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Made polytope example figure more understandable

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Andreas Tsouchlos 2023-02-19 15:45:15 +01:00
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1 changed files with 136 additions and 139 deletions
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latex/thesis/chapters/decoding_techniques.tex
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@ -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 $\text{poly}\left( \mathcal{C} \right) $, i.e. the constraints for the
equivalent linear program to exact \ac{ML} decoding - only valid codewords are equivalent linear program to exact \ac{ML} decoding - only valid codewords are
feasible solutions. 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 Their intersection, the relaxed codeword polytope $\overline{Q}$, is shown in
figure \ref{fig:dec:poly:relaxed}. figure \ref{fig:dec:poly:relaxed}.
It can be seen that the relaxed codeword polytope $\overline{Q}$ introduces 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 % Left side - codeword polytope
% %
\begin{subfigure}[c]{0.45\textwidth} \begin{subfigure}[b]{0.49\textwidth}
\centering \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} \begin{subfigure}{\textwidth}
\centering \centering
@ -338,7 +312,7 @@ The resulting formulation of the relaxed optimization problem is the following:%
draw, circle, inner sep=0pt, minimum size=4pt] draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25} \tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords] \begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
% Cube % Cube
\coordinate (p000) at (0, 0, 0); \coordinate (p000) at (0, 0, 0);
@ -389,143 +363,162 @@ The resulting formulation of the relaxed optimization problem is the following:%
% Right side - relaxed polytope % Right side - relaxed polytope
% %
% %
\begin{subfigure}[c]{0.45\textwidth} \begin{subfigure}[b]{0.49\textwidth}
\centering \centering
\begin{subfigure}{\textwidth} \begin{subfigure}{\textwidth}
\centering \centering
\tikzstyle{codeword} = [color=KITblue, fill=KITblue, \begin{minipage}{0.5\textwidth}
draw, circle, inner sep=0pt, minimum size=4pt] \centering
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25} \tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords] \begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
% Cube % Cube
\coordinate (p000) at (0, 0, 0); \coordinate (p000) at (0, 0, 0);
\coordinate (p001) at (0, 0, 2); \coordinate (p001) at (0, 0, 2);
\coordinate (p010) at (0, 2, 0); \coordinate (p010) at (0, 2, 0);
\coordinate (p011) at (0, 2, 2); \coordinate (p011) at (0, 2, 2);
\coordinate (p100) at (2, 0, 0); \coordinate (p100) at (2, 0, 0);
\coordinate (p101) at (2, 0, 2); \coordinate (p101) at (2, 0, 2);
\coordinate (p110) at (2, 2, 0); \coordinate (p110) at (2, 2, 0);
\coordinate (p111) at (2, 2, 2); \coordinate (p111) at (2, 2, 2);
\draw[] (p000) -- (p100); \draw[] (p000) -- (p100);
\draw[] (p100) -- (p101); \draw[] (p100) -- (p101);
\draw[] (p101) -- (p001); \draw[] (p101) -- (p001);
\draw[] (p001) -- (p000); \draw[] (p001) -- (p000);
\draw[dashed] (p010) -- (p110); \draw[dashed] (p010) -- (p110);
\draw[] (p110) -- (p111); \draw[] (p110) -- (p111);
\draw[] (p111) -- (p011); \draw[] (p111) -- (p011);
\draw[dashed] (p011) -- (p010); \draw[dashed] (p011) -- (p010);
\draw[dashed] (p000) -- (p010); \draw[dashed] (p000) -- (p010);
\draw[] (p100) -- (p110); \draw[] (p100) -- (p110);
\draw[] (p101) -- (p111); \draw[] (p101) -- (p111);
\draw[] (p001) -- (p011); \draw[] (p001) -- (p011);
% Polytope Vertices % Polytope Vertices
\node[codeword] (c000) at (p000) {}; \node[codeword] (c000) at (p000) {};
\node[codeword] (c101) at (p101) {}; \node[codeword] (c101) at (p101) {};
\node[codeword] (c110) at (p110) {}; \node[codeword] (c110) at (p110) {};
\node[codeword] (c011) at (p011) {}; \node[codeword] (c011) at (p011) {};
% Polytope Edges % Polytope Edges
\draw[line width=1pt, color=KITblue] (c000) -- (c101); \draw[line width=1pt, color=KITblue] (c000) -- (c101);
\draw[line width=1pt, color=KITblue] (c000) -- (c110); \draw[line width=1pt, color=KITblue] (c000) -- (c110);
\draw[line width=1pt, color=KITblue] (c000) -- (c011); \draw[line width=1pt, color=KITblue] (c000) -- (c011);
\draw[line width=1pt, color=KITblue] (c101) -- (c110); \draw[line width=1pt, color=KITblue] (c101) -- (c110);
\draw[line width=1pt, color=KITblue] (c101) -- (c011); \draw[line width=1pt, color=KITblue] (c101) -- (c011);
\draw[line width=1pt, color=KITblue] (c011) -- (c110); \draw[line width=1pt, color=KITblue] (c011) -- (c110);
% Polytope Annotations % Polytope Annotations
