<bibtex>\r
\r
+@preamble{\r
+ "\def\cprime{$'$} "\r
+}\r
+@book{ziemer1989,\r
+ AUTHOR = {Ziemer, William P.},\r
+ TITLE = {Weakly differentiable functions},\r
+ SERIES = {Graduate Texts in Mathematics},\r
+ VOLUME = {120},\r
+ PUBLISHER = {Springer-Verlag, New York},\r
+ YEAR = {1989},\r
+ PAGES = {xvi+308},\r
+ ISBN = {0-387-97017-7},\r
+ MRCLASS = {46E35},\r
+ MRNUMBER = {1014685 (91e:46046)},\r
+ MRREVIEWER = {V. M. Gol{\cprime}dshte{\u\i}n},\r
+ DOI = {10.1007/978-1-4612-1015-3},\r
+ URL = {http://dx.doi.org/10.1007/978-1-4612-1015-3},\r
+}\r
+\r
+@article{acar1994analysis,\r
+ AUTHOR = {Acar, R. and Vogel, C. R.},\r
+ TITLE = {Analysis of bounded variation penalty methods for ill-posed\r
+ problems},\r
+ JOURNAL = {Inverse Problems},\r
+ FJOURNAL = {Inverse Problems. An International Journal on the Theory and\r
+ Practice of Inverse Problems, Inverse Methods and Computerized\r
+ Inversion of Data},\r
+ VOLUME = {10},\r
+ YEAR = {1994},\r
+ NUMBER = {6},\r
+ PAGES = {1217--1229},\r
+ ISSN = {0266-5611},\r
+ CODEN = {INPEEY},\r
+ MRCLASS = {65J20 (47A50 49J45)},\r
+ MRNUMBER = {1306801 (95i:65092)},\r
+ URL = {http://stacks.iop.org/0266-5611/10/1217},\r
+}\r
+\r
+@article{weickert1999coherence,\r
+ title={Coherence-enhancing diffusion filtering},\r
+ author={Weickert, Joachim},\r
+ journal={International Journal of Computer Vision},\r
+ volume={31},\r
+ number={2-3},\r
+ pages={111--127},\r
+ year={1999},\r
+ publisher={Springer}\r
+}\r
+\r
+@book{jahne,\r
+ title={{D}igital {I}mage {P}rocessing},\r
+ author={J{\"a}hne, Bernd},\r
+ publisher={Springer-Verlag, Berlin Heidelberg},\r
+ year={2005}\r
+}\r
+\r
+@book {aubert2006proc,\r
+ AUTHOR = {Aubert, Gilles and Kornprobst, Pierre},\r
+ TITLE = {Mathematical problems in image processing},\r
+ SERIES = {Applied Mathematical Sciences},\r
+ VOLUME = {147},\r
+ EDITION = {Second},\r
+ PUBLISHER = {Springer, New York},\r
+ YEAR = {2006},\r
+ PAGES = {xxxii+377},\r
+ ISBN = {978-0387-32200-1; 0-387-32200-0},\r
+ MRCLASS = {94A08 (35B27 35J60 35L70 49J10 68T45)},\r
+ MRNUMBER = {2244145 (2007j:94004)},\r
+}\r
+\r
@book{do1976differential,\r
AUTHOR = {do Carmo, Manfredo P.},\r
TITLE = {Differential geometry of curves and surfaces},\r
doi={10.1007/978-0-387-92920-0_23},\r
title={Total Variation in Imaging},\r
url={http://dx.doi.org/10.1007/978-0-387-92920-0_23},\r
- publisher={Springer New York},\r
+ publisher={Springer, New York},\r
author={Caselles, V. and Chambolle, A. and Novaga, M.},\r
pages={1016-1057}\r
}\r
\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
$A(x)$ 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
\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,
\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}
\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
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
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}
\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
\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 $\TVA(u) \leq \int_{-\infty}^\infty
%\TVA(u^s) \, ds$.}
\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
\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, \\
\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}
\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
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}
\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}
\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
\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}
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
\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}
{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
projects / master / commitdiff