-\chapter{Concepts}\index{Concepts}%
-\setheader{{\it CHAPTER \thechapter}}{}{}{}{}{{\it CHAPTER \thechapter}}%
-\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}%
-
-\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
+\chapter{Concepts}
+\setheader{{\it CHAPTER \thechapter}}{}{}{\ctsimheadtitle}{}{{\it CHAPTER \thechapter}}
+\ctsimfooter
+
+\section{Conceptual Overview}\index{Conceptual overview}
+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. This projection data can be reconstructed using various
+user-controlled algorithms producing an image of the phantom
+object. These reconstructions can be visually and 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 primary objects that we need to be
+concerned with are the \helprefn{phantom}{conceptphantom} and the
+\helprefn{scanner}{conceptscanner}.
+
+\section{Phantoms}\label{conceptphantom}
+
+\ctsim\ uses geometrical objects to describe the object being
+scanned. A phantom is composed of one or more phantom elements.
+These elements are simple geometric shapes, specifically,
+rectangles, triangles, ellipses, sectors and segments. With these
+elements, the 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 with
+permission from G.T. Herman's publication\cite{HERMAN80}.
+
+\subsection{Phantom File}\label{phantomfile}\index{Phantom!File syntax}
+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.
+The first entry defines the type of the element, either
+\texttt{rectangle}, \texttt{ellipse}, \texttt{triangle},
+\texttt{sector}, or \texttt{segment}.
-{\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.
+For all phantom elements, \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.
+As opposed to the \texttt{r} and \texttt{a} fields, the \texttt{cx},
+\texttt{cy}, \texttt{dx} and \texttt{dy} fields have different
+meanings depending on the element type.
-\subsection{Phantom Elements}\label{phantomelements}\index{Concepts,Phantoms,Elements}
+
+\subsection{Phantom Elements}\label{phantomelements}\index{Phantom!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.
+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,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,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)}
+and 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 (???).
-
-\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}
-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}
+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 drawn between those two points and lies
+below the x-axis. The sector is then rotated and translated the
+same as a segment.
+
+\subsection{Phantom Size}\index{Phantom!Size}
+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 phantom size is
+0.101 in each direction.
+
+\section{Scanner}\label{conceptscanner}\index{Scanner!Concepts}%
+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. \ctsim,
+being a very flexible simulator, gives tremendous options in
+setting up the geometry for a scan.
+
+\subsection{Dimensions}
+The geometry for a scan starts with 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, the \emph{phantom diameter}. Remember, as mentioned
+above, the phantom dimensions are padded by 1\%.
+
+The other important geometry variables for scanning phantoms are
+the \emph{view diameter}, \emph{scan diameter}, \emph{focal
+length}, and \emph{center-detector length}. These variables are input into \ctsim\ in terms of
+ratios rather than absolute values.
+
+\subsubsection{Phantom Diameter}\index{Phantom!Diameter}
+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},} is given by the
+Pythagorean theorem and is
+\latexignore{\\\centerline{\emph{Pl x sqrt(2)}}\\}
+\latexonly{\begin{equation}p_d = p_l \sqrt{2}\end{equation}}
+CT scanners collect projections around a
+circle rather than a square. The diameter of this circle is
+the diameter of the boundary square \latexonly{$p_d$.}
+\latexignore{\emph{Pd}.}
+\latexonly{These relationships are diagrammed in figure~\ref{phantomgeomfig}.}
\begin{figure}
-\includegraphics[width=\textwidth]{ctsimfig1.eps}
-\caption{Geometry used for a 1st generation, parallel beam CT scanner.}
+\centerline{\image{8cm;0cm}{scangeometry.eps}}
+\latexonly{\caption{\label{phantomgeomfig} 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}
-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
+\subsubsection{View Diameter}\index{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} 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{\begin{equation}v_d = p_dv_r\end{equation}}
+\latexignore{\\\centerline{\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 analogous to inserting an object into the CT
+scanner that is larger than the scanner itself!
+
+\subsubsection{Scan Diameter}\index{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},}
+arises. The scan diameter,
+\latexonly{$s_d$,}\latexignore{\emph{Sd},} is the diameter over
+which x-rays are collected and is defined as
+\latexonly{\begin{equation}s_d =v_d s_r\end{equation}}
+\latexignore{\\\centerline{\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}\index{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{\begin{equation}f = (v_d / 2) f_r\end{equation}}
+\latexignore{\\\centerline{\emph{F = (Vd / 2) x FR}}}
+
+For parallel geometry scanning, the focal length doesn't matter.
+However, for 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} is
+physically impossible and it analagous to having the x-ray
+source inside of the \emph{view diameter}.
+
+\subsubsection{Center-Detector Length}\index{Center-Detector length}
+The \emph{center-detector length},
+\latexonly{$c$,}\latexignore{\emph{C},}
+is the distance from the center of
+the phantom to the center of the detector array. The center-detector length is set as a ratio,
+\latexonly{$c_r$,}\latexignore{\emph{CR},}
+of the view radius. The center-detector length is
+calculated as
+\latexonly{\begin{equation}f = (v_d / 2) c_r\end{equation}}
+\latexignore{\\\centerline{\emph{F = (Vd / 2) x CR}}}
+
+For parallel geometry scanning, the center-detector length doesn't matter.