\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $}; \node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
\node[color=KITblue, right=0.17cm of c101] {$\left( 1, 0, 1 \right) $}; \node[color=KITblue, right=0.17cm of c101] {$\left( 1, 0, 1 \right) $};
\node[color=KITblue, right=0cm of c110] {$\left( 1, 1, 0 \right) $}; \node[color=KITblue, right=0cm of c110] {$\left( 1, 1, 0 \right) $};
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $}; \node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
\end{tikzpicture}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
% Cube
\coordinate (p000) at (0, 0, 0);
\coordinate (p001) at (0, 0, 2);
\coordinate (p010) at (0, 2, 0);
\coordinate (p011) at (0, 2, 2);
\coordinate (p100) at (2, 0, 0);
\coordinate (p101) at (2, 0, 2);
\coordinate (p110) at (2, 2, 0);
\coordinate (p111) at (2, 2, 2);
\draw[] (p000) -- (p100);
\draw[] (p100) -- (p101);
\draw[] (p101) -- (p001);
\draw[] (p001) -- (p000);
\draw[dashed] (p010) -- (p110);
\draw[] (p110) -- (p111);
\draw[] (p111) -- (p011);
\draw[dashed] (p011) -- (p010);
\draw[dashed] (p000) -- (p010);
\draw[] (p100) -- (p110);
\draw[] (p101) -- (p111);
\draw[] (p001) -- (p011);
% Polytope Vertices
\node[codeword] (c000) at (p000) {};
\node[codeword] (c011) at (p011) {};
\node[codeword] (c100) at (p100) {};
\node[codeword] (c111) at (p111) {};
% Polytope Edges
\draw[line width=1pt, color=KITblue] (c000) -- (c011);
\draw[line width=1pt, color=KITblue] (c000) -- (c100);
\draw[line width=1pt, color=KITblue] (c100) -- (c111);
\draw[line width=1pt, color=KITblue] (c111) -- (c011);
% Polytope Annotations
\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
\node[color=KITblue, below=0cm of c100] {$\left( 1, 0, 0 \right) $};
\node[color=KITblue, above=0cm of c111] {$\left( 1, 1, 1 \right) $};
\end{tikzpicture}
\end{minipage}
\begin{tikzpicture}
\node[color=KITblue, align=center] at (-2,0)
{$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} \end{tikzpicture}
\caption{Local codeword polytope of check node\\ $j=1$
$\left( c_1 + c_2 + c_3 = 0 \right)$}
\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} \caption{Local codeword polytopes of the check nodes}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords] \label{fig:dec:poly:local}
% Cube
\coordinate (p000) at (0, 0, 0);
\coordinate (p001) at (0, 0, 2);
\coordinate (p010) at (0, 2, 0);
\coordinate (p011) at (0, 2, 2);
\coordinate (p100) at (2, 0, 0);
\coordinate (p101) at (2, 0, 2);
\coordinate (p110) at (2, 2, 0);
\coordinate (p111) at (2, 2, 2);
\draw[] (p000) -- (p100);
\draw[] (p100) -- (p101);
\draw[] (p101) -- (p001);
\draw[] (p001) -- (p000);
\draw[dashed] (p010) -- (p110);
\draw[] (p110) -- (p111);
\draw[] (p111) -- (p011);
\draw[dashed] (p011) -- (p010);
\draw[dashed] (p000) -- (p010);
\draw[] (p100) -- (p110);
\draw[] (p101) -- (p111);
\draw[] (p001) -- (p011);
% Polytope Vertices
\node[codeword] (c000) at (p000) {};
\node[codeword] (c011) at (p011) {};
\node[codeword] (c100) at (p100) {};
\node[codeword] (c111) at (p111) {};
% Polytope Edges
\draw[line width=1pt, color=KITblue] (c000) -- (c011);
\draw[line width=1pt, color=KITblue] (c000) -- (c100);
\draw[line width=1pt, color=KITblue] (c100) -- (c111);
\draw[line width=1pt, color=KITblue] (c111) -- (c011);
% Polytope Annotations
\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
\node[color=KITblue, below=0cm of c100] {$\left( 1, 0, 0 \right) $};
\node[color=KITblue, above=0cm of c111] {$\left( 1, 1, 1 \right) $};
\end{tikzpicture}
\caption{Local codeword polytope of check node\\ $j=2$
$\left( c_2 + c_3 = 0\right)$}
\label{fig:dec:poly:local2}
\end{subfigure}\\[1em] \end{subfigure}\\[1em]
\begin{subfigure}{\textwidth} \begin{subfigure}{\textwidth}
\centering \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, \tikzstyle{codeword} = [color=KITblue, fill=KITblue,
draw, circle, inner sep=0pt, minimum size=4pt] draw, circle, inner sep=0pt, minimum size=4pt]
\tikzstyle{pseudocodeword} = [color=KITred, fill=KITred, \tikzstyle{pseudocodeword} = [color=KITred, fill=KITred,
draw, circle, inner sep=0pt, minimum size=4pt] draw, circle, inner sep=0pt, minimum size=4pt]
\tdplotsetmaincoords{60}{25} \tdplotsetmaincoords{60}{25}
\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords] \begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
% Cube % Cube
\coordinate (p000) at (0, 0, 0); \coordinate (p000) at (0, 0, 0);
@ -578,7 +571,11 @@ The resulting formulation of the relaxed optimization problem is the following:%
\end{subfigure} \end{subfigure}
\caption{Visualization of the codeword polytope and the relaxed codeword \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} \label{fig:dec:poly}
\end{figure}% \end{figure}%
% %