~gerald-mwangi/+junk/Thesis

\newlabel{eq:KKTLieAlgebra}{{3.3}{54}{Noether's First Theorem: A Modern Version}{equation.3.0.3}{}}

\newlabel{eq:KKTLieAlgebra@cref}{{[equation][3][3]3.3}{54}}

\@writefile{toc}{\contentsline {section}{\numberline {3.1}The action of $\mathbb  {G}$ on Functionals}{55}{section.3.1}}

\newlabel{sec:GActionFunctionals}{{3.1}{55}{The action of $\mathbb {G}$ on Functionals}{section.3.1}{}}

\newlabel{eq:GOmegaPhiAction}{{3.4}{55}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.4}{}}

\newlabel{eq:GPhiAction}{{3.5}{55}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.5}{}}

\newlabel{eq:GOmegaAction}{{3.6}{55}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.6}{}}

\newlabel{eq:GOmegaPhiAlgebra}{{3.11}{56}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.11}{}}

\newlabel{eq:GOmegaPhiAlgebraDerivation}{{3.13}{56}{The action of $\mathcal {G}^{\Omega \phi }$ on $\functionspace $}{equation.3.1.13}{}}

\newlabel{eq:subdiffHi}{{3.27}{59}{The action of $\mathbb {G}^{\Omega \phi }$ on functionals of $\functionspace $}{equation.3.1.27}{}}

\newlabel{eq:subdiffHi@cref}{{[equation][27][3]3.27}{59}}

\newlabel{eq:flowSubdiffH}{{3.30}{59}{The action of $\mathbb {G}^{\Omega \phi }$ on functionals of $\functionspace $}{equation.3.1.30}{}}

\newlabel{eq:flowSubdiffH@cref}{{[equation][30][3]3.30}{59}}

\newlabel{eq:flowSubdiffE}{{3.32}{60}{The action of $\mathbb {G}^{\Omega \phi }$ on functionals of $\functionspace $}{equation.3.1.32}{}}

\newlabel{eq:flowSubdiffE@cref}{{[equation][32][3]3.32}{60}}

\newlabel{eq:flowEulerLagrange}{{3.33}{60}{The action of $\mathbb {G}^{\Omega \phi }$ on functionals of $\functionspace $}{equation.3.1.33}{}}

\newlabel{eq:flowEulerLagrange@cref}{{[equation][33][3]3.33}{60}}

\newlabel{eq:flowBending}{{3.34}{60}{The action of $\mathbb {G}^{\Omega \phi }$ on functionals of $\functionspace $}{equation.3.1.34}{}}

\newlabel{eq:flowBending@cref}{{[equation][34][3]3.34}{60}}

\newlabel{eq:trivialSymmetry}{{3.35}{60}{The action of $\mathbb {G}^{\Omega \phi }$ on functionals of $\functionspace $}{equation.3.1.35}{}}

\newlabel{eq:trivialSymmetry@cref}{{[equation][35][3]3.35}{60}}

\newlabel{eq:flowSubdiffENoether}{{3.36}{61}{Noether's First Theorem: A Modern Version}{equation.3.2.36}{}}

\newlabel{eq:flowSubdiffENoether@cref}{{[equation][36][3]3.36}{61}}

\newlabel{eq:flowEulerLagrangeNoether}{{3.37}{61}{Noether's First Theorem: A Modern Version}{equation.3.2.37}{}}

\newlabel{eq:flowEulerLagrangeNoether@cref}{{[equation][37][3]3.37}{61}}

\newlabel{eq:flowBendingNoether}{{3.38}{61}{Noether's First Theorem: A Modern Version}{equation.3.2.38}{}}

\newlabel{eq:flowBendingNoether@cref}{{[equation][38][3]3.38}{61}}

\newlabel{eq:flowDiffENoetherV}{{3.39}{61}{Noether's First Theorem: A Modern Version}{equation.3.2.39}{}}

\newlabel{eq:flowDiffENoetherV@cref}{{[equation][39][3]3.39}{61}}

\newlabel{eq:eulerLagrange}{{3.40}{61}{Noether's First Theorem: A Modern Version}{equation.3.2.40}{}}

\newlabel{eq:eulerLagrange@cref}{{[equation][40][3]3.40}{61}}

\newlabel{eq:GOmegaPhiAlgebraAction}{{3.41}{61}{Noether's First Theorem: A Modern Version}{equation.3.2.41}{}}

