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euler as graphs

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Robin Clark 2010-08-22 12:41:34 +01:00
parent 10751230f6
commit 3e97c0c9c1
2 changed files with 13 additions and 28 deletions
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39
eulerg/eulerg.tex
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@ -43,29 +43,7 @@ and write down set theory equations.
The interest here though, is to define relationships between the contours, that allow
processing and parsing of the diagram without resorting to extra area operations in the concerete plane.
\section{Defining `pair-wise intersection' and `enclosure'}
%\begin{figure}[htp]
% \begin{center}
% \subfigure[Original image]{\label{fig:edge-a}\includegraphics[scale=0.75]{toucan.eps}}
% \subfigure[After Laplace edge detection]{\label{fig:edge-b}\includegraphics[scale=0.75]{laplace_toucan.eps}} \\
% \subfigure[After Sobel edge detection]{\label{fig:edge-c}\includegraphics[scale=0.75]{sobel_toucan.eps}}
% \end{center}
% \caption{Various edge detection algorithms}
% \label{fig:edge}
%\end{figure}
%
%\begin{figure}[h]
% \centering
% \includegraphics[width=200pt,keepaspectratio=true]{./eulerg/eulerg1.jpg}
% % eulerg1.jpg: 513x215 pixel, 72dpi, 18.10x7.58 cm, bb=0 0 513 215
% \caption{An Euler Diagram showing enclosure and Pair-wise Intersection}
% \label{fig:eulerg1}
%\end{figure}
%
%
%
\section{Defining `pair-wise intersection' \\ and `enclosure'}
\begin{figure}[ht]
\centering
@ -165,6 +143,8 @@ $$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$
Enlcosure relationships are transitive.
\end{definition}
\pagebreak[1]
\section{Representing Euler Diagrams \\ as sets of relationships}
The diagram in figure \ref{fig:eulerg1} can be represented by the following relationships.
@ -185,7 +165,7 @@ and {\em enclosure} is transitive and {\pic} is not, we can represent
an {\em enclosure} relationship as a directed vertice and
{\pic} as non-directed.
\pagebreak[0]
\pagebreak[1]
\section{The {\pic}}
In graph theory a node is said to be reachable from another node
if you can start at the one node, travel via the edges
@ -271,7 +251,10 @@ Figure \ref{fig:eulerg_pic_g} only shows the {\pic}, but does not show the conto
enclosing $PIC1$. Figure \ref{fig:eulerg_pic_g_a}
shows contour A enclosing all elements in $PIC1$
\pagebreak[1]
\subsection{Enclosure and pair-wise \\ intersection in the graph}
Because enclose is a directed relationship and {\em pair-wise intersection} is non-directed
we can represent them both on the same graph, see figure \ref{fig:eulerg_pic_g_a}.
\begin{figure}[h]
\centering
\includegraphics[width=200pt,bb=0 0 330 162,keepaspectratio=true]{./eulerg/eulerg_pic_g_a.jpg}
@ -282,9 +265,11 @@ shows contour A enclosing all elements in $PIC1$
\pagebreak[1]
\subsection{Reducing clutter in the graph}
Because we know that a contour enclosing a contour within a {\pic} but not belonging
to it, encloses all elements of the {\pic}, see definition \ref{def:encpic}, we can draw this in a less cluttered way
Contour A encloses the pure intersection chain $PIC1$;
using definition \ref{def:encpic}, we can draw this in a less cluttered way
see figure \ref{fig:eulerg_pic_g_a_unc}.
We only need to show contour A enclosing one member of the {\pic} $PIC1$
in order to show that contour A encloses all contours in $PIC1$.
@ -297,7 +282,7 @@ see figure \ref{fig:eulerg_pic_g_a_unc}.
\end{figure}
\pagebreak[0]
\pagebreak[1]
\section{Reduction of searches \\ for available zones}
Another property of any {\pic} $P$, is that

2
thesis.tex
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@ -139,7 +139,7 @@ reference the MSC document and describe the Java extension classes.
Software documentation for fmmd tool.
\typeout{ ---------------- Euler Diagrams represented as graphs}
\chapter {Euler Diagrams Represented as graphs}
\chapter {Euler Diagrams \\ Represented as graphs}
\input{eulerg/eulerg}
\chapter{Fast Zone Discrimination Algorithm}