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

diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex
index 81dbf42b2da366baf1252f0b94369996c0d71fe4..c7c3ffd39350bd3a66b965e2c8aa25e2b5dcc8c9 100644 (file)
--- a/doc/ctsim-concepts.tex
+++ b/doc/ctsim-concepts.tex
@@ -1,43 +1,45 @@
 \chapter{Concepts}\index{Concepts}%
 \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.
+\setheader{{\it CHAPTER \thechapter}}{}{}{\ctsimheadtitle}{}{{\it CHAPTER \thechapter}}%
+\ctsimfooter%
+
+\section{Overview}\label{conceptoverview}\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. 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}.
 
 
 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}%
+\section{Phantoms}\label{conceptphantom}
+\subsection{Overview}\label{phantomoverview}\index{Phantom 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.
+\ctsim\ uses geometrical objects to describe the object being
+scanned. A phantom is composed a one or more phantom elements.
+These elements are simple geometric shapes, specifically,
+rectangles, triangles, ellipses, sectors and segments. With these
+elements, standard phantoms used in the CT literature can be
+constructed.  In fact, \ctsim\ provides a shortcut to load the
+published phantoms of Herman\cite{HERMAN80} and
+Shepp-Logan\cite{SHEPP74}. \ctsim\ also reads text files of
+user-defined phantoms.

 
 The types of phantom elements and their definitions are taken with
 permission from G.T. Herman's 1980 book\cite{HERMAN80}.
 
 
 The types of phantom elements and their definitions are taken with
 permission from G.T. Herman's 1980 book\cite{HERMAN80}.
 

-\subsection{Phantom File}\label{phantomfile}\index{Concepts,Phantoms,File}
+\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
-\rtfsp\texttt{rectangle}, \texttt{}, \texttt{triangle},
+The first entry defines the type of the element, either
+\rtfsp\texttt{rectangle}, \texttt{ellipse}, \texttt{triangle},

 \rtfsp\texttt{sector}, or \texttt{segment}. \texttt{cx},
 \rtfsp\texttt{cy}, \texttt{dx} and \texttt{dy} have different
 meanings depending on the element type.
 \rtfsp\texttt{sector}, or \texttt{segment}. \texttt{cx},
 \rtfsp\texttt{cy}, \texttt{dx} and \texttt{dy} have different
 meanings depending on the element type.


@@ -48,7 +50,7 @@ coefficient of the object. Where objects overlap, the attenuations
 of the overlapped objects are summed.
 
 
 of the overlapped objects are summed.
 
 

-\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


@@ -86,14 +88,14 @@ 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.
 
 below the x-axis. The sector is then rotated and translated the same
 as a segment.
 

-\subsection{Phantom Size}
+\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.
 
 The overall dimensions of the phantom are increased by 1\% above the
 specified sizes to avoid clipping due to round-off errors from
 sampling the polygons of the phantom elements.  So, if the phantom is
 defined as a rectangle of size 0.1 by 0.1, the actual phantom has
 extent 0.101 in each direction.
 

-\section{Scanner}\label{conceptscanner}\index{Concepts,Scanner}%
+\section{Scanner}\label{conceptscanner}\index{Scanner concepts}%

 \subsection{Dimensions}
 Understanding the scanning geometry is the most complicated aspect of
 using \ctsim. For real-world CT simulators, this is actually quite
 \subsection{Dimensions}
 Understanding the scanning geometry is the most complicated aspect of
 using \ctsim. For real-world CT simulators, this is actually quite


@@ -112,41 +114,41 @@ 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\%.
 
 \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.
+The other important geometry variables for scanning phantoms 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}
+\subsubsection{Phantom Diameter}\index{Phantom diameter}

 \begin{figure}
 $$\image{5cm;0cm}{scangeometry.eps}$$
 \begin{figure}
 $$\image{5cm;0cm}{scangeometry.eps}$$

-\caption{Phantom Geometry}
+\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},}
-\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}$$}
+The phantom diameter is automatically calculated by \ctsim\ from
+the phantom definition. The maximum of the phantom length and
+height is used to define the square that completely surrounds the
+phantom. Let \latexonly{$p_l$}\latexignore{\emph{Pl}} be the width
+and height of this square. The diameter of this boundary box,
+\latexonly{$p_d$,}\latexignore{\emph{Pd},} \rtfsp is then
+\latexignore{\\$$\emph{Pl x sqrt(2)}$$\\} \latexonly{$$p_d = p_l
+\sqrt{2}$$} CT scanners actually collect projections around a
+circle rather than a square. The diameter of this circle is also
+the diameter of the boundary square
+\latexonly{$p_d$.  These
+relationships are diagrammed in figure~\ref{phantomgeomfig}.}
+\latexignore{emph{Pd}.}
+
+\subsubsection{View Diameter}\index{View diameter}
+The \emph{view diameter} is the area that is being processed
+during scanning of phantoms as well as during rasterization of
+phantoms. By default, the \emph{view diameter} \rtfsp is set equal
+to the \emph{phantom diameter}. It may be useful, especially for
+experimental reasons, to process an area larger (and maybe even
+smaller) than the phantom. Thus, during rasterization or during
+projections, \ctsim\ will ask for a \emph{view ratio},
+\latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.} The \emph{view
+diameter} is then calculated as \latexonly{$$v_d = p_d
+v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}

 
 By using a
 \latexonly{$v_r$}\latexignore{\emph{VR}}
 
 By using a
 \latexonly{$v_r$}\latexignore{\emph{VR}}


@@ -157,20 +159,20 @@ This will lead to significant artifacts. Physically, this would
 be impossible and is analagous to inserting an object into the CT
 scanner that is larger than the scanner itself!
 
 be impossible and is analagous to inserting an object into the CT
 scanner that is larger than the scanner itself!
 

