\chapter{Concepts}\index{Concepts}%
\setheader{{\it CHAPTER \thechapter}}{}{}{}{}{{\it CHAPTER \thechapter}}%
-\setfooter{\thepage}{}{}{}{}{\thepage}%
+\setfooter{\thepage}{}{}{}{\small Version 0.2}{\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.
+The operation of \ctsim\ begins with the phantom object. A
+phantom object consists of geometric elements. A scanner is
+specified and the collection of x-ray data, or projections, is
+simulated. That projection data can be reconstructed using various
+user-controlled algorithms producing an image of the phantom
+object. This reconstruction can then be statistically compared to
+the original phantom 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'.
+object, but the two objects we need to be concerned with are the
+\emph{phantom} and the \emph{scanner}.
\section{Phantoms}\label{conceptphantom}\index{Concepts,Phantoms}%
\subsection{Overview}\label{phantomoverview}\index{Concepts,Phantoms,Overview}%
-\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.
+\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
+elements, standard phantoms used in the CT literature can be
+constructed. In fact, \ctsim\ provides a shortcut to load the
+published phantoms of Herman\cite{HERMAN80} and
+Shepp-Logan\cite{SHEPP74}. \ctsim\ also reads text files of
+user-defined phantoms.
-The types of phantom elements and their definitions are taken from Herman's 1980
-book\cite{HERMAN80}.
+The types of phantom elements and their definitions are taken with
+permission from G.T. 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
+element-type 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.
+The first entry defines the type of the element, either
+\rtfsp\texttt{rectangle}, \texttt{}, \texttt{triangle},
+\rtfsp\texttt{sector}, or \texttt{segment}. \texttt{cx},
+\rtfsp\texttt{cy}, \texttt{dx} and \texttt{dy} have different
+meanings depending on the element type.
+\rtfsp\texttt{r} is the rotation applied to the object in degrees
+counterclockwise, and \texttt{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.
+Ellipses use \texttt{dx} and \texttt{dy} to define the semi-major and
+semi-minor axis lengths, with the center of the ellipse at \texttt{cx}
+and \texttt{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.
+Rectangles use \texttt{cx} and \texttt{cy} to define the position of
+the center of the rectangle with respect to the origin. \texttt{dx}
+and \texttt{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.
+Triangles are drawn with the center of the base at \texttt{(cx,cy)},
+with a base half-width of \texttt{dx} and a height of \texttt{dy}.
+Rotations are then applied about the center of the base.
\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 (???).
+Segments are complex. They are the portion of an circle between a
+chord and the perimeter of the circle. \texttt{dy} sets the radius of
+the circle. Segments start with the center of the chord located at
+\texttt{(0,0)} and the chord horizontal. The half-width of the chord
+is set by \texttt{dx}. The portion of an circle lying below the chord
+is then added. The imaginary center of this circle is located at
+\texttt{(0,-dy)}. The segment is then rotated by \texttt{r} and then
+translated by \texttt{cx,cy}.
+
+\subsubsection{sector}
+Sectors are the like a ``pie slice'' from a circle. The radius of the
+circle is set by \texttt{dy}. Sectors are
+defined similarly to segments. In this case, though, a chord is not
+drawn. Instead, the lines are drawn from the origin of the circle
+\texttt{(0,-dy)} to the points \texttt{(-dx,0)} and \texttt{(dx,0)}.
+The perimeter of the circle is then draw between those two points
+below the x-axis. The sector is then rotated and translated the same
+as a segment.
\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.
+The overall dimensions of the phantom are increased by 1\% above the
+specified sizes to avoid clipping due to round-off errors from
+sampling the polygons of the phantom elements. So, if the phantom is
+defined as a rectangle of size 0.1 by 0.1, the actual phantom has
+extent 0.101 in each direction.
\section{Scanner}\label{conceptscanner}\index{Concepts,Scanner}%
-\subsection{Geometries}
-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
+\subsection{Dimensions}
+Understanding the scanning geometry is the most complicated aspect of
+using \ctsim. For real-world CT simulators, this is actually quite
+simple. The geometry is fixed by the manufacturer during the
+construction of the scanner and can not be changed. Conversely,
+real-world CT scanners can only take objects up to a fixed size.
+
+\ctsim, being a very flexible simulator,
+gives tremendous options in setting up the geometry for a scan.
+
+In general, the geometry for a scan all starts from the size of the
+phantom being scanned. This is because \ctsim\ allows for statistical
+comparisons between the original phantom image and it's reconstructions.
+Since CT scanners scan a circular area, the first important
+variable is the diameter of the circle surround the phantom, or the
+\emph{phantom diameter}. 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
-\latexonly{\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
-\latexonly{$\sqrt(2)$}
-\latexignore{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}
+The other important geometry variables for scanning objects are the
+\emph{view ratio}, \emph{scan ratio}, and \emph{focal length ratio}.