+A value of less than \texttt{1} is physically impossible and it analagous to
+having the detector array inside of the \emph{view diameter}.
+
+
+\subsection{Parallel Geometry}\label{geometryparallel}\index{Parallel geometry}\index{Scanner!Parallel}
+
+The simplest geometry, parallel, was used in first generation
+scanners. 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{Equilinear geometry}\index{Equiangular geometry}
+\index{Scanner!Equilinear}\index{Scanner!Equiangular}
+For both equilinear (second generation) and equiangular
+(third, fourth, and fifth generation) geometries,
+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.
+\latexonly{These configurations are shown in figure~\ref{divergentfig}.}
\begin{figure}
-\includegraphics[width=\textwidth]{ctsimfig2.eps}
-\caption{Equilinear and equiangular geometries.}
+\centerline{\image{10cm;0cm}{divergent.eps}}
+\latexonly{\caption{\label{divergentfig} 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
-\[
-\mathrm{dFocalPastPhm} = ({l_p}/{2}) (\sqrt{2}-1)
-\]
+
+\subsubsection{Fan Beam Angle}\index{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} as
+\latexignore{\centerline{\emph{alpha = 2 x asin (
+(Sd / 2) / f)}}}
+\latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1}
+((s_d / 2) / f)\end{equation}}
+\latexonly{This is illustrated in figure~\ref{alphacalcfig}.}
\begin{figure}
-\includegraphics[height=0.5\textheight]{ctsimfig3.eps}
-\caption{Equilinear and equiangluar geometry when focal length ratio =
-field of view ratio = 1.}
+\centerline{\image{10cm;0cm}{alphacalc.eps}}
+\latexonly{\caption{\label{alphacalcfig} 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} is usually fixed at the size of the phantom.
+
+To illustrate, the \emph{scan diameter} can be defined as
+\latexonly{\begin{equation}s_d = s_r v_r p_d\end{equation}}
+\latexignore{\\\centerline{\emph{Sd = Sr x Vr x Pd}}\\}
+
+Further, the \emph{focal length} can be defined as
+\latexonly{\begin{equation} f = f_r (v_r p_d / 2)\end{equation}}
+\latexignore{\\\centerline{\emph{F = FR x (VR x Pd)$$\\}}}
+
+Substituting these equations into \latexignore{the above
+equation,}\latexonly{equation~\ref{alphacalc},} We have,
+\latexonly{
+\begin{eqnarray}
+\alpha &=& 2\,\sin^{-1} \frac{\displaystyle s_r v_r p_d / 2}{\displaystyle f_r v_r (p_d / 2)} \nonumber \\
+&=& 2\,\sin^{-1} (s_r / f_r)
+\end{eqnarray}
+} \latexignore{\\\centerline{\emph{\alpha = 2 sin (Sr / Fr)}}\\}
+
+Since in normal scanning $s_r$ = 1, $\alpha$ depends only upon the
+\emph{focal length ratio} in normal scanning.
+
+\subsubsection{Detector Array Size}
+In general, you do not need to be concerned with the detector
+array size -- it is automatically calculated by \ctsim. For the
+particularly interested, this section explains how the detector
+array size is calculated.
+
+For parallel geometry, the detector length is simply the scan
+diameter.
+
+For divergent beam geometries, the size of the detector array also
+depends upon the \emph{focal length}: increasing the \emph{focal
+length} decreases the size of the detector array.
+
+For equiangular geometry, the detectors are equally spaced around a arc
+covering an angular distance of $\alpha$ as viewed from the source. When
+viewed from the center of the scanning, the angular distance is
+\latexonly{$$\pi + \alpha - 2 \, \cos^{-1} \Big( \frac{s_d / 2}{c} \Big)$$}
+\latexignore{\\\emph{pi + \alpha - 2 x acos ((Sd / 2) / C))}\\}
+The dotted circle
+\latexonly{in figure~\ref{equiangularfig}}
+indicates the positions of the detectors in this case.
+
\begin{figure}
-\includegraphics[width=\textwidth]{ctsimfig4.eps}
-\caption{Equilinear and equiangluar geometry when focal length ratio = 2
-and the field of view ratio = 1.}
+\centerline{\image{10cm;0cm}{equiangular.eps}}
+\latexonly{\caption{\label{equiangularfig}Equiangular 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.}
+For equilinear geometry, the detectors are equally spaced along a straight
+line. The detector length depends upon
+\latexonly{$\alpha$}\latexignore{\emph{alpha}} and the \emph{focal
+length}. This length,
+\latexonly{$d_l$,}\latexignore{Dl,} is calculated as
+\latexonly{\begin{equation} d_l = 2\,(f + c) \tan (\alpha / 2)\end{equation}}
+\latexignore{\\\centerline{\emph{2 x (F + C) x tan(\alpha/2)}}}
+\latexonly{This geometry is shown in figure~\ref{equilinearfig}.}
+\begin{figure}
+\centerline{\image{10cm;0cm}{equilinear.eps}}
+\latexonly{\caption{\label{equilinearfig} Equilinear geometry}}
\end{figure}
-\section{Reconstruction}\label{conceptreconstruction}\index{Concepts,Reconstruction}%
-\subsection{Overview}
-\subsection{Filtered Backprojection}
+\section{Reconstruction}\label{conceptreconstruction}\index{Reconstruction overview}%
+
\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.
