X-Git-Url: http://git.kpe.io/?p=ctsim.git;a=blobdiff_plain;f=doc%2Fctsim-concepts.tex;h=e8de3431d4cf6806ef53901dbdf2f8d411fb2f21;hp=32d60c83bd67b2f180637f1be1c3c09e93c9538b;hb=54d83ac5be392640ae8d65f3398e445c61741aaf;hpb=512a128ce9bc1d4a4477df791d2a1e63148752af

diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex
index 32d60c8..e8de343 100644
--- a/doc/ctsim-concepts.tex
+++ b/doc/ctsim-concepts.tex
@@ -125,28 +125,28 @@ 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},}
-is then
+\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{\emph{Pd}.}
+\latexonly{$p_d$.}\latexignore{\rtfsp\emph{Pd}.}
 These relationships are diagrammed in figure 2.1.
 
 \subsubsection{View Diameter}
 The \emph{view diameter} is the area that is being processed during scanning of phantoms as
 well as during rasterization of phantoms. By default, the \emph{view diameter}
-is set equal to the \emph{phantom diameter}. It may be useful, especially for 
+\rtfsp is set equal to the \emph{phantom diameter}. It may be useful, especially for 
 experimental reasons, to process an area larger (and maybe even smaller) than
 the phantom. Thus, during rasterization or during projections, \ctsim\ will
 ask for a \emph{view ratio},
-\latexonly{$v_{R}$.}\latexignore{\emph{VR}.}
+\latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.}
 The \emph{view diameter} is then set as
-\latexonly{$$v_d = p_d v_{R}$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
+\latexonly{$$v_d = p_d v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
 
 By using a 
-\latexonly{$v_{R}$}\latexignore{\emph{VR}}
+\latexonly{$v_r$}\latexignore{\emph{VR}}
 less than 1, \ctsim\ will allow
 for a \emph{view diameter} less than 
 \emph{phantom diameter}.
@@ -158,11 +158,11 @@ scanner that is larger than the scanner itself!
 By default, the entire \emph{view diameter} is scanned. For experimental
 purposes, it may be desirable to scan an area either larger or smaller than
 the \emph{view diameter}. Thus, the concept of \emph{scan ratio}
-\latexonly{$s_{R}$}\latexignore{\emph{SR}}
+\latexonly{$s_r$}\latexignore{\emph{SR}}
 is born. The scan diameter
 \latexonly{$s_d$}\latexignore{\emph{Sd}}
 is the diameter over which x-rays are collected and is defined as
-\latexonly{$$s_d = v_d s_{R}$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\}
+\latexonly{$$s_d = v_d s_r$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\}
 By default and for all ordinary scanning, the \emph{scan ratio} is to
 \texttt{1}. If the \emph{scan ratio} is less than \texttt{1},
 you can expect significant artifacts.
@@ -172,10 +172,10 @@ The \emph{focal length},
 \latexonly{$f$,}\latexignore{\emph{F},}
 is the distance of the X-ray source to the center of
 the phantom. The focal length is set as a ratio,
-\latexonly{$f_{R}$,}\latexignore{\emph{FR},}
+\latexonly{$f_r$,}\latexignore{\emph{FR},}
 of the view radius. Focal length is
 calculated as
-\latexonly{$$f = (v_d / 2) f_R$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$}
+\latexonly{$$f = (v_d / 2) f_r$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$}
 
 For parallel geometry scanning, the focal length doesn't matter. However, 
 divergent geometry scanning (equilinear and equiangular), the \emph{focal
@@ -213,10 +213,9 @@ See figure 2.2.
 For these divergent beam geometries, the \emph{fan beam angle} needs 
 to be calculated. For real-world CT scanners, this is fixed at the
 time of manufacture. \ctsim, however, calculates the \emph{fan beam angle},
-\latexonly{$\alpha$,}\latexignore{\emph{alpha},} 
-from the diameter of the \emph{scan diameter} and the \emph{focal length}
+$\alpha$ from the diameter of the \emph{scan diameter} and the \emph{focal length}
 \latexignore{\\$$\emph{alpha = 2 x asin ( (Sd / 2) / f)}$$\\}
-\latexonly{$$\alpha = 2 \sin^{-1} ((s_d / 2) / f)$$}
+\latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1} ((s_d / 2) / f)\end{equation}}
 This is illustrated in figure 2.3.
 \begin{figure}
 \image{10cm;0cm}{alphacalc.eps}
@@ -231,11 +230,19 @@ there are significant artifacts. The primary way to manage the
 \emph{fan beam angle} is by varying the \emph{focal length} since the
 \emph{scan diameter} by the size of the phantom.
 
-$$s_d = p_d v_R s_R$$
-If $v_r = 1$ and $s_R = 1$, then $s_d = p_d$. Further, $f = f_R v_R (p_d / 2)$
-Plugging these equations into the above equation,
-$$\alpha = 2\,\sin^{-1} \frac{p_d / 2}{f_R (p_d / 2)}$$
-$$\alpha = 2\,\sin^{-1} (1 / f_R)$$
+To illustrate, the \emph{scan diameter} can be defined as
+\latexonly{$$s_d = v_r s_r p_d$$}\latexignore{\\$$Sd = Vr x Sr x Pd$$\\}
+
+If $v_r = 1$ and $s_r = 1$, then $s_d = p_d$. Further, $f = f_r v_r (p_d / 2)$
+Plugging these equations into 
+\latexignore{the above equation,}\latexonly{equation~\ref{alphacalc},}
+We have,
+\latexonly{
+\begin{eqnarray}
+\alpha &= 2\,\sin^{-1} \frac{p_d / 2}{f_r (p_d / 2)} \nonumber \\
+&= 2\,\sin^{-1} (1 / f_r)
+\end{eqnarray}
+}
 
 Thus, $\alpha$ depends only upon the \emph{focal length ratio}.