\section{Graph cut formulation}
-\fixme{Short introduction to graphs. Notation. What is a cut, what is a
- minimal s-t-cut. How do we get from the discrete energy to the graph
- and why does finding the graph cut result in minimizing the energy
- of the current label. Boykov and Kolmogorov.}
+
+\fixme{Short intro here maybe? Something about what this section is all
+about.}
\subsection{Flow networks}
Using the notation of \cite{cormen2009introduction} we will denote a
\subsubsection{Dinitz'}
\fixme{I did implement this, but it is maybe not very central to the
-project/report.}
+project/report, since I use an other algorithm in the final
+implementation. Maybe a comparison would be nice though.}
\subsection{Graph representable energy functions}
The next step is to find a way to create special graph such that we can
\end{equation}
\end{theorem}
-The following theorem proved by Kolmogorov and Zabih in
+The following theorem is proved by Kolmogorov and Zabih in
\cite{kolmogorov2002energy} and will be crucial in our graph
construction.
\begin{theorem}[Additivity]
summands of \fixme{(?????)} and add them together to create our final
graph.
-\subsection{Graph construction}
+\subsection{Network construction}
+We will construct a network in such a way that if a variable
+$u^\lambda_x$ ends up on the $s$-side of the cut, we set $u^\lambda_x =
+0$, and if it ends up on the $t$-side, we set $u^\lambda_x = 1$. This is
+of course an arbitrary choice, but something we have to keep in mind
+through the rest of the section nonetheless.
+
For our neighboring relation in (\fixme{ref}) on the form
\begin{equation}
E^{x,y}(u^\lambda_x, u^\lambda_y) =
we have
\begin{equation}
\begin{aligned}
- E^{x,y}(0, 0) &= w_{x,y} \cdot 0, \\
- E^{x,y}(0, 1) &= w_{x,y} \cdot 1, \\
- E^{x,y}(1, 0) &= w_{x,y} \cdot 1, \\
- E^{x,y}(1, 1) &= w_{x,y} \cdot 0,
+ E^{x,y}(0, 0) &= w_{xy} \cdot 0, \\
+ E^{x,y}(0, 1) &= w_{xy} \cdot 1, \\
+ E^{x,y}(1, 0) &= w_{xy} \cdot 1, \\
+ E^{x,y}(1, 1) &= w_{xy} \cdot 0,
\end{aligned}
\label{eq:neigh_energies}
\end{equation}
representable function in \cite{kolmogorov2002energy}. Since our
energies in \eqref{eq:neigh_energies} are especially simple, the
construction and presentation is simplified.
-\begin{figure}
+
+The fidelity term of our energy function in (\fixme{REF}) simplifies to
+\begin{align}
+ E^x(0) &=
+ N_x(\lambda + 1) -
+ N_x(\lambda)
+ \label{eq:fidelity_energy0} \\
+ E^x(1) &= 0
+ \label{eq:fidelity_energy1}
+\end{align}
+where $E^x(0)$ might end up being positive or negative depending on
+$\lambda$ and the pixel value $v_x$. Figure \ref{fig:norm_subgraph}
+shows how a network can be constructed to represent the \fixme{fidelity}
+term of \eqref{eq:fidelity_energy0}.
+
+Figure \ref{fig:neigh_subgraph} shows how a network can be constructed
+to represent the term $E^x(u^\lambda_x, u^\lambda_y)$ of (\fixme{ref}).
+Table \ref{tab:neigh_energy} shows the energies, and also the cut
+capacities of the different possible variable value combinations, making
+it easy to verify that these two network constructions actually
+represent our energy function.
+\begin{table}
\centering
-\begin{tikzpicture}[scale=1.5]
- \node[vertex] (s) at (0, 0) {s};
- \node[vertex] (u) at (-1,-2) {u};
- \node[vertex] (v) at (1, -2) {v};
- \node[vertex] (t) at (0, -4) {t};
+ \caption{The possible configurations of the variables in the term
+ $E^{x,y}(u^\lambda_x, u^\lambda_y)$, the corresponding energy, and
+ the cut capacity for the two alternative network constructions shown in
+ Figure \ref{fig:neigh_subgraph}.
