From 15f606112dff2ee9bf65af6fdb82d430bdc312fe Mon Sep 17 00:00:00 2001 From: =?utf8?q?Bj=C3=B8rn=20Rustad?= Date: Mon, 26 Jan 2015 18:26:52 +0100 Subject: [PATCH] FINISHING UP --- bib.bib | 72 ++++++++++- fig/factory/contrast.m | 4 +- introduction.tex | 19 ++- main.tex | 4 +- results.tex | 42 ++++--- theory.tex | 268 ++++++++++++++++++++++++++--------------- 6 files changed, 278 insertions(+), 131 deletions(-) diff --git a/bib.bib b/bib.bib index 7ea0468..07b195a 100644 --- a/bib.bib +++ b/bib.bib @@ -1,5 +1,75 @@ +@preamble{ + "\def\cprime{$'$} " +} +@book{ziemer1989, + AUTHOR = {Ziemer, William P.}, + TITLE = {Weakly differentiable functions}, + SERIES = {Graduate Texts in Mathematics}, + VOLUME = {120}, + PUBLISHER = {Springer-Verlag, New York}, + YEAR = {1989}, + PAGES = {xvi+308}, + ISBN = {0-387-97017-7}, + MRCLASS = {46E35}, + MRNUMBER = {1014685 (91e:46046)}, + MRREVIEWER = {V. M. Gol{\cprime}dshte{\u\i}n}, + DOI = {10.1007/978-1-4612-1015-3}, + URL = {http://dx.doi.org/10.1007/978-1-4612-1015-3}, +} + +@article{acar1994analysis, + AUTHOR = {Acar, R. and Vogel, C. R.}, + TITLE = {Analysis of bounded variation penalty methods for ill-posed + problems}, + JOURNAL = {Inverse Problems}, + FJOURNAL = {Inverse Problems. An International Journal on the Theory and + Practice of Inverse Problems, Inverse Methods and Computerized + Inversion of Data}, + VOLUME = {10}, + YEAR = {1994}, + NUMBER = {6}, + PAGES = {1217--1229}, + ISSN = {0266-5611}, + CODEN = {INPEEY}, + MRCLASS = {65J20 (47A50 49J45)}, + MRNUMBER = {1306801 (95i:65092)}, + URL = {http://stacks.iop.org/0266-5611/10/1217}, +} + +@article{weickert1999coherence, + title={Coherence-enhancing diffusion filtering}, + author={Weickert, Joachim}, + journal={International Journal of Computer Vision}, + volume={31}, + number={2-3}, + pages={111--127}, + year={1999}, + publisher={Springer} +} + +@book{jahne, + title={{D}igital {I}mage {P}rocessing}, + author={J{\"a}hne, Bernd}, + publisher={Springer-Verlag, Berlin Heidelberg}, + year={2005} +} + +@book {aubert2006proc, + AUTHOR = {Aubert, Gilles and Kornprobst, Pierre}, + TITLE = {Mathematical problems in image processing}, + SERIES = {Applied Mathematical Sciences}, + VOLUME = {147}, + EDITION = {Second}, + PUBLISHER = {Springer, New York}, + YEAR = {2006}, + PAGES = {xxxii+377}, + ISBN = {978-0387-32200-1; 0-387-32200-0}, + MRCLASS = {94A08 (35B27 35J60 35L70 49J10 68T45)}, + MRNUMBER = {2244145 (2007j:94004)}, +} + @book{do1976differential, AUTHOR = {do Carmo, Manfredo P.}, TITLE = {Differential geometry of curves and surfaces}, @@ -118,7 +188,7 @@ doi={10.1007/978-0-387-92920-0_23}, title={Total Variation in Imaging}, url={http://dx.doi.org/10.1007/978-0-387-92920-0_23}, - publisher={Springer New York}, + publisher={Springer, New York}, author={Caselles, V. and Chambolle, A. and Novaga, M.}, pages={1016-1057} } diff --git a/fig/factory/contrast.m b/fig/factory/contrast.m index 699603c..deb783b 100644 --- a/fig/factory/contrast.m +++ b/fig/factory/contrast.m @@ -1,5 +1,5 @@ -tv = imread('finger/r_p2_n32_b9000_g1000000000000_r10_s3.png'); -atv = imread('finger/r_p2_n32_b9000_g150_r10_s3.png'); +tv = imread('finger/r_p2_n32_b9000_g1000000000000_r10_s3.pgm'); +atv = imread('finger/r_p2_n32_b9000_g150_r10_s3.pgm'); hold on; plot(atv(100,:),'r'); diff --git a/introduction.tex b/introduction.tex index f7c2ac5..fdb00cf 100644 --- a/introduction.tex +++ b/introduction.tex @@ -66,9 +66,10 @@ the restoration performance. \fixme{Rating: 7/10} There are numerous methods in image restoration, and since we do not -have time nor space to discuss them all, we will focus on the ones +have time nor space to discuss them all, this short overview will focus on the ones related to the anisotropic total variation method considered later in -this thesis. \fixme{Maybe refer to some overview book?