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\BOOKMARK [1][-]{section*.3}{Acknowledgements}{}% 1
\BOOKMARK [0][-]{chapter.1}{Introduction}{}% 2
\BOOKMARK [0][-]{chapter.2}{Background}{}% 3
\BOOKMARK [1][-]{section.2.1}{Gibbs Random Fields}{chapter.2}% 4
\BOOKMARK [1][-]{section.2.2}{Convex Optimization}{chapter.2}% 5
\BOOKMARK [2][-]{subsection.2.2.1}{The Proximal Operator}{section.2.2}% 6
\BOOKMARK [2][-]{subsection.2.2.2}{Fenchel Duality}{section.2.2}% 7
\BOOKMARK [2][-]{subsection.2.2.3}{Primal Dual Splitting}{section.2.2}% 8
\BOOKMARK [1][-]{section.2.3}{Principle of Least Action}{chapter.2}% 9
\BOOKMARK [1][-]{section.2.4}{Image De-Noising}{chapter.2}% 10
\BOOKMARK [1][-]{section.2.5}{Lie Groups and the Noether Theorem}{chapter.2}% 11
\BOOKMARK [2][-]{subsection.2.5.1}{Motivation 1, the problem}{section.2.5}% 12
\BOOKMARK [2][-]{subsection.2.5.2}{Motivation 2, the solution}{section.2.5}% 13
\BOOKMARK [1][-]{section.2.6}{Lie Groups}{chapter.2}% 14
\BOOKMARK [2][-]{subsection.2.6.1}{The Group G=TSO\(2\)}{section.2.6}% 15
\BOOKMARK [2][-]{subsection.2.6.2}{The action of G on Functionals}{section.2.6}% 16
\BOOKMARK [1][-]{section.2.7}{Noether's First Theorem}{chapter.2}% 17
\BOOKMARK [2][-]{subsection.2.7.1}{Noethers Theorems}{section.2.7}% 18
\BOOKMARK [2][-]{subsection.2.7.2}{Noether's First Theorem: A Modern Version}{section.2.7}% 19
\BOOKMARK [2][-]{subsection.2.7.3}{Pure Spacial Symmetries}{section.2.7}% 20
\BOOKMARK [1][-]{section.2.8}{Current usage of Noethers Theorems}{chapter.2}% 21
\BOOKMARK [1][-]{section.2.9}{Total Variation}{chapter.2}% 22
\BOOKMARK [2][-]{subsection.2.9.1}{The Mean Curvature of Total Variation}{section.2.9}% 23
\BOOKMARK [1][-]{section.2.10}{Optical Flow}{chapter.2}% 24
\BOOKMARK [2][-]{subsection.2.10.1}{Uni-Modal Optical Flow}{section.2.10}% 25
\BOOKMARK [2][-]{subsection.2.10.2}{Multi-Modal Optical Flow}{section.2.10}% 26
\BOOKMARK [1][-]{section.2.11}{Image Fusion}{chapter.2}% 27
\BOOKMARK [0][-]{chapter.3}{Noether's First Theorem: A Modern Version}{}% 28
\BOOKMARK [1][-]{section.3.1}{The action of G on Functionals}{chapter.3}% 29
\BOOKMARK [1][-]{section.3.2}{Noether's First Theorem: A Modern Version}{chapter.3}% 30
\BOOKMARK [2][-]{subsection.3.2.1}{Pure Spacial Symmetries}{section.3.2}% 31
\BOOKMARK [0][-]{chapter.4}{Linearized Priors}{}% 32
\BOOKMARK [1][-]{section.4.1}{The Linear Structure Tensor}{chapter.4}% 33
\BOOKMARK [1][-]{section.4.2}{Structure Tensor Based Prior}{chapter.4}% 34
\BOOKMARK [1][-]{section.4.3}{Geometrical Optical Flow Model}{chapter.4}% 35
\BOOKMARK [1][-]{section.4.4}{Multi-Modal Optical Flow with Differing Resolutions}{chapter.4}% 36
\BOOKMARK [1][-]{section.4.5}{Localization}{chapter.4}% 37
