% Title: glps_renderer figure
% Creator: GL2PS 1.3.8, (C) 1999-2012 C. Geuzaine
% For: Octave
-% CreationDate: Thu Dec 11 15:10:44 2014
+% CreationDate: Tue Dec 16 17:17:42 2014
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method. We will look at the functional we want to minimize, its
different forms and briefly discuss its well-posedness. Through the
coarea formula, the anisotropic total variation is rewritten as an
-integral of the perimeter of all the level sets.
+integral of the perimeter of all the level sets of the image.
Later, the Cauchy--Crofton formula is introduced to make it feasible to
calculate the perimeter of these level sets. All of this leads up to the
\section{Anisotropic total variation}
+\fixme{Rating: 8/10}
+
The method considered will build on the total variation regularization
method of Section~\ref{sec:total_variation}. From anisotropic diffusion
in Section~\ref{sec:anisotropic_diffusion} we borrow the idea of making
\TVA(u) = \int_\Omega \sqrt{\nabla u(x)^T A(x) \nabla u(x)} \, dx
\label{eq:aniso_tv_sqrt}
\end{equation}
-for all $u \in C^1(\Omega)$. If $A(x)$ is the identity
-matrix we get the regular total variation found in
-Definition~\ref{def:tv}. One problem with the regular total variation
-method is that when reducing the total variation, it will also try to
-reduce the variation over known edges in the image, which can lead to
-contrast loss, especially in thin details. By controlling $A(x)$ such
-that $\nabla u(x)$ is weighted down across known edges, we hope to
-retain the regularization properties of the method while reducing some
-of the negative effects. If the variation across an edge is ``ignored''
-by the energy functional, there is no gain in reducing the height of the
-edge as before.
+for all $u \in C^1(\Omega)$. If $A(x)$ is the identity matrix we get the
+regular total variation found in \eqref{eq:first_min_presentation}. One
+problem with the regular total variation method is that when reducing
+the total variation, it will also try to reduce the variation over known
+edges in the image, which can lead to contrast loss, especially in thin
+details. By controlling $A(x)$ such that $\nabla u(x)$ is weighted down
+across known edges, we hope to retain the regularization properties of
+the method while reducing some of the negative effects. If the variation
+across an edge is ``ignored'' by the energy functional, there is no gain
+in reducing the height of the edge as before.
+
+Note that $u(x)$ and $A(x)$ are always dependent on the position in the
+image $x$, but we will sometimes drop writing the $x$, when no
+misunderstandings are possible.
As we will not always be working with differentiable images, we extend
-the definition of the total variation functional to the space of
-functions of bounded variation $\BV(\Omega)$. Being symmetric positive
-definite, the matrix $A(x)$ can be factored into two symmetric matrices
-as $A(x) = \Ahalf(x) \Ahalf(x)$. We can then write
+the definition of the total variation functional. Being symmetric
+positive definite, the matrix $A(x)$ can be factored into two symmetric
+matrices as $A(x) = \Ahalf(x) \Ahalf(x)$. We can then write
\begin{align}
\TVA(u) &= \int_\Omega \abs{\Ahalf \nabla u} \, dx \\
&= \sup_{\abs{\xi(x)}
\subsection{Anisotropy tensor}
\label{sec:anisotropy_tensor}
+\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 symmetric positive
definite, and we have some wishes for its properties. We would first and
-foremost like it to down-weight the effect of $\nabla u$ in
-\eqref{eq:aniso_tv_sqrt} across true edges, while maintaining normal
-regularization properties in smooth sections.
+foremost like it to down-weight $\nabla u$ in \eqref{eq:aniso_tv_sqrt}
+across true edges, while maintaining normal regularization properties in
+smooth sections.
By true edges we mean that that we do not want the tensor to be
sensitive to noise in the image, and thus find edges where there are
structures, like corners and textures, which is why we introduce
the structure tensor
\begin{equation}
- S_\rho(x) := K_\rho * \left( \nabla f_\sigma(x) \otimes
- \nabla f_\sigma(x) \right).
+ S_\rho(x) := K_\rho * \big( \nabla f_\sigma(x) \otimes
+ \nabla f_\sigma(x) \big).