+ }
+ \begin{tabular}{cccc}
+ \hline
+ Configuration & $E^{x,y}(u^\lambda_x, u^\lambda_y)$
+ & Alt. 1 cut cap. & Alt. 2 cut cap. \\
+ \hline
+ $(0, 0)$ & $0$ & $w_{xy}$ & $0$ \\
+ $(0, 1)$ & $w_{xy}$ & $2w_{xy}$ & $w_{xy}$ \\
+ $(1, 0)$ & $w_{xy}$ & $2w_{xy}$ & $w_{xy}$ \\
+ $(1, 1)$ & $0$ & $w_{xy}$ & $0$
+ \end{tabular}
+ \label{tab:neigh_energy}
+\end{table}
- \path[edge] (s) -- node[weight] {$w_{x,y}$} (u);
- \path[edge] (u) -- node[weight] {$2w_{x,y}$} (v);
- \path[edge] (v) -- node[weight] {$w_{w,y}$} (t);
-\end{tikzpicture}
-\caption{\fixme{Why can't we subtract $w_{x,y}$ from the whole graph
-here? AND CHOOSE u v or x y....}
-\label{fig:neigh_subgraph}
+\begin{figure}
+ \centering
+ \begin{subfigure}[t]{0.45\textwidth}
+ \centering
+ \begin{tikzpicture}[scale=2.0]
+ \node[vertex] (s) at (0, 0) {s};
+ \node[vertex] (u) at (0, -1) {$u^\lambda_x$};
+ \node[vertex] (t) at (0, -2) {t};
+
+ \path[edge] (u) -- node[weight noslope] {$E^x(0)$} (t);
+ \end{tikzpicture}
+ %\caption{Network when $E^x(0) \geq 0$.}
+ \caption{The network when $E^x(0) > 0$, with constant equal to
+ 0.
+ }
+ \label{fig:norm_subgraph_pos}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.45\textwidth}
+ \centering
+ \begin{tikzpicture}[scale=2.0]
+ \node[vertex] (s) at (0, 0) {s};
+ \node[vertex] (u) at (0, -1) {$u^\lambda_x$};
+ \node[vertex] (t) at (0, -2) {t};
+
+ \path[edge] (s) -- node[weight noslope] {$-E^x(0)$} (u);
+ \end{tikzpicture}
+ \caption{Network when $E^x(0) < 0$, with constant equal to
+ $-E^x(0)$.
+ }
+ \label{fig:norm_subgraph_neg}
+ \end{subfigure}
+ \caption{For the two possible configurations $u^\lambda_x \in \{0,
+ 1\}$, the energies in our energy function (\fixme{ref}) are
+ $\{E^x(0), 0\}$. The minimal $s$-$t$-cuts obtaining these
+ configurations are $(S, T) = (\{s, u^\lambda_x\}, \{t\})$ and
+ $(S, T) = (\{s\}, \{u^\lambda_x, t\})$. They have capacity equal
+ to the corresponding energy \emph{plus} the constant (as
+ permitted in \fixme{theorem}).
+ }
+\label{fig:norm_subgraph}
\end{figure}
\begin{figure}
\centering
-\begin{tikzpicture}[scale=2.0]
- \node[vertex] (s) at (0, 0) {s};
- \node[vertex] (u) at (0, -1) {u};
- \node[vertex] (t) at (0, -2) {t};
+ \begin{subfigure}[t]{0.4\textwidth}
+ \centering
+ \begin{tikzpicture}[scale=1.5]
+ \node[vertex] (s) at (0, 0) {s};
+ \node[vertex] (u) at (-1,-2) {$u^\lambda_x$};
+ \node[vertex] (v) at (1, -2) {$u^\lambda_y$};
+ \node[vertex] (t) at (0, -4) {t};
+
+ \path[edge] (s) -- node[weight] {$w_{xy}$} (u);
+ \path[edge] (u) -- node[weight] {$2w_{xy}$} (v);
+ \path[edge] (v) -- node[weight] {$w_{xy}$} (t);
+ \end{tikzpicture}
+ \caption{Representing $E^{x,y}(u^\lambda_x, u^\lambda_y)$ with a
+ constant term of $w_{xy}$.}
+ \label{fig:neigh_subgraph_alt1}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.4\textwidth}
+ \centering
+ \begin{tikzpicture}[scale=1.5]
+ \node[vertex] (s) at (0, 0) {s};
+ \node[vertex] (u) at (-1,-2) {$u^\lambda_x$};
+ \node[vertex] (v) at (1, -2) {$u^\lambda_y$};
+ \node[vertex] (t) at (0, -4) {t};
+
+ \path[edge] ([yshift=1.5pt]u.east) -- node[weight around]
+ {$w_{xy}$}
+ ([yshift=1.5pt]v.west);
+ \path[edge] ([yshift=-1.5pt]v.west) -- node[weight around]
+ {$w_{xy}$}
+ ([yshift=-1.5pt]u.east);
+ \end{tikzpicture}
+ \caption{Representing $E^{x,y}(u^\lambda_x, u^\lambda_y)$ with a
+ constant term of 0.}
+ \label{fig:neigh_subgraph_alt2}
+ \end{subfigure}
+ \caption{Two alternative ways of constructing a network representing
+ the energy term $E^{x,y}(u^\lambda_x, u^\lambda_y)$.
