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  • Committer: gerald.mwangi at gmx
  • Date: 2018-04-04 18:55:14 UTC
  • Revision ID: gerald.mwangi@gmx.de-20180404185514-j00p97m1ueqssvqq
work on background finishing5

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\contentsline {subsubsection}{\nonumberline Cross Correlation}{48}{section*.15}
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\contentsline {section}{\numberline {2.9}Image Fusion}{50}{section.2.9}
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\contentsline {chapter}{\numberline {3}Noether's First Theorem: A Modern Version}{54}{chapter.3}
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\contentsline {section}{\numberline {3.1}The action of $\mathbb {G}$ on Functionals}{55}{section.3.1}
 
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\contentsline {section}{\numberline {3.1}The action of $\mathbb {G}^{\Omega \phi }$ on Functionals}{55}{section.3.1}
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\contentsline {section}{\numberline {3.2}Noether's First Theorem: A Modern Version}{60}{section.3.2}
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\contentsline {subsection}{\numberline {3.2.1}Pure spatial Symmetries}{63}{subsection.3.2.1}
 
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\contentsline {subsection}{\numberline {3.2.1}Pure spatial Symmetries}{62}{subsection.3.2.1}
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\contentsline {section}{\numberline {3.3}Embedding Geometrical Constraints into Prior Energies}{64}{section.3.3}
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\contentsline {chapter}{\numberline {4}Linearized Priors}{69}{chapter.4}
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\contentsline {section}{\numberline {4.1}The Linear Structure Tensor}{69}{section.4.1}
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\contentsline {section}{\numberline {4.2}Structure Tensor Based Prior}{71}{section.4.2}
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\contentsline {section}{\numberline {4.3}Geometrical Optical Flow Model}{72}{section.4.3}
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\contentsline {section}{\numberline {4.4}Multi-Modal Optical Flow with Differing Resolutions}{73}{section.4.4}
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\contentsline {section}{\numberline {4.5}Localization}{76}{section.4.5}
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\contentsline {section}{\numberline {4.6}The Multigrid Newton algorithm}{77}{section.4.6}
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\contentsline {section}{\numberline {4.7}Results}{79}{section.4.7}
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\contentsline {subsection}{\numberline {4.7.1}Uni-Modal Data}{80}{subsection.4.7.1}
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\contentsline {subsection}{\numberline {4.7.2}Rubber Whale Sequence}{81}{subsection.4.7.2}
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\contentsline {subsection}{\numberline {4.7.3}Hydrangea Sequence}{83}{subsection.4.7.3}
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\contentsline {subsection}{\numberline {4.7.4}Estimation of the Scale Difference $\ensuremath {{\sigma ^{sc}}}$}{84}{subsection.4.7.4}
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\contentsline {subsection}{\numberline {4.7.5}Real Multimodal Optical Flow Data}{85}{subsection.4.7.5}
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\contentsline {subsection}{\numberline {4.7.6}Eigenvalue analysis and the stabilization parameter $\lambda _2$}{93}{subsection.4.7.6}
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\contentsline {subsection}{\numberline {4.7.7}Summary}{94}{subsection.4.7.7}
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\contentsline {chapter}{\numberline {5}The Extended Least Action Algorithm}{98}{chapter.5}
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\contentsline {subsection}{\numberline {5.0.8}Newtonian Minimization}{101}{subsection.5.0.8}
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\contentsline {subsection}{\numberline {5.0.9}The dynamics of the level-sets $S$}{102}{subsection.5.0.9}
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\contentsline {subsubsection}{\nonumberline Dynamics of the normal vector $\ensuremath {{\bm {n}}}_S$}{103}{section*.41}
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\contentsline {subsubsection}{\nonumberline Dynamics of the tangential vector to $S$}{104}{section*.43}
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\contentsline {section}{\numberline {5.1}The Extended Least Action Algorithm}{107}{section.5.1}
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\contentsline {subsubsection}{\nonumberline The Curvature Operator $\ensuremath {{\bm {K}}}$}{109}{section*.45}
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\contentsline {subsection}{\numberline {5.1.1}Image De-noising}{113}{subsection.5.1.1}
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\contentsline {subsubsection}{\nonumberline Analysis Method}{114}{section*.48}
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\contentsline {subsubsection}{\nonumberline Total Variation based Image De-Noising}{115}{section*.49}
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\contentsline {subsubsection}{\nonumberline Structure Tensor Prior}{117}{section*.52}
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\contentsline {section}{\numberline {5.2}summary}{121}{section.5.2}
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\contentsline {chapter}{\numberline {6}Conclusions}{124}{chapter.6}
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\contentsline {section}{\numberline {6.1}Outlook}{128}{section.6.1}
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\contentsline {chapter}{\numberline {A}Smooth Manifolds}{129}{appendix.A}
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\contentsline {subsection}{\numberline {A.0.1}Topological Spaces}{129}{subsection.A.0.1}
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\contentsline {subsection}{\numberline {A.0.2}Smooth Manifolds}{132}{subsection.A.0.2}
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\contentsline {section}{\numberline {A.1}The Tangent Space $T_p M$}{134}{section.A.1}
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\contentsline {subsection}{\numberline {A.1.1}The Push-Forward}{136}{subsection.A.1.1}
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\contentsline {section}{\numberline {A.2}The Basis of $T_p M$}{139}{section.A.2}
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\contentsline {section}{\numberline {A.3}Vector Fields}{141}{section.A.3}
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\contentsline {section}{\numberline {A.4}Push-Forwards on $\mathcal {T}(M)$}{143}{section.A.4}
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\contentsline {section}{\numberline {A.5}Integral Curves and Flows}{145}{section.A.5}
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\contentsline {subsection}{\numberline {A.5.1}The Lie Derivative}{148}{subsection.A.5.1}
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\contentsline {chapter}{\numberline {B}Lie Groups}{151}{appendix.B}
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\contentsline {section}{\numberline {B.1}The Prolonged Action}{151}{section.B.1}
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\contentsline {section}{\numberline {B.2}Geometrical Meaning of the Commutator $\ensuremath {\left [{\cdot ,\cdot }\right ]}$}{152}{section.B.2}
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\contentsline {section}{\numberline {B.3}Derivation Of Noethers Theorem}{153}{section.B.3}
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\contentsline {subsection}{\numberline {B.3.1}Connection between $\ensuremath {{\mathbf {B}}}_m$, $\ensuremath {{\mathbf {W}}}_m$ and $\ensuremath {\left [{\mathcal {E}}\right ]}$}{156}{subsection.B.3.1}
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\contentsline {chapter}{\numberline {C}The Bending Algebra}{158}{appendix.C}
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\contentsline {section}{\numberline {C.1}The curvature operator}{158}{section.C.1}
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\contentsline {section}{\numberline {C.2}TV Image Denoising, supplementary results}{161}{section.C.2}
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\contentsline {chapter}{\numberline {D}Multimodal Optical Flow}{166}{appendix.D}
80
 
