\nomenclature{$\lnorm{\xi}_A$}{The norm $\sup_x (\xi^T A
\xi)^{\sfrac{1}{2}}$}%
\nomenclature{$\lnorm{\eta}_A^*$}{The norm $\sup_x (\eta^T A^{-1}
-\eta)^{\sfrac{1}{2}}$.}%
+\eta)^{\sfrac{1}{2}}$}%
present the formal definition of the anisotropic total variation.
\begin{definition}[Anisotropic total variation]
For a function $u \in L^2(\Omega)$ and a continuous symmetric
\label{def:extended_tv}
\end{definition}
\nomenclature{$\TVA(u)$}{Anisotropic total variation of image $u$ given
-the anisotropy tensor $A$.}%
+the anisotropy tensor $A$}%
With this extended definition, we have arrived at a minimization problem
where we seek to find a minimizer of the functional
\end{aligned}
\label{eq:sigma_construction}
\end{equation}
-\nomenclature{$\omega$}{Anisotropy parameter.}
+\nomenclature{$\omega$}{Anisotropy parameter}
Thus the eigenvectors of $A(x)$ and $S_\rho(x)$ are equal, while the
eigenvalues are different. A visualization of the two tensors can be
seen in Figure~\ref{fig:tensor_viz} where the two tensors are shown at
\end{equation}
\end{definition}
\nomenclature{$\PerA(U;\Omega)$}{Anisotropic perimeter of set $U$ using
-anisotropy tensor $A$.}%
+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 for
sufficiently nice level sets be calculated in the following way
Cauchy--Crofton formula. Note that the measure $d\mathcal{L}$ is
invariant under rotations.
\nomenclature{$\mathcal{L}$}{The set of all straight lines in the
-plane.}%
+plane}%
\nomenclature{$\ell_{\phi, \rho}$}{A line given by the angle of the
-normal $\phi$ and the distance to origin $\rho$.}%
+normal $\phi$ and the distance to origin $\rho$}%
\nomenclature{$\ell_{\nu, \rho}$}{A line given by a tangent vector $\nu$
-and the distance to origin $\rho$.}%
+and the distance to origin $\rho$}%
\begin{theorem}[The Euclidean Cauchy--Crofton formula]
Given a differentiable curve $C$ in $\mathbb{R}^2$, the length of
this curve $\abs{C}$ is related to the set of lines $\mathcal{L}$ as
product of two vectors $a$ and $b$ in a point $x$ is calculated as
$\langle a, b\rangle_M = \langle a, M(x) b \rangle$, then the length of
a curve $\gamma$ parametrized by some parameter $t$ becomes
-\nomenclature{$M(x)$}{A metric tensor.}%
-\nomenclature{$\abs{C}$}{The length of the curve $C$.}%
+\nomenclature{$M(x)$}{A metric tensor}%
+\nomenclature{$\abs{C}$}{The length of the curve $C$}%
\nomenclature{$\abs{C}_M$}{The length of the curve $C$ calculated using
-the metric tensor $M$.}%
+the metric tensor $M$}%
\begin{equation}
\abs{\gamma}_M = \int_\gamma \sqrt{\langle \dot{\gamma},
M\big(\gamma(t)\big) \, \dot{\gamma} \rangle} \, dt.
transformation $F : \mathcal{L} \to \mathcal{L}$, which maps
$\ell_{\phi, \rho} \mapsto \Mhalf \ell_{\phi, \rho}$.
\nomenclature{$J_M(\ell_{\phi, \rho})$}{Jacobian of the coordinate
- transformation induced by the metric tensor $M$.}
+ transformation induced by the metric tensor $M$}
We will now compute the Jacobian $J_M(\ell_{\phi, \rho})$. As $M\in
\mathbb{R}^{2\times2}$ is symmetric, so is $M^{\sfrac{1}{2}}$, and
\begin{equation}
N_x(k) = \abs{k - f_x}^2
\end{equation}
-\nomenclature{$N_x(k)$}{Helper function $N_x(k) = \abs{k-f_x}^2$.}%
+\nomenclature{$N_x(k)$}{Helper function $N_x(k) = \abs{k-f_x}^2$}%
which is the value of the fidelity term if we were to give $u_x$ a value
of $k$. This allows us write
\begin{equation}
Figure~\ref{fig:line_neigh} gives all lines going through the point considered.
Figure~\ref{fig:line_family} shows all lines of a given family, i.e.\
lines having the same angle parameter $\phi$.