} +this thesis. See \cite{jahne} and \cite{aubert2006proc} for more +background on image processing in general. In this chapter, and also in the rest of the thesis we will assume that we are given an image $f : \Omega \to \mathbb{R}$ where $\Omega$ is a @@ -201,12 +202,9 @@ both location and direction dependent. \subsection{Anisotropic diffusion} \label{sec:anisotropic_diffusion} -\fixme{Rating: 7/10, update figure, and refer to it. Maybe even a figure -showing the structure and anisotropy tensors as ellipses along an edge?} - -The diffusivity is made directionally dependent by introducing a diffusion -\emph{tensor} $A(u)$ such that the initial boundary value problem -becomes +The diffusivity is made directionally dependent by introducing a +diffusion \emph{tensor} $A(u)$ such that the initial boundary value +problem becomes \begin{equation} \left\{ \begin{alignedat}{10} @@ -396,9 +394,8 @@ restoration. \label{fig:grad_tv} \end{figure} -\fixme{First time we mention level values.} Thin objects and corners may also suffer from contrast loss since -bringing them closer to their surroundings in level value reduces the +bringing them closer to their surroundings reduces the total variation. An example of this is shown in Figure~\ref{fig:tv_example}, where a not particularly noisy fingerprint image has been strongly regularized. The original black and white levels @@ -503,5 +500,3 @@ And if in addition the cut is minimal, we know that the edges cut have minimal weight, and can prove that the $u^\lambda$ produced minimizes the functional in \eqref{eq:total_energy}. -\fixme{Wow. Such handwavy.} - diff --git a/main.tex b/main.tex index 4d911f9..3e70375 100644 --- a/main.tex +++ b/main.tex @@ -74,6 +74,7 @@ \usepackage{algpseudocode} \newtheorem{theorem}{Theorem} +\newtheorem{proposition}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{definition}{Definition} @@ -122,7 +123,8 @@ Lololol. \section*{Preface} -Bleep bloop. Acknowledements? +Bleep bloop. Acknowledements? Takk til Henrik for sjokolademelka, og +Kine for god muffins. % Romerske tall på alt før selve rapporten starter er pent. \pagenumbering{roman} diff --git a/results.tex b/results.tex index f3b43fe..708a902 100644 --- a/results.tex +++ b/results.tex @@ -466,11 +466,13 @@ circumference of circles of different radii. Note that the circumference approximated is that of an actual continuous circle $r = x^2 + y^2$ and not a discrete representation. +\section{Restoration} + \begin{figure} \centering \begin{subfigure}[t]{0.30\textwidth} \centering - \includegraphics[width=\textwidth]{fig/fingern.png} + \includegraphics[width=\textwidth]{fig/factory/finger/n_q300.png} \caption{% Noisy fingerprint. } @@ -478,46 +480,52 @@ not a discrete representation. ~ \begin{subfigure}[t]{0.30\textwidth} \centering - \includegraphics[width=\textwidth]{fig/fingertv.png} + \includegraphics[width=\textwidth]{fig/factory/finger/r_p2_n32_b9000_g1000000000000_r10_s3.png} \caption{% - Restored without much anisotropy. + Restored without anisotropy. } \label{fig:finger_contrast_tv} \end{subfigure} ~ \begin{subfigure}[t]{0.30\textwidth} \centering - \includegraphics[width=\textwidth]{fig/fingera.png} + \includegraphics[width=\textwidth]{fig/factory/finger/r_p2_n32_b9000_g150_r10_s3.png} \caption{% Restored with much anisotropy. } \label{fig:finger_contrast_atv} \end{subfigure} \caption{% - The different stages in the restoration of a part of the Lena - test image. + A noisy