\label{eq:s_def}
\end{equation}
\end{figure}
First consider the tensor $S_0(x) = \nabla f_\sigma(x) \otimes
-\nabla f_\sigma(x)$. It obviously contains the same information
-as the edge detector itself. Its eigenvalues will be 0 and $\labs{\nabla
-f_\sigma(x)}^2$ with corresponding eigenvectors $v_1$ and $v_2$
-perpendicular and parallel to $\nabla f_\sigma(x)$ respectively.
-Figure~\ref{fig:edges} shows that the largest eigenvalue of the
-structure tensor is a good edge detector.
+\nabla f_\sigma(x)$. It obviously contains no more information
+than the edge detector itself. Its eigenvalues will be 0 and
+$\labs{\nabla f_\sigma(x)}^2$ with corresponding eigenvectors $v_1$ and
+$v_2$ perpendicular and parallel to $\nabla f_\sigma(x)$ respectively.
We are also interested in identifying and being sensitive to features in
-a neighborhood around the point, such as corners or curved edges and
+a neighborhood around the point $x$, such as corners or curved edges and
\fixme{coherent structures}. This is why we introduce the convolution
with $K_\rho$, which is done component-wise. The parameter $\rho$, called
the \emph{integration scale} thus controls the size of the neighborhood
expression for the eigenvalues
\begin{equation}
\lambda = \frac{1}{2} \left( s_{11} + s_{22} \pm \sqrt{(s_{11} -
- s_{22})^2 + 4 s_{12}^2} \right)
+ s_{22})^2 + 4 s_{12}^2} \right).
\label{eq:s_eigenvalues}
\end{equation}
\lambda_2 \approx 0$, while smooth areas would give $\lambda_1 \approx
\lambda_2 \approx 0$. In corners we have variation in the direction of
$v_1$ but also perpendicular to $v_1$, so we will have $\lambda_1
-\approx \lambda_2 \gg 0$.
-
-These eigenvalues indicate that the product $\nabla u^T S_\rho(x) \nabla
-u$ would amplify the effect of $\nabla u$ across edges, the opposite of
-what we want.
+\approx \lambda_2 \gg 0$. Thus the quantity $(\lambda_1 - \lambda_2)^2$
+will be large around edges and small in smooth or non-coherent areas.
We begin our anisotropy tensor construction by eigendecomposing the
structure tensor as
diagonal, while $U(x)$ is a rotation matrix and has the eigenvectors of
$S_\rho(x)$ as its columns.
-The eigen-decomposition gave us
+The eigen-decomposition give us
\begin{equation}
\Lambda(x) = \begin{pmatrix}
\lambda_1 & 0 \\
\end{aligned}
\label{eq:sigma_construction}
\end{equation}
-This way, $\sigma_1 \in (0, 1]$.
-The rotation is kept, while the size of the eigenvalues are changed.
-This should be visualized, with a figure showing the length and
-direction of the eigenvalues in the area around an edge.
+This way, $\sigma_1 \in (0, 1]$. The rotation is kept, while the size of
+the eigenvalues are changed.
+
+\fixme{%
+ Discuss $\omega$. Discuss other tensor choices, the coherency thing
+ from Weickert. Discuss that this is not optimal in corners.
+}
Note that in smooth parts of the image we have $\sigma_1 \approx
\sigma_2 = 1$ and the anisotropic total variation is close to the
regular total variation.
+\fixme{%
+ This should be visualized, with a figure showing the length and
+ direction of the eigenvalues in the area around an edge.
+}
+
\fixme{%
The numerical problems should be discussed somewhere but maybe not
- here. The eigenvalues are extracted such that
+ here.
}
\section{Well-posedness}
+\fixme{Rating: 7/10, can give more references maybe, and be more
+specific on the ``problems.''}
+
The theory of existence and uniqueness for these kinds of variational
methods is a minefield of more or less subtle problems. Even if we
restrict ourself to a nice space such as $L^2(\Omega)$ we will at some
The basic things we ask of our functional
\begin{equation}
- F(u) = \int_\Omega (u - v)^2 + \beta \, \TVA(u)
+ 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
ourself to $L^2(\Omega)$, and leave the extension to $\BV(\Omega)$ to
-someone else \fixme{ref}.
