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\select@language {english}
\contentsline {section}{Acknowledgements}{I}{section*.3}
\contentsline {chapter}{\numberline {1}Introduction}{1}{chapter.1}
\contentsline {chapter}{\numberline {2}Background}{6}{chapter.2}
\contentsline {section}{\numberline {2.1}Gibbs Random Fields}{6}{section.2.1}
\contentsline {section}{\numberline {2.2}Convex Optimization}{7}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}The Proximal Operator}{10}{subsection.2.2.1}
\contentsline {subsection}{\numberline {2.2.2}Fenchel Duality}{13}{subsection.2.2.2}
\contentsline {subsection}{\numberline {2.2.3}Primal Dual Splitting}{21}{subsection.2.2.3}
\contentsline {section}{\numberline {2.3}Principle of Least Action}{24}{section.2.3}
\contentsline {section}{\numberline {2.4}Image De-Noising}{25}{section.2.4}
\contentsline {section}{\numberline {2.5}Lie Groups and the Noether Theorem}{28}{section.2.5}
\contentsline {subsection}{\numberline {2.5.1}Motivation 1, the problem}{28}{subsection.2.5.1}
\contentsline {subsection}{\numberline {2.5.2}Motivation 2, the solution}{30}{subsection.2.5.2}
\contentsline {section}{\numberline {2.6}Lie Groups}{33}{section.2.6}
\contentsline {subsection}{\numberline {2.6.1}The Group $\mathbb {G}=\mathbb {T}\times SO(2)$}{36}{subsection.2.6.1}
\contentsline {subsection}{\numberline {2.6.2}The action of $\mathbb {G}$ on Functionals}{37}{subsection.2.6.2}
\contentsline {section}{\numberline {2.7}Noether's First Theorem}{43}{section.2.7}
\contentsline {subsubsection}{\nonumberline Embedding Geometrical Constraints into Prior Energies}{43}{section*.10}
\contentsline {subsection}{\numberline {2.7.1}Noethers Theorems}{46}{subsection.2.7.1}
\contentsline {subsubsection}{\nonumberline Kepler's Two Body Problem}{47}{section*.11}
\contentsline {subsection}{\numberline {2.7.2}Noether's First Theorem: A Modern Version}{49}{subsection.2.7.2}
\contentsline {subsection}{\numberline {2.7.3}Pure Spacial Symmetries}{51}{subsection.2.7.3}
\contentsline {section}{\numberline {2.8}Current usage of Noethers Theorems}{52}{section.2.8}
\contentsline {section}{\numberline {2.9}Total Variation}{52}{section.2.9}
\contentsline {subsection}{\numberline {2.9.1}The Mean Curvature of Total Variation}{55}{subsection.2.9.1}
\contentsline {section}{\numberline {2.10}Optical Flow}{56}{section.2.10}
\contentsline {subsection}{\numberline {2.10.1}Uni-Modal Optical Flow}{58}{subsection.2.10.1}
\contentsline {subsection}{\numberline {2.10.2}Multi-Modal Optical Flow}{59}{subsection.2.10.2}
\contentsline {subsubsection}{\nonumberline Mutual Information}{60}{section*.13}
\contentsline {subsubsection}{\nonumberline Correlation Ratio}{61}{section*.14}
\contentsline {subsubsection}{\nonumberline Cross Correlation}{61}{section*.15}
\contentsline {section}{\numberline {2.11}Image Fusion}{63}{section.2.11}
\contentsline {chapter}{\numberline {3}Noether's First Theorem: A Modern Version}{67}{chapter.3}
\contentsline {section}{\numberline {3.1}The action of $\mathbb {G}$ on Functionals}{67}{section.3.1}
\contentsline {section}{\numberline {3.2}Embedding Geometrical Constraints into Prior Energies}{73}{section.3.2}
\contentsline {section}{\numberline {3.3}Noether's First Theorem: A Modern Version}{76}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}Pure Spacial Symmetries}{78}{subsection.3.3.1}
\contentsline {chapter}{\numberline {4}Linearized Priors}{80}{chapter.4}
\contentsline {section}{\numberline {4.1}The Linear Structure Tensor}{80}{section.4.1}
\contentsline {section}{\numberline {4.2}Structure Tensor Based Prior}{82}{section.4.2}
\contentsline {section}{\numberline {4.3}Geometrical Optical Flow Model}{83}{section.4.3}
\contentsline {section}{\numberline {4.4}Multi-Modal Optical Flow with Differing Resolutions}{84}{section.4.4}
\contentsline {section}{\numberline {4.5}Localization}{87}{section.4.5}