-\nomenclature{$\mathcal{N}(x)$}{Neighborhood of pixel $x$.}%
+\nomenclature{$\mathcal{N}(x)$}{Neighborhood of pixel $x$}%
Thus not only have we discretized the set of lines, but each line is
made up of edges going from one grid point to the next. We will
flow network, but as all our graphs will be capacitated from this point,
we will just call them graphs and we write $G = (V, E, c)$.
\nomenclature{$G = (V,E,c)$}{A graph given by the set of vertices $V$,
-the set of edges $E$, and the capacity function $c$.}%
-\nomenclature{$c(u,v)$}{The capacity function $c : V \times V \to \left[0, \infty \right)$.}%
+the set of edges $E$, and the capacity function $c$}%
+\nomenclature{$c(u,v)$}{The capacity function $c : V \times V \to \left[0, \infty \right)$}%
There are two special vertices in the graph, the source $s$ and the
sink $t$. Contrary to other vertices, which can neither produce nor
unlimited amount of flow. The most basic problem in graph flow theory
is the question of how much flow it is possible to send through the
graph from the source to the sink.
-\nomenclature{$s$}{The source vertex.}%
-\nomenclature{$t$}{The sink vertex.}%
+\nomenclature{$s$}{The source vertex}%
+\nomenclature{$t$}{The sink vertex}%
What we seek in our final graph is a minimum $s$-$t$-cut, a ``minimal''
line through the graph that cuts a set of edges and
which represents the amount of flow which disappears in vertex
$u$. Equivalent to the preflow conservation constraint is stating that
$e(u) \geq 0$ for all vertices $u \in V - \{s,t\}$.
-\nomenclature{$e(u)$}{Excess in vertex $u$.}%
+\nomenclature{$e(u)$}{Excess in vertex $u$}%
%The idea of the algorithm is to maintain a height map of the
%vertices in the graph where connected vertices can not have a large
on the length from $u$ to $t$ in $G_f$ which is why it is also often
called a distance labeling.
\nomenclature{$d(u)$}{Height map or distance labeling $d : V \to
-\mathbb{N}$.}%
+\mathbb{N}$}%
A vertex $u$ is \emph{active} if $u \in V - \{s,t\}$, it has positive
excess ($e(u) > 0$) and $d(u) < N$. These are the vertices we want to
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
rectangular, open domain. Because of limitations in the numerical method
-\nomenclature{$f$}{Original, noisy image.}%
-\nomenclature{$\Omega$}{Rectangular, open domain.}%
+\nomenclature{$f$}{Original, noisy image}%
+\nomenclature{$\Omega$}{Rectangular, open domain}%
used, the codomain is $\mathbb{R}$ and we are thus restricted to
monochrome, or grayscale images. Such images are produced in large
numbers by for example ultrasound, X-ray and MRI machines.
We will assume that the given image $f$ is a combination of an
underlying, actual image $u^*$, and some noise $\delta$. The simplest
-\nomenclature{$\delta$}{Noise.}%
+\nomenclature{$\delta$}{Noise}%
model is additive noise where the assumption is that $f = u^* + \delta$.
-\nomenclature{$u^*$}{Usually unknown, actual image without noise.}%
+\nomenclature{$u^*$}{Usually unknown, actual image without noise}%
There is also multiplicative noise where $f = u^* \cdot \delta$. An
other much seen noise type is salt and pepper noise, which is when black
and white pixels randomly appear in the image.
y^2}{2\sigma^2} \right).
\label{eq:gaussian_function}
\end{equation}
-\nomenclature{$K_\sigma(x,y)$}{The Gaussian kernel.}%
+\nomenclature{$K_\sigma(x,y)$}{The Gaussian kernel}%
In the discrete setting where the image consists of a grid of pixels,
the Gaussian blur amounts to calculating each pixel in the output image
as a weighted average of its neighboring pixels in the input image.
\end{aligned}
\right.