fingerprint restored using both isotropic and + anisotropic total variation. All parameters are kept constant + except for $\omega$, the anisotropy parameter. } \label{fig:finger_contrast} \end{figure} We have claimed that the anisotropic total variation will reduce some of the contrast loss one can encounter with regular total variation -regularization. An example of this can be seen in -Figure~\ref{fig:finger_contrast} where a portion of a fingerprint with -added Gaussian noise has been restored in two different ways. We see -that when the anisotropy $\omega$ is increased, more of the contrast between dark -and light portions of the fingerprint is retained. This can also be seen -in Figure~\ref{fig:contrast_plot} where a one-dimensional slice has been +regularization. In Figure~\ref{fig:finger_contrast} a fingerprint with +additional Gaussian noise has been restored with two different +anisotropy parameters. There is an obvious contrast difference in the +contrast of the two restored images, which can also be seen in +Figure~\ref{fig:contrast_plot} where a one-dimensional slice has been taken through Figure~\ref{fig:finger_contrast_tv} -and~\ref{fig:finger_contrast_atv}. Please note that this can be seen as -cheating. The anisotropy tensor, with one eigenvalue equal to 1, and the -other less or equal to 1, means that when all else being equal, the -amount of restoration will be reduced. +and~\ref{fig:finger_contrast_atv}. It is however not obvious what this +tells us about the quality of the anisotropic algorithm. As previously +mentioned, increasing the anisotropy (decreasing $\omega$) means +decreasing the amount of regularization done. And less contrast loss is +an expected outcome of decreasing the regularization, disregarding the +anisotropy. \begin{figure} \centering \input{fig/factory/contrast} - \caption{Showing some contrast loss.} + \caption{% + A one-dimensional slice through the restored fingerprint images + of Figure~\ref{fig:finger_contrast}, showing a difference in the + contrast. + } \label{fig:contrast_plot} \end{figure} diff --git a/theory.tex b/theory.tex index 1b20230..f504259 100644 --- a/theory.tex +++ b/theory.tex @@ -103,15 +103,6 @@ affects our numerical solution method. \fixme{Rating: 6/10} -\fixme{% - We also discuss whether to use the noisy image (yes, probably), - or the smoothed image (implicitly, complicated, possible - iteratively) in the structure tensor. -} - -\fixme{Using $\abs{v}$ for vector length has apparently spread from the -initial total variation $\abs{\nabla u}$.} - There are many possible choices for the anisotropy tensor $A(x)$. Our constraints are that we have assumed it to be continuous and symmetric positive definite, and we have some wishes for its properties. We would @@ -232,36 +223,35 @@ In smooth areas, $\sigma_1 \approx 1$ and $A(x)$ will be close to the identity matrix, while $A(x)$ will reduce the effect of $\nabla u$ across edges. -The parameter $\omega$ controls the amount of anisotropy in the method, -such that if it is very large we are left with the identity matrix. - Around corners $A(x)$ will be close to the identity matrix, which gives regularization similar to smooth areas. This is one possible down-side of this tensor choice. +The parameter $\omega$ controls the amount of anisotropy in the method, +such that if it is very large we are left with the identity matrix. Note +that changing the parameter $\omega$ implicitly affects how much +restoration is done. For an image $u$, decreasing $\omega$ will, all +else being equal, decrease the lowest eigenvalue of $A(x)$ and in turn +decrease the anisotropic total variation $\TVA(u)$. + For the case where $\lambda_1 = \lambda_2$, the $U(x)$ in our decomposition is not well-defined. This is not a problem though, since $\Sigma(x)$ will be the identity matrix, so any orthogonal matrix will suffice for $U(x)$. -\fixme{% - Discuss $\omega$. Discuss other tensor choices, the coherency thing - from Weickert. Discuss that this is not optimal in corners. -} +See \cite{weickert1999coherence} for a different tensor construction, +made to enhance flow structures in the image, relevant in for example +fingerprint analysis. + +\fixme{argue that $A$ is continuous?} \begin{figure} \input{fig/tensor_viz} \end{figure} -\fixme{% - The numerical problems should be discussed somewhere but maybe not - here. -} - \section{Well-posedness} -\fixme{Rating: 7/10, can give more references maybe, and be more -specific on the ``problems.''} +\fixme{Rating: 7/10} The theory of existence and uniqueness for these kinds of variational methods is a minefield of more or less subtle problems. Even if we @@ -275,11 +265,21 @@ The basic things we ask of our functional \begin{equation} F(u) = \int_\Omega (u - f)^2 + \beta \, \TVA(u) \end{equation} -are lower semicontinuity -and coercivity for existence, and convexity for uniqueness. We restrict +to have a \emph{well-posed} problem are lower semicontinuity +and coercivity for existence, convexity for uniqueness and stability. We restrict ourself to $L^2(\Omega)$ which makes sense with our fidelity term, assuming that $f \in L^2(\Omega)$ initially. +We consider the weak topology, as it will allow us to arrive at an +existence result relatively easily. +We say that a sequence $f_n$ in $L^2(\Omega)$ converges weakly to $f$ if +\begin{equation} + \lim_{n \to \infty} \int_\Omega f_n \, \xi \, dx = \int_\Omega f \, + \xi \, dx +\end{equation} +for all $\xi \in L^2(\Omega)$ and we write $f_n \rightharpoonup f$. A +weakly convergent sequence is a sequence that converges in the weak topology. + \subsection{Convexity} We start with convexity as it is the easiest to show. Being quadratic, @@ -311,13 +311,20 @@ given the existence of a minimizer, implies uniqueness. \subsection{Coercivity} -\fixme{Sequential coercivity?} - Coercivity relates to how the functional behaves when the norm of the -image $u$ tends to infinity. +image $u$ tends to infinity. What we need in order to conclude with existence +is sequential coercivity. Thus we need all level sets $F^\alpha = \{ u \in +L^2(\Omega) : F(u) \leq \alpha \}$ to be \emph{sequentially +pre-compact}, meaning that all sequences in the set contain a +subsequence converging to an element of the closure of the set. + It is obvious from the fidelity term that for some fixed $f \in L^2(\Omega)$, if $\lnorm{u}_{L^2} \to \infty$ then $F(u) \to \infty$. -This tells us something about the locality of potential minimizers. +This implies that all the level sets $F^\alpha$ are bounded. +Since $L^2(\Omega)$ is a Hilbert space, all bounded sequences contain a +weakly convergent subsequence. Thus all the level sets $F^\alpha$ are +weakly sequentially pre-compact, since all sequences in these sets have +a subsequence weakly converging to a point in the closure of the set. \begin{figure} \input{fig/lower_semicont} @@ -325,8 +332,6 @@ This tells us something about the locality of potential minimizers. \subsection{Lower semi-continuity} -\fixme{Find something else than $f$ for the example.