@cref}{{[equation][14][3]3.14}{69}}
971
\newlabel{eq:Xbasis}{{3.18}{70}{Dual Basis}{equation.3.1.18}{}}
972
\newlabel{eq:Xbasis@cref}{{[equation][18][3]3.18}{70}}
973
\newlabel{eq:XStarBasis}{{3.19}{70}{Dual Basis}{equation.3.1.19}{}}
974
\newlabel{eq:XStarBasis@cref}{{[equation][19][3]3.19}{70}}
975
\newlabel{eq:dualBasis}{{3.20}{70}{Dual Basis}{equation.3.1.20}{}}
976
\newlabel{eq:dualBasis@cref}{{[equation][20][3]3.20}{70}}
977
\newlabel{eq:dualBasisProof1}{{3.22}{70}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.22}{}}
978
\newlabel{eq:dualBasisProof1@cref}{{[equation][22][3]3.22}{70}}
979
\newlabel{eq:dualBasisProof2}{{3.24}{70}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.24}{}}
980
\newlabel{eq:dualBasisProof2@cref}{{[equation][24][3]3.24}{70}}
981
\newlabel{eq:derivativeE}{{3.28}{71}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.28}{}}
982
\newlabel{eq:derivativeE@cref}{{[equation][28][3]3.28}{71}}
983
\newlabel{eq:flowSubdiffF}{{3.29}{71}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.29}{}}
984
\newlabel{eq:flowSubdiffF@cref}{{[equation][29][3]3.29}{71}}
985
\newlabel{eq:subdiffHi}{{3.31}{71}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.31}{}}
986
\newlabel{eq:subdiffHi@cref}{{[equation][31][3]3.31}{71}}
987
\newlabel{eq:flowSubdiffH}{{3.34}{72}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.34}{}}
988
\newlabel{eq:flowSubdiffH@cref}{{[equation][34][3]3.34}{72}}
989
\newlabel{eq:flowSubdiffE}{{3.36}{72}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.36}{}}
990
\newlabel{eq:flowSubdiffE@cref}{{[equation][36][3]3.36}{72}}
991
\newlabel{eq:lieAlgebraPhiGeneral}{{3.39}{72}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.39}{}}
992
\newlabel{eq:lieAlgebraPhiGeneral@cref}{{[equation][39][3]3.39}{72}}
993
\newlabel{eq:lieAlgebraPhi}{{3.42}{73}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.42}{}}
994
\newlabel{eq:lieAlgebraPhi@cref}{{[equation][42][3]3.42}{73}}
995
\newlabel{eq:flowSubdiffFphi}{{3.46}{73}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.46}{}}
996
\newlabel{eq:flowSubdiffFphi@cref}{{[equation][46][3]3.46}{73}}
997
\newlabel{eq:flowSubdiffHphi}{{3.47}{73}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.47}{}}
998
\newlabel{eq:flowSubdiffHphi@cref}{{[equation][47][3]3.47}{73}}
999
\newlabel{eq:flowSubdiffEPhi}{{3.49}{74}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.49}{}}
1000
\newlabel{eq:flowSubdiffEPhi@cref}{{[equation][49][3]3.49}{74}}
1001
\@writefile{toc}{\contentsline {section}{\numberline {3.2}Embedding Geometrical Constraints into Prior Energies}{74}{section.3.2}}
1002
\newlabel{eq:priorMinimmizers}{{3.52}{74}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.52}{}}
1003
\newlabel{eq:priorMinimmizers@cref}{{[equation][52][3]3.52}{74}}
1004
\newlabel{eq:priorMinimizersG}{{3.53}{74}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.53}{}}
1005
\newlabel{eq:priorMinimizersG@cref}{{[equation][53][3]3.53}{74}}
1000
1006
\citation{BigunBook,FerraroTransfInvLieGroup}
1001
\newlabel{eq:levelSetVF}{{3.48}{74}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.48}{}}
1002
\newlabel{eq:levelSetVF@cref}{{[equation][48][3]3.48}{74}}
1003
\newlabel{eq:linLevelSet}{{3.49}{74}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.49}{}}
1004
\newlabel{eq:linLevelSet@cref}{{[equation][49][3]3.49}{74}}
1005
\@writefile{brf}{\backcite{BigunBook}{{74}{3.2}{equation.3.2.49}}}
1006
\@writefile{brf}{\backcite{FerraroTransfInvLieGroup}{{74}{3.2}{equation.3.2.49}}}
1007
\newlabel{eq:diffEqHarmonicFunctions}{{3.50}{74}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.50}{}}
1008
\newlabel{eq:diffEqHarmonicFunctions@cref}{{[equation][50][3]3.50}{74}}
1009
\newlabel{eq:linearDomain}{{3.51}{75}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.51}{}}
1010
\newlabel{eq:linearDomain@cref}{{[equation][51][3]3.51}{75}}
1011
\newlabel{eq:generalLevelSet}{{3.52}{75}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.52}{}}
1012
\newlabel{eq:generalLevelSet@cref}{{[equation][52][3]3.52}{75}}
1013
\newlabel{eq:cauchyRiemann}{{3.53}{75}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.53}{}}
1014
\newlabel{eq:cauchyRiemann@cref}{{[equation][53][3]3.53}{75}}
1015
\newlabel{eq:minimizerSetInv}{{3.54}{75}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.54}{}}
1016
\newlabel{eq:minimizerSetInv@cref}{{[equation][54][3]3.54}{75}}
1017
\newlabel{eq:constPrior}{{3.55}{75}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.55}{}}
1018
\newlabel{eq:constPrior@cref}{{[equation][55][3]3.55}{75}}
1007
\newlabel{eq:levelSetVF}{{3.54}{75}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.54}{}}
1008
\newlabel{eq:levelSetVF@cref}{{[equation][54][3]3.54}{75}}
1009
\newlabel{eq:linLevelSet}{{3.55}{75}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.55}{}}
1010
\newlabel{eq:linLevelSet@cref}{{[equation][55][3]3.55}{75}}
1011
\@writefile{brf}{\backcite{BigunBook}{{75}{3.2}{equation.3.2.55}}}
1012
\@writefile{brf}{\backcite{FerraroTransfInvLieGroup}{{75}{3.2}{equation.3.2.55}}}
1013
\newlabel{eq:diffEqHarmonicFunctions}{{3.56}{75}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.56}{}}
1014
\newlabel{eq:diffEqHarmonicFunctions@cref}{{[equation][56][3]3.56}{75}}
1015
\newlabel{eq:linearDomain}{{3.57}{75}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.57}{}}
1016
\newlabel{eq:linearDomain@cref}{{[equation][57][3]3.57}{75}}
1017
\newlabel{eq:generalLevelSet}{{3.58}{76}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.58}{}}
1018
\newlabel{eq:generalLevelSet@cref}{{[equation][58][3]3.58}{76}}
1019
\newlabel{eq:cauchyRiemann}{{3.59}{76}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.59}{}}
1020
\newlabel{eq:cauchyRiemann@cref}{{[equation][59][3]3.59}{76}}
1021
\newlabel{eq:minimizerSetInv}{{3.60}{76}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.60}{}}
1022
\newlabel{eq:minimizerSetInv@cref}{{[equation][60][3]3.60}{76}}
1023
\newlabel{eq:constPrior}{{3.61}{76}{Embedding Geometrical Constraints into Prior Energies}{equation.3.2.61}{}}
1024
\newlabel{eq:constPrior@cref}{{[equation][61][3]3.61}{76}}
1019
1025
\citation{NoetherTheroemEng,MansfieldInvarCalc}
1020
1026
\citation{kuypers2005klassische}
1021
\@writefile{toc}{\contentsline {section}{\numberline {3.3}Noether's First Theorem: A Modern Version}{76}{section.3.3}}
1022
\newlabel{sec:noetherModern}{{3.3}{76}{Noether's First Theorem: A Modern Version}{section.3.3}{}}
1023
\newlabel{sec:noetherModern@cref}{{[section][3][3]3.3}{76}}
1024
\newlabel{eq:NoetherEnergyInt}{{3.57}{76}{Noether's First Theorem: A Modern Version}{equation.3.3.57}{}}
1025
\newlabel{eq:NoetherEnergyInt@cref}{{[equation][57][3]3.57}{76}}
1026
\newlabel{eq:NoetherEnergyInfTrans}{{3.58}{76}{Noether's First Theorem: A Modern Version}{equation.3.3.58}{}}
1027
\newlabel{eq:NoetherEnergyInfTrans@cref}{{[equation][58][3]3.58}{76}}
1028
\newlabel{eq:eulerLagrange2}{{3.59}{76}{Noether's First Theorem: A Modern Version}{equation.3.3.59}{}}
1029
\newlabel{eq:eulerLagrange2@cref}{{[equation][59][3]3.59}{76}}
1030
\@writefile{brf}{\backcite{NoetherTheroemEng}{{76}{3.3}{equation.3.3.59}}}
1031
\@writefile{brf}{\backcite{MansfieldInvarCalc}{{76}{3.3}{equation.3.3.59}}}
1032
\@writefile{brf}{\backcite{kuypers2005klassische}{{76}{3.3}{equation.3.3.59}}}
1033
\newlabel{eq:volumeElement2}{{3.60}{77}{Noether's First Theorem: A Modern Version}{equation.3.3.60}{}}
1034
\newlabel{eq:volumeElement2@cref}{{[equation][60][3]3.60}{77}}
1035
\newlabel{eq:volumeElement}{{3.61}{77}{Noether's First Theorem: A Modern Version}{equation.3.3.61}{}}
1036
\newlabel{eq:volumeElement@cref}{{[equation][61][3]3.61}{77}}
1037
\newlabel{eq:VeExpansion}{{3.62}{77}{Noether's First Theorem: A Modern Version}{equation.3.3.62}{}}
1038
\newlabel{eq:VeExpansion@cref}{{[equation][62][3]3.62}{77}}
1039
\newlabel{eq:noetherVariation2}{{3.63}{77}{Noether's First Theorem: A Modern Version}{equation.3.3.63}{}}
1040
\newlabel{eq:noetherVariation2@cref}{{[equation][63][3]3.63}{77}}
1041
\newlabel{eq:bendingBackground}{{3.64}{77}{Noether's First Theorem: A Modern Version}{equation.3.3.64}{}}
1042
\newlabel{eq:bendingBackground@cref}{{[equation][64][3]3.64}{77}}
1043
\newlabel{eq:noetherVariation}{{3.65}{77}{Noether's First Theorem: A Modern Version}{equation.3.3.65}{}}
1044
\newlabel{eq:noetherVariation@cref}{{[equation][65][3]3.65}{77}}
1045
\newlabel{eq:noetherPureIntensTrans}{{3.66}{77}{Noether's First Theorem: A Modern Version}{equation.3.3.66}{}}
1046
\newlabel{eq:noetherPureIntensTrans@cref}{{[equation][66][3]3.66}{77}}
1047
\newlabel{eq:divergenceFreeVectors}{{3.67}{77}{Noether's First Theorem: A Modern Version}{equation.3.3.67}{}}
1048
\newlabel{eq:divergenceFreeVectors@cref}{{[equation][67][3]3.67}{77}}
1049
\newlabel{eq:noetherDivergence}{{3.68}{78}{Noether's First Theorem: A Modern Version}{equation.3.3.68}{}}
1050
\newlabel{eq:noetherDivergence@cref}{{[equation][68][3]3.68}{78}}
1051
\newlabel{eq:NoetherEnergyInfTransSymm}{{3.69}{78}{Noether's First Theorem: A Modern Version}{equation.3.3.69}{}}
1052
\newlabel{eq:NoetherEnergyInfTransSymm@cref}{{[equation][69][3]3.69}{78}}
1053
\newlabel{eq:eulerLagrangeEq}{{3.70}{78}{Noether's First Theorem: A Modern Version}{equation.3.3.70}{}}
1054
\newlabel{eq:eulerLagrangeEq@cref}{{[equation][70][3]3.70}{78}}
1055
\newlabel{eq:noetherDivergenceCanonMomentum}{{3.71}{78}{Noether's First Theorem: A Modern Version}{equation.3.3.71}{}}
1056
\newlabel{eq:noetherDivergenceCanonMomentum@cref}{{[equation][71][3]3.71}{78}}
1057
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.1}Pure Spacial Symmetries}{78}{subsection.3.3.1}}
1058
\newlabel{sec:pureSpacialSymmetry}{{3.3.1}{78}{Pure Spacial Symmetries}{subsection.3.3.1}{}}
1059
\newlabel{sec:pureSpacialSymmetry@cref}{{[subsection][1][3,3]3.3.1}{78}}
1060
\newlabel{eq:pureSpacialSymmetry}{{3.72}{78}{Pure Spacial Symmetries}{equation.3.3.72}{}}
1061
\newlabel{eq:pureSpacialSymmetry@cref}{{[equation][72][3]3.72}{78}}
1062
\newlabel{eq:pureSpacialSymmetry2}{{3.73}{78}{Pure Spacial Symmetries}{equation.3.3.73}{}}
1063
\newlabel{eq:pureSpacialSymmetry2@cref}{{[equation][73][3]3.73}{78}}
1064
\newlabel{eq:pureSpacialSymmetryCanonMomentum}{{3.74}{78}{Pure Spacial Symmetries}{equation.3.3.74}{}}
1065
\newlabel{eq:pureSpacialSymmetryCanonMomentum@cref}{{[equation][74][3]3.74}{78}}
1027
\@writefile{toc}{\contentsline {section}{\numberline {3.3}Noether's First Theorem: A Modern Version}{77}{section.3.3}}
1028
\newlabel{sec:noetherModern}{{3.3}{77}{Noether's First Theorem: A Modern Version}{section.3.3}{}}
1029
\newlabel{sec:noetherModern@cref}{{[section][3][3]3.3}{77}}
1030
\newlabel{eq:NoetherEnergyInt}{{3.63}{77}{Noether's First Theorem: A Modern Version}{equation.3.3.63}{}}
1031
\newlabel{eq:NoetherEnergyInt@cref}{{[equation][63][3]3.63}{77}}
1032
\newlabel{eq:NoetherEnergyInfTrans}{{3.64}{77}{Noether's First Theorem: A Modern Version}{equation.3.3.64}{}}
1033
\newlabel{eq:NoetherEnergyInfTrans@cref}{{[equation][64][3]3.64}{77}}
1034
\newlabel{eq:eulerLagrange2}{{3.65}{77}{Noether's First Theorem: A Modern Version}{equation.3.3.65}{}}
1035
\newlabel{eq:eulerLagrange2@cref}{{[equation][65][3]3.65}{77}}
1036
\@writefile{brf}{\backcite{NoetherTheroemEng}{{77}{3.3}{equation.3.3.65}}}
1037
\@writefile{brf}{\backcite{MansfieldInvarCalc}{{77}{3.3}{equation.3.3.65}}}
1038
\@writefile{brf}{\backcite{kuypers2005klassische}{{77}{3.3}{equation.3.3.65}}}
1039
\newlabel{eq:volumeElement2}{{3.66}{77}{Noether's First Theorem: A Modern Version}{equation.3.3.66}{}}
1040
\newlabel{eq:volumeElement2@cref}{{[equation][66][3]3.66}{77}}
1041
\newlabel{eq:volumeElement}{{3.67}{78}{Noether's First Theorem: A Modern Version}{equation.3.3.67}{}}
1042
\newlabel{eq:volumeElement@cref}{{[equation][67][3]3.67}{78}}
1043
\newlabel{eq:VeExpansion}{{3.68}{78}{Noether's First Theorem: A Modern Version}{equation.3.3.68}{}}
1044
\newlabel{eq:VeExpansion@cref}{{[equation][68][3]3.68}{78}}
1045
\newlabel{eq:noetherVariation2}{{3.69}{78}{Noether's First Theorem: A Modern Version}{equation.3.3.69}{}}
1046
\newlabel{eq:noetherVariation2@cref}{{[equation][69][3]3.69}{78}}
1047
\newlabel{eq:bendingBackground}{{3.70}{78}{Noether's First Theorem: A Modern Version}{equation.3.3.70}{}}
1048
\newlabel{eq:bendingBackground@cref}{{[equation][70][3]3.70}{78}}
1049
\newlabel{eq:noetherVariation}{{3.71}{78}{Noether's First Theorem: A Modern Version}{equation.3.3.71}{}}
1050
\newlabel{eq:noetherVariation@cref}{{[equation][71][3]3.71}{78}}
1051
\newlabel{eq:noetherPureIntensTrans}{{3.72}{78}{Noether's First Theorem: A Modern Version}{equation.3.3.72}{}}
1052
\newlabel{eq:noetherPureIntensTrans@cref}{{[equation][72][3]3.72}{78}}
1053
\newlabel{eq:divergenceFreeVectors}{{3.73}{78}{Noether's First Theorem: A Modern Version}{equation.3.3.73}{}}
1054
\newlabel{eq:divergenceFreeVectors@cref}{{[equation][73][3]3.73}{78}}
1055
\newlabel{eq:noetherDivergence}{{3.74}{78}{Noether's First Theorem: A Modern Version}{equation.3.3.74}{}}
1056
\newlabel{eq:noetherDivergence@cref}{{[equation][74][3]3.74}{78}}
1057
\newlabel{eq:NoetherEnergyInfTransSymm}{{3.75}{79}{Noether's First Theorem: A Modern Version}{equation.3.3.75}{}}
1058
\newlabel{eq:NoetherEnergyInfTransSymm@cref}{{[equation][75][3]3.75}{79}}
1059
\newlabel{eq:eulerLagrangeEq}{{3.76}{79}{Noether's First Theorem: A Modern Version}{equation.3.3.76}{}}
1060
\newlabel{eq:eulerLagrangeEq@cref}{{[equation][76][3]3.76}{79}}
1061
\newlabel{eq:noetherDivergenceCanonMomentum}{{3.77}{79}{Noether's First Theorem: A Modern Version}{equation.3.3.77}{}}
1062
\newlabel{eq:noetherDivergenceCanonMomentum@cref}{{[equation][77][3]3.77}{79}}
1063
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3.1}Pure Spacial Symmetries}{79}{subsection.3.3.1}}
1064
\newlabel{sec:pureSpacialSymmetry}{{3.3.1}{79}{Pure Spacial Symmetries}{subsection.3.3.1}{}}
1065
\newlabel{sec:pureSpacialSymmetry@cref}{{[subsection][1][3,3]3.3.1}{79}}
1066
\newlabel{eq:pureSpacialSymmetry}{{3.78}{79}{Pure Spacial Symmetries}{equation.3.3.78}{}}
1067
\newlabel{eq:pureSpacialSymmetry@cref}{{[equation][78][3]3.78}{79}}
1068
\newlabel{eq:pureSpacialSymmetry2}{{3.79}{79}{Pure Spacial Symmetries}{equation.3.3.79}{}}
1069
\newlabel{eq:pureSpacialSymmetry2@cref}{{[equation][79][3]3.79}{79}}
1070
\newlabel{eq:pureSpacialSymmetryCanonMomentum}{{3.80}{79}{Pure Spacial Symmetries}{equation.3.3.80}{}}
1071
\newlabel{eq:pureSpacialSymmetryCanonMomentum@cref}{{[equation][80][3]3.80}{79}}
1066
1072
\citation{Bigun1987}
1067
\@writefile{toc}{\contentsline {chapter}{\numberline {4}Linearized Priors}{80}{chapter.4}}
1073
\@writefile{toc}{\contentsline {chapter}{\numberline {4}Linearized Priors}{81}{chapter.4}}
1068
1074
\@writefile{lof}{\addvspace {10\p@ }}
1069
1075
\@writefile{lot}{\addvspace {10\p@ }}
1070
1076
\@writefile{lol}{\addvspace {10\p@ }}
1071
1077
\@writefile{loa}{\addvspace {10\p@ }}
1072
\newlabel{chap:GeometricalPrior}{{4}{80}{Linearized Priors}{chapter.4}{}}
1073
\newlabel{chap:GeometricalPrior@cref}{{[chapter][4][]4}{80}}
1074
\@writefile{toc}{\contentsline {section}{\numberline {4.1}The Linear Structure Tensor}{80}{section.4.1}}
1075
\newlabel{eq:vectorTransAlgebra}{{4.1}{80}{The Linear Structure Tensor}{equation.4.1.1}{}}
1076
\newlabel{eq:vectorTransAlgebra@cref}{{[equation][1][4]4.1}{80}}
1077
\newlabel{eq:linearLevelSet}{{4.3}{80}{The Linear Structure Tensor}{equation.4.1.3}{}}
1078
\newlabel{eq:linearLevelSet@cref}{{[equation][3][4]4.3}{80}}
1079
\@writefile{brf}{\backcite{Bigun1987}{{80}{4.1}{equation.4.1.3}}}
1080
\newlabel{eq:StructTensLeastSquaresEnergy}{{4.4}{80}{The Linear Structure Tensor}{equation.4.1.4}{}}
1081
\newlabel{eq:StructTensLeastSquaresEnergy@cref}{{[equation][4][4]4.4}{80}}
1082
\newlabel{eq:structtensDef}{{4.5}{80}{The Linear Structure Tensor}{equation.4.1.5}{}}
1083
\newlabel{eq:structtensDef@cref}{{[equation][5][4]4.5}{80}}
1084
\newlabel{eq:structtensEigenvalues}{{4.6}{81}{The Linear Structure Tensor}{equation.4.1.6}{}}
1085
\newlabel{eq:structtensEigenvalues@cref}{{[equation][6][4]4.6}{81}}
1086
\newlabel{eq:structtensRot}{{4.7}{81}{The Linear Structure Tensor}{equation.4.1.7}{}}
1087
\newlabel{eq:structtensRot@cref}{{[equation][7][4]4.7}{81}}
1088
\newlabel{eq:commutatorStructens}{{4.9}{81}{The Linear Structure Tensor}{equation.4.1.9}{}}
1089
\newlabel{eq:commutatorStructens@cref}{{[equation][9][4]4.9}{81}}
1090
\newlabel{eq:changeEigenvectors}{{4.10}{81}{The Linear Structure Tensor}{equation.4.1.10}{}}
1091
\newlabel{eq:changeEigenvectors@cref}{{[equation][10][4]4.10}{81}}
1092
\newlabel{eq:changeStructtens}{{4.11}{81}{The Linear Structure Tensor}{equation.4.1.11}{}}
1093
\newlabel{eq:changeStructtens@cref}{{[equation][11][4]4.11}{81}}
1094
\@writefile{toc}{\contentsline {section}{\numberline {4.2}Structure Tensor Based Prior}{82}{section.4.2}}
1095
\newlabel{sec:structureTensPrior}{{4.2}{82}{Structure Tensor Based Prior}{section.4.2}{}}
1096
\newlabel{sec:structureTensPrior@cref}{{[section][2][4]4.2}{82}}
1097
\newlabel{eq:structureTensorRotation}{{4.12}{82}{Structure Tensor Based Prior}{equation.4.2.12}{}}
1098
\newlabel{eq:structureTensorRotation@cref}{{[equation][12][4]4.12}{82}}
1099
\newlabel{eq:structtensPrior}{{4.14}{82}{Structure Tensor Based Prior}{equation.4.2.14}{}}
1100
\newlabel{eq:structtensPrior@cref}{{[equation][14][4]4.14}{82}}
1101
\newlabel{eq:structtensPriorRot}{{4.15}{82}{Structure Tensor Based Prior}{equation.4.2.15}{}}
1102
\newlabel{eq:structtensPriorRot@cref}{{[equation][15][4]4.15}{82}}
1103
\newlabel{eq:traceTransf}{{4.16}{82}{Structure Tensor Based Prior}{equation.4.2.16}{}}
1104
\newlabel{eq:traceTransf@cref}{{[equation][16][4]4.16}{82}}
1078
\newlabel{chap:GeometricalPrior}{{4}{81}{Linearized Priors}{chapter.4}{}}
1079
\newlabel{chap:GeometricalPrior@cref}{{[chapter][4][]4}{81}}
1080
\@writefile{toc}{\contentsline {section}{\numberline {4.1}The Linear Structure Tensor}{81}{section.4.1}}
1081
\newlabel{eq:vectorTransAlgebra}{{4.1}{81}{The Linear Structure Tensor}{equation.4.1.1}{}}
1082
\newlabel{eq:vectorTransAlgebra@cref}{{[equation][1][4]4.1}{81}}
1083
\newlabel{eq:linearLevelSet}{{4.3}{81}{The Linear Structure Tensor}{equation.4.1.3}{}}
1084
\newlabel{eq:linearLevelSet@cref}{{[equation][3][4]4.3}{81}}
1085
\@writefile{brf}{\backcite{Bigun1987}{{81}{4.1}{equation.4.1.3}}}
1086
\newlabel{eq:StructTensLeastSquaresEnergy}{{4.4}{81}{The Linear Structure Tensor}{equation.4.1.4}{}}
1087
\newlabel{eq:StructTensLeastSquaresEnergy@cref}{{[equation][4][4]4.4}{81}}
1088
\newlabel{eq:structtensDef}{{4.5}{81}{The Linear Structure Tensor}{equation.4.1.5}{}}
1089
\newlabel{eq:structtensDef@cref}{{[equation][5][4]4.5}{81}}
1090
\newlabel{eq:structtensEigenvalues}{{4.6}{82}{The Linear Structure Tensor}{equation.4.1.6}{}}
1091
\newlabel{eq:structtensEigenvalues@cref}{{[equation][6][4]4.6}{82}}
1092
\newlabel{eq:structtensRot}{{4.7}{82}{The Linear Structure Tensor}{equation.4.1.7}{}}
1093
\newlabel{eq:structtensRot@cref}{{[equation][7][4]4.7}{82}}
1094
\newlabel{eq:commutatorStructens}{{4.9}{82}{The Linear Structure Tensor}{equation.4.1.9}{}}
1095
\newlabel{eq:commutatorStructens@cref}{{[equation][9][4]4.9}{82}}
1096
\newlabel{eq:changeEigenvectors}{{4.10}{82}{The Linear Structure Tensor}{equation.4.1.10}{}}
1097
\newlabel{eq:changeEigenvectors@cref}{{[equation][10][4]4.10}{82}}
1098
\newlabel{eq:changeStructtens}{{4.11}{82}{The Linear Structure Tensor}{equation.4.1.11}{}}
1099
\newlabel{eq:changeStructtens@cref}{{[equation][11][4]4.11}{82}}
1100
\@writefile{toc}{\contentsline {section}{\numberline {4.2}Structure Tensor Based Prior}{83}{section.4.2}}
1101
\newlabel{sec:structureTensPrior}{{4.2}{83}{Structure Tensor Based Prior}{section.4.2}{}}
1102
\newlabel{sec:structureTensPrior@cref}{{[section][2][4]4.2}{83}}
1103
\newlabel{eq:structureTensorRotation}{{4.12}{83}{Structure Tensor Based Prior}{equation.4.2.12}{}}
1104
\newlabel{eq:structureTensorRotation@cref}{{[equation][12][4]4.12}{83}}
1105
\newlabel{eq:structtensPrior}{{4.14}{83}{Structure Tensor Based Prior}{equation.4.2.14}{}}
1106
\newlabel{eq:structtensPrior@cref}{{[equation][14][4]4.14}{83}}
1107
\newlabel{eq:structtensPriorRot}{{4.15}{83}{Structure Tensor Based Prior}{equation.4.2.15}{}}
1108
\newlabel{eq:structtensPriorRot@cref}{{[equation][15][4]4.15}{83}}
1109
\newlabel{eq:traceTransf}{{4.16}{83}{Structure Tensor Based Prior}{equation.4.2.16}{}}
1110
\newlabel{eq:traceTransf@cref}{{[equation][16][4]4.16}{83}}
1105
1111
\citation{MaesMutualInformationRegistration}
1106
1112
\citation{Roche98CorrelRatio}
1107
1113
\citation{RocheUnifyingMaxLikelihoodRegistration}
1108
1114
\citation{HardieSpacialImageResEnhancement}
1109
1115
\citation{HardieSpacialImageResEnhancement}
1110
\newlabel{eq:structtensPriorRotInv}{{4.17}{83}{Structure Tensor Based Prior}{equation.4.2.17}{}}
1111
\newlabel{eq:structtensPriorRotInv@cref}{{[equation][17][4]4.17}{83}}
1112
\@writefile{toc}{\contentsline {section}{\numberline {4.3}Geometrical Optical Flow Model}{83}{section.4.3}}
1113
\newlabel{chap:GeomOptFlowModel}{{4.3}{83}{Geometrical Optical Flow Model}{section.4.3}{}}
1114
\newlabel{chap:GeomOptFlowModel@cref}{{[section][3][4]4.3}{83}}
1115
\@writefile{brf}{\backcite{MaesMutualInformationRegistration}{{83}{4.3}{section.4.3}}}
1116
\@writefile{brf}{\backcite{Roche98CorrelRatio}{{83}{4.3}{section.4.3}}}
1117
\@writefile{brf}{\backcite{RocheUnifyingMaxLikelihoodRegistration}{{83}{4.3}{section.4.3}}}
1118
\@writefile{brf}{\backcite{HardieSpacialImageResEnhancement}{{83}{4.3}{section.4.3}}}
1119
\@writefile{brf}{\backcite{HardieSpacialImageResEnhancement}{{83}{4.3}{section.4.3}}}
1116
\newlabel{eq:structtensPriorRotInv}{{4.17}{84}{Structure Tensor Based Prior}{equation.4.2.17}{}}
1117
\newlabel{eq:structtensPriorRotInv@cref}{{[equation][17][4]4.17}{84}}
1118
\@writefile{toc}{\contentsline {section}{\numberline {4.3}Geometrical Optical Flow Model}{84}{section.4.3}}
1119
\newlabel{chap:GeomOptFlowModel}{{4.3}{84}{Geometrical Optical Flow Model}{section.4.3}{}}
1120
\newlabel{chap:GeomOptFlowModel@cref}{{[section][3][4]4.3}{84}}
1121
\@writefile{brf}{\backcite{MaesMutualInformationRegistration}{{84}{4.3}{section.4.3}}}
1122
\@writefile{brf}{\backcite{Roche98CorrelRatio}{{84}{4.3}{section.4.3}}}
1123
\@writefile{brf}{\backcite{RocheUnifyingMaxLikelihoodRegistration}{{84}{4.3}{section.4.3}}}
1124
\@writefile{brf}{\backcite{HardieSpacialImageResEnhancement}{{84}{4.3}{section.4.3}}}
1125
\@writefile{brf}{\backcite{HardieSpacialImageResEnhancement}{{84}{4.3}{section.4.3}}}
1120
1126
\citation{HardieSpacialImageResEnhancement}
1121
\newlabel{fig:multiModalSetupDiffRes}{{4.1a}{84}{Subfigure 4 4.1a}{subfigure.4.1.1}{}}
1127
\newlabel{fig:multiModalSetupDiffRes}{{4.1a}{85}{Subfigure 4 4.1a}{subfigure.4.1.1}{}}
1122
1128
\newlabel{sub@fig:multiModalSetupDiffRes}{{(a)}{a}{Subfigure 4 4.1a\relax }{subfigure.4.1.1}{}}
1123
\newlabel{fig:multiModalSetupDiffRes@cref}{{[subfigure][1][4,1]4.1a}{84}}
1124
\newlabel{fig:multiModalSetupDiffResIvsc}{{4.1b}{84}{Subfigure 4 4.1b}{subfigure.4.1.2}{}}
1129
\newlabel{fig:multiModalSetupDiffRes@cref}{{[subfigure][1][4,1]4.1a}{85}}
1130
\newlabel{fig:multiModalSetupDiffResIvsc}{{4.1b}{85}{Subfigure 4 4.1b}{subfigure.4.1.2}{}}
1125
1131
\newlabel{sub@fig:multiModalSetupDiffResIvsc}{{(b)}{b}{Subfigure 4 4.1b\relax }{subfigure.4.1.2}{}}
1126
\newlabel{fig:multiModalSetupDiffResIvsc@cref}{{[subfigure][2][4,1]4.1b}{84}}
1127
\newlabel{fig:multiModalSetupDiffResYtcLow}{{4.1c}{84}{Subfigure 4 4.1c}{subfigure.4.1.3}{}}
1132
\newlabel{fig:multiModalSetupDiffResIvsc@cref}{{[subfigure][2][4,1]4.1b}{85}}
1133
\newlabel{fig:multiModalSetupDiffResYtcLow}{{4.1c}{85}{Subfigure 4 4.1c}{subfigure.4.1.3}{}}
1128
1134
\newlabel{sub@fig:multiModalSetupDiffResYtcLow}{{(c)}{c}{Subfigure 4 4.1c\relax }{subfigure.4.1.3}{}}
1129
\newlabel{fig:multiModalSetupDiffResYtcLow@cref}{{[subfigure][3][4,1]4.1c}{84}}
1130
\@writefile{lof}{\contentsline {figure}{\numberline {4.1}{\ignorespaces \Cref {fig:multiModalSetupDiffRes} shows the setup of a thermographic camera (TC), $C_{tc}$, and a visual spectrum camera (VSC), $C_{vsc}$, recording an object $O$. \Cref {fig:multiModalSetupDiffResIvsc} shows the image $I$ which is recorded by $C_{vsc}$ and \cref {fig:multiModalSetupDiffResYtcLow} the lower resolution image $y$ recorded by $C_{tc}$. The solid line cone of $C_{tc}$ in \cref {fig:multiModalSetupDiffRes} which is small compared to the cone of $C_{vsc}$ indicates the low resolution of the TC compared to that of the VSC. The dotted cone indicates the high resolution of the image $Y$, which is jointly estimated together with the optical flow $\ensuremath {\ensuremath {{\bm {d}}}}$ (the mapping between $I$ and $y$) by the model in eq.~(\ref {eq:YtcToIvscDMapping}) \relax }}{84}{figure.caption.19}}
1131
\newlabel{fig:multiModalSetupDiffResSetup}{{4.1}{84}{\Figref {fig:multiModalSetupDiffRes} shows the setup of a thermographic camera (TC), $C_{tc}$, and a visual spectrum camera (VSC), $C_{vsc}$, recording an object $O$. \Figref {fig:multiModalSetupDiffResIvsc} shows the image $I$ which is recorded by $C_{vsc}$ and \figref {fig:multiModalSetupDiffResYtcLow} the lower resolution image $y$ recorded by $C_{tc}$. The solid line cone of $C_{tc}$ in \figref {fig:multiModalSetupDiffRes} which is small compared to the cone of $C_{vsc}$ indicates the low resolution of the TC compared to that of the VSC. The dotted cone indicates the high resolution of the image $Y$, which is jointly estimated together with the optical flow $\vd $ (the mapping between $I$ and $y$) by the model in \eqref {eq:YtcToIvscDMapping} \relax }{figure.caption.19}{}}
1132
\newlabel{fig:multiModalSetupDiffResSetup@cref}{{[figure][1][4]4.1}{84}}
1133
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {Schematic}}}{84}{subfigure.1.1}}
1134
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$I$}}}{84}{subfigure.1.2}}
1135
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$y$}}}{84}{subfigure.1.3}}
1136
\@writefile{toc}{\contentsline {section}{\numberline {4.4}Multi-Modal Optical Flow with Differing Resolutions}{84}{section.4.4}}
1137
\newlabel{sec:ImageFusionDisparity}{{4.4}{84}{Multi-Modal Optical Flow with Differing Resolutions}{section.4.4}{}}
1138
\newlabel{sec:ImageFusionDisparity@cref}{{[section][4][4]4.4}{84}}
1139
\@writefile{brf}{\backcite{HardieSpacialImageResEnhancement}{{84}{4.4}{figure.caption.19}}}
1140
\newlabel{eq:warpedIvsc}{{4.18}{84}{Multi-Modal Optical Flow with Differing Resolutions}{equation.4.4.18}{}}
1141
\newlabel{eq:warpedIvsc@cref}{{[equation][18][4]4.18}{84}}
1142
\newlabel{eq:multiResSimMeasure}{{4.19}{84}{Multi-Modal Optical Flow with Differing Resolutions}{equation.4.4.19}{}}
1143
\newlabel{eq:multiResSimMeasure@cref}{{[equation][19][4]4.19}{84}}
1144
\newlabel{eq:YtcToLowMapping}{{4.20}{85}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.20}{}}
1145
\newlabel{eq:YtcToLowMapping@cref}{{[equation][20][4]4.20}{85}}
1146
\newlabel{eq:YtcToIvscMapping}{{4.21}{85}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.21}{}}
1147
\newlabel{eq:YtcToIvscMapping@cref}{{[equation][21][4]4.21}{85}}
1148
\newlabel{eq:condVarYtc}{{4.22}{85}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.22}{}}
1149
\newlabel{eq:condVarYtc@cref}{{[equation][22][4]4.22}{85}}
1150
\newlabel{eq:condMeanYtc}{{4.23}{85}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.23}{}}
1151
\newlabel{eq:condMeanYtc@cref}{{[equation][23][4]4.23}{85}}
1152
\newlabel{eq:YtcToLowMapping2}{{4.24}{85}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.24}{}}
1153
\newlabel{eq:YtcToLowMapping2@cref}{{[equation][24][4]4.24}{85}}
1154
\newlabel{eq:YtcToIvscDMapping}{{4.25}{85}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.25}{}}
1155
\newlabel{eq:YtcToIvscDMapping@cref}{{[equation][25][4]4.25}{85}}
1156
\newlabel{fig:line02}{{4.2a}{86}{Subfigure 4 4.2a}{subfigure.4.2.1}{}}
1135
\newlabel{fig:multiModalSetupDiffResYtcLow@cref}{{[subfigure][3][4,1]4.1c}{85}}
1136
\@writefile{lof}{\contentsline {figure}{\numberline {4.1}{\ignorespaces \Cref {fig:multiModalSetupDiffRes} shows the setup of a thermographic camera (TC), $C_{tc}$, and a visual spectrum camera (VSC), $C_{vsc}$, recording an object $O$. \Cref {fig:multiModalSetupDiffResIvsc} shows the image $I$ which is recorded by $C_{vsc}$ and \cref {fig:multiModalSetupDiffResYtcLow} the lower resolution image $y$ recorded by $C_{tc}$. The solid line cone of $C_{tc}$ in \cref {fig:multiModalSetupDiffRes} which is small compared to the cone of $C_{vsc}$ indicates the low resolution of the TC compared to that of the VSC. The dotted cone indicates the high resolution of the image $Y$, which is jointly estimated together with the optical flow $\ensuremath {\ensuremath {{\bm {d}}}}$ (the mapping between $I$ and $y$) by the model in eq.~(\ref {eq:YtcToIvscDMapping}) \relax }}{85}{figure.caption.20}}
1137
\newlabel{fig:multiModalSetupDiffResSetup}{{4.1}{85}{\Figref {fig:multiModalSetupDiffRes} shows the setup of a thermographic camera (TC), $C_{tc}$, and a visual spectrum camera (VSC), $C_{vsc}$, recording an object $O$. \Figref {fig:multiModalSetupDiffResIvsc} shows the image $I$ which is recorded by $C_{vsc}$ and \figref {fig:multiModalSetupDiffResYtcLow} the lower resolution image $y$ recorded by $C_{tc}$. The solid line cone of $C_{tc}$ in \figref {fig:multiModalSetupDiffRes} which is small compared to the cone of $C_{vsc}$ indicates the low resolution of the TC compared to that of the VSC. The dotted cone indicates the high resolution of the image $Y$, which is jointly estimated together with the optical flow $\vd $ (the mapping between $I$ and $y$) by the model in \eqref {eq:YtcToIvscDMapping} \relax }{figure.caption.20}{}}
1138
\newlabel{fig:multiModalSetupDiffResSetup@cref}{{[figure][1][4]4.1}{85}}
1139
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {Schematic}}}{85}{subfigure.1.1}}
1140
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$I$}}}{85}{subfigure.1.2}}
1141
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$y$}}}{85}{subfigure.1.3}}
1142
\@writefile{toc}{\contentsline {section}{\numberline {4.4}Multi-Modal Optical Flow with Differing Resolutions}{85}{section.4.4}}
1143
\newlabel{sec:ImageFusionDisparity}{{4.4}{85}{Multi-Modal Optical Flow with Differing Resolutions}{section.4.4}{}}
1144
\newlabel{sec:ImageFusionDisparity@cref}{{[section][4][4]4.4}{85}}
1145
\@writefile{brf}{\backcite{HardieSpacialImageResEnhancement}{{85}{4.4}{figure.caption.20}}}
1146
\newlabel{eq:warpedIvsc}{{4.18}{85}{Multi-Modal Optical Flow with Differing Resolutions}{equation.4.4.18}{}}
1147
\newlabel{eq:warpedIvsc@cref}{{[equation][18][4]4.18}{85}}
1148
\newlabel{eq:multiResSimMeasure}{{4.19}{85}{Multi-Modal Optical Flow with Differing Resolutions}{equation.4.4.19}{}}
1149
\newlabel{eq:multiResSimMeasure@cref}{{[equation][19][4]4.19}{85}}
1150
\newlabel{eq:YtcToLowMapping}{{4.20}{86}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.20}{}}
1151
\newlabel{eq:YtcToLowMapping@cref}{{[equation][20][4]4.20}{86}}
1152
\newlabel{eq:YtcToIvscMapping}{{4.21}{86}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.21}{}}
1153
\newlabel{eq:YtcToIvscMapping@cref}{{[equation][21][4]4.21}{86}}
1154
\newlabel{eq:condVarYtc}{{4.22}{86}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.22}{}}
1155
\newlabel{eq:condVarYtc@cref}{{[equation][22][4]4.22}{86}}
1156
\newlabel{eq:condMeanYtc}{{4.23}{86}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.23}{}}
1157
\newlabel{eq:condMeanYtc@cref}{{[equation][23][4]4.23}{86}}
1158
\newlabel{eq:YtcToLowMapping2}{{4.24}{86}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.24}{}}
1159
\newlabel{eq:YtcToLowMapping2@cref}{{[equation][24][4]4.24}{86}}
1160
\newlabel{eq:YtcToIvscDMapping}{{4.25}{86}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.25}{}}
1161
\newlabel{eq:YtcToIvscDMapping@cref}{{[equation][25][4]4.25}{86}}
1162
\newlabel{fig:line02}{{4.2a}{87}{Subfigure 4 4.2a}{subfigure.4.2.1}{}}
1157
1163
\newlabel{sub@fig:line02}{{(a)}{a}{Subfigure 4 4.2a\relax }{subfigure.4.2.1}{}}
1158
\newlabel{fig:line02@cref}{{[subfigure][1][4,2]4.2a}{86}}
1159
\newlabel{fig:line1blurred2}{{4.2b}{86}{Subfigure 4 4.2b}{subfigure.4.2.2}{}}
1164
\newlabel{fig:line02@cref}{{[subfigure][1][4,2]4.2a}{87}}
1165
\newlabel{fig:line1blurred2}{{4.2b}{87}{Subfigure 4 4.2b}{subfigure.4.2.2}{}}
1160
1166
\newlabel{sub@fig:line1blurred2}{{(b)}{b}{Subfigure 4 4.2b\relax }{subfigure.4.2.2}{}}
1161
\newlabel{fig:line1blurred2@cref}{{[subfigure][2][4,2]4.2b}{86}}
1162
\newlabel{fig:line0warpedWithScaleDiff}{{4.2c}{86}{Subfigure 4 4.2c}{subfigure.4.2.3}{}}
1167
\newlabel{fig:line1blurred2@cref}{{[subfigure][2][4,2]4.2b}{87}}
1168
\newlabel{fig:line0warpedWithScaleDiff}{{4.2c}{87}{Subfigure 4 4.2c}{subfigure.4.2.3}{}}
1163
1169
\newlabel{sub@fig:line0warpedWithScaleDiff}{{(c)}{c}{Subfigure 4 4.2c\relax }{subfigure.4.2.3}{}}
1164
\newlabel{fig:line0warpedWithScaleDiff@cref}{{[subfigure][3][4,2]4.2c}{86}}
1165
\newlabel{fig:flowWithScaleDiff}{{4.2d}{86}{Subfigure 4 4.2d}{subfigure.4.2.4}{}}
1170
\newlabel{fig:line0warpedWithScaleDiff@cref}{{[subfigure][3][4,2]4.2c}{87}}
1171
\newlabel{fig:flowWithScaleDiff}{{4.2d}{87}{Subfigure 4 4.2d}{subfigure.4.2.4}{}}
1166
1172
\newlabel{sub@fig:flowWithScaleDiff}{{(d)}{d}{Subfigure 4 4.2d\relax }{subfigure.4.2.4}{}}
1167
\newlabel{fig:flowWithScaleDiff@cref}{{[subfigure][4][4,2]4.2d}{86}}
1168
\@writefile{lof}{\contentsline {figure}{\numberline {4.2}{\ignorespaces \Cref {fig:line02} shows a synthetic high resolution image $I^{syn}$. In \cref {fig:line1blurred2} we show a low resolution image $y^{syn}$. $y^{syn}$ is computed by convolution of $I^{syn}$ with Gaussian $G_\ensuremath {{\sigma ^{sc}}}$ with standard deviation $\ensuremath {{\sigma ^{sc}}}=5$ and translated by $10$ pixels relative to $I^{syn}$. \Cref {fig:flowWithScaleDiff} shows the flow $\ensuremath {\ensuremath {{\bm {d}}}}$ computed with the model in eq.~(\ref {eq:flowDataTerm2}), which incorporates knowledge of the scale difference between $y^{syn}$ and $I^{syn}$ and \cref {fig:line0warpedWithScaleDiff} show the warped image $I_{\ensuremath {\ensuremath {{\bm {d}}}}}$\relax }}{86}{figure.caption.21}}
1169
\newlabel{fig:scaleDiffProblemWithScale}{{4.2}{86}{\Figref {fig:line02} shows a synthetic high resolution image $I^{syn}$. In \figref {fig:line1blurred2} we show a low resolution image $y^{syn}$. $y^{syn}$ is computed by convolution of $I^{syn}$ with Gaussian $G_\scalediff $ with standard deviation $\scalediff =5$ and translated by $10$ pixels relative to $I^{syn}$. \Figref {fig:flowWithScaleDiff} shows the flow $\vd $ computed with the model in \eqref {eq:flowDataTerm2}, which incorporates knowledge of the scale difference between $y^{syn}$ and $I^{syn}$ and \figref {fig:line0warpedWithScaleDiff} show the warped image $I_{\vd }$\relax }{figure.caption.21}{}}
1170
\newlabel{fig:scaleDiffProblemWithScale@cref}{{[figure][2][4]4.2}{86}}
1171
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{86}{subfigure.2.1}}
1172
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{86}{subfigure.2.2}}
1173
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{86}{subfigure.2.3}}
1174
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{86}{subfigure.2.4}}
1175
\newlabel{eq:thermoSolutionOptflow}{{4.26}{86}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.26}{}}
1176
\newlabel{eq:thermoSolutionOptflow@cref}{{[equation][26][4]4.26}{86}}
1177
\newlabel{eq:flowDataTerm2}{{4.27}{86}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.27}{}}
1178
\newlabel{eq:flowDataTerm2@cref}{{[equation][27][4]4.27}{86}}
1179
\newlabel{eq:globalIntensFactor}{{4.28}{86}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.28}{}}
1180
\newlabel{eq:globalIntensFactor@cref}{{[equation][28][4]4.28}{86}}
1181
\newlabel{eq:linearRelationyI}{{4.30}{86}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.30}{}}
1182
\newlabel{eq:linearRelationyI@cref}{{[equation][30][4]4.30}{86}}
1183
\@writefile{toc}{\contentsline {section}{\numberline {4.5}Localization}{87}{section.4.5}}
1184
\newlabel{sec:localSimilarityMeasure}{{4.5}{87}{Localization}{section.4.5}{}}
1185
\newlabel{sec:localSimilarityMeasure@cref}{{[section][5][4]4.5}{87}}
1186
\newlabel{eq:globalRelation}{{4.32}{87}{Localization}{equation.4.5.32}{}}
1187
\newlabel{eq:globalRelation@cref}{{[equation][32][4]4.32}{87}}
1188
\newlabel{eq:localRelation}{{4.33}{87}{Localization}{equation.4.5.33}{}}
1189
\newlabel{eq:localRelation@cref}{{[equation][33][4]4.33}{87}}
1190
\newlabel{eq:locCovariance}{{4.34}{88}{Localization}{equation.4.5.34}{}}
1191
\newlabel{eq:locCovariance@cref}{{[equation][34][4]4.34}{88}}
1192
\newlabel{eq:likelihoodWindow}{{4.35}{88}{Localization}{equation.4.5.35}{}}
1193
\newlabel{eq:likelihoodWindow@cref}{{[equation][35][4]4.35}{88}}
1194
\newlabel{eq:condVarYtcLocal}{{4.36}{88}{Localization}{equation.4.5.36}{}}
1195
\newlabel{eq:condVarYtcLocal@cref}{{[equation][36][4]4.36}{88}}
1196
\newlabel{eq:localFfactor}{{4.37}{88}{Localization}{equation.4.5.37}{}}
1197
\newlabel{eq:localFfactor@cref}{{[equation][37][4]4.37}{88}}
1198
\newlabel{eq:flowDataTermLocal}{{4.38}{88}{Localization}{equation.4.5.38}{}}
1199
\newlabel{eq:flowDataTermLocal@cref}{{[equation][38][4]4.38}{88}}
1200
\@writefile{toc}{\contentsline {section}{\numberline {4.6}The Multigrid Newton algorithm}{88}{section.4.6}}
1201
\newlabel{eq:optFlowModelST}{{4.39}{88}{The Multigrid Newton algorithm}{equation.4.6.39}{}}
1202
\newlabel{eq:optFlowModelST@cref}{{[equation][39][4]4.39}{88}}
1203
\newlabel{eq:optFlowModelTV}{{4.40}{88}{The Multigrid Newton algorithm}{equation.4.6.40}{}}
1204
\newlabel{eq:optFlowModelTV@cref}{{[equation][40][4]4.40}{88}}
1205
\newlabel{item:FlowAlgoWhileCond}{{6}{89}{The Multigrid Newton algorithm}{ALG@line.6}{}}
1206
\newlabel{item:FlowAlgoWhileCond@cref}{{[line][6][]6}{89}}
1207
\newlabel{item:FlowAlgoLinearStep}{{9}{89}{The Multigrid Newton algorithm}{ALG@line.9}{}}
1208
\newlabel{item:FlowAlgoLinearStep@cref}{{[line][9][]9}{89}}
1209
\@writefile{loa}{\contentsline {algorithm}{\numberline {2}{\ignorespaces Multigrid Optical Flow (MOF)\relax }}{89}{algorithm.2}}
1210
\newlabel{alg:MultigridOpticalFlow}{{2}{89}{Multigrid Optical Flow (MOF)\relax }{algorithm.2}{}}
1211
\newlabel{alg:MultigridOpticalFlow@cref}{{[algorithm][2][]2}{89}}
1212
\newlabel{eq:structtensPriorFuncderiv}{{4.41}{89}{The Multigrid Newton algorithm}{equation.4.6.41}{}}
1213
\newlabel{eq:structtensPriorFuncderiv@cref}{{[equation][41][4]4.41}{89}}
1173
\newlabel{fig:flowWithScaleDiff@cref}{{[subfigure][4][4,2]4.2d}{87}}
1174
\@writefile{lof}{\contentsline {figure}{\numberline {4.2}{\ignorespaces \Cref {fig:line02} shows a synthetic high resolution image $I^{syn}$. In \cref {fig:line1blurred2} we show a low resolution image $y^{syn}$. $y^{syn}$ is computed by convolution of $I^{syn}$ with Gaussian $G_\ensuremath {{\sigma ^{sc}}}$ with standard deviation $\ensuremath {{\sigma ^{sc}}}=5$ and translated by $10$ pixels relative to $I^{syn}$. \Cref {fig:flowWithScaleDiff} shows the flow $\ensuremath {\ensuremath {{\bm {d}}}}$ computed with the model in eq.~(\ref {eq:flowDataTerm2}), which incorporates knowledge of the scale difference between $y^{syn}$ and $I^{syn}$ and \cref {fig:line0warpedWithScaleDiff} show the warped image $I_{\ensuremath {\ensuremath {{\bm {d}}}}}$\relax }}{87}{figure.caption.22}}
1175
\newlabel{fig:scaleDiffProblemWithScale}{{4.2}{87}{\Figref {fig:line02} shows a synthetic high resolution image $I^{syn}$. In \figref {fig:line1blurred2} we show a low resolution image $y^{syn}$. $y^{syn}$ is computed by convolution of $I^{syn}$ with Gaussian $G_\scalediff $ with standard deviation $\scalediff =5$ and translated by $10$ pixels relative to $I^{syn}$. \Figref {fig:flowWithScaleDiff} shows the flow $\vd $ computed with the model in \eqref {eq:flowDataTerm2}, which incorporates knowledge of the scale difference between $y^{syn}$ and $I^{syn}$ and \figref {fig:line0warpedWithScaleDiff} show the warped image $I_{\vd }$\relax }{figure.caption.22}{}}
