pixel is taken from the set $\mathcal{L} = \{0, \hdots, L-1\}$. This is
a reasonable assumption for grayscale images.
+\subsubsection{Regularization term}
Chambolle discretizes the total variation as
\begin{equation}
\TV(u) = \sum_{i,j} \sqrt{
\centering
\begin{subfigure}[b]{0.4\textwidth}
\centering
- \begin{tikzpicture}[scale=1]
- \node[vertex] (x) at (0, 0) {x};
- \node[vertex] (y) at (2, 0) {y};
- \node[vertex] (a) at (0, 2) {};
- \node[vertex] (b) at (0, -2) {};
- \node[vertex] (c) at (-2, 0) {};
-
- \path[edge] (x) -- (y);
- \path[edge] (x) -- (a);
- \path[edge] (x) -- (b);
- \path[edge] (x) -- (c);
+ \begin{tikzpicture}[scale=.7]
+ \foreach \x in {0,...,4} {
+ \foreach \y in {0,...,4} {
+ \node[small vertex] (\x\y) at (\x, \y) {};
+ }
+ }
+ \node[small vertex,draw] (a) at (2, 2) {};
+ \node[small vertex,draw] (b) at (2, 3) {};
+ \node[small vertex,draw] (c) at (2, 1) {};
+ \node[small vertex,draw] (d) at (3, 2) {};
+ \node[small vertex,draw] (e) at (1, 2) {};
+
+ \path[edge] (a) -- (b);
+ \path[edge] (a) -- (c);
+ \path[edge] (a) -- (d);
+ \path[edge] (a) -- (e);
+
+ \path (00) [edge, <->, anchor=center] -- node[anchor=south] {h} (10) ;
\end{tikzpicture}
\caption{Size four neighborhood.}
\label{fig:c4_grid}
~
\begin{subfigure}[b]{0.4\textwidth}
\centering
- \begin{tikzpicture}[scale=1]
-
- \node[vertex] (1) at (0, 0) {x};
- \node[vertex] (2) at (-2, 0) {};
- \node[vertex] (3) at ( 2, 0) {};
- \node[vertex] (4) at (0, -2) {};
- \node[vertex] (5) at (0, 2) {};
- \node[vertex] (6) at (-2, -2) {};
- \node[vertex] (7) at (-2, 2) {};
- \node[vertex] (8) at ( 2, 2) {y};
- \node[vertex] (9) at ( 2, -2) {};
-
- \path[edge] (1) -- (2);
- \path[edge] (1) -- (3);
- \path[edge] (1) -- (4);
- \path[edge] (1) -- (5);
- \path[edge] (1) -- (6);
- \path[edge] (1) -- (7);
- \path[edge] (1) -- (8);
- \path[edge] (1) -- (9);
-
- \draw (0,0) +(0:1cm) arc (0:45:1cm);
- \path (0,0) ++(22.5:.75cm) node{$\alpha$};
+ \begin{tikzpicture}[scale=.7]
+ \foreach \x in {0,...,4} {
+ \foreach \y in {0,...,4} {
+ \node[small vertex] (\x\y) at (\x, \y) {};
+ }
+ }
+
+ \path[dash] (2, 2) -- (5,2);
+ \path[dash] (2, 2) -- (5,5);
+ \path[dash] (2, 2) -- (2,5.5);
+
+ \draw (2,2) +(0:2.5cm) arc (0:45:2.5cm);
+ \path (2,2) ++(22.5:3cm) node {$\phi_{xy}$};
+
+ \draw (2,2) +(45:3.3cm) arc (45:90:3.3cm);
+ \path (2,2) ++(67.5:3.8cm) node{$\Delta\phi_{xy}$};
+
+ \node[small vertex,draw] (a) at (2, 2) {x};
+ \node[small vertex,draw] (b) at (2, 3) {};
+ \node[small vertex,draw] (c) at (2, 1) {};
+ \node[small vertex,draw] (d) at (3, 2) {};
+ \node[small vertex,draw] (e) at (1, 2) {};
+ \node[small vertex,draw] (f) at (3, 3) {y};
+ \node[small vertex,draw] (g) at (1, 1) {};
+ \node[small vertex,draw] (h) at (1, 3) {};
+ \node[small vertex,draw] (i) at (3, 1) {};
+
+ \path[edge] (a) -- (b);
+ \path[edge] (a) -- (c);
+ \path[edge] (a) -- (d);
+ \path[edge] (a) -- (e);
+ \path[edge] (a) -- (f);
+ \path[edge] (a) -- (g);
+ \path[edge] (a) -- (h);
+ \path[edge] (a) -- (i);
\end{tikzpicture}
\caption{Size eight neighborhood.}
- \label{fig:c8_grid}
+ \label{fig:c4_grid}
\end{subfigure}
\caption{Two common neighborhood stencils.}
\label{fig:common_neighborhoods}
\end{figure}
-Recall that we managed to write the total variation as an integral over
-the range of level values in \eqref{eq:coarea_formula}. Here, we
+
+We also need to introduce the notion of pixel neighborhoods. A pixel in
+the image has a neighborhood which consists of the pixels around it, but
+not itself. We denote the neighborhood of pixel $x$ by $\mathcal{N}(x)$
+and we will call the line going from pixel $x$ to pixel $y \in
+\mathcal{N}(x)$ the \emph{edge} between $x$ and $y$. Two common
+neighborhood stencils are shown in Figure
+\ref{fig:common_neighborhoods}.
