New level set figure

diff --git a/fig/img_decomp.tex b/fig/img_decomp.tex
new file mode 100644 (file)
index 0000000..27a56ec
--- /dev/null
+++ b/fig/img_decomp.tex
@@ -0,0 +1,136 @@
+\centering
+\begin{subfigure}[t]{0.4\textwidth}
+    \centering
+    \begin{tikzpicture}[scale=1,every node/.style={minimum size=1cm},on grid]
+
+        % Note the numbers here are reversed!
+        \def\mylist{
+            {0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
+            {0, 0, 1, 1, 1, 1, 1, 0, 0, 0},
+            {0, 1, 1, 2, 2, 2, 1, 0, 1, 0},
+            {0, 1, 2, 2, 3, 2, 1, 1, 1, 0},
+            {0, 1, 2, 3, 3, 2, 1, 0, 1, 0},
+            {0, 1, 2, 3, 3, 2, 1, 0, 1, 0},
+            {0, 1, 2, 2, 2, 2, 1, 1, 1, 0},
+            {0, 1, 1, 1, 1, 1, 1, 1, 0, 0},
+            {0, 0, 1, 0, 0, 1, 1, 0, 0, 0},
+            {0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
+        }
+
+        \fill[white,fill opacity=.7] (0,0) rectangle (5,5);
+        \draw[black,very thick] (0,0) rectangle (5,5);
+        \foreach[count=\i] \row in \mylist
+        \foreach[count=\j] \pix in \row {
+            \ifnum\pix=1\fill[black!33,fill opacity=.9] (\i*.5-.5,\j*.5-.5) rectangle
+            (\i*.5,\j*.5);\fi
+            \ifnum\pix=2\fill[black!66,fill opacity=.9] (\i*.5-.5,\j*.5-.5) rectangle
+            (\i*.5,\j*.5);\fi
+            \ifnum\pix=3\fill[black,fill opacity=.9] (\i*.5-.5,\j*.5-.5) rectangle
+            (\i*.5,\j*.5);\fi
+        }
+        \draw[step=5mm, black] (0,0) grid (5,5);
+        \draw (2.25, 0.25) node {$3$};
+        \draw (2.25, 0.75) node {$2$};
+        \draw (2.25, 1.25) node {$1$};
+        \draw (2.25, 1.75) node[white] {$0$};
+    \end{tikzpicture}
+    \caption{An example gray-scale image with four different levels.}
+\end{subfigure}
+~
+\begin{subfigure}[t]{0.4\textwidth}
+    \centering
+
+    \begin{tikzpicture}[scale=.5,every node/.style={minimum size=1cm},on grid]
+
+        \def\mylist{
+            {0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
+            {0, 0, 1, 1, 1, 1, 1, 0, 0, 0},
+            {0, 1, 1, 2, 2, 2, 1, 0, 1, 0},
+            {0, 1, 2, 2, 3, 2, 1, 1, 1, 0},
+            {0, 1, 2, 3, 3, 2, 1, 0, 1, 0},
+            {0, 1, 2, 3, 3, 2, 1, 0, 1, 0},
+            {0, 1, 2, 2, 2, 2, 1, 1, 1, 0},
+            {0, 1, 1, 1, 1, 1, 1, 1, 0, 0},
+            {0, 0, 1, 0, 0, 1, 1, 0, 0, 0},
+            {0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
+        }
+
+
+        \begin{scope}[
+                yshift=0,every node/.append style={
+                yslant=0.5,xslant=-1},yslant=0.5,xslant=-1
+            ]
+            \fill[white,fill opacity=.7] (0,0) rectangle (5,5);
+            \draw[black,very thick] (0,0) rectangle (5,5);
+
+            \fill[black,fill opacity=.9] (0,0) rectangle (5,5);
+
+            \draw[step=5mm, black] (0,0) grid (5,5);
+        \end{scope}
+
+        \begin{scope}[
+                yshift=50,every node/.append style={
+                yslant=0.5,xslant=-1},yslant=0.5,xslant=-1
+            ]
+            \fill[white,fill opacity=.7] (0,0) rectangle (5,5);
+            \draw[black,very thick] (0,0) rectangle (5,5);
+
+            \foreach[count=\i] \row in \mylist
+            \foreach[count=\j] \pix in \row {
+                \ifnum\pix<1\else\fill[black,fill opacity=.9] (\i*.5-.5,\j*.5-.5) rectangle
+                (\i*.5,\j*.5);\fi
+            }
+
+            \draw[step=5mm, black] (0,0) grid (5,5);
+        \end{scope}
+
+        \begin{scope}[
+                yshift=100,every node/.append style={
+                yslant=0.5,xslant=-1},yslant=0.5,xslant=-1
+            ]
+            \fill[white,fill opacity=.7] (0,0) rectangle (5,5);
+            \draw[black,very thick] (0,0) rectangle (5,5);
+
+            \foreach[count=\i] \row in \mylist
+            \foreach[count=\j] \pix in \row {
+                \ifnum\pix<2\else\fill[black,fill opacity=.9] (\i*.5-.5,\j*.5-.5) rectangle
+                (\i*.5,\j*.5);\fi
+            }
+
+            \draw[step=5mm, black] (0,0) grid (5,5);
+        \end{scope}
+
+        \begin{scope}[
+                yshift=150,every node/.append style={
+                yslant=0.5,xslant=-1},yslant=0.5,xslant=-1
+            ]
+            \fill[white,fill opacity=.7] (0,0) rectangle (5,5);
+            \draw[black,very thick] (0,0) rectangle (5,5);
+
+            \foreach[count=\i] \row in \mylist
+            \foreach[count=\j] \pix in \row {
+                \ifnum\pix=3\fill[black,fill opacity=.9] (\i*.5-.5,\j*.5-.5) rectangle
+                (\i*.5,\j*.5);\fi
+            }
+
+            \draw[step=5mm, black] (0,0) grid (5,5);
+        \end{scope}
+
+        \tikzstyle{blopp} = [
+            rectangle,anchor=west, text width=1.8cm
+        ]
+
+        \draw[-latex,thick] (5.2,1.5) node[blopp]
+            {$u^3 = \idfun_{u \leq 3}$} to[out=180,in=-50] (4,2);
+        \draw[-latex,thick] (5.2,3.25) node[blopp]{$u^2$}
+            to[out=180,in=-50] (4,3.75);
+        \draw[-latex,thick] (5.2,5) node[blopp]{$u^1$}
+            to[out=180,in=-50] (4,5.5);
+        \draw[-latex,thick] (5.2,6.75) node[blopp]{$u^0$}
+            to[out=180,in=-50] (4,7.25);
+
+    \end{tikzpicture}
+    \caption{The thresholded images, which are in
+    $\{0,1\}^{\abs{\Omega}}$.}
+\end{subfigure} 
+

