r502: no message

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

diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex
index 1d2d01bfe27a061aaf2b9d7fdec461f200feec94..32d60c83bd67b2f180637f1be1c3c09e93c9538b 100644 (file)
--- a/doc/ctsim-concepts.tex
+++ b/doc/ctsim-concepts.tex
@@ -181,6 +181,7 @@ For parallel geometry scanning, the focal length doesn't matter. However,
 divergent geometry scanning (equilinear and equiangular), the \emph{focal
 length ratio} should be set at \texttt{2} or more to avoid artifacts. 
 
+       
 \subsection{Parallel Geometry}\label{geometryparallel}\index{Concepts,Scanner,Geometries,Parallel}
 
 As mentioned above, the focal length is not used in this simple
@@ -192,6 +193,7 @@ the default values of \texttt{1} for the \emph{view diameter ratio} and
 the \emph{scan diameter ratio}. If values of less than \texttt{1} are
 used for these two variables, significant distortions will occur.
 
+
 \subsection{Divergent Geometries}\label{geometrydivergent}\index{Concepts,Scanner,Geometries,Divergent}
 \subsubsection{Overview}
 Next consider the case of equilinear (second generation) and equiangular
@@ -206,13 +208,14 @@ See figure 2.2.
 \caption{Equilinear and equiangular geometries.}
 \end{figure}
 
+
 \subsubsection{Fan Beam Angle}
 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}
-\latexignore{\\$$\emph{alpha = 2 x asin ( (Sd / 2) / F)}$$\\}
+\latexignore{\\$$\emph{alpha = 2 x asin ( (Sd / 2) / f)}$$\\}
 \latexonly{$$\alpha = 2 \sin^{-1} ((s_d / 2) / f)$$}
 This is illustrated in figure 2.3.
 \begin{figure}
@@ -220,13 +223,22 @@ This is illustrated in figure 2.3.
 \caption{Calculation of $\alpha$}
 \end{figure}
 
+
 Empiric testing with \ctsim\ shows that for very large \emph{fan beam angles},
 greater than approximately 
-\latexonly{$120^{\circ}$,}\latexignore{120 degrees,}
+\latexonly{$120^\circ$,}\latexignore{120 degrees,}
 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,
+$$\alpha = 2\,\sin^{-1} \frac{p_d / 2}{f_R (p_d / 2)}$$
+$$\alpha = 2\,\sin^{-1} (1 / f_R)$$
+
+Thus, $\alpha$ depends only upon the \emph{focal length ratio}.
+
 \subsubsection{Detector Array Size}
 In general, you do not need to be concerned with the detector array
 size. It is automatically calculated by \ctsim.  The size of the
@@ -257,6 +269,7 @@ and the \emph{focal length}. It is calculated as
 \end{figure}
 An example of the this geometry is in figure 2.5.
 
+
 \subsubsection{Examples of Geometry Settings}
 Consider increasing the focal length ratio to two leaving the
 field of view ratio as 1,  as in  Figure 4.  Now the detectors array is
@@ -301,4 +314,3 @@ filters for this purpose.
 Backprojection is the process of ``smearing'' the filtered projections over
 the reconstructing image. Various levels of interpolation can be specified.
 In general, the trade-off is between quality and execution time.
-