\chapter{Methods in image restoration}
-\fixme{Rating: 5/10}
+\fixme{Rating: 7/10}
There are numerous methods in image restoration, and since we do not
have time nor space to discuss them all, we will focus on the ones
related to the anisotropic total variation method considered later in
-this thesis.
+this thesis. \fixme{Maybe refer to some overview book?}
In this chapter, and also in the rest of the thesis we will assume that
-we are given an image $f$ on a rectangular, open domain $\Omega$. The
-space in which $f$ resides in will vary, but since we are looking at
-methods for image restoration, we assume that it includes some kind of
-noise.
-
-There are different kinds of noise models for different situations and
-applications, but we will assume that the given image $f$ is a
-combination of an actual image $u^*$, and some noise $\delta$. There is
-additive noise where the assumption is that $f = u^* + \delta$, and
-there is also multiplicative noise where $f = u^* \cdot \delta$.
-
-In the real world the noise might be more complex of course, and even
-come from a combination of sources. Depending on the application, the
-goal might even not be to find $u^*$, but rather an output which
-fulfills certain smoothness or regularity properties. In any case, we
-will denote the output of the methods $u$, and loosely discuss their
-properties.
+we are given an image $f : \Omega \to \mathbb{R}$ where $\Omega$ is a
+rectangular, open domain. Because of limitations in the numerical method
+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.
+
+The space in which the image $f$ resides in will vary, but since we are
+looking at image restoration methods, we assume that it includes some
+kind of noise. Depending on the application and how the image is
+obtained, one might construct different models describing different
+types of noise.
+
+We will assume that the given image $f$ is a combination of an
+underlying, actual image $u^*$, and some noise $\delta$. The simplest
+model is additive noise where the assumption is that $f = u^* + \delta$.
+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, independently of the
+actual image $u^*$.
+
+These are only models, and in the real world the noise might be more
+complex, and even come from a combination of sources. Depending on the
+application, the goal might not even be to recover $u^*$, but rather to
+obtain an output which fulfills certain smoothness or regularity
+properties. In any case, we will continue using $f$ as the input and $u$
+as the output in the description of our methods.
\section{Diffusion filtering}
Diffusion filtering is a broad group of filtering and restoration methods
based on physical diffusion processes. The basic idea is to take the
noisy image as the initial value of some diffusion process, and then let it
-evolve for some time. The most well-known method is probably the
-Gaussian filter or Gaussian blur, in which one convolves the image with
-the Gaussian function
+evolve for some time. The best known method is probably the Gaussian
+filter or Gaussian blur, in which one convolves the image with the
+Gaussian function
\begin{equation}
K_\sigma(x,y) = \frac{1}{2\pi\sigma^2} \exp \left( - \frac{x^2 +
y^2}{2\sigma^2} \right).
directions, which corresponds to zero flux boundary conditions.
By basic Fourier analysis it is possible to show that the Gaussian
-filter is a low-pass filter which attenuates high frequencies. This and
-some further theory can be found in Weickert's book on anisotropic
-diffusion \cite{weickert1998anisotropic}.
+filter is a low-pass filter which attenuates high frequencies. Further
+theory can be found in Weickert's book on anisotropic diffusion
+\cite{weickert1998anisotropic}.
The main concern with the Gaussian filter is that it will, in addition
to smoothing out possible noise, remove details from the image, which
In the theory of the heat equation one can introduce a \emph{thermal
diffusivity} $\alpha$ such that the equation becomes
\begin{equation}
- \begin{cases}
- \partial_t u &= \diver \big( \alpha(u) \nabla u\big) \\
- u |_{t=0} &= f
- \end{cases}
+ \left\{
+ \begin{aligned}
+ \partial_t u &= \diver \big( \alpha(\nabla u) \nabla u\big), \\
+ u |_{t=0} &= f.
+ \end{aligned}
+ \right.
\end{equation}
-\nomenclature{$\alpha(u)$}{Thermal diffusivity scalar.}%
-The thermal diffusivity $\alpha$ is material dependent, and can also
+\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
restoration context by controlling the diffusivity in different parts of
\frac{\abs{\nabla u}^2}{\lambda^2}} \right).
\label{eq:perona_malik}
\end{equation}
-The thermal diffusivity $\alpha(u) = (1 + \sfrac{\abs{\nabla u}^2}{
+The thermal diffusivity $\alpha(\nabla u) = (1 + \sfrac{\abs{\nabla u}^2}{
\lambda^2})^{-1}$ varies from $1$ in smooth areas to $0$ as the
norm of the gradient $\abs{\nabla u}$ grows.
\end{equation}
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
-forward Euler time-step in the solution of this partial differential
-equation corresponds to the Euler--Lagrange equation of the variational
+to the total variation formulation presented later. One forward Euler
+time-step in the solution of this partial differential equation
+corresponds to the Euler--Lagrange equation of the variational
formulation.
Note that we follow Weickert's terminology when it comes to the
distinction between non-linear and anisotropic diffusion methods. The
Perona--Malik equation, and other diffusion equations with
-non-homogenous diffusivities, are often by others called anisotropic,
-since the diffusion will be different in different directions.
-We will name these methods non-linear and spare the anisotropy
-term for the ``real'' anisotropic methods. These are methods where the
-diffusivity is a tensor, and thus both location and direction dependent.
+non-homogenous diffusivities, are often by others called anisotropic, as
+the diffusivity depends on the location. We will name these methods
+non-linear and spare the anisotropy term for the ``real'' anisotropic
+methods. These are methods where the diffusivity is a tensor, and thus
+both location and direction dependent.
\subsection{Anisotropic diffusion}
\label{sec:anisotropic_diffusion}
\begin{equation}
\left\{
\begin{alignedat}{10}
- &\partial_t u &{}={}& \diver \big(A(u) \nabla u\big)
+ \partial_t u &{}={} \diver \big(A(u) \nabla u\big)
&\quad \text{ on }& \Omega \times (0, \infty),\\
- &u|_{t=0} &{}={}& f
+ u|_{t=0} &{}={} f
&\text{ on }& \Omega,\\
- &A(u) \nabla u \cdot \nu &{}={}& 0
+ A(u) \nabla u \cdot \nu &{}={} 0
&\text{ on }& \partial \Omega \times (0, \infty),
\end{alignedat}
\right.
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 suggests in
-\cite{weickert1998anisotropic} constructing $A(u)$ based on the
+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.}%
\begin{equation}
prove well-posedness, regularity and an extremum principle of the
problem \eqref{eq:aniso_diff} as done in \cite{weickert1998anisotropic}.
-Even if the diffusivity tensor was introduced to make our method more
-conserving of edges, the solution of \eqref{eq:aniso_diff} will still be
+Even if the diffusivity tensor was introduced to reduce the amount of
+smoothing across edges, the solution of \eqref{eq:aniso_diff} will still be
infinitely differentiable \cite{weickert1998anisotropic}, i.e.\ $u(T) \in
C^\infty(\Omega)$ for $T > 0$. Thus we have no real discontinuities, and
no real edges in our solution.
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