+someone else \fixme{Ref, and is it really an extension? More like a
+constriction?}
\subsection{Convexity}
-We start with convexity as it is the easiest to show. The fidelity term
-of our functional
+We start with convexity as it is the easiest to show. Being quadratic,
+the fidelity term of our functional
\begin{equation}
- \int_\Omega (u - v)^2 \, dx
+ \int_\Omega (u - f)^2 \, dx
\end{equation}
is obviously strictly convex. It can be shown by expanding and
rearranging the strict convexity condition
\begin{equation}
- \int_\Omega (\lambda x + (1-\lambda)y - v)^2 \, dx < \lambda
- \int_\Omega (x - v)^2 \, dx + (1 - \lambda) \int_\Omega (y - v)^2 \,
+ \int_\Omega (\lambda x + (1-\lambda)y - f)^2 \, dx < \lambda
+ \int_\Omega (x - f)^2 \, dx + (1 - \lambda) \int_\Omega (y - f)^2 \,
dx
\end{equation}
to obtain that it is equivalent to
\TVA(u) = \sup_{\norm{\xi}_A^* \leq 1} \int_\Omega u \diver \xi \,
dx
\end{equation}
-can be thought of as - and has the properties of - a norm, and is therefore
-convex. The sum of the two is thus strictly convex, which, given the
-existence of a minimizer, implies uniqueness.
+can be thought of as -- and has the properties of -- a norm, and is
+therefore convex. The sum of the two is thus strictly convex, which,
+given the existence of a minimizer, implies uniqueness.
\subsection{Coercivity}
We need coercivity to show that we cannot go further and further away to
obtain a better and better solution. This means that $\lnorm{u}_{L^2} \to
\infty$ should imply that $F(u) \to \infty$, which is obvious from the
-fidelity term for some fixed $v \in L^2(\Omega)$.
+fidelity term for some fixed $f \in L^2(\Omega)$.
From the coercivity we can conclude that we should be able to find some
near-minimal solutions somewhere in $L^2(\Omega)$ without ``going too far
\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
necessarily continuous with respect to the underlying topology. In
other words, in these spaces, there is a difference between sequential
continuity and topological continuity. Topological continuity implies
-sequential continuity, but not the other way. One way to get around this
-would be to consider topological \emph{nets}, an extension of sequences,
-but we will stick to proving sequential lower semi-continuity and
-referring to further theory. For further reading on the theory of
-sequential versus topological continuity see for example Megginson's
-book on Banach space theory \cite{megginson}.
+sequential continuity, but not the other way around. One way to get
+around this would be to consider topological \emph{nets}, an extension
+of sequences, but for simplicity, and because it might not add much to
+the understanding of the restoration method, we will stick to proving
+sequential lower semi-continuity and referring to further theory. For
+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?}
where all the $F_i$ are sequentially weakly lower semi-continuous, then $F$
is sequentially weakly lower semi-continuous, meaning that for any sequence
$u_k \rightharpoonup u$ we have $F(u) \leq \liminf_k F(u_k)$.
+ \label{lem:sup_semi_cont}
\end{lemma}
\begin{proof}
For any sequence $u_k \rightharpoonup u$ in $L^2(\Omega)$ we have
\end{proof}
From our functional in \eqref{eq:first_anisotropic_functional}, we first
-consider the fidelity term
+consider the fidelity term, and rewrite it as a supremum
\begin{equation}
\int_\Omega (u - v)^2 \, dx
%= \sup_{\substack{\xi \in L^2(\Omega) \\ \norm{\xi}_{L^2} \leq
\end{equation}
As the map $u \mapsto \int_\Omega (u - v) \xi\, dx$ is continuous in the
weak topology, the fidelity term is then a supremum of weakly continuous
-functionals, and is thus sequentially lower semi-continuous.
+functionals, and is thus by Lemma~\ref{lem:sup_semi_cont} sequentially
+lower semi-continuous.