\newlabel{eq:GOmegaPhiAlgebraAction@cref}{{[equation][41][3]3.41}{61}}

\newlabel{eq:flowDiffENoether}{{3.42}{61}{Noether's First Theorem: A Modern Version}{equation.3.2.42}{}}

\newlabel{eq:flowDiffENoether@cref}{{[equation][42][3]3.42}{61}}

\newlabel{eq:flowDiffENoetherBending}{{3.43}{61}{Noether's First Theorem: A Modern Version}{equation.3.2.43}{}}

\newlabel{eq:flowDiffENoetherBending@cref}{{[equation][43][3]3.43}{61}}

\newlabel{eq:noetherDivergence}{{3.44}{61}{Noether's First Theorem: A Modern Version}{equation.3.2.44}{}}

\newlabel{eq:noetherDivergence@cref}{{[equation][44][3]3.44}{61}}

\newlabel{eq:divergenceFreeVectors}{{3.45}{62}{Noether's First Theorem: A Modern Version}{equation.3.2.45}{}}

\newlabel{eq:divergenceFreeVectors@cref}{{[equation][45][3]3.45}{62}}

\newlabel{eq:flowDiffENoether2}{{3.46}{62}{Noether's First Theorem: A Modern Version}{equation.3.2.46}{}}

\newlabel{eq:flowDiffENoether2@cref}{{[equation][46][3]3.46}{62}}

\newlabel{eq:flowDiffENoetherInvariant}{{3.47}{62}{Noether's First Theorem: A Modern Version}{equation.3.2.47}{}}

\newlabel{eq:flowDiffENoetherInvariant@cref}{{[equation][47][3]3.47}{62}}

\newlabel{eq:noetherMutualIndep}{{3.48}{62}{Noether's First Theorem: A Modern Version}{equation.3.2.48}{}}

\newlabel{eq:noetherMutualIndep@cref}{{[equation][48][3]3.48}{62}}

\newlabel{eq:flowDiffENoetherInvariantBending}{{3.49}{62}{Noether's First Theorem: A Modern Version}{equation.3.2.49}{}}

\newlabel{eq:flowDiffENoetherInvariantBending@cref}{{[equation][49][3]3.49}{62}}

\@writefile{toc}{\contentsline {subsection}{\numberline {3.2.1}Pure spatial Symmetries}{63}{subsection.3.2.1}}

\newlabel{sec:pureSpacialSymmetry}{{3.2.1}{63}{Pure spatial Symmetries}{subsection.3.2.1}{}}

\newlabel{sec:pureSpacialSymmetry@cref}{{[subsection][1][3,2]3.2.1}{63}}

\newlabel{eq:pureSpacialAlgebra}{{3.50}{63}{Pure spatial Algebra $\mathcal {G}^\Omega $}{equation.3.2.50}{}}

\newlabel{eq:pureSpacialAlgebra@cref}{{[equation][50][3]3.50}{63}}

\newlabel{eq:pureSpacialAlgebraQphi}{{3.51}{63}{Pure spatial Symmetries}{equation.3.2.51}{}}

\newlabel{eq:pureSpacialAlgebraQphi@cref}{{[equation][51][3]3.51}{63}}

\newlabel{eq:noetherPureSpacialSymmetryQ}{{3.52}{63}{Pure spatial Symmetries}{equation.3.2.52}{}}

\newlabel{eq:noetherPureSpacialSymmetryQ@cref}{{[equation][52][3]3.52}{63}}

\newlabel{eq:pureSpacialSymmetryCanonMomentum}{{3.53}{63}{Pure spatial Symmetries}{equation.3.2.53}{}}

\newlabel{eq:pureSpacialSymmetryCanonMomentum@cref}{{[equation][53][3]3.53}{63}}

\newlabel{eq:l2NonTrivInv}{{3.54}{64}{Symmetries of the prior $E^{prior}_{L_2}(\nabla \phi )$}{equation.3.2.54}{}}

\newlabel{eq:l2NonTrivInv@cref}{{[equation][54][3]3.54}{64}}

\@writefile{brf}{\backcite{Bigun1987}{{64}{3.2.1}{equation.3.2.55}}}

\@writefile{brf}{\backcite{BigunBook}{{64}{3.2.1}{equation.3.2.55}}}

\@writefile{brf}{\backcite{BigunSymmetryGenStructTensor}{{64}{3.2.1}{equation.3.2.55}}}

\newlabel{eq:priorMinimmizers}{{3.56}{64}{Embedding Geometrical Constraints into Prior Energies}{equation.3.3.56}{}}