1176
\newlabel{fig:scaleDiffProblemWithScale@cref}{{[figure][2][4]4.2}{87}}
1177
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{87}{subfigure.2.1}}
1178
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{87}{subfigure.2.2}}
1179
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{87}{subfigure.2.3}}
1180
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{87}{subfigure.2.4}}
1181
\newlabel{eq:thermoSolutionOptflow}{{4.26}{87}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.26}{}}
1182
\newlabel{eq:thermoSolutionOptflow@cref}{{[equation][26][4]4.26}{87}}
1183
\newlabel{eq:flowDataTerm2}{{4.27}{87}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.27}{}}
1184
\newlabel{eq:flowDataTerm2@cref}{{[equation][27][4]4.27}{87}}
1185
\newlabel{eq:globalIntensFactor}{{4.28}{87}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.28}{}}
1186
\newlabel{eq:globalIntensFactor@cref}{{[equation][28][4]4.28}{87}}
1187
\newlabel{eq:linearRelationyI}{{4.30}{87}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.30}{}}
1188
\newlabel{eq:linearRelationyI@cref}{{[equation][30][4]4.30}{87}}
1189
\@writefile{toc}{\contentsline {section}{\numberline {4.5}Localization}{88}{section.4.5}}
1190
\newlabel{sec:localSimilarityMeasure}{{4.5}{88}{Localization}{section.4.5}{}}
1191
\newlabel{sec:localSimilarityMeasure@cref}{{[section][5][4]4.5}{88}}
1192
\newlabel{eq:globalRelation}{{4.32}{88}{Localization}{equation.4.5.32}{}}
1193
\newlabel{eq:globalRelation@cref}{{[equation][32][4]4.32}{88}}
1194
\newlabel{eq:localRelation}{{4.33}{88}{Localization}{equation.4.5.33}{}}
1195
\newlabel{eq:localRelation@cref}{{[equation][33][4]4.33}{88}}
1196
\newlabel{eq:locCovariance}{{4.34}{89}{Localization}{equation.4.5.34}{}}
1197
\newlabel{eq:locCovariance@cref}{{[equation][34][4]4.34}{89}}
1198
\newlabel{eq:likelihoodWindow}{{4.35}{89}{Localization}{equation.4.5.35}{}}
1199
\newlabel{eq:likelihoodWindow@cref}{{[equation][35][4]4.35}{89}}
1200
\newlabel{eq:condVarYtcLocal}{{4.36}{89}{Localization}{equation.4.5.36}{}}
1201
\newlabel{eq:condVarYtcLocal@cref}{{[equation][36][4]4.36}{89}}
1202
\newlabel{eq:localFfactor}{{4.37}{89}{Localization}{equation.4.5.37}{}}
1203
\newlabel{eq:localFfactor@cref}{{[equation][37][4]4.37}{89}}
1204
\newlabel{eq:flowDataTermLocal}{{4.38}{89}{Localization}{equation.4.5.38}{}}
1205
\newlabel{eq:flowDataTermLocal@cref}{{[equation][38][4]4.38}{89}}
1206
\@writefile{toc}{\contentsline {section}{\numberline {4.6}The Multigrid Newton algorithm}{89}{section.4.6}}
1207
\newlabel{eq:optFlowModelST}{{4.39}{89}{The Multigrid Newton algorithm}{equation.4.6.39}{}}
1208
\newlabel{eq:optFlowModelST@cref}{{[equation][39][4]4.39}{89}}
1209
\newlabel{eq:optFlowModelTV}{{4.40}{89}{The Multigrid Newton algorithm}{equation.4.6.40}{}}
1210
\newlabel{eq:optFlowModelTV@cref}{{[equation][40][4]4.40}{89}}
1211
\newlabel{item:FlowAlgoWhileCond}{{6}{90}{The Multigrid Newton algorithm}{ALG@line.6}{}}
1212
\newlabel{item:FlowAlgoWhileCond@cref}{{[line][6][]6}{90}}
1213
\newlabel{item:FlowAlgoLinearStep}{{9}{90}{The Multigrid Newton algorithm}{ALG@line.9}{}}
1214
\newlabel{item:FlowAlgoLinearStep@cref}{{[line][9][]9}{90}}
1215
\@writefile{loa}{\contentsline {algorithm}{\numberline {2}{\ignorespaces Multigrid Optical Flow (MOF)\relax }}{90}{algorithm.2}}
1216
\newlabel{alg:MultigridOpticalFlow}{{2}{90}{Multigrid Optical Flow (MOF)\relax }{algorithm.2}{}}
1217
\newlabel{alg:MultigridOpticalFlow@cref}{{[algorithm][2][]2}{90}}
1218
\newlabel{eq:structtensPriorFuncderiv}{{4.41}{90}{The Multigrid Newton algorithm}{equation.4.6.41}{}}
1219
\newlabel{eq:structtensPriorFuncderiv@cref}{{[equation][41][4]4.41}{90}}
1214
1220
\citation{Middleburry}
1215
\newlabel{eq:structtensPriorStable}{{4.42}{90}{The Multigrid Newton algorithm}{equation.4.6.42}{}}
1216
\newlabel{eq:structtensPriorStable@cref}{{[equation][42][4]4.42}{90}}
1217
\newlabel{eq:optFlowModelSTStable}{{4.43}{90}{The Multigrid Newton algorithm}{equation.4.6.43}{}}
1218
\newlabel{eq:optFlowModelSTStable@cref}{{[equation][43][4]4.43}{90}}
1219
\@writefile{toc}{\contentsline {section}{\numberline {4.7}Results}{90}{section.4.7}}
1220
\@writefile{brf}{\backcite{Middleburry}{{90}{4.7}{section.4.7}}}
1221
\newlabel{fig:rubberWhale}{{4.3a}{91}{Subfigure 4 4.3a}{subfigure.4.3.1}{}}
1221
\newlabel{eq:structtensPriorStable}{{4.42}{91}{The Multigrid Newton algorithm}{equation.4.6.42}{}}
1222
\newlabel{eq:structtensPriorStable@cref}{{[equation][42][4]4.42}{91}}
1223
\newlabel{eq:optFlowModelSTStable}{{4.43}{91}{The Multigrid Newton algorithm}{equation.4.6.43}{}}
1224
\newlabel{eq:optFlowModelSTStable@cref}{{[equation][43][4]4.43}{91}}
1225
\@writefile{toc}{\contentsline {section}{\numberline {4.7}Results}{91}{section.4.7}}
1226
\@writefile{brf}{\backcite{Middleburry}{{91}{4.7}{section.4.7}}}
1227
\newlabel{fig:rubberWhale}{{4.3a}{92}{Subfigure 4 4.3a}{subfigure.4.3.1}{}}
1222
1228
\newlabel{sub@fig:rubberWhale}{{(a)}{a}{Subfigure 4 4.3a\relax }{subfigure.4.3.1}{}}
1223
\newlabel{fig:rubberWhale@cref}{{[subfigure][1][4,3]4.3a}{91}}
1224
\newlabel{fig:rubberWhale-flow}{{4.3b}{91}{Subfigure 4 4.3b}{subfigure.4.3.2}{}}
1229
\newlabel{fig:rubberWhale@cref}{{[subfigure][1][4,3]4.3a}{92}}
1230
\newlabel{fig:rubberWhale-flow}{{4.3b}{92}{Subfigure 4 4.3b}{subfigure.4.3.2}{}}
1225
1231
\newlabel{sub@fig:rubberWhale-flow}{{(b)}{b}{Subfigure 4 4.3b\relax }{subfigure.4.3.2}{}}
1226
\newlabel{fig:rubberWhale-flow@cref}{{[subfigure][2][4,3]4.3b}{91}}
1227
\newlabel{fig:rubberWhale-flowL1}{{4.3c}{91}{Subfigure 4 4.3c}{subfigure.4.3.3}{}}
1232
\newlabel{fig:rubberWhale-flow@cref}{{[subfigure][2][4,3]4.3b}{92}}
1233
\newlabel{fig:rubberWhale-flowL1}{{4.3c}{92}{Subfigure 4 4.3c}{subfigure.4.3.3}{}}
1228
1234
\newlabel{sub@fig:rubberWhale-flowL1}{{(c)}{c}{Subfigure 4 4.3c\relax }{subfigure.4.3.3}{}}
1229
\newlabel{fig:rubberWhale-flowL1@cref}{{[subfigure][3][4,3]4.3c}{91}}
1230
\newlabel{fig:rubberWhale-gt2}{{4.3d}{91}{Subfigure 4 4.3d}{subfigure.4.3.4}{}}
1235
\newlabel{fig:rubberWhale-flowL1@cref}{{[subfigure][3][4,3]4.3c}{92}}
1236
\newlabel{fig:rubberWhale-gt2}{{4.3d}{92}{Subfigure 4 4.3d}{subfigure.4.3.4}{}}
1231
1237
\newlabel{sub@fig:rubberWhale-gt2}{{(d)}{d}{Subfigure 4 4.3d\relax }{subfigure.4.3.4}{}}
1232
\newlabel{fig:rubberWhale-gt2@cref}{{[subfigure][4][4,3]4.3d}{91}}
1233
\@writefile{lof}{\contentsline {figure}{\numberline {4.3}{\ignorespaces Rubberwhale Sequence: Figure \ref {fig:rubberWhale} shows one frame of the sequence. \cref {fig:rubberWhale-flow} shows the estimated optical flow $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$, \cref {fig:rubberWhale-flowL1} the flow $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$ and \cref {fig:rubberWhale-gt2} shows the provided ground truth\relax }}{91}{figure.caption.22}}
1234
\newlabel{fig:rubberWhaleSeq2}{{4.3}{91}{Rubberwhale Sequence: Figure \ref {fig:rubberWhale} shows one frame of the sequence. \figref {fig:rubberWhale-flow} shows the estimated optical flow $\optimalflowST $, \figref {fig:rubberWhale-flowL1} the flow $\optimalflowTV $ and \figref {fig:rubberWhale-gt2} shows the provided ground truth\relax }{figure.caption.22}{}}
1235
\newlabel{fig:rubberWhaleSeq2@cref}{{[figure][3][4]4.3}{91}}
1236
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{91}{subfigure.3.1}}
1237
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$}}}{91}{subfigure.3.2}}
1238
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$}}}{91}{subfigure.3.3}}
1239
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{91}{subfigure.3.4}}
1240
\newlabel{eq:optFlowModelSTLocal}{{4.44}{91}{Results}{equation.4.7.44}{}}
1241
\newlabel{eq:optFlowModelSTLocal@cref}{{[equation][44][4]4.44}{91}}
1242
\newlabel{eq:optFlowModelTVLocal}{{4.45}{91}{Results}{equation.4.7.45}{}}
1243
\newlabel{eq:optFlowModelTVLocal@cref}{{[equation][45][4]4.45}{91}}
1244
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.1}Uni-Modal Data}{91}{subsection.4.7.1}}
1245
\newlabel{sec:uniModalOpticalFlow}{{4.7.1}{91}{Uni-Modal Data}{subsection.4.7.1}{}}
1246
\newlabel{sec:uniModalOpticalFlow@cref}{{[subsection][1][4,7]4.7.1}{91}}
1247
\newlabel{eq:EPEDefinition}{{4.46}{91}{Uni-Modal Data}{equation.4.7.46}{}}
1248
\newlabel{eq:EPEDefinition@cref}{{[equation][46][4]4.46}{91}}
1249
\newlabel{fig:hydrangea}{{4.4a}{92}{Subfigure 4 4.4a}{subfigure.4.4.1}{}}
1238
\newlabel{fig:rubberWhale-gt2@cref}{{[subfigure][4][4,3]4.3d}{92}}
1239
\@writefile{lof}{\contentsline {figure}{\numberline {4.3}{\ignorespaces Rubberwhale Sequence: Figure \ref {fig:rubberWhale} shows one frame of the sequence. \cref {fig:rubberWhale-flow} shows the estimated optical flow $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$, \cref {fig:rubberWhale-flowL1} the flow $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$ and \cref {fig:rubberWhale-gt2} shows the provided ground truth\relax }}{92}{figure.caption.23}}
1240
\newlabel{fig:rubberWhaleSeq2}{{4.3}{92}{Rubberwhale Sequence: Figure \ref {fig:rubberWhale} shows one frame of the sequence. \figref {fig:rubberWhale-flow} shows the estimated optical flow $\optimalflowST $, \figref {fig:rubberWhale-flowL1} the flow $\optimalflowTV $ and \figref {fig:rubberWhale-gt2} shows the provided ground truth\relax }{figure.caption.23}{}}
1241
\newlabel{fig:rubberWhaleSeq2@cref}{{[figure][3][4]4.3}{92}}
1242
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{92}{subfigure.3.1}}
1243
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$}}}{92}{subfigure.3.2}}
1244
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$}}}{92}{subfigure.3.3}}
1245
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{92}{subfigure.3.4}}
1246
\newlabel{eq:optFlowModelSTLocal}{{4.44}{92}{Results}{equation.4.7.44}{}}
1247
\newlabel{eq:optFlowModelSTLocal@cref}{{[equation][44][4]4.44}{92}}
1248
\newlabel{eq:optFlowModelTVLocal}{{4.45}{92}{Results}{equation.4.7.45}{}}
1249
\newlabel{eq:optFlowModelTVLocal@cref}{{[equation][45][4]4.45}{92}}
1250
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.1}Uni-Modal Data}{92}{subsection.4.7.1}}
1251
\newlabel{sec:uniModalOpticalFlow}{{4.7.1}{92}{Uni-Modal Data}{subsection.4.7.1}{}}
1252
\newlabel{sec:uniModalOpticalFlow@cref}{{[subsection][1][4,7]4.7.1}{92}}
1253
\newlabel{eq:EPEDefinition}{{4.46}{92}{Uni-Modal Data}{equation.4.7.46}{}}
1254
\newlabel{eq:EPEDefinition@cref}{{[equation][46][4]4.46}{92}}
1255
\newlabel{fig:hydrangea}{{4.4a}{93}{Subfigure 4 4.4a}{subfigure.4.4.1}{}}
1250
1256
\newlabel{sub@fig:hydrangea}{{(a)}{a}{Subfigure 4 4.4a\relax }{subfigure.4.4.1}{}}
1251
\newlabel{fig:hydrangea@cref}{{[subfigure][1][4,4]4.4a}{92}}
1252
\newlabel{fig:hydrangea-flow}{{4.4b}{92}{Subfigure 4 4.4b}{subfigure.4.4.2}{}}
1257
\newlabel{fig:hydrangea@cref}{{[subfigure][1][4,4]4.4a}{93}}
1258
\newlabel{fig:hydrangea-flow}{{4.4b}{93}{Subfigure 4 4.4b}{subfigure.4.4.2}{}}
1253
1259
\newlabel{sub@fig:hydrangea-flow}{{(b)}{b}{Subfigure 4 4.4b\relax }{subfigure.4.4.2}{}}
1254
\newlabel{fig:hydrangea-flow@cref}{{[subfigure][2][4,4]4.4b}{92}}
1255
\newlabel{fig:hydrangea-flowL1}{{4.4c}{92}{Subfigure 4 4.4c}{subfigure.4.4.3}{}}
1260
\newlabel{fig:hydrangea-flow@cref}{{[subfigure][2][4,4]4.4b}{93}}
1261
\newlabel{fig:hydrangea-flowL1}{{4.4c}{93}{Subfigure 4 4.4c}{subfigure.4.4.3}{}}
1256
1262
\newlabel{sub@fig:hydrangea-flowL1}{{(c)}{c}{Subfigure 4 4.4c\relax }{subfigure.4.4.3}{}}
1257
\newlabel{fig:hydrangea-flowL1@cref}{{[subfigure][3][4,4]4.4c}{92}}
1258
\newlabel{fig:hydrangea-gt2}{{4.4d}{92}{Subfigure 4 4.4d}{subfigure.4.4.4}{}}
1263
\newlabel{fig:hydrangea-flowL1@cref}{{[subfigure][3][4,4]4.4c}{93}}
1264
\newlabel{fig:hydrangea-gt2}{{4.4d}{93}{Subfigure 4 4.4d}{subfigure.4.4.4}{}}
1259
1265
\newlabel{sub@fig:hydrangea-gt2}{{(d)}{d}{Subfigure 4 4.4d\relax }{subfigure.4.4.4}{}}
1260
\newlabel{fig:hydrangea-gt2@cref}{{[subfigure][4][4,4]4.4d}{92}}
1261
\@writefile{lof}{\contentsline {figure}{\numberline {4.4}{\ignorespaces Hydrangea Sequence: Figure \ref {fig:hydrangea} shows one frame of the sequence. \cref {fig:hydrangea-flow} shows the estimated optical flow $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$, \cref {fig:hydrangea-flowL1} the flow $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$ and \cref {fig:hydrangea-gt2} shows the provided ground truth\relax }}{92}{figure.caption.23}}
1262
\newlabel{fig:hydrangeaSeq2}{{4.4}{92}{Hydrangea Sequence: Figure \ref {fig:hydrangea} shows one frame of the sequence. \figref {fig:hydrangea-flow} shows the estimated optical flow $\optimalflowST $, \figref {fig:hydrangea-flowL1} the flow $\optimalflowTV $ and \figref {fig:hydrangea-gt2} shows the provided ground truth\relax }{figure.caption.23}{}}
1263
\newlabel{fig:hydrangeaSeq2@cref}{{[figure][4][4]4.4}{92}}
1264
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{92}{subfigure.4.1}}
1265
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$}}}{92}{subfigure.4.2}}
1266
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$}}}{92}{subfigure.4.3}}
1267
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{92}{subfigure.4.4}}
1268
\newlabel{eq:meanCurv2}{{4.47}{92}{Uni-Modal Data}{equation.4.7.47}{}}
1269
\newlabel{eq:meanCurv2@cref}{{[equation][47][4]4.47}{92}}
1270
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.2}Rubber Whale Sequence}{92}{subsection.4.7.2}}
1271
\newlabel{sec:rubberWhale}{{4.7.2}{92}{Rubber Whale Sequence}{subsection.4.7.2}{}}
1272
\newlabel{sec:rubberWhale@cref}{{[subsection][2][4,7]4.7.2}{92}}
1273
\newlabel{fig:EPEToCurvST7}{{4.5a}{93}{Subfigure 4 4.5a}{subfigure.4.5.1}{}}
1266
\newlabel{fig:hydrangea-gt2@cref}{{[subfigure][4][4,4]4.4d}{93}}
1267
\@writefile{lof}{\contentsline {figure}{\numberline {4.4}{\ignorespaces Hydrangea Sequence: Figure \ref {fig:hydrangea} shows one frame of the sequence. \cref {fig:hydrangea-flow} shows the estimated optical flow $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$, \cref {fig:hydrangea-flowL1} the flow $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$ and \cref {fig:hydrangea-gt2} shows the provided ground truth\relax }}{93}{figure.caption.24}}
1268
\newlabel{fig:hydrangeaSeq2}{{4.4}{93}{Hydrangea Sequence: Figure \ref {fig:hydrangea} shows one frame of the sequence. \figref {fig:hydrangea-flow} shows the estimated optical flow $\optimalflowST $, \figref {fig:hydrangea-flowL1} the flow $\optimalflowTV $ and \figref {fig:hydrangea-gt2} shows the provided ground truth\relax }{figure.caption.24}{}}
1269
\newlabel{fig:hydrangeaSeq2@cref}{{[figure][4][4]4.4}{93}}
1270
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{93}{subfigure.4.1}}
1271
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$}}}{93}{subfigure.4.2}}
1272
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$}}}{93}{subfigure.4.3}}
1273
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{93}{subfigure.4.4}}
1274
\newlabel{eq:meanCurv2}{{4.47}{93}{Uni-Modal Data}{equation.4.7.47}{}}
1275
\newlabel{eq:meanCurv2@cref}{{[equation][47][4]4.47}{93}}
1276
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.2}Rubber Whale Sequence}{93}{subsection.4.7.2}}
1277
\newlabel{sec:rubberWhale}{{4.7.2}{93}{Rubber Whale Sequence}{subsection.4.7.2}{}}
1278
\newlabel{sec:rubberWhale@cref}{{[subsection][2][4,7]4.7.2}{93}}
1279
\newlabel{fig:EPEToCurvST7}{{4.5a}{94}{Subfigure 4 4.5a}{subfigure.4.5.1}{}}
1274
1280
\newlabel{sub@fig:EPEToCurvST7}{{(a)}{a}{Subfigure 4 4.5a\relax }{subfigure.4.5.1}{}}
1275
\newlabel{fig:EPEToCurvST7@cref}{{[subfigure][1][4,5]4.5a}{93}}
1276
\newlabel{fig:EPEToCurvST9}{{4.5b}{93}{Subfigure 4 4.5b}{subfigure.4.5.2}{}}
1281
\newlabel{fig:EPEToCurvST7@cref}{{[subfigure][1][4,5]4.5a}{94}}
1282
\newlabel{fig:EPEToCurvST9}{{4.5b}{94}{Subfigure 4 4.5b}{subfigure.4.5.2}{}}
1277
1283
\newlabel{sub@fig:EPEToCurvST9}{{(b)}{b}{Subfigure 4 4.5b\relax }{subfigure.4.5.2}{}}
1278
\newlabel{fig:EPEToCurvST9@cref}{{[subfigure][2][4,5]4.5b}{93}}
1279
\newlabel{fig:EPEToCurvST11}{{4.5c}{93}{Subfigure 4 4.5c}{subfigure.4.5.3}{}}
1284
\newlabel{fig:EPEToCurvST9@cref}{{[subfigure][2][4,5]4.5b}{94}}
1285
\newlabel{fig:EPEToCurvST11}{{4.5c}{94}{Subfigure 4 4.5c}{subfigure.4.5.3}{}}
1280
1286
\newlabel{sub@fig:EPEToCurvST11}{{(c)}{c}{Subfigure 4 4.5c\relax }{subfigure.4.5.3}{}}
1281
\newlabel{fig:EPEToCurvST11@cref}{{[subfigure][3][4,5]4.5c}{93}}
1282
\newlabel{fig:EPEToCurvTV}{{4.5d}{93}{Subfigure 4 4.5d}{subfigure.4.5.4}{}}
1287
\newlabel{fig:EPEToCurvST11@cref}{{[subfigure][3][4,5]4.5c}{94}}
1288
\newlabel{fig:EPEToCurvTV}{{4.5d}{94}{Subfigure 4 4.5d}{subfigure.4.5.4}{}}
1283
1289
\newlabel{sub@fig:EPEToCurvTV}{{(d)}{d}{Subfigure 4 4.5d\relax }{subfigure.4.5.4}{}}
1284
\newlabel{fig:EPEToCurvTV@cref}{{[subfigure][4][4,5]4.5d}{93}}
1285
\@writefile{lof}{\contentsline {figure}{\numberline {4.5}{\ignorespaces EPE to level-set curvature: Figures \ref {fig:EPEToCurvST7} to \ref {fig:EPEToCurvTV} show plots of the EPE (eq.~(\ref {eq:EPEDefinition})) against the curvature $\kappa $ (eq.~(\ref {eq:meanCurv2})) for the rubber whale sequence (\cref {fig:rubberWhaleSeq2}). Figures \ref {fig:EPEToCurvST7} to \ref {fig:EPEToCurvST11} show the results for the structure tensor model $E_{ST}$ and \cref {fig:EPEToCurvTV} the result for the TV model $E_{TV}$. The curvature $\kappa $ was split into $40$ bins and the height of the bars is the average EPE per curvature bin. \relax }}{93}{figure.caption.24}}
1286
\newlabel{fig:EPEToCurv}{{4.5}{93}{EPE to level-set curvature: Figures \ref {fig:EPEToCurvST7} to \ref {fig:EPEToCurvTV} show plots of the EPE (\eqref {eq:EPEDefinition}) against the curvature $\kappa $ (\eqref {eq:meanCurv2}) for the rubber whale sequence (\figref {fig:rubberWhaleSeq2}). Figures \ref {fig:EPEToCurvST7} to \ref {fig:EPEToCurvST11} show the results for the structure tensor model $E_{ST}$ and \figref {fig:EPEToCurvTV} the result for the TV model $E_{TV}$. The curvature $\kappa $ was split into $40$ bins and the height of the bars is the average EPE per curvature bin. \relax }{figure.caption.24}{}}
1287
\newlabel{fig:EPEToCurv@cref}{{[figure][5][4]4.5}{93}}
1288
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{93}{subfigure.5.1}}
1289
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{93}{subfigure.5.2}}
1290
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{93}{subfigure.5.3}}
1291
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{93}{subfigure.5.4}}
1292
\@writefile{lot}{\contentsline {table}{\numberline {4.1}{\ignorespaces EPE for different filter-sizes $\sigma _{ST}$ for the model $E^g_{ST}$ (eq.~(\ref {eq:optFlowModelST})) and for the TV model $E^g_{TV}$ (eq.~(\ref {eq:optFlowModelTV})). The value shown in the column \textsl {Median EPE} is the median EPE per ROI. The median per ROI was chosen over the average EPE per ROI due to its robustness towards outlier EPE values. The EPE values for the model $E^g_{ST}$ decrease with increasing structure tensor filtersizes $\sigma _{ST}$. However the general trend is that the ROI's with high curvatures $\kappa $ (\textsl {Wheel} and \textsl {Shell}) tend to have higher EPE values then the ROI's with low curvatures (\textsl {Fence} and \textsl {Box Edge}). \relax }}{93}{table.caption.25}}
1293
\newlabel{tab:rw-epeAE}{{4.1}{93}{EPE for different filter-sizes $\sigma _{ST}$ for the model $E^g_{ST}$ (\eqref {eq:optFlowModelST}) and for the TV model $E^g_{TV}$ (\eqref {eq:optFlowModelTV}). The value shown in the column \textsl {Median EPE} is the median EPE per ROI. The median per ROI was chosen over the average EPE per ROI due to its robustness towards outlier EPE values. The EPE values for the model $E^g_{ST}$ decrease with increasing structure tensor filtersizes $\sigma _{ST}$. However the general trend is that the ROI's with high curvatures $\kappa $ (\textsl {Wheel} and \textsl {Shell}) tend to have higher EPE values then the ROI's with low curvatures (\textsl {Fence} and \textsl {Box Edge}). \relax }{table.caption.25}{}}
1294
\newlabel{tab:rw-epeAE@cref}{{[table][1][4]4.1}{93}}
1295
\newlabel{fig:EPEToCurvST7hyd}{{4.6a}{94}{Subfigure 4 4.6a}{subfigure.4.6.1}{}}
1290
\newlabel{fig:EPEToCurvTV@cref}{{[subfigure][4][4,5]4.5d}{94}}
1291
\@writefile{lof}{\contentsline {figure}{\numberline {4.5}{\ignorespaces EPE to level-set curvature: Figures \ref {fig:EPEToCurvST7} to \ref {fig:EPEToCurvTV} show plots of the EPE (eq.~(\ref {eq:EPEDefinition})) against the curvature $\kappa $ (eq.~(\ref {eq:meanCurv2})) for the rubber whale sequence (\cref {fig:rubberWhaleSeq2}). Figures \ref {fig:EPEToCurvST7} to \ref {fig:EPEToCurvST11} show the results for the structure tensor model $E_{ST}$ and \cref {fig:EPEToCurvTV} the result for the TV model $E_{TV}$. The curvature $\kappa $ was split into $40$ bins and the height of the bars is the average EPE per curvature bin. \relax }}{94}{figure.caption.25}}
1292
\newlabel{fig:EPEToCurv}{{4.5}{94}{EPE to level-set curvature: Figures \ref {fig:EPEToCurvST7} to \ref {fig:EPEToCurvTV} show plots of the EPE (\eqref {eq:EPEDefinition}) against the curvature $\kappa $ (\eqref {eq:meanCurv2}) for the rubber whale sequence (\figref {fig:rubberWhaleSeq2}). Figures \ref {fig:EPEToCurvST7} to \ref {fig:EPEToCurvST11} show the results for the structure tensor model $E_{ST}$ and \figref {fig:EPEToCurvTV} the result for the TV model $E_{TV}$. The curvature $\kappa $ was split into $40$ bins and the height of the bars is the average EPE per curvature bin. \relax }{figure.caption.25}{}}
1293
\newlabel{fig:EPEToCurv@cref}{{[figure][5][4]4.5}{94}}
1294
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{94}{subfigure.5.1}}
1295
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{94}{subfigure.5.2}}
1296
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{94}{subfigure.5.3}}
1297
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{94}{subfigure.5.4}}
1298
\@writefile{lot}{\contentsline {table}{\numberline {4.1}{\ignorespaces EPE for different filter-sizes $\sigma _{ST}$ for the model $E^g_{ST}$ (eq.~(\ref {eq:optFlowModelST})) and for the TV model $E^g_{TV}$ (eq.~(\ref {eq:optFlowModelTV})). The value shown in the column \textsl {Median EPE} is the median EPE per ROI. The median per ROI was chosen over the average EPE per ROI due to its robustness towards outlier EPE values. The EPE values for the model $E^g_{ST}$ decrease with increasing structure tensor filtersizes $\sigma _{ST}$. However the general trend is that the ROI's with high curvatures $\kappa $ (\textsl {Wheel} and \textsl {Shell}) tend to have higher EPE values then the ROI's with low curvatures (\textsl {Fence} and \textsl {Box Edge}). \relax }}{94}{table.caption.26}}
1299
\newlabel{tab:rw-epeAE}{{4.1}{94}{EPE for different filter-sizes $\sigma _{ST}$ for the model $E^g_{ST}$ (\eqref {eq:optFlowModelST}) and for the TV model $E^g_{TV}$ (\eqref {eq:optFlowModelTV}). The value shown in the column \textsl {Median EPE} is the median EPE per ROI. The median per ROI was chosen over the average EPE per ROI due to its robustness towards outlier EPE values. The EPE values for the model $E^g_{ST}$ decrease with increasing structure tensor filtersizes $\sigma _{ST}$. However the general trend is that the ROI's with high curvatures $\kappa $ (\textsl {Wheel} and \textsl {Shell}) tend to have higher EPE values then the ROI's with low curvatures (\textsl {Fence} and \textsl {Box Edge}). \relax }{table.caption.26}{}}
1300
\newlabel{tab:rw-epeAE@cref}{{[table][1][4]4.1}{94}}
1301
\newlabel{fig:EPEToCurvST7hyd}{{4.6a}{95}{Subfigure 4 4.6a}{subfigure.4.6.1}{}}
1296
1302
\newlabel{sub@fig:EPEToCurvST7hyd}{{(a)}{a}{Subfigure 4 4.6a\relax }{subfigure.4.6.1}{}}
1297
\newlabel{fig:EPEToCurvST7hyd@cref}{{[subfigure][1][4,6]4.6a}{94}}
1298
\newlabel{fig:EPEToCurvST9hyd}{{4.6b}{94}{Subfigure 4 4.6b}{subfigure.4.6.2}{}}
1303
\newlabel{fig:EPEToCurvST7hyd@cref}{{[subfigure][1][4,6]4.6a}{95}}
1304
\newlabel{fig:EPEToCurvST9hyd}{{4.6b}{95}{Subfigure 4 4.6b}{subfigure.4.6.2}{}}
1299
1305
\newlabel{sub@fig:EPEToCurvST9hyd}{{(b)}{b}{Subfigure 4 4.6b\relax }{subfigure.4.6.2}{}}
1300
\newlabel{fig:EPEToCurvST9hyd@cref}{{[subfigure][2][4,6]4.6b}{94}}
1301
\newlabel{fig:EPEToCurvST11hyd}{{4.6c}{94}{Subfigure 4 4.6c}{subfigure.4.6.3}{}}
1306
\newlabel{fig:EPEToCurvST9hyd@cref}{{[subfigure][2][4,6]4.6b}{95}}
1307
\newlabel{fig:EPEToCurvST11hyd}{{4.6c}{95}{Subfigure 4 4.6c}{subfigure.4.6.3}{}}
1302
1308
\newlabel{sub@fig:EPEToCurvST11hyd}{{(c)}{c}{Subfigure 4 4.6c\relax }{subfigure.4.6.3}{}}
1303
\newlabel{fig:EPEToCurvST11hyd@cref}{{[subfigure][3][4,6]4.6c}{94}}
1304
\newlabel{fig:EPEToCurvTVhyd}{{4.6d}{94}{Subfigure 4 4.6d}{subfigure.4.6.4}{}}
1309
\newlabel{fig:EPEToCurvST11hyd@cref}{{[subfigure][3][4,6]4.6c}{95}}
1310
\newlabel{fig:EPEToCurvTVhyd}{{4.6d}{95}{Subfigure 4 4.6d}{subfigure.4.6.4}{}}
1305
1311
\newlabel{sub@fig:EPEToCurvTVhyd}{{(d)}{d}{Subfigure 4 4.6d\relax }{subfigure.4.6.4}{}}
1306
\newlabel{fig:EPEToCurvTVhyd@cref}{{[subfigure][4][4,6]4.6d}{94}}
1307
\@writefile{lof}{\contentsline {figure}{\numberline {4.6}{\ignorespaces EPE to level-set curvature: Figures \ref {fig:EPEToCurvST7hyd} to \ref {fig:EPEToCurvTVhyd} show plots of the EPE (eq.~(\ref {eq:EPEDefinition})) against the curvature $\kappa $ (eq.~(\ref {eq:meanCurv2})) for the hydrangea sequence (\cref {fig:hydrangeaSeq2}). Figures \ref {fig:EPEToCurvST7hyd} to \ref {fig:EPEToCurvST11hyd} show the results for the structure tensor model $E_{ST}$ and \cref {fig:EPEToCurvTV} the result for the TV model $E_{TV}$. The curvature $\kappa $ was split into $40$ bins and the height of the bars is the average EPE per curvature bin. \relax }}{94}{figure.caption.26}}
1308
\newlabel{fig:EPEToCurvHyd}{{4.6}{94}{EPE to level-set curvature: Figures \ref {fig:EPEToCurvST7hyd} to \ref {fig:EPEToCurvTVhyd} show plots of the EPE (\eqref {eq:EPEDefinition}) against the curvature $\kappa $ (\eqref {eq:meanCurv2}) for the hydrangea sequence (\figref {fig:hydrangeaSeq2}). Figures \ref {fig:EPEToCurvST7hyd} to \ref {fig:EPEToCurvST11hyd} show the results for the structure tensor model $E_{ST}$ and \figref {fig:EPEToCurvTV} the result for the TV model $E_{TV}$. The curvature $\kappa $ was split into $40$ bins and the height of the bars is the average EPE per curvature bin. \relax }{figure.caption.26}{}}
1309
\newlabel{fig:EPEToCurvHyd@cref}{{[figure][6][4]4.6}{94}}
1310
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{94}{subfigure.6.1}}
1311
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{94}{subfigure.6.2}}
1312
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{94}{subfigure.6.3}}
1313
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{94}{subfigure.6.4}}
1314
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.3}Hydrangea Sequence}{94}{subsection.4.7.3}}
1315
\newlabel{sec:hydrangea}{{4.7.3}{94}{Hydrangea Sequence}{subsection.4.7.3}{}}
1316
\newlabel{sec:hydrangea@cref}{{[subsection][3][4,7]4.7.3}{94}}
1317
\newlabel{fig:synthMultModalRWFrame10}{{4.7a}{95}{Subfigure 4 4.7a}{subfigure.4.7.1}{}}
1312
\newlabel{fig:EPEToCurvTVhyd@cref}{{[subfigure][4][4,6]4.6d}{95}}
1313
\@writefile{lof}{\contentsline {figure}{\numberline {4.6}{\ignorespaces EPE to level-set curvature: Figures \ref {fig:EPEToCurvST7hyd} to \ref {fig:EPEToCurvTVhyd} show plots of the EPE (eq.~(\ref {eq:EPEDefinition})) against the curvature $\kappa $ (eq.~(\ref {eq:meanCurv2})) for the hydrangea sequence (\cref {fig:hydrangeaSeq2}). Figures \ref {fig:EPEToCurvST7hyd} to \ref {fig:EPEToCurvST11hyd} show the results for the structure tensor model $E_{ST}$ and \cref {fig:EPEToCurvTV} the result for the TV model $E_{TV}$. The curvature $\kappa $ was split into $40$ bins and the height of the bars is the average EPE per curvature bin. \relax }}{95}{figure.caption.27}}
1314
\newlabel{fig:EPEToCurvHyd}{{4.6}{95}{EPE to level-set curvature: Figures \ref {fig:EPEToCurvST7hyd} to \ref {fig:EPEToCurvTVhyd} show plots of the EPE (\eqref {eq:EPEDefinition}) against the curvature $\kappa $ (\eqref {eq:meanCurv2}) for the hydrangea sequence (\figref {fig:hydrangeaSeq2}). Figures \ref {fig:EPEToCurvST7hyd} to \ref {fig:EPEToCurvST11hyd} show the results for the structure tensor model $E_{ST}$ and \figref {fig:EPEToCurvTV} the result for the TV model $E_{TV}$. The curvature $\kappa $ was split into $40$ bins and the height of the bars is the average EPE per curvature bin. \relax }{figure.caption.27}{}}
1315
\newlabel{fig:EPEToCurvHyd@cref}{{[figure][6][4]4.6}{95}}
1316
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{95}{subfigure.6.1}}
1317
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{95}{subfigure.6.2}}
1318
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{95}{subfigure.6.3}}
1319
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{95}{subfigure.6.4}}
1320
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.3}Hydrangea Sequence}{95}{subsection.4.7.3}}
1321
\newlabel{sec:hydrangea}{{4.7.3}{95}{Hydrangea Sequence}{subsection.4.7.3}{}}
1322
\newlabel{sec:hydrangea@cref}{{[subsection][3][4,7]4.7.3}{95}}
1323
\newlabel{fig:synthMultModalRWFrame10}{{4.7a}{96}{Subfigure 4 4.7a}{subfigure.4.7.1}{}}
1318
1324
\newlabel{sub@fig:synthMultModalRWFrame10}{{(a)}{a}{Subfigure 4 4.7a\relax }{subfigure.4.7.1}{}}
1319
\newlabel{fig:synthMultModalRWFrame10@cref}{{[subfigure][1][4,7]4.7a}{95}}
1320
\newlabel{fig:synthMultModalRWYCoAlScale2Frame10}{{4.7b}{95}{Subfigure 4 4.7b}{subfigure.4.7.2}{}}
1325
\newlabel{fig:synthMultModalRWFrame10@cref}{{[subfigure][1][4,7]4.7a}{96}}
1326
\newlabel{fig:synthMultModalRWYCoAlScale2Frame10}{{4.7b}{96}{Subfigure 4 4.7b}{subfigure.4.7.2}{}}
1321
1327
\newlabel{sub@fig:synthMultModalRWYCoAlScale2Frame10}{{(b)}{b}{Subfigure 4 4.7b\relax }{subfigure.4.7.2}{}}
1322
\newlabel{fig:synthMultModalRWYCoAlScale2Frame10@cref}{{[subfigure][2][4,7]4.7b}{95}}
1323
\newlabel{fig:synthMultModalRWYCoAlScale4Frame10}{{4.7c}{95}{Subfigure 4 4.7c}{subfigure.4.7.3}{}}
1328
\newlabel{fig:synthMultModalRWYCoAlScale2Frame10@cref}{{[subfigure][2][4,7]4.7b}{96}}
1329
\newlabel{fig:synthMultModalRWYCoAlScale4Frame10}{{4.7c}{96}{Subfigure 4 4.7c}{subfigure.4.7.3}{}}
1324
1330
\newlabel{sub@fig:synthMultModalRWYCoAlScale4Frame10}{{(c)}{c}{Subfigure 4 4.7c\relax }{subfigure.4.7.3}{}}
1325
\newlabel{fig:synthMultModalRWYCoAlScale4Frame10@cref}{{[subfigure][3][4,7]4.7c}{95}}
1326
\@writefile{lof}{\contentsline {figure}{\numberline {4.7}{\ignorespaces Synthesized multi-modal data. This data simulates the camera arrangement in \cref {fig:multiModalCoAligned}. The image $I$ in \cref {fig:synthMultModalRWFrame10} is from the rubberwhale data set in \cref {fig:rubberWhaleSeq2}. Figures \ref {fig:synthMultModalRWYCoAlScale2Frame10} and \ref {fig:synthMultModalRWYCoAlScale4Frame10} show the image $y_{{\ensuremath {{\sigma ^{sc}}}_{test}}}$ (eq.~(\ref {eq:synthMultiModalCoAlignedYLow})) at the scales $\ensuremath {{\sigma ^{sc}}}_{test}=2$ and $\ensuremath {{\sigma ^{sc}}}_{test}=4$\relax }}{95}{figure.caption.27}}
1327
\newlabel{fig:synthMultModalRWYCoAl}{{4.7}{95}{Synthesized multi-modal data. This data simulates the camera arrangement in \figref {fig:multiModalCoAligned}. The image $I$ in \figref {fig:synthMultModalRWFrame10} is from the rubberwhale data set in \figref {fig:rubberWhaleSeq2}. Figures \ref {fig:synthMultModalRWYCoAlScale2Frame10} and \ref {fig:synthMultModalRWYCoAlScale4Frame10} show the image $y_{{\scalediff _{test}}}$ (\eqref {eq:synthMultiModalCoAlignedYLow}) at the scales $\scalediff _{test}=2$ and $\scalediff _{test}=4$\relax }{figure.caption.27}{}}
1328
\newlabel{fig:synthMultModalRWYCoAl@cref}{{[figure][7][4]4.7}{95}}
1329
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$I$}}}{95}{subfigure.7.1}}
1330
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$y_2$}}}{95}{subfigure.7.2}}
1331
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$y_4$}}}{95}{subfigure.7.3}}
1332
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.4}Estimation of the Scale Difference $\ensuremath {{\sigma ^{sc}}}$}{95}{subsection.4.7.4}}
1333
\newlabel{sec:synthMultiModalScaleDiff}{{4.7.4}{95}{Estimation of the Scale Difference $\scalediff $}{subsection.4.7.4}{}}
1334
\newlabel{sec:synthMultiModalScaleDiff@cref}{{[subsection][4][4,7]4.7.4}{95}}
1335
\newlabel{eq:synthMultiModalCoAlignedYHigh}{{4.49}{95}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.49}{}}
1336
\newlabel{eq:synthMultiModalCoAlignedYHigh@cref}{{[equation][49][4]4.49}{95}}
1337
\newlabel{eq:synthMultiModalCoAlignedYLow}{{4.50}{95}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.50}{}}
1338
\newlabel{eq:synthMultiModalCoAlignedYLow@cref}{{[equation][50][4]4.50}{95}}
1331
\newlabel{fig:synthMultModalRWYCoAlScale4Frame10@cref}{{[subfigure][3][4,7]4.7c}{96}}
1332
\@writefile{lof}{\contentsline {figure}{\numberline {4.7}{\ignorespaces Synthesized multi-modal data. This data simulates the camera arrangement in \cref {fig:multiModalCoAligned}. The image $I$ in \cref {fig:synthMultModalRWFrame10} is from the rubberwhale data set in \cref {fig:rubberWhaleSeq2}. Figures \ref {fig:synthMultModalRWYCoAlScale2Frame10} and \ref {fig:synthMultModalRWYCoAlScale4Frame10} show the image $y_{{\ensuremath {{\sigma ^{sc}}}_{test}}}$ (eq.~(\ref {eq:synthMultiModalCoAlignedYLow})) at the scales $\ensuremath {{\sigma ^{sc}}}_{test}=2$ and $\ensuremath {{\sigma ^{sc}}}_{test}=4$\relax }}{96}{figure.caption.28}}
1333
\newlabel{fig:synthMultModalRWYCoAl}{{4.7}{96}{Synthesized multi-modal data. This data simulates the camera arrangement in \figref {fig:multiModalCoAligned}. The image $I$ in \figref {fig:synthMultModalRWFrame10} is from the rubberwhale data set in \figref {fig:rubberWhaleSeq2}. Figures \ref {fig:synthMultModalRWYCoAlScale2Frame10} and \ref {fig:synthMultModalRWYCoAlScale4Frame10} show the image $y_{{\scalediff _{test}}}$ (\eqref {eq:synthMultiModalCoAlignedYLow}) at the scales $\scalediff _{test}=2$ and $\scalediff _{test}=4$\relax }{figure.caption.28}{}}
1334
\newlabel{fig:synthMultModalRWYCoAl@cref}{{[figure][7][4]4.7}{96}}
1335
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$I$}}}{96}{subfigure.7.1}}
1336
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$y_2$}}}{96}{subfigure.7.2}}
1337
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$y_4$}}}{96}{subfigure.7.3}}
1338
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.4}Estimation of the Scale Difference $\ensuremath {{\sigma ^{sc}}}$}{96}{subsection.4.7.4}}
1339
\newlabel{sec:synthMultiModalScaleDiff}{{4.7.4}{96}{Estimation of the Scale Difference $\scalediff $}{subsection.4.7.4}{}}
1340
\newlabel{sec:synthMultiModalScaleDiff@cref}{{[subsection][4][4,7]4.7.4}{96}}
1341
\newlabel{eq:synthMultiModalCoAlignedYHigh}{{4.49}{96}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.49}{}}
1342
\newlabel{eq:synthMultiModalCoAlignedYHigh@cref}{{[equation][49][4]4.49}{96}}
1343
\newlabel{eq:synthMultiModalCoAlignedYLow}{{4.50}{96}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.50}{}}
1344
\newlabel{eq:synthMultiModalCoAlignedYLow@cref}{{[equation][50][4]4.50}{96}}
1339
1345
\citation{ferreiraCFRP,khanCFRP,tehraniCFRP,FanCFRP}
1340
1346
\citation{FanCFRP}
1341
1347
\citation{wuLockIn,SpiessbergerFusionLockin,meolaLockIn}
1342
\newlabel{fig:EdataCoAlScale2}{{4.8a}{96}{Subfigure 4 4.8a}{subfigure.4.8.1}{}}
1348
\newlabel{fig:EdataCoAlScale2}{{4.8a}{97}{Subfigure 4 4.8a}{subfigure.4.8.1}{}}
1343
1349
\newlabel{sub@fig:EdataCoAlScale2}{{(a)}{a}{Subfigure 4 4.8a\relax }{subfigure.4.8.1}{}}
1344
\newlabel{fig:EdataCoAlScale2@cref}{{[subfigure][1][4,8]4.8a}{96}}
1345
\newlabel{fig:EdataCoAlScale4}{{4.8b}{96}{Subfigure 4 4.8b}{subfigure.4.8.2}{}}
1350
\newlabel{fig:EdataCoAlScale2@cref}{{[subfigure][1][4,8]4.8a}{97}}
1351
\newlabel{fig:EdataCoAlScale4}{{4.8b}{97}{Subfigure 4 4.8b}{subfigure.4.8.2}{}}
1346
1352
\newlabel{sub@fig:EdataCoAlScale4}{{(b)}{b}{Subfigure 4 4.8b\relax }{subfigure.4.8.2}{}}
1347
\newlabel{fig:EdataCoAlScale4@cref}{{[subfigure][2][4,8]4.8b}{96}}
1348
\@writefile{lof}{\contentsline {figure}{\numberline {4.8}{\ignorespaces Figures \ref {fig:EdataCoAlScale2} and \ref {fig:EdataCoAlScale4} show plots the similarity measure $E^{data}_{y,I}(\ensuremath {{\sigma ^{sc}}},\ensuremath {\ensuremath {{\bm {d}}}})$ for the cases $y=y_2$, $\ensuremath {{\sigma ^{sc}}}_{test}=2$, and $y=y_4$, $\ensuremath {{\sigma ^{sc}}}_{test}=4$. We can observe that $E^{data}_{y,I}(\ensuremath {{\sigma ^{sc}}},\ensuremath {\ensuremath {{\bm {d}}}})$ is minimal with respect to $\ensuremath {{\sigma ^{sc}}}$ at the correct scales $\ensuremath {{\sigma ^{sc}}}_{test}$\relax }}{96}{figure.caption.28}}
1349
\newlabel{fig:EdataCoAlScales}{{4.8}{96}{Figures \ref {fig:EdataCoAlScale2} and \ref {fig:EdataCoAlScale4} show plots the similarity measure $E^{data}_{y,I}(\scalediff ,\vd )$ for the cases $y=y_2$, $\scalediff _{test}=2$, and $y=y_4$, $\scalediff _{test}=4$. We can observe that $E^{data}_{y,I}(\scalediff ,\vd )$ is minimal with respect to $\scalediff $ at the correct scales $\scalediff _{test}$\relax }{figure.caption.28}{}}
1350
\newlabel{fig:EdataCoAlScales@cref}{{[figure][8][4]4.8}{96}}
1351
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$y=y_2$}}}{96}{subfigure.8.1}}
1352
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$y=y_4$}}}{96}{subfigure.8.2}}
1353
\newlabel{eq:flowDataTerm3}{{4.51}{96}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.51}{}}
1354
\newlabel{eq:flowDataTerm3@cref}{{[equation][51][4]4.51}{96}}
1355
\newlabel{eq:globalIntensFactor2}{{4.52}{96}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.52}{}}
1356
\newlabel{eq:globalIntensFactor2@cref}{{[equation][52][4]4.52}{96}}
1357
\newlabel{eq:likelihood2}{{4.54}{96}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.54}{}}
1358
\newlabel{eq:likelihood2@cref}{{[equation][54][4]4.54}{96}}
1359
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.5}Real Multimodal Optical Flow Data}{96}{subsection.4.7.5}}
1360
\newlabel{sec:realMultiModalOpticalFlow}{{4.7.5}{96}{Real Multimodal Optical Flow Data}{subsection.4.7.5}{}}
1361
\newlabel{sec:realMultiModalOpticalFlow@cref}{{[subsection][5][4,7]4.7.5}{96}}
1362
\newlabel{fig:multiModalVSC2}{{4.9a}{97}{Subfigure 4 4.9a}{subfigure.4.9.1}{}}
1353
\newlabel{fig:EdataCoAlScale4@cref}{{[subfigure][2][4,8]4.8b}{97}}
1354
\@writefile{lof}{\contentsline {figure}{\numberline {4.8}{\ignorespaces Figures \ref {fig:EdataCoAlScale2} and \ref {fig:EdataCoAlScale4} show plots the similarity measure $E^{data}_{y,I}(\ensuremath {{\sigma ^{sc}}},\ensuremath {\ensuremath {{\bm {d}}}})$ for the cases $y=y_2$, $\ensuremath {{\sigma ^{sc}}}_{test}=2$, and $y=y_4$, $\ensuremath {{\sigma ^{sc}}}_{test}=4$. We can observe that $E^{data}_{y,I}(\ensuremath {{\sigma ^{sc}}},\ensuremath {\ensuremath {{\bm {d}}}})$ is minimal with respect to $\ensuremath {{\sigma ^{sc}}}$ at the correct scales $\ensuremath {{\sigma ^{sc}}}_{test}$\relax }}{97}{figure.caption.29}}
1355
\newlabel{fig:EdataCoAlScales}{{4.8}{97}{Figures \ref {fig:EdataCoAlScale2} and \ref {fig:EdataCoAlScale4} show plots the similarity measure $E^{data}_{y,I}(\scalediff ,\vd )$ for the cases $y=y_2$, $\scalediff _{test}=2$, and $y=y_4$, $\scalediff _{test}=4$. We can observe that $E^{data}_{y,I}(\scalediff ,\vd )$ is minimal with respect to $\scalediff $ at the correct scales $\scalediff _{test}$\relax }{figure.caption.29}{}}
1356
\newlabel{fig:EdataCoAlScales@cref}{{[figure][8][4]4.8}{97}}
1357
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$y=y_2$}}}{97}{subfigure.8.1}}
1358
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$y=y_4$}}}{97}{subfigure.8.2}}
1359
\newlabel{eq:flowDataTerm3}{{4.51}{97}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.51}{}}
1360
\newlabel{eq:flowDataTerm3@cref}{{[equation][51][4]4.51}{97}}
1361
\newlabel{eq:globalIntensFactor2}{{4.52}{97}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.52}{}}