For the regularization term the approach is similar. With our extended
definition from \eqref{eq:extended_tv}, we have
\end{equation}
This is again is a
supremum of weakly continuous functionals. Thus the regularization term
-is also sequentially weakly lower semi-continuous.
+is by Lemma~\ref{lem:sup_semi_cont} also 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}}.
+\cite{scherzer2008variational}} to conclude that we have existence.
\section{Anisotropic coarea formula}
+\fixme{Rating: 6.5/10, especially the part after the proof needs some
+work.}
+
+\fixme{More flow between the sections. Somehow.}
+
The anisotropic coarea formula we will present here allows us, as in the
-Euclidean case, to write the anisotropic total variation as an integral
-over the levels of the image. First we define the thresholded image at
-level $s$.
+Euclidean case \fixme{we removed the Euclidean one}, to write the
+anisotropic total variation as an integral over the levels of the image.
+First we define the thresholded image at level $s$.
\begin{definition}[Thresholded image]
The thresholded image at level $s$ is the function
\begin{equation}
\end{equation}
\label{def:thresholded_image}
\end{definition}
-This will be used throughout the rest of the thesis and allows us to
-write a non-negative image $u \geq 0$ as an integral over all the layers
+This will be used throughout the rest of the thesis. Note that given the
+thresholded image for every level, we are able to reconstruct the image
+as
+\begin{equation}
+ u(x) = \sup \left\{ s : u^s(x) = 1 \right\}.
+\end{equation}
+The thresholded image definition also allows us to write a non-negative
+image $u \geq 0$ as an integral over all the layers
\begin{equation}
u = \int_0^\infty u^s \, ds.
\label{eq:positive_int}
\end{equation}
Note that \eqref{eq:positive_int} only holds for a non-negative image,
-something which has to be worked around in the proof of the coarea
-formula. For the proof we will avoid measure theory and follow a proof
-given in \cite{olsson2009extending}.
+something which has to be worked around in the proof of the anisotropic
+coarea formula. For the proof we will avoid measure theory and follow a
+proof given in \cite{olsson2009extending}.
\begin{figure}
\input{fig/eta_r}
\begin{proof}
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.
+ arguments but will not be considered here. \fixme{ref maybe}
\paragraph{First we prove that $\TVA(u) \leq \int_{-\infty}^\infty
\TVA(u^s) \, ds$.}
For $u \leq 0$ we use that $\TVA(-v) = \TVA(v)$ and that $\TVA(c + v) =
\TVA(v)$ for any constant $c$. Note that $-u \geq 0$ and that its
thresholded image $(-u)^s$ will be exactly the opposite of $u^{-s}$,
- that is, $(-u)^s = 1 - u^{-s}$. This allows us to show that
+ that is $(-u)^s = 1 - u^{-s}$. This allows us to show that
\begin{equation}
\begin{aligned}
\TVA(u) &= \TVA(-u) \leq \int_0^\infty \TVA \big( (-u)^r
\end{equation}
and note that $m(\infty) = \TVA(u)$. Since $m(t)$ is non-decreasing
with $t$, we can apply the existence theorems of Lebesgue \cite[Thm.\
- 17.12, 18.14]{hewstrom} to conclude that $m'(t)$ exists almost
+ 17.12, 18.14]{hewstrom} to conclude that $m\prime(t)$ exists almost
everywhere and that the following inequality holds:
\begin{equation}
- \int_{-\infty}^\infty m'(t)\, dt \leq m(\infty) - m(-\infty) =
+ \int_{-\infty}^\infty m\prime(t)\, dt \leq m(\infty) - m(-\infty) =
\TVA(u).