-\subsubsection{Scan Diameter}
-By default, the entire \emph{view diameter} is scanned. For experimental
-purposes, it may be desirable to scan an area either larger or smaller than
-the \emph{view diameter}. Thus, the concept of \emph{scan ratio}
-\latexonly{$s_r$}\latexignore{\emph{SR}}
-is born. The scan diameter
-\latexonly{$s_d$}\latexignore{\emph{Sd}}
-is the diameter over which x-rays are collected and is defined as
-\latexonly{$$s_d = v_d s_r$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\}
-By default and for all ordinary scanning, the \emph{scan ratio} is to
-\texttt{1}. If the \emph{scan ratio} is less than \texttt{1},
-you can expect significant artifacts.
-
-\subsubsection{Focal Length}
+\subsubsection{Scan Diameter}\index{Scan diameter}
+By default, the entire \emph{view diameter} is scanned. For
+experimental purposes, it may be desirable to scan an area either
+larger or smaller than the \emph{view diameter}. Thus, the concept
+of \emph{scan ratio}, \latexonly{$s_r$,}\latexignore{\emph{SR},}
+is arises. The scan diameter
+\latexonly{$s_d$}\latexignore{\emph{Sd}} is the diameter over
+which x-rays are collected and is defined as \latexonly{$$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}\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 \emph{focal length},
 \latexonly{$f$,}\latexignore{\emph{F},}
 is the distance of the X-ray source to the center of


@@ -180,12 +182,15 @@ of the view radius. Focal length is
 calculated as
 \latexonly{$$f = (v_d / 2) f_r$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$}
 
 calculated as
 \latexonly{$$f = (v_d / 2) f_r$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$}
 

-For parallel geometry scanning, the focal length doesn't matter. However,
-divergent geometry scanning (equilinear and equiangular), the \emph{focal
-length ratio} should be set at \texttt{2} or more to avoid artifacts.
+For parallel geometry scanning, the focal length doesn't matter.
+However, divergent geometry scanning (equilinear and equiangular),
+the \emph{focal length ratio} should be set at \texttt{2} or more
+to avoid artifacts. Moreover, a value of less than \texttt{1} is
+physically impossible and it analagous to have having the x-ray
+source inside of the \emph{view diameter}.

 
 
 
 

-\subsection{Parallel Geometry}\label{geometryparallel}\index{Concepts,Scanner,Geometries,Parallel}
+\subsection{Parallel Geometry}\label{geometryparallel}\index{Parallel Geometry}

 
 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
 
 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


@@ -197,7 +202,7 @@ values of less than \texttt{1} are used for these two variables,
 significant distortions will occur.
 
 
 significant distortions will occur.
 
 

-\subsection{Divergent Geometries}\label{geometrydivergent}\index{Concepts,Scanner,Geometries,Divergent}
+\subsection{Divergent Geometries}\label{geometrydivergent}\index{Divergent geometry}

 \subsubsection{Overview}
 Next consider the case of equilinear (second generation) and equiangular
 (third, fourth, and fifth generation) geometries. In these cases,
 \subsubsection{Overview}
 Next consider the case of equilinear (second generation) and equiangular
 (third, fourth, and fifth generation) geometries. In these cases,


@@ -205,24 +210,26 @@ 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.
 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{See figure~\ref{divergentfig}.}

 \begin{figure}
 \image{10cm;0cm}{divergent.eps}
 \begin{figure}
 \image{10cm;0cm}{divergent.eps}

-\caption{Equilinear and equiangular geometries.}
+\caption{\label{divergentfig} Equilinear and equiangular geometries.}

 \end{figure}
 
 
 \subsubsection{Fan Beam Angle}
 \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)}$$\\}
-\latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1} ((s_d / 2) / f)\end{equation}}
-This is illustrated in figure 2.3.
+For these divergent beam geometries, the \emph{fan beam angle}
+needs to be calculated. For real-world CT scanners, this is fixed
+at the time of manufacture. \ctsim, however, calculates the
+\emph{fan beam angle}, $\alpha$, from the \emph{scan diameter} and
+the \emph{focal length} \latexignore{\\$$\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~\ref{alphacalcfig}.}

 \begin{figure}
 \image{10cm;0cm}{alphacalc.eps}
 \begin{figure}
 \image{10cm;0cm}{alphacalc.eps}

-\caption{Calculation of $\alpha$}
+\caption{\label{alphacalcfig} Calculation of $\alpha$}

