r537: no message

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

diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex
index 319ef48ca2fd277fc8a2c785dc06a403b89e9b34..c7c3ffd39350bd3a66b965e2c8aa25e2b5dcc8c9 100644 (file)
--- a/doc/ctsim-concepts.tex
+++ b/doc/ctsim-concepts.tex
@@ -1,8 +1,8 @@
 \chapter{Concepts}\index{Concepts}%
 \chapter{Concepts}\index{Concepts}%

-\setheader{{\it CHAPTER \thechapter}}{}{}{}{}{{\it CHAPTER \thechapter}}%
-\setfooter{\thepage}{}{}{}{\small Manual v0.2}{\thepage}%
+\setheader{{\it CHAPTER \thechapter}}{}{}{\ctsimheadtitle}{}{{\it CHAPTER \thechapter}}%
+\ctsimfooter%

 
 

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


@@ -16,8 +16,8 @@ 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}.
 
 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.
 
 \ctsim\ uses geometrical objects to describe the object being
 scanned. A phantom is composed a one or more phantom elements.


@@ -32,14 +32,14 @@ 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}
 The first entry defines the type of the element, either
 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, either

-\rtfsp\texttt{rectangle}, \texttt{}, \texttt{triangle},
+\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.


@@ -50,7 +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


@@ -88,14 +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


@@ -114,32 +114,31 @@ 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}$$}
-CT scanners actually collect projections around a circle rather than a
-square. The diameter of this circle is also the diameter of the boundary
-square
-\latexonly{$p_d$.}\latexignore{\rtfsp\emph{Pd}.}
-These relationships are diagrammed in figure 2.1.
-
-\subsubsection{View Diameter}
+The phantom diameter is automatically calculated by \ctsim\ from
+the phantom definition. The maximum of the phantom length and
+height is used to define the square that completely surrounds the
+phantom. Let \latexonly{$p_l$}\latexignore{\emph{Pl}} be the width
+and height of this square. The diameter of this boundary box,
+\latexonly{$p_d$,}\latexignore{\emph{Pd},} \rtfsp is then
+\latexignore{\\$$\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
 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


@@ -160,12 +159,12 @@ 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}
+\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},}
 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
+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}$$\\} By default and
 \latexonly{$s_d$}\latexignore{\emph{Sd}} is the diameter over
 which x-rays are collected and is defined as \latexonly{$$s_d =
 v_d s_r$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\} By default and


@@ -173,7 +172,7 @@ 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.
 
 If the \emph{scan ratio} is less than \texttt{1}, you can expect
 significant artifacts.
 

-\subsubsection{Focal Length}
+\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


@@ -186,13 +185,12 @@ calculated as
 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
 For parallel geometry scanning, the focal length doesn't matter.
 However, divergent geometry scanning (equilinear and equiangular),
 the \emph{focal length ratio} should be set at \texttt{2} or more

-to avoid artifacts. Moreover, a value of less than \texttt{1},
-though it can be given to \ctsim, is physically impossible and it
-analagous to have having the x-ray source with the \emph{view
-diameter}.
+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


@@ -204,7 +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,


@@ -212,10 +210,10 @@ 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}
 
 
 \end{figure}
 
 


@@ -227,10 +225,11 @@ at the time of manufacture. \ctsim, however, calculates the
 the \emph{focal length} \latexignore{\\$$\emph{alpha = 2 x asin (
 (Sd / 2) / f)}$$\\}
 \latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1}
 the \emph{focal length} \latexignore{\\$$\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.
+((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}
 
 


@@ -261,9 +260,9 @@ Since in normal scanning $s_r$ = 1, $\alpha$ depends only upon the
 
 \subsubsection{Detector Array Size}
 In general, you do not need to be concerned with the detector
 
 \subsubsection{Detector Array Size}
 In general, you do not need to be concerned with the detector

-array size. It is automatically calculated by \ctsim. For those
-interested, this section explains how the detector array size is
-calculated.
+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.


@@ -279,26 +278,26 @@ covering an angular distance of
 circle in
 \begin{figure}
 \image{10cm;0cm}{equiangular.eps}
 circle in
 \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
 \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)}}
 
 For equilinear geometry, the detectors are space along a straight
 line. The length of the line depends upon
 \latexonly{$\alpha$}\latexignore{\emph{alpha}} and the \emph{focal
 length}. It is calculated as \latexonly{$4\,f \tan (\alpha / 2)$}
 \latexignore{\emph{4 x F x tan(\alpha/2)}}

-\begin{figure}
+\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


@@ -307,7 +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.


@@ -338,3 +337,23 @@ multiple filters for this purpose.
 Backprojection is the process of ``smearing'' the filtered
 projections over the reconstructing image. Various levels of
 interpolation can be specified.
 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}
+}