\label{eq:tva_geq_mder}
\end{equation}
$\eta_r$ with our image $u$ and Green's identity we obtain
\begin{equation}
\int_\Omega - \eta_r(u) \diver \xi \, dx
- = \int_\Omega \eta_r'(u) \nabla u\cdot \xi \, dx
+ = \int_\Omega \eta_r\prime(u) \nabla u\cdot \xi \, dx
= \frac{1}{r} \int_{\{ s \leq u \leq s + r \}} \nabla u\cdot \xi
\, dx,
\end{equation}
As the limit of the left-hand side when $r \to 0$ exists almost
everywhere, suppose it exists at $s \in \mathbb{R}$, then
\begin{equation}
- m'(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
$\norm{\xi}_A^* \leq 1$, we get from the extended total variation
The anisotropic total variation of the thresholded images occurring in
the anisotropic coarea formula are very much related to the size of the
boundary of the level set, as the only variation in a characteristic
-function, occurs at the boundary of the set. This is why we introduce
+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
\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
+work.}
+
\begin{figure}
\input{fig/line_param}
\end{figure}
a number of interesting integral formulas. Several of them fall in a
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 a curve by
-counting the times it intersects line in the set of all lines.
-Intuitively, a long curve will intersect more lines.
+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. Intuitively, a long curve will intersect more lines.
We write $\mathcal{L}$ for the set of all straight lines in the plane,
and parametrize them as shown in Figure~\ref{fig:line_param}. Thus a
$\ell_{\phi, \rho} = \ell_{\nu, \rho}$ where $\nu$ is a unit vector
along the line, i.e.\ $\nu = (-\sin \phi, \cos \phi)^T$. By defining the
measure on this set $d\mathcal{L} = \dpdr$ we are ready to introduce the
-Cauchy--Crofton formula, which gives us a way to calculate the length of
-a curve by looking at the measure of the set of lines that intersect the
-curve. Note that the measure $d\mathcal{L}$ is invariant under rigid
-motions, meaning combinations of translations and rotations.
+Cauchy--Crofton formula. Note that the measure $d\mathcal{L}$ is
+invariant under rigid motions, meaning combinations of translations and
+rotations.
\nomenclature{$\mathcal{L}$}{The set of all straight lines in the
plane.}%
\nomenclature{$\ell_{\phi, \rho}$}{A line given by the angle of the
\emph{normal vector} around the boundary, while the anisotropic curve
length in \eqref{eq:riemannian_length} is the integral of the norm of
the \emph{tangent vector} of the curve. The normal and tangent vector
-are always perpendicular, so all we need is a 90\textdegree{} rotation.
-If $P$ is a 90\textdegree{} rotation matrix we have
+are always perpendicular, so all we need is a 90\textdegree{} rotation,
+assuming that the tangent vector is unit length. It will be if it arises
+from an arc length parametrization of the boundary. If $P$ is a
+90\textdegree{} rotation matrix we have
\begin{equation}
\begin{aligned}
\PerA(U; \Omega) &= \int_{\partial U} \sqrt{\langle
\label{eq:per_to_length1}
\end{equation}
We simplify the equation by defining the metric tensor $M(x) = P A(x)
-P^T$ and let $\gamma = \partial U$ be an arclength parametrization of
+P^T$ and letting $\gamma = \partial U$ be an arclength parametrization of
the boundary of $U$. Observe that a 90\textdegree{} rotation of the
-normal $\nu_{\partial U}$ gives us the tangent $\pm \dot{\gamma}$.
+normal $\nu_{\partial U}$ gives us the tangent $\dot{\gamma}$.
Inserting this into \eqref{eq:per_to_length1} we get
\begin{equation}
\PerA(U; \Omega) = \int_\gamma \sqrt{\langle \dot{\gamma}, M(x) \,
\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 $\Omega$.
+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}
the structure tensor $S_\rho(x)$. The extreme value theorem \fixme{ref}
states that a continuous real-valued function on a nonempty compact
space is bounded above. Thus the eigenvalues $\lambda_1$ and $\lambda_2$ of $S_\rho(x)$
-are bounded from above and by the construction in
+are bounded above and by the construction in
\eqref{eq:sigma_construction}, the smallest eigenvalue of
our anisotropy tensor $A(x)$ is bounded away from zero as
\begin{equation}
amounts to switching the two eigenvalues $\sigma_1$ and $\sigma_2$ in
$\Sigma$.