\newlabel{eq:priorMinimmizers@cref}{{[equation][56][3]3.56}{64}}

\newlabel{eq:priorMinimizersG}{{3.57}{65}{Embedding Geometrical Constraints into Prior Energies}{equation.3.3.57}{}}

\newlabel{eq:priorMinimizersG@cref}{{[equation][57][3]3.57}{65}}

\newlabel{eq:priorEnergyAlone}{{3.58}{65}{Embedding Geometrical Constraints into Prior Energies}{equation.3.3.58}{}}

\newlabel{eq:priorEnergyAlone@cref}{{[equation][58][3]3.58}{65}}

\newlabel{eq:priorMinimmizersCommBasis}{{3.59}{65}{Embedding Geometrical Constraints into Prior Energies}{equation.3.3.59}{}}

\newlabel{eq:priorMinimmizersCommBasis@cref}{{[equation][59][3]3.59}{65}}

\newlabel{eq:divergenceFreeDualVariable}{{3.60}{65}{Embedding Geometrical Constraints into Prior Energies}{equation.3.3.60}{}}

\newlabel{prop:constanceDualSpace}{{4}{65}{Constance of the Dual Space}{proposition.4}{}}

\newlabel{prop:constanceDualSpace@cref}{{[proposition][4][]4}{65}}

\newlabel{eq:divergenceFreeDualVariable}{{3.62}{65}{Constance of the Dual Space}{equation.3.3.62}{}}

\newlabel{eq:divergenceFreeDualVariable@cref}{{[equation][62][3]3.62}{65}}

\newlabel{eq:levelSetDualVariable3}{{3.63}{66}{Embedding Geometrical Constraints into Prior Energies}{equation.3.3.63}{}}

\newlabel{eq:levelSetDualVariable3@cref}{{[equation][63][3]3.63}{66}}

\newlabel{eq:levelSetVF2}{{3.64}{66}{Level-set}{equation.3.3.64}{}}

\newlabel{eq:levelSetVF2@cref}{{[equation][64][3]3.64}{66}}

\newlabel{eq:levelSetVF3}{{3.65}{66}{Level-set}{equation.3.3.65}{}}

\newlabel{eq:levelSetVF3@cref}{{[equation][65][3]3.65}{66}}

\@writefile{brf}{\backcite{BigunBook}{{66}{3.3}{equation.3.3.65}}}

\@writefile{brf}{\backcite{FerraroTransfInvLieGroup}{{66}{3.3}{equation.3.3.65}}}

\newlabel{eq:levelSetVFCommutativeBasis}{{3.66}{66}{Existence of $\phi ^Z$}{equation.3.3.66}{}}

\newlabel{eq:levelSetVFCommutativeBasis@cref}{{[equation][66][3]3.66}{66}}

\newlabel{eq:translationAlgebra}{{3.70}{67}{Embedding Geometrical Constraints into Prior Energies}{equation.3.3.70}{}}

\newlabel{eq:translationAlgebra@cref}{{[equation][70][3]3.70}{67}}

\newlabel{eq:linLevelSet}{{3.71}{67}{Embedding Geometrical Constraints into Prior Energies}{equation.3.3.71}{}}

\newlabel{eq:linLevelSet@cref}{{[equation][71][3]3.71}{67}}

\@writefile{brf}{\backcite{Bigun1987}{{67}{3.3}{equation.3.3.72}}}

\@writefile{brf}{\backcite{BigunSymmetryGenStructTensor}{{67}{3.3}{equation.3.3.72}}}

\@writefile{brf}{\backcite{BigunBook}{{67}{3.3}{equation.3.3.72}}}

\@writefile{toc}{\contentsline {chapter}{\numberline {4}Linearized Priors}{69}{chapter.4}}

\newlabel{chap:GeometricalPrior}{{4}{69}{Linearized Priors}{chapter.4}{}}

\newlabel{chap:GeometricalPrior@cref}{{[chapter][4][]4}{69}}

\@writefile{toc}{\contentsline {section}{\numberline {4.1}The Linear Structure Tensor}{69}{section.4.1}}

\newlabel{eq:vectorTransAlgebra}{{4.1}{69}{The Linear Structure Tensor}{equation.4.1.1}{}}

\newlabel{eq:vectorTransAlgebra@cref}{{[equation][1][4]4.1}{69}}

\newlabel{eq:linearLevelSet}{{4.3}{69}{The Linear Structure Tensor}{equation.4.1.3}{}}

