r591: Added Center-Detector length to scanning and reconstruction

[ctsim.git] / doc / ctsim-concepts.tex

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-\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.
-
-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}
+\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}
 element-type cx cy dx dy r a
 \end{verbatim}
 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 {\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 \texttt{dx} and \texttt{dy} to define the semi-major and
 
 \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}
 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}
 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
 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}
 
 \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 from polygonal sampling.  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{Dimensions}
+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
 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 diameter}, \emph{scan diameter}, and \emph{focal length}.
-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}
 \begin{figure}

-$$\image{5cm;0cm}{scangeometry.eps}$$
-\caption{Phantom Geometry}
+\centerline{\image{8cm;0cm}{scangeometry.eps}}
+\latexonly{\caption{\label{phantomgeomfig} Phantom Geometry}}

 \end{figure}
 \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},}
-is then
-\latexignore{\\$$\emph{Pl x sqrt(2)}$$\\}
-\latexonly{$$p_d = p_l \sqrt{2}$$}
-CT scanners actually collect projections around a circle rather than a
-square. The diameter of this circle is also the diameter of the boundary
-square
-\latexonly{$p_d$.}\latexignore{\emph{Pd}.}
-These relationships are diagrammed in figure 2.1.
-
-\subsubsection{View Diameter}
-The \emph{view diameter} is the area that is being processed during scanning of phantoms as
-well as during rasterization of phantoms. By default, the \emph{view diameter}
-is set equal to the \emph{phantom diameter}. It may be useful, especially for 
-experimental reasons, to process an area larger (and maybe even smaller) than
-the phantom. Thus, during rasterization or during projections, \ctsim\ will
-ask for a \emph{view ratio},
-\latexonly{$v_{R}$.}\latexignore{\emph{VR}.}
-The \emph{view diameter} is then set as
-\latexonly{$$v_d = p_d v_{R}$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
-
-By using a 
-\latexonly{$v_{R}$}\latexignore{\emph{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}}

 less than 1, \ctsim\ will allow
 less than 1, \ctsim\ will allow

-for a \emph{view diameter} less than 
+for a \emph{view diameter} less than

 \emph{phantom diameter}.
 This will lead to significant artifacts. Physically, this would
 \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!
 
 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
 the phantom. The focal length is set as a ratio,
 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},}
+\latexonly{$f_r$,}\latexignore{\emph{FR},}

 of the view radius. Focal length is
 calculated as
 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}

 
 

-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 diameter ratio} and
-the \emph{scan diameter ratio}. If values of less than \texttt{1} are
-used for these two variables, significant distortions will occur.
+\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{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.
 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}
 \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}
 
 
 \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},
-\latexonly{$\alpha$,}\latexignore{\emph{alpha},} 
-from the diameter of the \emph{scan diameter} and the \emph{focal length}
-\latexignore{\\$$\emph{alpha = 2 x asin ( (Sd / 2) / F)}$$\\}
-\latexonly{$$\alpha = 2 \sin^{-1} ((s_d / 2) / f)$$}
-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}
 \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}
 
 
 Empiric testing with \ctsim\ shows that for very large \emph{fan beam angles},
 \end{figure}
 
 
 Empiric testing with \ctsim\ shows that for very large \emph{fan beam angles},

-greater than approximately 
+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
 \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{\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}
 
 \subsubsection{Detector Array Size}

-In general, you do not need to be concerned with the detector array
-size. It is automatically calculated by \ctsim.  The size of the
-detector array depends upon the \emph{focal length} and the 
-\emph{scan diameter}. In general, increasing the \emph{focal length}
-decreases the size of the detector array and 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{$\alpha$.}\latexignore{\emph{alpha}.}
-The dotted circle in
+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}
 \begin{figure}

-\image{10cm;0cm}{equiangular.eps}
-\caption{Equiangluar geometry}
+\centerline{\image{10cm;0cm}{equiangular.eps}}
+\latexonly{\caption{\label{equiangularfig}Equiangular geometry}}

 \end{figure}
 \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}
 \begin{figure}

-\image{10cm;0cm}{equilinear.eps}
-\caption{Equilinear geometry}
+\centerline{\image{10cm;0cm}{equilinear.eps}}
+\latexonly{\caption{\label{equilinearfig} Equilinear geometry}}

 \end{figure}
 \end{figure}

-An example of the this geometry is in figure 2.5.
-

 
 

-\subsubsection{Examples of Geometry Settings}
-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 center of the phantom to the source, and
-the detector array is adjusted to give an angular coverage to include the
-whole phantom.

 
 

+\section{Reconstruction}\label{conceptreconstruction}\index{Reconstruction overview}%

 
 

-
-\section{Reconstruction}\label{conceptreconstruction}\index{Concepts,Reconstruction}%
-\subsection{Overview}

 \subsection{Direct Inverse Fourier}
 \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
 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.
 
 spatial domain.
 

-\subsection{Filtered Backprojection}
+\subsection{Filtered Backprojection}\index{Filtered backprojection}\index{Symmetric multiprocessing}\index{SMP}

 The technique is comprised of two sequential steps:
 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.
+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}
 
 \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
 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
+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.
 
 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}
 
 \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}