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[ctsim.git] / doc / ctsim-concepts.tex

diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex
index 417c47a90087bffcbcfe86c0872408f83d234e1a..ff76a9e52768e5ee2c1f07d2b5bd2c4df1985f2d 100644 (file)
--- 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.
 
 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}.
 
 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}}\\}
 
 \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}
 
 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
 \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
 
 \subsubsection{Detector Array Size}
 In general, you do not need to be concerned with the detector


@@ -346,21 +347,26 @@ 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
 \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.}
+%
+%Tex2RTF can not handle the any subscripts or superscripts for the inner summation unless
+% have a space character before the \sum
+\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{\[\label{requation}r = \frac{ \displaystyle \sum_{i=1}^{n}{ \sum_{j=1}^{m}{|p_{i,j} - r_{i,j}|}}}{ \displaystyle \sum_{i=1}^{n}{ \sum_{j=1}^{m}{|p_{i,j}|}}}\]}
+\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{\[\label{bigpequation}P_{k,l} = \textstyle \frac{1}{4} (p_{2k,2l} + p_{2k+1,2l} + p_{2k,2l+l} + p_{2k+1,2l+1})\]}
+\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}}