\newlabel{eq:linearLevelSet@cref}{{[equation][3][4]4.3}{69}}

\@writefile{brf}{\backcite{Bigun1987}{{69}{4.1}{equation.4.1.3}}}

\newlabel{eq:StructTensLeastSquaresEnergy}{{4.4}{69}{The Linear Structure Tensor}{equation.4.1.4}{}}

\newlabel{eq:StructTensLeastSquaresEnergy@cref}{{[equation][4][4]4.4}{69}}

\newlabel{eq:structtensDef}{{4.5}{69}{The Linear Structure Tensor}{equation.4.1.5}{}}

\newlabel{eq:structtensDef@cref}{{[equation][5][4]4.5}{69}}

\newlabel{eq:structtensEigenvalues}{{4.6}{70}{The Linear Structure Tensor}{equation.4.1.6}{}}

\newlabel{eq:structtensEigenvalues@cref}{{[equation][6][4]4.6}{70}}

\newlabel{eq:structtensRot}{{4.7}{70}{The Linear Structure Tensor}{equation.4.1.7}{}}

\newlabel{eq:structtensRot@cref}{{[equation][7][4]4.7}{70}}

\newlabel{eq:commutatorStructens}{{4.9}{70}{The Linear Structure Tensor}{equation.4.1.9}{}}

\newlabel{eq:commutatorStructens@cref}{{[equation][9][4]4.9}{70}}

\newlabel{eq:changeEigenvectors}{{4.10}{70}{The Linear Structure Tensor}{equation.4.1.10}{}}

\newlabel{eq:changeEigenvectors@cref}{{[equation][10][4]4.10}{70}}

\newlabel{eq:changeStructtens}{{4.11}{70}{The Linear Structure Tensor}{equation.4.1.11}{}}

\newlabel{eq:changeStructtens@cref}{{[equation][11][4]4.11}{70}}

\@writefile{toc}{\contentsline {section}{\numberline {4.2}Structure Tensor Based Prior}{71}{section.4.2}}

\newlabel{sec:structureTensPrior}{{4.2}{71}{Structure Tensor Based Prior}{section.4.2}{}}

\newlabel{sec:structureTensPrior@cref}{{[section][2][4]4.2}{71}}

\newlabel{eq:structureTensorRotation}{{4.12}{71}{Structure Tensor Based Prior}{equation.4.2.12}{}}

\newlabel{eq:structureTensorRotation@cref}{{[equation][12][4]4.12}{71}}

\newlabel{eq:structtensPrior}{{4.14}{71}{Structure Tensor Based Prior}{equation.4.2.14}{}}

\newlabel{eq:structtensPrior@cref}{{[equation][14][4]4.14}{71}}

\newlabel{eq:structtensPriorRot}{{4.15}{71}{Structure Tensor Based Prior}{equation.4.2.15}{}}

\newlabel{eq:structtensPriorRot@cref}{{[equation][15][4]4.15}{71}}

\newlabel{eq:traceTransf}{{4.16}{71}{Structure Tensor Based Prior}{equation.4.2.16}{}}

\newlabel{eq:traceTransf@cref}{{[equation][16][4]4.16}{71}}

\newlabel{eq:structtensPriorRotInv}{{4.17}{72}{Structure Tensor Based Prior}{equation.4.2.17}{}}

\newlabel{eq:structtensPriorRotInv@cref}{{[equation][17][4]4.17}{72}}

\@writefile{toc}{\contentsline {section}{\numberline {4.3}Geometrical Optical Flow Model}{72}{section.4.3}}

\newlabel{chap:GeomOptFlowModel}{{4.3}{72}{Geometrical Optical Flow Model}{section.4.3}{}}

\newlabel{chap:GeomOptFlowModel@cref}{{[section][3][4]4.3}{72}}

\@writefile{brf}{\backcite{MaesMutualInformationRegistration}{{72}{4.3}{section.4.3}}}

\@writefile{brf}{\backcite{Roche98CorrelRatio}{{72}{4.3}{section.4.3}}}

\@writefile{brf}{\backcite{RocheUnifyingMaxLikelihoodRegistration}{{72}{4.3}{section.4.3}}}

\@writefile{brf}{\backcite{HardieSpacialImageResEnhancement}{{72}{4.3}{section.4.3}}}

\@writefile{brf}{\backcite{HardieSpacialImageResEnhancement}{{72}{4.3}{section.4.3}}}

\newlabel{fig:multiModalSetupDiffRes}{{4.1a}{73}{Subfigure 4 4.1a}{subfigure.4.1.1}{}}

