equal to the flow from $s$ to $t$.
\end{proof}
+We will see later that with the gap relabeling heuristic, there will
+always be a gap at label $k = N - 1$ such that we can construct our cut
+by taking $S = \{ u : d(u) \geq N \}$.
+
\fixme{Sketch of proof, could be nicer, shorter (split up?) and more
rigorous maybe.}
largest distance label.
\fixme{Running time and reference.}
-
\subsubsection{Heuristics}
Different heuristics exists that can speed up the algorithm
considerably. Being heuristics, they are not guaranteed to work, and
Hence, both the preflow $f$ and distance labeling $d$ are valid.
\end{proof}
+When running the push-relabel algorithm with the gap heuristic, we can
+be sure that there will never be a node $u$ with label $d(u) = N-1$ at
+the end of the algorithm, i.e.\ we know that there will always be a gap
+at label $N-1$. Using the same reasoning as in Theorem \fixme{ref}, if
+there was a vertex with label $N-1$, there would only be $N-3$ possibly
+having labels in $[1, N-2]$, so a gap must exist somewhere in that
+interval. When using the gap relabeling heuristic, such a gap can no
+exist, so we can conclude that there is no vertex with label $N-1$.
+
+Using Theorem \fixme{ref} we can then conclude that the sets $S = \{ u
+\in V : d(u) \geq N\}$ and $T = V - S$ form a minimum cut of the network.
+
+\fixme{Ugh, what a mess. Lemma maybe?}
+
\subsection{Parametric push-relabel algorithm}
Now we have an algorithm for finding a minimum $s$-$t$-cut in a network,
so let's return to the network constructed in Section \fixme{ref}. For
gives us the thresholded image $u^\lambda$. These can then hopefully be
stacked together to form the final image $u$.
+\subsubsection{Network reuse}
Solving \fixme{??} separate minimum cut problems seems like a lot of
work, but when using the push-relabel algorithm we will, if we do things
in the right order, be able to reuse the network when going from one
label to the next.
-Going back to Equation \fixme{??} and the sub-network representations in
-Figure \fixme{??} and Figure \fixme{??} we know that only edges from
-sub-networks representing the fidelity term depend on our level
-parameter $\lambda$. From Figure \fixme{move closer??} we see that the
-energy term $E^x(0)$ increases monotonically with increasing $\lambda$
-parameter. Let $u, v \in V - \{s, t\}$. Since the edges in Figure
-\fixme{??} now are the only ones depending on $\lambda$, the following
-is true for \emph{decreasing} values of $\lambda$
+\begin{figure}
+ \input{fig/norm_evolution}
+\end{figure}
+
+Going back to the network representations in Figure
+\ref{fig:norm_subgraph} and Figure \ref{fig:neigh_subgraph} we know that
+only the capacity of edges from sub-networks representing the fidelity
+term depend on our level parameter $\lambda$. From the expression in
+\eqref{eq:fidelity_energy0}, visualized in Figure
+\ref{fig:norm_evolution} we see that the energy term $E^x(0)$ increases
+monotonically with increasing $\lambda$ parameter. Let $u, v \in V -
+\{s, t\}$. Since the edges in Figure \ref{fig:norm_subgraph} now are the
+only ones depending on $\lambda$, the following is true for
+\emph{decreasing} values of $\lambda$
\begin{description}
\item[Edges from $s$ to $u$]
As seen in Figure \fixme{ref} the capacity of these edges will
decrease monotonically with decreasing $\lambda$ parameter.
\end{description}
-\subsubsection{Bleep bloop}
+After running the push-relabel algorithm for $\lambda = k$, we are left
+with a network $G = (V, E, c)$, a preflow $f$ and a labeling $d$. To
+obtain the network for $\lambda = k-1$ we have to change the capacity of
+two different kinds of edges, and this is done in the following way
+\begin{description}
+ \item[Edges from $s$ to $u$]
+ The capacity $c(s, u)$ is increased, and the flow is set to be
+ equal to the capacity $f(s, u) = c(s, u)$. The vertex $u$ might
+ have an increased excess $e(u)$, which might in turn make it
+ active.
+ \item[Edges from $v$ to $t$]
+ The capacity $c(v, t)$ is decreased. If it is decreased to a
+ value below the current flow value, we set $f(v, t) = c(v, t)$
+ which will decrease the excess of the sink $t$, and increase the
+ excess of $v$.
+\end{description}
+The preflow and capacity constraints are trivially satisfied.
+
+None of these actions will create new edges in the residual network, and
+we do not change the labeling $d$, so the labeling constraints are also
+satisfied in the new network.
+
+Through this procedure we have easily created the network for $\lambda =
+k-1$, and the distance labels remain the same. As these labels always
+increase monotonically, we have a head start compared to if we had reset
+the flow and labels.
+
+\subsubsection{Output image construction}
+\fixme{Mention earlier that with gap relabeling, we gap at N.}
+
+We mentioned already in section \ref{sec:total_energy} that in order to
+be able to construct our output image $u$, the thresholded images
+$u^\lambda$ would have to stack one on top of the other \fixme{as shown
+in Figure ??}. We will now show that through reuse of the distance
+labels from the last iteration of the push-relabel algorithm, we can
+guarantee that it is possible to stack the thresholded images.
+
+Consider two subsequent runs of the push-relabel algorithm, for labels
+$\lambda = k$ and $\lambda = k-1$ ending with distance labelings $d^k$
+and $d^{k-1}$ respectively. We already know that the distance
+labels $d$ are monotonically increasing. This means that the set $\{ u
+\in V : d(u) \geq N \}$ is increasing in size, more precisely, we have
+the inclusion
+\begin{equation}
+ \{ u \in V : d^k(u) \geq N \} \subseteq \{ u \in V : d^{k-1}(u) \geq
+ N \}
+\end{equation}
+
+We then construct our output image $u$ by giving each pixel the value
+\begin{equation}
+ u_x = \min \{ \lambda \in [0, L-1] : u^\lambda_x = 1 \}.
+\end{equation}
+This marks the end of the description of the implemented algorithm, but
+we will further discuss some possible improvements, and also look at
+results when using the method on different kinds of pictures, with
+different kinds of noise.
+
+\subsubsection{Divide and conquer}
+\fixme{Write something here.}
+
+\subsection{Implementation}
+\cite{opencv_library}
+
+\subsection{Bleep bloop}
+
+Maybe we should delete the $u$-$t$ edge when the vertex reaches a level
+> 1?
The nodes are allowed to have a positive excess but we still follow the
capacity constraints. The nodes also have a labeling, which has to be
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