+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. This is due primarily
+to interpolation occurring in the frequency domain rather than the
+spatial domain.
+
+\subsection{Filtered Backprojection}\index{Filtered backprojection}\index{Symmetric multiprocessing}\index{SMP}
+The technique is comprised of two sequential steps:
+filtering projections followed by backprojecting the filtered projections. Though
+these two steps are sequential, each view position can be processed independently.
+
+\subsubsection{Parallel Computer Processing}\index{Parallel processing}
+Since each view can be processed independently, filtered backprojection is amendable to
+parallel processing. Indeed, this has been used in commercial scanners to speed reconstruction.
+This parallelism is exploited both in the \ctsim\ graphical shell and
+in the \helpref{LAM}{ctsimtextlam} version of \ctsimtext. \ctsim\ can distribute it's workload
+amongst multiple processors working in parallel.
+
+The graphical shell will automatically take advantage of multiple CPU's when
+running on a \emph{Symmetric Multiprocessing}
+computer. Dual-CPU computers are commonly available which provide a near doubling
+in reconstruction speeds. \ctsim, though, has no limits on the number of CPU's
+that can be used in parallel. The \emph{LAM} version
+of \ctsimtext\ is designed to work in a cluster of computers.
+This has been testing with a cluster of 16 computers in a
+\urlref{Beowulf-class}{http://www.beowulf.org} cluster with excellent
+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
+The first step in filtered backprojection reconstructions is the filtering
+of each projection. The projections for a each 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.
-\subsubsection{Backprojection of filtered projections}
\ No newline at end of file
+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. Various levels of
+interpolation can be specified.
+
+\section{Image Comparison}\label{conceptimagecompare}\index{Image!Comparison}
+Images can be compared statistically. Three measurements can be calculated
+by \ctsim. They are taken from the standard measurements used by
+Herman\cite{HERMAN80}. They are:
+
+\begin{itemize}\itemsep=0pt
+\item[]\textbf{$d$}\quad The normalized root mean squared distance measure.
+\item[]\textbf{$r$}\quad The normalized mean absolute distance measure.
+\item[]\textbf{$e$}\quad The worst case distance measure over a \latexonly{$2\times2$}\latexignore{\emph{2 x 2}} pixel area.
+\end{itemize}
+
+These measurements are defined in equations \ref{dequation} through \ref{bigrequation}.
+In these equations, $p$ denotes the phantom image, $r$ denotes the reconstruction
+image, and $\bar{p}$ denotes the average pixel value of $p$. Each of the images have a
+size of $m \times n$. In equation \ref{eequation} $[n/2]$ and $[m/2]$ denote the largest
+integers less than $n/2$ and $m/2$, respectively.
+
+\latexignore{These formulas are shown in the print documentation of \ctsim.}
+%
+%Tex2RTF can not handle the any subscripts or superscripts for the inner summation unless
+% have a space character before the \sum
+\latexonly{\begin{equation}\label{dequation} d =\sqrt{\frac{\displaystyle \sum_{i=1}^{n}{ \sum_{j=1}^{m}{(p_{i,j} - r_{i,j})^2}}}{\displaystyle \sum_{i=1}^{n}{ \sum_{j=1}^{m}{(p_{i,j} - \bar{p})^2}}}}\end{equation}}
+\latexonly{\begin{equation}\label{requation}r = \frac{ \displaystyle \sum_{i=1}^{n}{ \sum_{j=1}^{m}{|p_{i,j} - r_{i,j}|}}}{ \displaystyle \sum_{i=1}^{n}{ \sum_{j=1}^{m}{|p_{i,j}|}}}\end{equation}}
+\latexonly{\begin{equation}\label{eequation}e = \max_{1 \le k \le [n/2] \atop 1 \le l \le [m/2]}(|P_{k,l} - R_{k,l}|)\end{equation}}
+\latexonly{where}
+\latexonly{\begin{equation}\label{bigpequation}P_{k,l} = \textstyle \frac{1}{4} (p_{2k,2l} + p_{2k+1,2l} + p_{2k,2l+l} + p_{2k+1,2l+1})\end{equation}}
+\latexonly{\begin{equation}\label{bigrequation}R_{k,l} = \textstyle \frac{1}{4} (r_{2k,2l} + r_{2k+1,2l} + r_{2k,2l+1} + r_{2k+1,2l+1})\end{equation}}
+\begin{comment}
+\end{comment}
projects / ctsim.git / blobdiff