\end{equation}
-\nomenclature{$\alpha(\nabla u)$}{Scalar thermal diffusivity.}%
+\nomenclature{$\alpha(\nabla u)$}{Scalar thermal diffusivity}%
The thermal diffusivity $\alpha(\nabla u) = \alpha(x, \nabla u)$ is material dependent, and can also
vary throughout the object. It specifies how well heat travels through
the specific point in the object. We can make use of this in the image
\label{eq:aniso_diff}
\end{equation}
\nomenclature{$A(u)$}{Thermal diffusivity tensor, or anisotropy
-tensor.}%
+tensor}%
where $\nu$ is the outer normal of $\Omega$. The tensor $A(u)$
is constructed such as to diminish the effect of $\nabla
u$ across what we believe to be edges in the image. This way, there will
also be less diffusion through these edges. Weickert
\cite{weickert1998anisotropic} suggests constructing $A(u)$ based on the
edge estimator $\nabla u_\sigma$ where
-\nomenclature{$\nabla u_\sigma$}{Edge estimator.}%
+\nomenclature{$\nabla u_\sigma$}{Edge estimator}%
\begin{equation}
u_\sigma := K_\sigma * \tilde{u}
\end{equation}
\nomenclature{$u_\sigma$}{Edge estimator, image $u$ smoothed with a Gaussian of
-parameter $\sigma$.}%
+parameter $\sigma$}%
and $\tilde{u}$ is an extension of $u$ from $\Omega$ to $\mathbb{R}^2$
\nomenclature{$\tilde{u}$}{Symmetric extension of $u$ from $\Omega$ to
$\mathbb{R}^2$}%
\begin{equation}
S_\rho(x) := K_\rho * (\nabla u_\sigma \otimes \nabla u_\sigma),
\end{equation}
-\nomenclature{$S_\rho(x)$}{Structure tensor.}%
+\nomenclature{$S_\rho(x)$}{Structure tensor}%
where the convolution with the Gaussian function $K_\rho$ is done
component-wise. The anisotropy tensor $A(u)$ can then be constructed
based on the eigenvectors and eigenvalues of $S_\rho(x)$.
\label{eq:first_min_presentation}
\end{gathered}
\end{equation}
-\nomenclature{$F(u)$}{Variational functional on $u$.}%
\nomenclature{$L^p(\Omega)$}{Real functions $f$ on $\Omega$ for which
-$\int_\Omega \labs{f}^p < \infty$.}%
-\nomenclature{$\beta$}{Regularization parameter.}%
+$\int_\Omega \labs{f}^p < \infty$}%
+\nomenclature{$\beta$}{Regularization parameter}%
where $p$ is normally taken to be 1 or 2.
The fidelity term penalizes images $u$ that are far from the original
image $f$. The
$\Omega$.
\label{def:tv}
\end{definition}
-\nomenclature{$\TV(u)$}{Total variation of the image $u$.}%
+\nomenclature{$\TV(u)$}{Total variation of the image $u$}%
\nomenclature{$C^\infty_c\left(\Omega, \mathbb{R}^2\right)$}{The space of smooth
functions from $\Omega$ to $\mathbb{R}^2$ with compact support in
-$\Omega$.}%
-\nomenclature{$\varphi$}{Test function.}%
+$\Omega$}%
+\nomenclature{$\varphi$}{Test function}%
Note that since $\Omega$ is open and bounded, the test functions
$\varphi$ vanish on the boundary of $\Omega$. Thus no variation is
\end{equation}
\end{definition}
\nomenclature{$\BV(\Omega)$}{Functions of bounded variation in
-$\Omega$.}%
+$\Omega$}%
Our optimization problem has thus become
\begin{equation}
\min_{u \in \BV(\Omega)} \int_\Omega \abs{u - v}^p \, dx + \beta \,
In the discrete setting our image consists of pixels, and is represented
by a function $u : \mathcal{G} \to \mathcal{P}$ where $\mathcal{G}$ is a
-\nomenclature{$\mathcal{G}$}{Regular grid of pixels over $\Omega$.}%
-\nomenclature{$\mathcal{P}$}{Discrete set of pixel values, or levels.}%
+\nomenclature{$\mathcal{G}$}{Regular grid of pixels over $\Omega$}%
+\nomenclature{$\mathcal{P}$}{Discrete set of pixel values, or levels}%
regular grid over $\Omega$, and $\mathcal{P} = \{0, \hdots, L-1\}$ is
the discrete set of pixel values, or \emph{levels}. We denote the value
in pixel $x$ as $u(x) = u_x$.
$\{ u > \lambda\}$, defined as the set $\{ x \in \Omega : u_x > \lambda
\}$. The thresholded image $u^\lambda$, an indicator function, is then
defined as
-\nomenclature{$u^\lambda$}{Thresholded image at level $\lambda$.}%
+\nomenclature{$u^\lambda$}{Thresholded image at level $\lambda$}%
\begin{equation}
u^\lambda = \idfun_{u > \lambda}.
\end{equation}
Here, $\idfun_E$ signifies the characteristic function of the set $E$,
-\nomenclature{$\idfun_E$}{Characteristic function of the set $E$.}%
+\nomenclature{$\idfun_E$}{Characteristic function of the set $E$}%
the function which is equal to one in every point in $E$, and zero
elsewhere.
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