+\fixme{Small summary, complete functional, tensor construction.}
\chapter{Discrete formulation}
The whole transformation from the initial functional in
-\eqref{eq:first_anisotropic_functional} through the coarea formula and
-the Cauchy--Crofton formula was motivated by the discretization which
-will be described here. The coarea formula allows us to minimize the
-functional for each level separately, while the Cauchy--Crofton formula
-gives a feasible way of calculating the perimeter of each level set.
+\eqref{eq:first_anisotropic_functional} through the anisotropic coarea
+formula and the Cauchy--Crofton formula was motivated by the
+discretization which will be described here. The anisotropic coarea
+formula allows us to minimize the functional for each level separately,
+while the Cauchy--Crofton formula gives a feasible way of calculating
+the perimeter of each level set.
The integrals we had in \fixme{ref} will be approximated by discrete
sums, and we will be careful to discretize in a way which is consistent
\section{Discretization}
-We assume that our discrete images are given on a grid $\mathcal{G}$,
+\fixme{Rating: 4/10}
+
+\fixme{Sometimes I write uniform grid, and sometimes regular. Uniform is
+probably better.}
+
+We assume that our discrete images are given on a uniform grid $\mathcal{G}$,
where each discrete point is called a pixel and we further assume that
-each \emph{pixel} takes a value in the set of levels $\mathcal{P} = \{0,
-\hdots, L-1\}$. This is a reasonable assumption for grayscale images.
-(\fixme{repeated}). We now want to discretize the energy function in
-\eqref{eq:continuous_energy}. In addition, we want to decompose the
-energyfunction as a sum over all the levels $\mathcal{P}$.
+each pixel takes a value in the set of levels $\mathcal{P} = \{0,
+\hdots, L-1\}$. This is a reasonable assumption for digital grayscale
+images. We now want to discretize the energy function in
+\eqref{eq:continuous_energy}.
\subsection{Fidelity term}
+\fixme{Rating: 7/10}
+
Since it is not affected by our introduction of the anisotropy tensor,
the fidelity term can be discretized as in my project work
\cite{project}. For some pixel position $x \in \mathcal{G}$ and some
\big( N_x(\lambda + 1) - N_x(\lambda) \big) \, u_x^\lambda + N_x(0).
\end{equation}
As our domain is discretized uniformly, we drop the constant
-$\Delta x$, and absorb it into our parameter $\beta$. Note that since
+$\Delta x$, and absorb it into our parameter $\beta$ of
+\eqref{eq:first_anisotropic_functional}. Note that since
our image takes values in $\mathcal{P} = \{0, \hdots, L-1\}$, the
thresholded image $u^{L-1}$ is equal to zero everywhere.
\subsection{Regularization term}
\label{sec:disc_regularization}
+\fixme{Rating: 6/10}
+
Discretizing the regularization term is more challenging. We introduce
the discrete levels to get
\begin{equation}
\, \Delta \lambda.
\label{eq:per_approx1}
\end{equation}
-Note that we will later ignore the $\Delta \lambda$ difference, as we can
-just absorb it into the $\beta$ parameter of
+As with the $\Delta x$ difference, we can absorb the $\Delta \lambda$
+difference into the $\beta$ parameter of
\eqref{eq:first_anisotropic_functional}. The perimeter is then
calculated using a discretized version of the Cauchy--Crofton formula
introduced in Theorem~\ref{thm:riemannian_cauchy_crofton}. Again, we
value of 0 or 1 on this grid, as shown in Figure \fixme{ref}.
The choice of our discrete set of lines $\mathcal{L}_D$ is important, as
-it will decide the accuracy of our approximation in
-\eqref{eq:cauchy_crofton_approx1}. We need some sensible restrictions on
+it will decide the accuracy of our approximation.
+We need some sensible restrictions on
the set $\mathcal{L}_D$ to simplify the further discussion. All lines
intersect at least two grid points, and from the periodicity of our grid
they thus intersect an infinite number of grid points. This puts some
\fixme{maybe with a $w_{xy}$ definition}
-\fixme{edges here are a bit different from edges later}
-
\subsubsection{Consistency}
+\fixme{Rating: 6/10}
+
Consistency relates to how well a solution to the continuous problem
fits in the discretized equation, in other words, whether the
discretized equation approximates the continuous one.
projects / master / commitdiff