~gerald-mwangi/+junk/Thesis : contents of main.aux at revision 230

~gerald-mwangi/+junk/Thesis : (revision 230)
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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 }}{98}{figure.caption.30}}
\newlabel{fig:multiModalTCVSC2}{{4.9}{98}{\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.30}{}}
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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 }}{99}{figure.caption.31}}
\newlabel{fig:EdataSigmaAdependence}{{4.10}{99}{\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.31}{}}
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\newlabel{eq:flowDataTermLocal2@cref}{{[equation][56][4]4.56}{99}}
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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 }}{100}{figure.caption.32}}
\newlabel{fig:multModalFlowResult}{{4.11}{100}{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.32}{}}
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\newlabel{eq:trueOptScale@cref}{{[equation][58][4]4.58}{100}}
\citation{WassermanAllStatistics}
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\newlabel{eq:localRelation2@cref}{{[equation][61][4]4.61}{101}}
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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 }}{102}{figure.caption.33}}
\newlabel{fig:chiSqPValue}{{4.12}{102}{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.33}{}}
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\newlabel{eq:chiSqSatis@cref}{{[equation][68][4]4.68}{103}}
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\newlabel{eq:linConstraint@cref}{{[equation][69][4]4.69}{103}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5.2}{\ignorespaces This figure shows a transformation of the level-set $S$ to $S^\prime $ along the vector $\ensuremath  {{\bm  {W}}}_m(\ensuremath  {\ensuremath  {{\bm  {x}}}})$. The region $\mathcal  {A}\subset \Omega $ is the region a section of $S$ traverses as it is shifted along $\ensuremath  {{\bm  {W}}}_m$ to the end position $S^\prime $. If the divergence of $\ensuremath  {{\bm  {W}}}_m$ vanishes, this means that the incoming flux of $\ensuremath  {{\bm  {W}}}_m$ equals the outgoing flux (both indicated by the red arrows), $\ensuremath  {{\bm  {W}}}_m\delimiter "026A30C _{S}=\ensuremath  {{\bm  {W}}}_m\delimiter "026A30C _{S^\prime }$\relax }}{115}{figure.caption.41}}
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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 }}{124}{figure.caption.46}}
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\citation{Middleburry}
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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 }}{127}{figure.caption.49}}
\newlabel{fig:ArmyEnergy}{{5.6}{127}{\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.49}{}}
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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 }}{128}{figure.caption.50}}
\newlabel{fig:ArmyCurvature}{{5.7}{128}{\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.50}{}}
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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 }}{129}{figure.caption.52}}
\newlabel{fig:MeanEnergyST}{{5.8}{129}{\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.52}{}}
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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 }}{130}{figure.caption.53}}
\newlabel{fig:StdDevEnergyST}{{5.9}{130}{\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.53}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5.10}{\ignorespaces Study of the dependency the mean energy $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ and the mean curvature $\delimiter "426830A \ensuremath  {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $ on the window size $\sigma _{ST}$ of the structure tensor prior $E^{prior}_{ST}$. \Cref  {fig:EnergyWsizeGnaST} shows the mean energy $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ per iteration $k\geq 100$ for various $\sigma _{ST}$ and \cref  {fig:CurvatureWsizeGnaST} the mean curvature $\delimiter "426830A \ensuremath  {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $, also for various $\sigma _{ST}$. Figures \ref  {fig:InitialEnergyGnaST} and \ref  {fig:InitialCurvatureGnaST} show the initial energy $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ and the initial curvature $\delimiter "426830A \ensuremath  {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $ for $k=0$. In \cref  {fig:EnergyWsizeGnaST} we can see that for smaller $\sigma _{ST}$ the energy $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ converges to lower values. Conversely for larger window sizes $\sigma _{ST}$ the mean energy profiles $\delimiter "426830A E^k\delimiter "526930B _{ELAA}$ per $\sigma _{ST}$ converge. In \cref  {fig:CurvatureWsizeGnaST} we observe a similar behavior for the curvature $\delimiter "426830A \ensuremath  {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $: For small $\sigma _{ST}$ the curvature $\delimiter "426830A \ensuremath  {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $ is comparatively large. As $\sigma _{ST}$ rises the profile of $\delimiter "426830A \ensuremath  {\left \delimiter 69645069 K \right \delimiter 86422285 }\delimiter "526930B $ per $\sigma _{ST}$ converge, albeit at lower values. Figures \ref  {fig:InitialEnergyGnaST} and \ref  {fig:InitialCurvatureGnaST} show that the initial energy and the initial curvature for $\sigma _{ST}=3$ have half the values then for the larger window sizes $\sigma _{ST}=13\cdots  63$\relax }}{131}{figure.caption.54}}
\newlabel{fig:WsizeGnaST}{{5.10}{131}{Study of the dependency the mean energy $\langle E^k\rangle _{ELAA}$ and the mean curvature $\langle \norm {K}\rangle $ on the window size $\sigma _{ST}$ of the structure tensor prior $E^{prior}_{ST}$. \Figref {fig:EnergyWsizeGnaST} shows the mean energy $\langle E^k\rangle _{ELAA}$ per iteration $k\geq 100$ for various $\sigma _{ST}$ and \figref {fig:CurvatureWsizeGnaST} the mean curvature $\langle \norm {K}\rangle $, also for various $\sigma _{ST}$. Figures \ref {fig:InitialEnergyGnaST} and \ref {fig:InitialCurvatureGnaST} show the initial energy $\langle E^k\rangle _{ELAA}$ and the initial curvature $\langle \norm {K}\rangle $ for $k=0$. In \figref {fig:EnergyWsizeGnaST} we can see that for smaller $\sigma _{ST}$ the energy $\langle E^k\rangle _{ELAA}$ converges to lower values. Conversely for larger window sizes $\sigma _{ST}$ the mean energy profiles $\langle E^k\rangle _{ELAA}$ per $\sigma _{ST}$ converge. In \figref {fig:CurvatureWsizeGnaST} we observe a similar behavior for the curvature $\langle \norm {K}\rangle $: For small $\sigma _{ST}$ the curvature $\langle \norm {K}\rangle $ is comparatively large. As $\sigma _{ST}$ rises the profile of $\langle \norm {K}\rangle $ per $\sigma _{ST}$ converge, albeit at lower values. Figures \ref {fig:InitialEnergyGnaST} and \ref {fig:InitialCurvatureGnaST} show that the initial energy and the initial curvature for $\sigma _{ST}=3$ have half the values then for the larger window sizes $\sigma _{ST}=13\cdots 63$\relax }{figure.caption.54}{}}
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