Made polytope example figure more understandable
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@ -270,8 +270,8 @@ Figure \ref{fig:dec:poly:exact} shows the codeword polytope
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$\text{poly}\left( \mathcal{C} \right) $, i.e. the constraints for the
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equivalent linear program to exact \ac{ML} decoding - only valid codewords are
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feasible solutions.
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Figures \ref{fig:dec:poly:local1} and \ref{fig:dec:poly:local2} show the local
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codeword polytopes of each check node.
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Figure \ref{fig:dec:poly:local} shows the local codeword polytope of each check
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node.
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Their intersection, the relaxed codeword polytope $\overline{Q}$, is shown in
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figure \ref{fig:dec:poly:relaxed}.
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It can be seen that the relaxed codeword polytope $\overline{Q}$ introduces
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@ -302,35 +302,9 @@ The resulting formulation of the relaxed optimization problem is the following:%
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% Left side - codeword polytope
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%
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\begin{subfigure}[c]{0.45\textwidth}
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\begin{subfigure}[b]{0.49\textwidth}
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\centering
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\begin{subfigure}{\textwidth}
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\centering
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\begin{align*}
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\boldsymbol{H} &=
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\begin{bmatrix}
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1 & 1 & 1\\
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0 & 1 & 1
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\end{bmatrix}\\[1em]
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\mathcal{C} &= \left\{
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\begin{bmatrix}
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0\\
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0\\
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0
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\end{bmatrix},
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\begin{bmatrix}
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0\\
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1\\
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1
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\end{bmatrix}
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\right\}
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\end{align*}
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\caption{Definition of the visualized code}
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\label{fig:dec:poly:code_def}
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\end{subfigure} \\[7em]
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\begin{subfigure}{\textwidth}
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\centering
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@ -338,7 +312,7 @@ The resulting formulation of the relaxed optimization problem is the following:%
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{25}
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\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
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\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
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% Cube
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\coordinate (p000) at (0, 0, 0);
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@ -389,143 +363,162 @@ The resulting formulation of the relaxed optimization problem is the following:%
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% Right side - relaxed polytope
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%
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%
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\begin{subfigure}[c]{0.45\textwidth}
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\begin{subfigure}[b]{0.49\textwidth}
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\centering
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\begin{subfigure}{\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\begin{minipage}{0.5\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{25}
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\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
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% Cube
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\tdplotsetmaincoords{60}{25}
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\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
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% Cube
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\coordinate (p000) at (0, 0, 0);
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\coordinate (p001) at (0, 0, 2);
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\coordinate (p010) at (0, 2, 0);
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\coordinate (p011) at (0, 2, 2);
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\coordinate (p100) at (2, 0, 0);
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\coordinate (p101) at (2, 0, 2);
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\coordinate (p110) at (2, 2, 0);
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\coordinate (p111) at (2, 2, 2);
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\coordinate (p000) at (0, 0, 0);
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\coordinate (p001) at (0, 0, 2);
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\coordinate (p010) at (0, 2, 0);
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\coordinate (p011) at (0, 2, 2);
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\coordinate (p100) at (2, 0, 0);
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\coordinate (p101) at (2, 0, 2);
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\coordinate (p110) at (2, 2, 0);
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\coordinate (p111) at (2, 2, 2);
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\draw[] (p000) -- (p100);
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\draw[] (p100) -- (p101);
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\draw[] (p101) -- (p001);
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\draw[] (p001) -- (p000);
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\draw[] (p000) -- (p100);
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\draw[] (p100) -- (p101);
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\draw[] (p101) -- (p001);
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\draw[] (p001) -- (p000);
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\draw[dashed] (p010) -- (p110);
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\draw[] (p110) -- (p111);
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\draw[] (p111) -- (p011);