1362
\newlabel{eq:globalIntensFactor2@cref}{{[equation][52][4]4.52}{97}}
1363
\newlabel{eq:likelihood2}{{4.54}{97}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.54}{}}
1364
\newlabel{eq:likelihood2@cref}{{[equation][54][4]4.54}{97}}
1365
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.5}Real Multimodal Optical Flow Data}{97}{subsection.4.7.5}}
1366
\newlabel{sec:realMultiModalOpticalFlow}{{4.7.5}{97}{Real Multimodal Optical Flow Data}{subsection.4.7.5}{}}
1367
\newlabel{sec:realMultiModalOpticalFlow@cref}{{[subsection][5][4,7]4.7.5}{97}}
1368
\newlabel{fig:multiModalVSC2}{{4.9a}{98}{Subfigure 4 4.9a}{subfigure.4.9.1}{}}
1363
1369
\newlabel{sub@fig:multiModalVSC2}{{(a)}{a}{Subfigure 4 4.9a\relax }{subfigure.4.9.1}{}}
1364
\newlabel{fig:multiModalVSC2@cref}{{[subfigure][1][4,9]4.9a}{97}}
1365
\newlabel{fig:multiModalTC2}{{4.9b}{97}{Subfigure 4 4.9b}{subfigure.4.9.2}{}}
1370
\newlabel{fig:multiModalVSC2@cref}{{[subfigure][1][4,9]4.9a}{98}}
1371
\newlabel{fig:multiModalTC2}{{4.9b}{98}{Subfigure 4 4.9b}{subfigure.4.9.2}{}}
1366
1372
\newlabel{sub@fig:multiModalTC2}{{(b)}{b}{Subfigure 4 4.9b\relax }{subfigure.4.9.2}{}}
1367
\newlabel{fig:multiModalTC2@cref}{{[subfigure][2][4,9]4.9b}{97}}
1368
\newlabel{fig:multiModalHisto2}{{4.9c}{97}{Subfigure 4 4.9c}{subfigure.4.9.3}{}}
1373
\newlabel{fig:multiModalTC2@cref}{{[subfigure][2][4,9]4.9b}{98}}
1374
\newlabel{fig:multiModalHisto2}{{4.9c}{98}{Subfigure 4 4.9c}{subfigure.4.9.3}{}}
1369
1375
\newlabel{sub@fig:multiModalHisto2}{{(c)}{c}{Subfigure 4 4.9c\relax }{subfigure.4.9.3}{}}
1370
\newlabel{fig:multiModalHisto2@cref}{{[subfigure][3][4,9]4.9c}{97}}
1371
\@writefile{lof}{\contentsline {figure}{\numberline {4.9}{\ignorespaces \cref {fig:multiModalVSC2} shows an image from a visual spectrum camera (VSC). The object recorded is a carbon-fiber reinforced polymer (CFRP). \Cref {fig:multiModalTC2} shows an image of the same CFRP recorded with a thermographic camera (TC). The TC is sensitive in the infra-red domain, thus higher intensities in \cref {fig:multiModalTC2} correspond to warmer objects (the CFRP) and lower intensities to colder objects (the background). As in \cref {fig:multiModalSetupDiffRes} the optical centers of the VSC and the TC are physically separated so the problem that is being addressed is that of finding the optical flow field $\ensuremath {\ensuremath {{\bm {d}}}(\ensuremath {{\bm {x}}})}$ (see eq.~(\ref {eq:opticalFlowDef})) which maps every pixel in the TC image to the corresponding pixel in the VSC image. \Cref {fig:multiModalHisto2} shows the joint histogram of the VSC and TC image. It shows a complex mapping of the intensities of \cref {fig:multiModalVSC2} to those of \cref {fig:multiModalTC2} indicating that a linearity assumption between the TC and the VSC is not valid\relax }}{97}{figure.caption.29}}
1372
\newlabel{fig:multiModalTCVSC2}{{4.9}{97}{\figref {fig:multiModalVSC2} shows an image from a visual spectrum camera (VSC). The object recorded is a carbon-fiber reinforced polymer (CFRP). \Figref {fig:multiModalTC2} shows an image of the same CFRP recorded with a thermographic camera (TC). The TC is sensitive in the infra-red domain, thus higher intensities in \figref {fig:multiModalTC2} correspond to warmer objects (the CFRP) and lower intensities to colder objects (the background). As in \figref {fig:multiModalSetupDiffRes} the optical centers of the VSC and the TC are physically separated so the problem that is being addressed is that of finding the optical flow field $\vdx $ (see \eqref {eq:opticalFlowDef}) which maps every pixel in the TC image to the corresponding pixel in the VSC image. \Figref {fig:multiModalHisto2} shows the joint histogram of the VSC and TC image. It shows a complex mapping of the intensities of \figref {fig:multiModalVSC2} to those of \figref {fig:multiModalTC2} indicating that a linearity assumption between the TC and the VSC is not valid\relax }{figure.caption.29}{}}
1373
\newlabel{fig:multiModalTCVSC2@cref}{{[figure][9][4]4.9}{97}}
1374
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{97}{subfigure.9.1}}
1375
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{97}{subfigure.9.2}}
1376
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{97}{subfigure.9.3}}
1377
\@writefile{brf}{\backcite{ferreiraCFRP}{{97}{4.7.5}{figure.caption.29}}}
1378
\@writefile{brf}{\backcite{khanCFRP}{{97}{4.7.5}{figure.caption.29}}}
1379
\@writefile{brf}{\backcite{tehraniCFRP}{{97}{4.7.5}{figure.caption.29}}}
1380
\@writefile{brf}{\backcite{FanCFRP}{{97}{4.7.5}{figure.caption.29}}}
1381
\@writefile{brf}{\backcite{FanCFRP}{{97}{4.7.5}{figure.caption.29}}}
1382
\@writefile{brf}{\backcite{wuLockIn}{{97}{4.7.5}{figure.caption.29}}}
1383
\@writefile{brf}{\backcite{SpiessbergerFusionLockin}{{97}{4.7.5}{figure.caption.29}}}
1384
\@writefile{brf}{\backcite{meolaLockIn}{{97}{4.7.5}{figure.caption.29}}}
1385
\newlabel{fig:EdatalocA21}{{4.10a}{98}{Subfigure 4 4.10a}{subfigure.4.10.1}{}}
1376
\newlabel{fig:multiModalHisto2@cref}{{[subfigure][3][4,9]4.9c}{98}}
1377
\@writefile{lof}{\contentsline {figure}{\numberline {4.9}{\ignorespaces \cref {fig:multiModalVSC2} shows an image from a visual spectrum camera (VSC). The object recorded is a carbon-fiber reinforced polymer (CFRP). \Cref {fig:multiModalTC2} shows an image of the same CFRP recorded with a thermographic camera (TC). The TC is sensitive in the infra-red domain, thus higher intensities in \cref {fig:multiModalTC2} correspond to warmer objects (the CFRP) and lower intensities to colder objects (the background). As in \cref {fig:multiModalSetupDiffRes} the optical centers of the VSC and the TC are physically separated so the problem that is being addressed is that of finding the optical flow field $\ensuremath {\ensuremath {{\bm {d}}}(\ensuremath {{\bm {x}}})}$ (see eq.~(\ref {eq:opticalFlowDef})) which maps every pixel in the TC image to the corresponding pixel in the VSC image. \Cref {fig:multiModalHisto2} shows the joint histogram of the VSC and TC image. It shows a complex mapping of the intensities of \cref {fig:multiModalVSC2} to those of \cref {fig:multiModalTC2} indicating that a linearity assumption between the TC and the VSC is not valid\relax }}{98}{figure.caption.30}}
1378
\newlabel{fig:multiModalTCVSC2}{{4.9}{98}{\figref {fig:multiModalVSC2} shows an image from a visual spectrum camera (VSC). The object recorded is a carbon-fiber reinforced polymer (CFRP). \Figref {fig:multiModalTC2} shows an image of the same CFRP recorded with a thermographic camera (TC). The TC is sensitive in the infra-red domain, thus higher intensities in \figref {fig:multiModalTC2} correspond to warmer objects (the CFRP) and lower intensities to colder objects (the background). As in \figref {fig:multiModalSetupDiffRes} the optical centers of the VSC and the TC are physically separated so the problem that is being addressed is that of finding the optical flow field $\vdx $ (see \eqref {eq:opticalFlowDef}) which maps every pixel in the TC image to the corresponding pixel in the VSC image. \Figref {fig:multiModalHisto2} shows the joint histogram of the VSC and TC image. It shows a complex mapping of the intensities of \figref {fig:multiModalVSC2} to those of \figref {fig:multiModalTC2} indicating that a linearity assumption between the TC and the VSC is not valid\relax }{figure.caption.30}{}}
1379
\newlabel{fig:multiModalTCVSC2@cref}{{[figure][9][4]4.9}{98}}
1380
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{98}{subfigure.9.1}}
1381
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{98}{subfigure.9.2}}
1382
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{98}{subfigure.9.3}}
1383
\@writefile{brf}{\backcite{ferreiraCFRP}{{98}{4.7.5}{figure.caption.30}}}
1384
\@writefile{brf}{\backcite{khanCFRP}{{98}{4.7.5}{figure.caption.30}}}
1385
\@writefile{brf}{\backcite{tehraniCFRP}{{98}{4.7.5}{figure.caption.30}}}
1386
\@writefile{brf}{\backcite{FanCFRP}{{98}{4.7.5}{figure.caption.30}}}
1387
\@writefile{brf}{\backcite{FanCFRP}{{98}{4.7.5}{figure.caption.30}}}
1388
\@writefile{brf}{\backcite{wuLockIn}{{98}{4.7.5}{figure.caption.30}}}
1389
\@writefile{brf}{\backcite{SpiessbergerFusionLockin}{{98}{4.7.5}{figure.caption.30}}}
1390
\@writefile{brf}{\backcite{meolaLockIn}{{98}{4.7.5}{figure.caption.30}}}
1391
\newlabel{fig:EdatalocA21}{{4.10a}{99}{Subfigure 4 4.10a}{subfigure.4.10.1}{}}
1386
1392
\newlabel{sub@fig:EdatalocA21}{{(a)}{a}{Subfigure 4 4.10a\relax }{subfigure.4.10.1}{}}
1387
\newlabel{fig:EdatalocA21@cref}{{[subfigure][1][4,10]4.10a}{98}}
1388
\newlabel{fig:scaleOverA}{{4.10b}{98}{Subfigure 4 4.10b}{subfigure.4.10.2}{}}
1393
\newlabel{fig:EdatalocA21@cref}{{[subfigure][1][4,10]4.10a}{99}}
1394
\newlabel{fig:scaleOverA}{{4.10b}{99}{Subfigure 4 4.10b}{subfigure.4.10.2}{}}
1389
1395
\newlabel{sub@fig:scaleOverA}{{(b)}{b}{Subfigure 4 4.10b\relax }{subfigure.4.10.2}{}}
1390
\newlabel{fig:scaleOverA@cref}{{[subfigure][2][4,10]4.10b}{98}}
1391
\newlabel{fig:EdatalocOverA}{{4.10c}{98}{Subfigure 4 4.10c}{subfigure.4.10.3}{}}
1396
\newlabel{fig:scaleOverA@cref}{{[subfigure][2][4,10]4.10b}{99}}
1397
\newlabel{fig:EdatalocOverA}{{4.10c}{99}{Subfigure 4 4.10c}{subfigure.4.10.3}{}}
1392
1398
\newlabel{sub@fig:EdatalocOverA}{{(c)}{c}{Subfigure 4 4.10c\relax }{subfigure.4.10.3}{}}
1393
\newlabel{fig:EdatalocOverA@cref}{{[subfigure][3][4,10]4.10c}{98}}
1394
\@writefile{lof}{\contentsline {figure}{\numberline {4.10}{\ignorespaces \Cref {fig:EdatalocA21}: Plot $E^{data,l}_{y_{tc},I_{vsc}}(\ensuremath {{\sigma ^{sc}}},a,\ensuremath {{\bm {0}}})$ over the PSF scale difference $\ensuremath {{\sigma ^{sc}}}$ for the images $y_{tc}$ and $I_{vsc}$ in \cref {fig:multiModalTCVSC2} for the window size $a=25$. \Cref {fig:scaleOverA} shows the minimum scale $\ensuremath {{\sigma ^{sc}}}_{min}$ defined in eq.~(\ref {eq:scaleMin}) as a function over the window size $a$ and \cref {fig:EdatalocOverA} the similarity measure $E^{data,l}_{min}$ (eq.~(\ref {eq:EdataScaleMin})) over $a$. The minimum scale $\ensuremath {{\sigma ^{sc}}}_{min}$ increases or stays constant but does not decrease for larger window sizes $a$. The window size $a=21$ marks a sweet spot where $\ensuremath {{\sigma ^{sc}}}_{min}(21)= \ensuremath {\sigma ^{sc,\star }}=3$ while $E^{data,l}_{min}(21)$ is comparatively minimal.\relax }}{98}{figure.caption.30}}
1395
\newlabel{fig:EdataSigmaAdependence}{{4.10}{98}{\Figref {fig:EdatalocA21}: Plot $E^{data,l}_{y_{tc},I_{vsc}}(\scalediff ,a,\vector {0})$ over the PSF scale difference $\scalediff $ for the images $y_{tc}$ and $I_{vsc}$ in \figref {fig:multiModalTCVSC2} for the window size $a=25$. \Figref {fig:scaleOverA} shows the minimum scale $\scalediff _{min}$ defined in \eqref {eq:scaleMin} as a function over the window size $a$ and \figref {fig:EdatalocOverA} the similarity measure $E^{data,l}_{min}$ (\eqref {eq:EdataScaleMin}) over $a$. The minimum scale $\scalediff _{min}$ increases or stays constant but does not decrease for larger window sizes $a$. The window size $a=21$ marks a sweet spot where $\scalediff _{min}(21)= \optscalediff =3$ while $E^{data,l}_{min}(21)$ is comparatively minimal.\relax }{figure.caption.30}{}}
1396
\newlabel{fig:EdataSigmaAdependence@cref}{{[figure][10][4]4.10}{98}}
1397
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{98}{subfigure.10.1}}
1398
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{98}{subfigure.10.2}}
1399
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{98}{subfigure.10.3}}
1400
\newlabel{eq:flowDataTermLocal2}{{4.56}{98}{Real Multimodal Optical Flow Data}{equation.4.7.56}{}}
1401
\newlabel{eq:flowDataTermLocal2@cref}{{[equation][56][4]4.56}{98}}
1402
\newlabel{fig:flowST}{{4.11a}{99}{Subfigure 4 4.11a}{subfigure.4.11.1}{}}
1399
\newlabel{fig:EdatalocOverA@cref}{{[subfigure][3][4,10]4.10c}{99}}
1400
\@writefile{lof}{\contentsline {figure}{\numberline {4.10}{\ignorespaces \Cref {fig:EdatalocA21}: Plot $E^{data,l}_{y_{tc},I_{vsc}}(\ensuremath {{\sigma ^{sc}}},a,\ensuremath {{\bm {0}}})$ over the PSF scale difference $\ensuremath {{\sigma ^{sc}}}$ for the images $y_{tc}$ and $I_{vsc}$ in \cref {fig:multiModalTCVSC2} for the window size $a=25$. \Cref {fig:scaleOverA} shows the minimum scale $\ensuremath {{\sigma ^{sc}}}_{min}$ defined in eq.~(\ref {eq:scaleMin}) as a function over the window size $a$ and \cref {fig:EdatalocOverA} the similarity measure $E^{data,l}_{min}$ (eq.~(\ref {eq:EdataScaleMin})) over $a$. The minimum scale $\ensuremath {{\sigma ^{sc}}}_{min}$ increases or stays constant but does not decrease for larger window sizes $a$. The window size $a=21$ marks a sweet spot where $\ensuremath {{\sigma ^{sc}}}_{min}(21)= \ensuremath {\sigma ^{sc,\star }}=3$ while $E^{data,l}_{min}(21)$ is comparatively minimal.\relax }}{99}{figure.caption.31}}
1401
\newlabel{fig:EdataSigmaAdependence}{{4.10}{99}{\Figref {fig:EdatalocA21}: Plot $E^{data,l}_{y_{tc},I_{vsc}}(\scalediff ,a,\vector {0})$ over the PSF scale difference $\scalediff $ for the images $y_{tc}$ and $I_{vsc}$ in \figref {fig:multiModalTCVSC2} for the window size $a=25$. \Figref {fig:scaleOverA} shows the minimum scale $\scalediff _{min}$ defined in \eqref {eq:scaleMin} as a function over the window size $a$ and \figref {fig:EdatalocOverA} the similarity measure $E^{data,l}_{min}$ (\eqref {eq:EdataScaleMin}) over $a$. The minimum scale $\scalediff _{min}$ increases or stays constant but does not decrease for larger window sizes $a$. The window size $a=21$ marks a sweet spot where $\scalediff _{min}(21)= \optscalediff =3$ while $E^{data,l}_{min}(21)$ is comparatively minimal.\relax }{figure.caption.31}{}}
1402
\newlabel{fig:EdataSigmaAdependence@cref}{{[figure][10][4]4.10}{99}}
1403
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{99}{subfigure.10.1}}
1404
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{99}{subfigure.10.2}}
1405
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{99}{subfigure.10.3}}
1406
\newlabel{eq:flowDataTermLocal2}{{4.56}{99}{Real Multimodal Optical Flow Data}{equation.4.7.56}{}}
1407
\newlabel{eq:flowDataTermLocal2@cref}{{[equation][56][4]4.56}{99}}
1408
\newlabel{fig:flowST}{{4.11a}{100}{Subfigure 4 4.11a}{subfigure.4.11.1}{}}
1403
1409
\newlabel{sub@fig:flowST}{{(a)}{a}{Subfigure 4 4.11a\relax }{subfigure.4.11.1}{}}
1404
\newlabel{fig:flowST@cref}{{[subfigure][1][4,11]4.11a}{99}}
1405
\newlabel{fig:flowTV}{{4.11b}{99}{Subfigure 4 4.11b}{subfigure.4.11.2}{}}
1410
\newlabel{fig:flowST@cref}{{[subfigure][1][4,11]4.11a}{100}}
1411
\newlabel{fig:flowTV}{{4.11b}{100}{Subfigure 4 4.11b}{subfigure.4.11.2}{}}
1406
1412
\newlabel{sub@fig:flowTV}{{(b)}{b}{Subfigure 4 4.11b\relax }{subfigure.4.11.2}{}}
1407
\newlabel{fig:flowTV@cref}{{[subfigure][2][4,11]4.11b}{99}}
1408
\@writefile{lof}{\contentsline {figure}{\numberline {4.11}{\ignorespaces Resulting optical flows of the local models $E^l_{ST}$ ($\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$, eq.~(\ref {eq:optFlowModelSTLocal})) and $E^l_{TV}$ ($\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$, eq.~(\ref {eq:optFlowModelTVLocal})). We can see that the structure tensor prior in the model $E^l_{ST}$ fails to isotropically smooth the optical flow $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$ in the regions where the images $y_{tc}$ and $I_{vsc}$ are predominantly homogeneous. In these regions the TV model $E^l_{TV}$ excels due to the $L_1$ piecewise smoothing term in eq.~(\ref {eq:TVSplit}). \relax }}{99}{figure.caption.31}}
1409
\newlabel{fig:multModalFlowResult}{{4.11}{99}{Resulting optical flows of the local models $E^l_{ST}$ ($\optimalflowST $, \eqref {eq:optFlowModelSTLocal}) and $E^l_{TV}$ ($\optimalflowTV $, \eqref {eq:optFlowModelTVLocal}). We can see that the structure tensor prior in the model $E^l_{ST}$ fails to isotropically smooth the optical flow $\optimalflowST $ in the regions where the images $y_{tc}$ and $I_{vsc}$ are predominantly homogeneous. In these regions the TV model $E^l_{TV}$ excels due to the $L_1$ piecewise smoothing term in \eqref {eq:TVSplit}. \relax }{figure.caption.31}{}}
1410
\newlabel{fig:multModalFlowResult@cref}{{[figure][11][4]4.11}{99}}
1411
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$}}}{99}{subfigure.11.1}}
1412
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$}}}{99}{subfigure.11.2}}
1413
\newlabel{eq:viewAnglesEqual}{{4.57}{99}{Real Multimodal Optical Flow Data}{equation.4.7.57}{}}
1414
\newlabel{eq:viewAnglesEqual@cref}{{[equation][57][4]4.57}{99}}
1415
\newlabel{eq:trueOptScale}{{4.58}{99}{Real Multimodal Optical Flow Data}{equation.4.7.58}{}}
1416
\newlabel{eq:trueOptScale@cref}{{[equation][58][4]4.58}{99}}
1413
\newlabel{fig:flowTV@cref}{{[subfigure][2][4,11]4.11b}{100}}
1414
\@writefile{lof}{\contentsline {figure}{\numberline {4.11}{\ignorespaces Resulting optical flows of the local models $E^l_{ST}$ ($\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$, eq.~(\ref {eq:optFlowModelSTLocal})) and $E^l_{TV}$ ($\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$, eq.~(\ref {eq:optFlowModelTVLocal})). We can see that the structure tensor prior in the model $E^l_{ST}$ fails to isotropically smooth the optical flow $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$ in the regions where the images $y_{tc}$ and $I_{vsc}$ are predominantly homogeneous. In these regions the TV model $E^l_{TV}$ excels due to the $L_1$ piecewise smoothing term in eq.~(\ref {eq:TVSplit}). \relax }}{100}{figure.caption.32}}
1415
\newlabel{fig:multModalFlowResult}{{4.11}{100}{Resulting optical flows of the local models $E^l_{ST}$ ($\optimalflowST $, \eqref {eq:optFlowModelSTLocal}) and $E^l_{TV}$ ($\optimalflowTV $, \eqref {eq:optFlowModelTVLocal}). We can see that the structure tensor prior in the model $E^l_{ST}$ fails to isotropically smooth the optical flow $\optimalflowST $ in the regions where the images $y_{tc}$ and $I_{vsc}$ are predominantly homogeneous. In these regions the TV model $E^l_{TV}$ excels due to the $L_1$ piecewise smoothing term in \eqref {eq:TVSplit}. \relax }{figure.caption.32}{}}
1416
\newlabel{fig:multModalFlowResult@cref}{{[figure][11][4]4.11}{100}}
1417
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$}}}{100}{subfigure.11.1}}
1418
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$}}}{100}{subfigure.11.2}}
1419
\newlabel{eq:viewAnglesEqual}{{4.57}{100}{Real Multimodal Optical Flow Data}{equation.4.7.57}{}}
1420
\newlabel{eq:viewAnglesEqual@cref}{{[equation][57][4]4.57}{100}}
1421
\newlabel{eq:trueOptScale}{{4.58}{100}{Real Multimodal Optical Flow Data}{equation.4.7.58}{}}
1422
\newlabel{eq:trueOptScale@cref}{{[equation][58][4]4.58}{100}}
1417
1423
\citation{WassermanAllStatistics}
1418
\newlabel{eq:scaleMin}{{4.59}{100}{Real Multimodal Optical Flow Data}{equation.4.7.59}{}}
1419
\newlabel{eq:scaleMin@cref}{{[equation][59][4]4.59}{100}}
1420
\newlabel{eq:EdataScaleMin}{{4.60}{100}{Real Multimodal Optical Flow Data}{equation.4.7.60}{}}
1421
\newlabel{eq:EdataScaleMin@cref}{{[equation][60][4]4.60}{100}}
1422
\newlabel{eq:localRelation2}{{4.61}{100}{Real Multimodal Optical Flow Data}{equation.4.7.61}{}}
1423
\newlabel{eq:localRelation2@cref}{{[equation][61][4]4.61}{100}}
1424
\newlabel{eq:optFlowModelLocalST}{{4.62}{100}{Real Multimodal Optical Flow Data}{equation.4.7.62}{}}
1425
\newlabel{eq:optFlowModelLocalST@cref}{{[equation][62][4]4.62}{100}}
1426
\newlabel{eq:optFlowModelLocalTV}{{4.63}{100}{Real Multimodal Optical Flow Data}{equation.4.7.63}{}}
1427
\newlabel{eq:optFlowModelLocalTV@cref}{{[equation][63][4]4.63}{100}}
1428
\@writefile{brf}{\backcite{WassermanAllStatistics}{{100}{4.7.5}{equation.4.7.63}}}
1429
\newlabel{fig:chiSqNoFlow}{{4.12a}{101}{Subfigure 4 4.12a}{subfigure.4.12.1}{}}
1424
\newlabel{eq:scaleMin}{{4.59}{101}{Real Multimodal Optical Flow Data}{equation.4.7.59}{}}
1425
\newlabel{eq:scaleMin@cref}{{[equation][59][4]4.59}{101}}
1426
\newlabel{eq:EdataScaleMin}{{4.60}{101}{Real Multimodal Optical Flow Data}{equation.4.7.60}{}}
1427
\newlabel{eq:EdataScaleMin@cref}{{[equation][60][4]4.60}{101}}
1428
\newlabel{eq:localRelation2}{{4.61}{101}{Real Multimodal Optical Flow Data}{equation.4.7.61}{}}
1429
\newlabel{eq:localRelation2@cref}{{[equation][61][4]4.61}{101}}
1430
\newlabel{eq:optFlowModelLocalST}{{4.62}{101}{Real Multimodal Optical Flow Data}{equation.4.7.62}{}}
1431
\newlabel{eq:optFlowModelLocalST@cref}{{[equation][62][4]4.62}{101}}
1432
\newlabel{eq:optFlowModelLocalTV}{{4.63}{101}{Real Multimodal Optical Flow Data}{equation.4.7.63}{}}
1433
\newlabel{eq:optFlowModelLocalTV@cref}{{[equation][63][4]4.63}{101}}
1434
\@writefile{brf}{\backcite{WassermanAllStatistics}{{101}{4.7.5}{equation.4.7.63}}}
1435
\newlabel{fig:chiSqNoFlow}{{4.12a}{102}{Subfigure 4 4.12a}{subfigure.4.12.1}{}}
1430
1436
\newlabel{sub@fig:chiSqNoFlow}{{(a)}{a}{Subfigure 4 4.12a\relax }{subfigure.4.12.1}{}}
1431
\newlabel{fig:chiSqNoFlow@cref}{{[subfigure][1][4,12]4.12a}{101}}
1432
\newlabel{fig:chiSqSTFlow}{{4.12b}{101}{Subfigure 4 4.12b}{subfigure.4.12.2}{}}
1437
\newlabel{fig:chiSqNoFlow@cref}{{[subfigure][1][4,12]4.12a}{102}}
1438
\newlabel{fig:chiSqSTFlow}{{4.12b}{102}{Subfigure 4 4.12b}{subfigure.4.12.2}{}}
1433
1439
\newlabel{sub@fig:chiSqSTFlow}{{(b)}{b}{Subfigure 4 4.12b\relax }{subfigure.4.12.2}{}}
1434
\newlabel{fig:chiSqSTFlow@cref}{{[subfigure][2][4,12]4.12b}{101}}
1435
\newlabel{fig:chiSqTVFlow}{{4.12c}{101}{Subfigure 4 4.12c}{subfigure.4.12.3}{}}
1440
\newlabel{fig:chiSqSTFlow@cref}{{[subfigure][2][4,12]4.12b}{102}}
1441
\newlabel{fig:chiSqTVFlow}{{4.12c}{102}{Subfigure 4 4.12c}{subfigure.4.12.3}{}}
1436
1442
\newlabel{sub@fig:chiSqTVFlow}{{(c)}{c}{Subfigure 4 4.12c\relax }{subfigure.4.12.3}{}}
1437
\newlabel{fig:chiSqTVFlow@cref}{{[subfigure][3][4,12]4.12c}{101}}
1438
\@writefile{lof}{\contentsline {figure}{\numberline {4.12}{\ignorespaces Comparison of the p-values (eq.~(\ref {eq:pearsonPValue})) for the hypotheses (eq.~(\ref {eq:linConstraint})) $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {{\bm {0}}}}$ (\cref {fig:chiSqNoFlow}), $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}}$ (\cref {fig:chiSqSTFlow}) and $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}}$ (\cref {fig:chiSqTVFlow}). The p-values where computed for windows $\mathcal {A}_{\ensuremath {\ensuremath {{\bm {x}}}}_0}$ around each pixel $\ensuremath {\ensuremath {{\bm {x}}}}_0\in \Omega $ and plotted over the binned values of the gradient $\nabla y$. All three diagrams show high p-values for gradients $\nabla y\approx 0$ indicating that the structureless areas in the data in \cref {fig:multiModalTCVSC2} obey the linear relation in eq.~(\ref {eq:linConstraint}) regardless of the optical flow $\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}$. For higher values of the gradient $\nabla y$ the hypothesis $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {{\bm {0}}}}$ in \cref {fig:chiSqNoFlow} fails as expected since the p-values tend to zero. The p-values at higher gradients for the hypotheses $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}}$ (\cref {fig:chiSqSTFlow}) and $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}}$ (\cref {fig:chiSqTVFlow}) are significantly higher then for $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {{\bm {0}}}}$ with $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}}$ having the highest p-values meaning that the total variation model $E^l_{TV}$ in eq.~(\ref {eq:optFlowModelTVLocal}) best fulfills the linearity hypothesis in eq.~(\ref {eq:linConstraint}). \relax }}{101}{figure.caption.32}}
1439
\newlabel{fig:chiSqPValue}{{4.12}{101}{Comparison of the p-values (\eqref {eq:pearsonPValue}) for the hypotheses (\eqref {eq:linConstraint}) $H_{\hat {\vd }=\vector {0}}$ (\figref {fig:chiSqNoFlow}), $H_{\hat {\vd }=\optimalflowST }$ (\figref {fig:chiSqSTFlow}) and $H_{\hat {\vd }=\optimalflowTV }$ (\figref {fig:chiSqTVFlow}). The p-values where computed for windows $\mathcal {A}_{\vx _0}$ around each pixel $\vx _0\in \Omega $ and plotted over the binned values of the gradient $\nabla y$. All three diagrams show high p-values for gradients $\nabla y\approx 0$ indicating that the structureless areas in the data in \figref {fig:multiModalTCVSC2} obey the linear relation in \eqref {eq:linConstraint} regardless of the optical flow $\hat {\vd }$. For higher values of the gradient $\nabla y$ the hypothesis $H_{\hat {\vd }=\vector {0}}$ in \figref {fig:chiSqNoFlow} fails as expected since the p-values tend to zero. The p-values at higher gradients for the hypotheses $H_{\hat {\vd }=\optimalflowST }$ (\figref {fig:chiSqSTFlow}) and $H_{\hat {\vd }=\optimalflowTV }$ (\figref {fig:chiSqTVFlow}) are significantly higher then for $H_{\hat {\vd }=\vector {0}}$ with $H_{\hat {\vd }=\optimalflowTV }$ having the highest p-values meaning that the total variation model $E^l_{TV}$ in \eqref {eq:optFlowModelTVLocal} best fulfills the linearity hypothesis in \eqref {eq:linConstraint}. \relax }{figure.caption.32}{}}
1440
\newlabel{fig:chiSqPValue@cref}{{[figure][12][4]4.12}{101}}
1441
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{101}{subfigure.12.1}}
1442
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{101}{subfigure.12.2}}
1443
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{101}{subfigure.12.3}}
1444
\newlabel{eq:chiSqV}{{4.64}{101}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.64}{}}
1445
\newlabel{eq:chiSqV@cref}{{[equation][64][4]4.64}{101}}
1446
\newlabel{eq:pearsonCDF}{{4.65}{101}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.65}{}}
1447
\newlabel{eq:pearsonCDF@cref}{{[equation][65][4]4.65}{101}}
1448
\newlabel{eq:pearsonPValue}{{4.66}{102}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.66}{}}
1449
\newlabel{eq:pearsonPValue@cref}{{[equation][66][4]4.66}{102}}
1450
\newlabel{eq:chiSqVObs}{{4.67}{102}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.67}{}}
1451
\newlabel{eq:chiSqVObs@cref}{{[equation][67][4]4.67}{102}}
1452
\newlabel{eq:chiSqSatis}{{4.68}{102}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.68}{}}
1453
\newlabel{eq:chiSqSatis@cref}{{[equation][68][4]4.68}{102}}
1454
\newlabel{eq:linConstraint}{{4.69}{102}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.69}{}}
1455
\newlabel{eq:linConstraint@cref}{{[equation][69][4]4.69}{102}}
1456
\newlabel{eq:pearsonLocalObservation}{{4.70}{102}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.70}{}}
1457
\newlabel{eq:pearsonLocalObservation@cref}{{[equation][70][4]4.70}{102}}
1458
\newlabel{fig:subfigEdevisroi-184-397-histo}{{4.13a}{103}{Subfigure 4 4.13a}{subfigure.4.13.1}{}}
1443
\newlabel{fig:chiSqTVFlow@cref}{{[subfigure][3][4,12]4.12c}{102}}
1444
\@writefile{lof}{\contentsline {figure}{\numberline {4.12}{\ignorespaces Comparison of the p-values (eq.~(\ref {eq:pearsonPValue})) for the hypotheses (eq.~(\ref {eq:linConstraint})) $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {{\bm {0}}}}$ (\cref {fig:chiSqNoFlow}), $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}}$ (\cref {fig:chiSqSTFlow}) and $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}}$ (\cref {fig:chiSqTVFlow}). The p-values where computed for windows $\mathcal {A}_{\ensuremath {\ensuremath {{\bm {x}}}}_0}$ around each pixel $\ensuremath {\ensuremath {{\bm {x}}}}_0\in \Omega $ and plotted over the binned values of the gradient $\nabla y$. All three diagrams show high p-values for gradients $\nabla y\approx 0$ indicating that the structureless areas in the data in \cref {fig:multiModalTCVSC2} obey the linear relation in eq.~(\ref {eq:linConstraint}) regardless of the optical flow $\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}$. For higher values of the gradient $\nabla y$ the hypothesis $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {{\bm {0}}}}$ in \cref {fig:chiSqNoFlow} fails as expected since the p-values tend to zero. The p-values at higher gradients for the hypotheses $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}}$ (\cref {fig:chiSqSTFlow}) and $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}}$ (\cref {fig:chiSqTVFlow}) are significantly higher then for $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {{\bm {0}}}}$ with $H_{\mathaccentV {hat}05E{\ensuremath {\ensuremath {{\bm {d}}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}}$ having the highest p-values meaning that the total variation model $E^l_{TV}$ in eq.~(\ref {eq:optFlowModelTVLocal}) best fulfills the linearity hypothesis in eq.~(\ref {eq:linConstraint}). \relax }}{102}{figure.caption.33}}
1445
\newlabel{fig:chiSqPValue}{{4.12}{102}{Comparison of the p-values (\eqref {eq:pearsonPValue}) for the hypotheses (\eqref {eq:linConstraint}) $H_{\hat {\vd }=\vector {0}}$ (\figref {fig:chiSqNoFlow}), $H_{\hat {\vd }=\optimalflowST }$ (\figref {fig:chiSqSTFlow}) and $H_{\hat {\vd }=\optimalflowTV }$ (\figref {fig:chiSqTVFlow}). The p-values where computed for windows $\mathcal {A}_{\vx _0}$ around each pixel $\vx _0\in \Omega $ and plotted over the binned values of the gradient $\nabla y$. All three diagrams show high p-values for gradients $\nabla y\approx 0$ indicating that the structureless areas in the data in \figref {fig:multiModalTCVSC2} obey the linear relation in \eqref {eq:linConstraint} regardless of the optical flow $\hat {\vd }$. For higher values of the gradient $\nabla y$ the hypothesis $H_{\hat {\vd }=\vector {0}}$ in \figref {fig:chiSqNoFlow} fails as expected since the p-values tend to zero. The p-values at higher gradients for the hypotheses $H_{\hat {\vd }=\optimalflowST }$ (\figref {fig:chiSqSTFlow}) and $H_{\hat {\vd }=\optimalflowTV }$ (\figref {fig:chiSqTVFlow}) are significantly higher then for $H_{\hat {\vd }=\vector {0}}$ with $H_{\hat {\vd }=\optimalflowTV }$ having the highest p-values meaning that the total variation model $E^l_{TV}$ in \eqref {eq:optFlowModelTVLocal} best fulfills the linearity hypothesis in \eqref {eq:linConstraint}. \relax }{figure.caption.33}{}}
1446
\newlabel{fig:chiSqPValue@cref}{{[figure][12][4]4.12}{102}}
1447
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{102}{subfigure.12.1}}
1448
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{102}{subfigure.12.2}}
1449
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{102}{subfigure.12.3}}
1450
\newlabel{eq:chiSqV}{{4.64}{102}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.64}{}}
1451
\newlabel{eq:chiSqV@cref}{{[equation][64][4]4.64}{102}}
1452
\newlabel{eq:pearsonCDF}{{4.65}{102}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.65}{}}
1453
\newlabel{eq:pearsonCDF@cref}{{[equation][65][4]4.65}{102}}
1454
\newlabel{eq:pearsonPValue}{{4.66}{103}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.66}{}}
1455
\newlabel{eq:pearsonPValue@cref}{{[equation][66][4]4.66}{103}}
1456
\newlabel{eq:chiSqVObs}{{4.67}{103}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.67}{}}
1457
\newlabel{eq:chiSqVObs@cref}{{[equation][67][4]4.67}{103}}
1458
\newlabel{eq:chiSqSatis}{{4.68}{103}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.68}{}}
1459
\newlabel{eq:chiSqSatis@cref}{{[equation][68][4]4.68}{103}}
1460
\newlabel{eq:linConstraint}{{4.69}{103}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.69}{}}
1461
\newlabel{eq:linConstraint@cref}{{[equation][69][4]4.69}{103}}
1462
\newlabel{eq:pearsonLocalObservation}{{4.70}{103}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.70}{}}
1463
\newlabel{eq:pearsonLocalObservation@cref}{{[equation][70][4]4.70}{103}}
1464
\newlabel{fig:subfigEdevisroi-184-397-histo}{{4.13a}{104}{Subfigure 4 4.13a}{subfigure.4.13.1}{}}
1459
1465
\newlabel{sub@fig:subfigEdevisroi-184-397-histo}{{(a)}{a}{Subfigure 4 4.13a\relax }{subfigure.4.13.1}{}}
1460
\newlabel{fig:subfigEdevisroi-184-397-histo@cref}{{[subfigure][1][4,13]4.13a}{103}}
1461
\newlabel{fig:subfigEdevisroi-184-397-im0}{{4.13b}{103}{Subfigure 4 4.13b}{subfigure.4.13.2}{}}
1466
\newlabel{fig:subfigEdevisroi-184-397-histo@cref}{{[subfigure][1][4,13]4.13a}{104}}
1467
\newlabel{fig:subfigEdevisroi-184-397-im0}{{4.13b}{104}{Subfigure 4 4.13b}{subfigure.4.13.2}{}}
1462
1468
\newlabel{sub@fig:subfigEdevisroi-184-397-im0}{{(b)}{b}{Subfigure 4 4.13b\relax }{subfigure.4.13.2}{}}
1463
\newlabel{fig:subfigEdevisroi-184-397-im0@cref}{{[subfigure][2][4,13]4.13b}{103}}
1464
\newlabel{fig:subfigEdevisroi-184-397-im0warped}{{4.13c}{103}{Subfigure 4 4.13c}{subfigure.4.13.3}{}}
1469
\newlabel{fig:subfigEdevisroi-184-397-im0@cref}{{[subfigure][2][4,13]4.13b}{104}}
1470
\newlabel{fig:subfigEdevisroi-184-397-im0warped}{{4.13c}{104}{Subfigure 4 4.13c}{subfigure.4.13.3}{}}
1465
1471
\newlabel{sub@fig:subfigEdevisroi-184-397-im0warped}{{(c)}{c}{Subfigure 4 4.13c\relax }{subfigure.4.13.3}{}}
1466
\newlabel{fig:subfigEdevisroi-184-397-im0warped@cref}{{[subfigure][3][4,13]4.13c}{103}}
1467
\newlabel{fig:subfigEdevisroi-184-397-thermo}{{4.13d}{103}{Subfigure 4 4.13d}{subfigure.4.13.4}{}}
1472
\newlabel{fig:subfigEdevisroi-184-397-im0warped@cref}{{[subfigure][3][4,13]4.13c}{104}}
1473
\newlabel{fig:subfigEdevisroi-184-397-thermo}{{4.13d}{104}{Subfigure 4 4.13d}{subfigure.4.13.4}{}}
1468
1474
\newlabel{sub@fig:subfigEdevisroi-184-397-thermo}{{(d)}{d}{Subfigure 4 4.13d\relax }{subfigure.4.13.4}{}}
1469
\newlabel{fig:subfigEdevisroi-184-397-thermo@cref}{{[subfigure][4][4,13]4.13d}{103}}
1470
\newlabel{fig:subfigEdevisroiLone-184-397-histo}{{4.13e}{103}{Subfigure 4 4.13e}{subfigure.4.13.5}{}}
1475
\newlabel{fig:subfigEdevisroi-184-397-thermo@cref}{{[subfigure][4][4,13]4.13d}{104}}
1476
\newlabel{fig:subfigEdevisroiLone-184-397-histo}{{4.13e}{104}{Subfigure 4 4.13e}{subfigure.4.13.5}{}}
1471
1477
\newlabel{sub@fig:subfigEdevisroiLone-184-397-histo}{{(e)}{e}{Subfigure 4 4.13e\relax }{subfigure.4.13.5}{}}
1472
\newlabel{fig:subfigEdevisroiLone-184-397-histo@cref}{{[subfigure][5][4,13]4.13e}{103}}
1473
\newlabel{fig:subfigEdevisroiLone-184-397-im0}{{4.13f}{103}{Subfigure 4 4.13f}{subfigure.4.13.6}{}}
1478
\newlabel{fig:subfigEdevisroiLone-184-397-histo@cref}{{[subfigure][5][4,13]4.13e}{104}}
1479
\newlabel{fig:subfigEdevisroiLone-184-397-im0}{{4.13f}{104}{Subfigure 4 4.13f}{subfigure.4.13.6}{}}
1474
1480
\newlabel{sub@fig:subfigEdevisroiLone-184-397-im0}{{(f)}{f}{Subfigure 4 4.13f\relax }{subfigure.4.13.6}{}}
1475
\newlabel{fig:subfigEdevisroiLone-184-397-im0@cref}{{[subfigure][6][4,13]4.13f}{103}}
1476
\newlabel{fig:subfigEdevisroiLone-184-397-im0warped}{{4.13g}{103}{Subfigure 4 4.13g}{subfigure.4.13.7}{}}
1481
\newlabel{fig:subfigEdevisroiLone-184-397-im0@cref}{{[subfigure][6][4,13]4.13f}{104}}
1482
\newlabel{fig:subfigEdevisroiLone-184-397-im0warped}{{4.13g}{104}{Subfigure 4 4.13g}{subfigure.4.13.7}{}}
1477
1483
\newlabel{sub@fig:subfigEdevisroiLone-184-397-im0warped}{{(g)}{g}{Subfigure 4 4.13g\relax }{subfigure.4.13.7}{}}
1478
\newlabel{fig:subfigEdevisroiLone-184-397-im0warped@cref}{{[subfigure][7][4,13]4.13g}{103}}
1479
\newlabel{fig:subfigEdevisroiLone-184-397-thermo}{{4.13h}{103}{Subfigure 4 4.13h}{subfigure.4.13.8}{}}
1484
\newlabel{fig:subfigEdevisroiLone-184-397-im0warped@cref}{{[subfigure][7][4,13]4.13g}{104}}
1485
\newlabel{fig:subfigEdevisroiLone-184-397-thermo}{{4.13h}{104}{Subfigure 4 4.13h}{subfigure.4.13.8}{}}
1480
1486
\newlabel{sub@fig:subfigEdevisroiLone-184-397-thermo}{{(h)}{h}{Subfigure 4 4.13h\relax }{subfigure.4.13.8}{}}
1481
\newlabel{fig:subfigEdevisroiLone-184-397-thermo@cref}{{[subfigure][8][4,13]4.13h}{103}}
1482
\@writefile{lof}{\contentsline {figure}{\numberline {4.13}{\ignorespaces Comparison of region of interests (ROI) of size $a^\star =21$. Figures \ref {fig:subfigEdevisroi-184-397-im0} and \ref {fig:subfigEdevisroiLone-184-397-im0} show a ROI of $I_{vsc}$ and \ref {fig:subfigEdevisroi-184-397-thermo} and \ref {fig:subfigEdevisroiLone-184-397-thermo} the corresponding ROI of the image $y_{tc}$. Figures \ref {fig:subfigEdevisroi-184-397-im0warped} and \ref {fig:subfigEdevisroiLone-184-397-im0warped} show \cref {fig:subfigEdevisroi-184-397-im0} warped by the flows $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$ and $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$. \ref {fig:subfigEdevisroi-184-397-histo} and \ref {fig:subfigEdevisroiLone-184-397-histo} show the histograms between \ref {fig:subfigEdevisroi-184-397-thermo} and the filtered roi's $\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle I$}\mathaccent "0365{I}_{vsc,\ensuremath {\ensuremath {{\bm {d}}}}}=W_{\ensuremath {\sigma ^{sc,\star }}}\star I_{vsc,\ensuremath {\ensuremath {{\bm {d}}}}}$ \relax }}{103}{figure.caption.34}}
1483
\newlabel{fig:multimodalRoi}{{4.13}{103}{Comparison of region of interests (ROI) of size $a^\star =21$. Figures \ref {fig:subfigEdevisroi-184-397-im0} and \ref {fig:subfigEdevisroiLone-184-397-im0} show a ROI of $I_{vsc}$ and \ref {fig:subfigEdevisroi-184-397-thermo} and \ref {fig:subfigEdevisroiLone-184-397-thermo} the corresponding ROI of the image $y_{tc}$. Figures \ref {fig:subfigEdevisroi-184-397-im0warped} and \ref {fig:subfigEdevisroiLone-184-397-im0warped} show \figref {fig:subfigEdevisroi-184-397-im0} warped by the flows $\optimalflowST $ and $\optimalflowTV $. \ref {fig:subfigEdevisroi-184-397-histo} and \ref {fig:subfigEdevisroiLone-184-397-histo} show the histograms between \ref {fig:subfigEdevisroi-184-397-thermo} and the filtered roi's $\tilde {I}_{vsc,\vd }=W_{\optscalediff }\star I_{vsc,\vd }$ \relax }{figure.caption.34}{}}
1484
\newlabel{fig:multimodalRoi@cref}{{[figure][13][4]4.13}{103}}
1485
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{103}{subfigure.13.1}}
1486
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{103}{subfigure.13.2}}
1487
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\ensuremath {\ensuremath {{\bm {d}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$}}}{103}{subfigure.13.3}}
1488
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{103}{subfigure.13.4}}
1489
\@writefile{lof}{\contentsline {subfigure}{\numberline{(e)}{\ignorespaces {}}}{103}{subfigure.13.5}}
1490
\@writefile{lof}{\contentsline {subfigure}{\numberline{(f)}{\ignorespaces {}}}{103}{subfigure.13.6}}
1491
\@writefile{lof}{\contentsline {subfigure}{\numberline{(g)}{\ignorespaces {$\ensuremath {\ensuremath {{\bm {d}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$}}}{103}{subfigure.13.7}}
1492
\@writefile{lof}{\contentsline {subfigure}{\numberline{(h)}{\ignorespaces {}}}{103}{subfigure.13.8}}
1493
\newlabel{fig:QregEigVal3}{{4.14a}{104}{Subfigure 4 4.14a}{subfigure.4.14.1}{}}
1487
\newlabel{fig:subfigEdevisroiLone-184-397-thermo@cref}{{[subfigure][8][4,13]4.13h}{104}}
1488
\@writefile{lof}{\contentsline {figure}{\numberline {4.13}{\ignorespaces Comparison of region of interests (ROI) of size $a^\star =21$. Figures \ref {fig:subfigEdevisroi-184-397-im0} and \ref {fig:subfigEdevisroiLone-184-397-im0} show a ROI of $I_{vsc}$ and \ref {fig:subfigEdevisroi-184-397-thermo} and \ref {fig:subfigEdevisroiLone-184-397-thermo} the corresponding ROI of the image $y_{tc}$. Figures \ref {fig:subfigEdevisroi-184-397-im0warped} and \ref {fig:subfigEdevisroiLone-184-397-im0warped} show \cref {fig:subfigEdevisroi-184-397-im0} warped by the flows $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$ and $\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$. \ref {fig:subfigEdevisroi-184-397-histo} and \ref {fig:subfigEdevisroiLone-184-397-histo} show the histograms between \ref {fig:subfigEdevisroi-184-397-thermo} and the filtered roi's $\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle I$}\mathaccent "0365{I}_{vsc,\ensuremath {\ensuremath {{\bm {d}}}}}=W_{\ensuremath {\sigma ^{sc,\star }}}\star I_{vsc,\ensuremath {\ensuremath {{\bm {d}}}}}$ \relax }}{104}{figure.caption.35}}
1489
