From: Bjørn Rustad <bjorn@rustad.me>
Date: Fri, 28 Mar 2014 10:21:44 +0000 (+0100)
Subject: Fixes to the methods chapter
X-Git-Url: http://git.rustad.me/?a=commitdiff_plain;h=99991c980545cb243bbb541ee99ca1171e602f50;p=prosjektoppgave

Fixes to the methods chapter
---

diff --git a/methods.tex b/methods.tex
index 227317a..dc2a6f3 100644
--- a/methods.tex
+++ b/methods.tex
@@ -1,4 +1,9 @@
 \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
@@ -7,29 +12,33 @@ method where the image is convolved with the Gaussian function
     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.
@@ -47,29 +56,33 @@ has some theoretical problems related to well-posedness, for more
 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