\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
-\eqref{first_min_presentation}. One problem with the regular total
+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 $A(x)$
where we seek to find a minimizer of the functional
\begin{equation}
F(u) = \int_\Omega (u - f)^2 \, dx + \beta \, \TVA(u)
+ \label{eq:first_anisotropic_functional}
\end{equation}
The question is then how to construct this anisotropy tensor $A(x)$ to
get the improvements we hope for, and how this affects our numerical
Edges can be found in many different ways, but as suggested by Weickert
in his book on Anisotropic Diffusion \cite{weickert1998anisotropic}, and
briefly mentioned in Section \ref{sec:anisotropic_diffusion}, a good
-starting point is the structure tensor
+starting point is the \emph{edge detector} $\nabla \tilde{f}_\sigma$ and
+the structure tensor
\begin{equation}
S_\rho(x) := K_\rho * \left( \nabla \tilde{f}_\sigma(x) \otimes
\nabla \tilde{f}_\sigma(x) \right),
control how sensitive the edge detector $\nabla \tilde{f}_\sigma$ is to
noise and small details.
+\begin{figure}
+ \centering
+ \includegraphics[width=0.4\textwidth]{fig/edges.png}
+ \caption{
+ The largest eigenvalue of the structure tensor is $\abs{\nabla
+ \tilde{f}_\sigma(x)}^2$ and is as we see a good edge detector.
+ }
+ \label{fig:edges}
+\end{figure}
+
First consider the tensor $S_0(x) = \nabla \tilde{f}_\sigma(x) \otimes
\nabla \tilde{f}_\sigma(x)$. It obviously contains the same information
as the edge detector itself. Its eigenvalues will be 0 and $\labs{\nabla
\tilde{f}_\sigma(x)}^2$ with corresponding eigenvectors $v_1$ and $v_2$
perpendicular and parallel to $\nabla \tilde{f}_\sigma(x)$ repectively.
+Figure \ref{fig:edges} shows that the largest eigenvalue of the
+structure tensor is a good edge detector.
We are also interested in identifying and being sensitive to features in
a neighborhood around the point, such as corners or curved edges and
Being a smoothed combination of symmetric positive semi-definite
matrices, $S_\rho(x)$ can easily be verified to be symmetric positive
semi-definite itself. We order its two real eigenvalues $\lambda_1 \geq
-\lambda_2$ with corresponding eigenvectors $v_1$ and $v_2$. The vector
+\lambda_2$ with corresponding eigenvectors $v_1$ and $v_2$. From the
+characteristic polynomial of $S_\rho(x) = (\begin{smallmatrix} s_{11} &
+s_{12} \\ s_{12} & s_{22} \end{smallmatrix})$ we obtain a closed form
+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)
+ \label{eq:s_eigenvalues}
+\end{equation}
+
+The vector
$v_1$ will then indicate the direction of most variation in the
neighborhood. From this we see that an edge would give $\lambda_1 \gg
\lambda_2 \approx 0$, while smooth areas would give $\lambda_1 \approx
$v_1$ but also perpendicular to $v_1$, so we will have $\lambda_1
\approx \lambda_2 \gg 0$.
-These eigenvalues indicate that the product $\sqrt{\nabla u^T S_\rho(x)
-\nabla u}$ would amplify the effect of $\nabla u$ across edges, the
-opposite of what we want.
+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.
We begin our anisotropy tensor construction by eigendecomposing the
structure tensor as
\end{equation}
To reverse the effect of $S_\rho(x)$ such that $A(x)$ actually weighs down
the norm across edges, we need $\sigma_1 \leq \sigma_2$, so we construct
-\begin{align}
- \sigma_1 &= \left(1 + \frac{(\lambda_1 -
- \lambda_2)^2}{\gamma^2}\right)^{-1}, \\
- \sigma_2 &= 1.
-\end{align}
+\begin{equation}
+ \begin{aligned}
+ \sigma_1 &= \left(1 +
+ \frac{(\lambda_1 - \lambda_2)^2}{\gamma^2}\right)^{-1}, \\
+ \sigma_2 &= 1.