\newlabel{fig:multiModalSetupDiffRes@cref}{{[subfigure][1][4,1]4.1a}{73}}

\newlabel{fig:multiModalSetupDiffResIvsc}{{4.1b}{73}{Subfigure 4 4.1b}{subfigure.4.1.2}{}}

\newlabel{fig:multiModalSetupDiffResIvsc@cref}{{[subfigure][2][4,1]4.1b}{73}}

\newlabel{fig:multiModalSetupDiffResYtcLow}{{4.1c}{73}{Subfigure 4 4.1c}{subfigure.4.1.3}{}}

\newlabel{fig:multiModalSetupDiffResYtcLow@cref}{{[subfigure][3][4,1]4.1c}{73}}

\@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 }}{73}{figure.caption.21}}

\newlabel{fig:multiModalSetupDiffResSetup}{{4.1}{73}{\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.21}{}}

\newlabel{fig:multiModalSetupDiffResSetup@cref}{{[figure][1][4]4.1}{73}}

\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {Schematic}}}{73}{subfigure.1.1}}

\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {$I$}}}{73}{subfigure.1.2}}

\@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {$y$}}}{73}{subfigure.1.3}}

\@writefile{toc}{\contentsline {section}{\numberline {4.4}Multi-Modal Optical Flow with Differing Resolutions}{73}{section.4.4}}

\newlabel{sec:ImageFusionDisparity}{{4.4}{73}{Multi-Modal Optical Flow with Differing Resolutions}{section.4.4}{}}

\newlabel{sec:ImageFusionDisparity@cref}{{[section][4][4]4.4}{73}}

\@writefile{brf}{\backcite{HardieSpacialImageResEnhancement}{{73}{4.4}{figure.caption.21}}}

\newlabel{eq:warpedIvsc}{{4.18}{73}{Multi-Modal Optical Flow with Differing Resolutions}{equation.4.4.18}{}}

\newlabel{eq:warpedIvsc@cref}{{[equation][18][4]4.18}{73}}

\newlabel{eq:multiResSimMeasure}{{4.19}{73}{Multi-Modal Optical Flow with Differing Resolutions}{equation.4.4.19}{}}

\newlabel{eq:multiResSimMeasure@cref}{{[equation][19][4]4.19}{73}}

\newlabel{eq:YtcToLowMapping}{{4.20}{74}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.20}{}}

\newlabel{eq:YtcToLowMapping@cref}{{[equation][20][4]4.20}{74}}

\newlabel{eq:YtcToIvscMapping}{{4.21}{74}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.21}{}}

\newlabel{eq:YtcToIvscMapping@cref}{{[equation][21][4]4.21}{74}}

\newlabel{eq:condVarYtc}{{4.22}{74}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.22}{}}

\newlabel{eq:condVarYtc@cref}{{[equation][22][4]4.22}{74}}

\newlabel{eq:condMeanYtc}{{4.23}{74}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.23}{}}

\newlabel{eq:condMeanYtc@cref}{{[equation][23][4]4.23}{74}}

\newlabel{eq:YtcToLowMapping2}{{4.24}{74}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.24}{}}

\newlabel{eq:YtcToLowMapping2@cref}{{[equation][24][4]4.24}{74}}

\newlabel{eq:YtcToIvscDMapping}{{4.25}{74}{Computation of the similarity measure $E^{data}_{y,I}$}{equation.4.4.25}{}}

\newlabel{eq:YtcToIvscDMapping@cref}{{[equation][25][4]4.25}{74}}

\newlabel{fig:line02}{{4.2a}{75}{Subfigure 4 4.2a}{subfigure.4.2.1}{}}

\newlabel{fig:line02@cref}{{[subfigure][1][4,2]4.2a}{75}}

\newlabel{fig:line1blurred2}{{4.2b}{75}{Subfigure 4 4.2b}{subfigure.4.2.2}{}}

\newlabel{fig:line1blurred2@cref}{{[subfigure][2][4,2]4.2b}{75}}

\newlabel{fig:line0warpedWithScaleDiff}{{4.2c}{75}{Subfigure 4 4.2c}{subfigure.4.2.3}{}}

\newlabel{fig:line0warpedWithScaleDiff@cref}{{[subfigure][3][4,2]4.2c}{75}}

\newlabel{fig:flowWithScaleDiff}{{4.2d}{75}{Subfigure 4 4.2d}{subfigure.4.2.4}{}}

\newlabel{fig:flowWithScaleDiff@cref}{{[subfigure][4][4,2]4.2d}{75}}

\@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 }}{75}{figure.caption.23}}

