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}.
\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
\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}}
projects / ctsim.git / blobdiff