-% Dette eksempelet er laget for article-dokumentklassen. Hvis
-% skriver i 'book'-dokumentklassen vil du kanskje bytte ut
-% \section med \chapter, \subsection med \section, osv...
-
\section{Total variation}
-The method of total variation was initially introduced to image
-restoration by L. Rudin, S. Osher, and E. Fatemi in
-\cite{rudin1992nonlinear}, but is also used in numerous other
-applications. One of its strong theoretical points is its ability to
-recover edges, compared to other restoration algorithms which might
-smooth over edges in the original image. The method can be formulated as
-minimizing the energy function
+The image restoration method considered in this project tries to find an
+image $u$ which is close to the captured image $v$, while it at the same
+time has a low \emph{total variation} by solving
\begin{equation}
- E_v(u) =
- \int_\Omega \abs{u - v}^p
- + \beta \TV(u)
- \label{eq:energy_function}
+ \min_u \int_\Omega \abs{u - v}^p + \beta \int_\Omega \abs{\nabla u}.
+ \label{eq:first_min_presentation}
\end{equation}
-where $v$ is the original image, and $\Omega$ is the image domain, in
-our case an open subset of $\mathbb{R}^2$. Normally, $p$ is chosen to be
-1 or 2.
+How we can integrate the gradient of an image which can contain
+discontinuities will be clarified later, but first we will look at how
+this model can be grounded in spatial statistics; \fixme{an important
+field in image analysis.}
\subsection{Probabilistic background}
-\fixme{We discuss pixels here already?}
\fixme{The actual expression for the total variation has not been
introduced yet!}
so-called maximum a posteriori estimation (MAP for short) of spatial
statistics.
-We introduce the likelihood $p(u \mid v) \cdot p(u)$ and try to maximize
-this in order to obtain our output image $u$. This is equivalent to the
-formulation
+There are many ways to model noise in an image, and different models
+take different physical phenomena into account. We will in this brief
+introduction only consider the simplest model, namely \emph{additive
+noise}, which given an image $u$ results in a captured image
+\begin{equation}
+ v = u + \epsilon.
+\end{equation}
+
+For a captured noisy image $v$ and we want to recover $u$, or something
+as close to $u$ as possible, so we introduce the likelihood function
\begin{equation}
- \min_u \big( - \ln p(u \mid v) - \ln p(u) \big).
+ p(u \mid v) = \frac{p(v \mid u) \cdot p(u)}{p(v)}
+\end{equation}
+using Bayes' theorem. Maximizing this will give us the image $u$ which
+is most probable given our noisy image $v$. The \emph{prior} $p(u)$
+represents how probable we think it is to obtain an image like $u$ in
+the set of all obtainable images, and $p(v \mid u)$ is our noise model,
+how we think a noisy image $v$ is distributed given an image $u$.
+
+Since $p(v)$ is constant, and the logarithm is a monotonically
+increasing function, the maximization problem is equivalent to
+\begin{equation}
+ \min_u \big( - \ln p(v \mid u) - \ln p(u) \big).
\label{eq:minimize_neg_log}
\end{equation}
-If the noise term $p(u \mid v)$ is modeled to be independently and
-identically distributed Gaussian noise on each pixel, we have
+For discrete images consisting of pixels, we denote use $x$ and $y$ to
+denote pixel positions such that $u_x$ and $u_y$ are the values of those
+pixels, not to be confused with partial derivatives $\partial_x u$ and
+$\partial_y u$. If the noise term $p(v \mid u)$ is modeled to be
+independently and identically distributed Gaussian noise on each pixel,
+we have
\begin{equation}
- p(u \mid v) \propto
+ p(v \mid u) \propto
\prod_x \exp \left(-\frac{(u_x - v_x)^2}{2\sigma^2} \right).
+ \label{eq:gaussian_noise}
\end{equation}
-The prior $p(u)$ represents how probable we think it is to obtain an
-image like $u$. This probability depends on what kinds of images are
+
+The prior probability $p(u)$ depends on what kinds of images are
considered, and how they are obtained. In the theory of Markov random
fields, one lets the probability of a pixel value depend only on some
-neighborhood $\mathcal{N}$. One common choice for the relation between
-neighboring pixels is a Laplace distribution
+neighborhood $\mathcal{N}$. A common choice for the relation between
+neighboring pixels is the Laplace distribution
\begin{equation}
p(u) \propto
\prod_x \prod_{y \in \mathcal{N}(x)} \exp \left(
- \frac{\abs{u_x - u_y}}{b}
\right).
+ \label{eq:laplace_prior}
\end{equation}
Consult for example \cite{scherzer2008variational} for further reading
-on different possibilities for the prior probability.
+on noise models and different possibilities for the prior probability.
-Taking the logarithm in \eqref{eq:minimize_neg_log} we obtain
+Inserting the prior \eqref{eq:laplace_prior} and the noise model
+\eqref{eq:gaussian_noise} into \eqref{eq:minimize_neg_log} we obtain
\begin{equation}
\min_u \left(
\sum_x \abs{u_x - v_x}^2
- \beta \sum_x \sum_{y \in \mathcal{N}(x)} \abs{u_x - u_y}
\right).
