URL = {http://dx.doi.org/10.1137/0218003},\r
}\r
\r
+@article {grasmair2010anisotropic,\r
+ AUTHOR = {Grasmair, Markus and Lenzen, Frank},\r
+ TITLE = {Anisotropic total variation filtering},\r
+ JOURNAL = {Appl. Math. Optim.},\r
+ FJOURNAL = {Applied Mathematics and Optimization. An International Journal\r
+ with Applications to Stochastics},\r
+ VOLUME = {62},\r
+ YEAR = {2010},\r
+ NUMBER = {3},\r
+ PAGES = {323--339},\r
+ ISSN = {0095-4616},\r
+ CODEN = {AMOMBN},\r
+ MRCLASS = {49Q20 (35R30 94A08)},\r
+ MRNUMBER = {2727338 (2011h:49064)},\r
+ MRREVIEWER = {Ilya S. Molchanov},\r
+ DOI = {10.1007/s00245-010-9105-x},\r
+ URL = {http://dx.doi.org/10.1007/s00245-010-9105-x},\r
+}\r
+\r
</bibtex>\r
it fails. Anisotropic filtering might introduce structures that do not
exist in the original image. Some example images are possible here.
+We consider the initial boundary value problem
+\begin{align}
+ \begin{cases}
+ \partial_t u &= \mathrm{div} \big(A(u) \nabla u\big)
+ \text{ on } \Omega \times (0, \infty),\\
+ u(x, 0) &= f(x)
+ \text{ on } \Omega,\\
+ A(u) \nabla u \cdot \nu &= 0
+ \text{ on } \Gamma \times (0, \infty).
+ \end{cases}
+\end{align}
+where $\nu$ is the outer normal of $\Omega$ and $A(u)$ is our image
+dependent diffusion tensor. Say something about $\Gamma$, if it hasn't
+been described earlier.
+\fixme{Fix the alignment.}
+
+We define the smoothed image
\begin{equation}
- \partial_t u = \mathrm{div} \big(A(u) \cdot \nabla u\big).
+ u_\sigma(x,t) := \big(K_\sigma * \tilde{u}(\cdot,t)\big)(x)
\end{equation}
+(after we introduce the Gaussian kernel $K_\sigma$ of course), and we
+also have to say something about $\tilde{u}$ being the extension of $u$
+in $\mathbb{R}^2$.
+
+We then convolve again, componentwise, to obtain the structure tensor
+\begin{equation}
+ J_\rho(\nabla u_\sigma) := K_\rho * \left( \nabla u_\sigma \otimes
+ \nabla u_\sigma \right).
+\end{equation}
+We then have to say something about what the $\sigma$ and $\rho$
+actually control, and how the eigenvectors of this tensor behaves. Maybe
+say something about existence and uniqueness as well. Weickert does not
+say much about how to proceed from the structure tensor, at least not in
+Chapter 2, but we should definitely say something about that.
Main source here is Weickert.
u^\lambda = \idfun_{u \leq \lambda}.
\label{eq:level_set}
\end{equation}
+\fixme{Decide upon which way this inequality should go, as it varies in
+the literature.}
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 $\lambda$.
but we need to fill in some text at least), and add some general
references.}
+Now write a little bit about how one might solve this? Just as a small
+introduction for the things to come.
+
+\subsection{Primal--dual approach}
+
+Maybe, maybe not.
+
+\subsection{Graph cut approach}
+
+Just a little bit to show how it works maybe. Describe briefly the
+intuition of it all, how we split the problem into one problem on each
+level, and how these are solved using graph cuts. Describe what the
+graph cuts give us, and how it is combined back into an image.
+
-\chapter{Theory}
+\chapter{Continous formulation}
This is the chapter containing all the theory of the presented
restoration method. The order in which everything is presented is not
of total variation, and the space of functions of bounded variation. The
order is not clear to me, yet!
+\begin{equation}
+ \int_\Omega \left( u - v \right)^2
+ + \alpha \int_\Omega \nabla u(x)^T A(u) \nabla u(x)
+\end{equation}
+\fixme{To differential or not to differential.}
+Describe how this is similar to a Euler time step in a related equation
+\cite{grasmair2010anisotropic}. But how do we calculate the gradients in
+this case? According to Grasmair the $u$ lives in $W^{1,1}(\Omega)$ (at
+least when the square root is involved), and the functional has to be
+extended to $\BV(\Omega)$. Maybe this $dDu$ thing from the other paper
+is important, have to understand what's happening there.
+
\section{Anisotropy tensor}
We discuss what kind of properties we want from the anisotropy tensor,
literature. This is one of the most important parts of the theory
chapter, I guess.
-\section{Discretization}
+\chapter{Discrete formulation}
This is where we discretize! It will also be an important chapter, as
the choices and possibilities are endless. The easiest way out is to
on the anisotropy. This can work, but size has to be discussed (and
tested later). Then the stencil shape can be adjusted as well.
+\section{Discretization}
+
\section{Graph cut formulation}
\subsection{Networks}
\subsection{Network construction}
-\section{Maximum flow approach}
+\chapter{Maximum flow approach}
+
+\section{Flow networks}
+
+\section{Augmenting flow algorithms}
+
+\subsection{Ford--Fulkerson}
+
+\subsection{Edmonds--Karp}
+
+\section{Push--relabel algorithm}
-\subsection{Flow networks}
+\subsection{Network reuse}
-\subsection{Augmenting flow algorithms}
+\section{Boykov--Kolmogorov algorithm}
-\subsection{Push--relabel algorithm}
+\section{Performance comparison}
-\subsection{Boykov--Kolmogorov algorithm}
+Maybe.
projects / master / commitdiff