X-Git-Url: http://git.kpe.io/?p=ctsim.git;a=blobdiff_plain;f=doc%2Fctsim-concepts.tex;h=86a94bccae69761ea5e5370600f1e50940b85da3;hp=81dbf42b2da366baf1252f0b94369996c0d71fe4;hb=c953cbb6ffc2fd50e736230f4e6976a025983cff;hpb=eb4b8ecaf864329867c9d68c5911d2a2673d8a04

diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex
index 81dbf42..86a94bc 100644
--- a/doc/ctsim-concepts.tex
+++ b/doc/ctsim-concepts.tex
@@ -1,152 +1,157 @@
-\chapter{Concepts}\index{Concepts}%
-\setheader{{\it CHAPTER \thechapter}}{}{}{}{}{{\it CHAPTER \thechapter}}%
-\setfooter{\thepage}{}{}{}{}{\thepage}%
-
-\section{Overview}\label{conceptoverview}\index{Concepts,Overview}%
-The operation of \ctsim\ begins with the phantom object.  A phantom
-object consists of geometric elements.  A scanner is specified and the
-projection data simulated.  Finally 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
-\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 elements, standard phantoms used in the CT literature can
-be constructed.  In fact, \ctsim\ provides a shortcut to load the
-published phantoms of Herman and Shepp-Logan.  \ctsim\ also reads text
-files of user-defined phantoms.
+\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 1980 book\cite{HERMAN80}.
+permission from G.T. Herman's publication\cite{HERMAN80}.
 
-\subsection{Phantom File}\label{phantomfile}\index{Concepts,Phantoms,File}
+\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}
 element-type cx cy dx dy r a
 \end{verbatim}
-The first entry defines the type of the element, one of
-\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.
+The first entry defines the type of the element, either
+\texttt{rectangle}, \texttt{ellipse}, \texttt{triangle},
+\texttt{sector}, or \texttt{segment}.
 
-\rtfsp\texttt{r} is the rotation applied to the object in degrees
-counterclockwise, and \texttt{a} is the X-ray attenuation
+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 \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
+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 \texttt{cx} and \texttt{cy} to define the position of
+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 center of the base at \texttt{(cx,cy)},
-with a base half-width of \texttt{dx} and a height of \texttt{dy}.
+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 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}.
+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}
+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 actual phantom has
-extent 0.101 in each direction.
+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{Concepts,Scanner}%
-\subsection{Dimensions}
+\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. 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 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}
+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}
-$$\image{5cm;0cm}{scangeometry.eps}$$
-\caption{Phantom Geometry}
+\centerline{\image{8cm;0cm}{scangeometry.eps}}
+\latexonly{\caption{\label{phantomgeomfig} Phantom Geometry}}
 \end{figure}
-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}$$}
-CT scanners actually collect projections around a circle rather than a
-square. The diameter of this circle is also the diameter of the boundary
-square
-\latexonly{$p_d$.}\latexignore{\rtfsp\emph{Pd}.}
-These relationships are diagrammed in figure 2.1.
-
-\subsubsection{View Diameter}
-The \emph{view diameter} is the area that is being processed during scanning of phantoms as
-well as during rasterization of phantoms. By default, the \emph{view diameter}
-\rtfsp is set equal to the \emph{phantom diameter}. It may be useful, especially for
-experimental reasons, to process an area larger (and maybe even smaller) than
-the phantom. Thus, during rasterization or during projections, \ctsim\ will
-ask for a \emph{view ratio},
-\latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.}
-The \emph{view diameter} is then set as
-\latexonly{$$v_d = p_d v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
+
+\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}}
@@ -154,23 +159,25 @@ 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
+be impossible and is analogous 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}$$\\}
-By default and for all ordinary scanning, the \emph{scan ratio} is to
-\texttt{1}. If the \emph{scan ratio} is less than \texttt{1},
-you can expect significant artifacts.
-
-\subsubsection{Focal Length}
+\subsubsection{Scan Diameter}\index{Scan diameter}
+By default, the entire \emph{view diameter} is scanned. For
+experimental purposes, it may be desirable to scan an area either
+larger or smaller than the \emph{view diameter}. Thus, the concept
+of \emph{scan ratio}, \latexonly{$s_r$,}\latexignore{\emph{SR},}
+arises. The scan diameter,
+\latexonly{$s_d$,}\latexignore{\emph{Sd},} is the diameter over
+which x-rays are collected and is defined as
+\latexonly{\begin{equation}s_d =v_d s_r\end{equation}}
+\latexignore{\\\centerline{\emph{Sd = Vd x SR}}\\}
+By default and
+for all ordinary scanning, the \emph{scan ratio} is to \texttt{1}.
+If the \emph{scan ratio} is less than \texttt{1}, you can expect
+significant artifacts.
+
+\subsubsection{Focal Length}\index{Focal length}
 The \emph{focal length},
 \latexonly{$f$,}\latexignore{\emph{F},}
 is the distance of the X-ray source to the center of
@@ -178,16 +185,36 @@ the phantom. The focal length is set as a ratio,
 \latexonly{$f_r$,}\latexignore{\emph{FR},}
 of the view radius. Focal length is
 calculated as
-\latexonly{$$f = (v_d / 2) f_r$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$}
+\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 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.
+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{Concepts,Scanner,Geometries,Parallel}
+\subsection{Parallel Geometry}\label{geometryparallel}\index{Parallel geometry}\index{Scanner!Parallel}
 