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\draw[dashed] (p011) -- (p010);
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\draw[dashed] (p010) -- (p110);
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\draw[] (p110) -- (p111);
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\draw[] (p111) -- (p011);
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\draw[dashed] (p011) -- (p010);
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\draw[dashed] (p000) -- (p010);
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\draw[] (p100) -- (p110);
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\draw[] (p101) -- (p111);
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\draw[] (p001) -- (p011);
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\draw[dashed] (p000) -- (p010);
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\draw[] (p100) -- (p110);
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\draw[] (p101) -- (p111);
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\draw[] (p001) -- (p011);
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% Polytope Vertices
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% Polytope Vertices
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\node[codeword] (c000) at (p000) {};
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\node[codeword] (c101) at (p101) {};
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\node[codeword] (c110) at (p110) {};
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\node[codeword] (c011) at (p011) {};
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\node[codeword] (c000) at (p000) {};
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\node[codeword] (c101) at (p101) {};
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\node[codeword] (c110) at (p110) {};
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\node[codeword] (c011) at (p011) {};
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% Polytope Edges
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% Polytope Edges
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\draw[line width=1pt, color=KITblue] (c000) -- (c101);
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\draw[line width=1pt, color=KITblue] (c000) -- (c110);
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\draw[line width=1pt, color=KITblue] (c000) -- (c011);
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\draw[line width=1pt, color=KITblue] (c000) -- (c101);
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\draw[line width=1pt, color=KITblue] (c000) -- (c110);
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\draw[line width=1pt, color=KITblue] (c000) -- (c011);
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\draw[line width=1pt, color=KITblue] (c101) -- (c110);
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\draw[line width=1pt, color=KITblue] (c101) -- (c011);
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\draw[line width=1pt, color=KITblue] (c101) -- (c110);
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\draw[line width=1pt, color=KITblue] (c101) -- (c011);
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\draw[line width=1pt, color=KITblue] (c011) -- (c110);
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\draw[line width=1pt, color=KITblue] (c011) -- (c110);
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% Polytope Annotations
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% Polytope Annotations
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\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, right=0.17cm of c101] {$\left( 1, 0, 1 \right) $};
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\node[color=KITblue, right=0cm of c110] {$\left( 1, 1, 0 \right) $};
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, right=0.17cm of c101] {$\left( 1, 0, 1 \right) $};
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\node[color=KITblue, right=0cm of c110] {$\left( 1, 1, 0 \right) $};
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\end{tikzpicture}
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\end{minipage}%
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\begin{minipage}{0.5\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{25}
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\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
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% Cube
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\coordinate (p000) at (0, 0, 0);
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\coordinate (p001) at (0, 0, 2);
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\coordinate (p010) at (0, 2, 0);
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\coordinate (p011) at (0, 2, 2);
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\coordinate (p100) at (2, 0, 0);
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\coordinate (p101) at (2, 0, 2);
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\coordinate (p110) at (2, 2, 0);
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\coordinate (p111) at (2, 2, 2);
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\draw[] (p000) -- (p100);
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\draw[] (p100) -- (p101);
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\draw[] (p101) -- (p001);
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\draw[] (p001) -- (p000);
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\draw[dashed] (p010) -- (p110);
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\draw[] (p110) -- (p111);
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\draw[] (p111) -- (p011);
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\draw[dashed] (p011) -- (p010);
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\draw[dashed] (p000) -- (p010);
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\draw[] (p100) -- (p110);
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\draw[] (p101) -- (p111);
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\draw[] (p001) -- (p011);
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% Polytope Vertices
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\node[codeword] (c000) at (p000) {};
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\node[codeword] (c011) at (p011) {};
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\node[codeword] (c100) at (p100) {};
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\node[codeword] (c111) at (p111) {};
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% Polytope Edges
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\draw[line width=1pt, color=KITblue] (c000) -- (c011);
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\draw[line width=1pt, color=KITblue] (c000) -- (c100);
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\draw[line width=1pt, color=KITblue] (c100) -- (c111);
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\draw[line width=1pt, color=KITblue] (c111) -- (c011);
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% Polytope Annotations
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\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\node[color=KITblue, below=0cm of c100] {$\left( 1, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c111] {$\left( 1, 1, 1 \right) $};
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\end{tikzpicture}
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\end{minipage}
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\begin{tikzpicture}