+ }
+ \label{fig:neigh_subgraph}
+\end{figure}
- \path[edge] (s) -- node[weight noslope] {$E^u(0)$} (u);
-\end{tikzpicture}
-\caption{Something something $E^u(0)$ is always positive and $E^u(1)$ is
-always zero.}
-\label{fig:norm_subgraph}
+\begin{figure}
+ \centering
+ \begin{subfigure}[t]{0.4\textwidth}
+ \centering
+ \begin{tikzpicture}
+ \draw[->] (0,0) -- (4,0) node[right] {$\lambda$};
+ \draw[->] (0,-2) -- (0,2) node[above] {$E^x(0)$};
+ \draw[line,domain=0:1] plot ({\x},{-1});
+ \draw[line,domain=1:2,dashed] plot ({\x},{2*\x-3});
+ \draw[line,domain=2:4] plot ({\x},{1});
+
+ \draw (1,2pt) -- (1,-2pt) node[
+ font=\small,
+ anchor=south
+ ]
+ {$v_x-1$};
+
+ \draw (2,2pt) -- (2,-2pt) node[
+ font=\small,
+ anchor=north
+ ]
+ {$v_x$};
+
+ \end{tikzpicture}
+ \caption{$L^1$ fidelity term.}
+ \label{fig:l1_norm_evolution}
+ \end{subfigure}
+ ~
+ \begin{subfigure}[t]{0.4\textwidth}
+ \centering
+ \begin{tikzpicture}
+ \draw[->] (0,0) -- (4,0) node[right] {$\lambda$};
+ \draw[->] (0,-2) -- (0,2) node[above] {$E^x(0)$};
+ \draw[line,domain=0:4] plot ({\x},{.8*\x - 1.6});
+
+ \draw (2.5,2pt) -- (2.5,-2pt) node[
+ font=\small,
+ anchor=north
+ ]
+ {$v_x$};
+
+ \draw (1.5,2pt) -- (1.5,-2pt) node[
+ font=\small,
+ anchor=south
+ ]
+ {$v_x-1$};
+
+ \end{tikzpicture}
+ \caption{$L^2$ fidelity term.}
+ \label{fig:l2_norm_evolution}
+ \end{subfigure}
+ \caption{Two figures showing how the fidelity energy term $E^x(0)$
+ in (\fixme{REF}) increases monotonically with $\lambda$.}
+ \label{fig:norm_evolution}
\end{figure}
\fixme{Make it clear what happens when a node lands on the $T$-side
of jumps in the resulting image $u$ is contained in the set of jumps in
the original image $v$ are also presented.
-\fixme{\cite{caselles2011total} has a nice introduction to total
- variation. It describes the foundations, the spaces, and some
- methods, but mostly continuous. Could include some more theory here,
- like the coarea stuff, and the number of discontinuities?}
-
-\fixme{Write something about the different methods that exist?}
-
\subsection{Discrete problem}
Since digital images are given on a discrete grid, with values taken
from a discrete and finite set of levels, we want to discretize
\phi_{xy}$, and $d_{xy}$ go to zero. In other words, if the grid size
decreases and we at the same time increase the density of the
neighborhood.
-\fixme{Maybe move (and expand) this somewhere else? Where the grids are
-discussed for example? Also, the section is a bit messy.}
\fixme{BAD TRANSITION}
\big) + N(0) \\
&= \sum_{\lambda=0}^{L-2} \big(
N(\lambda + 1) - N(\lambda)
- \big) \mathbbm{1}_{\lambda < k} + N(0) \\
- &= \sum_{\lambda=0}^{L-2} \big(
- N(\lambda + 1) - N(\lambda)
- \big) (1 - u^\lambda) + N(0).
+ \big) \mathbbm{1}_{\lambda < k} + N(0)
\end{aligned}
\end{equation}
-Using this we rewrite \eqref{eq:norm_discrete_int} and obtain
+Since $\mathbbm{1}_{\lambda < u_x} = (1 - u^\lambda_x)$ we rewrite
+\eqref{eq:norm_discrete_int} and obtain
\begin{equation}
+ \sum_x N_x(u_x) =
\sum_{\lambda=0}^{L-2} \sum_x
\big(
N_x(\lambda + 1) -
\quad \forall x \in S.
\end{equation}
-\fixme{Want to use $u$ and $v$ as pixels/nodes, what to use as image?
-Since $s$ and $t$ normally is the source and sink.}
+\fixme{So, $u$ and $v$ are images, but also nodes in the flow-section?
+$x$ and $y$ are pixels. $s$ is the source, and $t$ is the sink?}
\fixme{Some like to call one term the fidelity term, and the other the
regularization term?}
-\fixme{\cite{boykov2003computing} motivates the discrete estimation of
- the perimeter of a set, and it makes sense. The size of the
- perimeter should be proportional to the number of edges it crosses,
- which is why we sum up $\abs{u^\lambda_s - u^\lambda_t}$ (which is
- only different from zero at the cut).
-}
-
-\fixme{How do we get to the discrete formulation. Why is it OK to do
- what we do with the gradient. How do we decompose the energy
- function into different levels and why is it OK to optimize each
- level separately. How do we choose the different neighbourhoods, and
- what could this mean for the result. Darbon and Sigelle.
- \cite{chan2011numerical} describes different methods for using total
- variation, and especially how one can use it in the discrete
- setting. It also goes into the min-cut method. Yay. Good book.
+\fixme{
+ Why is it OK to optimize each
+ level separately? Why does it work?
}
projects / prosjektoppgave / commitdiff