 \end{figure}
 
 
 \end{figure}
 
 


@@ -236,60 +243,61 @@ there are significant artifacts. The primary way to manage the
 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$$\\}
 
 To illustrate, the \emph{scan diameter} can be defined as
 \latexonly{$$s_d = s_r v_r p_d$$}\latexignore{\\$$Sd = Sr x Vr x Pd$$\\}
 

-Further, $f$ can be defined as
-\latexonly{$$f = f_r (v_r p_d / 2)$$}
-Plugging these equations into
-\latexignore{the above equation,}\latexonly{equation~\ref{alphacalc},}
-We have,
+Further, $f$ can be defined as \latexonly{$$f = f_r (v_r p_d /
+2)$$}\latexignore{\\$$F = FR x (VR x Pd)$$\\}
+
+Substituting these equations into \latexignore{the above
+equation,}\latexonly{equation~\ref{alphacalc},} We have,

 \latexonly{
 \begin{eqnarray}
 \alpha &= 2\,\sin^{-1} \frac{s_r v_r p_d / 2}{f_r v_r (p_d / 2)} \nonumber \\
 &= 2\,\sin^{-1} (s_r / f_r)
 \end{eqnarray}
 \latexonly{
 \begin{eqnarray}
 \alpha &= 2\,\sin^{-1} \frac{s_r v_r p_d / 2}{f_r v_r (p_d / 2)} \nonumber \\
 &= 2\,\sin^{-1} (s_r / f_r)
 \end{eqnarray}

-}
+} \latexignore{\\$$\alpha = 2 sin (Sr / Fr$$\\}

 
 

-Since in normal scanning $s_r = 1$, $\alpha$ depends only upon the \emph{focal length ratio}.
+Since in normal scanning $s_r$ = 1, $\alpha$ depends only upon the
+\emph{focal length ratio}.

 
 \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.
+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
 diameter.
 
 
 For parallel geometry, the detector length is equal to 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 while increasing
+the \emph{scan diameter} increases the detector array size.

 
 

-For equiangular geometry, the detectors are spaced around a
-circle covering an angular distance of
-\latexonly{$\alpha$.}\latexignore{\emph{alpha}.}
-The dotted circle in
+For equiangular geometry, the detectors are spaced around a circle
+covering an angular distance of
+\latexonly{$2\,\alpha$.}\latexignore{\emph{2 \alpha}.} The dotted
+circle in

 \begin{figure}
 \image{10cm;0cm}{equiangular.eps}
 \begin{figure}
 \image{10cm;0cm}{equiangular.eps}

-\caption{Equiangluar geometry}
+\caption{\label{equiangularfig}Equiangular geometry}

 \end{figure}
 \end{figure}

-figure 2.4 indicates the positions of the detectors in this case.
+figure~\ref{equiangularfig} 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
 
 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)$$\\}
-\begin{figure}
+\latexonly{$\alpha$}\latexignore{\emph{alpha}} and the \emph{focal
+length}. It is calculated as \latexonly{$4\,f \tan (\alpha / 2)$}
+\latexignore{\emph{4 x F x tan(\alpha/2)}}
+\begin{figure}\label{equilinearfig}

 \image{10cm;0cm}{equilinear.eps}
 \image{10cm;0cm}{equilinear.eps}

-\caption{Equilinear geometry}
+\caption{\label{equilinearfig} Equilinear geometry}

 \end{figure}
 \end{figure}

-An example of the this geometry is in figure 2.5.
+\latexonly{This geometry is shown in figure~\ref{equilinearfig}.}

 
 
 \subsubsection{Examples of Geometry Settings}
 
 
 
 
 \subsubsection{Examples of Geometry Settings}
 
 

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

 \subsection{Overview}
 \subsection{Direct Inverse Fourier}
 This method is not currently implemented in \ctsim, however it is
 \subsection{Overview}
 \subsection{Direct Inverse Fourier}
 This method is not currently implemented in \ctsim, however it is


@@ -298,7 +306,7 @@ accurate as filtered backprojection. The difference is due primarily
 because interpolation occurs in the frequency domain rather than the
 spatial domain.
 
 because interpolation occurs in the frequency domain rather than the
 spatial domain.
 

-\subsection{Filtered Backprojection}
+\subsection{Filtered Backprojection}\index{Filtered backprojection}

 The technique is comprised of two sequential steps:
 filtering projections and then backprojecting the filtered projections. Though
 these two steps are sequential, each view position can be processed individually.
 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.


@@ -320,12 +328,32 @@ 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
 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}
 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}\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}.
+$d$ is the standard error, $e$ is the maximum error, and
+$r$ is the maximum error of a 2 by 2 pixel area.
+
+To compare two images, $A$ and $B$, each of which has $n$ columns and $m$ rows,
+these values are calculated as below.
+
+
+\latexonly{
+\begin{equation}
+d = \frac{\sum_{i=0}^{n}{\sum_{j=0}^{m}{(A_{ij} - B_{ij})^2}}}{m n}
+\end{equation}
+\begin{equation}
+r = \max(|A_{ij} - B{ij}|)
+\end{equation}
+}