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 \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 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: 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{SHEPP77} 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 centreing 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.  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 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}
+
 
 \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 absolute(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 absolute(x). The other two methods perform an fourier
+transform of the projection data and multiply that by the absolute(x) filter and
+then perform an inverse fourier transform.
+
+\item{Backprojection of filtered projections}
\ No newline at end of file