} - The lower semicontinuity is the most tricky part, and this is where we will take some shortcuts. Lower semicontinuity for a functional $F$ at a point $u$ means that at points $u_\epsilon$ close to $u$, the functional @@ -350,19 +355,7 @@ further reading on the theory of sequential versus topological continuity see for example Megginson's book on Banach space theory \cite{megginson}. -\fixme{We consider the weak topology. Because it is convenient? No, -apparently to get the existence proof to work, but it does not say -anything about weakness in the book? We do however have sequential -coercivity in the assumptions of the Theorem, which might be a problem.} - -We say that a sequence $f_n$ in $L^2(\Omega)$ converges weakly to $f$ if -\begin{equation} - \lim_{n \to \infty} \int_\Omega f_n \, \xi \, dx = \int_\Omega f \, - \xi \, dx -\end{equation} -for all $\xi \in L^2(\Omega)$ and we write $f_n \rightharpoonup f$. An -important property is that all weakly convergent -sequences also converge in the weak topology. Also, the mapping $u +The mapping $u \mapsto \int_\Omega u \, \xi \, dx$ is weakly continuous for all $\xi \in L^2(\Omega)$. Note that when we write weakly continuous it is not a weaker version of continuity, but rather continuity in the weak @@ -441,9 +434,85 @@ semi-continuous functional since and thus our functional is sequentially weakly lower semi-continuous. The usual ways of going from coercivity and lower semicontinuity to -existence do not work in infinite dimensions. But with our coercivity -and sequential lower semi-continuity we can apply \fixme{Theorem 5.1 in -\cite{scherzer2008variational}} to conclude that we have existence. +existence do not work in infinite dimensions. But with sequential +coercivity and sequential lower semi-continuity in the weak topology we +can conclude that we have existence from \cite[Theorem +5.1]{scherzer2008variational}. + +\subsection{Stability} + +The last property normally required for well-posedness is stability. +Thus the question is how the solutions behave when we perturb the +problem. We stick to $L^2(\Omega)$ and adopt a proof given by Acar and +Vogel in \cite{acar1994analysis}. + +\begin{proposition} + Consider a sequence of perturbed problems + \begin{equation} + \min_{u \in L^2(\Omega)} F_n(u). + \end{equation} + Assume that each of the functionals $F_n$ is weakly sequentially + lower-semicontinuous, has a unique minimizer $u_n$, and in addition that they are + uniformly coercive in the sense that for any sequence $v_n \in + L^2(\Omega)$ + \begin{equation} + \lim_{n \to \infty} F_n(v_n) = \infty \quad \text{whenever } \, \lim_{n \to + \infty} \norm{v_n}_{L^2} = \infty. + \label{eq:fn_coercive} + \end{equation} + Assume also that the problems are consistent in the sense that $F_n \to + F$ uniformly, i.e.\ given an $R > 0$ and $\epsilon > 0$ there exists an + $N$ such that + \begin{equation} + \abs{F_n(u) - F(u)} < \epsilon \quad \text{whenever } \, n \geq N \text{ and + } \norm{u}_{L^2} \leq R. + \label{eq:fn_concistent} + \end{equation} + The problem is then stable with respect to the perturbations $F_n$ in + the sense that if $u^*$ is a minimizer of $F$ we have + \begin{equation} + u_n \rightharpoonup u^*. + \label{eq:un_u_weak} + \end{equation} +\end{proposition} +\begin{proof} + As $u_n$ is a minimizer of $F_n$ we have + \begin{equation} + F_n(u_n) \leq F_n(u^*) + \end{equation} + and using \eqref{eq:fn_concistent} we get + \begin{equation} + %\lim F_n(u_n) \leq \limsup F_n(u_n) \leq \limsup F_n(u^*) + %= F(u^*) < \infty. + \lim_{n \to \infty} F_n(u_n) \leq \lim_{n \to \infty} F_n(u^*) = + F(u^*) < \infty. + \end{equation} + From the coercivity in \eqref{eq:fn_coercive} we obtain that + the sequence $u_n$ is bounded in $L^2(\Omega)$ and thus contains a + weakly convergent subsequence. + + Assume that the convergence in \eqref{eq:un_u_weak} does not hold, + and