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+ makeindex main.nlo -s nomencl.ist -o main.nls
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\end{tikzpicture}
\caption{
- A visual argument that $\delta^2 = \Delta \rho \norm{e}$. If extended
- to the whole plane, there will be the same amount of blue squares as
- red squares, so their area must be equal.
+ A visual argument showing that $\delta^2 = \Delta \rho \norm{e}$. If
+ extended to the whole plane, there will be the same amount of blue
+ squares as red rectangles, so their areas must be equal.
}
\label{fig:area_proof}
about the different noise models, and common assumptions concerning the
noise.
+\fixme{introduce u and f as images}
+\fixme{introduce Omega}
+
\section{Diffusion filtering}
Diffusion filtering is a broad group of filtering or restoration methods
y^2}{2\sigma^2} \right).
\label{eq:gaussian_function}
\end{equation}
+\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.
\subsection{Non-linear diffusion}
In the theory of the heat equation one can introduce a \emph{thermal
-diffusivity}
-$\alpha$ such that
+diffusivity} $\alpha$ such that
\begin{equation}
\begin{cases}
\partial_t u &= \diver \big( \alpha(u) \nabla u\big) \\
u |_{t=0} &= f
\end{cases}
\end{equation}
-The thermal diffusivity $\alpha$ is material dependant, 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 restoration context by
-controlling the diffusivity in different parts of the image, where our
-goal is to reduce noise without loosing image detail. Optimally,
-we would like there to be a lot of diffusion in smooth parts of the
-image, and not so much in areas with a lot of details.
+\nomenclature{$\alpha(u)$}{Thermal diffusivity scalar.}%
+The thermal diffusivity $\alpha$ 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
+restoration context by controlling the diffusivity in different parts of
+the image, where our goal is to reduce noise without loosing image
+detail. Optimally, we would like there to be a lot of diffusion in
+smooth parts of the image, and not so much in areas with a lot of
+details.
One much-studied non-linear diffusion equation is the Perona--Malik
equation
contribute to an increase in the size of $\abs{\nabla u}$, the general
assumption is that this is a good measure of where the details are in
the image. This particular form of the thermal diffusivity has been
-shown to be related to how brightness is percepted by the human visual
+shown to be related to how brightness is perceived by the human visual
system. The model has some theoretical problems related to
well-posedness, for more information see \cite{weickert1998anisotropic}.
Perona--Malik method, and other methods with varying scalar
diffusivities, are often called anisotropic, but we will call them
non-linear, and spare the anisotropic term for methods where the
-diffusivity is a tensor, and thus both location and direction dependant.
+diffusivity is a tensor, and thus both location and direction dependent.
A different kind of non-linear diffusion model is the total variation
flow which can be formulated as
where the diffusivity has a similar effect of reducing the diffusion in
areas of high variation. As the name suggests this model can be related
to the variational total variation formulation presented later. One
-discrete time-step in the solution of this PDE corresponds to the
-Euler--Lagrange equation of the variational formulation.
+discrete time-step in the solution of this partial differential equation
+corresponds to the Euler--Lagrange equation of the variational
+formulation.
Common for all of these are that we control the amount of diffusion done
in each point of the image, based somehow on the image itself. But we
-can go further by not only making the diffusivity location dependant,
-but also direction dependant. This allows us to have have more
+can go further by not only making the diffusivity location dependent,
+but also direction dependent. This allows us to have have more
diffusion in directions with little variation. As mentioned, we will
use the term anisotropic diffusion, meaning ``not uniform in all
directions.''
\subsection{Anisotropic diffusion}
\label{sec:anisotropic_diffusion}
-The diffusivity is made directionally dependant by introducing a diffusion
+The diffusivity is made directionally dependent by introducing a diffusion
\emph{tensor} $A(u)$ such that the initial boundary value problem
becomes
\begin{equation}
\end{cases}
\label{eq:aniso_diff}
\end{equation}
+\nomenclature{$A(u)$}{Thermal diffusivity tensor, or anisotropy
+tensor.}%
\fixme{triple column alignment?}
where $\nu$ is the outer normal of $\Omega$. The tensor $A(u)$ would
normally be constructed such as to diminish the effect of $\nabla
also be less diffusion through these edges. Weickert suggests in
\cite{weickert1998anisotropic} constructing $A(u)$ based on the
edge estimator $\nabla u_\sigma$ where
+\nomenclature{$\nabla u_\sigma$}{Edge estimator.}%
\begin{equation}
u_\sigma := K_\sigma * \tilde{u}
\end{equation}
+\nomenclature{$u_\sigma$}{Image $u$ smoothed with a Gaussian of
+parameter $\sigma$.}%
and $\tilde{u}$ is an extension of $u$ from $\Omega$ to $\mathbb{R}^2$
made by for example symmetrically extending $u$ across the edges.