-As mentioned above, the focal length is not used in this simple
+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
@@ -197,32 +224,35 @@ 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. In these cases,
+\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 2.2.
+\latexonly{These configurations are shown in figure~\ref{divergentfig}.}
 \begin{figure}
-\image{10cm;0cm}{divergent.eps}
-\caption{Equilinear and equiangular geometries.}
+\centerline{\image{10cm;0cm}{divergent.eps}}
+\latexonly{\caption{\label{divergentfig} Equilinear and equiangular geometries.}}
 \end{figure}
 
 
-\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.
+\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}
-\image{10cm;0cm}{alphacalc.eps}
-\caption{Calculation of $\alpha$}
+\centerline{\image{10cm;0cm}{alphacalc.eps}}
+\latexonly{\caption{\label{alphacalcfig} Calculation of $\alpha$}}
 \end{figure}
 
 
@@ -231,86 +261,103 @@ 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.
+\emph{scan diameter} is usually fixed at 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$$\\}
+\latexonly{\begin{equation}s_d = s_r v_r p_d\end{equation}}
+\latexignore{\\\centerline{\emph{Sd = Sr x Vr x Pd}}\\}
 
-Further, $f$ can be defined as
-\latexonly{$$f = f_r (v_r p_d / 2)$$}
-Plugging these equations into
-\latexignore{the above equation,}\latexonly{equation~\ref{alphacalc},}
-We have,
+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{s_r v_r p_d / 2}{f_r v_r (p_d / 2)} \nonumber \\
-&= 2\,\sin^{-1} (s_r / f_r)
+\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}.
+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.
+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 equal to the scan
+For parallel geometry, the detector length is simply the scan
 diameter.
 
-For divergent beam geometrys, 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 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.
 
-For equiangular geometry, the detectors are spaced around a
-circle covering an angular distance of
-\latexonly{$\alpha$.}\latexignore{\emph{alpha}.}
-The dotted circle in
 \begin{figure}
-\image{10cm;0cm}{equiangular.eps}
-\caption{Equiangluar geometry}
+\centerline{\image{10cm;0cm}{equiangular.eps}}
+\latexonly{\caption{\label{equiangularfig}Equiangular geometry}}
 \end{figure}
-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{$$\mathrm{detLengh} = 4\,f \tan (\alpha / 2)$$}
-\latexignore{\\$$\emph{detLength} = 4 x F x tan(alpha/2)$$\\}
+
+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}
-\image{10cm;0cm}{equilinear.eps}
-\caption{Equilinear geometry}
+\centerline{\image{10cm;0cm}{equilinear.eps}}
+\latexonly{\caption{\label{equilinearfig} 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{Reconstruction overview}%
 
-\section{Reconstruction}\label{conceptreconstruction}\index{Concepts,Reconstruction}%
-\subsection{Overview}
 \subsection{Direct Inverse Fourier}
-This method is not currently implemented in \ctsim, however it is
+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
+accurate as filtered backprojection. This is due primarily
+to interpolation occurring in the frequency domain rather than the
 spatial domain.
 
-\subsection{Filtered Backprojection}
+\subsection{Filtered Backprojection}\index{Filtered backprojection}\index{Symmetric multiprocessing}\index{SMP}
 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.
-
-\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
+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
+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.
 
@@ -320,12 +367,42 @@ 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
+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.  In general, the trade-off is between quality and execution
-time.
+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}