diff --git a/fig/level_sets_tab.tex b/fig/level_sets_tab.tex
new file mode 100644 (file)
index 0000000..137c844
--- /dev/null
+++ b/fig/level_sets_tab.tex
@@ -0,0 +1,27 @@
+\centering
+\begin{tikzpicture}[
+        baseline=(M.center),
+        bg1/.style={fill=black!10},
+        bg2/.style={fill=black!25},
+        bg3/.style={fill=black!40},
+        bg4/.style={fill=black!55},
+        bg5/.style={fill=black!70},
+    ]
+    \matrix (M) [
+        matrix of nodes,
+        %column sep={6pt,between origins},
+        %row sep={6pt,between origins}
+    ] 
+    {
+        \node (A) {}; & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
+                      & 1 & 1 & 1 & 1 & 1 & 1 & 1 &   & 1 & 1 & 1 & 1 \\
+                      & 1 & 1 &   & 1 & 1 & 1 &   &   &   & 1 & 1 & 1 \\
+                      & 1 &   &   &   & 1 & 1 &   &   &   &   & 1 & 1 \\
+        \node (L) {$\mathcal{L}$}; & 1 &   &   &   &   &   &   &   &   &   &   & 1 \\
+                      & |[bg5]| 5 & |[bg3]| 3 & |[bg2]| 2 & |[bg3]| 3 & |[bg4]| 4 & |[bg4]| 4 & |[bg2]| 2 & |[bg1]| 1 & |[bg2]| 2 & |[bg3]| 3 & |[bg4]| 4 & |[bg5]| 5 \\
+    };
+
+    \draw[->] (L) -- (A);
+\end{tikzpicture}
+\caption{The level sets of a one-dimensional image.}
+\label{fig:level_sets}

diff --git a/total.tex b/total.tex
index cf95388ac7cca62e73b0ff7908e67547967c66e2..42be4f55b79cebd7def52ba30c05c75d5718be5b 100644 (file)
--- a/total.tex
+++ b/total.tex
@@ -250,33 +250,7 @@ it does not decompose into a sum over the levels of the image, which we
 need for our graph composition. 
 
 \begin{figure}
-    \centering
-    \begin{tikzpicture}[
-        baseline=(M.center),
-        bg1/.style={fill=black!10},
-        bg2/.style={fill=black!25},
-        bg3/.style={fill=black!40},
-        bg4/.style={fill=black!55},
-        bg5/.style={fill=black!70},
-    ]
-    \matrix (M) [
-        matrix of nodes,
-        %column sep={6pt,between origins},
-        %row sep={6pt,between origins}
-    ] 
-{
-    \node (A) {}; & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
-                  & 1 & 1 & 1 & 1 & 1 & 1 & 1 &   & 1 & 1 & 1 & 1 \\
-                  & 1 & 1 &   & 1 & 1 & 1 &   &   &   & 1 & 1 & 1 \\
-                  & 1 &   &   &   & 1 & 1 &   &   &   &   & 1 & 1 \\
-    \node (L) {$\mathcal{L}$}; & 1 &   &   &   &   &   &   &   &   &   &   & 1 \\
-                               & |[bg5]| 5 & |[bg3]| 3 & |[bg2]| 2 & |[bg3]| 3 & |[bg4]| 4 & |[bg4]| 4 & |[bg2]| 2 & |[bg1]| 1 & |[bg2]| 2 & |[bg3]| 3 & |[bg4]| 4 & |[bg5]| 5 \\
-};
-
-\draw[->] (L) -- (A);
-\end{tikzpicture}
-\caption{The level sets of a one-dimensional image.}
-\label{fig:level_sets}
+    \input{fig/img_decomp}
 \end{figure}
 
 Following the notation used in \cite{darbon2006image}, we let $u_x$
@@ -459,6 +433,8 @@ Since $\mathbbm{1}_{\lambda < u_x} = (1 - u^\lambda_x)$ we rewrite
 \fixme{$E_\lambda^x$}
 
 \subsubsection{Total energy}
+\label{sec: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