\newlabel{fig:scaleDiffProblemWithScale}{{4.2}{75}{\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.23}{}}

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\@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 }}{84}{figure.caption.29}}

\newlabel{fig:synthMultModalRWYCoAl}{{4.7}{84}{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.29}{}}

\newlabel{fig:synthMultModalRWYCoAl@cref}{{[figure][7][4]4.7}{84}}

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\newlabel{sec:synthMultiModalScaleDiff}{{4.7.4}{84}{Estimation of the Scale Difference $\scalediff $}{subsection.4.7.4}{}}

\newlabel{sec:synthMultiModalScaleDiff@cref}{{[subsection][4][4,7]4.7.4}{84}}

\newlabel{eq:synthMultiModalCoAlignedYHigh}{{4.49}{84}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.49}{}}

\newlabel{eq:synthMultiModalCoAlignedYHigh@cref}{{[equation][49][4]4.49}{84}}

\newlabel{eq:synthMultiModalCoAlignedYLow}{{4.50}{84}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.50}{}}

\newlabel{eq:synthMultiModalCoAlignedYLow@cref}{{[equation][50][4]4.50}{84}}

\newlabel{fig:EdataCoAlScale2}{{4.8a}{85}{Subfigure 4 4.8a}{subfigure.4.8.1}{}}

\newlabel{fig:EdataCoAlScale2@cref}{{[subfigure][1][4,8]4.8a}{85}}

\newlabel{fig:EdataCoAlScale4}{{4.8b}{85}{Subfigure 4 4.8b}{subfigure.4.8.2}{}}

\newlabel{fig:EdataCoAlScale4@cref}{{[subfigure][2][4,8]4.8b}{85}}

\@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 }}{85}{figure.caption.30}}

\newlabel{fig:EdataCoAlScales}{{4.8}{85}{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.30}{}}

\newlabel{fig:EdataCoAlScales@cref}{{[figure][8][4]4.8}{85}}

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\newlabel{eq:flowDataTerm3}{{4.51}{85}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.51}{}}

\newlabel{eq:flowDataTerm3@cref}{{[equation][51][4]4.51}{85}}

\newlabel{eq:globalIntensFactor2}{{4.52}{85}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.52}{}}

\newlabel{eq:globalIntensFactor2@cref}{{[equation][52][4]4.52}{85}}

\newlabel{eq:likelihood2}{{4.54}{85}{Estimation of the Scale Difference $\scalediff $}{equation.4.7.54}{}}

\newlabel{eq:likelihood2@cref}{{[equation][54][4]4.54}{85}}

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\newlabel{sec:realMultiModalOpticalFlow@cref}{{[subsection][5][4,7]4.7.5}{85}}

\newlabel{fig:multiModalVSC2}{{4.9a}{86}{Subfigure 4 4.9a}{subfigure.4.9.1}{}}

\newlabel{fig:multiModalVSC2@cref}{{[subfigure][1][4,9]4.9a}{86}}

\newlabel{fig:multiModalTC2}{{4.9b}{86}{Subfigure 4 4.9b}{subfigure.4.9.2}{}}

\newlabel{fig:multiModalTC2@cref}{{[subfigure][2][4,9]4.9b}{86}}

\newlabel{fig:multiModalHisto2}{{4.9c}{86}{Subfigure 4 4.9c}{subfigure.4.9.3}{}}

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\@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 }}{86}{figure.caption.31}}

\newlabel{fig:multiModalTCVSC2}{{4.9}{86}{\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.31}{}}

\newlabel{fig:multiModalTCVSC2@cref}{{[figure][9][4]4.9}{86}}

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\newlabel{fig:EdatalocA21}{{4.10a}{87}{Subfigure 4 4.10a}{subfigure.4.10.1}{}}

\newlabel{fig:EdatalocA21@cref}{{[subfigure][1][4,10]4.10a}{87}}

\newlabel{fig:scaleOverA}{{4.10b}{87}{Subfigure 4 4.10b}{subfigure.4.10.2}{}}

\newlabel{fig:scaleOverA@cref}{{[subfigure][2][4,10]4.10b}{87}}

\newlabel{fig:EdatalocOverA}{{4.10c}{87}{Subfigure 4 4.10c}{subfigure.4.10.3}{}}

\newlabel{fig:EdatalocOverA@cref}{{[subfigure][3][4,10]4.10c}{87}}

\@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 }}{87}{figure.caption.32}}