denote the weakly convergent subsequence by $u_{n_j} + \rightharpoonup \bar{u} \neq u^*$. By the weak sequential lower + semicontinuity we have + \begin{equation} + \begin{aligned} + F(\bar{u}) &\leq \liminf_{j \to \infty} F\left(u_{n_j}\right) \\ + &= \liminf_{j \to \infty} \left(F\left(u_{n_j}\right) - + F_{n_j}\left(u_{n_j}\right)\right) + \liminf_{j \to \infty} + F_{n_j}\left(u_{n_j}\right) \\ + &= \liminf_{j \to \infty} F_{n_j}\left(u_{n_j}\right) \\ + &\leq F(u^*) + \end{aligned} + \end{equation} + which contradicts the uniqueness of $u^*$ and thus we can conclude + that the solutions of the perturbed problems converge $u_n + \rightharpoonup \bar{u} = u^*$. +\end{proof} + +Note that these are properties of the continuous problem, and that the +discretizations and numerical methods used for approximating a solution +can have its own issues with consistency, convergence and stability. \section{Anisotropic coarea formula} @@ -452,9 +521,11 @@ work.} \fixme{More flow between the sections. Somehow.} -The anisotropic coarea formula we will present here allows us, as in the -Euclidean case \fixme{we removed the Euclidean one}, to write the -anisotropic total variation as an integral over the levels of the image. +The anisotropic coarea formula we will present here allows us to write +the anisotropic total variation as an integral over the levels of the +image. For a similar presentation of the regular coarea formula for all +$f \in \BV(\Omega)$ see \cite{evans1991measure}. + First we define the thresholded image at level $s$. \begin{definition}[Thresholded image] The thresholded image at level $s$ is the function @@ -509,11 +580,11 @@ result from measure theory. \end{theorem} For a proof and further background on measure theory and Lebesgue integration theory see for example \cite{bartle1995elements}. -\begin{proof}[Proof of anisotropic coarea formula.] +\begin{proof}[Proof of the anisotropic coarea formula.] Assume that $u \in C^1(\Omega) \cap \BV(\Omega)$. The extension to - all functions $u \in \BV(\Omega)$ can be made by approximation - arguments but will not be considered here \fixme{theorem 5.3.3 in - ziemer}. + all functions $u \in \BV(\Omega)$ will not be considered here, but + for the case of regular total variation see \cite[Theorem + 5.3.3]{ziemer1989}. %\paragraph{First we prove that $\TVA(u) \leq \int_{-\infty}^\infty %\TVA(u^s) \, ds$.} @@ -586,6 +657,7 @@ integration theory see for example \cite{bartle1995elements}. \begin{equation} m(t) = \int_{\{ x \in \Omega : u(x) \leq t\}} \norm{\nabla u}_A \, dx, + \label{eq:mdef} \end{equation} and note that $m(\infty) = \TVA(u)$ and $m(-\infty) = 0$. Since $m(t)$ is non-decreasing with $t$, we can apply the existence @@ -603,9 +675,9 @@ integration theory see for example \cite{bartle1995elements}. \eta_r(t) = \begin{cases} 0 & \text{if } t < s, \\ (t - s)/r & \text{if } s \leq t < s + r, \\ - 1 & \text{if } t > s + r, + 1 & \text{if } t \geq s + r, \end{cases} - & \qquad + & \quad \eta_r\prime(t) = \begin{cases} 0 & \text{if } t < s, \\ 1 & \text{if } s < t < s + r, \\ @@ -620,14 +692,17 @@ integration theory see for example \cite{bartle1995elements}. \begin{equation} \int_\Omega - \eta_r(u) \diver \xi \, dx = \int_\Omega \eta_r\prime(u) \nabla u\cdot \xi \, dx - = \frac{1}{r} \int_{\{ s < u \leq s + r \}} \nabla u\cdot \xi + = \frac{1}{r} \int_{\{ s < u < s + r \}} \nabla u\cdot \xi \, dx, + \label{eq:eta_greens} \end{equation} for all vector fields $\xi \in C_c^\infty(\Omega, \mathbb{R}^2)$. - The measure of $\{ x : u(x) = s \text{ and } \nabla u(x) \neq - 0\}$ is zero for all $s$ following from \cite[Corollary \rom{1}, - Section 