\newlabel{fig:multimodalRoi}{{4.13}{104}{Comparison of region of interests (ROI) of size $a^\star =21$. Figures \ref {fig:subfigEdevisroi-184-397-im0} and \ref {fig:subfigEdevisroiLone-184-397-im0} show a ROI of $I_{vsc}$ and \ref {fig:subfigEdevisroi-184-397-thermo} and \ref {fig:subfigEdevisroiLone-184-397-thermo} the corresponding ROI of the image $y_{tc}$. Figures \ref {fig:subfigEdevisroi-184-397-im0warped} and \ref {fig:subfigEdevisroiLone-184-397-im0warped} show \figref {fig:subfigEdevisroi-184-397-im0} warped by the flows $\optimalflowST $ and $\optimalflowTV $. \ref {fig:subfigEdevisroi-184-397-histo} and \ref {fig:subfigEdevisroiLone-184-397-histo} show the histograms between \ref {fig:subfigEdevisroi-184-397-thermo} and the filtered roi's $\tilde {I}_{vsc,\vd }=W_{\optscalediff }\star I_{vsc,\vd }$ \relax }{figure.caption.35}{}}
1490
\newlabel{fig:multimodalRoi@cref}{{[figure][13][4]4.13}{104}}
1491
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{104}{subfigure.13.1}}
1492
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{104}{subfigure.13.2}}
1493
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\ensuremath {\ensuremath {{\bm {d}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{ST}^{\star }}$}}}{104}{subfigure.13.3}}
1494
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{104}{subfigure.13.4}}
1495
\@writefile{lof}{\contentsline {subfigure}{\numberline{(e)}{\ignorespaces {}}}{104}{subfigure.13.5}}
1496
\@writefile{lof}{\contentsline {subfigure}{\numberline{(f)}{\ignorespaces {}}}{104}{subfigure.13.6}}
1497
\@writefile{lof}{\contentsline {subfigure}{\numberline{(g)}{\ignorespaces {$\ensuremath {\ensuremath {{\bm {d}}}}=\ensuremath {\ensuremath {\ensuremath {{\bm {d}}}}_{TV}^{\star }}$}}}{104}{subfigure.13.7}}
1498
\@writefile{lof}{\contentsline {subfigure}{\numberline{(h)}{\ignorespaces {}}}{104}{subfigure.13.8}}
1499
\newlabel{fig:QregEigVal3}{{4.14a}{105}{Subfigure 4 4.14a}{subfigure.4.14.1}{}}
1494
1500
\newlabel{sub@fig:QregEigVal3}{{(a)}{a}{Subfigure 4 4.14a\relax }{subfigure.4.14.1}{}}
1495
\newlabel{fig:QregEigVal3@cref}{{[subfigure][1][4,14]4.14a}{104}}
1496
\newlabel{fig:QregEigVal6}{{4.14b}{104}{Subfigure 4 4.14b}{subfigure.4.14.2}{}}
1501
\newlabel{fig:QregEigVal3@cref}{{[subfigure][1][4,14]4.14a}{105}}
1502
\newlabel{fig:QregEigVal6}{{4.14b}{105}{Subfigure 4 4.14b}{subfigure.4.14.2}{}}
1497
1503
\newlabel{sub@fig:QregEigVal6}{{(b)}{b}{Subfigure 4 4.14b\relax }{subfigure.4.14.2}{}}
1498
\newlabel{fig:QregEigVal6@cref}{{[subfigure][2][4,14]4.14b}{104}}
1499
\newlabel{fig:QregEigVal9}{{4.14c}{104}{Subfigure 4 4.14c}{subfigure.4.14.3}{}}
1504
\newlabel{fig:QregEigVal6@cref}{{[subfigure][2][4,14]4.14b}{105}}
1505
\newlabel{fig:QregEigVal9}{{4.14c}{105}{Subfigure 4 4.14c}{subfigure.4.14.3}{}}
1500
1506
\newlabel{sub@fig:QregEigVal9}{{(c)}{c}{Subfigure 4 4.14c\relax }{subfigure.4.14.3}{}}
1501
\newlabel{fig:QregEigVal9@cref}{{[subfigure][3][4,14]4.14c}{104}}
1502
\@writefile{lof}{\contentsline {figure}{\numberline {4.14}{\ignorespaces The largest eigenvalue $\sigma ^k_Q$ of $Q^{reg}$ plotted over the iterations $k$ for three values of $\lambda _2$ in eq.~(\ref {eq:structtensPriorStable}). Initially we have $\sigma ^k_Q\approx 8\lambda _2$ which is the eigenvalue of the $L_2$ term in eq.~(\ref {eq:structtensPriorStable}). For $\lambda _2=10^{-3}$ we see that $\sigma ^k_Q$ slowly rises for increasing iterations $k$ until at $k\approx 40$ a sudden jump occurs and $\sigma ^k_Q$ begins to decrease. This is the regime where the structure tensor prior $E^{prior}_{ST}$ begins to act an-isotropically. For smaller values of $\lambda _2$ (figures \ref {fig:QregEigVal6} and \ref {fig:QregEigVal9}) the jump occurs sooner indicating quicker an-isotropic behavior of $E^{prior}_{ST}$. \relax }}{104}{figure.caption.35}}
1503
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$\lambda _2=10^{-3}$}}}{104}{subfigure.14.1}}
1504
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\lambda _2=10^{-6}$}}}{104}{subfigure.14.2}}
1505
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\lambda _2=10^{-9}$}}}{104}{subfigure.14.3}}
1506
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.6}Eigenvalue analysis and the stabilization parameter $\lambda _2$}{104}{subsection.4.7.6}}
1507
\newlabel{sec:EigenValueSTPrior}{{4.7.6}{104}{Eigenvalue analysis and the stabilization parameter $\lambda _2$}{subsection.4.7.6}{}}
1508
\newlabel{sec:EigenValueSTPrior@cref}{{[subsection][6][4,7]4.7.6}{104}}
1507
\newlabel{fig:QregEigVal9@cref}{{[subfigure][3][4,14]4.14c}{105}}
1508
\@writefile{lof}{\contentsline {figure}{\numberline {4.14}{\ignorespaces The largest eigenvalue $\sigma ^k_Q$ of $Q^{reg}$ plotted over the iterations $k$ for three values of $\lambda _2$ in eq.~(\ref {eq:structtensPriorStable}). Initially we have $\sigma ^k_Q\approx 8\lambda _2$ which is the eigenvalue of the $L_2$ term in eq.~(\ref {eq:structtensPriorStable}). For $\lambda _2=10^{-3}$ we see that $\sigma ^k_Q$ slowly rises for increasing iterations $k$ until at $k\approx 40$ a sudden jump occurs and $\sigma ^k_Q$ begins to decrease. This is the regime where the structure tensor prior $E^{prior}_{ST}$ begins to act an-isotropically. For smaller values of $\lambda _2$ (figures \ref {fig:QregEigVal6} and \ref {fig:QregEigVal9}) the jump occurs sooner indicating quicker an-isotropic behavior of $E^{prior}_{ST}$. \relax }}{105}{figure.caption.36}}
1509
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$\lambda _2=10^{-3}$}}}{105}{subfigure.14.1}}
1510
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\lambda _2=10^{-6}$}}}{105}{subfigure.14.2}}
1511
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\lambda _2=10^{-9}$}}}{105}{subfigure.14.3}}
1512
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.6}Eigenvalue analysis and the stabilization parameter $\lambda _2$}{105}{subsection.4.7.6}}
1513
\newlabel{sec:EigenValueSTPrior}{{4.7.6}{105}{Eigenvalue analysis and the stabilization parameter $\lambda _2$}{subsection.4.7.6}{}}
1514
\newlabel{sec:EigenValueSTPrior@cref}{{[subsection][6][4,7]4.7.6}{105}}
1509
1515
\citation{Bigun1987,BigunBook}
1510
\newlabel{fig:bnorm3}{{4.15a}{105}{Subfigure 4 4.15a}{subfigure.4.15.1}{}}
1516
\newlabel{fig:bnorm3}{{4.15a}{106}{Subfigure 4 4.15a}{subfigure.4.15.1}{}}
1511
1517
\newlabel{sub@fig:bnorm3}{{(a)}{a}{Subfigure 4 4.15a\relax }{subfigure.4.15.1}{}}
1512
\newlabel{fig:bnorm3@cref}{{[subfigure][1][4,15]4.15a}{105}}
1513
\newlabel{fig:bnorm6}{{4.15b}{105}{Subfigure 4 4.15b}{subfigure.4.15.2}{}}
1518
\newlabel{fig:bnorm3@cref}{{[subfigure][1][4,15]4.15a}{106}}
1519
\newlabel{fig:bnorm6}{{4.15b}{106}{Subfigure 4 4.15b}{subfigure.4.15.2}{}}
1514
1520
\newlabel{sub@fig:bnorm6}{{(b)}{b}{Subfigure 4 4.15b\relax }{subfigure.4.15.2}{}}
1515
\newlabel{fig:bnorm6@cref}{{[subfigure][2][4,15]4.15b}{105}}
1516
\newlabel{fig:bnorm9}{{4.15c}{105}{Subfigure 4 4.15c}{subfigure.4.15.3}{}}
1521
\newlabel{fig:bnorm6@cref}{{[subfigure][2][4,15]4.15b}{106}}
1522
\newlabel{fig:bnorm9}{{4.15c}{106}{Subfigure 4 4.15c}{subfigure.4.15.3}{}}
1517
1523
\newlabel{sub@fig:bnorm9}{{(c)}{c}{Subfigure 4 4.15c\relax }{subfigure.4.15.3}{}}
1518
\newlabel{fig:bnorm9@cref}{{[subfigure][3][4,15]4.15c}{105}}
1519
\@writefile{lof}{\contentsline {figure}{\numberline {4.15}{\ignorespaces The residual vector $\ensuremath {{\bm {b}}}$ plotted over the iterations $k$ for three values of $\lambda _2$ in eq.~(\ref {eq:structtensPriorStable}). While the norm of $\ensuremath {{\bm {b}}}$ is approximately equal for $\lambda _2=10^{-3}$ and $\lambda _2=10^{-6}$, it is an order of magnitude higher for $\lambda _2=10^{-9}$. This indicates a numerical instability of the MOF algorithm for $\lambda _2=10^{-9}$ \relax }}{105}{figure.caption.36}}
1520
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$\lambda _2=10^{-3}$}}}{105}{subfigure.15.1}}
1521
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\lambda _2=10^{-6}$}}}{105}{subfigure.15.2}}
1522
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\lambda _2=10^{-9}$}}}{105}{subfigure.15.3}}
1523
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.7}Summary}{105}{subsection.4.7.7}}
1524
\newlabel{sec:OptFlowsummary}{{4.7.7}{105}{Summary}{subsection.4.7.7}{}}
1525
\newlabel{sec:OptFlowsummary@cref}{{[subsection][7][4,7]4.7.7}{105}}
1526
\@writefile{brf}{\backcite{Bigun1987}{{105}{4.7.7}{subsection.4.7.7}}}
1527
\@writefile{brf}{\backcite{BigunBook}{{105}{4.7.7}{subsection.4.7.7}}}
1528
\newlabel{eq:summaryGlobalLin}{{4.71}{105}{Summary}{equation.4.7.71}{}}
1529
\newlabel{eq:summaryGlobalLin@cref}{{[equation][71][4]4.71}{105}}
1524
\newlabel{fig:bnorm9@cref}{{[subfigure][3][4,15]4.15c}{106}}
1525
\@writefile{lof}{\contentsline {figure}{\numberline {4.15}{\ignorespaces The residual vector $\ensuremath {{\bm {b}}}$ plotted over the iterations $k$ for three values of $\lambda _2$ in eq.~(\ref {eq:structtensPriorStable}). While the norm of $\ensuremath {{\bm {b}}}$ is approximately equal for $\lambda _2=10^{-3}$ and $\lambda _2=10^{-6}$, it is an order of magnitude higher for $\lambda _2=10^{-9}$. This indicates a numerical instability of the MOF algorithm for $\lambda _2=10^{-9}$ \relax }}{106}{figure.caption.37}}
1526
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$\lambda _2=10^{-3}$}}}{106}{subfigure.15.1}}
1527
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\lambda _2=10^{-6}$}}}{106}{subfigure.15.2}}
1528
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\lambda _2=10^{-9}$}}}{106}{subfigure.15.3}}
1529
\@writefile{toc}{\contentsline {subsection}{\numberline {4.7.7}Summary}{106}{subsection.4.7.7}}
1530
\newlabel{sec:OptFlowsummary}{{4.7.7}{106}{Summary}{subsection.4.7.7}{}}
1531
\newlabel{sec:OptFlowsummary@cref}{{[subsection][7][4,7]4.7.7}{106}}
1532
\@writefile{brf}{\backcite{Bigun1987}{{106}{4.7.7}{subsection.4.7.7}}}
1533
\@writefile{brf}{\backcite{BigunBook}{{106}{4.7.7}{subsection.4.7.7}}}
1534
\newlabel{eq:summaryGlobalLin}{{4.71}{106}{Summary}{equation.4.7.71}{}}
1535
\newlabel{eq:summaryGlobalLin@cref}{{[equation][71][4]4.71}{106}}
1530
1536
\citation{Middleburry}
1531
\newlabel{eq:sumoptFlowModelST}{{4.72}{106}{Summary}{equation.4.7.72}{}}
1532
\newlabel{eq:sumoptFlowModelST@cref}{{[equation][72][4]4.72}{106}}
1533
\newlabel{eq:sumoptFlowModelTV}{{4.73}{106}{Summary}{equation.4.7.73}{}}
1534
\newlabel{eq:sumoptFlowModelTV@cref}{{[equation][73][4]4.73}{106}}
1535
\@writefile{brf}{\backcite{Middleburry}{{106}{4.7.7}{equation.4.7.73}}}
1536
\newlabel{eq:summaryLocalRelation}{{4.74}{107}{Summary}{equation.4.7.74}{}}
1537
\newlabel{eq:summaryLocalRelation@cref}{{[equation][74][4]4.74}{107}}
1538
\newlabel{eq:sumoptFlowModelSTLocal}{{4.75}{107}{Summary}{equation.4.7.75}{}}
1539
\newlabel{eq:sumoptFlowModelSTLocal@cref}{{[equation][75][4]4.75}{107}}
1540
\newlabel{eq:sumoptFlowModelTVLocal}{{4.76}{107}{Summary}{equation.4.7.76}{}}
1541
\newlabel{eq:sumoptFlowModelTVLocal@cref}{{[equation][76][4]4.76}{107}}
1542
\@writefile{toc}{\contentsline {chapter}{\numberline {5}The Extended Least Action Algorithm}{109}{chapter.5}}
1537
\newlabel{eq:sumoptFlowModelST}{{4.72}{107}{Summary}{equation.4.7.72}{}}
1538
\newlabel{eq:sumoptFlowModelST@cref}{{[equation][72][4]4.72}{107}}
1539
\newlabel{eq:sumoptFlowModelTV}{{4.73}{107}{Summary}{equation.4.7.73}{}}
1540
\newlabel{eq:sumoptFlowModelTV@cref}{{[equation][73][4]4.73}{107}}
1541
\@writefile{brf}{\backcite{Middleburry}{{107}{4.7.7}{equation.4.7.73}}}
1542
\newlabel{eq:summaryLocalRelation}{{4.74}{108}{Summary}{equation.4.7.74}{}}
1543
\newlabel{eq:summaryLocalRelation@cref}{{[equation][74][4]4.74}{108}}
1544
\newlabel{eq:sumoptFlowModelSTLocal}{{4.75}{108}{Summary}{equation.4.7.75}{}}
1545
\newlabel{eq:sumoptFlowModelSTLocal@cref}{{[equation][75][4]4.75}{108}}
1546
\newlabel{eq:sumoptFlowModelTVLocal}{{4.76}{108}{Summary}{equation.4.7.76}{}}
1547
\newlabel{eq:sumoptFlowModelTVLocal@cref}{{[equation][76][4]4.76}{108}}
1548
\@writefile{toc}{\contentsline {chapter}{\numberline {5}The Extended Least Action Algorithm}{110}{chapter.5}}
1543
1549
\@writefile{lof}{\addvspace {10\p@ }}
1544
1550
\@writefile{lot}{\addvspace {10\p@ }}
1545
1551
\@writefile{lol}{\addvspace {10\p@ }}
1546
1552
\@writefile{loa}{\addvspace {10\p@ }}
1547
\newlabel{chap:GeneralizedNewtonAlgorithm}{{5}{109}{The Extended Least Action Algorithm}{chapter.5}{}}
1548
\newlabel{chap:GeneralizedNewtonAlgorithm@cref}{{[chapter][5][]5}{109}}
1549
\newlabel{eq:totEnergyGenNewton}{{5.1}{109}{The Extended Least Action Algorithm}{equation.5.0.1}{}}
1550
\newlabel{eq:totEnergyGenNewton@cref}{{[equation][1][5]5.1}{109}}
1551
\newlabel{eq:eulerLagrangeGRF2}{{5.2}{109}{The Extended Least Action Algorithm}{equation.5.0.2}{}}
1552
\newlabel{eq:eulerLagrangeGRF2@cref}{{[equation][2][5]5.2}{109}}
1553
\newlabel{eq:pureSpacialSymmetryCanonMomentum2}{{5.4}{109}{The Extended Least Action Algorithm}{equation.5.0.4}{}}
1554
\newlabel{eq:pureSpacialSymmetryCanonMomentum2@cref}{{[equation][4][5]5.4}{109}}
1555
\newlabel{fig:GNAMotivCurvImage}{{5.1a}{110}{Subfigure 5 5.1a}{subfigure.5.1.1}{}}
1553
\newlabel{chap:GeneralizedNewtonAlgorithm}{{5}{110}{The Extended Least Action Algorithm}{chapter.5}{}}
1554
\newlabel{chap:GeneralizedNewtonAlgorithm@cref}{{[chapter][5][]5}{110}}
1555
\newlabel{eq:totEnergyGenNewton}{{5.1}{110}{The Extended Least Action Algorithm}{equation.5.0.1}{}}
1556
\newlabel{eq:totEnergyGenNewton@cref}{{[equation][1][5]5.1}{110}}
1557
\newlabel{eq:eulerLagrangeGRF2}{{5.2}{110}{The Extended Least Action Algorithm}{equation.5.0.2}{}}
1558
\newlabel{eq:eulerLagrangeGRF2@cref}{{[equation][2][5]5.2}{110}}
1559
\newlabel{eq:pureSpacialSymmetryCanonMomentum2}{{5.4}{110}{The Extended Least Action Algorithm}{equation.5.0.4}{}}
1560
\newlabel{eq:pureSpacialSymmetryCanonMomentum2@cref}{{[equation][4][5]5.4}{110}}
1561
\newlabel{fig:GNAMotivCurvImage}{{5.1a}{111}{Subfigure 5 5.1a}{subfigure.5.1.1}{}}
1556
1562
\newlabel{sub@fig:GNAMotivCurvImage}{{(a)}{a}{Subfigure 5 5.1a\relax }{subfigure.5.1.1}{}}
1557
\newlabel{fig:GNAMotivCurvImage@cref}{{[subfigure][1][5,1]5.1a}{110}}
1558
\newlabel{fig:GNAMotivCoordFrame}{{5.1b}{110}{Subfigure 5 5.1b}{subfigure.5.1.2}{}}
1563
\newlabel{fig:GNAMotivCurvImage@cref}{{[subfigure][1][5,1]5.1a}{111}}
1564
\newlabel{fig:GNAMotivCoordFrame}{{5.1b}{111}{Subfigure 5 5.1b}{subfigure.5.1.2}{}}
1559
1565
\newlabel{sub@fig:GNAMotivCoordFrame}{{(b)}{b}{Subfigure 5 5.1b\relax }{subfigure.5.1.2}{}}
1560
\newlabel{fig:GNAMotivCoordFrame@cref}{{[subfigure][2][5,1]5.1b}{110}}
1561
\newlabel{fig:GNAMotivCurvImageStraight}{{5.1c}{110}{Subfigure 5 5.1c}{subfigure.5.1.3}{}}
1566
\newlabel{fig:GNAMotivCoordFrame@cref}{{[subfigure][2][5,1]5.1b}{111}}
1567
\newlabel{fig:GNAMotivCurvImageStraight}{{5.1c}{111}{Subfigure 5 5.1c}{subfigure.5.1.3}{}}
1562
1568
\newlabel{sub@fig:GNAMotivCurvImageStraight}{{(c)}{c}{Subfigure 5 5.1c\relax }{subfigure.5.1.3}{}}
1563
\newlabel{fig:GNAMotivCurvImageStraight@cref}{{[subfigure][3][5,1]5.1c}{110}}
1564
\newlabel{fig:GNAMotivCoordFrameStraight}{{5.1d}{110}{Subfigure 5 5.1d}{subfigure.5.1.4}{}}
1569
\newlabel{fig:GNAMotivCurvImageStraight@cref}{{[subfigure][3][5,1]5.1c}{111}}
1570
\newlabel{fig:GNAMotivCoordFrameStraight}{{5.1d}{111}{Subfigure 5 5.1d}{subfigure.5.1.4}{}}
1565
1571
\newlabel{sub@fig:GNAMotivCoordFrameStraight}{{(d)}{d}{Subfigure 5 5.1d\relax }{subfigure.5.1.4}{}}
1566
\newlabel{fig:GNAMotivCoordFrameStraight@cref}{{[subfigure][4][5,1]5.1d}{110}}
1567
\@writefile{lof}{\contentsline {figure}{\numberline {5.1}{\ignorespaces \Cref {fig:GNAMotivCurvImage} shows an image $\phi _0$ with parabolic level-sets according to eq.~(\ref {eq:GNAMotivLevelSet}). The white line indicates the level-sets $S_{\phi _0,c}$ with $39<c<43$. In \cref {fig:GNAMotivCoordFrame} the coordinate frame $\Omega _0$ is shown together with the level-sets $S_{\phi _0,c}$. \Cref {fig:GNAMotivCurvImageStraight} shows the warped image $\phi _0(\ensuremath {T^B_t}\circ \ensuremath {\ensuremath {{\bm {x}}}})$ and \cref {fig:GNAMotivCoordFrameStraight} the transformed coordinate frame $\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \Omega $}\mathaccent "0365{\Omega }=\ensuremath {T^B_t}\circ \Omega _0$. $\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \Omega $}\mathaccent "0365{\Omega }$ has been deformed by the algorithm in eq.~(\ref {eq:MotivSimpleAlgo}) in such a way that the level-set $S_{\phi _0,c}$ (indicated by the black line) appears to be straight and hence it is identified with the linear domain $\Omega ^{\epsilon }$ of the TV prior $E^{prior}_{TV}\left (\nabla \phi \right )$. \relax }}{110}{figure.caption.37}}
1568
\newlabel{fig:GNAMotivCurvImagesWithLevelSet}{{5.1}{110}{\Figref {fig:GNAMotivCurvImage} shows an image $\phi _0$ with parabolic level-sets according to \eqref {eq:GNAMotivLevelSet}. The white line indicates the level-sets $S_{\phi _0,c}$ with $39<c<43$. In \figref {fig:GNAMotivCoordFrame} the coordinate frame $\Omega _0$ is shown together with the level-sets $S_{\phi _0,c}$. \Figref {fig:GNAMotivCurvImageStraight} shows the warped image $\phi _0(\omegadeform \circ \vx )$ and \figref {fig:GNAMotivCoordFrameStraight} the transformed coordinate frame $\tilde {\Omega }=\omegadeform \circ \Omega _0$. $\tilde {\Omega }$ has been deformed by the algorithm in \eqref {eq:MotivSimpleAlgo} in such a way that the level-set $S_{\phi _0,c}$ (indicated by the black line) appears to be straight and hence it is identified with the linear domain $\Omega ^{\epsilon }$ of the TV prior $E^{prior}_{TV}\brackets {\nabla \phi }$. \relax }{figure.caption.37}{}}
1569
\newlabel{fig:GNAMotivCurvImagesWithLevelSet@cref}{{[figure][1][5]5.1}{110}}
1570
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$\phi _0(\ensuremath {\ensuremath {{\bm {x}}}})$}}}{110}{subfigure.1.1}}
1571
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\Omega _0$}}}{110}{subfigure.1.2}}
1572
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \phi $}\mathaccent "0365{\phi }(\ensuremath {\ensuremath {{\bm {x}}}})=\phi _0(\ensuremath {T^B_t}\circ \ensuremath {\ensuremath {{\bm {x}}}})$}}}{110}{subfigure.1.3}}
1573
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {$\Omega ^{\epsilon }=\ensuremath {T^B_t}\circ \Omega _0$}}}{110}{subfigure.1.4}}
1574
\newlabel{sec:GNABasicIdea}{{5}{110}{The Basic Idea}{section*.38}{}}
1575
\newlabel{sec:GNABasicIdea@cref}{{[chapter][5][]5}{110}}
1576
\newlabel{eq:MotivBendingFlow}{{5.6}{111}{The Basic Idea}{equation.5.0.6}{}}
1577
\newlabel{eq:MotivBendingFlow@cref}{{[equation][6][5]5.6}{111}}
1578
\newlabel{eq:MotivBendingFlowIntegration}{{5.7}{111}{The Basic Idea}{equation.5.0.7}{}}
1579
\newlabel{eq:MotivBendingFlowIntegration@cref}{{[equation][7][5]5.7}{111}}
1580
\newlabel{eq:GNAMotivLevelSet}{{5.9}{111}{The Basic Idea}{equation.5.0.9}{}}
1581
\newlabel{eq:GNAMotivLevelSet@cref}{{[equation][9][5]5.9}{111}}
1572
\newlabel{fig:GNAMotivCoordFrameStraight@cref}{{[subfigure][4][5,1]5.1d}{111}}
1573
\@writefile{lof}{\contentsline {figure}{\numberline {5.1}{\ignorespaces \Cref {fig:GNAMotivCurvImage} shows an image $\phi _0$ with parabolic level-sets according to eq.~(\ref {eq:GNAMotivLevelSet}). The white line indicates the level-sets $S_{\phi _0,c}$ with $39<c<43$. In \cref {fig:GNAMotivCoordFrame} the coordinate frame $\Omega _0$ is shown together with the level-sets $S_{\phi _0,c}$. \Cref {fig:GNAMotivCurvImageStraight} shows the warped image $\phi _0(\ensuremath {T^B_t}\circ \ensuremath {\ensuremath {{\bm {x}}}})$ and \cref {fig:GNAMotivCoordFrameStraight} the transformed coordinate frame $\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \Omega $}\mathaccent "0365{\Omega }=\ensuremath {T^B_t}\circ \Omega _0$. $\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \Omega $}\mathaccent "0365{\Omega }$ has been deformed by the algorithm in eq.~(\ref {eq:MotivSimpleAlgo}) in such a way that the level-set $S_{\phi _0,c}$ (indicated by the black line) appears to be straight and hence it is identified with the linear domain $\Omega ^{\epsilon }$ of the TV prior $E^{prior}_{TV}\left (\nabla \phi \right )$. \relax }}{111}{figure.caption.38}}
1574
\newlabel{fig:GNAMotivCurvImagesWithLevelSet}{{5.1}{111}{\Figref {fig:GNAMotivCurvImage} shows an image $\phi _0$ with parabolic level-sets according to \eqref {eq:GNAMotivLevelSet}. The white line indicates the level-sets $S_{\phi _0,c}$ with $39<c<43$. In \figref {fig:GNAMotivCoordFrame} the coordinate frame $\Omega _0$ is shown together with the level-sets $S_{\phi _0,c}$. \Figref {fig:GNAMotivCurvImageStraight} shows the warped image $\phi _0(\omegadeform \circ \vx )$ and \figref {fig:GNAMotivCoordFrameStraight} the transformed coordinate frame $\tilde {\Omega }=\omegadeform \circ \Omega _0$. $\tilde {\Omega }$ has been deformed by the algorithm in \eqref {eq:MotivSimpleAlgo} in such a way that the level-set $S_{\phi _0,c}$ (indicated by the black line) appears to be straight and hence it is identified with the linear domain $\Omega ^{\epsilon }$ of the TV prior $E^{prior}_{TV}\brackets {\nabla \phi }$. \relax }{figure.caption.38}{}}
1575
\newlabel{fig:GNAMotivCurvImagesWithLevelSet@cref}{{[figure][1][5]5.1}{111}}
1576
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {$\phi _0(\ensuremath {\ensuremath {{\bm {x}}}})$}}}{111}{subfigure.1.1}}
1577
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\Omega _0$}}}{111}{subfigure.1.2}}
1578
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle \phi $}\mathaccent "0365{\phi }(\ensuremath {\ensuremath {{\bm {x}}}})=\phi _0(\ensuremath {T^B_t}\circ \ensuremath {\ensuremath {{\bm {x}}}})$}}}{111}{subfigure.1.3}}
1579
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {$\Omega ^{\epsilon }=\ensuremath {T^B_t}\circ \Omega _0$}}}{111}{subfigure.1.4}}
1580
\newlabel{sec:GNABasicIdea}{{5}{111}{The Basic Idea}{section*.39}{}}
1581
\newlabel{sec:GNABasicIdea@cref}{{[chapter][5][]5}{111}}
1582
\newlabel{eq:MotivBendingFlow}{{5.6}{112}{The Basic Idea}{equation.5.0.6}{}}
1583
\newlabel{eq:MotivBendingFlow@cref}{{[equation][6][5]5.6}{112}}
1584
\newlabel{eq:MotivBendingFlowIntegration}{{5.7}{112}{The Basic Idea}{equation.5.0.7}{}}
1585
\newlabel{eq:MotivBendingFlowIntegration@cref}{{[equation][7][5]5.7}{112}}
1586
\newlabel{eq:GNAMotivLevelSet}{{5.9}{112}{The Basic Idea}{equation.5.0.9}{}}
1587
\newlabel{eq:GNAMotivLevelSet@cref}{{[equation][9][5]5.9}{112}}
1582
1588
\citation{FieguthStatImProc}
1583
\newlabel{eq:MotivSimpleAlgo}{{5.10}{112}{The Basic Idea}{equation.5.0.10}{}}
1584
\newlabel{eq:MotivSimpleAlgo@cref}{{[equation][10][5]5.10}{112}}
1585
\@writefile{toc}{\contentsline {subsection}{\numberline {5.0.8}Newtonian Minimization}{112}{subsection.5.0.8}}
1586
\@writefile{brf}{\backcite{FieguthStatImProc}{{112}{5.0.8}{subsection.5.0.8}}}
1587
\newlabel{eq:eulerFlow}{{5.11}{112}{Newtonian Minimization}{equation.5.0.11}{}}
1588
\newlabel{eq:eulerFlow@cref}{{[equation][11][5]5.11}{112}}
1589
\newlabel{eq:steepestDescentInitialUpdate}{{5.13}{112}{Newtonian Minimization}{equation.5.0.13}{}}
1590
\newlabel{eq:steepestDescentInitialUpdate@cref}{{[equation][13][5]5.13}{112}}
1591
\newlabel{eq:steepestDescentInitialUpdate2}{{5.16}{113}{Newtonian Minimization}{equation.5.0.16}{}}
1592
\newlabel{eq:steepestDescentInitialUpdate2@cref}{{[equation][16][5]5.16}{113}}
1593
\@writefile{toc}{\contentsline {subsection}{\numberline {5.0.9}The dynamics of the level-sets $S$}{113}{subsection.5.0.9}}
1594
\newlabel{eq:noetherVariationChap4}{{5.18}{113}{The dynamics of the level-sets $S$}{equation.5.0.18}{}}
1595
\newlabel{eq:noetherVariationChap4@cref}{{[equation][18][5]5.18}{113}}
1596
\newlabel{eq:noetherPureIntensTransChap4}{{5.19}{113}{The dynamics of the level-sets $S$}{equation.5.0.19}{}}
1597
\newlabel{eq:noetherPureIntensTransChap4@cref}{{[equation][19][5]5.19}{113}}
1598
\newlabel{eq:noetherVariationPureSpacial}{{5.20}{113}{The dynamics of the level-sets $S$}{equation.5.0.20}{}}
1599
\newlabel{eq:noetherVariationPureSpacial@cref}{{[equation][20][5]5.20}{113}}
1600
\@writefile{lof}{\contentsline {figure}{\numberline {5.2}{\ignorespaces This figure shows a transformation of the level-set $S$ to $S^\prime $ along the vector $\ensuremath {{\bm {W}}}_m(\ensuremath {\ensuremath {{\bm {x}}}})$. The region $\mathcal {A}\subset \Omega $ is the region a section of $S$ traverses as it is shifted along $\ensuremath {{\bm {W}}}_m$ to the end position $S^\prime $. If the divergence of $\ensuremath {{\bm {W}}}_m$ vanishes, this means that the incoming flux of $\ensuremath {{\bm {W}}}_m$ equals the outgoing flux (both indicated by the red arrows), $\ensuremath {{\bm {W}}}_m\delimiter "026A30C _{S}=\ensuremath {{\bm {W}}}_m\delimiter "026A30C _{S^\prime }$\relax }}{114}{figure.caption.40}}
1601
\newlabel{fig:divergenceLevelSetShift}{{5.2}{114}{This figure shows a transformation of the level-set $S$ to $S^\prime $ along the vector $\vector {W}_m(\vx )$. The region $\mathcal {A}\subset \Omega $ is the region a section of $S$ traverses as it is shifted along $\vector {W}_m$ to the end position $S^\prime $. If the divergence of $\vector {W}_m$ vanishes, this means that the incoming flux of $\vector {W}_m$ equals the outgoing flux (both indicated by the red arrows), $\vector {W}_m\vert _{S}=\vector {W}_m\vert _{S^\prime }$\relax }{figure.caption.40}{}}
1602
\newlabel{fig:divergenceLevelSetShift@cref}{{[figure][2][5]5.2}{114}}
1603
\newlabel{eq:divergenceSource}{{5.21}{114}{The dynamics of the level-sets $S$}{equation.5.0.21}{}}
1604
\newlabel{eq:divergenceSource@cref}{{[equation][21][5]5.21}{114}}
1605
\@writefile{toc}{\contentsline {subsubsection}{\nonumberline Dynamics of the normal vector $\ensuremath {{\bm {n}}}_S$}{114}{section*.39}}
1606
\newlabel{eq:divergenceSource2}{{5.22}{114}{Dynamics of the normal vector $\vector {n}_S$}{equation.5.0.22}{}}
1607
\newlabel{eq:divergenceSource2@cref}{{[equation][22][5]5.22}{114}}
1608
\newlabel{eq:gaussLaw}{{5.23}{114}{Dynamics of the normal vector $\vector {n}_S$}{equation.5.0.23}{}}
1609
\newlabel{eq:gaussLaw@cref}{{[equation][23][5]5.23}{114}}
1610
\newlabel{eq:gaussLaw2}{{5.24}{114}{Dynamics of the normal vector $\vector {n}_S$}{equation.5.0.24}{}}
1611
\newlabel{eq:gaussLaw2@cref}{{[equation][24][5]5.24}{114}}
1612
\newlabel{eq:gaussLawSurface}{{5.25}{115}{Dynamics of the normal vector $\vector {n}_S$}{equation.5.0.25}{}}
1613
\newlabel{eq:gaussLawSurface@cref}{{[equation][25][5]5.25}{115}}
1614
\@writefile{toc}{\contentsline {subsubsection}{\nonumberline Dynamics of the tangential vector to $S$}{115}{section*.41}}
1615
\newlabel{sec:dynamicsTangential}{{5.0.9}{115}{Dynamics of the tangential vector to $S$}{section*.41}{}}
1616
\newlabel{sec:dynamicsTangential@cref}{{[subsection][9][5,0]5.0.9}{115}}
1617
\newlabel{eq:BlevelSet}{{5.27}{115}{Dynamics of the tangential vector to $S$}{equation.5.0.27}{}}
1618
\newlabel{eq:BlevelSet@cref}{{[equation][27][5]5.27}{115}}
1619
\newlabel{eq:newtonLevelSetMotivation}{{5.28}{115}{Dynamics of the tangential vector to $S$}{equation.5.0.28}{}}
1620
\newlabel{eq:newtonLevelSetMotivation@cref}{{[equation][28][5]5.28}{115}}
1621
\newlabel{eq:newtonLevelSetEnergyInvariant}{{5.29}{115}{Dynamics of the tangential vector to $S$}{equation.5.0.29}{}}
1622
\newlabel{eq:newtonLevelSetEnergyInvariant@cref}{{[equation][29][5]5.29}{115}}
1623
\newlabel{fig:proofBendingEnergy}{{5.3a}{116}{Subfigure 5 5.3a}{subfigure.5.3.1}{}}
1589
\newlabel{eq:MotivSimpleAlgo}{{5.10}{113}{The Basic Idea}{equation.5.0.10}{}}
1590
\newlabel{eq:MotivSimpleAlgo@cref}{{[equation][10][5]5.10}{113}}
1591
\@writefile{toc}{\contentsline {subsection}{\numberline {5.0.8}Newtonian Minimization}{113}{subsection.5.0.8}}
1592
\@writefile{brf}{\backcite{FieguthStatImProc}{{113}{5.0.8}{subsection.5.0.8}}}
1593
\newlabel{eq:eulerFlow}{{5.11}{113}{Newtonian Minimization}{equation.5.0.11}{}}
1594
\newlabel{eq:eulerFlow@cref}{{[equation][11][5]5.11}{113}}
1595
\newlabel{eq:steepestDescentInitialUpdate}{{5.13}{113}{Newtonian Minimization}{equation.5.0.13}{}}
1596
\newlabel{eq:steepestDescentInitialUpdate@cref}{{[equation][13][5]5.13}{113}}
1597
\newlabel{eq:steepestDescentInitialUpdate2}{{5.16}{114}{Newtonian Minimization}{equation.5.0.16}{}}
1598
\newlabel{eq:steepestDescentInitialUpdate2@cref}{{[equation][16][5]5.16}{114}}
1599
\@writefile{toc}{\contentsline {subsection}{\numberline {5.0.9}The dynamics of the level-sets $S$}{114}{subsection.5.0.9}}
1600
\newlabel{eq:noetherVariationChap4}{{5.18}{114}{The dynamics of the level-sets $S$}{equation.5.0.18}{}}
1601
\newlabel{eq:noetherVariationChap4@cref}{{[equation][18][5]5.18}{114}}
1602
\newlabel{eq:noetherPureIntensTransChap4}{{5.19}{114}{The dynamics of the level-sets $S$}{equation.5.0.19}{}}
1603
\newlabel{eq:noetherPureIntensTransChap4@cref}{{[equation][19][5]5.19}{114}}
1604
\newlabel{eq:noetherVariationPureSpacial}{{5.20}{114}{The dynamics of the level-sets $S$}{equation.5.0.20}{}}
1605
\newlabel{eq:noetherVariationPureSpacial@cref}{{[equation][20][5]5.20}{114}}
1606
\@writefile{lof}{\contentsline {figure}{\numberline {5.2}{\ignorespaces This figure shows a transformation of the level-set $S$ to $S^\prime $ along the vector $\ensuremath {{\bm {W}}}_m(\ensuremath {\ensuremath {{\bm {x}}}})$. The region $\mathcal {A}\subset \Omega $ is the region a section of $S$ traverses as it is shifted along $\ensuremath {{\bm {W}}}_m$ to the end position $S^\prime $. If the divergence of $\ensuremath {{\bm {W}}}_m$ vanishes, this means that the incoming flux of $\ensuremath {{\bm {W}}}_m$ equals the outgoing flux (both indicated by the red arrows), $\ensuremath {{\bm {W}}}_m\delimiter "026A30C _{S}=\ensuremath {{\bm {W}}}_m\delimiter "026A30C _{S^\prime }$\relax }}{115}{figure.caption.41}}
1607
\newlabel{fig:divergenceLevelSetShift}{{5.2}{115}{This figure shows a transformation of the level-set $S$ to $S^\prime $ along the vector $\vector {W}_m(\vx )$. The region $\mathcal {A}\subset \Omega $ is the region a section of $S$ traverses as it is shifted along $\vector {W}_m$ to the end position $S^\prime $. If the divergence of $\vector {W}_m$ vanishes, this means that the incoming flux of $\vector {W}_m$ equals the outgoing flux (both indicated by the red arrows), $\vector {W}_m\vert _{S}=\vector {W}_m\vert _{S^\prime }$\relax }{figure.caption.41}{}}
1608
\newlabel{fig:divergenceLevelSetShift@cref}{{[figure][2][5]5.2}{115}}
1609
\newlabel{eq:divergenceSource}{{5.21}{115}{The dynamics of the level-sets $S$}{equation.5.0.21}{}}
1610
\newlabel{eq:divergenceSource@cref}{{[equation][21][5]5.21}{115}}
1611
\@writefile{toc}{\contentsline {subsubsection}{\nonumberline Dynamics of the normal vector $\ensuremath {{\bm {n}}}_S$}{115}{section*.40}}
1612
\newlabel{eq:divergenceSource2}{{5.22}{115}{Dynamics of the normal vector $\vector {n}_S$}{equation.5.0.22}{}}
1613
\newlabel{eq:divergenceSource2@cref}{{[equation][22][5]5.22}{115}}
1614
\newlabel{eq:gaussLaw}{{5.23}{115}{Dynamics of the normal vector $\vector {n}_S$}{equation.5.0.23}{}}
1615
\newlabel{eq:gaussLaw@cref}{{[equation][23][5]5.23}{115}}
1616
\newlabel{eq:gaussLaw2}{{5.24}{115}{Dynamics of the normal vector $\vector {n}_S$}{equation.5.0.24}{}}
1617
\newlabel{eq:gaussLaw2@cref}{{[equation][24][5]5.24}{115}}
1618
\newlabel{eq:gaussLawSurface}{{5.25}{116}{Dynamics of the normal vector $\vector {n}_S$}{equation.5.0.25}{}}
1619
\newlabel{eq:gaussLawSurface@cref}{{[equation][25][5]5.25}{116}}
1620
\@writefile{toc}{\contentsline {subsubsection}{\nonumberline Dynamics of the tangential vector to $S$}{116}{section*.42}}
1621
\newlabel{sec:dynamicsTangential}{{5.0.9}{116}{Dynamics of the tangential vector to $S$}{section*.42}{}}
1622
\newlabel{sec:dynamicsTangential@cref}{{[subsection][9][5,0]5.0.9}{116}}
1623
\newlabel{eq:BlevelSet}{{5.27}{116}{Dynamics of the tangential vector to $S$}{equation.5.0.27}{}}
1624
\newlabel{eq:BlevelSet@cref}{{[equation][27][5]5.27}{116}}
1625
\newlabel{eq:newtonLevelSetMotivation}{{5.28}{116}{Dynamics of the tangential vector to $S$}{equation.5.0.28}{}}
1626
\newlabel{eq:newtonLevelSetMotivation@cref}{{[equation][28][5]5.28}{116}}
1627
\newlabel{eq:newtonLevelSetEnergyInvariant}{{5.29}{116}{Dynamics of the tangential vector to $S$}{equation.5.0.29}{}}
1628
\newlabel{eq:newtonLevelSetEnergyInvariant@cref}{{[equation][29][5]5.29}{116}}
1629
\newlabel{fig:proofBendingEnergy}{{5.3a}{117}{Subfigure 5 5.3a}{subfigure.5.3.1}{}}
1624
1630
\newlabel{sub@fig:proofBendingEnergy}{{(a)}{a}{Subfigure 5 5.3a\relax }{subfigure.5.3.1}{}}
1625
\newlabel{fig:proofBendingEnergy@cref}{{[subfigure][1][5,3]5.3a}{116}}
1626
\@writefile{lof}{\contentsline {figure}{\numberline {5.3}{\ignorespaces \Cref {fig:proofBendingEnergy} shows an image $\phi $ in which a group of level-sets with $47<\phi <53$ (indicated by the white area) all converge into one point $P$ at the top of the image. The line sections $S_{1,2}$ and $T_{1,2}$ enclose the region $\mathcal {R}_B$ in eq.~(\ref {eq:divLevelSetB}) and eq.~(\ref {eq:divLevelSetBT}). The normal vectors $\ensuremath {{\bm {n}}}_{S_{1,2}}$ of lines $S_{1,2}$ are orthogonal to $\ensuremath {{\bm {b}}}_S$, hence the corresponding line integrals on the right hand side of eq.~(\ref {eq:divLevelSetB}) vanish. In contrast the normal vectors $\ensuremath {{\bm {n}}}_{T_{1,2}}$ of lines $T_{1,2}$ are parallel to $\ensuremath {{\bm {b}}}_S$ so that the corresponding line integrals on the right hand side of eq.~(\ref {eq:divLevelSetB}) do not vanish \relax }}{116}{figure.caption.42}}
1627
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{116}{subfigure.3.1}}
1628
\newlabel{eq:divergenceSourceB}{{5.30}{116}{Dynamics of the tangential vector to $S$}{equation.5.0.30}{}}
1629
\newlabel{eq:divergenceSourceB@cref}{{[equation][30][5]5.30}{116}}
1630
\newlabel{eq:divBVanish}{{5.31}{116}{Dynamics of the tangential vector to $S$}{equation.5.0.31}{}}
1631
\newlabel{eq:divBVanish@cref}{{[equation][31][5]5.31}{116}}
1632
\newlabel{eq:divergenceFreeVectorsNewton}{{5.32}{116}{Dynamics of the tangential vector to $S$}{equation.5.0.32}{}}
1633
\newlabel{eq:divergenceFreeVectorsNewton@cref}{{[equation][32][5]5.32}{116}}
1634
\newlabel{eq:divergenceFreeVectorsNewton2}{{5.33}{116}{Dynamics of the tangential vector to $S$}{equation.5.0.33}{}}
1635
\newlabel{eq:divergenceFreeVectorsNewton2@cref}{{[equation][33][5]5.33}{116}}
1636
\newlabel{eq:divergenceFreeVectorsNewton3}{{5.34}{117}{Dynamics of the tangential vector to $S$}{equation.5.0.34}{}}
1637
\newlabel{eq:divergenceFreeVectorsNewton3@cref}{{[equation][34][5]5.34}{117}}
1638
\newlabel{eq:divLevelSetB}{{5.35}{117}{Dynamics of the tangential vector to $S$}{equation.5.0.35}{}}
1639
\newlabel{eq:divLevelSetB@cref}{{[equation][35][5]5.35}{117}}
1640
\newlabel{eq:bendingGauge}{{5.36}{117}{Dynamics of the tangential vector to $S$}{equation.5.0.36}{}}
1641
\newlabel{eq:bendingGauge@cref}{{[equation][36][5]5.36}{117}}
1642
\newlabel{eq:divLevelSetBT}{{5.37}{117}{Dynamics of the tangential vector to $S$}{equation.5.0.37}{}}
1643
\newlabel{eq:divLevelSetBT@cref}{{[equation][37][5]5.37}{117}}
1644
\newlabel{eq:divLevelSetBT2}{{5.38}{117}{Dynamics of the tangential vector to $S$}{equation.5.0.38}{}}
1645
\newlabel{eq:divLevelSetBT2@cref}{{[equation][38][5]5.38}{117}}
1646
\newlabel{eq:newtonLevelSetEnergyNonInvariant}{{5.39}{118}{Dynamics of the tangential vector to $S$}{equation.5.0.39}{}}
1647
\newlabel{eq:newtonLevelSetEnergyNonInvariant@cref}{{[equation][39][5]5.39}{118}}
1648
\newlabel{eq:flowMotivation}{{5.40}{118}{Dynamics of the tangential vector to $S$}{equation.5.0.40}{}}
1649
\newlabel{eq:flowMotivation@cref}{{[equation][40][5]5.40}{118}}
1650
\@writefile{toc}{\contentsline {section}{\numberline {5.1}The Extended Least Action Algorithm}{118}{section.5.1}}
1651
\newlabel{eq:noetherVariation3}{{5.43}{118}{The Extended Least Action Algorithm}{equation.5.1.43}{}}
1652
\newlabel{eq:noetherVariation3@cref}{{[equation][43][5]5.43}{118}}
1653
\newlabel{eq:eulerLagrange3}{{5.45}{119}{The Extended Least Action Algorithm}{equation.5.1.45}{}}
1654
\newlabel{eq:eulerLagrange3@cref}{{[equation][45][5]5.45}{119}}
1655
\newlabel{eq:pureSpacialSymmetry3}{{5.46}{119}{The Extended Least Action Algorithm}{equation.5.1.46}{}}
1656
\newlabel{eq:pureSpacialSymmetry3@cref}{{[equation][46][5]5.46}{119}}
1657
\newlabel{eq:firstOrderSpacialUpdate}{{5.49}{119}{The Extended Least Action Algorithm}{equation.5.1.49}{}}
1658
\newlabel{eq:firstOrderSpacialUpdate@cref}{{[equation][49][5]5.49}{119}}
1659
\newlabel{eq:bendingOperator}{{5.50}{119}{The Extended Least Action Algorithm}{equation.5.1.50}{}}
1660
\newlabel{eq:bendingOperator@cref}{{[equation][50][5]5.50}{119}}
1661
\newlabel{eq:levelSetNewton}{{5.51}{119}{The Extended Least Action Algorithm}{equation.5.1.51}{}}
1662
\newlabel{eq:levelSetNewton@cref}{{[equation][51][5]5.51}{119}}
1663
\newlabel{eq:diffusionProcess}{{5.52}{119}{The Extended Least Action Algorithm}{equation.5.1.52}{}}
1664
\newlabel{eq:diffusionProcess@cref}{{[equation][52][5]5.52}{119}}
1665
\newlabel{eq:eulerFlow2}{{5.53}{120}{The Extended Least Action Algorithm}{equation.5.1.53}{}}
1666
\newlabel{eq:eulerFlow2@cref}{{[equation][53][5]5.53}{120}}
1667
\newlabel{eq:diffusionProcess3}{{5.54}{120}{The Extended Least Action Algorithm}{equation.5.1.54}{}}
1668
\newlabel{eq:diffusionProcess3@cref}{{[equation][54][5]5.54}{120}}
1669
\@writefile{toc}{\contentsline {subsubsection}{\nonumberline The Curvature Operator $\ensuremath {{\bm {K}}}$}{120}{section*.43}}
1670
\newlabel{eq:GNAdivergenceP}{{5.57}{120}{The Curvature Operator $\vector {K}$}{equation.5.1.57}{}}
1671
\newlabel{eq:GNAdivergenceP@cref}{{[equation][57][5]5.57}{120}}
1672
\newlabel{fig:curvatureOperator1}{{5.4a}{121}{Subfigure 5 5.4a}{subfigure.5.4.1}{}}
1631
\newlabel{fig:proofBendingEnergy@cref}{{[subfigure][1][5,3]5.3a}{117}}
1632
\@writefile{lof}{\contentsline {figure}{\numberline {5.3}{\ignorespaces \Cref {fig:proofBendingEnergy} shows an image $\phi $ in which a group of level-sets with $47<\phi <53$ (indicated by the white area) all converge into one point $P$ at the top of the image. The line sections $S_{1,2}$ and $T_{1,2}$ enclose the region $\mathcal {R}_B$ in eq.~(\ref {eq:divLevelSetB}) and eq.~(\ref {eq:divLevelSetBT}). The normal vectors $\ensuremath {{\bm {n}}}_{S_{1,2}}$ of lines $S_{1,2}$ are orthogonal to $\ensuremath {{\bm {b}}}_S$, hence the corresponding line integrals on the right hand side of eq.~(\ref {eq:divLevelSetB}) vanish. In contrast the normal vectors $\ensuremath {{\bm {n}}}_{T_{1,2}}$ of lines $T_{1,2}$ are parallel to $\ensuremath {{\bm {b}}}_S$ so that the corresponding line integrals on the right hand side of eq.~(\ref {eq:divLevelSetB}) do not vanish \relax }}{117}{figure.caption.43}}
1633
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{117}{subfigure.3.1}}
1634
\newlabel{eq:divergenceSourceB}{{5.30}{117}{Dynamics of the tangential vector to $S$}{equation.5.0.30}{}}
1635
\newlabel{eq:divergenceSourceB@cref}{{[equation][30][5]5.30}{117}}
1636
\newlabel{eq:divBVanish}{{5.31}{117}{Dynamics of the tangential vector to $S$}{equation.5.0.31}{}}
1637
\newlabel{eq:divBVanish@cref}{{[equation][31][5]5.31}{117}}
1638
\newlabel{eq:divergenceFreeVectorsNewton}{{5.32}{117}{Dynamics of the tangential vector to $S$}{equation.5.0.32}{}}
1639
\newlabel{eq:divergenceFreeVectorsNewton@cref}{{[equation][32][5]5.32}{117}}
1640
\newlabel{eq:divergenceFreeVectorsNewton2}{{5.33}{117}{Dynamics of the tangential vector to $S$}{equation.5.0.33}{}}
1641
\newlabel{eq:divergenceFreeVectorsNewton2@cref}{{[equation][33][5]5.33}{117}}
1642
\newlabel{eq:divergenceFreeVectorsNewton3}{{5.34}{118}{Dynamics of the tangential vector to $S$}{equation.5.0.34}{}}
1643
\newlabel{eq:divergenceFreeVectorsNewton3@cref}{{[equation][34][5]5.34}{118}}
1644
\newlabel{eq:divLevelSetB}{{5.35}{118}{Dynamics of the tangential vector to $S$}{equation.5.0.35}{}}
1645
\newlabel{eq:divLevelSetB@cref}{{[equation][35][5]5.35}{118}}
1646
\newlabel{eq:bendingGauge}{{5.36}{118}{Dynamics of the tangential vector to $S$}{equation.5.0.36}{}}
1647
\newlabel{eq:bendingGauge@cref}{{[equation][36][5]5.36}{118}}
1648
