absolute values.
\subsubsection{Phantom Diameter}
+\begin{figure}
+$$\image{5cm;0cm}{scangeometry.eps}$$
+\caption{Phantom Geometry}
+\end{figure}
The maximum of the phantom length and height is used to define the square
that completely surrounds the phantom. Let
\latexonly{$P_l$}\latexignore{\emph{Pl}}
square. The diameter of this circle is also the diameter of the boundary
square
\latexonly{$P_d$.}\latexignore{\emph{Pd}.}
-These relationships are diagrammed in figure 1.
+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
length ratio} should be set at \texttt{2} or more to avoid artifacts.
\subsection{Parallel Geometry}\label{geometryparallel}\index{Concepts,Scanner,Geometries,Parallel}
-\begin{figure}
-\image{10cm;0cm}{ctsimfig1.eps}
-\caption{Geometry used for a 1st generation, parallel beam CT scanner}\label{fistgenfig}
-\end{figure}
As mentioned above, the focal length is not used in this simple
geometry. The detector array is set to
In the equilinear mode, a single
source produces a fan beam which is read by a linear array of detectors. If
the detectors occupy an arc of a circle, then the geometry is equiangular.
-See figure 2.
+See figure 2.2.
\begin{figure}
\image{10cm;0cm}{ctsimfig2.eps}
\caption{Equilinear and equiangular geometries.}
\end{figure}
\subsubsection{Fan Beam Angle}
-For these divergent beam geometries, the angle of the fan beam needs
+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 fan beam angle,
+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_l)}$$\\}
\latexonly{$$\alpha = 2 \sin^{-1} ((S_d / 2) / F_l)$$}
-This is illustrated in figure 3.
+This is illustrated in figure 2.3.
\begin{figure}
\image{10cm;0cm}{alphacalc.eps}
\caption{Calculation of $\alpha$}
\end{figure}
-If this quantity is less than or equal to zero, then at least for some
-projections the source is inside the phantom. Perhaps a figure will help at
-this point. Consider first the case where $f_{vR} = f_{lR} =1 $, figure 3. The
-square in the figure bounds the phantom and has sides $l_p$. For this case
-then,
-\latexonly{$$f_l=\sqrt{2}l_p/2 = l_p/\sqrt{2}$$,
-$$f_v = \sqrt{2}l_p$$,
-and
-$$d_{hs} = {l_p}/{2}$$.
-Then
-$$\mathrm{dFocalPastPhm} = ({l_p}/{2}) (\sqrt{2}-1)$$
-}
-\begin{figure}
-\image{5cm;0cm}{ctsimfig3.eps}
-\caption{Equilinear and equiangluar geometry when focal length ratio =
-field of view ratio = 1.}
-\end{figure}
-The angle $\alpha$ is now defined as shown in figure 3, and the detector
-length is adjusted to subtend the angle $2\alpha$ as shown. Note that the
-size of the detector array may have changed and the field of view is not
-used.
-For a circular array of detectors, the detectors are spaced around a
-circle covering an angular distance of $2\alpha$. The dotted circle in
-figure 3 indicates the positions of the detectors in this case. Note that
-detectors at the ends of the range would not be illuminated by the source.
-
-Now, consider increasing the focal length ratio to two leaving the
+Empiric testing with \ctsim\ shows that for very large \emph{fan beam angles},
+greater than approximately
+\latexonly{$120^{\circ}$,}\latexignore{120 degrees,}
+there are significant artifacts. The primary way to manage the
+\emph{fan beam angle} is by varying the \emph{focal length} since the
+\emph{scan diameter} by the size of the phantom.
+
+\subsubsection{Detector Array Size}
+In general, you do not need to be concerned with the detector array
+size. It is automatically calculated by \ctsim. The size of the
+detector array depends upon the \emph{focal length} and the
+\emph{scan diameter}. In general, increasing the \emph{focal length}
+decreases the size of the detector array and increasing the \emph{scan
+diameter} increases the detector array size.
+
+For equiangular geometry, the detectors are spaced around a
+circle covering an angular distance of
+\latexonly{$\alpha$.}\latexignore{\emph{alpha}.}
+The dotted circle in
+figure 2.4 indicates the positions of the detectors in this case.
+
+For equilinear geometry, the detectors are space along a straight
+line. The length of the line depends upon
+\latexonly{$\alpha$}\latexignore{\emph{alpha}}
+and the \emph{focal length}. It is calculated as
+\latexonly{$$\mathrm{detLengh} = 4\,F_l \tan (\alpha / 2)$$}
+\latexignore{\\$$\emph{detLength} = 4 x Fl x tan(alpha/2)$$\\}
+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
denser, and the real field of view is closer to that specified, but note
again that the field of view is not used. Instead, the focal length is
\caption{Equilinear and equiangluar geometry when focal length ratio = 4.}
\end{figure}
+
+
\section{Reconstruction}\label{conceptreconstruction}\index{Concepts,Reconstruction}%
\subsection{Overview}
\subsection{Direct Inverse Fourier}
projects / ctsim.git / commitdiff