X-Git-Url: http://git.kpe.io/?p=ctsim.git;a=blobdiff_plain;f=doc%2Fctsim-concepts.tex;fp=doc%2Fctsim-concepts.tex;h=3f0dab0ff558207cc4a9411e5a90b35027811ea2;hp=21f94ddec9e999d3d39ea3f09613f51bbf824f0d;hb=e5c753200f28fdf5542a48051ad79c7c7dfbb299;hpb=464c5028a9dd12b75fc05ea80fdf124af2bcbb01

diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex
index 21f94dd..3f0dab0 100644
--- a/doc/ctsim-concepts.tex
+++ b/doc/ctsim-concepts.tex
@@ -122,22 +122,21 @@ ratios rather than absolute values.
 \subsubsection{Phantom Diameter}
 \begin{figure}
 $$\image{5cm;0cm}{scangeometry.eps}$$
-\caption{Phantom Geometry}
+\caption{\label{phantomgeomfig} Phantom Geometry}
 \end{figure}
-The phantom diameter is automatically calculated by \ctsim\ from the
-phantom definition. The maximum of the phantom length and height is
-used to define the square that completely surrounds the phantom. Let
-\latexonly{$p_l$}\latexignore{\emph{Pl}}
-be the width and height of this square. The diameter of this boundary box,
-\latexonly{$p_d$,}\latexignore{\emph{Pd},}
-\rtfsp is then
-\latexignore{\\$$\emph{Pl x sqrt(2)}$$\\}
-\latexonly{$$p_d = p_l \sqrt{2}$$}
-CT scanners actually collect projections around a circle rather than a
-square. The diameter of this circle is also the diameter of the boundary
-square
-\latexonly{$p_d$.}\latexignore{\rtfsp\emph{Pd}.}
-These relationships are diagrammed in figure 2.1.
+The phantom diameter is automatically calculated by \ctsim\ from
+the phantom definition. The maximum of the phantom length and
+height is used to define the square that completely surrounds the
+phantom. Let \latexonly{$p_l$}\latexignore{\emph{Pl}} be the width
+and height of this square. The diameter of this boundary box,
+\latexonly{$p_d$,}\latexignore{\emph{Pd},} \rtfsp is then
+\latexignore{\\$$\emph{Pl x sqrt(2)}$$\\} \latexonly{$$p_d = p_l
+\sqrt{2}$$} CT scanners actually collect projections around a
+circle rather than a square. The diameter of this circle is also
+the diameter of the boundary square
+\latexonly{$p_d$.  These
+relationships are diagrammed in figure~\ref{phantomgeomfig}.}
+\latexignore{emph{Pd}.}
 
 \subsubsection{View Diameter}
 The \emph{view diameter} is the area that is being processed
@@ -211,10 +210,10 @@ the x-ray beams diverge from a single source to a detector array.
 In the equilinear mode, a single
 source produces a fan beam which is read by a linear array of detectors.  If
 the detectors occupy an arc of a circle, then the geometry is equiangular.
-See figure 2.2.
+\latexonly{See figure~\ref{divergentfig}.}
 \begin{figure}
 \image{10cm;0cm}{divergent.eps}
-\caption{Equilinear and equiangular geometries.}
+\caption{\label{divergentfig} Equilinear and equiangular geometries.}
 \end{figure}
 
 
@@ -226,10 +225,11 @@ at the time of manufacture. \ctsim, however, calculates the
 the \emph{focal length} \latexignore{\\$$\emph{alpha = 2 x asin (
 (Sd / 2) / f)}$$\\}
 \latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1}
-((s_d / 2) / f)\end{equation}} This is illustrated in figure 2.3.
+((s_d / 2) / f)\end{equation}
+ This is illustrated in figure~\ref{alphacalcfig}.}
 \begin{figure}
 \image{10cm;0cm}{alphacalc.eps}
-\caption{Calculation of $\alpha$}
+\caption{\label{alphacalcfig} Calculation of $\alpha$}
 \end{figure}
 
 
@@ -276,10 +276,11 @@ For equiangular geometry, the detectors are spaced around a circle
 covering an angular distance of
 \latexonly{$2\,\alpha$.}\latexignore{\emph{2 \alpha}.} The dotted
 circle in
-\begin{figure}\label{equiangularfig}
-\image{10cm;0cm}{equiangular.eps} \caption{Equiangular geometry}
+\begin{figure}
+\image{10cm;0cm}{equiangular.eps}
+\caption{\label{equiangularfig}Equiangular geometry}
 \end{figure}
-figure 2.4 indicates the positions of the detectors in this case.
+figure~\ref{equiangularfig} indicates the positions of the detectors in this case.
 
 For equilinear geometry, the detectors are space along a straight
 line. The length of the line depends upon
@@ -288,9 +289,9 @@ length}. It is calculated as \latexonly{$4\,f \tan (\alpha / 2)$}
 \latexignore{\emph{4 x F x tan(\alpha/2)}}
 \begin{figure}\label{equilinearfig}
 \image{10cm;0cm}{equilinear.eps}
-\caption{Equilinear geometry}
+\caption{\label{equilinearfig} Equilinear geometry}
 \end{figure}
-This geometry is shown in figure~2.5.
+\latexonly{This geometry is shown in figure~\ref{equilinearfig}.}
 
 
 \subsubsection{Examples of Geometry Settings}
@@ -336,3 +337,20 @@ multiple filters for this purpose.
 Backprojection is the process of ``smearing'' the filtered
 projections over the reconstructing image. Various levels of
 interpolation can be specified.
+
+\section{Image Comparison}
+Images can be compared statistically. Three measurements can be calculated
+by \ctsim. They are taken from the standard measurements used by
+Herman\cite{HERMAN80}.
+$d$ is the standard error, $e$ is the maximum error, and
+$r$ is the maximum error of a 2 by 2 pixel area.
+
+To compare two images, $A$ and $B$, each of which has $n$ columns and $m$ rows,
+these values are calculated as below.
+
+\begin{equation}
+d = \frac{\sum_{i=0}^{n}{\sum_{j=0}^{m}{(A_{ij} - B_{ij})^2}}}{m n}
+\end{equation}
+\begin{equation}
+r = max(|A_{ij} - B{ij}|)
+\end{equation}