3.1.2]{evans1991measure}, and thus no problems arise there. - Assuming that $\norm{\xi}_A^* \leq 1$ we have + The measure of $\{ x : u(x) = \lambda \text{ and } \nabla u(x) \neq + 0\}$ is zero for all $\lambda$ following from \cite[Corollary \rom{1}, + Section 3.1.2]{evans1991measure}. Thus we can ignore the sets $\{ u + = s\}$ and $\{ u = s + r \}$. + Assuming that $\norm{\xi}_A^* \leq 1$ we obtain from \eqref{eq:mdef} + and \eqref{eq:eta_greens} that \begin{equation} \begin{aligned} \frac{m(s+r) - m(s)}{r} @@ -640,17 +715,23 @@ integration theory see for example \cite{bartle1995elements}. \label{eq:m_ineq_sr} \end{equation} As the limit when $r \to 0$ of the left-hand side exists almost - everywhere, suppose it exists at $s \in \mathbb{R}$. We apply - Lebesgue's dominated convergence theorem on the right-hand side in - \eqref{eq:m_ineq_sr}, using that $\abs{\eta_r(u) \diver \xi} \leq - \abs{u^s \diver \xi}$. As $\xi \in C^\infty_c(\Omega, - \mathbb{R}^2)$, it is bounded by the extreme value theorem, and thus - $\abs{u^s \diver \xi}$ is integrable. From \eqref{eq:m_ineq_sr} we - then obtain + everywhere, suppose it exists at $s \in \mathbb{R}$. The integrand + on the right-hand side $-\eta_r(u) \diver \xi$ approaches $-u^s + \diver \xi$ pointwise almost everywhere. We apply + Lebesgue's dominated convergence theorem + using that $\abs{\eta_r(u) \diver \xi} \leq + \abs{u^s \diver \xi}$ and that $\xi \in C^\infty_c(\Omega, + \mathbb{R}^2)$ is bounded by the extreme value theorem. Thus + $\abs{u^s \diver \xi}$ is integrable and + \begin{equation} + \int_\Omega -\eta_r(u) + \diver \xi \, dx \to \int_\Omega -u^s \diver \xi \, dx + \end{equation} + From \eqref{eq:m_ineq_sr} we then obtain \begin{equation} - m\prime(s) \geq - \int_\Omega u^s \diver \xi \, dx + m\prime(s) \geq - \int_\Omega u^s \diver \xi \, dx. \end{equation} - since $\eta_r(u) \to u^s$ when $r \to 0$. As this holds for any + As this holds for any $\norm{\xi}_A^* \leq 1$, we get from the extended total variation definition in \eqref{eq:extended_tv} that $m'(s) \geq \TVA(u_s)$ almost everywhere and conclude using \eqref{eq:tva_geq_mder} that @@ -670,7 +751,7 @@ function occurs at the boundary of the set. This is why we introduce the following definition of the anisotropic set perimeter. \begin{definition}[The anisotropic set perimeter] Given an anisotropy tensor $A$ the anisotropic perimeter of a set - $U$ is defined as + $U$ in $\Omega$ is defined as \begin{equation} \PerA(U;\Omega) = \TVA(\idfun_U). \end{equation} @@ -678,8 +759,8 @@ the following definition of the anisotropic set perimeter. \nomenclature{$\PerA(u;\Omega)$}{Anisotropic perimeter of set $U$ using anisotropy tensor $A$.}% The anisotropic set perimeter is not like the regular set perimeter and -does not measure the length of the boundary of the set, but it can be -calculated in the following way +does not measure the length of the boundary of the set, but it can for +sufficiently nice level sets be calculated in the following way \begin{equation} \begin{aligned} \PerA(\{ u > s \}; \Omega) &= \TVA(u^s) \\ &= \sup_{\norm{\xi}_A^* \leq 1} @@ -690,24 +771,20 @@ calculated in the following way \cdot \xi \, dt \\ &= \sup_{\norm{\eta} \leq 1} \int_{\partial \{ u > s\} } \nu_s \cdot \Ahalf \eta \, dt \\ - &= \int_{\partial \{ u > s\} } \Ahalf \nu_s - \cdot \frac{\Ahalf \nu_s}{\norm{\Ahalf \nu_s}} \, dt \\ &= \int_{\partial \{ u > s\} } \sqrt{\nu_s A \nu_s} \, dt. \end{aligned} \label{eq:perimeter_calc} \end{equation} -Here, $\nu_s$ is the unit normal of the level set $\{ u > s \}$ and by -applying the divergence theorem we have assumed that the boundary is -\fixme{piecewise smooth}, which holds for almost all level sets if $u$ is -differentiable. A consideration of exterior normals and perimeters of -level sets of any function $u \in \BV(\Omega)$ will not be considered -here, but can be found in for example \fixme{ziemer 5.4.1 and 5.5.1, in -the non-anisotropic case.