\setfooter{\thepage}{}{}{}{}{\thepage}%
\section{Overview}\label{conceptoverview}\index{Concepts,Overview}%
+In \ctsim, a phantom object, or a geometrical description of the object
+of a CT study is constructed and an image can be created. Then a
+scanner geometry can be specified, and the projection data simulated.
+Finally that projection data can be reconstructed using various user
+controlled algorithms producing an image of the phantom or study object.
+
+In order to use \ctsim\ effectively, some knowledge of how \ctsim\ works
+and the approach taken is required. \ctsim\ deals with a variety of
+object, but the two objects we need to be concerned with are the 'phantom' and
+the 'scanner'.
\section{Phantoms}\label{conceptphantom}\index{Concepts,Phantoms}%
\subsection{Overview}\label{phantomoverview}\index{Concepts,Phantoms,Overview}%
-\subsection{Phantom Elements}\label{phantoelements}\index{Concepts,Phantoms,Elements}
+
+\ctsim\ uses geometrical objects to
+describe the object being scanned. A phantom is composed a one or more
+phantom elements. These elements are simple geometric shapes,
+specifically, rectangles, triangles, ellipses, sectors and segments.
+With these the standard phantoms used in the CT literature (the Herman
+and the Shepp-Logan) can be constructed. In fact
+\ctsim\ provides a shortcut to construct those phantoms for you. It also
+allows you to write a file in which the composition of your own phantom is
+described.
+
+The types of phantom elements and their definitions are taken from Herman's 1980
+book\cite{HERMAN80}.
+
+\subsection{Phantom File}\label{phantomfile}\index{Concepts,Phantoms,File}
+Each line in the text file describes an element of the
+phantom. Each line contains seven entries, in the following form:
+\begin{verbatim}
+item cx cy dx dy r a
+\end{verbatim}
+The first entry defines the type of the element, one
+of {\tt rectangle}, {\tt ellipse}, {\tt triangle}, {\tt sector}, or {\tt segment}.
+{\tt cx}, {\tt cy}, {\tt dx} and {\tt dy} have different meanings depending on the element type.
+
+{\tt r} is the rotation applied to the object in degrees counterclockwise,
+and {\tt a} is the X-ray attenuation coefficient of the object.
+Where objects overlap, the attenuations of the overlapped objects are summed.
+
+
+
+\subsection{Phantom Elements}\label{phantomelements}\index{Concepts,Phantoms,Elements}
+
\subsubsection{ellipse}
+Ellipses use dx and dy to define the semi-major and semi-minor axis lengths,
+with the centre of the ellipse at cx and cy. Of note, the commonly used
+phantom described by Shepp and Logan\cite{SHEPP74} uses only ellipses.
+
\subsubsection{rectangle}
+Rectangles use
+cx and cy to define the position of the centre of the rectangle with respect
+to the origin. dx and dy are the half-width and half-height of the
+rectangle.
+
\subsubsection{triangle}
+Triangles are drawn with the centre of the base at cx,cy, with a base
+width of 2*dx in x direction, and a height of dy. Rotations are then
+applied about the origin.
+
\subsubsection{sector}
+It appears that dx and dy
+define the end points of a radius of the sector, from which the radius and
+the angle of the two arms of the sector are calculated. But then
+orientation and centering of the sector don't make much sense yet.
+
\subsubsection{segment}
+Segments are the segments of a circle between a chord and the
+perimeter of the circle. This also isn't clear to me, but it appears that
+perhaps the distance from chord to circle perimeter, and circle radius is
+defined by dx and dy. Chord is always horizontal through the origin, then
+translated and then rotated (???).
+
+\subsection{Phantom Size}
+Also note that the overall dimensions of the phantom are increased by 1\%
+above the specified sizes to avoid clipping due to round-off errors.
+So, if the phantom is defined as
+a rectangle of size 0.1 by 0.1, the actual phantom has extent $\pm$0.101 in
+each direction.
\section{Scanner}\label{conceptscanner}\index{Concepts,Scanner}%
\subsection{Geometries}
-\subsection{Focal Length}
-\subsection{Field of View}
+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 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}}
+\latexignore{sqrt(2) * $p_l$.}
+This relationship can be seen in figure 1.
+
+\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 $p_d$\\}
+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
+\latexonly{\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,
+\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}
+}
+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}
+\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}
+
\section{Reconstruction}\label{conceptreconstruction}\index{Concepts,Reconstruction}%
+\subsection{Overview}
\subsection{Filtered Backprojection}
+\subsection{Direct Inverse Fourier}
+This method is not currently implemented in \ctsim, however it is planned for a
+future release. This method does not give as accurate result as filtered
+backprojection mostly due to interpolation occuring in the frequency domain rather
+than the spatial domain. The technique is comprised of two sequential steps:
+filtering projections and then backprojecting the filtered projections. Though
+these two steps are sequential, each view position can be processed individually.
+This parallelism is exploited in the MPI versions of \ctsim\ where the data from
+all the views are spread about amongst all of the processors. This has been testing
+in a 16-CPU cluster with good results.
+
+\subsubsection{Filter projections}
+The projections for a single view have their frequency data multipled by
+a filter of $|w|$. \ctsim\ permits four different ways to accomplish this
+filtering. Two of the methods use convolution of the projection data with the
+inverse fourier transform of $|w|$. The other two methods perform an fourier
+transform of the projection data and multiply that by the $|w|$ filter and
+then perform an inverse fourier transform.
+
+Though multiplying by $|w|$ gives the sharpest reconstructions, in practice, superior results are obtained by mutiplying the $|w|$ filter by
+another filter that attenuates the higher frequencies. \ctsim\ has multiple
+filters for this purpose.
+
+\subsubsection{Backprojection of filtered projections}
+Backprojection is the process of ``smearing'' the filtered projections over
+the reconstructing image.
\ No newline at end of file
projects / ctsim.git / blobdiff