+These variables are all input into \ctsim\ in terms of ratios rather
+than absolute values.
+
+\subsubsection{Phantom Diameter}
\begin{figure}
-\includegraphics[width=\textwidth]{ctsimfig1.eps}
-\caption{Geometry used for a 1st generation, parallel beam CT scanner.}
+$$\image{5cm;0cm}{scangeometry.eps}$$
+\caption{Phantom Geometry}
\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}
+The phantom diameter is automatically calculated by \ctsim\ from the
+phantom definition. 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}}
+be the width and height of this square. The diameter of this boundary box,
+\latexonly{$p_d$,}\latexignore{\emph{Pd},}
+\rtfsp is then
+\latexignore{\\$$\emph{Pl x sqrt(2)}$$\\}
+\latexonly{$$p_d = p_l \sqrt{2}$$}
+CT scanners actually collect projections around a circle rather than a
+square. The diameter of this circle is also the diameter of the boundary
+square
+\latexonly{$p_d$.}\latexignore{\rtfsp\emph{Pd}.}
+These relationships are diagrammed in figure 2.1.
+
+\subsubsection{View Diameter}
+The \emph{view diameter} is the area that is being processed
+during scanning of phantoms as well as during rasterization of
+phantoms. By default, the \emph{view diameter} \rtfsp is set equal
+to the \emph{phantom diameter}. It may be useful, especially for
+experimental reasons, to process an area larger (and maybe even
+smaller) than the phantom. Thus, during rasterization or during
+projections, \ctsim\ will ask for a \emph{view ratio},
+\latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.} The \emph{view
+diameter} is then calculated as \latexonly{$$v_d = p_d
+v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
+
+By using a
+\latexonly{$v_r$}\latexignore{\emph{VR}}
+less than 1, \ctsim\ will allow
+for a \emph{view diameter} less than
+\emph{phantom diameter}.
+This will lead to significant artifacts. Physically, this would
+be impossible and is analagous to inserting an object into the CT
+scanner that is larger than the scanner itself!
+
+\subsubsection{Scan Diameter}
+By default, the entire \emph{view diameter} is scanned. For
+experimental purposes, it may be desirable to scan an area either
+larger or smaller than the \emph{view diameter}. Thus, the concept
+of \emph{scan ratio}, \latexonly{$s_r$,}\latexignore{\emph{SR},}
+is born. The scan diameter
+\latexonly{$s_d$}\latexignore{\emph{Sd}} is the diameter over
+which x-rays are collected and is defined as \latexonly{$$s_d =
+v_d s_r$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\} By default and
+for all ordinary scanning, the \emph{scan ratio} is to \texttt{1}.
+If the \emph{scan ratio} is less than \texttt{1}, you can expect
+significant artifacts.
+
+\subsubsection{Focal Length}
+The \emph{focal length},
+\latexonly{$f$,}\latexignore{\emph{F},}
+is the distance of the X-ray source to the center of
+the phantom. The focal length is set as a ratio,
+\latexonly{$f_r$,}\latexignore{\emph{FR},}
+of the view radius. Focal length is
+calculated as
+\latexonly{$$f = (v_d / 2) f_r$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$}
+
+For parallel geometry scanning, the focal length doesn't matter.
+However, divergent geometry scanning (equilinear and equiangular),
+the \emph{focal length ratio} should be set at \texttt{2} or more
+to avoid artifacts. Moreover, a value of less than \texttt{1},
+though it can be given to \ctsim, is physically impossible and it
+analagous to have having the x-ray source with the \emph{view
+diameter}.
+
+
+\subsection{Parallel Geometry}\label{geometryparallel}\index{Concepts,Scanner,Geometries,Parallel}
+
+As mentioned above, the focal length is not used in this simple
+geometry. The detector array is set to be the same size as the
+\emph{scan diameter}. For optimal scanning in this geometry, the
+\emph{scan diameter} should be equal to the \emph{phantom
+diameter}. This is accomplished by using the default values of
+\texttt{1} for the \emph{view ratio} and the \emph{scan ratio}. If
+values of less than \texttt{1} are used for these two variables,
+significant distortions will occur.
+
+
+\subsection{Divergent Geometries}\label{geometrydivergent}\index{Concepts,Scanner,Geometries,Divergent}
+\subsubsection{Overview}
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
+(third, fourth, and fifth generation) geometries. In these cases,
+the x-ray beams diverge from a single source to a detector array.