\BOOKMARK [1][-]{section.4.6}{The Multigrid Newton algorithm}{chapter.4}% 38
\BOOKMARK [1][-]{section.4.7}{Results}{chapter.4}% 39
\BOOKMARK [2][-]{subsection.4.7.1}{Uni-Modal Data}{section.4.7}% 40
\BOOKMARK [2][-]{subsection.4.7.2}{Rubber Whale Sequence}{section.4.7}% 41
\BOOKMARK [2][-]{subsection.4.7.3}{Hydrangea Sequence}{section.4.7}% 42
\BOOKMARK [2][-]{subsection.4.7.4}{Estimation of the Scale Difference sc}{section.4.7}% 43
\BOOKMARK [2][-]{subsection.4.7.5}{Real Multimodal Optical Flow Data}{section.4.7}% 44
\BOOKMARK [2][-]{subsection.4.7.6}{Eigenvalue analysis and the stabilization parameter 2}{section.4.7}% 45
\BOOKMARK [2][-]{subsection.4.7.7}{Summary}{section.4.7}% 46
\BOOKMARK [0][-]{chapter.5}{The Extended Least Action Algorithm}{}% 47
\BOOKMARK [1][-]{subsection.5.0.8}{Newtonian Minimization}{chapter.5}% 48
\BOOKMARK [2][-]{subsection.5.0.9}{The dynamics of the level-sets S}{subsection.5.0.8}% 49
\BOOKMARK [1][-]{section.5.1}{The Extended Least Action Algorithm}{chapter.5}% 50
\BOOKMARK [2][-]{subsection.5.1.1}{Image De-noising}{section.5.1}% 51
\BOOKMARK [1][-]{section.5.2}{summary}{chapter.5}% 52
\BOOKMARK [0][-]{chapter.6}{Conclusions}{}% 53
\BOOKMARK [1][-]{section.6.1}{Outlook}{chapter.6}% 54
\BOOKMARK [0][-]{appendix.A}{Smooth Manifolds}{}% 55
\BOOKMARK [1][-]{subsection.A.0.1}{Topological Spaces}{appendix.A}% 56
\BOOKMARK [2][-]{subsection.A.0.2}{Smooth Manifolds}{subsection.A.0.1}% 57
\BOOKMARK [1][-]{section.A.1}{The Tangent Space Tp M}{appendix.A}% 58
\BOOKMARK [2][-]{subsection.A.1.1}{The Push-Forward}{section.A.1}% 59
\BOOKMARK [1][-]{section.A.2}{The Basis of Tp M}{appendix.A}% 60
\BOOKMARK [1][-]{section.A.3}{Vector Fields}{appendix.A}% 61
\BOOKMARK [1][-]{section.A.4}{Push-Forwards on T\(M\)}{appendix.A}% 62
\BOOKMARK [1][-]{section.A.5}{Integral Curves and Flows}{appendix.A}% 63
\BOOKMARK [2][-]{subsection.A.5.1}{The Lie Derivative}{section.A.5}% 64
\BOOKMARK [0][-]{appendix.B}{Lie Groups}{}% 65
\BOOKMARK [1][-]{section.B.1}{The Prolonged Action}{appendix.B}% 66
\BOOKMARK [1][-]{section.B.2}{Geometrical Meaning of the Commutator [,]}{appendix.B}% 67
\BOOKMARK [1][-]{section.B.3}{Derivation Of Noethers Theorem}{appendix.B}% 68
\BOOKMARK [2][-]{subsection.B.3.1}{Connection between Bm, Wm and [E]}{section.B.3}% 69
\BOOKMARK [0][-]{appendix.C}{The Bending Algebra}{}% 70
\BOOKMARK [1][-]{section.C.1}{The curvature operator}{appendix.C}% 71
\BOOKMARK [1][-]{section.C.2}{TV Image Denoising, supplementary results}{appendix.C}% 72
\BOOKMARK [0][-]{appendix.D}{Multimodal Optical Flow}{}% 73
\BOOKMARK [0][-]{appendix*.59}{Bibliography}{}% 74
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