\newlabel{fig:EdataSigmaAdependence}{{4.10}{87}{\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.32}{}}

\newlabel{fig:EdataSigmaAdependence@cref}{{[figure][10][4]4.10}{87}}

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\newlabel{eq:flowDataTermLocal2}{{4.56}{87}{Real Multimodal Optical Flow Data}{equation.4.7.56}{}}

\newlabel{eq:flowDataTermLocal2@cref}{{[equation][56][4]4.56}{87}}

\newlabel{fig:flowST}{{4.11a}{88}{Subfigure 4 4.11a}{subfigure.4.11.1}{}}

\newlabel{fig:flowST@cref}{{[subfigure][1][4,11]4.11a}{88}}

\newlabel{fig:flowTV}{{4.11b}{88}{Subfigure 4 4.11b}{subfigure.4.11.2}{}}

\newlabel{fig:flowTV@cref}{{[subfigure][2][4,11]4.11b}{88}}

\@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 }}{88}{figure.caption.33}}

\newlabel{fig:multModalFlowResult}{{4.11}{88}{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.33}{}}

\newlabel{fig:multModalFlowResult@cref}{{[figure][11][4]4.11}{88}}

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\newlabel{eq:viewAnglesEqual}{{4.57}{88}{Real Multimodal Optical Flow Data}{equation.4.7.57}{}}

\newlabel{eq:viewAnglesEqual@cref}{{[equation][57][4]4.57}{88}}

\newlabel{eq:trueOptScale}{{4.58}{88}{Real Multimodal Optical Flow Data}{equation.4.7.58}{}}

\newlabel{eq:trueOptScale@cref}{{[equation][58][4]4.58}{88}}

\newlabel{eq:scaleMin}{{4.59}{89}{Real Multimodal Optical Flow Data}{equation.4.7.59}{}}

\newlabel{eq:scaleMin@cref}{{[equation][59][4]4.59}{89}}

\newlabel{eq:EdataScaleMin}{{4.60}{89}{Real Multimodal Optical Flow Data}{equation.4.7.60}{}}

\newlabel{eq:EdataScaleMin@cref}{{[equation][60][4]4.60}{89}}

\newlabel{eq:localRelation2}{{4.61}{89}{Real Multimodal Optical Flow Data}{equation.4.7.61}{}}

\newlabel{eq:localRelation2@cref}{{[equation][61][4]4.61}{89}}

\newlabel{eq:optFlowModelLocalST}{{4.62}{89}{Real Multimodal Optical Flow Data}{equation.4.7.62}{}}

\newlabel{eq:optFlowModelLocalST@cref}{{[equation][62][4]4.62}{89}}

\newlabel{eq:optFlowModelLocalTV}{{4.63}{89}{Real Multimodal Optical Flow Data}{equation.4.7.63}{}}

\newlabel{eq:optFlowModelLocalTV@cref}{{[equation][63][4]4.63}{89}}

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\newlabel{fig:chiSqNoFlow}{{4.12a}{90}{Subfigure 4 4.12a}{subfigure.4.12.1}{}}

\newlabel{fig:chiSqNoFlow@cref}{{[subfigure][1][4,12]4.12a}{90}}

\newlabel{fig:chiSqSTFlow}{{4.12b}{90}{Subfigure 4 4.12b}{subfigure.4.12.2}{}}

\newlabel{fig:chiSqSTFlow@cref}{{[subfigure][2][4,12]4.12b}{90}}

\newlabel{fig:chiSqTVFlow}{{4.12c}{90}{Subfigure 4 4.12c}{subfigure.4.12.3}{}}

\newlabel{fig:chiSqTVFlow@cref}{{[subfigure][3][4,12]4.12c}{90}}

\@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 }}{90}{figure.caption.34}}

\newlabel{fig:chiSqPValue}{{4.12}{90}{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.34}{}}

\newlabel{fig:chiSqPValue@cref}{{[figure][12][4]4.12}{90}}

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\newlabel{eq:chiSqV}{{4.64}{90}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.64}{}}

\newlabel{eq:chiSqV@cref}{{[equation][64][4]4.64}{90}}

\newlabel{eq:pearsonCDF}{{4.65}{90}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.65}{}}

\newlabel{eq:pearsonCDF@cref}{{[equation][65][4]4.65}{90}}

\newlabel{eq:pearsonPValue}{{4.66}{91}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.66}{}}

\newlabel{eq:pearsonPValue@cref}{{[equation][66][4]4.66}{91}}

\newlabel{eq:chiSqVObs}{{4.67}{91}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.67}{}}