\newlabel{eq:divLevelSetBT}{{5.37}{118}{Dynamics of the tangential vector to $S$}{equation.5.0.37}{}}
1649
\newlabel{eq:divLevelSetBT@cref}{{[equation][37][5]5.37}{118}}
1650
\newlabel{eq:divLevelSetBT2}{{5.38}{118}{Dynamics of the tangential vector to $S$}{equation.5.0.38}{}}
1651
\newlabel{eq:divLevelSetBT2@cref}{{[equation][38][5]5.38}{118}}
1652
\newlabel{eq:newtonLevelSetEnergyNonInvariant}{{5.39}{119}{Dynamics of the tangential vector to $S$}{equation.5.0.39}{}}
1653
\newlabel{eq:newtonLevelSetEnergyNonInvariant@cref}{{[equation][39][5]5.39}{119}}
1654
\newlabel{eq:flowMotivation}{{5.40}{119}{Dynamics of the tangential vector to $S$}{equation.5.0.40}{}}
1655
\newlabel{eq:flowMotivation@cref}{{[equation][40][5]5.40}{119}}
1656
\@writefile{toc}{\contentsline {section}{\numberline {5.1}The Extended Least Action Algorithm}{119}{section.5.1}}
1657
\newlabel{eq:noetherVariation3}{{5.43}{119}{The Extended Least Action Algorithm}{equation.5.1.43}{}}
1658
\newlabel{eq:noetherVariation3@cref}{{[equation][43][5]5.43}{119}}
1659
\newlabel{eq:eulerLagrange3}{{5.45}{120}{The Extended Least Action Algorithm}{equation.5.1.45}{}}
1660
\newlabel{eq:eulerLagrange3@cref}{{[equation][45][5]5.45}{120}}
1661
\newlabel{eq:pureSpacialSymmetry3}{{5.46}{120}{The Extended Least Action Algorithm}{equation.5.1.46}{}}
1662
\newlabel{eq:pureSpacialSymmetry3@cref}{{[equation][46][5]5.46}{120}}
1663
\newlabel{eq:firstOrderSpacialUpdate}{{5.49}{120}{The Extended Least Action Algorithm}{equation.5.1.49}{}}
1664
\newlabel{eq:firstOrderSpacialUpdate@cref}{{[equation][49][5]5.49}{120}}
1665
\newlabel{eq:bendingOperator}{{5.50}{120}{The Extended Least Action Algorithm}{equation.5.1.50}{}}
1666
\newlabel{eq:bendingOperator@cref}{{[equation][50][5]5.50}{120}}
1667
\newlabel{eq:levelSetNewton}{{5.51}{120}{The Extended Least Action Algorithm}{equation.5.1.51}{}}
1668
\newlabel{eq:levelSetNewton@cref}{{[equation][51][5]5.51}{120}}
1669
\newlabel{eq:diffusionProcess}{{5.52}{120}{The Extended Least Action Algorithm}{equation.5.1.52}{}}
1670
\newlabel{eq:diffusionProcess@cref}{{[equation][52][5]5.52}{120}}
1671
\newlabel{eq:eulerFlow2}{{5.53}{121}{The Extended Least Action Algorithm}{equation.5.1.53}{}}
1672
\newlabel{eq:eulerFlow2@cref}{{[equation][53][5]5.53}{121}}
1673
\newlabel{eq:diffusionProcess3}{{5.54}{121}{The Extended Least Action Algorithm}{equation.5.1.54}{}}
1674
\newlabel{eq:diffusionProcess3@cref}{{[equation][54][5]5.54}{121}}
1675
\@writefile{toc}{\contentsline {subsubsection}{\nonumberline The Curvature Operator $\ensuremath {{\bm {K}}}$}{121}{section*.44}}
1676
\newlabel{eq:GNAdivergenceP}{{5.57}{121}{The Curvature Operator $\vector {K}$}{equation.5.1.57}{}}
1677
\newlabel{eq:GNAdivergenceP@cref}{{[equation][57][5]5.57}{121}}
1678
\newlabel{fig:curvatureOperator1}{{5.4a}{122}{Subfigure 5 5.4a}{subfigure.5.4.1}{}}
1673
1679
\newlabel{sub@fig:curvatureOperator1}{{(a)}{a}{Subfigure 5 5.4a\relax }{subfigure.5.4.1}{}}
1674
\newlabel{fig:curvatureOperator1@cref}{{[subfigure][1][5,4]5.4a}{121}}
1675
\newlabel{fig:curvatureOperator2}{{5.4b}{121}{Subfigure 5 5.4b}{subfigure.5.4.2}{}}
1680
\newlabel{fig:curvatureOperator1@cref}{{[subfigure][1][5,4]5.4a}{122}}
1681
\newlabel{fig:curvatureOperator2}{{5.4b}{122}{Subfigure 5 5.4b}{subfigure.5.4.2}{}}
1676
1682
\newlabel{sub@fig:curvatureOperator2}{{(b)}{b}{Subfigure 5 5.4b\relax }{subfigure.5.4.2}{}}
1677
\newlabel{fig:curvatureOperator2@cref}{{[subfigure][2][5,4]5.4b}{121}}
1678
\@writefile{lof}{\contentsline {figure}{\numberline {5.4}{\ignorespaces Effect of the diffusion $\ensuremath {\ensuremath {{\bm {x}}}}^\prime =\ensuremath {T^B_t}\circ \ensuremath {\ensuremath {{\bm {x}}}}$ (eq.~(\ref {eq:diffusionProcess})) on the canonical momentum $\ensuremath {{\bm {P}}}$. \Cref {fig:curvatureOperator1} shows a schematic picture of a region $\mathcal {R}_B\subset \Omega $ between two level-sets $S_1$ and $S_2$. The canonical momentum (the vectors on the level-sets $S_{1,2}$) is denoted by $\ensuremath {{\bm {P}}}_{S_{1,2}}$. $\ensuremath {{\bm {P}}}$ changes its orientation when shifted along the level-sets $S_1$ and $S_2$ since the norm of the curvature operator $\ensuremath {{\bm {K}}}$ (eq.~(\ref {eq:GNACurvatureDef})) is non zero. In \cref {fig:curvatureOperator2} the level-sets $S_{1,2}$ have been deformed according to $\ensuremath {\ensuremath {{\bm {x}}}}^\prime =\ensuremath {T^B_t}\circ \ensuremath {\ensuremath {{\bm {x}}}}$ such that the canonical momentum $\ensuremath {{\bm {P}}}$ becomes invariant with respect to shifts along $S_{1,2}$. In this case the norm of the curvature operator $\ensuremath {{\bm {K}}}$ vanishes\relax }}{121}{figure.caption.44}}
1679
\newlabel{fig:curvatureOperator}{{5.4}{121}{Effect of the diffusion $\vx ^\prime =\omegadeform \circ \vx $ (\eqref {eq:diffusionProcess}) on the canonical momentum $\vector {P}$. \Figref {fig:curvatureOperator1} shows a schematic picture of a region $\mathcal {R}_B\subset \Omega $ between two level-sets $S_1$ and $S_2$. The canonical momentum (the vectors on the level-sets $S_{1,2}$) is denoted by $\vector {P}_{S_{1,2}}$. $\vector {P}$ changes its orientation when shifted along the level-sets $S_1$ and $S_2$ since the norm of the curvature operator $\vector {K}$ (\eqref {eq:GNACurvatureDef}) is non zero. In \figref {fig:curvatureOperator2} the level-sets $S_{1,2}$ have been deformed according to $\vx ^\prime =\omegadeform \circ \vx $ such that the canonical momentum $\vector {P}$ becomes invariant with respect to shifts along $S_{1,2}$. In this case the norm of the curvature operator $\vector {K}$ vanishes\relax }{figure.caption.44}{}}
1680
\newlabel{fig:curvatureOperator@cref}{{[figure][4][5]5.4}{121}}
1681
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{121}{subfigure.4.1}}
1682
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{121}{subfigure.4.2}}
1683
\newlabel{eq:GNAdivergencePintegral}{{5.58}{121}{The Curvature Operator $\vector {K}$}{equation.5.1.58}{}}
1684
\newlabel{eq:GNAdivergencePintegral@cref}{{[equation][58][5]5.58}{121}}
1685
\newlabel{eq:GNACurvatureDef}{{5.59}{121}{The Curvature Operator $\vector {K}$}{equation.5.1.59}{}}
1686
\newlabel{eq:GNACurvatureDef@cref}{{[equation][59][5]5.59}{121}}
1687
\@writefile{loa}{\contentsline {algorithm}{\numberline {3}{\ignorespaces Basic Newton Algorithm (BNA)\relax }}{122}{algorithm.3}}
1688
\newlabel{alg:basicNewtonMethod}{{3}{122}{Basic Newton Algorithm (BNA)\relax }{algorithm.3}{}}
1689
\newlabel{alg:basicNewtonMethod@cref}{{[algorithm][3][]3}{122}}
1690
\@writefile{loa}{\contentsline {algorithm}{\numberline {4}{\ignorespaces Diffusion Algorithm (DA) \relax }}{122}{algorithm.4}}
1691
\newlabel{alg:warpOnlyMethod}{{4}{122}{Diffusion Algorithm (DA) \relax }{algorithm.4}{}}
1692
\newlabel{alg:warpOnlyMethod@cref}{{[algorithm][4][]4}{122}}
1693
\@writefile{loa}{\contentsline {algorithm}{\numberline {5}{\ignorespaces Extended Least Action Algorithm (ELAA)\relax }}{122}{algorithm.5}}
1694
\newlabel{alg:GeneralizedNewtonMethod}{{5}{122}{Extended Least Action Algorithm (ELAA)\relax }{algorithm.5}{}}
1695
\newlabel{alg:GeneralizedNewtonMethod@cref}{{[algorithm][5][]5}{122}}
1696
\newlabel{fig:Army}{{5.5a}{123}{Subfigure 5 5.5a}{subfigure.5.5.1}{}}
1683
\newlabel{fig:curvatureOperator2@cref}{{[subfigure][2][5,4]5.4b}{122}}
1684
\@writefile{lof}{\contentsline {figure}{\numberline {5.4}{\ignorespaces Effect of the diffusion $\ensuremath {\ensuremath {{\bm {x}}}}^\prime =\ensuremath {T^B_t}\circ \ensuremath {\ensuremath {{\bm {x}}}}$ (eq.~(\ref {eq:diffusionProcess})) on the canonical momentum $\ensuremath {{\bm {P}}}$. \Cref {fig:curvatureOperator1} shows a schematic picture of a region $\mathcal {R}_B\subset \Omega $ between two level-sets $S_1$ and $S_2$. The canonical momentum (the vectors on the level-sets $S_{1,2}$) is denoted by $\ensuremath {{\bm {P}}}_{S_{1,2}}$. $\ensuremath {{\bm {P}}}$ changes its orientation when shifted along the level-sets $S_1$ and $S_2$ since the norm of the curvature operator $\ensuremath {{\bm {K}}}$ (eq.~(\ref {eq:GNACurvatureDef})) is non zero. In \cref {fig:curvatureOperator2} the level-sets $S_{1,2}$ have been deformed according to $\ensuremath {\ensuremath {{\bm {x}}}}^\prime =\ensuremath {T^B_t}\circ \ensuremath {\ensuremath {{\bm {x}}}}$ such that the canonical momentum $\ensuremath {{\bm {P}}}$ becomes invariant with respect to shifts along $S_{1,2}$. In this case the norm of the curvature operator $\ensuremath {{\bm {K}}}$ vanishes\relax }}{122}{figure.caption.45}}
1685
\newlabel{fig:curvatureOperator}{{5.4}{122}{Effect of the diffusion $\vx ^\prime =\omegadeform \circ \vx $ (\eqref {eq:diffusionProcess}) on the canonical momentum $\vector {P}$. \Figref {fig:curvatureOperator1} shows a schematic picture of a region $\mathcal {R}_B\subset \Omega $ between two level-sets $S_1$ and $S_2$. The canonical momentum (the vectors on the level-sets $S_{1,2}$) is denoted by $\vector {P}_{S_{1,2}}$. $\vector {P}$ changes its orientation when shifted along the level-sets $S_1$ and $S_2$ since the norm of the curvature operator $\vector {K}$ (\eqref {eq:GNACurvatureDef}) is non zero. In \figref {fig:curvatureOperator2} the level-sets $S_{1,2}$ have been deformed according to $\vx ^\prime =\omegadeform \circ \vx $ such that the canonical momentum $\vector {P}$ becomes invariant with respect to shifts along $S_{1,2}$. In this case the norm of the curvature operator $\vector {K}$ vanishes\relax }{figure.caption.45}{}}
1686
\newlabel{fig:curvatureOperator@cref}{{[figure][4][5]5.4}{122}}
1687
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{122}{subfigure.4.1}}
1688
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{122}{subfigure.4.2}}
1689
\newlabel{eq:GNAdivergencePintegral}{{5.58}{122}{The Curvature Operator $\vector {K}$}{equation.5.1.58}{}}
1690
\newlabel{eq:GNAdivergencePintegral@cref}{{[equation][58][5]5.58}{122}}
1691
\newlabel{eq:GNACurvatureDef}{{5.59}{122}{The Curvature Operator $\vector {K}$}{equation.5.1.59}{}}
1692
\newlabel{eq:GNACurvatureDef@cref}{{[equation][59][5]5.59}{122}}
1693
\@writefile{loa}{\contentsline {algorithm}{\numberline {3}{\ignorespaces Basic Newton Algorithm (BNA)\relax }}{123}{algorithm.3}}
1694
\newlabel{alg:basicNewtonMethod}{{3}{123}{Basic Newton Algorithm (BNA)\relax }{algorithm.3}{}}
1695
\newlabel{alg:basicNewtonMethod@cref}{{[algorithm][3][]3}{123}}
1696
\@writefile{loa}{\contentsline {algorithm}{\numberline {4}{\ignorespaces Diffusion Algorithm (DA) \relax }}{123}{algorithm.4}}
1697
\newlabel{alg:warpOnlyMethod}{{4}{123}{Diffusion Algorithm (DA) \relax }{algorithm.4}{}}
1698
\newlabel{alg:warpOnlyMethod@cref}{{[algorithm][4][]4}{123}}
1699
\@writefile{loa}{\contentsline {algorithm}{\numberline {5}{\ignorespaces Extended Least Action Algorithm (ELAA)\relax }}{123}{algorithm.5}}
1700
\newlabel{alg:GeneralizedNewtonMethod}{{5}{123}{Extended Least Action Algorithm (ELAA)\relax }{algorithm.5}{}}
1701
\newlabel{alg:GeneralizedNewtonMethod@cref}{{[algorithm][5][]5}{123}}
1702
\newlabel{fig:Army}{{5.5a}{124}{Subfigure 5 5.5a}{subfigure.5.5.1}{}}
1697
1703
\newlabel{sub@fig:Army}{{(a)}{a}{Subfigure 5 5.5a\relax }{subfigure.5.5.1}{}}
1698
\newlabel{fig:Army@cref}{{[subfigure][1][5,5]5.5a}{123}}
1699
\newlabel{fig:ArmyNoise}{{5.5b}{123}{Subfigure 5 5.5b}{subfigure.5.5.2}{}}
1704
\newlabel{fig:Army@cref}{{[subfigure][1][5,5]5.5a}{124}}
1705
\newlabel{fig:ArmyNoise}{{5.5b}{124}{Subfigure 5 5.5b}{subfigure.5.5.2}{}}
1700
1706
\newlabel{sub@fig:ArmyNoise}{{(b)}{b}{Subfigure 5 5.5b\relax }{subfigure.5.5.2}{}}
1701
\newlabel{fig:ArmyNoise@cref}{{[subfigure][2][5,5]5.5b}{123}}
1702
\newlabel{fig:ArmyGNA}{{5.5c}{123}{Subfigure 5 5.5c}{subfigure.5.5.3}{}}
1707
\newlabel{fig:ArmyNoise@cref}{{[subfigure][2][5,5]5.5b}{124}}
1708
\newlabel{fig:ArmyGNA}{{5.5c}{124}{Subfigure 5 5.5c}{subfigure.5.5.3}{}}
1703
1709
\newlabel{sub@fig:ArmyGNA}{{(c)}{c}{Subfigure 5 5.5c\relax }{subfigure.5.5.3}{}}
1704
\newlabel{fig:ArmyGNA@cref}{{[subfigure][3][5,5]5.5c}{123}}
1705
\newlabel{fig:ArmyBNA}{{5.5d}{123}{Subfigure 5 5.5d}{subfigure.5.5.4}{}}
1710
\newlabel{fig:ArmyGNA@cref}{{[subfigure][3][5,5]5.5c}{124}}
1711
\newlabel{fig:ArmyBNA}{{5.5d}{124}{Subfigure 5 5.5d}{subfigure.5.5.4}{}}
1706
1712
\newlabel{sub@fig:ArmyBNA}{{(d)}{d}{Subfigure 5 5.5d\relax }{subfigure.5.5.4}{}}
1707
\newlabel{fig:ArmyBNA@cref}{{[subfigure][4][5,5]5.5d}{123}}
1708
\@writefile{lof}{\contentsline {figure}{\numberline {5.5}{\ignorespaces \Cref {fig:Army} shows a picture $\phi ^c$ of a person. $\phi ^c$ is taken to be free of noise. \Cref {fig:ArmyNoise} is a noise corrupted version of $\phi ^c$ in \cref {fig:Army}, $\phi ^d=\phi ^c+n$ where $n$ is iid Gaussian noise with a standard deviation $\sigma =100$. \Cref {fig:ArmyGNA} shows the result of the ELAA (alg. \ref {alg:GeneralizedNewtonMethod}) and \cref {fig:ArmyBNA} the result of the BNA (alg. \ref {alg:basicNewtonMethod})\relax }}{123}{figure.caption.45}}
1709
\newlabel{fig:ArmyTotal}{{5.5}{123}{\Figref {fig:Army} shows a picture $\phi ^c$ of a person. $\phi ^c$ is taken to be free of noise. \Figref {fig:ArmyNoise} is a noise corrupted version of $\phi ^c$ in \figref {fig:Army}, $\phi ^d=\phi ^c+n$ where $n$ is iid Gaussian noise with a standard deviation $\sigma =100$. \Figref {fig:ArmyGNA} shows the result of the ELAA (alg. \ref {alg:GeneralizedNewtonMethod}) and \figref {fig:ArmyBNA} the result of the BNA (alg. \ref {alg:basicNewtonMethod})\relax }{figure.caption.45}{}}
1710
\newlabel{fig:ArmyTotal@cref}{{[figure][5][5]5.5}{123}}
1711
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{123}{subfigure.5.1}}
1712
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{123}{subfigure.5.2}}
1713
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{123}{subfigure.5.3}}
1714
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{123}{subfigure.5.4}}
1715
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1.1}Image De-noising}{123}{subsection.5.1.1}}
1716
\newlabel{eq:cameraLikelihood2}{{5.60}{123}{Image De-noising}{equation.5.1.60}{}}
1717
\newlabel{eq:cameraLikelihood2@cref}{{[equation][60][5]5.60}{123}}
1718
\newlabel{eq:minimizationIO2}{{5.61}{123}{Image De-noising}{equation.5.1.61}{}}
1719
\newlabel{eq:minimizationIO2@cref}{{[equation][61][5]5.61}{123}}
1713
\newlabel{fig:ArmyBNA@cref}{{[subfigure][4][5,5]5.5d}{124}}
1714
\@writefile{lof}{\contentsline {figure}{\numberline {5.5}{\ignorespaces \Cref {fig:Army} shows a picture $\phi ^c$ of a person. $\phi ^c$ is taken to be free of noise. \Cref {fig:ArmyNoise} is a noise corrupted version of $\phi ^c$ in \cref {fig:Army}, $\phi ^d=\phi ^c+n$ where $n$ is iid Gaussian noise with a standard deviation $\sigma =100$. \Cref {fig:ArmyGNA} shows the result of the ELAA (alg. \ref {alg:GeneralizedNewtonMethod}) and \cref {fig:ArmyBNA} the result of the BNA (alg. \ref {alg:basicNewtonMethod})\relax }}{124}{figure.caption.46}}
1715
\newlabel{fig:ArmyTotal}{{5.5}{124}{\Figref {fig:Army} shows a picture $\phi ^c$ of a person. $\phi ^c$ is taken to be free of noise. \Figref {fig:ArmyNoise} is a noise corrupted version of $\phi ^c$ in \figref {fig:Army}, $\phi ^d=\phi ^c+n$ where $n$ is iid Gaussian noise with a standard deviation $\sigma =100$. \Figref {fig:ArmyGNA} shows the result of the ELAA (alg. \ref {alg:GeneralizedNewtonMethod}) and \figref {fig:ArmyBNA} the result of the BNA (alg. \ref {alg:basicNewtonMethod})\relax }{figure.caption.46}{}}
1716
\newlabel{fig:ArmyTotal@cref}{{[figure][5][5]5.5}{124}}
1717
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{124}{subfigure.5.1}}
1718
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{124}{subfigure.5.2}}
1719
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{124}{subfigure.5.3}}
1720
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{124}{subfigure.5.4}}
1721
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1.1}Image De-noising}{124}{subsection.5.1.1}}
1722
\newlabel{eq:cameraLikelihood2}{{5.60}{124}{Image De-noising}{equation.5.1.60}{}}
1723
\newlabel{eq:cameraLikelihood2@cref}{{[equation][60][5]5.60}{124}}
1724
\newlabel{eq:minimizationIO2}{{5.61}{124}{Image De-noising}{equation.5.1.61}{}}
1725
\newlabel{eq:minimizationIO2@cref}{{[equation][61][5]5.61}{124}}
1720
1726
\citation{Middleburry}
1721
\@writefile{loa}{\contentsline {algorithm}{\numberline {6}{\ignorespaces Image de-noising analysis\relax }}{124}{algorithm.6}}
1722
\newlabel{alg:ImageDenoisingAnalysis}{{6}{124}{Image de-noising analysis\relax }{algorithm.6}{}}
1723
\newlabel{alg:ImageDenoisingAnalysis@cref}{{[algorithm][6][]6}{124}}
1724
\newlabel{eq:DenoiseFunctional}{{5.62}{124}{Image De-noising}{equation.5.1.62}{}}
1725
\newlabel{eq:DenoiseFunctional@cref}{{[equation][62][5]5.62}{124}}
1726
\@writefile{toc}{\contentsline {subsubsection}{\nonumberline Analysis Method}{124}{section*.46}}
1727
\@writefile{brf}{\backcite{Middleburry}{{125}{5.1.1}{ALG@line.7}}}
1728
\@writefile{toc}{\contentsline {subsubsection}{\nonumberline Total Variation based Image De-Noising}{125}{section*.47}}
1729
\newlabel{eq:DenoiseFunctionalTV}{{5.64}{125}{Total Variation based Image De-Noising}{equation.5.1.64}{}}
1730
\newlabel{eq:DenoiseFunctionalTV@cref}{{[equation][64][5]5.64}{125}}
1731
\newlabel{eq:bendingTV}{{5.66}{125}{Total Variation based Image De-Noising}{equation.5.1.66}{}}
1732
\newlabel{eq:bendingTV@cref}{{[equation][66][5]5.66}{125}}
1733
\newlabel{fig:ArmyMeanEnergy}{{5.6a}{126}{Subfigure 5 5.6a}{subfigure.5.6.1}{}}
1727
\@writefile{loa}{\contentsline {algorithm}{\numberline {6}{\ignorespaces Image de-noising analysis\relax }}{125}{algorithm.6}}
1728
\newlabel{alg:ImageDenoisingAnalysis}{{6}{125}{Image de-noising analysis\relax }{algorithm.6}{}}
1729
\newlabel{alg:ImageDenoisingAnalysis@cref}{{[algorithm][6][]6}{125}}
1730
\newlabel{eq:DenoiseFunctional}{{5.62}{125}{Image De-noising}{equation.5.1.62}{}}
1731
\newlabel{eq:DenoiseFunctional@cref}{{[equation][62][5]5.62}{125}}
1732
\@writefile{toc}{\contentsline {subsubsection}{\nonumberline Analysis Method}{125}{section*.47}}
1733
\@writefile{brf}{\backcite{Middleburry}{{126}{5.1.1}{ALG@line.7}}}
1734
\@writefile{toc}{\contentsline {subsubsection}{\nonumberline Total Variation based Image De-Noising}{126}{section*.48}}
1735
\newlabel{eq:DenoiseFunctionalTV}{{5.64}{126}{Total Variation based Image De-Noising}{equation.5.1.64}{}}
1736
\newlabel{eq:DenoiseFunctionalTV@cref}{{[equation][64][5]5.64}{126}}
1737
\newlabel{eq:bendingTV}{{5.66}{126}{Total Variation based Image De-Noising}{equation.5.1.66}{}}
1738
\newlabel{eq:bendingTV@cref}{{[equation][66][5]5.66}{126}}
1739
\newlabel{fig:ArmyMeanEnergy}{{5.6a}{127}{Subfigure 5 5.6a}{subfigure.5.6.1}{}}
1734
1740
\newlabel{sub@fig:ArmyMeanEnergy}{{(a)}{a}{Subfigure 5 5.6a\relax }{subfigure.5.6.1}{}}
1735
\newlabel{fig:ArmyMeanEnergy@cref}{{[subfigure][1][5,6]5.6a}{126}}
1736
\newlabel{fig:ArmyStdDevEnergy}{{5.6b}{126}{Subfigure 5 5.6b}{subfigure.5.6.2}{}}
1741
\newlabel{fig:ArmyMeanEnergy@cref}{{[subfigure][1][5,6]5.6a}{127}}
1742
\newlabel{fig:ArmyStdDevEnergy}{{5.6b}{127}{Subfigure 5 5.6b}{subfigure.5.6.2}{}}
1737
1743
\newlabel{sub@fig:ArmyStdDevEnergy}{{(b)}{b}{Subfigure 5 5.6b\relax }{subfigure.5.6.2}{}}
1738
\newlabel{fig:ArmyStdDevEnergy@cref}{{[subfigure][2][5,6]5.6b}{126}}
1739
\@writefile{lof}{\contentsline {figure}{\numberline {5.6}{\ignorespaces \Cref {fig:ArmyMeanEnergy} shows the mean energy $\delimiter "426830A E^k\delimiter "526930B $ and \cref {fig:ArmyStdDevEnergy} the standard deviation $\sigma _{E^k}$ per iteration $k$ for the Army image in \cref {fig:Army}. The the ELAA (solid line) converges about twice as fast as the BNA (dashed line) according to \cref {fig:ArmyMeanEnergy}. The standard deviation $\sigma _{E^k}$ in \cref {fig:ArmyStdDevEnergy} converges approximately three times faster for the ELAA then for the BNA indicating that the ELAA is robuster to noise at every iteration $k$ \relax }}{126}{figure.caption.48}}
1740
\newlabel{fig:ArmyEnergy}{{5.6}{126}{\Figref {fig:ArmyMeanEnergy} shows the mean energy $\langle E^k\rangle $ and \figref {fig:ArmyStdDevEnergy} the standard deviation $\sigma _{E^k}$ per iteration $k$ for the Army image in \figref {fig:Army}. The the ELAA (solid line) converges about twice as fast as the BNA (dashed line) according to \figref {fig:ArmyMeanEnergy}. The standard deviation $\sigma _{E^k}$ in \figref {fig:ArmyStdDevEnergy} converges approximately three times faster for the ELAA then for the BNA indicating that the ELAA is robuster to noise at every iteration $k$ \relax }{figure.caption.48}{}}
1741
\newlabel{fig:ArmyEnergy@cref}{{[figure][6][5]5.6}{126}}
1742
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{126}{subfigure.6.1}}
1743
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{126}{subfigure.6.2}}
1744
\newlabel{eq:expCurv}{{5.67}{126}{Total Variation based Image De-Noising}{equation.5.1.67}{}}
1745
\newlabel{eq:expCurv@cref}{{[equation][67][5]5.67}{126}}
1746
\newlabel{eq:expCurvParameter}{{5.68}{126}{Total Variation based Image De-Noising}{equation.5.1.68}{}}
1747
\newlabel{eq:expCurvParameter@cref}{{[equation][68][5]5.68}{126}}
1748
\newlabel{fig:ArmyMeanCurvature}{{5.7a}{127}{Subfigure 5 5.7a}{subfigure.5.7.1}{}}
1744
\newlabel{fig:ArmyStdDevEnergy@cref}{{[subfigure][2][5,6]5.6b}{127}}
1745
\@writefile{lof}{\contentsline {figure}{\numberline {5.6}{\ignorespaces \Cref {fig:ArmyMeanEnergy} shows the mean energy $\delimiter "426830A E^k\delimiter "526930B $ and \cref {fig:ArmyStdDevEnergy} the standard deviation $\sigma _{E^k}$ per iteration $k$ for the Army image in \cref {fig:Army}. The the ELAA (solid line) converges about twice as fast as the BNA (dashed line) according to \cref {fig:ArmyMeanEnergy}. The standard deviation $\sigma _{E^k}$ in \cref {fig:ArmyStdDevEnergy} converges approximately three times faster for the ELAA then for the BNA indicating that the ELAA is robuster to noise at every iteration $k$ \relax }}{127}{figure.caption.49}}
1746
\newlabel{fig:ArmyEnergy}{{5.6}{127}{\Figref {fig:ArmyMeanEnergy} shows the mean energy $\langle E^k\rangle $ and \figref {fig:ArmyStdDevEnergy} the standard deviation $\sigma _{E^k}$ per iteration $k$ for the Army image in \figref {fig:Army}. The the ELAA (solid line) converges about twice as fast as the BNA (dashed line) according to \figref {fig:ArmyMeanEnergy}. The standard deviation $\sigma _{E^k}$ in \figref {fig:ArmyStdDevEnergy} converges approximately three times faster for the ELAA then for the BNA indicating that the ELAA is robuster to noise at every iteration $k$ \relax }{figure.caption.49}{}}
1747
\newlabel{fig:ArmyEnergy@cref}{{[figure][6][5]5.6}{127}}
1748
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{127}{subfigure.6.1}}
1749
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{127}{subfigure.6.2}}
1750
\newlabel{eq:expCurv}{{5.67}{127}{Total Variation based Image De-Noising}{equation.5.1.67}{}}
1751
\newlabel{eq:expCurv@cref}{{[equation][67][5]5.67}{127}}
1752
\newlabel{eq:expCurvParameter}{{5.68}{127}{Total Variation based Image De-Noising}{equation.5.1.68}{}}
1753
\newlabel{eq:expCurvParameter@cref}{{[equation][68][5]5.68}{127}}
1754
\newlabel{fig:ArmyMeanCurvature}{{5.7a}{128}{Subfigure 5 5.7a}{subfigure.5.7.1}{}}
1749
1755
\newlabel{sub@fig:ArmyMeanCurvature}{{(a)}{a}{Subfigure 5 5.7a\relax }{subfigure.5.7.1}{}}
1750
\newlabel{fig:ArmyMeanCurvature@cref}{{[subfigure][1][5,7]5.7a}{127}}
1751
\newlabel{fig:ArmyStdDevcurvature}{{5.7b}{127}{Subfigure 5 5.7b}{subfigure.5.7.2}{}}
1756
\newlabel{fig:ArmyMeanCurvature@cref}{{[subfigure][1][5,7]5.7a}{128}}
1757
\newlabel{fig:ArmyStdDevcurvature}{{5.7b}{128}{Subfigure 5 5.7b}{subfigure.5.7.2}{}}
1752
1758
\newlabel{sub@fig:ArmyStdDevcurvature}{{(b)}{b}{Subfigure 5 5.7b\relax }{subfigure.5.7.2}{}}
1753
\newlabel{fig:ArmyStdDevcurvature@cref}{{[subfigure][2][5,7]5.7b}{127}}
1754
\newlabel{fig:ArmyCurvatureFit}{{5.7c}{127}{Subfigure 5 5.7c}{subfigure.5.7.3}{}}
1759
\newlabel{fig:ArmyStdDevcurvature@cref}{{[subfigure][2][5,7]5.7b}{128}}
1760
\newlabel{fig:ArmyCurvatureFit}{{5.7c}{128}{Subfigure 5 5.7c}{subfigure.5.7.3}{}}
1755
1761
\newlabel{sub@fig:ArmyCurvatureFit}{{(c)}{c}{Subfigure 5 5.7c\relax }{subfigure.5.7.3}{}}
1756
\newlabel{fig:ArmyCurvatureFit@cref}{{[subfigure][3][5,7]5.7c}{127}}
1757
\@writefile{lof}{\contentsline {figure}{\numberline {5.7}{\ignorespaces \Cref {fig:ArmyMeanCurvature} shows the mean curvature $\delimiter "426830A \ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }^k\delimiter "526930B $ and \cref {fig:ArmyStdDevcurvature} the standard deviation $\sigma _{\ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }^k}$ per iteration $k$ for the Army image in \cref {fig:Army}. For the DA (dotted line), which only depends on the TV prior $E^{prior}_{TV}$, $\delimiter "426830A \ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }\delimiter "526930B $ has an exponential decay. For the ELAA (solid line) $\delimiter "426830A \ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }\delimiter "526930B $ drops faster then for the DA, until a point where the data term $E^{data}$ prohibits further smoothing of the level-sets $S$. Then $\delimiter "426830A \ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }\delimiter "526930B $ rises slightly and converges at a higher value. The BNA falls off slower then the ELAA and the DA and converging at a slightly higher value then the ELAA. The standard deviation $\sigma _{\ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }}$ is for both the ELAA and the BNA comparatively of equal order and small and two orders of magnitude smaller then $\delimiter "426830A \ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }\delimiter "526930B $. When comparing the ELAA and the BNA to the DA (dotted line) we can see that the data term $E^{data}$ has an impact on the noise distribution of the curvature $\ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }$ particularly at later iterations $k>100$. \Cref {fig:ArmyCurvatureFit} shows a fit of the exponential function in eq.~(\ref {eq:expCurv}) to the curvature of the DA algorithm. The difference between the DA (solid line) and the fit (dashed line) is of the order $10^{4}$, an order of magnitude smaller then $\ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }$ \relax }}{127}{figure.caption.49}}
1758
\newlabel{fig:ArmyCurvature}{{5.7}{127}{\Figref {fig:ArmyMeanCurvature} shows the mean curvature $\langle \norm {\vector {K}}^k\rangle $ and \figref {fig:ArmyStdDevcurvature} the standard deviation $\sigma _{\norm {\vector {K}}^k}$ per iteration $k$ for the Army image in \figref {fig:Army}. For the DA (dotted line), which only depends on the TV prior $E^{prior}_{TV}$, $\langle \norm {\vector {K}}\rangle $ has an exponential decay. For the ELAA (solid line) $\langle \norm {\vector {K}}\rangle $ drops faster then for the DA, until a point where the data term $E^{data}$ prohibits further smoothing of the level-sets $S$. Then $\langle \norm {\vector {K}}\rangle $ rises slightly and converges at a higher value. The BNA falls off slower then the ELAA and the DA and converging at a slightly higher value then the ELAA. The standard deviation $\sigma _{\norm {\vector {K}}}$ is for both the ELAA and the BNA comparatively of equal order and small and two orders of magnitude smaller then $\langle \norm {\vector {K}}\rangle $. When comparing the ELAA and the BNA to the DA (dotted line) we can see that the data term $E^{data}$ has an impact on the noise distribution of the curvature $\norm {\vector {K}}$ particularly at later iterations $k>100$. \Figref {fig:ArmyCurvatureFit} shows a fit of the exponential function in \eqref {eq:expCurv} to the curvature of the DA algorithm. The difference between the DA (solid line) and the fit (dashed line) is of the order $10^{4}$, an order of magnitude smaller then $\norm {\vector {K}}$ \relax }{figure.caption.49}{}}
1759
\newlabel{fig:ArmyCurvature@cref}{{[figure][7][5]5.7}{127}}
1760
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{127}{subfigure.7.1}}
1761
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{127}{subfigure.7.2}}
1762
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{127}{subfigure.7.3}}
1763
\@writefile{toc}{\contentsline {subsubsection}{\nonumberline Structure Tensor Prior}{127}{section*.50}}
1764
\newlabel{eq:DenoiseFunctionalST}{{5.70}{127}{Structure Tensor Prior}{equation.5.1.70}{}}
1765
\newlabel{eq:DenoiseFunctionalST@cref}{{[equation][70][5]5.70}{127}}
1766
\newlabel{eq:structtensPriorRotInv2}{{5.71}{127}{Structure Tensor Prior}{equation.5.1.71}{}}
1767
\newlabel{eq:structtensPriorRotInv2@cref}{{[equation][71][5]5.71}{127}}
1768
\newlabel{fig:ArmyMeanEnergyGnaDaST}{{5.8a}{128}{Subfigure 5 5.8a}{subfigure.5.8.1}{}}
1762
\newlabel{fig:ArmyCurvatureFit@cref}{{[subfigure][3][5,7]5.7c}{128}}
1763
\@writefile{lof}{\contentsline {figure}{\numberline {5.7}{\ignorespaces \Cref {fig:ArmyMeanCurvature} shows the mean curvature $\delimiter "426830A \ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }^k\delimiter "526930B $ and \cref {fig:ArmyStdDevcurvature} the standard deviation $\sigma _{\ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }^k}$ per iteration $k$ for the Army image in \cref {fig:Army}. For the DA (dotted line), which only depends on the TV prior $E^{prior}_{TV}$, $\delimiter "426830A \ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }\delimiter "526930B $ has an exponential decay. For the ELAA (solid line) $\delimiter "426830A \ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }\delimiter "526930B $ drops faster then for the DA, until a point where the data term $E^{data}$ prohibits further smoothing of the level-sets $S$. Then $\delimiter "426830A \ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }\delimiter "526930B $ rises slightly and converges at a higher value. The BNA falls off slower then the ELAA and the DA and converging at a slightly higher value then the ELAA. The standard deviation $\sigma _{\ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }}$ is for both the ELAA and the BNA comparatively of equal order and small and two orders of magnitude smaller then $\delimiter "426830A \ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }\delimiter "526930B $. When comparing the ELAA and the BNA to the DA (dotted line) we can see that the data term $E^{data}$ has an impact on the noise distribution of the curvature $\ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }$ particularly at later iterations $k>100$. \Cref {fig:ArmyCurvatureFit} shows a fit of the exponential function in eq.~(\ref {eq:expCurv}) to the curvature of the DA algorithm. The difference between the DA (solid line) and the fit (dashed line) is of the order $10^{4}$, an order of magnitude smaller then $\ensuremath {\left \delimiter 69645069 \ensuremath {{\bm {K}}} \right \delimiter 86422285 }$ \relax }}{128}{figure.caption.50}}
1764
\newlabel{fig:ArmyCurvature}{{5.7}{128}{\Figref {fig:ArmyMeanCurvature} shows the mean curvature $\langle \norm {\vector {K}}^k\rangle $ and \figref {fig:ArmyStdDevcurvature} the standard deviation $\sigma _{\norm {\vector {K}}^k}$ per iteration $k$ for the Army image in \figref {fig:Army}. For the DA (dotted line), which only depends on the TV prior $E^{prior}_{TV}$, $\langle \norm {\vector {K}}\rangle $ has an exponential decay. For the ELAA (solid line) $\langle \norm {\vector {K}}\rangle $ drops faster then for the DA, until a point where the data term $E^{data}$ prohibits further smoothing of the level-sets $S$. Then $\langle \norm {\vector {K}}\rangle $ rises slightly and converges at a higher value. The BNA falls off slower then the ELAA and the DA and converging at a slightly higher value then the ELAA. The standard deviation $\sigma _{\norm {\vector {K}}}$ is for both the ELAA and the BNA comparatively of equal order and small and two orders of magnitude smaller then $\langle \norm {\vector {K}}\rangle $. When comparing the ELAA and the BNA to the DA (dotted line) we can see that the data term $E^{data}$ has an impact on the noise distribution of the curvature $\norm {\vector {K}}$ particularly at later iterations $k>100$. \Figref {fig:ArmyCurvatureFit} shows a fit of the exponential function in \eqref {eq:expCurv} to the curvature of the DA algorithm. The difference between the DA (solid line) and the fit (dashed line) is of the order $10^{4}$, an order of magnitude smaller then $\norm {\vector {K}}$ \relax }{figure.caption.50}{}}
1765
\newlabel{fig:ArmyCurvature@cref}{{[figure][7][5]5.7}{128}}
1766
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{128}{subfigure.7.1}}
1767
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{128}{subfigure.7.2}}
1768
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{128}{subfigure.7.3}}
1769
\@writefile{toc}{\contentsline {subsubsection}{\nonumberline Structure Tensor Prior}{128}{section*.51}}
1770
\newlabel{eq:DenoiseFunctionalST}{{5.70}{128}{Structure Tensor Prior}{equation.5.1.70}{}}
1771
\newlabel{eq:DenoiseFunctionalST@cref}{{[equation][70][5]5.70}{128}}
1772
\newlabel{eq:structtensPriorRotInv2}{{5.71}{128}{Structure Tensor Prior}{equation.5.1.71}{}}
1773
\newlabel{eq:structtensPriorRotInv2@cref}{{[equation][71][5]5.71}{128}}
1774
\newlabel{fig:ArmyMeanEnergyGnaDaST}{{5.8a}{129}{Subfigure 5 5.8a}{subfigure.5.8.1}{}}
1769
1775
\newlabel{sub@fig:ArmyMeanEnergyGnaDaST}{{(a)}{a}{Subfigure 5 5.8a\relax }{subfigure.5.8.1}{}}
1770
\newlabel{fig:ArmyMeanEnergyGnaDaST@cref}{{[subfigure][1][5,8]5.8a}{128}}
1771
\newlabel{fig:ArmyMeanEnergyGnaST}{{5.8b}{128}{Subfigure 5 5.8b}{subfigure.5.8.2}{}}
1776
\newlabel{fig:ArmyMeanEnergyGnaDaST@cref}{{[subfigure][1][5,8]5.8a}{129}}
1777
\newlabel{fig:ArmyMeanEnergyGnaST}{{5.8b}{129}{Subfigure 5 5.8b}{subfigure.5.8.2}{}}
1772
1778
\newlabel{sub@fig:ArmyMeanEnergyGnaST}{{(b)}{b}{Subfigure 5 5.8b\relax }{subfigure.5.8.2}{}}
1773
\newlabel{fig:ArmyMeanEnergyGnaST@cref}{{[subfigure][2][5,8]5.8b}{128}}
1774
\newlabel{fig:ArmyMeanEnergyGnaBnaDiffST}{{5.8c}{128}{Subfigure 5 5.8c}{subfigure.5.8.3}{}}
1779
\newlabel{fig:ArmyMeanEnergyGnaST@cref}{{[subfigure][2][5,8]5.8b}{129}}
1780
\newlabel{fig:ArmyMeanEnergyGnaBnaDiffST}{{5.8c}{129}{Subfigure 5 5.8c}{subfigure.5.8.3}{}}
1775
1781
\newlabel{sub@fig:ArmyMeanEnergyGnaBnaDiffST}{{(c)}{c}{Subfigure 5 5.8c\relax }{subfigure.5.8.3}{}}
1776
\newlabel{fig:ArmyMeanEnergyGnaBnaDiffST@cref}{{[subfigure][3][5,8]5.8c}{128}}
1777
\@writefile{lof}{\contentsline {figure}{\numberline {5.8}{\ignorespaces \Cref {fig:ArmyMeanEnergyGnaDaST} shows the mean energy $\delimiter "426830A E\delimiter "526930B $ as a function of the iteration $k$ for the ELAA (solid line) and the DA (dotted line) for the structure tensor model. \Cref {fig:ArmyMeanEnergyGnaST} shows a close up of $\delimiter "426830A E\delimiter "526930B _{ELAA}$ for $k\geq 10$ and \cref {fig:ArmyMeanEnergyGnaBnaDiffST} shows the difference between $\delimiter "426830A E\delimiter "526930B _{BNA}$ and $\delimiter "426830A E\delimiter "526930B _{ELAA}$. From \cref {fig:ArmyMeanEnergyGnaST} we can see that the mean energy for the ELAA $\delimiter "426830A E\delimiter "526930B _{ELAA}$ is $2$ orders of magnitude smaller then the mean energy for the DA and by \cref {fig:ArmyMeanEnergyGnaBnaDiffST} only slightly smaller then $\delimiter "426830A E\delimiter "526930B _{BNA}$. Thus the effect of the diffusion process in eq.~(\ref {eq:diffusionProcess}) on the minimization of the energy $E$ in eq.~(\ref {eq:DenoiseFunctionalST}) is at most marginal\relax }}{128}{figure.caption.51}}
1778
\newlabel{fig:MeanEnergyST}{{5.8}{128}{\Figref {fig:ArmyMeanEnergyGnaDaST} shows the mean energy $\langle E\rangle $ as a function of the iteration $k$ for the ELAA (solid line) and the DA (dotted line) for the structure tensor model. \Figref {fig:ArmyMeanEnergyGnaST} shows a close up of $\langle E\rangle _{ELAA}$ for $k\geq 10$ and \figref {fig:ArmyMeanEnergyGnaBnaDiffST} shows the difference between $\langle E\rangle _{BNA}$ and $\langle E\rangle _{ELAA}$. From \figref {fig:ArmyMeanEnergyGnaST} we can see that the mean energy for the ELAA $\langle E\rangle _{ELAA}$ is $2$ orders of magnitude smaller then the mean energy for the DA and by \figref {fig:ArmyMeanEnergyGnaBnaDiffST} only slightly smaller then $\langle E\rangle _{BNA}$. Thus the effect of the diffusion process in \eqref {eq:diffusionProcess} on the minimization of the energy $E$ in \eqref {eq:DenoiseFunctionalST} is at most marginal\relax }{figure.caption.51}{}}
1779
\newlabel{fig:MeanEnergyST@cref}{{[figure][8][5]5.8}{128}}
1780
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{128}{subfigure.8.1}}
1781
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{128}{subfigure.8.2}}
1782
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{128}{subfigure.8.3}}
1783
\newlabel{eq:BlevelSetST}{{5.72}{128}{Structure Tensor Prior}{equation.5.1.72}{}}
1784
\newlabel{eq:BlevelSetST@cref}{{[equation][72][5]5.72}{128}}
1785
\newlabel{eq:bendingOperatorST0}{{5.73}{128}{Structure Tensor Prior}{equation.5.1.73}{}}
1786
\newlabel{eq:bendingOperatorST0@cref}{{[equation][73][5]5.73}{128}}
1787
\newlabel{fig:ArmyStdDevEnergyGnaDaST}{{5.9a}{129}{Subfigure 5 5.9a}{subfigure.5.9.1}{}}
1782
\newlabel{fig:ArmyMeanEnergyGnaBnaDiffST@cref}{{[subfigure][3][5,8]5.8c}{129}}
1783
\@writefile{lof}{\contentsline {figure}{\numberline {5.8}{\ignorespaces \Cref {fig:ArmyMeanEnergyGnaDaST} shows the mean energy $\delimiter "426830A E\delimiter "526930B $ as a function of the iteration $k$ for the ELAA (solid line) and the DA (dotted line) for the structure tensor model. \Cref {fig:ArmyMeanEnergyGnaST} shows a close up of $\delimiter "426830A E\delimiter "526930B _{ELAA}$ for $k\geq 10$ and \cref {fig:ArmyMeanEnergyGnaBnaDiffST} shows the difference between $\delimiter "426830A E\delimiter "526930B _{BNA}$ and $\delimiter "426830A E\delimiter "526930B _{ELAA}$. From \cref {fig:ArmyMeanEnergyGnaST} we can see that the mean energy for the ELAA $\delimiter "426830A E\delimiter "526930B _{ELAA}$ is $2$ orders of magnitude smaller then the mean energy for the DA and by \cref {fig:ArmyMeanEnergyGnaBnaDiffST} only slightly smaller then $\delimiter "426830A E\delimiter "526930B _{BNA}$. Thus the effect of the diffusion process in eq.~(\ref {eq:diffusionProcess}) on the minimization of the energy $E$ in eq.~(\ref {eq:DenoiseFunctionalST}) is at most marginal\relax }}{129}{figure.caption.52}}
1784
\newlabel{fig:MeanEnergyST}{{5.8}{129}{\Figref {fig:ArmyMeanEnergyGnaDaST} shows the mean energy $\langle E\rangle $ as a function of the iteration $k$ for the ELAA (solid line) and the DA (dotted line) for the structure tensor model. \Figref {fig:ArmyMeanEnergyGnaST} shows a close up of $\langle E\rangle _{ELAA}$ for $k\geq 10$ and \figref {fig:ArmyMeanEnergyGnaBnaDiffST} shows the difference between $\langle E\rangle _{BNA}$ and $\langle E\rangle _{ELAA}$. From \figref {fig:ArmyMeanEnergyGnaST} we can see that the mean energy for the ELAA $\langle E\rangle _{ELAA}$ is $2$ orders of magnitude smaller then the mean energy for the DA and by \figref {fig:ArmyMeanEnergyGnaBnaDiffST} only slightly smaller then $\langle E\rangle _{BNA}$. Thus the effect of the diffusion process in \eqref {eq:diffusionProcess} on the minimization of the energy $E$ in \eqref {eq:DenoiseFunctionalST} is at most marginal\relax }{figure.caption.52}{}}
1785
\newlabel{fig:MeanEnergyST@cref}{{[figure][8][5]5.8}{129}}
1786
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{129}{subfigure.8.1}}
1787
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{129}{subfigure.8.2}}
1788
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{129}{subfigure.8.3}}
1789
\newlabel{eq:BlevelSetST}{{5.72}{129}{Structure Tensor Prior}{equation.5.1.72}{}}
1790
\newlabel{eq:BlevelSetST@cref}{{[equation][72][5]5.72}{129}}
1791
\newlabel{eq:bendingOperatorST0}{{5.73}{129}{Structure Tensor Prior}{equation.5.1.73}{}}
1792
