X-Git-Url: http://git.kpe.io/?p=ctsim.git;a=blobdiff_plain;f=doc%2Fctsim-concepts.tex;h=e76a77f6c54cf4f9221c6661bff3466cea0eb572;hp=417c47a90087bffcbcfe86c0872408f83d234e1a;hb=82ea0c94394a5a175b260160760155a6686203a1;hpb=676f8753e1b3edd337240391855f34dde1af24fa

diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex
index 417c47a..e76a77f 100644
--- a/doc/ctsim-concepts.tex
+++ b/doc/ctsim-concepts.tex
@@ -11,8 +11,8 @@ user-controlled algorithms producing an image of the phantom
 object. These reconstructions can be visually and statistically
 compared to the original phantom object.
 
-In order to use \ctsim\ effectively, some knowledge of how \ctsim\
-works and the approach taken is required. \ctsim\ deals with a
+In order to use \ctsim\ effectively, some knowledge of how
+\ctsim\ works and the approach taken is required. \ctsim\ deals with a
 variety of object, but the two primary objects that we need to be
 concerned with are the \emph{phantom} and the \emph{scanner}.
 
@@ -249,20 +249,21 @@ To illustrate, the \emph{scan diameter} can be defined as
 \latexonly{\begin{equation}s_d = s_r v_r p_d\end{equation}}
 \latexignore{\\\centerline{\emph{Sd = Sr x Vr x Pd}}\\}
 
-Further, $f$ can be defined as \latexonly{$$f = f_r (v_r p_d /
-2)$$}\latexignore{\\$$F = FR x (VR x Pd)$$\\}
+Further, $f$ can be defined as
+\latexonly{\[f = f_r (v_r p_d / 2)\]}
+\latexignore{\\\centerline{\emph{F = FR x (VR x Pd)$$\\}}}
 
 Substituting these equations into \latexignore{the above
 equation,}\latexonly{equation~\ref{alphacalc},} We have,
 \latexonly{
 \begin{eqnarray}
-\alpha &= 2\,\sin^{-1} \frac{s_r v_r p_d / 2}{f_r v_r (p_d / 2)} \nonumber \\
-&= 2\,\sin^{-1} (s_r / f_r)
+\alpha &=& 2\,\sin^{-1} \frac{\displaystyle s_r v_r p_d / 2}{\displaystyle f_r v_r (p_d / 2)} \nonumber \\
+&=& 2\,\sin^{-1} (s_r / f_r)
 \end{eqnarray}
 } \latexignore{\\\centerline{\emph{\alpha = 2 sin (Sr / Fr)}}\\}
 
 Since in normal scanning $s_r$ = 1, $\alpha$ depends only upon the
-\emph{focal length ratio}.
+\emph{focal length ratio} in normal scanning.
 
 \subsubsection{Detector Array Size}
 In general, you do not need to be concerned with the detector
@@ -346,21 +347,23 @@ interpolation can be specified.
 \section{Image Comparison}\index{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 normalized root mean squared distance measure,
-$r$ is the normalized mean absolute distance measure,
-and $e$ is the worst case distance measure over a $2\times2$ area.
-
-To compare two images, $A$ and $B$, each of which has $n$ columns and $m$ rows,
-these values are calculated as below.
-
-
-\latexonly{
-\begin{equation}
-d = \sqrt{\frac{\sum_{i=1}^{n}{\sum_{j=1}^{m}{(A_{ij} - B_{ij})^2}}}
-               {\sum_{i=1}^{n}{\sum_{j=1}^{m}{(A_{ij} - A^{\_})^2}}}}
-\end{equation}
-\begin{equation}
-r = \max(|A_{ij} - B{ij}|)
-\end{equation}
-}
+Herman\cite{HERMAN80}. They are:
+\begin{description}
+\item[$d$] The normalized root mean squared distance measure.
+\item[$r$] The normalized mean absolute distance measure.
+\item[$e$] The worst case distance measure over a $2\times2$ area.
+\end{description}
+
+These measurements are defined in equations \ref{dequation} through \ref{bigrequation}.
+In these equations, $p$ denotes the phantom image, $r$ denotes the reconstruction
+image, and $\bar{p}$ denotes the average pixel value for $p$. Each of the images have a
+size of $m \times n$. In equation \ref{eequation} $[n/2]$ and $[m/2]$ denote the largest
+integers less than $n/2$ and $m/2$, respectively.
+
+\latexignore{These formulas are shown in the print documentation of \ctsim.}
+\latexonly{\begin{equation}\label{dequation}d = \sqrt{\frac{\displaystyle \sum_{i=1}^{n}{\sum_{j=1}^{m}{(p_{i,j} - r_{i,j})^2}}}               {\displaystyle \sum_{i=1}^{n}{\sum_{j=1}^{m}{(p_{i,j} - \bar{p})^2}}}}\end{equation}}
+\latexonly{\begin{equation}\label{requation}r = \frac{\displaystyle \sum_{i=1}^{n}{\sum_{j=1}^{m}{|p_{i,j} - r_{i,j}|}}}         {\displaystyle \sum_{i=1}^{n}{\sum_{j=1}^{m}{|p_{i,j}|}}}\end{equation}}
+\latexonly{\begin{equation}\label{eequation}e = \max_{1 \le k \le [n/2] \atop 1 \le l \le [m/2]}(|P_{k,l} - R_{k,l}|)\end{equation}}
+\latexonly{where}
+\latexonly{\begin{equation}\label{bigpequation}P_{k,l} = {\textstyle \frac{1}{4}} (p_{2k,2l} + p_{2k+1,2l} + p_{2k,2j+l} + p_{2k+1,2l+1})\end{equation}}
+\latexonly{\begin{equation}\label{bigrequation}R_{k,l} = \textstyle \frac{1}{4} (r_{2k,2l} + r_{2k+1,2l} + r_{2k,2l+1} + r_{2k+1,2l+1})\end{equation}}