} +Here, $\nu_s$ is the unit exterior normal of the level set $\{ u > s \}$. +Note that because of the compact support of $\xi$ in +Definition~\ref{def:extended_tv}, the parts of the boundary of $U$ that +overlap with the boundary of $\Omega$ will not be included in the +perimeter. -Further note that this is an integral of the anisotropic norm of the -unit normal vector. The length of the boundary would normally be -calculated by integrating the norm of the tangent vector. The connection -will be made after the Cauchy--Crofton formulas have been introduced. +Exterior normals and perimeters of +level sets of any function $u \in \BV(\Omega)$ will not be considered +here, but can for the isotropic case be found in for example +\cite[Section 5.4 and 5.5]{ziemer1989}. The anisotropic coarea formula allows us to transform the anisotropic total variation such that we are left with minimizing the following @@ -723,10 +800,7 @@ perimeter will be computed using a graph cut machinery. \section{Cauchy--Crofton formulas} -\fixme{Rating: 6/10, comment in the beginning that we are actually going -to introduce two formulas. The transition between the two can be -smoother as well. And we should refer to maybe Do Carmo, or sketch a -proof of the Euclidean one. The stuff after the proof could use some +\fixme{Rating: 7/10, the stuff after the proof could use some work.} \begin{figure} @@ -739,7 +813,8 @@ category often referred to as \emph{Cauchy--Crofton style formulas}, and give ways to measure geometric objects using the set of all lines in the plane. The formulas presented here will give a way to measure the length of a curve by counting the times it intersects line in the set of all -lines. +lines. The first formula will be for the isotropic case, and we will +use this to prove the anisotropic formula following it. We write $\mathcal{L}$ for the set of all lines in the plane, and parametrize them as shown in Figure~\ref{fig:line_param}. A @@ -1108,16 +1183,13 @@ If $P$ is a 90\textdegree{} rotation matrix we have \label{eq:per_to_length1} \end{equation} We simplify the equation by defining the metric tensor $M(x) = P A(x) -P^T$ and letting $\gamma = \partial U$ be an arclength parametrization of -the boundary of $U$ to obtain +P^T$ and letting $\gamma = \partial U \cap \Omega$ be an arclength parametrization of +the boundary of $U$ that does not overlap with the boundary of $\Omega$ \begin{equation} \PerA(U; \Omega) = \int_\gamma \sqrt{\langle \dot{\gamma}, M(x) \, - \dot{\gamma} \rangle }. + \dot{\gamma} \rangle } \, dt. \label{eq:per_to_length2} \end{equation} -Note that since $\Omega$ is open, we assume that $\gamma = \partial U -\subset \Omega$ and thus $\gamma$ can not include parts of the boundary -of $\Omega$. \fixme{rewrite} Now we make sure that all the assumptions of the Riemannian Cauchy--Crofton formula in Theorem~\ref{thm:riemannian_cauchy_crofton} @@ -1160,7 +1232,7 @@ perimeter in \eqref{eq:per_to_length2} as {2 \left( \nu^T \cdot M(x) \cdot \nu \right)^{\sfrac{3}{2}}} \, d\mathcal{L}(\ell_{\nu, \rho}) \, ds, \end{equation} -where $\gamma = \partial U$. +where $\gamma = \partial U \cap \Omega$. Note that $P$ does not affect the determinant, i.e.\ $\det A = \det PAP^T = \det M$, and from our decomposition in \eqref{eq:sigma_construction} we see that the transformation $PAP^T \to -- 2.47.3