+ \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
which proves that $F$ is sequentially weakly lower semi-continuous.
\end{proof}
-From our functional in \fixme{ref}, we first consider the fidelity term
+From our functional in \eqref{eq:first_anisotropic_functional}, we first
+consider the fidelity term
\begin{equation}
\int_\Omega (u - v)^2 \, dx
%= \sup_{\substack{\xi \in L^2(\Omega) \\ \norm{\xi}_{L^2} \leq
can be written as an integral over all the levels
\begin{equation}
\TVA(u) = \int_{-\infty}^\infty \TVA(u^s) \, ds.
+ \label{eq:anisotropic_coarea}
\end{equation}
\label{thm:anisotropic_coarea}
\end{theorem}
This coarea formula is our first step in transforming the anisotropic
total variation into an easily discretizisable expression.
-This is where we introduce the perimeter, or more specifically
+The anisotropic total variation of the thresholded images occuring 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
+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
\end{equation}
Because of the convolutions with the Gaussian function, this is a smooth
continuous map from $\bar{\Omega}$ to $\mathbb{R}^{2 \times 2}$. As we
-can see in \fixme{ref} the eigenvalues are the roots of a monic
-polynomial and thus depend continuously on the coefficients of the
-polynomial, which in turn are continuous functions of the elements in
+can see in \eqref{eq:s_eigenvalues}, the eigenvalues
+depend continuously on the coefficients of the
+elements in
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 $s_1$ and $s_2$ of $S_\rho(x)$
-are bounded from above and by construction the smallest eigenvalue of
+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
+\eqref{eq:sigma_construction}, the smallest eigenvalue of
our anisotropy tensor $A(x)$ is bounded away from zero as
\begin{equation}
\lambda_2 = \left(1
+ \frac{(s_1 - s_2)^2}{\gamma^2}\right)^{-1} \geq \left(1 +
- \frac{s_1^2}{\gamma^2}\right)^{-1} \geq Q > 0.
+ \frac{s_1^2}{\gamma^2}\right)^{-1} \geq k > 0.
\end{equation}
\fixme{bah, now gamma is a curve...}
-Hence, our metric tensor $M(x) = P A(x) P^T$ in
-\eqref{eq:per_to_length2} fulfills all the assumptions of Theorem
+Hence, our metric tensor $M(x) = P A(x) P^T$ and curve length
+calculation in
+\eqref{eq:per_to_length2} fulfill all the assumptions of Theorem
\ref{thm:riemannian_cauchy_crofton}. \fixme{did we argue for
-continuous?} Inserting the Cauchy--Crofton formula, we obtain the final
-expression for the continuous anisotropic total variation
+continuous?} Thus we can apply the Cauchy--Crofton formula of
+\eqref{eq:riemannian_cauchy_crofton} to calculate the curve length in
+\eqref{eq:per_to_length2} and through the anisotropic coarea formula in
+Theorem \ref{thm:anisotropic_coarea} obtain our final expression for the
+continuous anisotropic total variation
\begin{equation}
\TVA(u) = \int_{-\infty}^{\infty} \int_\mathcal{L} \sum_{x \in
\ell_{\nu, \rho} \cap \gamma_s } \frac{\det M(x)}
amounts to switching the two eigenvalues $\sigma_1$ and $\sigma_2$ in
$\Sigma$.
-The whole transformation from the initial formulation in \fixme{ref}
+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 wich will be described next. We will see that we
can minimize the energy functional for each level separately, and that
\subsection{Fidelity term}
-For some pixel position $x \in \mathcal{G}$ and some level value $k \in
+The fidelity term can be discretize as in my project work
+\cite{project}. For some pixel position $x \in \mathcal{G}$ and some level value $k \in
\mathcal{L}$, we define the following function
\begin{equation}
N_x(k) = \abs{k - v_x}^p
projects / master / commitdiff