- \label{eq:minimizing_probability}
+ \label{eq:minimizing_probability_l2}
\end{equation}
-If the noise had been modeled with the Laplace distribution
+If the noise is modeled by a Laplace distribution
\begin{equation}
p(u \mid v) \propto
\prod_x \exp \left( - \frac{\abs{u_x - v_x}}{b} \right)
\end{equation}
-we would have obtained the $L^1$-norm instead of the $L^2$-norm in
-\eqref{eq:minimizing_probability}.
+we obtain the $L^1$-norm instead of the $L^2$-norm,
+\begin{equation}
+ \min_u \left(
+ \sum_x \abs{u_x - v_x}
+ - \beta \sum_x \sum_{y \in \mathcal{N}(x)} \abs{u_x - u_y}
+ \right).
+ \label{eq:minimizing_probability_l1}
+\end{equation}
+We see that the two discrete problems
+\eqref{eq:minimizing_probability_l2} and
+\eqref{eq:minimizing_probability_l1} resemble the original formulation
+in \eqref{eq:first_min_presentation}. Before presenting the
+discretization, we will look at a more rigorous presentation of
+\eqref{eq:first_min_presentation}, and the definitions, spaces and
+theorems needed to treat it.
-\subsection{Functional analysis foundation}
-\fixme{Meh heading.}
+\subsection{Continuous formulation}
+The method of total variation was initially introduced to image
+restoration by L. Rudin, S. Osher, and E. Fatemi in
+\cite{rudin1992nonlinear}, but is also used in numerous other
+applications. One of its strong theoretical points is its ability to
+recover edges, compared to other restoration algorithms which might
+smooth over edges in the original image. The method can be formulated as
+minimizing the energy function
+\begin{equation}
+ E_v(u) =
+ \int_\Omega \abs{u - v}^p
+ + \beta \TV(u)
+ \label{eq:energy_function}
+\end{equation}
+where $v$ is the original image, and $\Omega$ is the image domain, in
+our case an open subset of $\mathbb{R}^2$. Normally, $p$ is chosen to be
+1 or 2, the two choices that correspond to the two different noise
+models in \eqref{eq:minimizing_probability_l1} and
+\eqref{eq:minimizing_probability_l2}.
-The term $\TV(u)$ is the total variation of the image, and is defined as
-follows:
+The term $\TV(u)$ is the total variation of the image, introduced
+earlier as $\int_\Omega \abs{\nabla u}$, and is defined as follows:
\begin{definition}[Total variation]
Given a function $u \in L^1(\Omega)$, the total variation of $u$,
often written $\int_\Omega \abs{Du}$, where the $D$ is the gradient
\begin{equation}
\TV(u)
= \int_\Omega \abs{Du}
- = \sup \left\{ \int_\Omega u \,\, \mathrm{div} \, \varphi :
+ = \sup \left\{ \int_\Omega u \cdot \mathrm{div} \, \varphi :
\varphi \in C^\infty_c\left(\Omega, \mathbb{R}^2\right),
\norm{\varphi}_{L^\infty(\Omega)} \leq 1
\right\}.
\end{equation}
The test functions $\varphi$ are taken from
- $C^\infty_c\left(\Omega, \mathbb{R}^N\right)$, the space of smooth
- functions from $\Omega$ to $\mathbb{R}^N$ with compact support.
+ $C^\infty_c\left(\Omega, \mathbb{R}^2\right)$, the space of smooth
+ functions from $\Omega$ to $\mathbb{R}^2$ with compact support.
\fixme{$\Omega$ open, bounded, Lipschitz boundary?}.
\end{definition}
-Minimizing the total variationn will smooth out differences in the
-image. The term $\int_\Omega \abs{u - v}^p$ of
-\eqref{eq:energy_function} will constrain the solution image $u$ to be
-close to the input image $v$. Combining the two will give us an image
-close to the original, but more regular. The parameter $\beta$ controls
-how strongly we want to regularize the image.
+Minimizing the total variation will smooth out differences in the image.
+The term $\int_\Omega \abs{u - v}^p$ of \eqref{eq:energy_function} will
+constrain the solution image $u$ to be close to the input image $v$.
+Combining the two will give us an image close to the original, but more
+regular. The parameter $\beta$ controls how strongly we want to
+regularize the image.
When discretizing the energy function \eqref{eq:energy_function} later
on, we want to decompose it as a sum over the different levels, or pixel
of jumps in the resulting image $u$ is contained in the set of jumps in
the original image $v$ are also presented.
-\subsection{Discrete problem}
+\subsection{Discrete formulation}
Since digital images are given on a discrete grid, with values taken
from a discrete and finite set of levels, we want to discretize
\eqref{eq:energy_perimeter}, and solve the resulting discrete problem.
projects / prosjektoppgave / commitdiff