Assuming we are at an edge in the image, the
\begin{equation}
S_\rho(x) := K_\rho * (\nabla u_\sigma \otimes \nabla u_\sigma),
\end{equation}
+\nomenclature{$S_\rho(x)$}{Structure tensor.}
where the convolution with the Gaussian function $K_\rho$ is done
-componentwise. The anisotropy tensor $A(u)$ can then be constructed
+component-wise. The anisotropy tensor $A(u)$ can then be constructed
based on the eigenvectors and eigenvalues of $S_\rho(x)$.
The structure tensor and its properties will be discussed further when
$\Omega$.
\label{def:tv}
\end{definition}
+\nomenclature{$\TV(u)$}{Total variation of image $u$.}%
Note that since $\Omega$ is open and bounded, the test functions
$\varphi$ vanish on the boundary of $\Omega$. This means we assume
\right\}.
\end{equation}
\end{definition}
+\nomenclature{$\BV(\Omega)$}{Functions of bounded variation in
+$\Omega$.}%
Our optimization problem has thus become
\begin{equation}
\min_{u \in \BV(\Omega)} \int_\Omega \abs{u - v}^p \, dx + \beta \,
solution.
As with anisotropic diffusion, the total variation method also has its
-strengths and weaknesses. Its main strenght is its ability to recover
+strengths and weaknesses. Its main strength is its ability to recover
edges in the input image. The total variation of a section only takes
-the absolute change into account, and does not favour gradual changes
+the absolute change into account, and does not favor gradual changes
over sharp discontinuities like the diffusion methods. This comes at a
cost though, and in some cases the method may introduce new edges that
were not in the original image, since flat sections of zero variation
\begin{equation}
u^\lambda = \idfun_{u > \lambda}.
\end{equation}
+\nomenclature{$u^\lambda$}{Thresholded image at level $\lambda$.}%
Here, $\idfun_E$ signifies the 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. The graph cut we find later will for each level $\lambda$
give us the thresholded image $u^\lambda$ which can then be combined to
\end{equation}
Our input images will be digital images given as pixels on a grid, and
-we will represent them as functions $u : \mathcal{G} \to \mathcal{L}$,
+we will represent them as functions $u : \mathcal{G} \to \mathcal{P}$,
where $\mathcal{G}$ is our regular discrete pixel grid, and
-$\mathcal{L} = \{0, \hdots, L-1\}$ is the set of discrete level values.
+$\mathcal{P} = \{0, \hdots, L-1\}$ is the set of discrete level values.
Through careful manipulation of the functional in
\eqref{eq:second_min_presentation} we obtain a discrete energy
functional decomposed as a sum over all the level values
\usepackage{subcaption}
\usepackage{xfrac}
\usepackage{listings}
+\usepackage[intoc]{nomencl}
\lstset{
% FIXME: Courier 9pt anbefales
\input{commands}
+\makenomenclature
+
% Selve dokumentet begynner:
\begin{document}
% Pass på at neste side ikke begynner på baksida av en annen side.
\cleardoublepage
+\printnomenclature[2.5cm]
+
+\cleardoublepage
+
% Arabisk (vanlige tall) sidenummerering. Starter på side 1 igjen.
\pagenumbering{arabic}
\end{equation}
\label{def:extended_tv}
\end{definition}
+\nomenclature{$\TVA(u)$}{Anisotropic total variation of image $u$, with
+anisotropy tensor $A(u)$.}%
With this extended definition, we have arrived at a minimization problem
where we seek to find a minimizer of the functional
We are also interested in identifying and being sensitive to features in
a neighborhood around the point, such as corners or curved edges and
\fixme{coherent structures}. This is why we introduce the convolution
-with $K_\rho$, which is done componentwise. The parameter $\rho$, called
+with $K_\rho$, which is done component-wise. The parameter $\rho$, called
the \emph{integration scale} thus controls the size of the neighborhood
which affects the structure tensor. It controls the size of the
structures we want our anisotropy tensor to be sensitive to.