\contentsline {section}{\numberline {4.6}The Multigrid Newton algorithm}{88}{section.4.6}
\contentsline {section}{\numberline {4.7}Results}{90}{section.4.7}
\contentsline {subsection}{\numberline {4.7.1}Uni-Modal Data}{91}{subsection.4.7.1}
\contentsline {subsection}{\numberline {4.7.2}Rubber Whale Sequence}{92}{subsection.4.7.2}
\contentsline {subsection}{\numberline {4.7.3}Hydrangea Sequence}{94}{subsection.4.7.3}
\contentsline {subsection}{\numberline {4.7.4}Estimation of the Scale Difference $\ensuremath {{\sigma ^{sc}}}$}{95}{subsection.4.7.4}
\contentsline {subsection}{\numberline {4.7.5}Real Multimodal Optical Flow Data}{96}{subsection.4.7.5}
\contentsline {subsection}{\numberline {4.7.6}Eigenvalue analysis and the stabilization parameter $\lambda _2$}{104}{subsection.4.7.6}
\contentsline {subsection}{\numberline {4.7.7}Summary}{105}{subsection.4.7.7}
\contentsline {chapter}{\numberline {5}The Extended Least Action Algorithm}{109}{chapter.5}
\contentsline {subsection}{\numberline {5.0.8}Newtonian Minimization}{112}{subsection.5.0.8}
\contentsline {subsection}{\numberline {5.0.9}The dynamics of the level-sets $S$}{113}{subsection.5.0.9}
\contentsline {subsubsection}{\nonumberline Dynamics of the normal vector $\ensuremath {{\bm {n}}}_S$}{114}{section*.39}
\contentsline {subsubsection}{\nonumberline Dynamics of the tangential vector to $S$}{115}{section*.41}
\contentsline {section}{\numberline {5.1}The Extended Least Action Algorithm}{118}{section.5.1}
\contentsline {subsubsection}{\nonumberline The Curvature Operator $\ensuremath {{\bm {K}}}$}{120}{section*.43}
\contentsline {subsection}{\numberline {5.1.1}Image De-noising}{123}{subsection.5.1.1}
\contentsline {subsubsection}{\nonumberline Analysis Method}{124}{section*.46}
\contentsline {subsubsection}{\nonumberline Total Variation based Image De-Noising}{125}{section*.47}
\contentsline {subsubsection}{\nonumberline Structure Tensor Prior}{127}{section*.50}
\contentsline {section}{\numberline {5.2}summary}{131}{section.5.2}
\contentsline {chapter}{\numberline {6}Conclusions}{134}{chapter.6}
\contentsline {section}{\numberline {6.1}Outlook}{138}{section.6.1}
\contentsline {chapter}{\numberline {A}Smooth Manifolds}{139}{appendix.A}
\contentsline {subsection}{\numberline {A.0.1}Topological Spaces}{139}{subsection.A.0.1}
\contentsline {subsection}{\numberline {A.0.2}Smooth Manifolds}{142}{subsection.A.0.2}
\contentsline {section}{\numberline {A.1}The Tangent Space $T_p M$}{144}{section.A.1}
\contentsline {subsection}{\numberline {A.1.1}The Push-Forward}{146}{subsection.A.1.1}
\contentsline {section}{\numberline {A.2}The Basis of $T_p M$}{149}{section.A.2}
\contentsline {section}{\numberline {A.3}Vector Fields}{151}{section.A.3}
\contentsline {section}{\numberline {A.4}Push-Forwards on $\mathcal {T}(M)$}{153}{section.A.4}
\contentsline {section}{\numberline {A.5}Integral Curves and Flows}{155}{section.A.5}
\contentsline {subsection}{\numberline {A.5.1}The Lie Derivative}{158}{subsection.A.5.1}
\contentsline {chapter}{\numberline {B}Lie Groups}{161}{appendix.B}
\contentsline {section}{\numberline {B.1}The Prolonged Action}{161}{section.B.1}
\contentsline {section}{\numberline {B.2}Geometrical Meaning of the Commutator $\ensuremath {\left [{\cdot ,\cdot }\right ]}$}{162}{section.B.2}
\contentsline {section}{\numberline {B.3}Derivation Of Noethers Theorem}{163}{section.B.3}
\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}
\contentsline {chapter}{\numberline {C}The Bending Algebra}{168}{appendix.C}
\contentsline {section}{\numberline {C.1}The curvature operator}{168}{section.C.1}
\contentsline {section}{\numberline {C.2}TV Image Denoising, supplementary results}{171}{section.C.2}
\contentsline {chapter}{\numberline {D}Multimodal Optical Flow}{176}{appendix.D}
\contentsline {chapter}{\nonumberline Bibliography}{179}{appendix*.59}
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