\newlabel{eq:bendingOperatorST0@cref}{{[equation][73][5]5.73}{129}}
1793
\newlabel{fig:ArmyStdDevEnergyGnaDaST}{{5.9a}{130}{Subfigure 5 5.9a}{subfigure.5.9.1}{}}
1788
1794
\newlabel{sub@fig:ArmyStdDevEnergyGnaDaST}{{(a)}{a}{Subfigure 5 5.9a\relax }{subfigure.5.9.1}{}}
1789
\newlabel{fig:ArmyStdDevEnergyGnaDaST@cref}{{[subfigure][1][5,9]5.9a}{129}}
1790
\newlabel{fig:ArmyStdDevEnergyGnaST}{{5.9b}{129}{Subfigure 5 5.9b}{subfigure.5.9.2}{}}
1795
\newlabel{fig:ArmyStdDevEnergyGnaDaST@cref}{{[subfigure][1][5,9]5.9a}{130}}
1796
\newlabel{fig:ArmyStdDevEnergyGnaST}{{5.9b}{130}{Subfigure 5 5.9b}{subfigure.5.9.2}{}}
1791
1797
\newlabel{sub@fig:ArmyStdDevEnergyGnaST}{{(b)}{b}{Subfigure 5 5.9b\relax }{subfigure.5.9.2}{}}
1792
\newlabel{fig:ArmyStdDevEnergyGnaST@cref}{{[subfigure][2][5,9]5.9b}{129}}
1793
\newlabel{fig:ArmyStdDevEnergyGnaBnaDiffST}{{5.9c}{129}{Subfigure 5 5.9c}{subfigure.5.9.3}{}}
1798
\newlabel{fig:ArmyStdDevEnergyGnaST@cref}{{[subfigure][2][5,9]5.9b}{130}}
1799
\newlabel{fig:ArmyStdDevEnergyGnaBnaDiffST}{{5.9c}{130}{Subfigure 5 5.9c}{subfigure.5.9.3}{}}
1794
1800
\newlabel{sub@fig:ArmyStdDevEnergyGnaBnaDiffST}{{(c)}{c}{Subfigure 5 5.9c\relax }{subfigure.5.9.3}{}}
1795
\newlabel{fig:ArmyStdDevEnergyGnaBnaDiffST@cref}{{[subfigure][3][5,9]5.9c}{129}}
1796
\@writefile{lof}{\contentsline {figure}{\numberline {5.9}{\ignorespaces \Cref {fig:ArmyStdDevEnergyGnaDaST} shows the standard deviation $\sigma _E$ as a function of the iteration $k$ for the ELAA (solid line) and the DA (dotted line) for the structure tensor model. \Cref {fig:ArmyStdDevEnergyGnaST} shows a close up of $\sigma _{E,ELAA}$ for $k\geq 10$ and \cref {fig:ArmyStdDevEnergyGnaBnaDiffST} shows the difference between $\sigma _{E,BNA}$ and $\sigma _{E,ELAA}$. We essentially see the same behavior for the standard deviation $\sigma _E$ as for the mean energy in \cref {fig:MeanEnergyST}: By \cref {fig:ArmyStdDevEnergyGnaST} the standard deviation energy for the ELAA $\sigma _{E,ELAA}$ is $1$ order of magnitude smaller that of the DA and by \cref {fig:ArmyStdDevEnergyGnaBnaDiffST} only slightly smaller then $\sigma _{E,BNA}$. Hence the diffusion process eq.~(\ref {eq:diffusionProcess}) has a marginal contribution to the statistical robustness of the minimizers of $E$ in eq.~(\ref {eq:DenoiseFunctionalST})\relax }}{129}{figure.caption.52}}
1797
\newlabel{fig:StdDevEnergyST}{{5.9}{129}{\Figref {fig:ArmyStdDevEnergyGnaDaST} shows the standard deviation $\sigma _E$ as a function of the iteration $k$ for the ELAA (solid line) and the DA (dotted line) for the structure tensor model. \Figref {fig:ArmyStdDevEnergyGnaST} shows a close up of $\sigma _{E,ELAA}$ for $k\geq 10$ and \figref {fig:ArmyStdDevEnergyGnaBnaDiffST} shows the difference between $\sigma _{E,BNA}$ and $\sigma _{E,ELAA}$. We essentially see the same behavior for the standard deviation $\sigma _E$ as for the mean energy in \figref {fig:MeanEnergyST}: By \figref {fig:ArmyStdDevEnergyGnaST} the standard deviation energy for the ELAA $\sigma _{E,ELAA}$ is $1$ order of magnitude smaller that of the DA and by \figref {fig:ArmyStdDevEnergyGnaBnaDiffST} only slightly smaller then $\sigma _{E,BNA}$. Hence the diffusion process \eqref {eq:diffusionProcess} has a marginal contribution to the statistical robustness of the minimizers of $E$ in \eqref {eq:DenoiseFunctionalST}\relax }{figure.caption.52}{}}
1798
\newlabel{fig:StdDevEnergyST@cref}{{[figure][9][5]5.9}{129}}
1799
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{129}{subfigure.9.1}}
1800
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{129}{subfigure.9.2}}
1801
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{129}{subfigure.9.3}}
1802
\newlabel{eq:bendingOperatorST1}{{5.74}{129}{Structure Tensor Prior}{equation.5.1.74}{}}
1803
\newlabel{eq:bendingOperatorST1@cref}{{[equation][74][5]5.74}{129}}
1804
\newlabel{eq:bendingOperatorST}{{5.75}{129}{Structure Tensor Prior}{equation.5.1.75}{}}
1805
\newlabel{eq:bendingOperatorST@cref}{{[equation][75][5]5.75}{129}}
1806
\newlabel{fig:EnergyWsizeGnaST}{{5.10a}{130}{Subfigure 5 5.10a}{subfigure.5.10.1}{}}
1801
\newlabel{fig:ArmyStdDevEnergyGnaBnaDiffST@cref}{{[subfigure][3][5,9]5.9c}{130}}
1802
\@writefile{lof}{\contentsline {figure}{\numberline {5.9}{\ignorespaces \Cref {fig:ArmyStdDevEnergyGnaDaST} shows the standard deviation $\sigma _E$ as a function of the iteration $k$ for the ELAA (solid line) and the DA (dotted line) for the structure tensor model. \Cref {fig:ArmyStdDevEnergyGnaST} shows a close up of $\sigma _{E,ELAA}$ for $k\geq 10$ and \cref {fig:ArmyStdDevEnergyGnaBnaDiffST} shows the difference between $\sigma _{E,BNA}$ and $\sigma _{E,ELAA}$. We essentially see the same behavior for the standard deviation $\sigma _E$ as for the mean energy in \cref {fig:MeanEnergyST}: By \cref {fig:ArmyStdDevEnergyGnaST} the standard deviation energy for the ELAA $\sigma _{E,ELAA}$ is $1$ order of magnitude smaller that of the DA and by \cref {fig:ArmyStdDevEnergyGnaBnaDiffST} only slightly smaller then $\sigma _{E,BNA}$. Hence the diffusion process eq.~(\ref {eq:diffusionProcess}) has a marginal contribution to the statistical robustness of the minimizers of $E$ in eq.~(\ref {eq:DenoiseFunctionalST})\relax }}{130}{figure.caption.53}}
1803
\newlabel{fig:StdDevEnergyST}{{5.9}{130}{\Figref {fig:ArmyStdDevEnergyGnaDaST} shows the standard deviation $\sigma _E$ as a function of the iteration $k$ for the ELAA (solid line) and the DA (dotted line) for the structure tensor model. \Figref {fig:ArmyStdDevEnergyGnaST} shows a close up of $\sigma _{E,ELAA}$ for $k\geq 10$ and \figref {fig:ArmyStdDevEnergyGnaBnaDiffST} shows the difference between $\sigma _{E,BNA}$ and $\sigma _{E,ELAA}$. We essentially see the same behavior for the standard deviation $\sigma _E$ as for the mean energy in \figref {fig:MeanEnergyST}: By \figref {fig:ArmyStdDevEnergyGnaST} the standard deviation energy for the ELAA $\sigma _{E,ELAA}$ is $1$ order of magnitude smaller that of the DA and by \figref {fig:ArmyStdDevEnergyGnaBnaDiffST} only slightly smaller then $\sigma _{E,BNA}$. Hence the diffusion process \eqref {eq:diffusionProcess} has a marginal contribution to the statistical robustness of the minimizers of $E$ in \eqref {eq:DenoiseFunctionalST}\relax }{figure.caption.53}{}}
1804
\newlabel{fig:StdDevEnergyST@cref}{{[figure][9][5]5.9}{130}}
1805
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{130}{subfigure.9.1}}
1806
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{130}{subfigure.9.2}}
1807
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{130}{subfigure.9.3}}
1808
\newlabel{eq:bendingOperatorST1}{{5.74}{130}{Structure Tensor Prior}{equation.5.1.74}{}}
1809
\newlabel{eq:bendingOperatorST1@cref}{{[equation][74][5]5.74}{130}}
1810
\newlabel{eq:bendingOperatorST}{{5.75}{130}{Structure Tensor Prior}{equation.5.1.75}{}}
1811
\newlabel{eq:bendingOperatorST@cref}{{[equation][75][5]5.75}{130}}
1812
\newlabel{fig:EnergyWsizeGnaST}{{5.10a}{131}{Subfigure 5 5.10a}{subfigure.5.10.1}{}}
1807
1813
\newlabel{sub@fig:EnergyWsizeGnaST}{{(a)}{a}{Subfigure 5 5.10a\relax }{subfigure.5.10.1}{}}
1808
\newlabel{fig:EnergyWsizeGnaST@cref}{{[subfigure][1][5,10]5.10a}{130}}
1809
\newlabel{fig:CurvatureWsizeGnaST}{{5.10b}{130}{Subfigure 5 5.10b}{subfigure.5.10.2}{}}
1814
\newlabel{fig:EnergyWsizeGnaST@cref}{{[subfigure][1][5,10]5.10a}{131}}
1815
\newlabel{fig:CurvatureWsizeGnaST}{{5.10b}{131}{Subfigure 5 5.10b}{subfigure.5.10.2}{}}
1810
1816
\newlabel{sub@fig:CurvatureWsizeGnaST}{{(b)}{b}{Subfigure 5 5.10b\relax }{subfigure.5.10.2}{}}
1811
\newlabel{fig:CurvatureWsizeGnaST@cref}{{[subfigure][2][5,10]5.10b}{130}}
1812
\newlabel{fig:InitialEnergyGnaST}{{5.10c}{130}{Subfigure 5 5.10c}{subfigure.5.10.3}{}}
1817
\newlabel{fig:CurvatureWsizeGnaST@cref}{{[subfigure][2][5,10]5.10b}{131}}
1818
\newlabel{fig:InitialEnergyGnaST}{{5.10c}{131}{Subfigure 5 5.10c}{subfigure.5.10.3}{}}
1813
1819
\newlabel{sub@fig:InitialEnergyGnaST}{{(c)}{c}{Subfigure 5 5.10c\relax }{subfigure.5.10.3}{}}
1814
\newlabel{fig:InitialEnergyGnaST@cref}{{[subfigure][3][5,10]5.10c}{130}}
1815
\newlabel{fig:InitialCurvatureGnaST}{{5.10d}{130}{Subfigure 5 5.10d}{subfigure.5.10.4}{}}
1820
\newlabel{fig:InitialEnergyGnaST@cref}{{[subfigure][3][5,10]5.10c}{131}}
1821
\newlabel{fig:InitialCurvatureGnaST}{{5.10d}{131}{Subfigure 5 5.10d}{subfigure.5.10.4}{}}
1816
1822
\newlabel{sub@fig:InitialCurvatureGnaST}{{(d)}{d}{Subfigure 5 5.10d\relax }{subfigure.5.10.4}{}}
1817
\newlabel{fig:InitialCurvatureGnaST@cref}{{[subfigure][4][5,10]5.10d}{130}}
1818
\@writefile{lof}{\contentsline {figure}{\numberline {5.10}{\ignorespaces Study of the dependency the mean energy $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ and the mean curvature $\delimiter "426830A \ensuremath {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $ on the window size $\sigma _{ST}$ of the structure tensor prior $E^{prior}_{ST}$. \Cref {fig:EnergyWsizeGnaST} shows the mean energy $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ per iteration $k\geq 100$ for various $\sigma _{ST}$ and \cref {fig:CurvatureWsizeGnaST} the mean curvature $\delimiter "426830A \ensuremath {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $, also for various $\sigma _{ST}$. Figures \ref {fig:InitialEnergyGnaST} and \ref {fig:InitialCurvatureGnaST} show the initial energy $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ and the initial curvature $\delimiter "426830A \ensuremath {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $ for $k=0$. In \cref {fig:EnergyWsizeGnaST} we can see that for smaller $\sigma _{ST}$ the energy $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ converges to lower values. Conversely for larger window sizes $\sigma _{ST}$ the mean energy profiles $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ per $\sigma _{ST}$ converge. In \cref {fig:CurvatureWsizeGnaST} we observe a similar behavior for the curvature $\delimiter "426830A \ensuremath {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $: For small $\sigma _{ST}$ the curvature $\delimiter "426830A \ensuremath {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $ is comparatively large. As $\sigma _{ST}$ rises the profile of $\delimiter "426830A \ensuremath {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $ per $\sigma _{ST}$ converge, albeit at lower values. Figures \ref {fig:InitialEnergyGnaST} and \ref {fig:InitialCurvatureGnaST} show that the initial energy and the initial curvature for $\sigma _{ST}=3$ have half the values then for the larger window sizes $\sigma _{ST}=13\cdots 63$\relax }}{130}{figure.caption.53}}
1819
\newlabel{fig:WsizeGnaST}{{5.10}{130}{Study of the dependency the mean energy $\langle E^k\rangle _{ELAA}$ and the mean curvature $\langle \norm {K}\rangle $ on the window size $\sigma _{ST}$ of the structure tensor prior $E^{prior}_{ST}$. \Figref {fig:EnergyWsizeGnaST} shows the mean energy $\langle E^k\rangle _{ELAA}$ per iteration $k\geq 100$ for various $\sigma _{ST}$ and \figref {fig:CurvatureWsizeGnaST} the mean curvature $\langle \norm {K}\rangle $, also for various $\sigma _{ST}$. Figures \ref {fig:InitialEnergyGnaST} and \ref {fig:InitialCurvatureGnaST} show the initial energy $\langle E^k\rangle _{ELAA}$ and the initial curvature $\langle \norm {K}\rangle $ for $k=0$. In \figref {fig:EnergyWsizeGnaST} we can see that for smaller $\sigma _{ST}$ the energy $\langle E^k\rangle _{ELAA}$ converges to lower values. Conversely for larger window sizes $\sigma _{ST}$ the mean energy profiles $\langle E^k\rangle _{ELAA}$ per $\sigma _{ST}$ converge. In \figref {fig:CurvatureWsizeGnaST} we observe a similar behavior for the curvature $\langle \norm {K}\rangle $: For small $\sigma _{ST}$ the curvature $\langle \norm {K}\rangle $ is comparatively large. As $\sigma _{ST}$ rises the profile of $\langle \norm {K}\rangle $ per $\sigma _{ST}$ converge, albeit at lower values. Figures \ref {fig:InitialEnergyGnaST} and \ref {fig:InitialCurvatureGnaST} show that the initial energy and the initial curvature for $\sigma _{ST}=3$ have half the values then for the larger window sizes $\sigma _{ST}=13\cdots 63$\relax }{figure.caption.53}{}}
1820
\newlabel{fig:WsizeGnaST@cref}{{[figure][10][5]5.10}{130}}
1821
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{130}{subfigure.10.1}}
1822
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{130}{subfigure.10.2}}
1823
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{130}{subfigure.10.3}}
1824
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{130}{subfigure.10.4}}
1825
\newlabel{eq:RelativeEnergyST}{{5.76}{131}{Structure Tensor Prior}{equation.5.1.76}{}}
1826
\newlabel{eq:RelativeEnergyST@cref}{{[equation][76][5]5.76}{131}}
1827
\@writefile{toc}{\contentsline {section}{\numberline {5.2}summary}{131}{section.5.2}}
1828
\newlabel{eq:totEnergyGenNewton2}{{5.77}{132}{summary}{equation.5.2.77}{}}
1829
\newlabel{eq:totEnergyGenNewton2@cref}{{[equation][77][5]5.77}{132}}
1830
\newlabel{eq:eulerLagrangeGRF3}{{5.78}{132}{summary}{equation.5.2.78}{}}
1831
\newlabel{eq:eulerLagrangeGRF3@cref}{{[equation][78][5]5.78}{132}}
1832
\newlabel{eq:diffusionProcess2}{{5.79}{132}{summary}{equation.5.2.79}{}}
1833
\newlabel{eq:diffusionProcess2@cref}{{[equation][79][5]5.79}{132}}
1823
\newlabel{fig:InitialCurvatureGnaST@cref}{{[subfigure][4][5,10]5.10d}{131}}
1824
\@writefile{lof}{\contentsline {figure}{\numberline {5.10}{\ignorespaces Study of the dependency the mean energy $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ and the mean curvature $\delimiter "426830A \ensuremath {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $ on the window size $\sigma _{ST}$ of the structure tensor prior $E^{prior}_{ST}$. \Cref {fig:EnergyWsizeGnaST} shows the mean energy $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ per iteration $k\geq 100$ for various $\sigma _{ST}$ and \cref {fig:CurvatureWsizeGnaST} the mean curvature $\delimiter "426830A \ensuremath {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $, also for various $\sigma _{ST}$. Figures \ref {fig:InitialEnergyGnaST} and \ref {fig:InitialCurvatureGnaST} show the initial energy $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ and the initial curvature $\delimiter "426830A \ensuremath {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $ for $k=0$. In \cref {fig:EnergyWsizeGnaST} we can see that for smaller $\sigma _{ST}$ the energy $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ converges to lower values. Conversely for larger window sizes $\sigma _{ST}$ the mean energy profiles $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ per $\sigma _{ST}$ converge. In \cref {fig:CurvatureWsizeGnaST} we observe a similar behavior for the curvature $\delimiter "426830A \ensuremath {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $: For small $\sigma _{ST}$ the curvature $\delimiter "426830A \ensuremath {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $ is comparatively large. As $\sigma _{ST}$ rises the profile of $\delimiter "426830A \ensuremath {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $ per $\sigma _{ST}$ converge, albeit at lower values. Figures \ref {fig:InitialEnergyGnaST} and \ref {fig:InitialCurvatureGnaST} show that the initial energy and the initial curvature for $\sigma _{ST}=3$ have half the values then for the larger window sizes $\sigma _{ST}=13\cdots 63$\relax }}{131}{figure.caption.54}}
1825
\newlabel{fig:WsizeGnaST}{{5.10}{131}{Study of the dependency the mean energy $\langle E^k\rangle _{ELAA}$ and the mean curvature $\langle \norm {K}\rangle $ on the window size $\sigma _{ST}$ of the structure tensor prior $E^{prior}_{ST}$. \Figref {fig:EnergyWsizeGnaST} shows the mean energy $\langle E^k\rangle _{ELAA}$ per iteration $k\geq 100$ for various $\sigma _{ST}$ and \figref {fig:CurvatureWsizeGnaST} the mean curvature $\langle \norm {K}\rangle $, also for various $\sigma _{ST}$. Figures \ref {fig:InitialEnergyGnaST} and \ref {fig:InitialCurvatureGnaST} show the initial energy $\langle E^k\rangle _{ELAA}$ and the initial curvature $\langle \norm {K}\rangle $ for $k=0$. In \figref {fig:EnergyWsizeGnaST} we can see that for smaller $\sigma _{ST}$ the energy $\langle E^k\rangle _{ELAA}$ converges to lower values. Conversely for larger window sizes $\sigma _{ST}$ the mean energy profiles $\langle E^k\rangle _{ELAA}$ per $\sigma _{ST}$ converge. In \figref {fig:CurvatureWsizeGnaST} we observe a similar behavior for the curvature $\langle \norm {K}\rangle $: For small $\sigma _{ST}$ the curvature $\langle \norm {K}\rangle $ is comparatively large. As $\sigma _{ST}$ rises the profile of $\langle \norm {K}\rangle $ per $\sigma _{ST}$ converge, albeit at lower values. Figures \ref {fig:InitialEnergyGnaST} and \ref {fig:InitialCurvatureGnaST} show that the initial energy and the initial curvature for $\sigma _{ST}=3$ have half the values then for the larger window sizes $\sigma _{ST}=13\cdots 63$\relax }{figure.caption.54}{}}
1826
\newlabel{fig:WsizeGnaST@cref}{{[figure][10][5]5.10}{131}}
1827
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {}}}{131}{subfigure.10.1}}
1828
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {}}}{131}{subfigure.10.2}}
1829
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {}}}{131}{subfigure.10.3}}
1830
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {}}}{131}{subfigure.10.4}}
1831
\newlabel{eq:RelativeEnergyST}{{5.76}{132}{Structure Tensor Prior}{equation.5.1.76}{}}
1832
\newlabel{eq:RelativeEnergyST@cref}{{[equation][76][5]5.76}{132}}
1833
\@writefile{toc}{\contentsline {section}{\numberline {5.2}summary}{132}{section.5.2}}
1834
\newlabel{eq:totEnergyGenNewton2}{{5.77}{133}{summary}{equation.5.2.77}{}}
1835
\newlabel{eq:totEnergyGenNewton2@cref}{{[equation][77][5]5.77}{133}}
1836
\newlabel{eq:eulerLagrangeGRF3}{{5.78}{133}{summary}{equation.5.2.78}{}}
1837
\newlabel{eq:eulerLagrangeGRF3@cref}{{[equation][78][5]5.78}{133}}
1838
\newlabel{eq:diffusionProcess2}{{5.79}{133}{summary}{equation.5.2.79}{}}
1839
\newlabel{eq:diffusionProcess2@cref}{{[equation][79][5]5.79}{133}}
1834
1840
\citation{NoetherTheoremDeu,NoetherTheroemEng}
1835
\@writefile{toc}{\contentsline {chapter}{\numberline {6}Conclusions}{134}{chapter.6}}
1841
\@writefile{toc}{\contentsline {chapter}{\numberline {6}Conclusions}{135}{chapter.6}}
1836
1842
\@writefile{lof}{\addvspace {10\p@ }}
1837
1843
\@writefile{lot}{\addvspace {10\p@ }}
1838
1844
\@writefile{lol}{\addvspace {10\p@ }}
1839
1845
\@writefile{loa}{\addvspace {10\p@ }}
1840
\@writefile{brf}{\backcite{NoetherTheoremDeu}{{134}{6}{chapter.6}}}
1841
\@writefile{brf}{\backcite{NoetherTheroemEng}{{134}{6}{chapter.6}}}
1842
\newlabel{eq:noetherTheoremConclusion}{{6.1}{134}{Conclusions}{equation.6.0.1}{}}
1843
\newlabel{eq:noetherTheoremConclusion@cref}{{[equation][1][6]6.1}{134}}
1846
\@writefile{brf}{\backcite{NoetherTheoremDeu}{{135}{6}{chapter.6}}}
1847
\@writefile{brf}{\backcite{NoetherTheroemEng}{{135}{6}{chapter.6}}}
1848
\newlabel{eq:noetherTheoremConclusion}{{6.1}{135}{Conclusions}{equation.6.0.1}{}}
1849
\newlabel{eq:noetherTheoremConclusion@cref}{{[equation][1][6]6.1}{135}}
1844
1850
\citation{Bigun1987,BigunBook}
1845
\newlabel{eq:noetherTheoremConclusionLeftInv}{{6.2}{135}{Conclusions}{equation.6.0.2}{}}
1846
\newlabel{eq:noetherTheoremConclusionLeftInv@cref}{{[equation][2][6]6.2}{135}}
1847
\@writefile{brf}{\backcite{Bigun1987}{{135}{6}{equation.6.0.2}}}
1848
\@writefile{brf}{\backcite{BigunBook}{{135}{6}{equation.6.0.2}}}
1851
\newlabel{eq:noetherTheoremConclusionLeftInv}{{6.2}{136}{Conclusions}{equation.6.0.2}{}}
1852
\newlabel{eq:noetherTheoremConclusionLeftInv@cref}{{[equation][2][6]6.2}{136}}
1853
\@writefile{brf}{\backcite{Bigun1987}{{136}{6}{equation.6.0.2}}}
1854
\@writefile{brf}{\backcite{BigunBook}{{136}{6}{equation.6.0.2}}}
1849
1855
\citation{FieguthStatImProc}
1850
\newlabel{eq:flowEulerConclusion}{{6.5}{136}{Conclusions}{equation.6.0.5}{}}
1851
\newlabel{eq:flowEulerConclusion@cref}{{[equation][5][6]6.5}{136}}
1852
\newlabel{eq:flowBabetteConclusion}{{6.6}{136}{Conclusions}{equation.6.0.6}{}}
1853
\newlabel{eq:flowBabetteConclusion@cref}{{[equation][6][6]6.6}{136}}
1854
\@writefile{brf}{\backcite{FieguthStatImProc}{{136}{6}{equation.6.0.6}}}
1855
\newlabel{eq:curvaturePriorConclusion}{{6.7}{136}{Conclusions}{equation.6.0.7}{}}
1856
\newlabel{eq:curvaturePriorConclusion@cref}{{[equation][7][6]6.7}{136}}
1857
\newlabel{eq:curvatureConclusion}{{6.8}{137}{Conclusions}{equation.6.0.8}{}}
1858
\newlabel{eq:curvatureConclusion@cref}{{[equation][8][6]6.8}{137}}
1859
\newlabel{eq:curvatureTVConclusion}{{6.9}{137}{Conclusions}{equation.6.0.9}{}}
1860
\newlabel{eq:curvatureTVConclusion@cref}{{[equation][9][6]6.9}{137}}
1856
\newlabel{eq:flowEulerConclusion}{{6.5}{137}{Conclusions}{equation.6.0.5}{}}
1857
\newlabel{eq:flowEulerConclusion@cref}{{[equation][5][6]6.5}{137}}
1858
\newlabel{eq:flowBabetteConclusion}{{6.6}{137}{Conclusions}{equation.6.0.6}{}}
1859
\newlabel{eq:flowBabetteConclusion@cref}{{[equation][6][6]6.6}{137}}
1860
\@writefile{brf}{\backcite{FieguthStatImProc}{{137}{6}{equation.6.0.6}}}
1861
\newlabel{eq:curvaturePriorConclusion}{{6.7}{137}{Conclusions}{equation.6.0.7}{}}
1862
\newlabel{eq:curvaturePriorConclusion@cref}{{[equation][7][6]6.7}{137}}
1863
\newlabel{eq:curvatureConclusion}{{6.8}{138}{Conclusions}{equation.6.0.8}{}}
1864
\newlabel{eq:curvatureConclusion@cref}{{[equation][8][6]6.8}{138}}
1865
\newlabel{eq:curvatureTVConclusion}{{6.9}{138}{Conclusions}{equation.6.0.9}{}}
1866
\newlabel{eq:curvatureTVConclusion@cref}{{[equation][9][6]6.9}{138}}
1861
1867
\citation{PeskinQFT}
1862
1868
\citation{misner1973gravitation}
1863
1869
\citation{rovelli2007quantum}
1864
1870
\citation{becker2006string}
1865
\@writefile{toc}{\contentsline {section}{\numberline {6.1}Outlook}{138}{section.6.1}}
1866
\@writefile{brf}{\backcite{PeskinQFT}{{138}{6.1}{section.6.1}}}
1867
\@writefile{brf}{\backcite{misner1973gravitation}{{138}{6.1}{section.6.1}}}
1868
\@writefile{brf}{\backcite{rovelli2007quantum}{{138}{6.1}{section.6.1}}}
1869
\@writefile{brf}{\backcite{becker2006string}{{138}{6.1}{section.6.1}}}
1871
\@writefile{toc}{\contentsline {section}{\numberline {6.1}Outlook}{139}{section.6.1}}
1872
\@writefile{brf}{\backcite{PeskinQFT}{{139}{6.1}{section.6.1}}}
1873
\@writefile{brf}{\backcite{misner1973gravitation}{{139}{6.1}{section.6.1}}}
1874
\@writefile{brf}{\backcite{rovelli2007quantum}{{139}{6.1}{section.6.1}}}
1875
\@writefile{brf}{\backcite{becker2006string}{{139}{6.1}{section.6.1}}}
1870
1876
\citation{LeeSmoothManifolds}
1871
\@writefile{toc}{\contentsline {chapter}{\numberline {A}Smooth Manifolds}{139}{appendix.A}}
1877
\@writefile{toc}{\contentsline {chapter}{\numberline {A}Smooth Manifolds}{140}{appendix.A}}
1872
1878
\@writefile{lof}{\addvspace {10\p@ }}
1873
1879
\@writefile{lot}{\addvspace {10\p@ }}
1874
1880
\@writefile{lol}{\addvspace {10\p@ }}
1875
1881
\@writefile{loa}{\addvspace {10\p@ }}
1876
\newlabel{App:SmoothManifolds}{{A}{139}{Smooth Manifolds}{appendix.A}{}}
1877
\newlabel{App:SmoothManifolds@cref}{{[appendix][1][2147483647]A}{139}}
1878
\@writefile{brf}{\backcite{LeeSmoothManifolds}{{139}{A}{appendix.A}}}
1879
\@writefile{toc}{\contentsline {subsection}{\numberline {A.0.1}Topological Spaces}{139}{subsection.A.0.1}}
1880
\newlabel{def:TopManifold}{{21}{140}{Topological Manifold}{definition.21}{}}
1881
\newlabel{def:TopManifold@cref}{{[definition][21][2147483647]21}{140}}
1882
\citation{LeeSmoothManifolds}
1883
\newlabel{lem:countBasis}{{10}{141}{Countable Basis of a topological Manifold}{lemma.10}{}}
1884
\newlabel{lem:countBasis@cref}{{[lemma][10][2147483647]10}{141}}
1885
\@writefile{brf}{\backcite{LeeSmoothManifolds}{{141}{A.0.1}{equation.A.0.1}}}
1886
\newlabel{enum:ConnectLocPath}{{1}{141}{Connectivity of a Manifold}{Item.3}{}}
1887
\newlabel{enum:ConnectLocPath@cref}{{[enumi][1][2147483647]1}{141}}
1888
\newlabel{enum:ConnectGlobPath}{{2}{141}{Connectivity of a Manifold}{Item.4}{}}
1889
\newlabel{enum:ConnectGlobPath@cref}{{[enumi][2][2147483647]2}{141}}
1890
\@writefile{toc}{\contentsline {subsection}{\numberline {A.0.2}Smooth Manifolds}{142}{subsection.A.0.2}}
1891
\newlabel{def:smoothAtlas}{{24}{142}{Atlas and Smooth Manifold}{definition.24}{}}
1892
\newlabel{def:smoothAtlas@cref}{{[definition][24][2147483647]24}{142}}
1893
\newlabel{eq:localCoordinates}{{A.4}{143}{Smooth Manifolds}{equation.A.0.4}{}}
1894
\newlabel{eq:localCoordinates@cref}{{[equation][4][2147483647,1]A.4}{143}}
1895
\newlabel{eq:coordinateTransform}{{A.6}{143}{Coordinate Transformation}{equation.A.0.6}{}}
1896
\newlabel{eq:coordinateTransform@cref}{{[equation][6][2147483647,1]A.6}{143}}
1897
\@writefile{toc}{\contentsline {section}{\numberline {A.1}The Tangent Space $T_p M$}{144}{section.A.1}}
1898
\newlabel{eq:derivationDef}{{A.7}{144}{Derivation}{equation.A.1.7}{}}
1899
\newlabel{eq:derivationDef@cref}{{[equation][7][2147483647,1]A.7}{144}}
1900
\newlabel{eq:derivationDef2}{{A.8}{144}{Derivation}{equation.A.1.8}{}}
1901
\newlabel{eq:derivationDef2@cref}{{[equation][8][2147483647,1]A.8}{144}}
1902
\newlabel{def:tangSpace}{{27}{144}{Tangential Space}{definition.27}{}}
1903
\newlabel{def:tangSpace@cref}{{[definition][27][2147483647]27}{144}}
1904
\newlabel{lem:tangentSpaceLinear}{{12}{145}{Linearity}{lemma.12}{}}
1905
\newlabel{lem:tangentSpaceLinear@cref}{{[lemma][12][2147483647]12}{145}}
1906
\newlabel{lem:derivationProp}{{13}{145}{Properties of Derivations}{lemma.13}{}}
1907
\newlabel{lem:derivationProp@cref}{{[lemma][13][2147483647]13}{145}}
1908
\newlabel{item:PropDeriv1}{{1}{145}{Properties of Derivations}{Item.5}{}}
1909
\newlabel{item:PropDeriv1@cref}{{[enumi][1][2147483647]1}{145}}
1910
\newlabel{item:PropDeriv2}{{2}{145}{Properties of Derivations}{Item.6}{}}
1911
\newlabel{item:PropDeriv2@cref}{{[enumi][2][2147483647]2}{145}}
1912
\newlabel{proof:PropDeriv1}{{A.13}{145}{The Tangent Space $T_p M$}{equation.A.1.13}{}}
1913
\newlabel{proof:PropDeriv1@cref}{{[equation][13][2147483647,1]A.13}{145}}
1914
\@writefile{toc}{\contentsline {subsection}{\numberline {A.1.1}The Push-Forward}{146}{subsection.A.1.1}}
1915
\newlabel{def:pushForward}{{28}{146}{Push-Forward}{definition.28}{}}
1916
\newlabel{def:pushForward@cref}{{[definition][28][2147483647]28}{146}}
1917
\newlabel{eq:pushForward}{{A.14}{146}{Push-Forward}{equation.A.1.14}{}}
1918
\newlabel{eq:pushForward@cref}{{[equation][14][2147483647,1]A.14}{146}}
1919
\newlabel{lem:pushForwardproperties}{{14}{146}{Properties of Push-Forwards}{lemma.14}{}}
1920
\newlabel{lem:pushForwardproperties@cref}{{[lemma][14][2147483647]14}{146}}
1921
\newlabel{item:linPushForward}{{1}{146}{Properties of Push-Forwards}{Item.7}{}}
1922
\newlabel{item:linPushForward@cref}{{[enumi][1][2147483647]1}{146}}
1923
\newlabel{item:chainPushForward}{{2}{146}{Properties of Push-Forwards}{Item.8}{}}
1924
\newlabel{item:chainPushForward@cref}{{[enumi][2][2147483647]2}{146}}
1925
\newlabel{item:identPushForward}{{3}{146}{Properties of Push-Forwards}{Item.9}{}}
1926
\newlabel{item:identPushForward@cref}{{[enumi][3][2147483647]3}{146}}
1927
\newlabel{item:diffeoPushForward}{{4}{146}{Properties of Push-Forwards}{Item.10}{}}
1928
\newlabel{item:diffeoPushForward@cref}{{[enumi][4][2147483647]4}{146}}
1929
\newlabel{proof:linPushForward}{{A.15}{146}{The Push-Forward}{equation.A.1.15}{}}
1930
\newlabel{proof:linPushForward@cref}{{[equation][15][2147483647,1]A.15}{146}}
1931
\newlabel{prop:equivRelation}{{5}{147}{Equivalence Relation}{proposition.5}{}}
1932
\newlabel{prop:equivRelation@cref}{{[proposition][5][2147483647]5}{147}}
1933
\newlabel{eq:equivalenceRelationDerivation}{{A.18}{147}{Equivalence Relation}{equation.A.1.18}{}}
1934
\newlabel{eq:equivalenceRelationDerivation@cref}{{[equation][18][2147483647,1]A.18}{147}}
1935
\citation{LeeSmoothManifolds}
1936
\newlabel{def:inclusionMap}{{29}{148}{Inclusion Map}{definition.29}{}}
1937
\newlabel{def:inclusionMap@cref}{{[definition][29][2147483647]29}{148}}
1938
\newlabel{prop:tangentialInclusion}{{6}{148}{Tangential Inclusion Map $\iota _\star $}{proposition.6}{}}
1939
\newlabel{prop:tangentialInclusion@cref}{{[proposition][6][2147483647]6}{148}}
1940
\@writefile{brf}{\backcite{LeeSmoothManifolds}{{148}{A.1.1}{equation.A.1.21}}}
1941
\newlabel{eq:inclMapSurjective}{{A.24}{148}{The Push-Forward}{equation.A.1.24}{}}
1942
\newlabel{eq:inclMapSurjective@cref}{{[equation][24][2147483647,1]A.24}{148}}
1943
\@writefile{toc}{\contentsline {section}{\numberline {A.2}The Basis of $T_p M$}{149}{section.A.2}}
1944
\newlabel{def:euclDirectionalDeriv}{{30}{149}{Euclidean Directional Derivative}{definition.30}{}}
1945
\newlabel{def:euclDirectionalDeriv@cref}{{[definition][30][2147483647]30}{149}}
1946
\newlabel{eq:euclDirectionalDeriv}{{A.27}{149}{Euclidean Directional Derivative}{equation.A.2.27}{}}
1947
\newlabel{eq:euclDirectionalDeriv@cref}{{[equation][27][2147483647,1]A.27}{149}}
1948
\newlabel{eq:euclDirectionMap}{{A.28}{149}{The Basis of $T_p M$}{equation.A.2.28}{}}
1949
\newlabel{eq:euclDirectionMap@cref}{{[equation][28][2147483647,1]A.28}{149}}
1950
\newlabel{lem:basisOfTRn}{{15}{150}{Basis of $T_a\mathbb {R}^n$}{lemma.15}{}}
1951
\newlabel{lem:basisOfTRn@cref}{{[lemma][15][2147483647]15}{150}}
1952
\newlabel{eq:directionalDerivative}{{A.32}{150}{}{equation.A.2.32}{}}
1953
\newlabel{eq:directionalDerivative@cref}{{[equation][32][2147483647,1]A.32}{150}}
1954
\newlabel{eq:vectorOperatorRep}{{A.33}{150}{The Basis of $T_p M$}{equation.A.2.33}{}}
1955
\newlabel{eq:vectorOperatorRep@cref}{{[equation][33][2147483647,1]A.33}{150}}
1956
\newlabel{eq:pushForwardCoordinates}{{A.35}{151}{The Basis of $T_p M$}{equation.A.2.35}{}}
1957
\newlabel{eq:pushForwardCoordinates@cref}{{[equation][35][2147483647,1]A.35}{151}}
1958
\newlabel{eq:coordinateTransformTp}{{A.36}{151}{The Basis of $T_p M$}{equation.A.2.36}{}}
1959
\newlabel{eq:coordinateTransformTp@cref}{{[equation][36][2147483647,1]A.36}{151}}
1960
\@writefile{toc}{\contentsline {section}{\numberline {A.3}Vector Fields}{151}{section.A.3}}
1961
\citation{LeeSmoothManifolds}
1962
\newlabel{eq:tangentialBundle}{{A.37}{152}{Tangential Bundle}{equation.A.3.37}{}}
1963
\newlabel{eq:tangentialBundle@cref}{{[equation][37][2147483647,1]A.37}{152}}
1964
\@writefile{brf}{\backcite{LeeSmoothManifolds}{{152}{A.3}{lemma.16}}}
1965
\newlabel{lem:smoothVectorFields}{{17}{153}{Smooth Vector Fields}{lemma.17}{}}
1966
\newlabel{lem:smoothVectorFields@cref}{{[lemma][17][2147483647]17}{153}}
1967
\@writefile{toc}{\contentsline {section}{\numberline {A.4}Push-Forwards on $\mathcal {T}(M)$}{153}{section.A.4}}
1968
\newlabel{eq:FRelated1}{{A.43}{153}{Push-Forwards on $\mathcal {T}(M)$}{equation.A.4.43}{}}
1969
\newlabel{eq:FRelated1@cref}{{[equation][43][2147483647,1]A.43}{153}}
1970
\newlabel{eq:FRelated2}{{A.44}{154}{$F$-Related}{equation.A.4.44}{}}
1971
\newlabel{eq:FRelated2@cref}{{[equation][44][2147483647,1]A.44}{154}}
1972
\newlabel{prop:vectorFieldPushForward}{{7}{154}{Push-Forward on $\mathcal {T}(M)$}{proposition.7}{}}
1973
\newlabel{prop:vectorFieldPushForward@cref}{{[proposition][7][2147483647]7}{154}}
1974
\newlabel{eq:FRelated3}{{A.45}{154}{Push-Forwards on $\mathcal {T}(M)$}{equation.A.4.45}{}}
1975
\newlabel{eq:FRelated3@cref}{{[equation][45][2147483647,1]A.45}{154}}
1976
\newlabel{eq:pushForwardCoordinatesVectorField}{{A.47}{154}{Push-Forwards on $\mathcal {T}(M)$}{equation.A.4.47}{}}
1977
\newlabel{eq:pushForwardCoordinatesVectorField@cref}{{[equation][47][2147483647,1]A.47}{154}}
1978
\@writefile{toc}{\contentsline {section}{\numberline {A.5}Integral Curves and Flows}{155}{section.A.5}}
1979
\newlabel{eq:fundamentalTheoremMotiv}{{A.48}{155}{Integral Curves and Flows}{equation.A.5.48}{}}
1980
\newlabel{eq:fundamentalTheoremMotiv@cref}{{[equation][48][2147483647,1]A.48}{155}}
1981
\newlabel{def:integralCurve}{{35}{155}{Integral Curve}{definition.35}{}}
1982
\newlabel{def:integralCurve@cref}{{[definition][35][2147483647]35}{155}}
1983
\newlabel{eq:integralCurveDerivative}{{A.50}{155}{Integral Curve}{equation.A.5.50}{}}
1984
\newlabel{eq:integralCurveDerivative@cref}{{[equation][50][2147483647,1]A.50}{155}}
1985
\newlabel{eq:integralCurveDerivative2}{{A.53}{156}{Integral Curves and Flows}{equation.A.5.53}{}}
1986
\newlabel{eq:integralCurveDerivative2@cref}{{[equation][53][2147483647,1]A.53}{156}}
1987
\newlabel{eq:integralCurveDerivativeDiffEq}{{A.54}{156}{Integral Curves and Flows}{equation.A.5.54}{}}
1988
\newlabel{eq:integralCurveDerivativeDiffEq@cref}{{[equation][54][2147483647,1]A.54}{156}}
1989
\citation{LeeSmoothManifolds}
1990
\newlabel{eq:curveFlow}{{A.57}{157}{Integral Curves and Flows}{equation.A.5.57}{}}
1991
\newlabel{eq:curveFlow@cref}{{[equation][57][2147483647,1]A.57}{157}}
1992
\newlabel{eq:diffeoFlow}{{A.58}{157}{Integral Curves and Flows}{equation.A.5.58}{}}
1993
\newlabel{eq:diffeoFlow@cref}{{[equation][58][2147483647,1]A.58}{157}}
1994
\newlabel{eq:flowDiffEq}{{A.59}{157}{Integral Curves and Flows}{equation.A.5.59}{}}
1995
\newlabel{eq:flowDiffEq@cref}{{[equation][59][2147483647,1]A.59}{157}}
1996
\newlabel{theorem:FundamentalTheoremOnFlows}{{3}{157}{Fundamental Theorem on Flows}{theorem.3}{}}
1997
\newlabel{theorem:FundamentalTheoremOnFlows@cref}{{[theorem][3][2147483647]3}{157}}
1882
\newlabel{App:SmoothManifolds}{{A}{140}{Smooth Manifolds}{appendix.A}{}}
1883
\newlabel{App:SmoothManifolds@cref}{{[appendix][1][2147483647]A}{140}}
1884
\@writefile{brf}{\backcite{LeeSmoothManifolds}{{140}{A}{appendix.A}}}
1885
\@writefile{toc}{\contentsline {subsection}{\numberline {A.0.1}Topological Spaces}{140}{subsection.A.0.1}}
1886
\newlabel{def:TopManifold}{{21}{141}{Topological Manifold}{definition.21}{}}
1887
\newlabel{def:TopManifold@cref}{{[definition][21][2147483647]21}{141}}
1888
\citation{LeeSmoothManifolds}
1889
\newlabel{lem:countBasis}{{10}{142}{Countable Basis of a topological Manifold}{lemma.10}{}}
1890
\newlabel{lem:countBasis@cref}{{[lemma][10][2147483647]10}{142}}
1891
\@writefile{brf}{\backcite{LeeSmoothManifolds}{{142}{A.0.1}{equation.A.0.1}}}
1892
\newlabel{enum:ConnectLocPath}{{1}{142}{Connectivity of a Manifold}{Item.3}{}}
1893
\newlabel{enum:ConnectLocPath@cref}{{[enumi][1][2147483647]1}{142}}
1894
\newlabel{enum:ConnectGlobPath}{{2}{142}{Connectivity of a Manifold}{Item.4}{}}
1895
\newlabel{enum:ConnectGlobPath@cref}{{[enumi][2][2147483647]2}{142}}
1896
\@writefile{toc}{\contentsline {subsection}{\numberline {A.0.2}Smooth Manifolds}{143}{subsection.A.0.2}}
1897
\newlabel{def:smoothAtlas}{{24}{143}{Atlas and Smooth Manifold}{definition.24}{}}
1898
\newlabel{def:smoothAtlas@cref}{{[definition][24][2147483647]24}{143}}
1899
\newlabel{eq:localCoordinates}{{A.4}{144}{Smooth Manifolds}{equation.A.0.4}{}}
1900
\newlabel{eq:localCoordinates@cref}{{[equation][4][2147483647,1]A.4}{144}}
1901
\newlabel{eq:coordinateTransform}{{A.6}{144}{Coordinate Transformation}{equation.A.0.6}{}}
1902
\newlabel{eq:coordinateTransform@cref}{{[equation][6][2147483647,1]A.6}{144}}
1903
\@writefile{toc}{\contentsline {section}{\numberline {A.1}The Tangent Space $T_p M$}{145}{section.A.1}}
1904
\newlabel{eq:derivationDef}{{A.7}{145}{Derivation}{equation.A.1.7}{}}
1905
\newlabel{eq:derivationDef@cref}{{[equation][7][2147483647,1]A.7}{145}}
1906
\newlabel{eq:derivationDef2}{{A.8}{145}{Derivation}{equation.A.1.8}{}}
1907
\newlabel{eq:derivationDef2@cref}{{[equation][8][2147483647,1]A.8}{145}}
1908
\newlabel{def:tangSpace}{{27}{145}{Tangential Space}{definition.27}{}}
1909
\newlabel{def:tangSpace@cref}{{[definition][27][2147483647]27}{145}}
1910
\newlabel{lem:tangentSpaceLinear}{{12}{146}{Linearity}{lemma.12}{}}
1911
\newlabel{lem:tangentSpaceLinear@cref}{{[lemma][12][2147483647]12}{146}}
1912
\newlabel{lem:derivationProp}{{13}{146}{Properties of Derivations}{lemma.13}{}}
1913
\newlabel{lem:derivationProp@cref}{{[lemma][13][2147483647]13}{146}}
1914
\newlabel{item:PropDeriv1}{{1}{146}{Properties of Derivations}{Item.5}{}}
1915
\newlabel{item:PropDeriv1@cref}{{[enumi][1][2147483647]1}{146}}
1916
\newlabel{item:PropDeriv2}{{2}{146}{Properties of Derivations}{Item.6}{}}
1917
\newlabel{item:PropDeriv2@cref}{{[enumi][2][2147483647]2}{146}}
1918
\newlabel{proof:PropDeriv1}{{A.13}{146}{The Tangent Space $T_p M$}{equation.A.1.13}{}}
1919
\newlabel{proof:PropDeriv1@cref}{{[equation][13][2147483647,1]A.13}{146}}
1920
\@writefile{toc}{\contentsline {subsection}{\numberline {A.1.1}The Push-Forward}{147}{subsection.A.1.1}}
1921
\newlabel{def:pushForward}{{28}{147}{Push-Forward}{definition.28}{}}
1922
\newlabel{def:pushForward@cref}{{[definition][28][2147483647]28}{147}}
1923
\newlabel{eq:pushForward}{{A.14}{147}{Push-Forward}{equation.A.1.14}{}}
1924
\newlabel{eq:pushForward@cref}{{[equation][14][2147483647,1]A.14}{147}}
1925
\newlabel{lem:pushForwardproperties}{{14}{147}{Properties of Push-Forwards}{lemma.14}{}}
1926
\newlabel{lem:pushForwardproperties@cref}{{[lemma][14][2147483647]14}{147}}
1927
\newlabel{item:linPushForward}{{1}{147}{Properties of Push-Forwards}{Item.7}{}}
1928
\newlabel{item:linPushForward@cref}{{[enumi][1][2147483647]1}{147}}
1929
\newlabel{item:chainPushForward}{{2}{147}{Properties of Push-Forwards}{Item.8}{}}
1930
\newlabel{item:chainPushForward@cref}{{[enumi][2][2147483647]2}{147}}
1931
\newlabel{item:identPushForward}{{3}{147}{Properties of Push-Forwards}{Item.9}{}}
1932
\newlabel{item:identPushForward@cref}{{[enumi][3][2147483647]3}{147}}
1933
\newlabel{item:diffeoPushForward}{{4}{147}{Properties of Push-Forwards}{Item.10}{}}
1934
\newlabel{item:diffeoPushForward@cref}{{[enumi][4][2147483647]4}{147}}
1935
\newlabel{proof:linPushForward}{{A.15}{147}{The Push-Forward}{equation.A.1.15}{}}
1936
\newlabel{proof:linPushForward@cref}{{[equation][15][2147483647,1]A.15}{147}}
1937
\newlabel{prop:equivRelation}{{5}{148}{Equivalence Relation}{proposition.5}{}}
1938
\newlabel{prop:equivRelation@cref}{{[proposition][5][2147483647]5}{148}}
1939
\newlabel{eq:equivalenceRelationDerivation}{{A.18}{148}{Equivalence Relation}{equation.A.1.18}{}}
1940
\newlabel{eq:equivalenceRelationDerivation@cref}{{[equation][18][2147483647,1]A.18}{148}}
1941
\citation{LeeSmoothManifolds}
1942
\newlabel{def:inclusionMap}{{29}{149}{Inclusion Map}{definition.29}{}}
1943
\newlabel{def:inclusionMap@cref}{{[definition][29][2147483647]29}{149}}
1944
\newlabel{prop:tangentialInclusion}{{6}{149}{Tangential Inclusion Map $\iota _\star $}{proposition.6}{}}
1945
\newlabel{prop:tangentialInclusion@cref}{{[proposition][6][2147483647]6}{149}}
1946
\@writefile{brf}{\backcite{LeeSmoothManifolds}{{149}{A.1.1}{equation.A.1.21}}}
1947
\newlabel{eq:inclMapSurjective}{{A.24}{149}{The Push-Forward}{equation.A.1.24}{}}
1948
\newlabel{eq:inclMapSurjective@cref}{{[equation][24][2147483647,1]A.24}{149}}
1949
\@writefile{toc}{\contentsline {section}{\numberline {A.2}The Basis of $T_p M$}{150}{section.A.2}}
1950
\newlabel{def:euclDirectionalDeriv}{{30}{150}{Euclidean Directional Derivative}{definition.30}{}}
1951
\newlabel{def:euclDirectionalDeriv@cref}{{[definition][30][2147483647]30}{150}}
1952
\newlabel{eq:euclDirectionalDeriv}{{A.27}{150}{Euclidean Directional Derivative}{equation.A.2.27}{}}
1953
\newlabel{eq:euclDirectionalDeriv@cref}{{[equation][27][2147483647,1]A.27}{150}}
1954
\newlabel{eq:euclDirectionMap}{{A.28}{150}{The Basis of $T_p M$}{equation.A.2.28}{}}
1955
\newlabel{eq:euclDirectionMap@cref}{{[equation][28][2147483647,1]A.28}{150}}
1956
\newlabel{lem:basisOfTRn}{{15}{151}{Basis of $T_a\mathbb {R}^n$}{lemma.15}{}}
1957
\newlabel{lem:basisOfTRn@cref}{{[lemma][15][2147483647]15}{151}}
1958
\newlabel{eq:directionalDerivative}{{A.32}{151}{}{equation.A.2.32}{}}
1959
\newlabel{eq:directionalDerivative@cref}{{[equation][32][2147483647,1]A.32}{151}}
1960
\newlabel{eq:vectorOperatorRep}{{A.33}{151}{The Basis of $T_p M$}{equation.A.2.33}{}}
1961
\newlabel{eq:vectorOperatorRep@cref}{{[equation][33][2147483647,1]A.33}{151}}
