something about the motivation, its well-posedness (or lack thereof) and
other alternatives. But we can control the diffusion even more!
+Perona--Malik
+\begin{equation}
+ \partial_t u = \mathrm{div} \left( \frac{\nabla u}{1 +
+ \frac{\abs{\nabla u}^2}{\lambda^2}} \right).
+ \label{eq:perona_malik}
+\end{equation}
+Total variation flow
+\begin{equation}
+ \partial_t u = \mathrm{div} \frac{\nabla u}{\abs{\nabla u}}.
+\end{equation}
+
\subsection{Anisotropic diffusion}
Section for methods introducing a diffusion \emph{tensor} $A$ that
varies around the image. This means we can control diffusion strength in
different directions. More specifically we can make the diffusion strong
-along edges, and weak across edges.
+along edges, and weak across edges. We should give a real theoretical
+background here, so we have something to base the next chapter on.
We must also say something about what this method does well, and where
it fails. Anisotropic filtering might introduce structures that do not
exist in the original image. Some example images are possible here.
+\begin{equation}
+ \partial_t u = \mathrm{div} \big(A(u) \cdot \nabla u\big).
+\end{equation}
+
+Main source here is Weickert.
+
\section{Total variation filtering}
Describe briefly the functional we minimize in this method, and
are different ways to do it, and that one of them involves finding a
minimum cut for each level.
+We could introduce the normal coarea and perimeter theory here so that
+we can refer to it when we introduce the anisotropic versions in the
+next chapter?
+
We also write something about the limitations of the method. It is good
at preserving sharp edges, but suffers from the so-called stair-casing
effect. Thin objects and corners might gradually lose contrast depending
these two methods, retaining their strenghts but eliminating their
weaknesses.
+\fixme{This is taken from the project.}
+
+First we introduce this.
+\begin{equation}
+ \min_u \int_\Omega \abs{u - v}^p + \beta \int_\Omega \abs{\nabla u}.
+ \label{eq:first_min_presentation}
+\end{equation}
+Then we say something about difficulties calculating the gradient and
+introduce the total variation.
+\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
+ taken in the distributional sense, is
+ \begin{equation}
+ \TV(u)
+ = \int_\Omega \abs{Du}
+ = \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
+ $C^\infty_c\left(\Omega, \mathbb{R}^2\right)$, the space of smooth
+ functions from $\Omega$ to $\mathbb{R}^2$ with compact support in
+ $\Omega$.
+\end{definition}
+
+Note that since $\Omega$ is open and bounded, the test functions
+$\varphi$ vanish on the boundary of $\Omega$ \fixme{which means we
+assume some boundary condition}. Next we introduce the space of
+functions with bounded variation.
+\begin{definition}[Functions of bounded variation]
+ The space of functions of bounded variation $\BV(\Omega)$ is the
+ space of functions $u \in L^1(\Omega)$ for which the total variation
+ is finite, i.e.,
+ \begin{equation}
+ \BV(\Omega) = \left\{ u \in L^1(\Omega) : \TV(u) < \infty
+ \right\}.
+ \end{equation}
+\end{definition}
+
+Before introducing the coarea formula, we need the formal definition of
+a set perimeter. Here, $\idfun_E$ signifies the characteristic function
+of the set $E$, the function which is equal to one in every point in
+$E$, and zero elsewhere.
+\begin{definition}[Set perimeter]
+ The perimeter of a set $E$ is defined as
+ \begin{equation}
+ P(E,\Omega) = \int_\Omega \abs{D \idfun_E}.
+ \end{equation}
+ A measurable set $E \subset \Omega$ is of finite perimeter in
+ $\Omega$ if $\idfun_E \in \BV(\Omega)$.
+\end{definition}
+\fixme{Something about reduced boundary and Hausdorff measures here?}
+The set perimeter behaves mostly as one might expect for ``nice'' sets,
+just note that if the boundary of $E$ overlaps with the boundary of
+$\Omega$, then $P(E,\Omega)$ will not include the overlapping parts of
+the boundary.
+
+For an image $u$ and a level $\lambda$ we denote the \emph{level set} by
+$\{u \leq \lambda\}$ defined as the set $\{ x \in \Omega : u_x \leq
+\lambda\}$. The thresholded images are defined as
+\begin{equation}
+ u^\lambda = \idfun_{u \leq \lambda}.
+ \label{eq:level_set}
+\end{equation}
+With all these definitions in place, we are ready to introduce the
+coarea formula, which will let us write the total variation as an
+integral over the range of the image levels $\lambda$.
+\begin{theorem}[The coarea formula]
+ Let $u \in BV(\Omega)$. Then for almost every $\lambda$ the set
+ $\{u \leq \lambda\}$ is of finite perimeter, and one has the coarea
+ formula
+ \begin{equation}
+ \TV(u)
+ = \int_\Omega \abs{D u}
+ = \int_\mathbb{R} P(\{u \leq \lambda\},\Omega) \, d\lambda
+ \label{eq:coarea_formula}
+ \end{equation}
+\end{theorem}
+Note that the equality in \eqref{eq:coarea_formula} holds for all $u \in
+L^1(\Omega)$, not only when the total variation is finite. We can now
+rewrite \eqref{eq:first_min_presentation} as
+\begin{equation}
+ E_v(u) =
+ \int_\Omega \abs{u - v}^p
+ + \beta \int_\mathbb{R} P(\{u \leq \lambda\},\Omega) \, d\lambda.
+ \label{eq:energy_perimeter}
+\end{equation}
+\fixme{Should this be a minimization problem or an energy function.}
+
+\fixme{Shorten this section (or maybe it's short enough, only two pages,
+but we need to fill in some text at least), and add some general
+references.}
+
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