The code has a Phantom object and a Scanner object. The geometry of the
Scanner is defined in part by the properties of the Phantom. In fact,
all dimensions are determined in terms of the phantom size, which is used
-as the standard length scale. Remember, as mentioned above, the
+as the standard length scale. Remember, as mentioned above, the
phantom dimensions are also padded by 1\%.
The maximum of the phantom length and height is used to define the square
that completely surrounds the phantom. Let $p_l$ be the width (also height)
of this square. The diameter of this boundary box, $p_d$ is then
-\latexonly{\begin{equation}p_d = \sqrt{2} (p_l/2)\end{equation}}
+\latexonly{$$p_d = \sqrt{2}(p_l)$$}
\latexignore{sqrt(2) * $p_l$.}
-This relationship can be seen in figure 1.
+This relationship can be seen in figure 1 with the parallel geometry.
\subsubsection{Focal Length \& Field of View}
The two important variables is the focal-length-ratio $f_lr$.
This is used along with $p_d$ to
define the focal length according to
-\latexonly{\begin{equation}f_l = f_lr p_d\end{equation}}
-\latexignore{$f_l$ = $f_lr$ x $p_d$\\}
+\latexonly{$$f_l = f_lr p_d$$}
+\latexignore{\\$f_l$ = $f_lr$ x $p_d$\\}
where
we consider the case of a first generation parallel beam CT scanner.
\subsubsection{Parallel Geometry}\label{geometryparallel}\index{Concepts,Scanner,Geometries,Parallel}
\begin{figure}
-\includegraphics[width=\textwidth]{ctsimfig1.eps}
-\caption{Geometry used for a 1st generation, parallel beam CT scanner.}
+\image{10cm;0cm}{ctsimfig1.eps}
+\caption{Geometry used for a 1st generation, parallel beam CT scanner}\label{fistgenfig}
\end{figure}
In figure 1A, the excursion of the source and detector need only be $l_p$,
the detectors occupy an arc of a circle, then the geometry is equiangular.
See figure 2.
\begin{figure}
-\includegraphics[width=\textwidth]{ctsimfig2.eps}
+\image{10cm;0cm}{ctsimfig2.eps}
\caption{Equilinear and equiangular geometries.}
\end{figure}
For these geometries, the following logic is executed: A variable dHalfSquare
$d_{hs}$ is defined as
-\latexonly{\begin{equation}
-d_{hs} = (f_v)/(2\sqrt{2}) = (l_p/2) f_{vR}
-\end{equation}}
+\latexonly{$$d_{hs} = (f_v)/(2\sqrt{2}) = (l_p/2) f_{vR}$$}
This is then subtracted from the focal length $f_l$ as calculated above, and
-assigned to a new variable $\mathrm{dFocalPastPhm} = f_l - d_{hs}$. Since $f_l$ and
+assigned to a new variable
+\latexonly{$\mathrm{dFocalPastPhm} = f_l - d_{hs}$}. Since $f_l$ and
$d_{hs}$ are derived from the phantom dimension and the input focal length and field of view ratios, one can write,
-\latexonly{
-\begin{equation}
-\mathrm{dFocalPastPhm} = f_l -d_{hs}
- = \sqrt{2}(l_p/2) f_{lR} - (l_p/2) f_{vR} = l_p(\sqrt{2}f_{lR} - f_{vR})
-\end{equation}
-}
+\latexonly{$$\mathrm{dFocalPastPhm} = f_l -d_{hs}
+ = \sqrt{2}(l_p/2) f_{lR} - (l_p/2) f_{vR} = l_p(\sqrt{2}f_{lR} - f_{vR})$$}
If this quantity is less than or equal to zero, then at least for some
projections the source is inside the phantom. Perhaps a figure will help at
this point. Consider first the case where $f_{vR} = f_{lR} =1 $, figure 3. The
square in the figure bounds the phantom and has sides $l_p$. For this case
then,
-\latexonly{
-\[
-f_l=\sqrt{2}l_p/2 = l_p/\sqrt{2},
-\]
-\[
-f_v = \sqrt{2}l_p,
-\]
+\latexonly{$$f_l=\sqrt{2}l_p/2 = l_p/\sqrt{2}$$,
+$$f_v = \sqrt{2}l_p$$,
and
-\[
-d_{hs} = {l_p}/{2}.
-\]
+$$d_{hs} = {l_p}/{2}$$.
Then
-\[
-\mathrm{dFocalPastPhm} = ({l_p}/{2}) (\sqrt{2}-1)
-\]
+$$\mathrm{dFocalPastPhm} = ({l_p}/{2}) (\sqrt{2}-1)$$
}
\begin{figure}
-\includegraphics[height=0.5\textheight]{ctsimfig3.eps}
+\image{5cm;0cm}{ctsimfig3.eps}
\caption{Equilinear and equiangluar geometry when focal length ratio =
field of view ratio = 1.}
\end{figure}
the detector array is adjusted to give an angular coverage to include the
whole phantom.
\begin{figure}
-\includegraphics[width=\textwidth]{ctsimfig4.eps}
+\image{10cm;0cm}{ctsimfig4.eps}
\caption{Equilinear and equiangluar geometry when focal length ratio = 2
and the field of view ratio = 1.}
\end{figure}
-Now consider a focal length ratio of 4 (figure 5). As expected, the angle
-$\alpha$ is smaller still. The dotted square is the bounding square of
-the phantom rotated by 45 degrees, corresponding to the geometry of a
-projection taken at that angle. Note that the fan beam now clips the top
-and bottom corners of the bounding square. This illustrates that one may
-still be clipping the phantom, despite \ctsim\'s best efforts. You have
-been warned.
\begin{figure}
-\includegraphics[width=\textwidth]{ctsimfig5.eps}
+\image{10cm;0cm}{ctsimfig5.eps}
\caption{Equilinear and equiangluar geometry when focal length ratio = 4.}
-
\end{figure}
-
\section{Reconstruction}\label{conceptreconstruction}\index{Concepts,Reconstruction}%
\subsection{Overview}
\subsection{Filtered Backprojection}
\subsubsection{Backprojection of filtered projections}
Backprojection is the process of ``smearing'' the filtered projections over
-the reconstructing image.
\ No newline at end of file
+the reconstructing image.
+
projects / ctsim.git / blobdiff