\contentsline {chapter}{\nonumberline Bibliography}{169}{appendix*.61}
 
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\contentsline {chapter}{\numberline {4}Linearized Priors}{68}{chapter.4}
 
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\contentsline {section}{\numberline {4.1}The Linear Structure Tensor}{68}{section.4.1}
 
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\contentsline {section}{\numberline {4.2}Structure Tensor Based Prior}{70}{section.4.2}
 
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\contentsline {section}{\numberline {4.3}Geometrical Optical Flow Model}{71}{section.4.3}
 
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\contentsline {section}{\numberline {4.4}Multi-Modal Optical Flow with Differing Resolutions}{72}{section.4.4}
 
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\contentsline {section}{\numberline {4.5}Localization}{75}{section.4.5}
 
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\contentsline {section}{\numberline {4.6}The Multigrid Newton algorithm}{76}{section.4.6}
 
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\contentsline {section}{\numberline {4.7}Results}{78}{section.4.7}
 
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\contentsline {subsection}{\numberline {4.7.1}Uni-Modal Data}{79}{subsection.4.7.1}
 
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\contentsline {subsection}{\numberline {4.7.2}Rubber Whale Sequence}{80}{subsection.4.7.2}
 
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\contentsline {subsection}{\numberline {4.7.3}Hydrangea Sequence}{82}{subsection.4.7.3}
 
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\contentsline {subsection}{\numberline {4.7.4}Estimation of the Scale Difference $\ensuremath {{\sigma ^{sc}}}$}{83}{subsection.4.7.4}
 
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\contentsline {subsection}{\numberline {4.7.5}Real Multimodal Optical Flow Data}{84}{subsection.4.7.5}
 
