\section{Methods in image restoration}
+There are numerous methods for restoring an image, and here we will
+briefly look at some of the more popular approaches. Given an original
+image $v$, the goal is to obtain a somehow denoised output image $u$.
+Which method to choose depends on what properties one wants the output
+image to have, and also on how the original image was obtained.
\subsection{Gaussian filtering}
Gaussian filtering, or Gaussian blur is a straight forward kind of
G_\sigma(x, y) = \frac{1}{2 \pi \sigma^2}\exp\left(-\frac{x^2 +
y^2}{2\sigma^2}\right).
\end{equation}
-In the discrete case this amounts to setting each pixel value as the
-weighted average of its neighbors in the original image. This will of
-course, in addition to smoothing out possible noise, add blur and remove
-details from the image.
+In the discrete case of an image consisting separate pixels, Gaussian
+blur amounts to setting each pixel value as the weighted average of its
+neighbors in the original image. This will, in addition to smoothing out
+possible noise, add blur and remove details from the image.
-The Gaussian function is also the fundamental solution of the heat
-equation $\partial_t u = \Delta u$, so convolving with it is equivalent to
-solving the heat equation with the original image as initial value. Care
-must of course be taken on the boundary, and it is normal to
-symmetrically extend the image in all directions.
+The Gaussian happens to be the fundamental solution of the heat
+equation $\partial_t u = \Delta u$. Convolving it with the original
+image $v$ is therefore equivalent to solving the heat equation with $v$
+as initial value, for some time $t > 0$ depending on $\sigma$. Care must
+be taken on the boundary, and one possibility is to symmetrically extend
+the image in all directions.
-By basic fourier analysis it is possible to show that the Gaussian
-filter is a low-pass filter which attenuates high frequencies
+By basic Fourier analysis it is possible to show that the Gaussian
+filter is a low-pass filter which attenuates high frequencies.
\fixme{Cite Weickert for example?}
\subsection{Anisotropic diffusion}
Since the Gaussian filter will blur out both noise and details of the
-image, it would be nice to somehow reduce the amount of blur it adds
-in areas with edges, or what we beleive to be edges, in the image. This
-can be done by controlling the thermal diffusivity $\alpha(u)$ of the
-image in the heat equation
+image, it would be nice to somehow reduce the amount of blur that is
+added in areas of the image with edges, or what we believe to be edges.
+This can be done by controlling the thermal diffusivity $\alpha(u)$ of
+the image in the heat equation
\begin{equation}
- \partial_t u = \mathrm{div} \big( a(u) \nabla u\big).
+ \begin{cases}
+ \partial_t u &= \mathrm{div} \big( a(u) \nabla u\big) \\
+ u|_{t=0} &= v.
+ \end{cases}
\end{equation}
The problem is then, how to detect the edges such that we can reduce
$\alpha(u)$ in those areas.
information see \cite{weickert1998anisotropic}.
A different kind of anisotropic diffusion model is the total variation
-flow model which is related to the model considered in this project.
-It can be formulated as
+flow model which can be formulated as
\begin{equation}
- \partial_t u = \mathrm{div} \frac{\nabla u}{\abs{\nabla u}}
+ \partial_t u = \mathrm{div} \frac{\nabla u}{\abs{\nabla u}}.
\end{equation}
-aaaaaaaaand?
+As the name suggests this model can be related to the total variation
+formulation considered in this project. One discrete time-step in the
+solution of this PDE corresponds to the Euler-Lagrange equation of the
+total variation minimization problem presented later.
-An other approach is to let the thermal diffusivity be a matrix $A(u)$
-in the equation
+Another more advanced approach is to let the thermal diffusivity be a
+tensor $A(u)$ in the equation
\begin{equation}
\partial_t u = \mathrm{div} \big(A(u) \cdot \nabla u\big).
\end{equation}
This way, it is possible to have a small diffusivity \emph{across}
edges, and at the same time a big diffusivity \emph{along} the edges.
+See for example \cite{weickert1998anisotropic} for more information on
+these diffusion tensor methods.
\subsection{Non-local means}
The non-local means method of image restoration takes into account the
fact that two separate parts of an image might be very similar, for
example in the case of a regular texture. One defines some neighborhood
for each pixel, and when restoring a pixel, one considers \emph{all}
-other pixels which has similiar neighborhoods as the current pixel, for
-some notion of similiar. One then averages all these pixels to obtain
-the new pixel value.
+other pixels whose neighborhoods are similar to the neighborhood of the
+current pixel, for some notion of similarity. One then averages all
+these pixels to obtain the new pixel value.
\subsection{Other filters}
A different filter much used in real-world image processing is the
projects / prosjektoppgave / commitdiff