Made polytope example figure more understandable

latex/thesis/chapters/decoding_techniques.tex

@@ -270,8 +270,8 @@ Figure \ref{fig:dec:poly:exact} shows the codeword polytope
 $\text{poly}\left( \mathcal{C} \right) $, i.e. the constraints for the
 equivalent linear program to exact \ac{ML} decoding - only valid codewords are
 feasible solutions.
-Figures \ref{fig:dec:poly:local1} and \ref{fig:dec:poly:local2} show the local
+Figure \ref{fig:dec:poly:local} shows the local codeword polytope of each check
-codeword polytopes of each check node.
+node.
 Their intersection, the relaxed codeword polytope $\overline{Q}$, is shown in
 figure \ref{fig:dec:poly:relaxed}.
 It can be seen that the relaxed codeword polytope $\overline{Q}$ introduces
@@ -302,35 +302,9 @@ The resulting formulation of the relaxed optimization problem is the following:%
     % Left side - codeword polytope
     %
 
-    \begin{subfigure}[c]{0.45\textwidth}
+    \begin{subfigure}[b]{0.49\textwidth}
         \centering
    
-        \begin{subfigure}{\textwidth}
-            \centering
-        
-            \begin{align*}
-                \boldsymbol{H} &=
-                    \begin{bmatrix}
-                        1 & 1 & 1\\
-                        0 & 1 & 1
-                    \end{bmatrix}\\[1em]
-                \mathcal{C} &= \left\{ 
-                    \begin{bmatrix}
-                        0\\
-                        0\\
-                        0
-                    \end{bmatrix},
-                    \begin{bmatrix}
-                        0\\
-                        1\\
-                        1
-                    \end{bmatrix}
-                \right\}
-            \end{align*}
-
-            \caption{Definition of the visualized code}
-            \label{fig:dec:poly:code_def}
-        \end{subfigure} \\[7em]
         \begin{subfigure}{\textwidth}
             \centering
 
@@ -338,7 +312,7 @@ The resulting formulation of the relaxed optimization problem is the following:%
                                 draw, circle, inner sep=0pt, minimum size=4pt]
 
             \tdplotsetmaincoords{60}{25}
-            \begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
+            \begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
                 % Cube
 
                 \coordinate (p000) at (0, 0, 0);
@@ -389,17 +363,20 @@ The resulting formulation of the relaxed optimization problem is the following:%
     % Right side - relaxed polytope
     %
 %
-    \begin{subfigure}[c]{0.45\textwidth}
+    \begin{subfigure}[b]{0.49\textwidth}
         \centering
     
         \begin{subfigure}{\textwidth}
             \centering
        
+            \begin{minipage}{0.5\textwidth}
+                \centering
+            
                 \tikzstyle{codeword} = [color=KITblue, fill=KITblue,
                                     draw, circle, inner sep=0pt, minimum size=4pt]
 
                 \tdplotsetmaincoords{60}{25}
-            \begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
+                \begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
                     % Cube
 
                     \coordinate (p000) at (0, 0, 0);
@@ -451,19 +428,15 @@ The resulting formulation of the relaxed optimization problem is the following:%
                     \node[color=KITblue, right=0cm of c110]    {$\left( 1, 1, 0 \right) $};
                     \node[color=KITblue, above=0cm of c011]    {$\left( 0, 1, 1 \right) $};
                 \end{tikzpicture}
-        
+            \end{minipage}%
-            \caption{Local codeword polytope of check node\\ $j=1$
+            \begin{minipage}{0.5\textwidth}
-                $\left( c_1 + c_2 + c_3 = 0 \right)$}
-            \label{fig:dec:poly:local1}
-        \end{subfigure} \\[1em]
-        \begin{subfigure}{\textwidth}
                 \centering
 
                 \tikzstyle{codeword} = [color=KITblue, fill=KITblue,
                                     draw, circle, inner sep=0pt, minimum size=4pt]
 
                 \tdplotsetmaincoords{60}{25}
-            \begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
+                \begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
                     % Cube
 
                     \coordinate (p000) at (0, 0, 0);
@@ -511,21 +484,41 @@ The resulting formulation of the relaxed optimization problem is the following:%
                     \node[color=KITblue, below=0cm of c100] {$\left( 1, 0, 0 \right) $};
                     \node[color=KITblue, above=0cm of c111] {$\left( 1, 1, 1 \right) $};
                 \end{tikzpicture}
+            \end{minipage}
  
-            \caption{Local codeword polytope of check node\\ $j=2$
+            \begin{tikzpicture}
-                $\left( c_2 + c_3 = 0\right)$}
+                \node[color=KITblue, align=center] at (-2,0)
-            \label{fig:dec:poly:local2}
+                    {$j=1$\\ $\left( c_1 + c_2+ c_3 = 0 \right) $};
+                \node[color=KITblue, align=center] at (2,0)
+                    {$j=2$\\ $\left(c_2 + c_3 = 0\right)$};
+            \end{tikzpicture}
+
+            \caption{Local codeword polytopes of the check nodes}
+            \label{fig:dec:poly:local}
         \end{subfigure}\\[1em]
         \begin{subfigure}{\textwidth}
             \centering
         
+            \begin{tikzpicture}
+                \draw (-2, 0) -- (2, 0);
+                \draw (-2, 0.5) -- (-2, 0);
+                \draw (2, 0.5) -- (2, 0);
+                
+                \node (intersection) at (0, -0.5) {Intersection};
+
+                \draw (0, 0) -- (intersection);
+                \draw[->] (intersection) -- (0, -1);
+            \end{tikzpicture}
+
+            \vspace{2mm}
+
             \tikzstyle{codeword} = [color=KITblue, fill=KITblue,
                                 draw, circle, inner sep=0pt, minimum size=4pt]
             \tikzstyle{pseudocodeword} = [color=KITred, fill=KITred,
                                 draw, circle, inner sep=0pt, minimum size=4pt]
 
             \tdplotsetmaincoords{60}{25}
-            \begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
+            \begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
                 % Cube
 
                 \coordinate (p000) at (0, 0, 0);
@@ -578,7 +571,11 @@ The resulting formulation of the relaxed optimization problem is the following:%
     \end{subfigure}
     
     \caption{Visualization of the codeword polytope and the relaxed codeword
-    polytope}
+        polytope for the code defined by the parity check matrix $\boldsymbol{H} =
+            \begin{bmatrix}
+                1 & 1 & 1\\
+                0 & 1 & 1
+            \end{bmatrix}$}
     \label{fig:dec:poly}
 \end{figure}%
 %