+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}
-\includegraphics[width=\textwidth]{ctsimfig2.eps}
+\image{10cm;0cm}{divergent.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
-\[
-\mathrm{dFocalPastPhm} = ({l_p}/{2}) (\sqrt{2}-1)
-\]
-}
+
+\subsubsection{Fan Beam Angle}
+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
+\emph{fan beam angle}, $\alpha$, from the \emph{scan diameter} and
+the \emph{focal length} \latexignore{\\$$\emph{alpha = 2 x asin (
+(Sd / 2) / f)}$$\\}
+\latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1}
+((s_d / 2) / f)\end{equation}} This is illustrated in figure 2.3.
\begin{figure}
-\includegraphics[height=0.5\textheight]{ctsimfig3.eps}
-\caption{Equilinear and equiangluar geometry when focal length ratio =
-field of view ratio = 1.}
+\image{10cm;0cm}{alphacalc.eps}
+\caption{Calculation of $\alpha$}
\end{figure}
-The angle $\alpha$ is now defined as shown in figure 3, and the detector
-length is adjusted to subtend the angle $2\alpha$ as shown. Note that the
-size of the detector array may have changed and the field of view is not
-used.
-For a circular array of detectors, the detectors are spaced around a
-circle covering an angular distance of $2\alpha$. The dotted circle in
-figure 3 indicates the positions of the detectors in this case. Note that
-detectors at the ends of the range would not be illuminated by the source.
-
-Now, consider increasing the focal length ratio to two leaving the
-field of view ratio as 1, as in Figure 4. Now the detectors array is
-denser, and the real field of view is closer to that specified, but note
-again that the field of view is not used. Instead, the focal length is
-used to give a distance from the centre of the phantom to the source, and
-the detector array is adjusted to give an angular coverage to include the
-whole phantom.
+
+
+Empiric testing with \ctsim\ shows that for very large \emph{fan beam angles},
+greater than approximately
+\latexonly{$120^\circ$,}\latexignore{120 degrees,}
+there are significant artifacts. The primary way to manage the
+\emph{fan beam angle} is by varying the \emph{focal length} since the
+\emph{scan diameter} by the size of the phantom.
+
+To illustrate, the \emph{scan diameter} can be defined as
+\latexonly{$$s_d = s_r v_r p_d$$}\latexignore{\\$$Sd = Sr x Vr x Pd$$\\}
+
+Further, $f$ can be defined as \latexonly{$$f = f_r (v_r p_d /
+2)$$}\latexignore{\\$$F = FR x (VR x Pd)$$\\}
+
+Substituting these equations into \latexignore{the above
+equation,}\latexonly{equation~\ref{alphacalc},} We have,
+\latexonly{
+\begin{eqnarray}
+\alpha &= 2\,\sin^{-1} \frac{s_r v_r p_d / 2}{f_r v_r (p_d / 2)} \nonumber \\
+&= 2\,\sin^{-1} (s_r / f_r)
+\end{eqnarray}
+} \latexignore{\\$$\alpha = 2 sin (Sr / Fr$$\\}
+
+Since in normal scanning $s_r$ = 1, $\alpha$ depends only upon the
+\emph{focal length ratio}.
+
+\subsubsection{Detector Array Size}
+In general, you do not need to be concerned with the detector
+array size. It is automatically calculated by \ctsim. For those
+interested, this section explains how the detector array size is
+calculated.
+
+For parallel geometry, the detector length is equal to the scan
+diameter.
+
+For divergent beam geometries, the size of the detector array also
+depends upon the \emph{focal length}. Increasing the \emph{focal
+length} decreases the size of the detector array while 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{$2\,\alpha$.}\latexignore{\emph{2 \alpha}.} The dotted
+circle in
\begin{figure}
-\includegraphics[width=\textwidth]{ctsimfig4.eps}
-\caption{Equilinear and equiangluar geometry when focal length ratio = 2
-and the field of view ratio = 1.}
+\image{10cm;0cm}{equiangular.eps}
+\caption{Equiangluar geometry}
\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.}
+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{$4\,f \tan (\alpha / 2)$}
+\latexignore{\emph{4 x F x tan(\alpha/2)}}
+\begin{figure}
+\image{10cm;0cm}{equilinear.eps}
+\caption{Equilinear geometry}
\end{figure}
+An example of the this geometry is in figure 2.5.
+
+
+\subsubsection{Examples of Geometry Settings}
\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:
+This method is not currently implemented in \ctsim, however it is
+planned for a future release. This method does not give results as
+accurate as filtered backprojection. The difference is due primarily
+because interpolation occurs in the frequency domain rather than the
+spatial domain.
+
+\subsection{Filtered Backprojection}
+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{Multiple Computer Processing}
+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
+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.
+Though multiplying by $|w|$ gives the sharpest reconstructions, in
+practice, superior results are obtained by reducing the higher
+frequencies. This is performed 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
+Backprojection is the process of ``smearing'' the filtered
+projections over the reconstructing image. Various levels of
+interpolation can be specified.
projects / ctsim.git / blobdiff