1962
\newlabel{eq:pushForwardCoordinates}{{A.35}{152}{The Basis of $T_p M$}{equation.A.2.35}{}}
1963
\newlabel{eq:pushForwardCoordinates@cref}{{[equation][35][2147483647,1]A.35}{152}}
1964
\newlabel{eq:coordinateTransformTp}{{A.36}{152}{The Basis of $T_p M$}{equation.A.2.36}{}}
1965
\newlabel{eq:coordinateTransformTp@cref}{{[equation][36][2147483647,1]A.36}{152}}
1966
\@writefile{toc}{\contentsline {section}{\numberline {A.3}Vector Fields}{152}{section.A.3}}
1967
\citation{LeeSmoothManifolds}
1968
\newlabel{eq:tangentialBundle}{{A.37}{153}{Tangential Bundle}{equation.A.3.37}{}}
1969
\newlabel{eq:tangentialBundle@cref}{{[equation][37][2147483647,1]A.37}{153}}
1970
\@writefile{brf}{\backcite{LeeSmoothManifolds}{{153}{A.3}{lemma.16}}}
1971
\newlabel{lem:smoothVectorFields}{{17}{154}{Smooth Vector Fields}{lemma.17}{}}
1972
\newlabel{lem:smoothVectorFields@cref}{{[lemma][17][2147483647]17}{154}}
1973
\@writefile{toc}{\contentsline {section}{\numberline {A.4}Push-Forwards on $\mathcal {T}(M)$}{154}{section.A.4}}
1974
\newlabel{eq:FRelated1}{{A.43}{154}{Push-Forwards on $\mathcal {T}(M)$}{equation.A.4.43}{}}
1975
\newlabel{eq:FRelated1@cref}{{[equation][43][2147483647,1]A.43}{154}}
1976
\newlabel{eq:FRelated2}{{A.44}{155}{$F$-Related}{equation.A.4.44}{}}
1977
\newlabel{eq:FRelated2@cref}{{[equation][44][2147483647,1]A.44}{155}}
1978
\newlabel{prop:vectorFieldPushForward}{{7}{155}{Push-Forward on $\mathcal {T}(M)$}{proposition.7}{}}
1979
\newlabel{prop:vectorFieldPushForward@cref}{{[proposition][7][2147483647]7}{155}}
1980
\newlabel{eq:FRelated3}{{A.45}{155}{Push-Forwards on $\mathcal {T}(M)$}{equation.A.4.45}{}}
1981
\newlabel{eq:FRelated3@cref}{{[equation][45][2147483647,1]A.45}{155}}
1982
\newlabel{eq:pushForwardCoordinatesVectorField}{{A.47}{155}{Push-Forwards on $\mathcal {T}(M)$}{equation.A.4.47}{}}
1983
\newlabel{eq:pushForwardCoordinatesVectorField@cref}{{[equation][47][2147483647,1]A.47}{155}}
1984
\@writefile{toc}{\contentsline {section}{\numberline {A.5}Integral Curves and Flows}{156}{section.A.5}}
1985
\newlabel{eq:fundamentalTheoremMotiv}{{A.48}{156}{Integral Curves and Flows}{equation.A.5.48}{}}
1986
\newlabel{eq:fundamentalTheoremMotiv@cref}{{[equation][48][2147483647,1]A.48}{156}}
1987
\newlabel{def:integralCurve}{{35}{156}{Integral Curve}{definition.35}{}}
1988
\newlabel{def:integralCurve@cref}{{[definition][35][2147483647]35}{156}}
1989
\newlabel{eq:integralCurveDerivative}{{A.50}{156}{Integral Curve}{equation.A.5.50}{}}
1990
\newlabel{eq:integralCurveDerivative@cref}{{[equation][50][2147483647,1]A.50}{156}}
1991
\citation{LeeSmoothManifolds}
1992
\newlabel{eq:integralCurveDerivative2}{{A.53}{157}{Integral Curves and Flows}{equation.A.5.53}{}}
1993
\newlabel{eq:integralCurveDerivative2@cref}{{[equation][53][2147483647,1]A.53}{157}}
1994
\newlabel{eq:integralCurveDerivativeDiffEq}{{A.54}{157}{Integral Curves and Flows}{equation.A.5.54}{}}
1995
\newlabel{eq:integralCurveDerivativeDiffEq@cref}{{[equation][54][2147483647,1]A.54}{157}}
1996
\newlabel{theorem:ODE}{{3}{157}{ODE Existence, Uniqueness and Smoothness}{theorem.3}{}}
1997
\newlabel{theorem:ODE@cref}{{[theorem][3][2147483647]3}{157}}
1998
1998
\@writefile{brf}{\backcite{LeeSmoothManifolds}{{157}{A.5}{theorem.3}}}
1999
\newlabel{eq:invariantVectorField}{{A.60}{158}{Integral Curves and Flows}{equation.A.5.60}{}}
2000
\newlabel{eq:invariantVectorField@cref}{{[equation][60][2147483647,1]A.60}{158}}
2001
\@writefile{toc}{\contentsline {subsection}{\numberline {A.5.1}The Lie Derivative}{158}{subsection.A.5.1}}
2002
\newlabel{eq:naiveLieDerivative}{{A.61}{158}{The Lie Derivative}{equation.A.5.61}{}}
2003
\newlabel{eq:naiveLieDerivative@cref}{{[equation][61][2147483647,1]A.61}{158}}
2004
\newlabel{eq:lieDerivative}{{A.62}{158}{Lie Derivative}{equation.A.5.62}{}}
2005
\newlabel{eq:lieDerivative@cref}{{[equation][62][2147483647,1]A.62}{158}}
2006
\newlabel{prop:Commutator}{{8}{159}{Commutator}{proposition.8}{}}
2007
\newlabel{prop:Commutator@cref}{{[proposition][8][2147483647]8}{159}}
2008
\newlabel{eq:commutatorApp}{{A.63}{159}{Commutator}{equation.A.5.63}{}}
2009
\newlabel{eq:commutatorApp@cref}{{[equation][63][2147483647,1]A.63}{159}}
2010
\newlabel{eq:lieDerivCommutator}{{A.64}{159}{Commutator}{equation.A.5.64}{}}
2011
\newlabel{eq:lieDerivCommutator@cref}{{[equation][64][2147483647,1]A.64}{159}}
2012
\newlabel{eq:fundamentalTheoremPartdLieDeriv}{{A.71}{160}{The Lie Derivative}{equation.A.5.71}{}}
2013
\newlabel{eq:fundamentalTheoremPartdLieDeriv@cref}{{[equation][71][2147483647,1]A.71}{160}}
1999
\newlabel{eq:curveFlow}{{A.57}{158}{Integral Curves and Flows}{equation.A.5.57}{}}
2000
\newlabel{eq:curveFlow@cref}{{[equation][57][2147483647,1]A.57}{158}}
2001
\newlabel{eq:diffeoFlow}{{A.58}{158}{Integral Curves and Flows}{equation.A.5.58}{}}
2002
\newlabel{eq:diffeoFlow@cref}{{[equation][58][2147483647,1]A.58}{158}}
2003
\newlabel{eq:flowDiffEq}{{A.59}{158}{Integral Curves and Flows}{equation.A.5.59}{}}
2004
\newlabel{eq:flowDiffEq@cref}{{[equation][59][2147483647,1]A.59}{158}}
2005
\newlabel{theorem:FundamentalTheoremOnFlows}{{4}{158}{Fundamental Theorem on Flows}{theorem.4}{}}
2006
\newlabel{theorem:FundamentalTheoremOnFlows@cref}{{[theorem][4][2147483647]4}{158}}
2007
\citation{LeeSmoothManifolds}
2008
\@writefile{brf}{\backcite{LeeSmoothManifolds}{{159}{A.5}{theorem.4}}}
2009
\newlabel{eq:invariantVectorField}{{A.60}{159}{Integral Curves and Flows}{equation.A.5.60}{}}
2010
\newlabel{eq:invariantVectorField@cref}{{[equation][60][2147483647,1]A.60}{159}}
2011
\@writefile{toc}{\contentsline {subsection}{\numberline {A.5.1}The Lie Derivative}{159}{subsection.A.5.1}}
2012
\newlabel{eq:naiveLieDerivative}{{A.61}{160}{The Lie Derivative}{equation.A.5.61}{}}
2013
\newlabel{eq:naiveLieDerivative@cref}{{[equation][61][2147483647,1]A.61}{160}}
2014
\newlabel{eq:lieDerivative}{{A.62}{160}{Lie Derivative}{equation.A.5.62}{}}
2015
\newlabel{eq:lieDerivative@cref}{{[equation][62][2147483647,1]A.62}{160}}
2016
\newlabel{prop:Commutator}{{8}{160}{Commutator}{proposition.8}{}}
2017
\newlabel{prop:Commutator@cref}{{[proposition][8][2147483647]8}{160}}
2018
\newlabel{eq:commutatorApp}{{A.63}{160}{Commutator}{equation.A.5.63}{}}
2019
\newlabel{eq:commutatorApp@cref}{{[equation][63][2147483647,1]A.63}{160}}
2020
\newlabel{eq:lieDerivCommutator}{{A.64}{160}{Commutator}{equation.A.5.64}{}}
2021
\newlabel{eq:lieDerivCommutator@cref}{{[equation][64][2147483647,1]A.64}{160}}
2022
\newlabel{eq:fundamentalTheoremPartdLieDeriv}{{A.71}{161}{The Lie Derivative}{equation.A.5.71}{}}
2023
\newlabel{eq:fundamentalTheoremPartdLieDeriv@cref}{{[equation][71][2147483647,1]A.71}{161}}
2014
2024
\citation{MansfieldInvarCalc,OlverSymmetry}
2015
\@writefile{toc}{\contentsline {chapter}{\numberline {B}Lie Groups}{161}{appendix.B}}
2016
\@writefile{lof}{\addvspace {10\p@ }}
2017
\@writefile{lot}{\addvspace {10\p@ }}
2018
\@writefile{lol}{\addvspace {10\p@ }}
2019
\@writefile{loa}{\addvspace {10\p@ }}
2020
\newlabel{sec:AppLieGroups}{{B}{161}{Lie Groups}{appendix.B}{}}
2021
\newlabel{sec:AppLieGroups@cref}{{[appendix][2][2147483647]B}{161}}
2022
\@writefile{toc}{\contentsline {section}{\numberline {B.1}The Prolonged Action}{161}{section.B.1}}
2023
\newlabel{sec:AppProlongedAction}{{B.1}{161}{The Prolonged Action}{section.B.1}{}}
2024
\newlabel{sec:AppProlongedAction@cref}{{[subappendix][1][2147483647,2]B.1}{161}}
2025
\newlabel{eq:AppProlongedAction}{{B.1}{161}{The Prolonged Action}{equation.B.1.1}{}}
2026
\newlabel{eq:AppProlongedAction@cref}{{[equation][1][2147483647,2]B.1}{161}}
2027
\@writefile{brf}{\backcite{MansfieldInvarCalc}{{161}{B.1}{equation.B.1.1}}}
2028
\@writefile{brf}{\backcite{OlverSymmetry}{{161}{B.1}{equation.B.1.1}}}
2029
\newlabel{eq:AppProlActionDeriv}{{B.5}{161}{The Prolonged Action}{equation.B.1.5}{}}
2030
\newlabel{eq:AppProlActionDeriv@cref}{{[equation][5][2147483647,2]B.5}{161}}
2031
\@writefile{toc}{\contentsline {section}{\numberline {B.2}Geometrical Meaning of the Commutator $\ensuremath {\left [{\cdot ,\cdot }\right ]}$}{162}{section.B.2}}
2032
\newlabel{sec:AppCommutator}{{B.2}{162}{Geometrical Meaning of the Commutator $\squarebrackets {\cdot ,\cdot }$}{section.B.2}{}}
2033
\newlabel{sec:AppCommutator@cref}{{[subappendix][2][2147483647,2]B.2}{162}}
2034
\newlabel{eq:AppCommutator}{{B.13}{162}{Geometrical Meaning of the Commutator $\squarebrackets {\cdot ,\cdot }$}{equation.B.2.13}{}}
2035
\newlabel{eq:AppCommutator@cref}{{[equation][13][2147483647,2]B.13}{162}}
2036
\newlabel{eq:AppComm1}{{B.17}{163}{Geometrical Meaning of the Commutator $\squarebrackets {\cdot ,\cdot }$}{equation.B.2.17}{}}
2037
\newlabel{eq:AppComm1@cref}{{[equation][17][2147483647,2]B.17}{163}}
2038
\newlabel{eq:AppComm2}{{B.18}{163}{Geometrical Meaning of the Commutator $\squarebrackets {\cdot ,\cdot }$}{equation.B.2.18}{}}
2039
\newlabel{eq:AppComm2@cref}{{[equation][18][2147483647,2]B.18}{163}}
2040
\@writefile{toc}{\contentsline {section}{\numberline {B.3}Derivation Of Noethers Theorem}{163}{section.B.3}}
2041
\newlabel{sec:AppNoether}{{B.3}{163}{Derivation Of Noethers Theorem}{section.B.3}{}}
2042
\newlabel{sec:AppNoether@cref}{{[subappendix][3][2147483647,2]B.3}{163}}
2043
\newlabel{eq:AppNoetherTotEnergy}{{B.19}{163}{Derivation Of Noethers Theorem}{equation.B.3.19}{}}
2044
\newlabel{eq:AppNoetherTotEnergy@cref}{{[equation][19][2147483647,2]B.19}{163}}
2045
\newlabel{eq:AppNoetherLieAlg}{{B.20}{163}{Derivation Of Noethers Theorem}{equation.B.3.20}{}}
2046
\newlabel{eq:AppNoetherLieAlg@cref}{{[equation][20][2147483647,2]B.20}{163}}
2047
\newlabel{eq:AppNoetherStatement1}{{B.21}{164}{Derivation Of Noethers Theorem}{equation.B.3.21}{}}
2048
\newlabel{eq:AppNoetherStatement1@cref}{{[equation][21][2147483647,2]B.21}{164}}
2049
\newlabel{eq:AppNoetherStatement2}{{B.22}{164}{Derivation Of Noethers Theorem}{equation.B.3.22}{}}
2050
\newlabel{eq:AppNoetherStatement2@cref}{{[equation][22][2147483647,2]B.22}{164}}
2051
\newlabel{eq:AppNoetherProof0}{{B.24}{164}{Derivation Of Noethers Theorem}{equation.B.3.24}{}}
2052
\newlabel{eq:AppNoetherProof0@cref}{{[equation][24][2147483647,2]B.24}{164}}
2053
\newlabel{eq:AppNoetherProof1}{{B.25}{164}{Derivation Of Noethers Theorem}{equation.B.3.25}{}}
2054
\newlabel{eq:AppNoetherProof1@cref}{{[equation][25][2147483647,2]B.25}{164}}
2055
\newlabel{eq:AppNoetherProof2}{{B.29}{165}{Derivation Of Noethers Theorem}{equation.B.3.29}{}}
2056
\newlabel{eq:AppNoetherProof2@cref}{{[equation][29][2147483647,2]B.29}{165}}
2057
\newlabel{eq:AppNoetherProof3}{{B.30}{165}{Derivation Of Noethers Theorem}{equation.B.3.30}{}}
2058
\newlabel{eq:AppNoetherProof3@cref}{{[equation][30][2147483647,2]B.30}{165}}
2059
\newlabel{eq:AppNoetherProof4}{{B.31}{165}{Derivation Of Noethers Theorem}{equation.B.3.31}{}}
2060
\newlabel{eq:AppNoetherProof4@cref}{{[equation][31][2147483647,2]B.31}{165}}
2061
\newlabel{eq:AppNoetherProof5}{{B.32}{165}{Derivation Of Noethers Theorem}{equation.B.3.32}{}}
2062
\newlabel{eq:AppNoetherProof5@cref}{{[equation][32][2147483647,2]B.32}{165}}
2063
\newlabel{eq:AppNoetherProof7}{{B.33}{165}{Derivation Of Noethers Theorem}{equation.B.3.33}{}}
2064
\newlabel{eq:AppNoetherProof7@cref}{{[equation][33][2147483647,2]B.33}{165}}
2065
\newlabel{eq:AppNoetherProof6}{{B.34}{165}{Derivation Of Noethers Theorem}{equation.B.3.34}{}}
2066
\newlabel{eq:AppNoetherProof6@cref}{{[equation][34][2147483647,2]B.34}{165}}
2067
\newlabel{eq:AppNoetherProof8}{{B.35}{165}{Derivation Of Noethers Theorem}{equation.B.3.35}{}}
2068
\newlabel{eq:AppNoetherProof8@cref}{{[equation][35][2147483647,2]B.35}{165}}
2069
\newlabel{eq:AppNoetherProof10}{{B.36}{166}{Derivation Of Noethers Theorem}{equation.B.3.36}{}}
2070
\newlabel{eq:AppNoetherProof10@cref}{{[equation][36][2147483647,2]B.36}{166}}
2071
\newlabel{eq:AppNoetherProof11}{{B.38}{166}{Derivation Of Noethers Theorem}{equation.B.3.38}{}}
2072
\newlabel{eq:AppNoetherProof11@cref}{{[equation][38][2147483647,2]B.38}{166}}
2073
\newlabel{eq:AppNoetherLieAlgExp}{{B.39}{166}{Derivation Of Noethers Theorem}{equation.B.3.39}{}}
2074
\newlabel{eq:AppNoetherLieAlgExp@cref}{{[equation][39][2147483647,2]B.39}{166}}
2075
\newlabel{eq:AppNoetherProof12}{{B.40}{166}{Derivation Of Noethers Theorem}{equation.B.3.40}{}}
2076
\newlabel{eq:AppNoetherProof12@cref}{{[equation][40][2147483647,2]B.40}{166}}
2077
\@writefile{toc}{\contentsline {subsection}{\numberline {B.3.1}Connection between $\ensuremath {{\mathbf {B}}}_m$, $\ensuremath {{\mathbf {W}}}_m$ and $\ensuremath {\left [{\mathcal {E}}\right ]}$}{166}{subsection.B.3.1}}
2078
\newlabel{eq:AppBendingNoetherCurrent1}{{B.42}{167}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.42}{}}
2079
\newlabel{eq:AppBendingNoetherCurrent1@cref}{{[equation][42][2147483647,2]B.42}{167}}
2080
\newlabel{eq:AppBendingNoetherCurrent2}{{B.43}{167}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.43}{}}
2081
\newlabel{eq:AppBendingNoetherCurrent2@cref}{{[equation][43][2147483647,2]B.43}{167}}
2082
\newlabel{eq:AppBendingNoetherCurrent3}{{B.44}{167}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.44}{}}
2083
\newlabel{eq:AppBendingNoetherCurrent3@cref}{{[equation][44][2147483647,2]B.44}{167}}
2084
\newlabel{eq:AppBendingNoetherCurrent4}{{B.45}{167}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.45}{}}
2085
\newlabel{eq:AppBendingNoetherCurrent4@cref}{{[equation][45][2147483647,2]B.45}{167}}
2086
\newlabel{eq:AppBendingNoetherCurrent5}{{B.46}{167}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.46}{}}
2087
\newlabel{eq:AppBendingNoetherCurrent5@cref}{{[equation][46][2147483647,2]B.46}{167}}
2088
\newlabel{eq:AppBendingNoetherCurrent6}{{B.47}{167}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.47}{}}
2089
\newlabel{eq:AppBendingNoetherCurrent6@cref}{{[equation][47][2147483647,2]B.47}{167}}
2090
\newlabel{eq:AppBendingNoetherCurrent7}{{B.48}{167}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.48}{}}
2091
\newlabel{eq:AppBendingNoetherCurrent7@cref}{{[equation][48][2147483647,2]B.48}{167}}
2092
\newlabel{eq:AppBendingLevelSet}{{B.49}{167}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.49}{}}
2093
\newlabel{eq:AppBendingLevelSet@cref}{{[equation][49][2147483647,2]B.49}{167}}
2094
\@writefile{toc}{\contentsline {chapter}{\numberline {C}The Bending Algebra}{168}{appendix.C}}
2095
\@writefile{lof}{\addvspace {10\p@ }}
2096
\@writefile{lot}{\addvspace {10\p@ }}
2097
\@writefile{lol}{\addvspace {10\p@ }}
2098
\@writefile{loa}{\addvspace {10\p@ }}
2099
\@writefile{toc}{\contentsline {section}{\numberline {C.1}The curvature operator}{168}{section.C.1}}
2100
\newlabel{sec:AppCurv}{{C.1}{168}{The curvature operator}{section.C.1}{}}
2101
\newlabel{sec:AppCurv@cref}{{[subappendix][1][2147483647,3]C.1}{168}}
2102
\newlabel{eq:AppDiffusionProcess}{{C.1}{168}{The curvature operator}{equation.C.1.1}{}}
2103
\newlabel{eq:AppDiffusionProcess@cref}{{[equation][1][2147483647,3]C.1}{168}}
2104
\newlabel{eq:AppEulerLagrangeGRF2}{{C.3}{168}{The curvature operator}{equation.C.1.3}{}}
2105
\newlabel{eq:AppEulerLagrangeGRF2@cref}{{[equation][3][2147483647,3]C.3}{168}}
2106
\newlabel{eq:DivPChange}{{C.6}{168}{The curvature operator}{equation.C.1.6}{}}
2107
\newlabel{eq:DivPChange@cref}{{[equation][6][2147483647,3]C.6}{168}}
2025
\@writefile{toc}{\contentsline {chapter}{\numberline {B}Lie Groups}{162}{appendix.B}}
2026
\@writefile{lof}{\addvspace {10\p@ }}
2027
\@writefile{lot}{\addvspace {10\p@ }}
2028
\@writefile{lol}{\addvspace {10\p@ }}
2029
\@writefile{loa}{\addvspace {10\p@ }}
2030
\newlabel{sec:AppLieGroups}{{B}{162}{Lie Groups}{appendix.B}{}}
2031
\newlabel{sec:AppLieGroups@cref}{{[appendix][2][2147483647]B}{162}}
2032
\@writefile{toc}{\contentsline {section}{\numberline {B.1}The Prolonged Action}{162}{section.B.1}}
2033
\newlabel{sec:AppProlongedAction}{{B.1}{162}{The Prolonged Action}{section.B.1}{}}
2034
\newlabel{sec:AppProlongedAction@cref}{{[subappendix][1][2147483647,2]B.1}{162}}
2035
\newlabel{eq:AppProlongedAction}{{B.1}{162}{The Prolonged Action}{equation.B.1.1}{}}
2036
\newlabel{eq:AppProlongedAction@cref}{{[equation][1][2147483647,2]B.1}{162}}
2037
\@writefile{brf}{\backcite{MansfieldInvarCalc}{{162}{B.1}{equation.B.1.1}}}
2038
\@writefile{brf}{\backcite{OlverSymmetry}{{162}{B.1}{equation.B.1.1}}}
2039
\newlabel{eq:AppProlActionDeriv}{{B.5}{162}{The Prolonged Action}{equation.B.1.5}{}}
2040
\newlabel{eq:AppProlActionDeriv@cref}{{[equation][5][2147483647,2]B.5}{162}}
2041
\@writefile{toc}{\contentsline {section}{\numberline {B.2}Geometrical Meaning of the Commutator $\ensuremath {\left [{\cdot ,\cdot }\right ]}$}{163}{section.B.2}}
2042
\newlabel{sec:AppCommutator}{{B.2}{163}{Geometrical Meaning of the Commutator $\squarebrackets {\cdot ,\cdot }$}{section.B.2}{}}
2043
\newlabel{sec:AppCommutator@cref}{{[subappendix][2][2147483647,2]B.2}{163}}
2044
\newlabel{eq:AppCommutator}{{B.13}{163}{Geometrical Meaning of the Commutator $\squarebrackets {\cdot ,\cdot }$}{equation.B.2.13}{}}
2045
\newlabel{eq:AppCommutator@cref}{{[equation][13][2147483647,2]B.13}{163}}
2046
\newlabel{eq:AppComm1}{{B.17}{164}{Geometrical Meaning of the Commutator $\squarebrackets {\cdot ,\cdot }$}{equation.B.2.17}{}}
2047
\newlabel{eq:AppComm1@cref}{{[equation][17][2147483647,2]B.17}{164}}
2048
\newlabel{eq:AppComm2}{{B.18}{164}{Geometrical Meaning of the Commutator $\squarebrackets {\cdot ,\cdot }$}{equation.B.2.18}{}}
2049
\newlabel{eq:AppComm2@cref}{{[equation][18][2147483647,2]B.18}{164}}
2050
\@writefile{toc}{\contentsline {section}{\numberline {B.3}Derivation Of Noethers Theorem}{164}{section.B.3}}
2051
\newlabel{sec:AppNoether}{{B.3}{164}{Derivation Of Noethers Theorem}{section.B.3}{}}
2052
\newlabel{sec:AppNoether@cref}{{[subappendix][3][2147483647,2]B.3}{164}}
2053
\newlabel{eq:AppNoetherTotEnergy}{{B.19}{164}{Derivation Of Noethers Theorem}{equation.B.3.19}{}}
2054
\newlabel{eq:AppNoetherTotEnergy@cref}{{[equation][19][2147483647,2]B.19}{164}}
2055
\newlabel{eq:AppNoetherLieAlg}{{B.20}{164}{Derivation Of Noethers Theorem}{equation.B.3.20}{}}
2056
\newlabel{eq:AppNoetherLieAlg@cref}{{[equation][20][2147483647,2]B.20}{164}}
2057
\newlabel{eq:AppNoetherStatement1}{{B.21}{165}{Derivation Of Noethers Theorem}{equation.B.3.21}{}}
2058
\newlabel{eq:AppNoetherStatement1@cref}{{[equation][21][2147483647,2]B.21}{165}}
2059
\newlabel{eq:AppNoetherStatement2}{{B.22}{165}{Derivation Of Noethers Theorem}{equation.B.3.22}{}}
2060
\newlabel{eq:AppNoetherStatement2@cref}{{[equation][22][2147483647,2]B.22}{165}}
2061
\newlabel{eq:AppNoetherProof0}{{B.24}{165}{Derivation Of Noethers Theorem}{equation.B.3.24}{}}
2062
\newlabel{eq:AppNoetherProof0@cref}{{[equation][24][2147483647,2]B.24}{165}}
2063
\newlabel{eq:AppNoetherProof1}{{B.25}{165}{Derivation Of Noethers Theorem}{equation.B.3.25}{}}
2064
\newlabel{eq:AppNoetherProof1@cref}{{[equation][25][2147483647,2]B.25}{165}}
2065
\newlabel{eq:AppNoetherProof2}{{B.29}{166}{Derivation Of Noethers Theorem}{equation.B.3.29}{}}
2066
\newlabel{eq:AppNoetherProof2@cref}{{[equation][29][2147483647,2]B.29}{166}}
2067
\newlabel{eq:AppNoetherProof3}{{B.30}{166}{Derivation Of Noethers Theorem}{equation.B.3.30}{}}
2068
\newlabel{eq:AppNoetherProof3@cref}{{[equation][30][2147483647,2]B.30}{166}}
2069
\newlabel{eq:AppNoetherProof4}{{B.31}{166}{Derivation Of Noethers Theorem}{equation.B.3.31}{}}
2070
\newlabel{eq:AppNoetherProof4@cref}{{[equation][31][2147483647,2]B.31}{166}}
2071
\newlabel{eq:AppNoetherProof5}{{B.32}{166}{Derivation Of Noethers Theorem}{equation.B.3.32}{}}
2072
\newlabel{eq:AppNoetherProof5@cref}{{[equation][32][2147483647,2]B.32}{166}}
2073
\newlabel{eq:AppNoetherProof7}{{B.33}{166}{Derivation Of Noethers Theorem}{equation.B.3.33}{}}
2074
\newlabel{eq:AppNoetherProof7@cref}{{[equation][33][2147483647,2]B.33}{166}}
2075
\newlabel{eq:AppNoetherProof6}{{B.34}{166}{Derivation Of Noethers Theorem}{equation.B.3.34}{}}
2076
\newlabel{eq:AppNoetherProof6@cref}{{[equation][34][2147483647,2]B.34}{166}}
2077
\newlabel{eq:AppNoetherProof8}{{B.35}{166}{Derivation Of Noethers Theorem}{equation.B.3.35}{}}
2078
\newlabel{eq:AppNoetherProof8@cref}{{[equation][35][2147483647,2]B.35}{166}}
2079
\newlabel{eq:AppNoetherProof10}{{B.36}{167}{Derivation Of Noethers Theorem}{equation.B.3.36}{}}
2080
\newlabel{eq:AppNoetherProof10@cref}{{[equation][36][2147483647,2]B.36}{167}}
2081
\newlabel{eq:AppNoetherProof11}{{B.38}{167}{Derivation Of Noethers Theorem}{equation.B.3.38}{}}
2082
\newlabel{eq:AppNoetherProof11@cref}{{[equation][38][2147483647,2]B.38}{167}}
2083
\newlabel{eq:AppNoetherLieAlgExp}{{B.39}{167}{Derivation Of Noethers Theorem}{equation.B.3.39}{}}
2084
\newlabel{eq:AppNoetherLieAlgExp@cref}{{[equation][39][2147483647,2]B.39}{167}}
2085
\newlabel{eq:AppNoetherProof12}{{B.40}{167}{Derivation Of Noethers Theorem}{equation.B.3.40}{}}
2086
\newlabel{eq:AppNoetherProof12@cref}{{[equation][40][2147483647,2]B.40}{167}}
2087
\@writefile{toc}{\contentsline {subsection}{\numberline {B.3.1}Connection between $\ensuremath {{\mathbf {B}}}_m$, $\ensuremath {{\mathbf {W}}}_m$ and $\ensuremath {\left [{\mathcal {E}}\right ]}$}{167}{subsection.B.3.1}}
2088
\newlabel{eq:AppBendingNoetherCurrent1}{{B.42}{168}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.42}{}}
2089
\newlabel{eq:AppBendingNoetherCurrent1@cref}{{[equation][42][2147483647,2]B.42}{168}}
2090
\newlabel{eq:AppBendingNoetherCurrent2}{{B.43}{168}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.43}{}}
2091
\newlabel{eq:AppBendingNoetherCurrent2@cref}{{[equation][43][2147483647,2]B.43}{168}}
2092
\newlabel{eq:AppBendingNoetherCurrent3}{{B.44}{168}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.44}{}}
2093
\newlabel{eq:AppBendingNoetherCurrent3@cref}{{[equation][44][2147483647,2]B.44}{168}}
2094
\newlabel{eq:AppBendingNoetherCurrent4}{{B.45}{168}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.45}{}}
2095
\newlabel{eq:AppBendingNoetherCurrent4@cref}{{[equation][45][2147483647,2]B.45}{168}}
2096
\newlabel{eq:AppBendingNoetherCurrent5}{{B.46}{168}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.46}{}}
2097
\newlabel{eq:AppBendingNoetherCurrent5@cref}{{[equation][46][2147483647,2]B.46}{168}}
2098
\newlabel{eq:AppBendingNoetherCurrent6}{{B.47}{168}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.47}{}}
2099
\newlabel{eq:AppBendingNoetherCurrent6@cref}{{[equation][47][2147483647,2]B.47}{168}}
2100
\newlabel{eq:AppBendingNoetherCurrent7}{{B.48}{168}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.48}{}}
2101
\newlabel{eq:AppBendingNoetherCurrent7@cref}{{[equation][48][2147483647,2]B.48}{168}}
2102
\newlabel{eq:AppBendingLevelSet}{{B.49}{168}{Connection between $\vectorheader {B}_m$, $\vectorheader {W}_m$ and $\squarebrackets {\mathcal {E}}$}{equation.B.3.49}{}}
2103
\newlabel{eq:AppBendingLevelSet@cref}{{[equation][49][2147483647,2]B.49}{168}}
2104
\@writefile{toc}{\contentsline {chapter}{\numberline {C}The Bending Algebra}{169}{appendix.C}}
2105
\@writefile{lof}{\addvspace {10\p@ }}
2106
\@writefile{lot}{\addvspace {10\p@ }}
2107
\@writefile{lol}{\addvspace {10\p@ }}
2108
\@writefile{loa}{\addvspace {10\p@ }}
2109
\@writefile{toc}{\contentsline {section}{\numberline {C.1}The curvature operator}{169}{section.C.1}}
2110
\newlabel{sec:AppCurv}{{C.1}{169}{The curvature operator}{section.C.1}{}}
2111
\newlabel{sec:AppCurv@cref}{{[subappendix][1][2147483647,3]C.1}{169}}
2112
\newlabel{eq:AppDiffusionProcess}{{C.1}{169}{The curvature operator}{equation.C.1.1}{}}
2113
\newlabel{eq:AppDiffusionProcess@cref}{{[equation][1][2147483647,3]C.1}{169}}
2114
\newlabel{eq:AppEulerLagrangeGRF2}{{C.3}{169}{The curvature operator}{equation.C.1.3}{}}
2115
\newlabel{eq:AppEulerLagrangeGRF2@cref}{{[equation][3][2147483647,3]C.3}{169}}
2116
\newlabel{eq:DivPChange}{{C.6}{169}{The curvature operator}{equation.C.1.6}{}}
2117
\newlabel{eq:DivPChange@cref}{{[equation][6][2147483647,3]C.6}{169}}
2108
2118
\citation{BrediesMathemBildverarbeitung}
2109
\newlabel{eq:DivPIntegral}{{C.7}{169}{The curvature operator}{equation.C.1.7}{}}
2110
\newlabel{eq:DivPIntegral@cref}{{[equation][7][2147483647,3]C.7}{169}}
2111
\newlabel{eq:DivPChange2}{{C.9}{169}{The curvature operator}{equation.C.1.9}{}}
2112
\newlabel{eq:DivPChange2@cref}{{[equation][9][2147483647,3]C.9}{169}}
2113
\@writefile{brf}{\backcite{BrediesMathemBildverarbeitung}{{169}{C.1}{equation.C.1.9}}}
2114
\newlabel{eq:DivPIntegral2}{{C.10}{169}{The curvature operator}{equation.C.1.10}{}}
2115
\newlabel{eq:DivPIntegral2@cref}{{[equation][10][2147483647,3]C.10}{169}}
2116
\newlabel{eq:DivPIntegral3}{{C.12}{169}{The curvature operator}{equation.C.1.12}{}}
2117
\newlabel{eq:DivPIntegral3@cref}{{[equation][12][2147483647,3]C.12}{169}}
2118
\newlabel{eq:DivPChange3}{{C.14}{169}{The curvature operator}{equation.C.1.14}{}}
2119
\newlabel{eq:DivPChange3@cref}{{[equation][14][2147483647,3]C.14}{169}}
2120
\newlabel{eq:CurvOperator}{{C.15}{170}{The curvature operator}{equation.C.1.15}{}}
2121
\newlabel{eq:CurvOperator@cref}{{[equation][15][2147483647,3]C.15}{170}}
2122
\newlabel{eq:AppCurveCoeff}{{C.17}{170}{The curvature operator}{equation.C.1.17}{}}
2123
\newlabel{eq:AppCurveCoeff@cref}{{[equation][17][2147483647,3]C.17}{170}}
2124
\@writefile{toc}{\contentsline {section}{\numberline {C.2}TV Image Denoising, supplementary results}{171}{section.C.2}}
2125
\newlabel{sec:AppTVSupplementary}{{C.2}{171}{TV Image Denoising, supplementary results}{section.C.2}{}}
2126
\newlabel{sec:AppTVSupplementary@cref}{{[subappendix][2][2147483647,3]C.2}{171}}
2127
\newlabel{fig:Grove}{{C.1a}{171}{Subfigure C C.1a}{subfigure.C.1.1}{}}
2119
\newlabel{eq:DivPIntegral}{{C.7}{170}{The curvature operator}{equation.C.1.7}{}}
2120
\newlabel{eq:DivPIntegral@cref}{{[equation][7][2147483647,3]C.7}{170}}
2121
\newlabel{eq:DivPChange2}{{C.9}{170}{The curvature operator}{equation.C.1.9}{}}
2122
\newlabel{eq:DivPChange2@cref}{{[equation][9][2147483647,3]C.9}{170}}
2123
\@writefile{brf}{\backcite{BrediesMathemBildverarbeitung}{{170}{C.1}{equation.C.1.9}}}
2124
\newlabel{eq:DivPIntegral2}{{C.10}{170}{The curvature operator}{equation.C.1.10}{}}
2125
\newlabel{eq:DivPIntegral2@cref}{{[equation][10][2147483647,3]C.10}{170}}
2126
\newlabel{eq:DivPIntegral3}{{C.12}{170}{The curvature operator}{equation.C.1.12}{}}
2127
\newlabel{eq:DivPIntegral3@cref}{{[equation][12][2147483647,3]C.12}{170}}
2128
\newlabel{eq:DivPChange3}{{C.14}{170}{The curvature operator}{equation.C.1.14}{}}
2129
\newlabel{eq:DivPChange3@cref}{{[equation][14][2147483647,3]C.14}{170}}
2130
\newlabel{eq:CurvOperator}{{C.15}{171}{The curvature operator}{equation.C.1.15}{}}
2131
\newlabel{eq:CurvOperator@cref}{{[equation][15][2147483647,3]C.15}{171}}
2132
\newlabel{eq:AppCurveCoeff}{{C.17}{171}{The curvature operator}{equation.C.1.17}{}}
2133
\newlabel{eq:AppCurveCoeff@cref}{{[equation][17][2147483647,3]C.17}{171}}
2134
\@writefile{toc}{\contentsline {section}{\numberline {C.2}TV Image Denoising, supplementary results}{172}{section.C.2}}
2135
\newlabel{sec:AppTVSupplementary}{{C.2}{172}{TV Image Denoising, supplementary results}{section.C.2}{}}
2136
\newlabel{sec:AppTVSupplementary@cref}{{[subappendix][2][2147483647,3]C.2}{172}}
2137
\newlabel{fig:Grove}{{C.1a}{172}{Subfigure C C.1a}{subfigure.C.1.1}{}}
2128
2138
\newlabel{sub@fig:Grove}{{(a)}{a}{Subfigure C C.1a\relax }{subfigure.C.1.1}{}}
2129
\newlabel{fig:Grove@cref}{{[subfigure][1][2147483647,3,1]C.1a}{171}}
2130
\newlabel{fig:Grove-Noise}{{C.1b}{171}{Subfigure C C.1b}{subfigure.C.1.2}{}}
2139
\newlabel{fig:Grove@cref}{{[subfigure][1][2147483647,3,1]C.1a}{172}}
2140
\newlabel{fig:Grove-Noise}{{C.1b}{172}{Subfigure C C.1b}{subfigure.C.1.2}{}}
2131
2141
\newlabel{sub@fig:Grove-Noise}{{(b)}{b}{Subfigure C C.1b\relax }{subfigure.C.1.2}{}}
2132
\newlabel{fig:Grove-Noise@cref}{{[subfigure][2][2147483647,3,1]C.1b}{171}}
2133
\newlabel{fig:Grove-GNA}{{C.1c}{171}{Subfigure C C.1c}{subfigure.C.1.3}{}}
2142
\newlabel{fig:Grove-Noise@cref}{{[subfigure][2][2147483647,3,1]C.1b}{172}}
2143
\newlabel{fig:Grove-GNA}{{C.1c}{172}{Subfigure C C.1c}{subfigure.C.1.3}{}}
2134
2144
\newlabel{sub@fig:Grove-GNA}{{(c)}{c}{Subfigure C C.1c\relax }{subfigure.C.1.3}{}}
2135
\newlabel{fig:Grove-GNA@cref}{{[subfigure][3][2147483647,3,1]C.1c}{171}}
2136
\newlabel{fig:Grove-BNA}{{C.1d}{171}{Subfigure C C.1d}{subfigure.C.1.4}{}}
2145
\newlabel{fig:Grove-GNA@cref}{{[subfigure][3][2147483647,3,1]C.1c}{172}}
2146
\newlabel{fig:Grove-BNA}{{C.1d}{172}{Subfigure C C.1d}{subfigure.C.1.4}{}}
2137
2147
\newlabel{sub@fig:Grove-BNA}{{(d)}{d}{Subfigure C C.1d\relax }{subfigure.C.1.4}{}}
2138
\newlabel{fig:Grove-BNA@cref}{{[subfigure][4][2147483647,3,1]C.1d}{171}}
2139
\newlabel{fig:Grove-MeanEnergy}{{C.1e}{171}{Subfigure C C.1e}{subfigure.C.1.5}{}}
2148
\newlabel{fig:Grove-BNA@cref}{{[subfigure][4][2147483647,3,1]C.1d}{172}}
2149
\newlabel{fig:Grove-MeanEnergy}{{C.1e}{172}{Subfigure C C.1e}{subfigure.C.1.5}{}}
2140
2150
\newlabel{sub@fig:Grove-MeanEnergy}{{(e)}{e}{Subfigure C C.1e\relax }{subfigure.C.1.5}{}}
2141
\newlabel{fig:Grove-MeanEnergy@cref}{{[subfigure][5][2147483647,3,1]C.1e}{171}}
2142
\newlabel{fig:Grove-StdDevEnergy}{{C.1f}{171}{Subfigure C C.1f}{subfigure.C.1.6}{}}
2151
\newlabel{fig:Grove-MeanEnergy@cref}{{[subfigure][5][2147483647,3,1]C.1e}{172}}
2152
\newlabel{fig:Grove-StdDevEnergy}{{C.1f}{172}{Subfigure C C.1f}{subfigure.C.1.6}{}}
2143
2153
\newlabel{sub@fig:Grove-StdDevEnergy}{{(f)}{f}{Subfigure C C.1f\relax }{subfigure.C.1.6}{}}
2144
\newlabel{fig:Grove-StdDevEnergy@cref}{{[subfigure][6][2147483647,3,1]C.1f}{171}}
2145
\newlabel{fig:Grove-MeanCurvature}{{C.1g}{171}{Subfigure C C.1g}{subfigure.C.1.7}{}}
2154
\newlabel{fig:Grove-StdDevEnergy@cref}{{[subfigure][6][2147483647,3,1]C.1f}{172}}
2155
\newlabel{fig:Grove-MeanCurvature}{{C.1g}{172}{Subfigure C C.1g}{subfigure.C.1.7}{}}
2146
2156
\newlabel{sub@fig:Grove-MeanCurvature}{{(g)}{g}{Subfigure C C.1g\relax }{subfigure.C.1.7}{}}
2147
\newlabel{fig:Grove-MeanCurvature@cref}{{[subfigure][7][2147483647,3,1]C.1g}{171}}
2148
\newlabel{fig:GroveCurvatureFit}{{C.1h}{171}{Subfigure C C.1h}{subfigure.C.1.8}{}}
2157
\newlabel{fig:Grove-MeanCurvature@cref}{{[subfigure][7][2147483647,3,1]C.1g}{172}}
2158
\newlabel{fig:GroveCurvatureFit}{{C.1h}{172}{Subfigure C C.1h}{subfigure.C.1.8}{}}
2149
2159
\newlabel{sub@fig:GroveCurvatureFit}{{(h)}{h}{Subfigure C C.1h\relax }{subfigure.C.1.8}{}}
2150
\newlabel{fig:GroveCurvatureFit@cref}{{[subfigure][8][2147483647,3,1]C.1h}{171}}
2151
\newlabel{fig:Evergreen}{{C.1i}{171}{Subfigure C C.1i}{subfigure.C.1.9}{}}
2160
\newlabel{fig:GroveCurvatureFit@cref}{{[subfigure][8][2147483647,3,1]C.1h}{172}}
2161
\newlabel{fig:Evergreen}{{C.1i}{172}{Subfigure C C.1i}{subfigure.C.1.9}{}}
2152
2162
\newlabel{sub@fig:Evergreen}{{(i)}{i}{Subfigure C C.1i\relax }{subfigure.C.1.9}{}}
2153
\newlabel{fig:Evergreen@cref}{{[subfigure][9][2147483647,3,1]C.1i}{171}}
2154
\newlabel{fig:Evergreen-Noise}{{C.1j}{171}{Subfigure C C.1j}{subfigure.C.1.10}{}}
2163
\newlabel{fig:Evergreen@cref}{{[subfigure][9][2147483647,3,1]C.1i}{172}}
2164
\newlabel{fig:Evergreen-Noise}{{C.1j}{172}{Subfigure C C.1j}{subfigure.C.1.10}{}}
2155
2165
\newlabel{sub@fig:Evergreen-Noise}{{(j)}{j}{Subfigure C C.1j\relax }{subfigure.C.1.10}{}}
2156
\newlabel{fig:Evergreen-Noise@cref}{{[subfigure][10][2147483647,3,1]C.1j}{171}}
2157
\newlabel{fig:Evergreen-GNA}{{C.1k}{171}{Subfigure C C.1k}{subfigure.C.1.11}{}}
2166
\newlabel{fig:Evergreen-Noise@cref}{{[subfigure][10][2147483647,3,1]C.1j}{172}}
2167
\newlabel{fig:Evergreen-GNA}{{C.1k}{172}{Subfigure C C.1k}{subfigure.C.1.11}{}}
2158
2168
\newlabel{sub@fig:Evergreen-GNA}{{(k)}{k}{Subfigure C C.1k\relax }{subfigure.C.1.11}{}}
2159
\newlabel{fig:Evergreen-GNA@cref}{{[subfigure][11][2147483647,3,1]C.1k}{171}}
2160
\newlabel{fig:Evergreen-BNA}{{C.1l}{171}{Subfigure C C.1l}{subfigure.C.1.12}{}}
2169
\newlabel{fig:Evergreen-GNA@cref}{{[subfigure][11][2147483647,3,1]C.1k}{172}}
2170
\newlabel{fig:Evergreen-BNA}{{C.1l}{172}{Subfigure C C.1l}{subfigure.C.1.12}{}}
2161
2171
\newlabel{sub@fig:Evergreen-BNA}{{(l)}{l}{Subfigure C C.1l\relax }{subfigure.C.1.12}{}}
2162
\newlabel{fig:Evergreen-BNA@cref}{{[subfigure][12][2147483647,3,1]C.1l}{171}}
2163
\newlabel{fig:Evergreen-MeanEnergy}{{C.1m}{171}{Subfigure C C.1m}{subfigure.C.1.13}{}}
2172
\newlabel{fig:Evergreen-BNA@cref}{{[subfigure][12][2147483647,3,1]C.1l}{172}}
2173
\newlabel{fig:Evergreen-MeanEnergy}{{C.1m}{172}{Subfigure C C.1m}{subfigure.C.1.13}{}}
2164
2174
\newlabel{sub@fig:Evergreen-MeanEnergy}{{(m)}{m}{Subfigure C C.1m\relax }{subfigure.C.1.13}{}}
2165
\newlabel{fig:Evergreen-MeanEnergy@cref}{{[subfigure][13][2147483647,3,1]C.1m}{171}}
2166
\newlabel{fig:Evergreen-StdDevEnergy}{{C.1n}{171}{Subfigure C C.1n}{subfigure.C.1.14}{}}
2175
\newlabel{fig:Evergreen-MeanEnergy@cref}{{[subfigure][13][2147483647,3,1]C.1m}{172}}
2176
\newlabel{fig:Evergreen-StdDevEnergy}{{C.1n}{172}{Subfigure C C.1n}{subfigure.C.1.14}{}}
2167
2177
\newlabel{sub@fig:Evergreen-StdDevEnergy}{{(n)}{n}{Subfigure C C.1n\relax }{subfigure.C.1.14}{}}
2168
\newlabel{fig:Evergreen-StdDevEnergy@cref}{{[subfigure][14][2147483647,3,1]C.1n}{171}}
2169
\newlabel{fig:Evergreen-MeanCurvature}{{C.1o}{171}{Subfigure C C.1o}{subfigure.C.1.15}{}}
2178
\newlabel{fig:Evergreen-StdDevEnergy@cref}{{[subfigure][14][2147483647,3,1]C.1n}{172}}
2179
\newlabel{fig:Evergreen-MeanCurvature}{{C.1o}{172}{Subfigure C C.1o}{subfigure.C.1.15}{}}
2170
2180
\newlabel{sub@fig:Evergreen-MeanCurvature}{{(o)}{o}{Subfigure C C.1o\relax }{subfigure.C.1.15}{}}
2171
\newlabel{fig:Evergreen-MeanCurvature@cref}{{[subfigure][15][2147483647,3,1]C.1o}{171}}
2172
\newlabel{fig:EvergreenCurvatureFit}{{C.1p}{171}{Subfigure C C.1p}{subfigure.C.1.16}{}}
2181
\newlabel{fig:Evergreen-MeanCurvature@cref}{{[subfigure][15][2147483647,3,1]C.1o}{172}}
2182
\newlabel{fig:EvergreenCurvatureFit}{{C.1p}{172}{Subfigure C C.1p}{subfigure.C.1.16}{}}
2173
2183
\newlabel{sub@fig:EvergreenCurvatureFit}{{(p)}{p}{Subfigure C C.1p\relax }{subfigure.C.1.16}{}}
2174
\newlabel{fig:EvergreenCurvatureFit@cref}{{[subfigure][16][2147483647,3,1]C.1p}{171}}
2175
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {input image}}}{171}{subfigure.1.1}}
2176
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\phi _0$}}}{171}{subfigure.1.2}}
2177
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{171}{subfigure.1.3}}
2178
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{171}{subfigure.1.4}}
2179
\@writefile{lof}{\contentsline {subfigure}{\numberline{(e)}{\ignorespaces {}}}{171}{subfigure.1.5}}
2180
\@writefile{lof}{\contentsline {subfigure}{\numberline{(f)}{\ignorespaces {}}}{171}{subfigure.1.6}}
2181
\@writefile{lof}{\contentsline {subfigure}{\numberline{(g)}{\ignorespaces {}}}{171}{subfigure.1.7}}
2182
\@writefile{lof}{\contentsline {subfigure}{\numberline{(h)}{\ignorespaces {}}}{171}{subfigure.1.8}}
2183
\@writefile{lof}{\contentsline {subfigure}{\numberline{(i)}{\ignorespaces {input image}}}{171}{subfigure.1.9}}
2184
\@writefile{lof}{\contentsline {subfigure}{\numberline{(j)}{\ignorespaces {$\phi _0$}}}{171}{subfigure.1.10}}
2185
\@writefile{lof}{\contentsline {subfigure}{\numberline{(k)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{171}{subfigure.1.11}}
2186
\@writefile{lof}{\contentsline {subfigure}{\numberline{(l)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{171}{subfigure.1.12}}
2187
\@writefile{lof}{\contentsline {subfigure}{\numberline{(m)}{\ignorespaces {}}}{171}{subfigure.1.13}}
2188
\@writefile{lof}{\contentsline {subfigure}{\numberline{(n)}{\ignorespaces {}}}{171}{subfigure.1.14}}
2189
\@writefile{lof}{\contentsline {subfigure}{\numberline{(o)}{\ignorespaces {}}}{171}{subfigure.1.15}}
2190
\@writefile{lof}{\contentsline {subfigure}{\numberline{(p)}{\ignorespaces {}}}{171}{subfigure.1.16}}
2191
\newlabel{fig:Dumptruck}{{C.1a}{172}{Subfigure C C.1a}{subfigure.C.1.1}{}}
2184
\newlabel{fig:EvergreenCurvatureFit@cref}{{[subfigure][16][2147483647,3,1]C.1p}{172}}
2185
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {input image}}}{172}{subfigure.1.1}}
2186
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\phi _0$}}}{172}{subfigure.1.2}}
2187
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{172}{subfigure.1.3}}
2188
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{172}{subfigure.1.4}}
2189
\@writefile{lof}{\contentsline {subfigure}{\numberline{(e)}{\ignorespaces {}}}{172}{subfigure.1.5}}
2190
\@writefile{lof}{\contentsline {subfigure}{\numberline{(f)}{\ignorespaces {}}}{172}{subfigure.1.6}}
2191
\@writefile{lof}{\contentsline {subfigure}{\numberline{(g)}{\ignorespaces {}}}{172}{subfigure.1.7}}
2192
\@writefile{lof}{\contentsline {subfigure}{\numberline{(h)}{\ignorespaces {}}}{172}{subfigure.1.8}}
2193
\@writefile{lof}{\contentsline {subfigure}{\numberline{(i)}{\ignorespaces {input image}}}{172}{subfigure.1.9}}
2194
\@writefile{lof}{\contentsline {subfigure}{\numberline{(j)}{\ignorespaces {$\phi _0$}}}{172}{subfigure.1.10}}
2195
\@writefile{lof}{\contentsline {subfigure}{\numberline{(k)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{172}{subfigure.1.11}}
2196
\@writefile{lof}{\contentsline {subfigure}{\numberline{(l)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{172}{subfigure.1.12}}
2197
\@writefile{lof}{\contentsline {subfigure}{\numberline{(m)}{\ignorespaces {}}}{172}{subfigure.1.13}}
2198
\@writefile{lof}{\contentsline {subfigure}{\numberline{(n)}{\ignorespaces {}}}{172}{subfigure.1.14}}
2199
\@writefile{lof}{\contentsline {subfigure}{\numberline{(o)}{\ignorespaces {}}}{172}{subfigure.1.15}}
2200
\@writefile{lof}{\contentsline {subfigure}{\numberline{(p)}{\ignorespaces {}}}{172}{subfigure.1.16}}
2201
\newlabel{fig:Dumptruck}{{C.1a}{173}{Subfigure C C.1a}{subfigure.C.1.1}{}}
2192
2202
\newlabel{sub@fig:Dumptruck}{{(a)}{a}{Subfigure C C.1a\relax }{subfigure.C.1.1}{}}
2193
\newlabel{fig:Dumptruck@cref}{{[subfigure][1][2147483647,3,1]C.1a}{172}}
2194
\newlabel{fig:Dumptruck-Noise}{{C.1b}{172}{Subfigure C C.1b}{subfigure.C.1.2}{}}
2203
\newlabel{fig:Dumptruck@cref}{{[subfigure][1][2147483647,3,1]C.1a}{173}}
2204
\newlabel{fig:Dumptruck-Noise}{{C.1b}{173}{Subfigure C C.1b}{subfigure.C.1.2}{}}
2195
2205
\newlabel{sub@fig:Dumptruck-Noise}{{(b)}{b}{Subfigure C C.1b\relax }{subfigure.C.1.2}{}}
2196
\newlabel{fig:Dumptruck-Noise@cref}{{[subfigure][2][2147483647,3,1]C.1b}{172}}
2197
\newlabel{fig:Dumptruck-GNA}{{C.1c}{172}{Subfigure C C.1c}{subfigure.C.1.3}{}}
2206
\newlabel{fig:Dumptruck-Noise@cref}{{[subfigure][2][2147483647,3,1]C.1b}{173}}
2207
\newlabel{fig:Dumptruck-GNA}{{C.1c}{173}{Subfigure C C.1c}{subfigure.C.1.3}{}}
2198
2208
\newlabel{sub@fig:Dumptruck-GNA}{{(c)}{c}{Subfigure C C.1c\relax }{subfigure.C.1.3}{}}
2199
\newlabel{fig:Dumptruck-GNA@cref}{{[subfigure][3][2147483647,3,1]C.1c}{172}}
2200
\newlabel{fig:Dumptruck-BNA}{{C.1d}{172}{Subfigure C C.1d}{subfigure.C.1.4}{}}
2209
\newlabel{fig:Dumptruck-GNA@cref}{{[subfigure][3][2147483647,3,1]C.1c}{173}}
2210
\newlabel{fig:Dumptruck-BNA}{{C.1d}{173}{Subfigure C C.1d}{subfigure.C.1.4}{}}
2201
2211
\newlabel{sub@fig:Dumptruck-BNA}{{(d)}{d}{Subfigure C C.1d\relax }{subfigure.C.1.4}{}}
2202
\newlabel{fig:Dumptruck-BNA@cref}{{[subfigure][4][2147483647,3,1]C.1d}{172}}
2203
\newlabel{fig:Dumptruck-MeanEnergy}{{C.1e}{172}{Subfigure C C.1e}{subfigure.C.1.5}{}}
2212
\newlabel{fig:Dumptruck-BNA@cref}{{[subfigure][4][2147483647,3,1]C.1d}{173}}
2213
\newlabel{fig:Dumptruck-MeanEnergy}{{C.1e}{173}{Subfigure C C.1e}{subfigure.C.1.5}{}}
2204
2214
\newlabel{sub@fig:Dumptruck-MeanEnergy}{{(e)}{e}{Subfigure C C.1e\relax }{subfigure.C.1.5}{}}
2205
\newlabel{fig:Dumptruck-MeanEnergy@cref}{{[subfigure][5][2147483647,3,1]C.1e}{172}}
2206
\newlabel{fig:Dumptruck-StdDevEnergy}{{C.1f}{172}{Subfigure C C.1f}{subfigure.C.1.6}{}}
2215
\newlabel{fig:Dumptruck-MeanEnergy@cref}{{[subfigure][5][2147483647,3,1]C.1e}{173}}
2216
\newlabel{fig:Dumptruck-StdDevEnergy}{{C.1f}{173}{Subfigure C C.1f}{subfigure.C.1.6}{}}