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\node[color=KITblue, align=center] at (-2,0)
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{$j=1$\\ $\left( c_1 + c_2+ c_3 = 0 \right) $};
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\node[color=KITblue, align=center] at (2,0)
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{$j=2$\\ $\left(c_2 + c_3 = 0\right)$};
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\end{tikzpicture}
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\caption{Local codeword polytope of check node\\ $j=1$
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$\left( c_1 + c_2 + c_3 = 0 \right)$}
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\label{fig:dec:poly:local1}
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\end{subfigure} \\[1em]
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\begin{subfigure}{\textwidth}
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\centering
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{25}
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\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
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% Cube
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\coordinate (p000) at (0, 0, 0);
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\coordinate (p001) at (0, 0, 2);
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\coordinate (p010) at (0, 2, 0);
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\coordinate (p011) at (0, 2, 2);
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\coordinate (p100) at (2, 0, 0);
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\coordinate (p101) at (2, 0, 2);
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\coordinate (p110) at (2, 2, 0);
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\coordinate (p111) at (2, 2, 2);
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\draw[] (p000) -- (p100);
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\draw[] (p100) -- (p101);
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\draw[] (p101) -- (p001);
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\draw[] (p001) -- (p000);
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\draw[dashed] (p010) -- (p110);
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\draw[] (p110) -- (p111);
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\draw[] (p111) -- (p011);
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\draw[dashed] (p011) -- (p010);
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\draw[dashed] (p000) -- (p010);
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\draw[] (p100) -- (p110);
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\draw[] (p101) -- (p111);
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\draw[] (p001) -- (p011);
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% Polytope Vertices
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\node[codeword] (c000) at (p000) {};
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\node[codeword] (c011) at (p011) {};
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\node[codeword] (c100) at (p100) {};
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\node[codeword] (c111) at (p111) {};
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% Polytope Edges
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\draw[line width=1pt, color=KITblue] (c000) -- (c011);
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\draw[line width=1pt, color=KITblue] (c000) -- (c100);
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\draw[line width=1pt, color=KITblue] (c100) -- (c111);
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\draw[line width=1pt, color=KITblue] (c111) -- (c011);
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% Polytope Annotations
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\node[color=KITblue, below=0cm of c000] {$\left( 0, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c011] {$\left( 0, 1, 1 \right) $};
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\node[color=KITblue, below=0cm of c100] {$\left( 1, 0, 0 \right) $};
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\node[color=KITblue, above=0cm of c111] {$\left( 1, 1, 1 \right) $};
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\end{tikzpicture}
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\caption{Local codeword polytope of check node\\ $j=2$
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$\left( c_2 + c_3 = 0\right)$}
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\label{fig:dec:poly:local2}
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\caption{Local codeword polytopes of the check nodes}
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\label{fig:dec:poly:local}
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\end{subfigure}\\[1em]
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\begin{subfigure}{\textwidth}
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\centering
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\begin{tikzpicture}
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\draw (-2, 0) -- (2, 0);
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\draw (-2, 0.5) -- (-2, 0);
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\draw (2, 0.5) -- (2, 0);
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\node (intersection) at (0, -0.5) {Intersection};
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\draw (0, 0) -- (intersection);
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\draw[->] (intersection) -- (0, -1);
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\end{tikzpicture}
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\vspace{2mm}
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\tikzstyle{codeword} = [color=KITblue, fill=KITblue,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tikzstyle{pseudocodeword} = [color=KITred, fill=KITred,
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draw, circle, inner sep=0pt, minimum size=4pt]
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\tdplotsetmaincoords{60}{25}
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\begin{tikzpicture}[scale=1, transform shape, tdplot_main_coords]
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\begin{tikzpicture}[scale=0.8, transform shape, tdplot_main_coords]
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% Cube
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\coordinate (p000) at (0, 0, 0);
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@ -578,7 +571,11 @@ The resulting formulation of the relaxed optimization problem is the following:%
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\end{subfigure}
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\caption{Visualization of the codeword polytope and the relaxed codeword
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polytope}
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polytope for the code defined by the parity check matrix $\boldsymbol{H} =
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\begin{bmatrix}
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1 & 1 & 1\\
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0 & 1 & 1
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\end{bmatrix}$}
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\label{fig:dec:poly}
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\end{figure}%
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%
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