+This is where things get tricky. There are two possible approaches. The
+simple approach would be to define the size of a phantom which is put at
+the centre of the scanner. The scanner would have it's bore size defined,
+or perhaps better, the field of view defined. Here, field of view would be
+the radius or diameter of the circular area from which data is collected
+and an image reconstructed. In a real CT scanner, if the object being
+scanned is larger than the field of view, you get image artifacts. And of
+course you can't stuff an object into a scanner if the object is larger
+than the bore! In this model, the scanner size or field of view would
+be used as the standard length scale.
+
+However, CTSim takes another approach. I believe this approach arose
+because the "image" of the phantom produced from the phantom description
+was being matched to the reconstruction image of the phantom. That is,
+the dimensions of the 'before' and 'after' images were being matched.
+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
+phantom dimensions are also padded by 1\%.
+
+The maximum of the phantom length and height is used as the phantom
+dimension, and one can think of a square bounding box of this size
+which completely contains the phantom. Let $l_p$ be the width (or height)
+of this square.
+
+\subsubsection{Focal Length & Field of View}
+The two other important variables are the field-of-view-ratio ($f_{vR}$)
+and the focal-length-ratio ($f_{lR}$). These are used along with $l_p$ to
+define the focal length and the field of view (not ratios) according to
+\begin{equation}
+f_l = \sqrt{2} (l_p/2)(f_{lR})= (l_p/\sqrt{2}) f_{lR}
+\end{equation}
+\begin{equation}
+f_v = \sqrt{2}l_p f_{vR}
+\end{equation}
+So the field of view ratio is specified in units of the phantom diameter,
+whereas the focal length is specified in units of the phantom radius. The
+factor of $\sqrt(2)$ can be understood if one refers to figure 1, 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.}
+\end{figure}
+
+In figure 1A, the excursion of the source and detector need only be $l_p$,
+the height (or width) of the phantom's bounding square. However, if the
+field of view were only $l_p$, then the projection shown in figure 1B
+would clip the corners of the phantom. By increasing the field of view by
+$\sqrt{2}$ the whole phantom is included in every projection. Of course,
+if the field-of-view ratio $f_{vR}$ is larger than 1, there is no problem.
+However, if $f_{vR}$ is less than one and thus the scanner is smaller than
+the phantom, then distortions will occur without warning from the program.
+
+The code also sets the detector length equal to the field of view in this
+case. The focal length is chosen to be $\sqrt{2}l_p$ so the phantom will
+fit between the source and detector at all rotation angles, when the focal
+length ratio is specified as 1. Again, what happens if the focal length
+ratio is chosen less than 1?
+
+The other thing to note is that in this code the detector array is set to
+be the same size as the field-of-view $f_v$, equation (2). So, one has to
+know the size of the phantom to specify a given scanner geometry with a
+given source-detector distance (or $f_l$ here) and a given range of
+excursion ($f_v$ here).
+
+\subsubsection{Divergent Geometries}\label{geometrydivergent}\index{Concepts,Scanner,Geometries,Divergent}
+Next consider the case of equilinear (second generation) and equiangular
+(third, fourth, and fifth generation) geometries.
+The parts of the code relevant to this
+discussion are the same for both modes. 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.
+\begin{figure}
+\includegraphics[width=\textwidth]{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
+\begin{equation}
+d_{hs} = (f_v)/(2\sqrt{2}) = (l_p/2) f_{vR}
+\end{equation}
+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
+$d_{hs}$ are derived from the phantom dimension and the input focal length and field of view ratios, one can write,
+\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}
+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,
+\[
+f_l=\sqrt{2}l_p/2 = l_p/\sqrt{2},
+\]
+\[
+f_v = \sqrt{2}l_p,
+\]
+and
+\[
+d_{hs} = {l_p}/{2}.
+\]
+Then
+\[
+\mathrm{dFocalPastPhm} = ({l_p}/{2}) (\sqrt{2}-1)
+\]
+\begin{figure}
+\includegraphics[height=0.5\textheight]{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
+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
+used to give a distance from the centre of the phantom to the source, and
+the detector array is adjusted to give an angular coverage to include the
+whole phantom.
+\begin{figure}
+\includegraphics[width=\textwidth]{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}
+\caption{Equilinear and equiangluar geometry when focal length ratio = 4.}
+
+\end{figure}
+
projects / ctsim.git / commitdiff