\newlabel{eq:chiSqVObs@cref}{{[equation][67][4]4.67}{91}}

\newlabel{eq:chiSqSatis}{{4.68}{91}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.68}{}}

\newlabel{eq:chiSqSatis@cref}{{[equation][68][4]4.68}{91}}

\newlabel{eq:linConstraint}{{4.69}{91}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.69}{}}

\newlabel{eq:linConstraint@cref}{{[equation][69][4]4.69}{91}}

\newlabel{eq:pearsonLocalObservation}{{4.70}{91}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.70}{}}

\newlabel{eq:pearsonLocalObservation@cref}{{[equation][70][4]4.70}{91}}

\newlabel{fig:subfigEdevisroi-184-397-histo}{{4.13a}{92}{Subfigure 4 4.13a}{subfigure.4.13.1}{}}

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\newlabel{fig:subfigEdevisroiLone-184-397-thermo}{{4.13h}{92}{Subfigure 4 4.13h}{subfigure.4.13.8}{}}

\newlabel{fig:subfigEdevisroiLone-184-397-thermo@cref}{{[subfigure][8][4,13]4.13h}{92}}

\@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 }}{92}{figure.caption.36}}

\newlabel{fig:multimodalRoi}{{4.13}{92}{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.36}{}}

\newlabel{fig:multimodalRoi@cref}{{[figure][13][4]4.13}{92}}

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\newlabel{fig:QregEigVal3}{{4.14a}{93}{Subfigure 4 4.14a}{subfigure.4.14.1}{}}

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\@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 }}{93}{figure.caption.37}}

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\@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 }}{94}{figure.caption.38}}

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\newlabel{eq:summaryGlobalLin@cref}{{[equation][71][4]4.71}{94}}

\newlabel{eq:sumoptFlowModelST}{{4.72}{95}{Summary}{equation.4.7.72}{}}

\newlabel{eq:sumoptFlowModelST@cref}{{[equation][72][4]4.72}{95}}

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\newlabel{eq:sumoptFlowModelSTLocal}{{4.75}{96}{Summary}{equation.4.7.75}{}}

\newlabel{eq:sumoptFlowModelSTLocal@cref}{{[equation][75][4]4.75}{96}}

\newlabel{eq:sumoptFlowModelTVLocal}{{4.76}{96}{Summary}{equation.4.7.76}{}}

\newlabel{eq:sumoptFlowModelTVLocal@cref}{{[equation][76][4]4.76}{96}}

\@writefile{toc}{\contentsline {chapter}{\numberline {5}The Extended Least Action Algorithm}{98}{chapter.5}}

\newlabel{chap:GeneralizedNewtonAlgorithm}{{5}{98}{The Extended Least Action Algorithm}{chapter.5}{}}

\newlabel{chap:GeneralizedNewtonAlgorithm@cref}{{[chapter][5][]5}{98}}

\newlabel{eq:totEnergyGenNewton}{{5.1}{98}{The Extended Least Action Algorithm}{equation.5.0.1}{}}

\newlabel{eq:totEnergyGenNewton@cref}{{[equation][1][5]5.1}{98}}

\newlabel{eq:eulerLagrangeGRF2}{{5.2}{98}{The Extended Least Action Algorithm}{equation.5.0.2}{}}

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\newlabel{eq:pureSpacialSymmetryCanonMomentum2}{{5.4}{98}{The Extended Least Action Algorithm}{equation.5.0.4}{}}

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\@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 }}{99}{figure.caption.39}}

\newlabel{fig:GNAMotivCurvImagesWithLevelSet}{{5.1}{99}{\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.39}{}}

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\@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 }}{116}{figure.caption.50}}

\newlabel{fig:ArmyEnergy}{{5.6}{116}{\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.50}{}}

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\@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 }}{117}{figure.caption.51}}

\newlabel{fig:ArmyCurvature}{{5.7}{117}{\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.51}{}}

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\newlabel{eq:DenoiseFunctionalST}{{5.70}{117}{Structure Tensor Prior}{equation.5.1.70}{}}

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\newlabel{eq:structtensPriorRotInv2@cref}{{[equation][71][5]5.71}{117}}

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\@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 }}{118}{figure.caption.53}}

\newlabel{fig:MeanEnergyST}{{5.8}{118}{\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.53}{}}

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\newlabel{eq:BlevelSetST}{{5.72}{118}{Structure Tensor Prior}{equation.5.1.72}{}}

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