A measurable set $U \subset \Omega$ is of finite anisotropic
perimeter in $\Omega$ if $\idfun_U \in \BV(\Omega)$. \fixme{nope?}
\end{definition}
+\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
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.
+\nomenclature{$\mathcal{L}$}{The set of all straight lines in the
+plane.}%
+\nomenclature{$\ell_{\phi, \rho}$}{A line given by the angle of the
+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 $\rho$.}%
\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{$\abs{C}_M$}{The length of the curve $C$ calculated using
+the metric tensor $M$.}%
\begin{equation}
\abs{\gamma}_M = \int_\gamma \sqrt{\langle \dot{\gamma},
M\big(\gamma(t)\big) \, \dot{\gamma} \rangle} \, dt.
where $J_M(\ell_{\phi, \rho})$ is the Jacobian of our coordinate
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 $M$.}
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
We assume that our discrete images are given on a 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{L} = \{0,
+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 $\mathcal{L}$.
+energyfunction as a sum over all the levels $\mathcal{P}$.
\subsection{Fidelity term}
the fidelity term can be discretized 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
+\mathcal{P}$, we define the following function
\begin{equation}
N_x(k) = \abs{k - v_x}^p
\end{equation}
\end{equation}
The reason we introduce the function $N_x(k)$ is that we want to apply
the following decomposition formula, which holds for any function $F(k)$
-taking values $k \in \mathcal{L}$:
+taking values $k \in \mathcal{P}$:
\begin{equation}
\begin{aligned}
F(k) &= \sum_{\lambda = 0}^{k-1} \big( F(\lambda + 1) -
\end{equation}
As our domain is discretized uniformly, we drop the constant
$\Delta x$, and absorb it into our parameter $\beta$. Note that since
-our image takes values in $\mathcal{L} = \{0, \hdots, L-1\}$, the
+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}
In the discrete setting our domain $\Omega$ is discretized as a regular
grid $\mathcal{G}$. Our image is then reduced to a function $u :
-\mathcal{G} \to \mathcal{L}$. \fixme{mathcal L is now two things.}
+\mathcal{G} \to \mathcal{P}$.
Moreover, the level sets $\{ u > \lambda\}$ will be functions taking the
value of 0 or 1 on this grid, as shown in Figure \fixme{ref}.
\emph{integration scale} $\rho$, and thus we expect the tensors $M(a)$
and $M(b)$ to be similar for edges $e$ of reasonably short length.
-\fixme{really? componentwise? will that not mess up the
-eigenvalues? sure, a bit, but it won't change consistency..}
+We also remark that using the min-max theorem, it is easy to conclude
+that the eigenvalues of the tensor approximation $M(e_{ab})$ is bounded
+below and above by the smallest and largest eigenvalues of $M(a)$ and
+$M(b)$. \fixme{Reference to min-max/Rayleigh}
\begin{figure}
\input{fig/area_proof}
\label{lem:delta_rho}
\end{lemma}
\begin{proof}
- Consider a line $\ell$ intersecting the point $(p, q)$ in the grid.
- The distance $\Delta \rho$ from this line $\ell$ to the next line
- can then be calculated as a minimum over the distance to all other
- lines.
-
- The given family of lines consists of edges in the grid, which we
- write $e = (\delta s, \delta t)^T$ where $s$ and $t$ are coprime
- such that $e$ does not intersect any other points than its two
- endpoints.
+ Consider a line $\ell$ intersecting the point $(p, q) \in
+ \mathbb{Z}^2$ in the grid.
+ The distance $\Delta \rho$ from this line $\ell$ to the neighboring
+ lines can then be calculated as a minimum over the distance to all
+ other lines.
+
+ The lines are split into edges $e = (\delta s, \delta t)^T$ where
+ $s, t \in \mathbb{Z}$ are coprime such that $e$ does not intersect
+ any other points than its two endpoints.
Let $(p\prime, q\prime)$ be an arbitrary point not on the line
$\ell$. We can then calculate
projects / master / commitdiff