+
+Recall that we managed to write the total variation as an integral of
+the perimeter of the level sets in \eqref{eq:coarea_formula}. We
estimate the perimeter of the level set $u^\lambda$ in the discrete
setting, such that the total variation can be written as a sum over the
-level values. We introduce a neighborhood?
-
-We estimate the perimeter in the discrete setting to write the total
-variation as
+level values
\begin{equation}
\TV(u)
= \sum_{\lambda = 0}^{L-2} P(u^\lambda, S)
= \sum_{\lambda = 0}^{L-2} \sum_{(x,y)} w_{xy}
- \abs{u_x^\lambda - u_y^\lambda},
+ \abs{u_x^\lambda - u_y^\lambda}.
\label{eq:tv_discrete_int}
\end{equation}
-where the sum over $(x,y)$ signifies neighborhood relation and the
-$w_{xy}$ is the weight of this relation. Intuitively, the perimeter of
-the level set $u^\lambda$ is proportional to the number of edges
-crossing this perimeter, as $\abs{u^\lambda_x - u^\lambda_y}$ only
-contributes to the sum when one node is 0 and the other is 1.
-\fixme{Describe what this sum over $(x,y)$ is actually over.}
-
-The sum only goes up to the level $L-2$ since $u^{L-1}$ is constant
-equal to 1 in every pixel of the image. Figure
-\ref{fig:common_neighborhoods} shows the two commonly used neighbordhood
-stencils, which will be discussed more thorougly in \fixme{Section ??}.
+The sum over $(x,y)$ signifies a sum over all pixels $x$ and $y$ that
+are in a neighborhood relation, and $w_{xy}$ is a weight parameter. The
+sum over $\lambda$ ends at $L-2$ since $u^{L-1}$ is equal to 1 in every
+pixel of the image.
+
+Intuitively, the perimeter of the level set $u^\lambda$ is proportional
+to the number of pixels at the boundary of the set. This is again
+proportional to the number of neighborhood relations crossing the
+boundary, and $\abs{u^\lambda_x - u^\lambda_y}$ only contributes to the
+sum when one node is in $u^\lambda$ and the other is not.
+
Boykov and Kolmogorov argue in \cite{boykov2003computing} that if the
weight is chosen as
\begin{equation}
\end{equation}
the discrete perimeter in \eqref{eq:tv_discrete_int} converges to the
continuous perimeter in \eqref{eq:coarea_formula}. Here, $h$ is the grid
-size and $d_{xy}$ is the euclidean distance of the edge. The $\Delta
-\phi_{xy}$ is the difference between the angle of this edge and the
-next edge, if the edges are sorted by increasing angles. These
+size and $d_{xy}$ is the euclidean distance of the edge from $x$ to $y$.
+The $\Delta \phi_{xy}$ is the difference between the angle of this edge
+and the next edge, if the edges are sorted by increasing angles. These
parameters are also shown in Figure \ref{fig:common_neighborhoods}.
Boykov and Kolmogorov prove we have convergence when all of $h$, $\Delta
-\phi_{xy}$, 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.
+\phi_{xy}$, and $d_{xy}$ go to zero.
-\fixme{BAD TRANSITION}
-
-For two binary variables $a$ and $b$ we can easily verify that $\abs{a -
-b} = a + b - 2 a b$. Using this we rewrite \eqref{eq:tv_discrete_int} to
-\begin{equation}
- \TV(u) = \sum_{\lambda = 0}^{L-2} \sum_{(x,y)} w_{xy}
- \left(
- \left(1 - 2 u_y^\lambda \right) u_x^\lambda + u_y^\lambda
- \right).
- \label{eq:tv_discrete}
-\end{equation}
+\fixme{IS THIS EVEN NECESSARY? For two binary variables $a$ and $b$ we
+ can easily verify that $\abs{a - b} = a + b - 2 a b$. Using this we
+ rewrite \eqref{eq:tv_discrete_int} to
+ \begin{equation}
+ \TV(u) = \sum_{\lambda = 0}^{L-2} \sum_{(x,y)} w_{xy}
+ \left(
+ \left(1 - 2 u_y^\lambda \right) u_x^\lambda + u_y^\lambda
+ \right).
+ \label{eq:tv_discrete}
+ \end{equation}
+}
+\subsubsection{Fidelity term}
+Now that the total variation is discretized, we need to take care of the
+fidelity term.
We define the following function for some pixel value $k$ and some pixel
position $x$ in the original image $v$, which is the value of the energy
if we were to color pixel $u$ with label $k$
\big)
(1 - u^\lambda_x) + N_x(0)
\end{equation}
+
+\subsubsection{Total energy}
We have now discretized the energy function and decomposed it into a sum
over all the levels $\lambda$. Ignoring the constant term $N_x(0)$ we
are left with
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