2207
2217
\newlabel{sub@fig:Dumptruck-StdDevEnergy}{{(f)}{f}{Subfigure C C.1f\relax }{subfigure.C.1.6}{}}
2208
\newlabel{fig:Dumptruck-StdDevEnergy@cref}{{[subfigure][6][2147483647,3,1]C.1f}{172}}
2209
\newlabel{fig:Dumptruck-MeanCurvature}{{C.1g}{172}{Subfigure C C.1g}{subfigure.C.1.7}{}}
2218
\newlabel{fig:Dumptruck-StdDevEnergy@cref}{{[subfigure][6][2147483647,3,1]C.1f}{173}}
2219
\newlabel{fig:Dumptruck-MeanCurvature}{{C.1g}{173}{Subfigure C C.1g}{subfigure.C.1.7}{}}
2210
2220
\newlabel{sub@fig:Dumptruck-MeanCurvature}{{(g)}{g}{Subfigure C C.1g\relax }{subfigure.C.1.7}{}}
2211
\newlabel{fig:Dumptruck-MeanCurvature@cref}{{[subfigure][7][2147483647,3,1]C.1g}{172}}
2212
\newlabel{fig:DumptruckCurvatureFit}{{C.1h}{172}{Subfigure C C.1h}{subfigure.C.1.8}{}}
2221
\newlabel{fig:Dumptruck-MeanCurvature@cref}{{[subfigure][7][2147483647,3,1]C.1g}{173}}
2222
\newlabel{fig:DumptruckCurvatureFit}{{C.1h}{173}{Subfigure C C.1h}{subfigure.C.1.8}{}}
2213
2223
\newlabel{sub@fig:DumptruckCurvatureFit}{{(h)}{h}{Subfigure C C.1h\relax }{subfigure.C.1.8}{}}
2214
\newlabel{fig:DumptruckCurvatureFit@cref}{{[subfigure][8][2147483647,3,1]C.1h}{172}}
2215
\newlabel{fig:Basketball}{{C.1i}{172}{Subfigure C C.1i}{subfigure.C.1.9}{}}
2224
\newlabel{fig:DumptruckCurvatureFit@cref}{{[subfigure][8][2147483647,3,1]C.1h}{173}}
2225
\newlabel{fig:Basketball}{{C.1i}{173}{Subfigure C C.1i}{subfigure.C.1.9}{}}
2216
2226
\newlabel{sub@fig:Basketball}{{(i)}{i}{Subfigure C C.1i\relax }{subfigure.C.1.9}{}}
2217
\newlabel{fig:Basketball@cref}{{[subfigure][9][2147483647,3,1]C.1i}{172}}
2218
\newlabel{fig:Basketball-Noise}{{C.1j}{172}{Subfigure C C.1j}{subfigure.C.1.10}{}}
2227
\newlabel{fig:Basketball@cref}{{[subfigure][9][2147483647,3,1]C.1i}{173}}
2228
\newlabel{fig:Basketball-Noise}{{C.1j}{173}{Subfigure C C.1j}{subfigure.C.1.10}{}}
2219
2229
\newlabel{sub@fig:Basketball-Noise}{{(j)}{j}{Subfigure C C.1j\relax }{subfigure.C.1.10}{}}
2220
\newlabel{fig:Basketball-Noise@cref}{{[subfigure][10][2147483647,3,1]C.1j}{172}}
2221
\newlabel{fig:Basketball-GNA}{{C.1k}{172}{Subfigure C C.1k}{subfigure.C.1.11}{}}
2230
\newlabel{fig:Basketball-Noise@cref}{{[subfigure][10][2147483647,3,1]C.1j}{173}}
2231
\newlabel{fig:Basketball-GNA}{{C.1k}{173}{Subfigure C C.1k}{subfigure.C.1.11}{}}
2222
2232
\newlabel{sub@fig:Basketball-GNA}{{(k)}{k}{Subfigure C C.1k\relax }{subfigure.C.1.11}{}}
2223
\newlabel{fig:Basketball-GNA@cref}{{[subfigure][11][2147483647,3,1]C.1k}{172}}
2224
\newlabel{fig:Basketball-BNA}{{C.1l}{172}{Subfigure C C.1l}{subfigure.C.1.12}{}}
2233
\newlabel{fig:Basketball-GNA@cref}{{[subfigure][11][2147483647,3,1]C.1k}{173}}
2234
\newlabel{fig:Basketball-BNA}{{C.1l}{173}{Subfigure C C.1l}{subfigure.C.1.12}{}}
2225
2235
\newlabel{sub@fig:Basketball-BNA}{{(l)}{l}{Subfigure C C.1l\relax }{subfigure.C.1.12}{}}
2226
\newlabel{fig:Basketball-BNA@cref}{{[subfigure][12][2147483647,3,1]C.1l}{172}}
2227
\newlabel{fig:Basketball-MeanEnergy}{{C.1m}{172}{Subfigure C C.1m}{subfigure.C.1.13}{}}
2236
\newlabel{fig:Basketball-BNA@cref}{{[subfigure][12][2147483647,3,1]C.1l}{173}}
2237
\newlabel{fig:Basketball-MeanEnergy}{{C.1m}{173}{Subfigure C C.1m}{subfigure.C.1.13}{}}
2228
2238
\newlabel{sub@fig:Basketball-MeanEnergy}{{(m)}{m}{Subfigure C C.1m\relax }{subfigure.C.1.13}{}}
2229
\newlabel{fig:Basketball-MeanEnergy@cref}{{[subfigure][13][2147483647,3,1]C.1m}{172}}
2230
\newlabel{fig:Basketball-StdDevEnergy}{{C.1n}{172}{Subfigure C C.1n}{subfigure.C.1.14}{}}
2239
\newlabel{fig:Basketball-MeanEnergy@cref}{{[subfigure][13][2147483647,3,1]C.1m}{173}}
2240
\newlabel{fig:Basketball-StdDevEnergy}{{C.1n}{173}{Subfigure C C.1n}{subfigure.C.1.14}{}}
2231
2241
\newlabel{sub@fig:Basketball-StdDevEnergy}{{(n)}{n}{Subfigure C C.1n\relax }{subfigure.C.1.14}{}}
2232
\newlabel{fig:Basketball-StdDevEnergy@cref}{{[subfigure][14][2147483647,3,1]C.1n}{172}}
2233
\newlabel{fig:Basketball-MeanCurvature}{{C.1o}{172}{Subfigure C C.1o}{subfigure.C.1.15}{}}
2242
\newlabel{fig:Basketball-StdDevEnergy@cref}{{[subfigure][14][2147483647,3,1]C.1n}{173}}
2243
\newlabel{fig:Basketball-MeanCurvature}{{C.1o}{173}{Subfigure C C.1o}{subfigure.C.1.15}{}}
2234
2244
\newlabel{sub@fig:Basketball-MeanCurvature}{{(o)}{o}{Subfigure C C.1o\relax }{subfigure.C.1.15}{}}
2235
\newlabel{fig:Basketball-MeanCurvature@cref}{{[subfigure][15][2147483647,3,1]C.1o}{172}}
2236
\newlabel{fig:BasketballCurvatureFit}{{C.1p}{172}{Subfigure C C.1p}{subfigure.C.1.16}{}}
2245
\newlabel{fig:Basketball-MeanCurvature@cref}{{[subfigure][15][2147483647,3,1]C.1o}{173}}
2246
\newlabel{fig:BasketballCurvatureFit}{{C.1p}{173}{Subfigure C C.1p}{subfigure.C.1.16}{}}
2237
2247
\newlabel{sub@fig:BasketballCurvatureFit}{{(p)}{p}{Subfigure C C.1p\relax }{subfigure.C.1.16}{}}
2238
\newlabel{fig:BasketballCurvatureFit@cref}{{[subfigure][16][2147483647,3,1]C.1p}{172}}
2239
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {input image}}}{172}{subfigure.1.1}}
2240
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\phi _0$}}}{172}{subfigure.1.2}}
2241
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{172}{subfigure.1.3}}
2242
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{172}{subfigure.1.4}}
2243
\@writefile{lof}{\contentsline {subfigure}{\numberline{(e)}{\ignorespaces {}}}{172}{subfigure.1.5}}
2244
\@writefile{lof}{\contentsline {subfigure}{\numberline{(f)}{\ignorespaces {}}}{172}{subfigure.1.6}}
2245
\@writefile{lof}{\contentsline {subfigure}{\numberline{(g)}{\ignorespaces {}}}{172}{subfigure.1.7}}
2246
\@writefile{lof}{\contentsline {subfigure}{\numberline{(h)}{\ignorespaces {}}}{172}{subfigure.1.8}}
2247
\@writefile{lof}{\contentsline {subfigure}{\numberline{(i)}{\ignorespaces {input image}}}{172}{subfigure.1.9}}
2248
\@writefile{lof}{\contentsline {subfigure}{\numberline{(j)}{\ignorespaces {$\phi _0$}}}{172}{subfigure.1.10}}
2249
\@writefile{lof}{\contentsline {subfigure}{\numberline{(k)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{172}{subfigure.1.11}}
2250
\@writefile{lof}{\contentsline {subfigure}{\numberline{(l)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{172}{subfigure.1.12}}
2251
\@writefile{lof}{\contentsline {subfigure}{\numberline{(m)}{\ignorespaces {}}}{172}{subfigure.1.13}}
2252
\@writefile{lof}{\contentsline {subfigure}{\numberline{(n)}{\ignorespaces {}}}{172}{subfigure.1.14}}
2253
\@writefile{lof}{\contentsline {subfigure}{\numberline{(o)}{\ignorespaces {}}}{172}{subfigure.1.15}}
2254
\@writefile{lof}{\contentsline {subfigure}{\numberline{(p)}{\ignorespaces {}}}{172}{subfigure.1.16}}
2255
\newlabel{fig:MiniCooper}{{C.1a}{173}{Subfigure C C.1a}{subfigure.C.1.1}{}}
2248
\newlabel{fig:BasketballCurvatureFit@cref}{{[subfigure][16][2147483647,3,1]C.1p}{173}}
2249
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {input image}}}{173}{subfigure.1.1}}
2250
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\phi _0$}}}{173}{subfigure.1.2}}
2251
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{173}{subfigure.1.3}}
2252
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{173}{subfigure.1.4}}
2253
\@writefile{lof}{\contentsline {subfigure}{\numberline{(e)}{\ignorespaces {}}}{173}{subfigure.1.5}}
2254
\@writefile{lof}{\contentsline {subfigure}{\numberline{(f)}{\ignorespaces {}}}{173}{subfigure.1.6}}
2255
\@writefile{lof}{\contentsline {subfigure}{\numberline{(g)}{\ignorespaces {}}}{173}{subfigure.1.7}}
2256
\@writefile{lof}{\contentsline {subfigure}{\numberline{(h)}{\ignorespaces {}}}{173}{subfigure.1.8}}
2257
\@writefile{lof}{\contentsline {subfigure}{\numberline{(i)}{\ignorespaces {input image}}}{173}{subfigure.1.9}}
2258
\@writefile{lof}{\contentsline {subfigure}{\numberline{(j)}{\ignorespaces {$\phi _0$}}}{173}{subfigure.1.10}}
2259
\@writefile{lof}{\contentsline {subfigure}{\numberline{(k)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{173}{subfigure.1.11}}
2260
\@writefile{lof}{\contentsline {subfigure}{\numberline{(l)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{173}{subfigure.1.12}}
2261
\@writefile{lof}{\contentsline {subfigure}{\numberline{(m)}{\ignorespaces {}}}{173}{subfigure.1.13}}
2262
\@writefile{lof}{\contentsline {subfigure}{\numberline{(n)}{\ignorespaces {}}}{173}{subfigure.1.14}}
2263
\@writefile{lof}{\contentsline {subfigure}{\numberline{(o)}{\ignorespaces {}}}{173}{subfigure.1.15}}
2264
\@writefile{lof}{\contentsline {subfigure}{\numberline{(p)}{\ignorespaces {}}}{173}{subfigure.1.16}}
2265
\newlabel{fig:MiniCooper}{{C.1a}{174}{Subfigure C C.1a}{subfigure.C.1.1}{}}
2256
2266
\newlabel{sub@fig:MiniCooper}{{(a)}{a}{Subfigure C C.1a\relax }{subfigure.C.1.1}{}}
2257
\newlabel{fig:MiniCooper@cref}{{[subfigure][1][2147483647,3,1]C.1a}{173}}
2258
\newlabel{fig:MiniCooper-Noise}{{C.1b}{173}{Subfigure C C.1b}{subfigure.C.1.2}{}}
2267
\newlabel{fig:MiniCooper@cref}{{[subfigure][1][2147483647,3,1]C.1a}{174}}
2268
\newlabel{fig:MiniCooper-Noise}{{C.1b}{174}{Subfigure C C.1b}{subfigure.C.1.2}{}}
2259
2269
\newlabel{sub@fig:MiniCooper-Noise}{{(b)}{b}{Subfigure C C.1b\relax }{subfigure.C.1.2}{}}
2260
\newlabel{fig:MiniCooper-Noise@cref}{{[subfigure][2][2147483647,3,1]C.1b}{173}}
2261
\newlabel{fig:MiniCooper-GNA}{{C.1c}{173}{Subfigure C C.1c}{subfigure.C.1.3}{}}
2270
\newlabel{fig:MiniCooper-Noise@cref}{{[subfigure][2][2147483647,3,1]C.1b}{174}}
2271
\newlabel{fig:MiniCooper-GNA}{{C.1c}{174}{Subfigure C C.1c}{subfigure.C.1.3}{}}
2262
2272
\newlabel{sub@fig:MiniCooper-GNA}{{(c)}{c}{Subfigure C C.1c\relax }{subfigure.C.1.3}{}}
2263
\newlabel{fig:MiniCooper-GNA@cref}{{[subfigure][3][2147483647,3,1]C.1c}{173}}
2264
\newlabel{fig:MiniCooper-BNA}{{C.1d}{173}{Subfigure C C.1d}{subfigure.C.1.4}{}}
2273
\newlabel{fig:MiniCooper-GNA@cref}{{[subfigure][3][2147483647,3,1]C.1c}{174}}
2274
\newlabel{fig:MiniCooper-BNA}{{C.1d}{174}{Subfigure C C.1d}{subfigure.C.1.4}{}}
2265
2275
\newlabel{sub@fig:MiniCooper-BNA}{{(d)}{d}{Subfigure C C.1d\relax }{subfigure.C.1.4}{}}
2266
\newlabel{fig:MiniCooper-BNA@cref}{{[subfigure][4][2147483647,3,1]C.1d}{173}}
2267
\newlabel{fig:MiniCooper-MeanEnergy}{{C.1e}{173}{Subfigure C C.1e}{subfigure.C.1.5}{}}
2276
\newlabel{fig:MiniCooper-BNA@cref}{{[subfigure][4][2147483647,3,1]C.1d}{174}}
2277
\newlabel{fig:MiniCooper-MeanEnergy}{{C.1e}{174}{Subfigure C C.1e}{subfigure.C.1.5}{}}
2268
2278
\newlabel{sub@fig:MiniCooper-MeanEnergy}{{(e)}{e}{Subfigure C C.1e\relax }{subfigure.C.1.5}{}}
2269
\newlabel{fig:MiniCooper-MeanEnergy@cref}{{[subfigure][5][2147483647,3,1]C.1e}{173}}
2270
\newlabel{fig:MiniCooper-StdDevEnergy}{{C.1f}{173}{Subfigure C C.1f}{subfigure.C.1.6}{}}
2279
\newlabel{fig:MiniCooper-MeanEnergy@cref}{{[subfigure][5][2147483647,3,1]C.1e}{174}}
2280
\newlabel{fig:MiniCooper-StdDevEnergy}{{C.1f}{174}{Subfigure C C.1f}{subfigure.C.1.6}{}}
2271
2281
\newlabel{sub@fig:MiniCooper-StdDevEnergy}{{(f)}{f}{Subfigure C C.1f\relax }{subfigure.C.1.6}{}}
2272
\newlabel{fig:MiniCooper-StdDevEnergy@cref}{{[subfigure][6][2147483647,3,1]C.1f}{173}}
2273
\newlabel{fig:MiniCooper-MeanCurvature}{{C.1g}{173}{Subfigure C C.1g}{subfigure.C.1.7}{}}
2282
\newlabel{fig:MiniCooper-StdDevEnergy@cref}{{[subfigure][6][2147483647,3,1]C.1f}{174}}
2283
\newlabel{fig:MiniCooper-MeanCurvature}{{C.1g}{174}{Subfigure C C.1g}{subfigure.C.1.7}{}}
2274
2284
\newlabel{sub@fig:MiniCooper-MeanCurvature}{{(g)}{g}{Subfigure C C.1g\relax }{subfigure.C.1.7}{}}
2275
\newlabel{fig:MiniCooper-MeanCurvature@cref}{{[subfigure][7][2147483647,3,1]C.1g}{173}}
2276
\newlabel{fig:MiniCooperCurvatureFit}{{C.1h}{173}{Subfigure C C.1h}{subfigure.C.1.8}{}}
2285
\newlabel{fig:MiniCooper-MeanCurvature@cref}{{[subfigure][7][2147483647,3,1]C.1g}{174}}
2286
\newlabel{fig:MiniCooperCurvatureFit}{{C.1h}{174}{Subfigure C C.1h}{subfigure.C.1.8}{}}
2277
2287
\newlabel{sub@fig:MiniCooperCurvatureFit}{{(h)}{h}{Subfigure C C.1h\relax }{subfigure.C.1.8}{}}
2278
\newlabel{fig:MiniCooperCurvatureFit@cref}{{[subfigure][8][2147483647,3,1]C.1h}{173}}
2279
\newlabel{fig:Beanbags}{{C.1i}{173}{Subfigure C C.1i}{subfigure.C.1.9}{}}
2288
\newlabel{fig:MiniCooperCurvatureFit@cref}{{[subfigure][8][2147483647,3,1]C.1h}{174}}
2289
\newlabel{fig:Beanbags}{{C.1i}{174}{Subfigure C C.1i}{subfigure.C.1.9}{}}
2280
2290
\newlabel{sub@fig:Beanbags}{{(i)}{i}{Subfigure C C.1i\relax }{subfigure.C.1.9}{}}
2281
\newlabel{fig:Beanbags@cref}{{[subfigure][9][2147483647,3,1]C.1i}{173}}
2282
\newlabel{fig:Beanbags-Noise}{{C.1j}{173}{Subfigure C C.1j}{subfigure.C.1.10}{}}
2291
\newlabel{fig:Beanbags@cref}{{[subfigure][9][2147483647,3,1]C.1i}{174}}
2292
\newlabel{fig:Beanbags-Noise}{{C.1j}{174}{Subfigure C C.1j}{subfigure.C.1.10}{}}
2283
2293
\newlabel{sub@fig:Beanbags-Noise}{{(j)}{j}{Subfigure C C.1j\relax }{subfigure.C.1.10}{}}
2284
\newlabel{fig:Beanbags-Noise@cref}{{[subfigure][10][2147483647,3,1]C.1j}{173}}
2285
\newlabel{fig:Beanbags-GNA}{{C.1k}{173}{Subfigure C C.1k}{subfigure.C.1.11}{}}
2294
\newlabel{fig:Beanbags-Noise@cref}{{[subfigure][10][2147483647,3,1]C.1j}{174}}
2295
\newlabel{fig:Beanbags-GNA}{{C.1k}{174}{Subfigure C C.1k}{subfigure.C.1.11}{}}
2286
2296
\newlabel{sub@fig:Beanbags-GNA}{{(k)}{k}{Subfigure C C.1k\relax }{subfigure.C.1.11}{}}
2287
\newlabel{fig:Beanbags-GNA@cref}{{[subfigure][11][2147483647,3,1]C.1k}{173}}
2288
\newlabel{fig:Beanbags-BNA}{{C.1l}{173}{Subfigure C C.1l}{subfigure.C.1.12}{}}
2297
\newlabel{fig:Beanbags-GNA@cref}{{[subfigure][11][2147483647,3,1]C.1k}{174}}
2298
\newlabel{fig:Beanbags-BNA}{{C.1l}{174}{Subfigure C C.1l}{subfigure.C.1.12}{}}
2289
2299
\newlabel{sub@fig:Beanbags-BNA}{{(l)}{l}{Subfigure C C.1l\relax }{subfigure.C.1.12}{}}
2290
\newlabel{fig:Beanbags-BNA@cref}{{[subfigure][12][2147483647,3,1]C.1l}{173}}
2291
\newlabel{fig:Beanbags-MeanEnergy}{{C.1m}{173}{Subfigure C C.1m}{subfigure.C.1.13}{}}
2300
\newlabel{fig:Beanbags-BNA@cref}{{[subfigure][12][2147483647,3,1]C.1l}{174}}
2301
\newlabel{fig:Beanbags-MeanEnergy}{{C.1m}{174}{Subfigure C C.1m}{subfigure.C.1.13}{}}
2292
2302
\newlabel{sub@fig:Beanbags-MeanEnergy}{{(m)}{m}{Subfigure C C.1m\relax }{subfigure.C.1.13}{}}
2293
\newlabel{fig:Beanbags-MeanEnergy@cref}{{[subfigure][13][2147483647,3,1]C.1m}{173}}
2294
\newlabel{fig:Beanbags-StdDevEnergy}{{C.1n}{173}{Subfigure C C.1n}{subfigure.C.1.14}{}}
2303
\newlabel{fig:Beanbags-MeanEnergy@cref}{{[subfigure][13][2147483647,3,1]C.1m}{174}}
2304
\newlabel{fig:Beanbags-StdDevEnergy}{{C.1n}{174}{Subfigure C C.1n}{subfigure.C.1.14}{}}
2295
2305
\newlabel{sub@fig:Beanbags-StdDevEnergy}{{(n)}{n}{Subfigure C C.1n\relax }{subfigure.C.1.14}{}}
2296
\newlabel{fig:Beanbags-StdDevEnergy@cref}{{[subfigure][14][2147483647,3,1]C.1n}{173}}
2297
\newlabel{fig:Beanbags-MeanCurvature}{{C.1o}{173}{Subfigure C C.1o}{subfigure.C.1.15}{}}
2306
\newlabel{fig:Beanbags-StdDevEnergy@cref}{{[subfigure][14][2147483647,3,1]C.1n}{174}}
2307
\newlabel{fig:Beanbags-MeanCurvature}{{C.1o}{174}{Subfigure C C.1o}{subfigure.C.1.15}{}}
2298
2308
\newlabel{sub@fig:Beanbags-MeanCurvature}{{(o)}{o}{Subfigure C C.1o\relax }{subfigure.C.1.15}{}}
2299
\newlabel{fig:Beanbags-MeanCurvature@cref}{{[subfigure][15][2147483647,3,1]C.1o}{173}}
2300
\newlabel{fig:BeanbagsCurvatureFit}{{C.1p}{173}{Subfigure C C.1p}{subfigure.C.1.16}{}}
2309
\newlabel{fig:Beanbags-MeanCurvature@cref}{{[subfigure][15][2147483647,3,1]C.1o}{174}}
2310
\newlabel{fig:BeanbagsCurvatureFit}{{C.1p}{174}{Subfigure C C.1p}{subfigure.C.1.16}{}}
2301
2311
\newlabel{sub@fig:BeanbagsCurvatureFit}{{(p)}{p}{Subfigure C C.1p\relax }{subfigure.C.1.16}{}}
2302
\newlabel{fig:BeanbagsCurvatureFit@cref}{{[subfigure][16][2147483647,3,1]C.1p}{173}}
2303
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {input image}}}{173}{subfigure.1.1}}
2304
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\phi _0$}}}{173}{subfigure.1.2}}
2305
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{173}{subfigure.1.3}}
2306
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{173}{subfigure.1.4}}
2307
\@writefile{lof}{\contentsline {subfigure}{\numberline{(e)}{\ignorespaces {}}}{173}{subfigure.1.5}}
2308
\@writefile{lof}{\contentsline {subfigure}{\numberline{(f)}{\ignorespaces {}}}{173}{subfigure.1.6}}
2309
\@writefile{lof}{\contentsline {subfigure}{\numberline{(g)}{\ignorespaces {}}}{173}{subfigure.1.7}}
2310
\@writefile{lof}{\contentsline {subfigure}{\numberline{(h)}{\ignorespaces {}}}{173}{subfigure.1.8}}
2311
\@writefile{lof}{\contentsline {subfigure}{\numberline{(i)}{\ignorespaces {input image}}}{173}{subfigure.1.9}}
2312
\@writefile{lof}{\contentsline {subfigure}{\numberline{(j)}{\ignorespaces {$\phi _0$}}}{173}{subfigure.1.10}}
2313
\@writefile{lof}{\contentsline {subfigure}{\numberline{(k)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{173}{subfigure.1.11}}
2314
\@writefile{lof}{\contentsline {subfigure}{\numberline{(l)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{173}{subfigure.1.12}}
2315
\@writefile{lof}{\contentsline {subfigure}{\numberline{(m)}{\ignorespaces {}}}{173}{subfigure.1.13}}
2316
\@writefile{lof}{\contentsline {subfigure}{\numberline{(n)}{\ignorespaces {}}}{173}{subfigure.1.14}}
2317
\@writefile{lof}{\contentsline {subfigure}{\numberline{(o)}{\ignorespaces {}}}{173}{subfigure.1.15}}
2318
\@writefile{lof}{\contentsline {subfigure}{\numberline{(p)}{\ignorespaces {}}}{173}{subfigure.1.16}}
2319
\newlabel{fig:Urban}{{C.1a}{174}{Subfigure C C.1a}{subfigure.C.1.1}{}}
2312
\newlabel{fig:BeanbagsCurvatureFit@cref}{{[subfigure][16][2147483647,3,1]C.1p}{174}}
2313
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {input image}}}{174}{subfigure.1.1}}
2314
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\phi _0$}}}{174}{subfigure.1.2}}
2315
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{174}{subfigure.1.3}}
2316
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{174}{subfigure.1.4}}
2317
\@writefile{lof}{\contentsline {subfigure}{\numberline{(e)}{\ignorespaces {}}}{174}{subfigure.1.5}}
2318
\@writefile{lof}{\contentsline {subfigure}{\numberline{(f)}{\ignorespaces {}}}{174}{subfigure.1.6}}
2319
\@writefile{lof}{\contentsline {subfigure}{\numberline{(g)}{\ignorespaces {}}}{174}{subfigure.1.7}}
2320
\@writefile{lof}{\contentsline {subfigure}{\numberline{(h)}{\ignorespaces {}}}{174}{subfigure.1.8}}
2321
\@writefile{lof}{\contentsline {subfigure}{\numberline{(i)}{\ignorespaces {input image}}}{174}{subfigure.1.9}}
2322
\@writefile{lof}{\contentsline {subfigure}{\numberline{(j)}{\ignorespaces {$\phi _0$}}}{174}{subfigure.1.10}}
2323
\@writefile{lof}{\contentsline {subfigure}{\numberline{(k)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{174}{subfigure.1.11}}
2324
\@writefile{lof}{\contentsline {subfigure}{\numberline{(l)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{174}{subfigure.1.12}}
2325
\@writefile{lof}{\contentsline {subfigure}{\numberline{(m)}{\ignorespaces {}}}{174}{subfigure.1.13}}
2326
\@writefile{lof}{\contentsline {subfigure}{\numberline{(n)}{\ignorespaces {}}}{174}{subfigure.1.14}}
2327
\@writefile{lof}{\contentsline {subfigure}{\numberline{(o)}{\ignorespaces {}}}{174}{subfigure.1.15}}
2328
\@writefile{lof}{\contentsline {subfigure}{\numberline{(p)}{\ignorespaces {}}}{174}{subfigure.1.16}}
2329
\newlabel{fig:Urban}{{C.1a}{175}{Subfigure C C.1a}{subfigure.C.1.1}{}}
2320
2330
\newlabel{sub@fig:Urban}{{(a)}{a}{Subfigure C C.1a\relax }{subfigure.C.1.1}{}}
2321
\newlabel{fig:Urban@cref}{{[subfigure][1][2147483647,3,1]C.1a}{174}}
2322
\newlabel{fig:Urban-Noise}{{C.1b}{174}{Subfigure C C.1b}{subfigure.C.1.2}{}}
2331
\newlabel{fig:Urban@cref}{{[subfigure][1][2147483647,3,1]C.1a}{175}}
2332
\newlabel{fig:Urban-Noise}{{C.1b}{175}{Subfigure C C.1b}{subfigure.C.1.2}{}}
2323
2333
\newlabel{sub@fig:Urban-Noise}{{(b)}{b}{Subfigure C C.1b\relax }{subfigure.C.1.2}{}}
2324
\newlabel{fig:Urban-Noise@cref}{{[subfigure][2][2147483647,3,1]C.1b}{174}}
2325
\newlabel{fig:Urban-GNA}{{C.1c}{174}{Subfigure C C.1c}{subfigure.C.1.3}{}}
2334
\newlabel{fig:Urban-Noise@cref}{{[subfigure][2][2147483647,3,1]C.1b}{175}}
2335
\newlabel{fig:Urban-GNA}{{C.1c}{175}{Subfigure C C.1c}{subfigure.C.1.3}{}}
2326
2336
\newlabel{sub@fig:Urban-GNA}{{(c)}{c}{Subfigure C C.1c\relax }{subfigure.C.1.3}{}}
2327
\newlabel{fig:Urban-GNA@cref}{{[subfigure][3][2147483647,3,1]C.1c}{174}}
2328
\newlabel{fig:Urban-BNA}{{C.1d}{174}{Subfigure C C.1d}{subfigure.C.1.4}{}}
2337
\newlabel{fig:Urban-GNA@cref}{{[subfigure][3][2147483647,3,1]C.1c}{175}}
2338
\newlabel{fig:Urban-BNA}{{C.1d}{175}{Subfigure C C.1d}{subfigure.C.1.4}{}}
2329
2339
\newlabel{sub@fig:Urban-BNA}{{(d)}{d}{Subfigure C C.1d\relax }{subfigure.C.1.4}{}}
2330
\newlabel{fig:Urban-BNA@cref}{{[subfigure][4][2147483647,3,1]C.1d}{174}}
2331
\newlabel{fig:Urban-MeanEnergy}{{C.1e}{174}{Subfigure C C.1e}{subfigure.C.1.5}{}}
2340
\newlabel{fig:Urban-BNA@cref}{{[subfigure][4][2147483647,3,1]C.1d}{175}}
2341
\newlabel{fig:Urban-MeanEnergy}{{C.1e}{175}{Subfigure C C.1e}{subfigure.C.1.5}{}}
2332
2342
\newlabel{sub@fig:Urban-MeanEnergy}{{(e)}{e}{Subfigure C C.1e\relax }{subfigure.C.1.5}{}}
2333
\newlabel{fig:Urban-MeanEnergy@cref}{{[subfigure][5][2147483647,3,1]C.1e}{174}}
2334
\newlabel{fig:Urban-StdDevEnergy}{{C.1f}{174}{Subfigure C C.1f}{subfigure.C.1.6}{}}
2343
\newlabel{fig:Urban-MeanEnergy@cref}{{[subfigure][5][2147483647,3,1]C.1e}{175}}
2344
\newlabel{fig:Urban-StdDevEnergy}{{C.1f}{175}{Subfigure C C.1f}{subfigure.C.1.6}{}}
2335
2345
\newlabel{sub@fig:Urban-StdDevEnergy}{{(f)}{f}{Subfigure C C.1f\relax }{subfigure.C.1.6}{}}
2336
\newlabel{fig:Urban-StdDevEnergy@cref}{{[subfigure][6][2147483647,3,1]C.1f}{174}}
2337
\newlabel{fig:Urban-MeanCurvature}{{C.1g}{174}{Subfigure C C.1g}{subfigure.C.1.7}{}}
2346
\newlabel{fig:Urban-StdDevEnergy@cref}{{[subfigure][6][2147483647,3,1]C.1f}{175}}
2347
\newlabel{fig:Urban-MeanCurvature}{{C.1g}{175}{Subfigure C C.1g}{subfigure.C.1.7}{}}
2338
2348
\newlabel{sub@fig:Urban-MeanCurvature}{{(g)}{g}{Subfigure C C.1g\relax }{subfigure.C.1.7}{}}
2339
\newlabel{fig:Urban-MeanCurvature@cref}{{[subfigure][7][2147483647,3,1]C.1g}{174}}
2340
\newlabel{fig:UrbanCurvatureFit}{{C.1h}{174}{Subfigure C C.1h}{subfigure.C.1.8}{}}
2349
\newlabel{fig:Urban-MeanCurvature@cref}{{[subfigure][7][2147483647,3,1]C.1g}{175}}
2350
\newlabel{fig:UrbanCurvatureFit}{{C.1h}{175}{Subfigure C C.1h}{subfigure.C.1.8}{}}
2341
2351
\newlabel{sub@fig:UrbanCurvatureFit}{{(h)}{h}{Subfigure C C.1h\relax }{subfigure.C.1.8}{}}
2342
\newlabel{fig:UrbanCurvatureFit@cref}{{[subfigure][8][2147483647,3,1]C.1h}{174}}
2343
\newlabel{fig:Schefflera}{{C.1i}{174}{Subfigure C C.1i}{subfigure.C.1.9}{}}
2352
\newlabel{fig:UrbanCurvatureFit@cref}{{[subfigure][8][2147483647,3,1]C.1h}{175}}
2353
\newlabel{fig:Schefflera}{{C.1i}{175}{Subfigure C C.1i}{subfigure.C.1.9}{}}
2344
2354
\newlabel{sub@fig:Schefflera}{{(i)}{i}{Subfigure C C.1i\relax }{subfigure.C.1.9}{}}
2345
\newlabel{fig:Schefflera@cref}{{[subfigure][9][2147483647,3,1]C.1i}{174}}
2346
\newlabel{fig:Schefflera-Noise}{{C.1j}{174}{Subfigure C C.1j}{subfigure.C.1.10}{}}
2355
\newlabel{fig:Schefflera@cref}{{[subfigure][9][2147483647,3,1]C.1i}{175}}
2356
\newlabel{fig:Schefflera-Noise}{{C.1j}{175}{Subfigure C C.1j}{subfigure.C.1.10}{}}
2347
2357
\newlabel{sub@fig:Schefflera-Noise}{{(j)}{j}{Subfigure C C.1j\relax }{subfigure.C.1.10}{}}
2348
\newlabel{fig:Schefflera-Noise@cref}{{[subfigure][10][2147483647,3,1]C.1j}{174}}
2349
\newlabel{fig:Schefflera-GNA}{{C.1k}{174}{Subfigure C C.1k}{subfigure.C.1.11}{}}
2358
\newlabel{fig:Schefflera-Noise@cref}{{[subfigure][10][2147483647,3,1]C.1j}{175}}
2359
\newlabel{fig:Schefflera-GNA}{{C.1k}{175}{Subfigure C C.1k}{subfigure.C.1.11}{}}
2350
2360
\newlabel{sub@fig:Schefflera-GNA}{{(k)}{k}{Subfigure C C.1k\relax }{subfigure.C.1.11}{}}
2351
\newlabel{fig:Schefflera-GNA@cref}{{[subfigure][11][2147483647,3,1]C.1k}{174}}
2352
\newlabel{fig:Schefflera-BNA}{{C.1l}{174}{Subfigure C C.1l}{subfigure.C.1.12}{}}
2361
\newlabel{fig:Schefflera-GNA@cref}{{[subfigure][11][2147483647,3,1]C.1k}{175}}
2362
\newlabel{fig:Schefflera-BNA}{{C.1l}{175}{Subfigure C C.1l}{subfigure.C.1.12}{}}
2353
2363
\newlabel{sub@fig:Schefflera-BNA}{{(l)}{l}{Subfigure C C.1l\relax }{subfigure.C.1.12}{}}
2354
\newlabel{fig:Schefflera-BNA@cref}{{[subfigure][12][2147483647,3,1]C.1l}{174}}
2355
\newlabel{fig:Schefflera-MeanEnergy}{{C.1m}{174}{Subfigure C C.1m}{subfigure.C.1.13}{}}
2364
\newlabel{fig:Schefflera-BNA@cref}{{[subfigure][12][2147483647,3,1]C.1l}{175}}
2365
\newlabel{fig:Schefflera-MeanEnergy}{{C.1m}{175}{Subfigure C C.1m}{subfigure.C.1.13}{}}
2356
2366
\newlabel{sub@fig:Schefflera-MeanEnergy}{{(m)}{m}{Subfigure C C.1m\relax }{subfigure.C.1.13}{}}
2357
\newlabel{fig:Schefflera-MeanEnergy@cref}{{[subfigure][13][2147483647,3,1]C.1m}{174}}
2358
\newlabel{fig:Schefflera-StdDevEnergy}{{C.1n}{174}{Subfigure C C.1n}{subfigure.C.1.14}{}}
2367
\newlabel{fig:Schefflera-MeanEnergy@cref}{{[subfigure][13][2147483647,3,1]C.1m}{175}}
2368
\newlabel{fig:Schefflera-StdDevEnergy}{{C.1n}{175}{Subfigure C C.1n}{subfigure.C.1.14}{}}
2359
2369
\newlabel{sub@fig:Schefflera-StdDevEnergy}{{(n)}{n}{Subfigure C C.1n\relax }{subfigure.C.1.14}{}}
2360
\newlabel{fig:Schefflera-StdDevEnergy@cref}{{[subfigure][14][2147483647,3,1]C.1n}{174}}
2361
\newlabel{fig:Schefflera-MeanCurvature}{{C.1o}{174}{Subfigure C C.1o}{subfigure.C.1.15}{}}
2370
\newlabel{fig:Schefflera-StdDevEnergy@cref}{{[subfigure][14][2147483647,3,1]C.1n}{175}}
2371
\newlabel{fig:Schefflera-MeanCurvature}{{C.1o}{175}{Subfigure C C.1o}{subfigure.C.1.15}{}}
2362
2372
\newlabel{sub@fig:Schefflera-MeanCurvature}{{(o)}{o}{Subfigure C C.1o\relax }{subfigure.C.1.15}{}}
2363
\newlabel{fig:Schefflera-MeanCurvature@cref}{{[subfigure][15][2147483647,3,1]C.1o}{174}}
2364
\newlabel{fig:ScheffleraCurvatureFit}{{C.1p}{174}{Subfigure C C.1p}{subfigure.C.1.16}{}}
2373
\newlabel{fig:Schefflera-MeanCurvature@cref}{{[subfigure][15][2147483647,3,1]C.1o}{175}}
2374
\newlabel{fig:ScheffleraCurvatureFit}{{C.1p}{175}{Subfigure C C.1p}{subfigure.C.1.16}{}}
2365
2375
\newlabel{sub@fig:ScheffleraCurvatureFit}{{(p)}{p}{Subfigure C C.1p\relax }{subfigure.C.1.16}{}}
2366
\newlabel{fig:ScheffleraCurvatureFit@cref}{{[subfigure][16][2147483647,3,1]C.1p}{174}}
2367
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {input image}}}{174}{subfigure.1.1}}
2368
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\phi _0$}}}{174}{subfigure.1.2}}
2369
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{174}{subfigure.1.3}}
2370
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{174}{subfigure.1.4}}
2371
\@writefile{lof}{\contentsline {subfigure}{\numberline{(e)}{\ignorespaces {}}}{174}{subfigure.1.5}}
2372
\@writefile{lof}{\contentsline {subfigure}{\numberline{(f)}{\ignorespaces {}}}{174}{subfigure.1.6}}
2373
\@writefile{lof}{\contentsline {subfigure}{\numberline{(g)}{\ignorespaces {}}}{174}{subfigure.1.7}}
2374
\@writefile{lof}{\contentsline {subfigure}{\numberline{(h)}{\ignorespaces {}}}{174}{subfigure.1.8}}
2375
\@writefile{lof}{\contentsline {subfigure}{\numberline{(i)}{\ignorespaces {input image}}}{174}{subfigure.1.9}}
2376
\@writefile{lof}{\contentsline {subfigure}{\numberline{(j)}{\ignorespaces {$\phi _0$}}}{174}{subfigure.1.10}}
2377
\@writefile{lof}{\contentsline {subfigure}{\numberline{(k)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{174}{subfigure.1.11}}
2378
\@writefile{lof}{\contentsline {subfigure}{\numberline{(l)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{174}{subfigure.1.12}}
2379
\@writefile{lof}{\contentsline {subfigure}{\numberline{(m)}{\ignorespaces {}}}{174}{subfigure.1.13}}
2380
\@writefile{lof}{\contentsline {subfigure}{\numberline{(n)}{\ignorespaces {}}}{174}{subfigure.1.14}}
2381
\@writefile{lof}{\contentsline {subfigure}{\numberline{(o)}{\ignorespaces {}}}{174}{subfigure.1.15}}
2382
\@writefile{lof}{\contentsline {subfigure}{\numberline{(p)}{\ignorespaces {}}}{174}{subfigure.1.16}}
2383
\newlabel{fig:Mequon}{{C.1a}{175}{Subfigure C C.1a}{subfigure.C.1.1}{}}
2376
\newlabel{fig:ScheffleraCurvatureFit@cref}{{[subfigure][16][2147483647,3,1]C.1p}{175}}
2377
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {input image}}}{175}{subfigure.1.1}}
2378
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\phi _0$}}}{175}{subfigure.1.2}}
2379
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{175}{subfigure.1.3}}
2380
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{175}{subfigure.1.4}}
2381
\@writefile{lof}{\contentsline {subfigure}{\numberline{(e)}{\ignorespaces {}}}{175}{subfigure.1.5}}
2382
\@writefile{lof}{\contentsline {subfigure}{\numberline{(f)}{\ignorespaces {}}}{175}{subfigure.1.6}}
2383
\@writefile{lof}{\contentsline {subfigure}{\numberline{(g)}{\ignorespaces {}}}{175}{subfigure.1.7}}
2384
\@writefile{lof}{\contentsline {subfigure}{\numberline{(h)}{\ignorespaces {}}}{175}{subfigure.1.8}}
2385
\@writefile{lof}{\contentsline {subfigure}{\numberline{(i)}{\ignorespaces {input image}}}{175}{subfigure.1.9}}
2386
\@writefile{lof}{\contentsline {subfigure}{\numberline{(j)}{\ignorespaces {$\phi _0$}}}{175}{subfigure.1.10}}
2387
\@writefile{lof}{\contentsline {subfigure}{\numberline{(k)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{175}{subfigure.1.11}}
2388
\@writefile{lof}{\contentsline {subfigure}{\numberline{(l)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{175}{subfigure.1.12}}
2389
\@writefile{lof}{\contentsline {subfigure}{\numberline{(m)}{\ignorespaces {}}}{175}{subfigure.1.13}}
2390
\@writefile{lof}{\contentsline {subfigure}{\numberline{(n)}{\ignorespaces {}}}{175}{subfigure.1.14}}
2391
\@writefile{lof}{\contentsline {subfigure}{\numberline{(o)}{\ignorespaces {}}}{175}{subfigure.1.15}}
2392
\@writefile{lof}{\contentsline {subfigure}{\numberline{(p)}{\ignorespaces {}}}{175}{subfigure.1.16}}
2393
\newlabel{fig:Mequon}{{C.1a}{176}{Subfigure C C.1a}{subfigure.C.1.1}{}}
2384
2394
\newlabel{sub@fig:Mequon}{{(a)}{a}{Subfigure C C.1a\relax }{subfigure.C.1.1}{}}
2385
\newlabel{fig:Mequon@cref}{{[subfigure][1][2147483647,3,1]C.1a}{175}}
2386
\newlabel{fig:Mequon-Noise}{{C.1b}{175}{Subfigure C C.1b}{subfigure.C.1.2}{}}
2395
\newlabel{fig:Mequon@cref}{{[subfigure][1][2147483647,3,1]C.1a}{176}}
2396
\newlabel{fig:Mequon-Noise}{{C.1b}{176}{Subfigure C C.1b}{subfigure.C.1.2}{}}
2387
2397
\newlabel{sub@fig:Mequon-Noise}{{(b)}{b}{Subfigure C C.1b\relax }{subfigure.C.1.2}{}}
2388
\newlabel{fig:Mequon-Noise@cref}{{[subfigure][2][2147483647,3,1]C.1b}{175}}
2389
\newlabel{fig:Mequon-GNA}{{C.1c}{175}{Subfigure C C.1c}{subfigure.C.1.3}{}}
2398
\newlabel{fig:Mequon-Noise@cref}{{[subfigure][2][2147483647,3,1]C.1b}{176}}
2399
\newlabel{fig:Mequon-GNA}{{C.1c}{176}{Subfigure C C.1c}{subfigure.C.1.3}{}}
2390
2400
\newlabel{sub@fig:Mequon-GNA}{{(c)}{c}{Subfigure C C.1c\relax }{subfigure.C.1.3}{}}
2391
\newlabel{fig:Mequon-GNA@cref}{{[subfigure][3][2147483647,3,1]C.1c}{175}}
2392
\newlabel{fig:Mequon-BNA}{{C.1d}{175}{Subfigure C C.1d}{subfigure.C.1.4}{}}
2401
\newlabel{fig:Mequon-GNA@cref}{{[subfigure][3][2147483647,3,1]C.1c}{176}}
2402
\newlabel{fig:Mequon-BNA}{{C.1d}{176}{Subfigure C C.1d}{subfigure.C.1.4}{}}
2393
2403
\newlabel{sub@fig:Mequon-BNA}{{(d)}{d}{Subfigure C C.1d\relax }{subfigure.C.1.4}{}}
2394
\newlabel{fig:Mequon-BNA@cref}{{[subfigure][4][2147483647,3,1]C.1d}{175}}
2395
\newlabel{fig:Mequon-MeanEnergy}{{C.1e}{175}{Subfigure C C.1e}{subfigure.C.1.5}{}}
2404
\newlabel{fig:Mequon-BNA@cref}{{[subfigure][4][2147483647,3,1]C.1d}{176}}
2405
\newlabel{fig:Mequon-MeanEnergy}{{C.1e}{176}{Subfigure C C.1e}{subfigure.C.1.5}{}}
2396
2406
\newlabel{sub@fig:Mequon-MeanEnergy}{{(e)}{e}{Subfigure C C.1e\relax }{subfigure.C.1.5}{}}
2397
\newlabel{fig:Mequon-MeanEnergy@cref}{{[subfigure][5][2147483647,3,1]C.1e}{175}}
2398
\newlabel{fig:Mequon-StdDevEnergy}{{C.1f}{175}{Subfigure C C.1f}{subfigure.C.1.6}{}}
2407
\newlabel{fig:Mequon-MeanEnergy@cref}{{[subfigure][5][2147483647,3,1]C.1e}{176}}
2408
\newlabel{fig:Mequon-StdDevEnergy}{{C.1f}{176}{Subfigure C C.1f}{subfigure.C.1.6}{}}
2399
2409
\newlabel{sub@fig:Mequon-StdDevEnergy}{{(f)}{f}{Subfigure C C.1f\relax }{subfigure.C.1.6}{}}
2400
\newlabel{fig:Mequon-StdDevEnergy@cref}{{[subfigure][6][2147483647,3,1]C.1f}{175}}
2401
\newlabel{fig:Mequon-MeanCurvature}{{C.1g}{175}{Subfigure C C.1g}{subfigure.C.1.7}{}}
2410
\newlabel{fig:Mequon-StdDevEnergy@cref}{{[subfigure][6][2147483647,3,1]C.1f}{176}}
2411
\newlabel{fig:Mequon-MeanCurvature}{{C.1g}{176}{Subfigure C C.1g}{subfigure.C.1.7}{}}
2402
2412
\newlabel{sub@fig:Mequon-MeanCurvature}{{(g)}{g}{Subfigure C C.1g\relax }{subfigure.C.1.7}{}}
2403
\newlabel{fig:Mequon-MeanCurvature@cref}{{[subfigure][7][2147483647,3,1]C.1g}{175}}
2404
\newlabel{fig:MequonCurvatureFit}{{C.1h}{175}{Subfigure C C.1h}{subfigure.C.1.8}{}}
2413
\newlabel{fig:Mequon-MeanCurvature@cref}{{[subfigure][7][2147483647,3,1]C.1g}{176}}
2414
\newlabel{fig:MequonCurvatureFit}{{C.1h}{176}{Subfigure C C.1h}{subfigure.C.1.8}{}}
2405
2415
\newlabel{sub@fig:MequonCurvatureFit}{{(h)}{h}{Subfigure C C.1h\relax }{subfigure.C.1.8}{}}
2406
\newlabel{fig:MequonCurvatureFit@cref}{{[subfigure][8][2147483647,3,1]C.1h}{175}}
2407
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {input image}}}{175}{subfigure.1.1}}
2408
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\phi _0$}}}{175}{subfigure.1.2}}
2409
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{175}{subfigure.1.3}}
2410
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{175}{subfigure.1.4}}
2411
\@writefile{lof}{\contentsline {subfigure}{\numberline{(e)}{\ignorespaces {}}}{175}{subfigure.1.5}}
2412
\@writefile{lof}{\contentsline {subfigure}{\numberline{(f)}{\ignorespaces {}}}{175}{subfigure.1.6}}
2413
\@writefile{lof}{\contentsline {subfigure}{\numberline{(g)}{\ignorespaces {}}}{175}{subfigure.1.7}}
2414
\@writefile{lof}{\contentsline {subfigure}{\numberline{(h)}{\ignorespaces {}}}{175}{subfigure.1.8}}
2415
\@writefile{toc}{\contentsline {chapter}{\numberline {D}Multimodal Optical Flow}{176}{appendix.D}}
2416
\newlabel{fig:MequonCurvatureFit@cref}{{[subfigure][8][2147483647,3,1]C.1h}{176}}
2417
\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {input image}}}{176}{subfigure.1.1}}
2418
\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$\phi _0$}}}{176}{subfigure.1.2}}
2419
\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$\phi ^\star _{ELAA}$}}}{176}{subfigure.1.3}}
2420
\@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {$\phi ^\star _{BNA}$}}}{176}{subfigure.1.4}}
2421
\@writefile{lof}{\contentsline {subfigure}{\numberline{(e)}{\ignorespaces {}}}{176}{subfigure.1.5}}
2422
\@writefile{lof}{\contentsline {subfigure}{\numberline{(f)}{\ignorespaces {}}}{176}{subfigure.1.6}}
2423
\@writefile{lof}{\contentsline {subfigure}{\numberline{(g)}{\ignorespaces {}}}{176}{subfigure.1.7}}
2424
\@writefile{lof}{\contentsline {subfigure}{\numberline{(h)}{\ignorespaces {}}}{176}{subfigure.1.8}}
2425
\@writefile{toc}{\contentsline {chapter}{\numberline {D}Multimodal Optical Flow}{177}{appendix.D}}
2416
2426
\@writefile{lof}{\addvspace {10\p@ }}
2417
2427
\@writefile{lot}{\addvspace {10\p@ }}
2418
2428
\@writefile{lol}{\addvspace {10\p@ }}
2419
2429
\@writefile{loa}{\addvspace {10\p@ }}
2420
\newlabel{sec:AppOptflow}{{D}{176}{Multimodal Optical Flow}{appendix.D}{}}
2421
\newlabel{sec:AppOptflow@cref}{{[appendix][4][2147483647]D}{176}}
2422
\newlabel{eq:AppNoiseModel}{{D.3}{176}{Multimodal Optical Flow}{equation.D.0.3}{}}
2423
\newlabel{eq:AppNoiseModel@cref}{{[equation][3][2147483647,4]D.3}{176}}
2424
\newlabel{eq:AppLogLikelihood}{{D.4}{176}{Multimodal Optical Flow}{equation.D.0.4}{}}
2425
\newlabel{eq:AppLogLikelihood@cref}{{[equation][4][2147483647,4]D.4}{176}}
2426
\newlabel{eq:AppJointGaussian}{{D.5}{177}{Multimodal Optical Flow}{equation.D.0.5}{}}
2427
\newlabel{eq:AppJointGaussian@cref}{{[equation][5][2147483647,4]D.5}{177}}
2428
\newlabel{eq:AppConditional}{{D.8}{177}{Multimodal Optical Flow}{equation.D.0.8}{}}
2429
\newlabel{eq:AppConditional@cref}{{[equation][8][2147483647,4]D.8}{177}}
2430
\newlabel{eq:AppPostEnergy}{{D.10}{177}{Multimodal Optical Flow}{equation.D.0.10}{}}
2431
\newlabel{eq:AppPostEnergy@cref}{{[equation][10][2147483647,4]D.10}{177}}
2432
\newlabel{eq:AppOptimalHighRes}{{D.11}{177}{Multimodal Optical Flow}{equation.D.0.11}{}}
2433
\newlabel{eq:AppOptimalHighRes@cref}{{[equation][11][2147483647,4]D.11}{177}}
2430
\newlabel{sec:AppOptflow}{{D}{177}{Multimodal Optical Flow}{appendix.D}{}}
2431
\newlabel{sec:AppOptflow@cref}{{[appendix][4][2147483647]D}{177}}
2432
\newlabel{eq:AppNoiseModel}{{D.3}{177}{Multimodal Optical Flow}{equation.D.0.3}{}}
2433
\newlabel{eq:AppNoiseModel@cref}{{[equation][3][2147483647,4]D.3}{177}}
2434
\newlabel{eq:AppLogLikelihood}{{D.4}{177}{Multimodal Optical Flow}{equation.D.0.4}{}}
2435
\newlabel{eq:AppLogLikelihood@cref}{{[equation][4][2147483647,4]D.4}{177}}
2436
\newlabel{eq:AppJointGaussian}{{D.5}{178}{Multimodal Optical Flow}{equation.D.0.5}{}}
2437
\newlabel{eq:AppJointGaussian@cref}{{[equation][5][2147483647,4]D.5}{178}}
2438
\newlabel{eq:AppConditional}{{D.8}{178}{Multimodal Optical Flow}{equation.D.0.8}{}}
2439
\newlabel{eq:AppConditional@cref}{{[equation][8][2147483647,4]D.8}{178}}
2440
\newlabel{eq:AppPostEnergy}{{D.10}{178}{Multimodal Optical Flow}{equation.D.0.10}{}}
2441
\newlabel{eq:AppPostEnergy@cref}{{[equation][10][2147483647,4]D.10}{178}}
2442
\newlabel{eq:AppOptimalHighRes}{{D.11}{178}{Multimodal Optical Flow}{equation.D.0.11}{}}
2443
\newlabel{eq:AppOptimalHighRes@cref}{{[equation][11][2147483647,4]D.11}{178}}
2434
2444
\citation{ZhangSpatialResEnhWavelet}
2435
2445
\bibdata{OpticalFlowPapers}
2436
\newlabel{eq:AppSimMeasure}{{D.14}{178}{Multimodal Optical Flow}{equation.D.0.14}{}}
2437
\newlabel{eq:AppSimMeasure@cref}{{[equation][14][2147483647,4]D.14}{178}}
2438
\@writefile{brf}{\backcite{ZhangSpatialResEnhWavelet}{{178}{D}{equation.D.0.14}}}
2439
\newlabel{eq:AppSimMeasure2}{{D.17}{178}{Multimodal Optical Flow}{equation.D.0.17}{}}
2440
\newlabel{eq:AppSimMeasure2@cref}{{[equation][17][2147483647,4]D.17}{178}}
2441
\newlabel{eq:AppflowDataTerm2}{{D.18}{178}{Multimodal Optical Flow}{equation.D.0.18}{}}
2442
\newlabel{eq:AppflowDataTerm2@cref}{{[equation][18][2147483647,4]D.18}{178}}
2443
\newlabel{eq:AppflowDataTermLocal}{{D.20}{178}{Multimodal Optical Flow}{equation.D.0.20}{}}
2444
\newlabel{eq:AppflowDataTermLocal@cref}{{[equation][20][2147483647,4]D.20}{178}}
2446
\newlabel{eq:AppSimMeasure}{{D.14}{179}{Multimodal Optical Flow}{equation.D.0.14}{}}
2447
\newlabel{eq:AppSimMeasure@cref}{{[equation][14][2147483647,4]D.14}{179}}
2448
\@writefile{brf}{\backcite{ZhangSpatialResEnhWavelet}{{179}{D}{equation.D.0.14}}}
2449
\newlabel{eq:AppSimMeasure2}{{D.17}{179}{Multimodal Optical Flow}{equation.D.0.17}{}}
2450
\newlabel{eq:AppSimMeasure2@cref}{{[equation][17][2147483647,4]D.17}{179}}
2451
\newlabel{eq:AppflowDataTerm2}{{D.18}{179}{Multimodal Optical Flow}{equation.D.0.18}{}}
2452
\newlabel{eq:AppflowDataTerm2@cref}{{[equation][18][2147483647,4]D.18}{179}}
2453
\newlabel{eq:AppflowDataTermLocal}{{D.20}{179}{Multimodal Optical Flow}{equation.D.0.20}{}}
2454
\newlabel{eq:AppflowDataTermLocal@cref}{{[equation][20][2147483647,4]D.20}{179}}
2445
2455
\bibcite{AmbrosioApproxFunctionalTConverg}{{1}{1990}{{Ambrosio and Tortorelli}}{{}}}
2446
2456
\bibcite{arrow1958studies}{{2}{1958}{{Arrow et~al.}}{{Arrow, Hurwicz, Uzawa, and Chenery}}}
2447
2457
\bibcite{Middleburry}{{3}{2011}{{Baker et~al.}}{{Baker, Scharstein, Lewis, Roth, Black, and Szeliski}}}
2453
2463
\bibcite{BlackAnandanRobustOpticalFlow}{{9}{1996}{{Black and Anandan}}{{}}}
2454
2464
\bibcite{bredies2008forward}{{10}{2008}{{Bredies}}{{}}}
2455
2465
\bibcite{BrediesMathemBildverarbeitung}{{11}{2011}{{Bredies and Lorenz}}{{}}}
2456
\@writefile{toc}{\contentsline {chapter}{\nonumberline Bibliography}{179}{appendix*.59}}
2466
\@writefile{toc}{\contentsline {chapter}{\nonumberline Bibliography}{180}{appendix*.60}}
2457
2467
\@writefile{lof}{\addvspace {10\p@ }}
2458
2468
\@writefile{lot}{\addvspace {10\p@ }}
2459
2469
\@writefile{lol}{\addvspace {10\p@ }}
Loggerhead is a web-based interface for Breezy
Version: 2.0.1