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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 }}{94}{figure.caption.29}}
\newlabel{fig:multiModalTCVSC2}{{4.9}{94}{\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}{}}
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\@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 }}{95}{figure.caption.30}}
\newlabel{fig:EdataSigmaAdependence}{{4.10}{95}{\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}{}}
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\@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 }}{96}{figure.caption.31}}
\newlabel{fig:multModalFlowResult}{{4.11}{96}{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}{}}
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\newlabel{eq:trueOptScale@cref}{{[equation][58][4]4.58}{96}}
\citation{WassermanAllStatistics}
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\@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 }}{98}{figure.caption.32}}
\newlabel{fig:chiSqPValue}{{4.12}{98}{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}{}}
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\newlabel{eq:chiSqVObs@cref}{{[equation][67][4]4.67}{99}}
\newlabel{eq:chiSqSatis}{{4.68}{99}{Pearson's $\chi ^{2}$ statistic}{equation.4.7.68}{}}
\newlabel{eq:chiSqSatis@cref}{{[equation][68][4]4.68}{99}}
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\newlabel{eq:linConstraint@cref}{{[equation][69][4]4.69}{99}}
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\newlabel{eq:pearsonLocalObservation@cref}{{[equation][70][4]4.70}{99}}
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\@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 }}{100}{figure.caption.34}}
\newlabel{fig:multimodalRoi}{{4.13}{100}{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}{}}
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\newlabel{sub@fig:curvatureOperator1}{{(a)}{a}{Subfigure 5 5.4a\relax }{subfigure.5.4.1}{}}
\newlabel{fig:curvatureOperator1@cref}{{[subfigure][1][5,4]5.4a}{118}}
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\newlabel{fig:curvatureOperator2@cref}{{[subfigure][2][5,4]5.4b}{118}}
\@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 }}{118}{figure.caption.44}}
\newlabel{fig:curvatureOperator}{{5.4}{118}{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}{}}
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\@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 }}{120}{figure.caption.45}}
\newlabel{fig:ArmyTotal}{{5.5}{120}{\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}{}}
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\citation{Middleburry}
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\newlabel{fig:ArmyMeanEnergy@cref}{{[subfigure][1][5,6]5.6a}{123}}
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\newlabel{sub@fig:ArmyStdDevEnergy}{{(b)}{b}{Subfigure 5 5.6b\relax }{subfigure.5.6.2}{}}
\newlabel{fig:ArmyStdDevEnergy@cref}{{[subfigure][2][5,6]5.6b}{123}}
\@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 }}{123}{figure.caption.48}}
\newlabel{fig:ArmyEnergy}{{5.6}{123}{\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}{}}
\newlabel{fig:ArmyEnergy@cref}{{[figure][6][5]5.6}{123}}
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\newlabel{eq:expCurvParameter@cref}{{[equation][68][5]5.68}{123}}
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\newlabel{sub@fig:ArmyMeanCurvature}{{(a)}{a}{Subfigure 5 5.7a\relax }{subfigure.5.7.1}{}}
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\newlabel{sub@fig:ArmyCurvatureFit}{{(c)}{c}{Subfigure 5 5.7c\relax }{subfigure.5.7.3}{}}
\newlabel{fig:ArmyCurvatureFit@cref}{{[subfigure][3][5,7]5.7c}{124}}
\@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 }}{124}{figure.caption.49}}
\newlabel{fig:ArmyCurvature}{{5.7}{124}{\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}{}}
\newlabel{fig:ArmyCurvature@cref}{{[figure][7][5]5.7}{124}}
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\newlabel{eq:DenoiseFunctionalST}{{5.70}{124}{Structure Tensor Prior}{equation.5.1.70}{}}
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\newlabel{eq:structtensPriorRotInv2}{{5.71}{124}{Structure Tensor Prior}{equation.5.1.71}{}}
\newlabel{eq:structtensPriorRotInv2@cref}{{[equation][71][5]5.71}{124}}
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\newlabel{sub@fig:ArmyMeanEnergyGnaST}{{(b)}{b}{Subfigure 5 5.8b\relax }{subfigure.5.8.2}{}}
\newlabel{fig:ArmyMeanEnergyGnaST@cref}{{[subfigure][2][5,8]5.8b}{125}}
\newlabel{fig:ArmyMeanEnergyGnaBnaDiffST}{{5.8c}{125}{Subfigure 5 5.8c}{subfigure.5.8.3}{}}
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\newlabel{fig:ArmyMeanEnergyGnaBnaDiffST@cref}{{[subfigure][3][5,8]5.8c}{125}}
\@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 }}{125}{figure.caption.51}}
\newlabel{fig:MeanEnergyST}{{5.8}{125}{\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}{}}
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\@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 }}{126}{figure.caption.52}}
\newlabel{fig:StdDevEnergyST}{{5.9}{126}{\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}{}}
\newlabel{fig:StdDevEnergyST@cref}{{[figure][9][5]5.9}{126}}
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\newlabel{eq:AppSimMeasure2@cref}{{[equation][17][2147483647,4]D.17}{175}}
\newlabel{eq:AppflowDataTerm2}{{D.18}{175}{Multimodal Optical Flow}{equation.D.0.18}{}}
\newlabel{eq:AppflowDataTerm2@cref}{{[equation][18][2147483647,4]D.18}{175}}
\newlabel{eq:AppflowDataTermLocal}{{D.20}{175}{Multimodal Optical Flow}{equation.D.0.20}{}}
\newlabel{eq:AppflowDataTermLocal@cref}{{[equation][20][2147483647,4]D.20}{175}}
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