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\contentsline {subsection}{\numberline {4.7.6}Eigenvalue analysis and the stabilization parameter $\lambda _2$}{92}{subsection.4.7.6}
 
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\contentsline {subsection}{\numberline {4.7.7}Summary}{93}{subsection.4.7.7}
 
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\contentsline {chapter}{\numberline {5}The Extended Least Action Algorithm}{97}{chapter.5}
 
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\contentsline {subsection}{\numberline {5.0.8}Newtonian Minimization}{100}{subsection.5.0.8}
 
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\contentsline {subsection}{\numberline {5.0.9}The dynamics of the level-sets $S$}{101}{subsection.5.0.9}
 
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\contentsline {subsubsection}{\nonumberline Dynamics of the normal vector $\ensuremath {{\bm {n}}}_S$}{102}{section*.41}
 
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\contentsline {subsubsection}{\nonumberline Dynamics of the tangential vector to $S$}{103}{section*.43}
 
52
\contentsline {section}{\numberline {5.1}The Extended Least Action Algorithm}{106}{section.5.1}
 
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\contentsline {subsubsection}{\nonumberline The Curvature Operator $\ensuremath {{\bm {K}}}$}{108}{section*.45}
 
54
\contentsline {subsection}{\numberline {5.1.1}Image De-noising}{112}{subsection.5.1.1}
 
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\contentsline {subsubsection}{\nonumberline Analysis Method}{113}{section*.48}
 
56
\contentsline {subsubsection}{\nonumberline Total Variation based Image De-Noising}{114}{section*.49}
 
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\contentsline {subsubsection}{\nonumberline Structure Tensor Prior}{116}{section*.52}
 
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\contentsline {section}{\numberline {5.2}summary}{120}{section.5.2}
 
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\contentsline {chapter}{\numberline {6}Conclusions}{123}{chapter.6}
 
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\contentsline {section}{\numberline {6.1}Outlook}{127}{section.6.1}
 
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\contentsline {chapter}{\numberline {A}Smooth Manifolds}{128}{appendix.A}
 
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\contentsline {subsection}{\numberline {A.0.1}Topological Spaces}{128}{subsection.A.0.1}
 
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\contentsline {subsection}{\numberline {A.0.2}Smooth Manifolds}{131}{subsection.A.0.2}
 
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\contentsline {section}{\numberline {A.1}The Tangent Space $T_p M$}{133}{section.A.1}
 
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\contentsline {subsection}{\numberline {A.1.1}The Push-Forward}{135}{subsection.A.1.1}
 
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\contentsline {section}{\numberline {A.2}The Basis of $T_p M$}{138}{section.A.2}
 
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\contentsline {section}{\numberline {A.3}Vector Fields}{140}{section.A.3}
 
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\contentsline {section}{\numberline {A.4}Push-Forwards on $\mathcal {T}(M)$}{142}{section.A.4}
 
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\contentsline {section}{\numberline {A.5}Integral Curves and Flows}{144}{section.A.5}
 
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\contentsline {subsection}{\numberline {A.5.1}The Lie Derivative}{147}{subsection.A.5.1}
 
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\contentsline {chapter}{\numberline {B}Lie Groups}{150}{appendix.B}
 
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\contentsline {section}{\numberline {B.1}The Prolonged Action}{150}{section.B.1}
 
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\contentsline {section}{\numberline {B.2}Geometrical Meaning of the Commutator $\ensuremath {\left [{\cdot ,\cdot }\right ]}$}{151}{section.B.2}
 
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\contentsline {section}{\numberline {B.3}Derivation Of Noethers Theorem}{152}{section.B.3}
 
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\contentsline {subsection}{\numberline {B.3.1}Connection between $\ensuremath {{\mathbf {B}}}_m$, $\ensuremath {{\mathbf {W}}}_m$ and $\ensuremath {\left [{\mathcal {E}}\right ]}$}{155}{subsection.B.3.1}
 
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\contentsline {chapter}{\numberline {C}The Bending Algebra}{157}{appendix.C}
 
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\contentsline {section}{\numberline {C.1}The curvature operator}{157}{section.C.1}
 
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\contentsline {section}{\numberline {C.2}TV Image Denoising, supplementary results}{160}{section.C.2}
 
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\contentsline {chapter}{\numberline {D}Multimodal Optical Flow}{165}{appendix.D}
 
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\contentsline {chapter}{\nonumberline Bibliography}{168}{appendix*.61}