X-Git-Url: http://git.kpe.io/?a=blobdiff_plain;f=doc%2Fctsim-concepts.tex;h=c7c3ffd39350bd3a66b965e2c8aa25e2b5dcc8c9;hb=3fe1c8e09462c4b44908fec9d309aa41908889c8;hp=9653a120ead80b7343711c9699b6a87b9dc600f9;hpb=dc60803b173bcf8719862fbed7e8652883e23f96;p=ctsim.git

diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex
index 9653a12..c7c3ffd 100644
--- a/doc/ctsim-concepts.tex
+++ b/doc/ctsim-concepts.tex
@@ -1,8 +1,8 @@
 \chapter{Concepts}\index{Concepts}%
-\setheader{{\it CHAPTER \thechapter}}{}{}{}{}{{\it CHAPTER \thechapter}}%
+\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
@@ -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}.
 
-\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.
@@ -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}.
 
-\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
-\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.
@@ -50,7 +50,7 @@ coefficient of the object. Where objects overlap, the attenuations
 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
@@ -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.
 
-\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.
 
-\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
@@ -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\%.
 
-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}
 $$\image{5cm;0cm}{scangeometry.eps}$$
-\caption{Phantom Geometry}
+\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}$$}
-CT scanners actually collect projections around a circle rather than a
-square. The diameter of this circle is also the diameter of the boundary
-square
-\latexonly{$p_d$.}\latexignore{\rtfsp\emph{Pd}.}
-These relationships are diagrammed in figure 2.1.
-
-\subsubsection{View Diameter}
+The phantom diameter is automatically calculated by \ctsim\ from
+the phantom definition. The maximum of the phantom length and
+height is used to define the square that completely surrounds the
+phantom. Let \latexonly{$p_l$}\latexignore{\emph{Pl}} be the width
+and height of this square. The diameter of this boundary box,
+\latexonly{$p_d$,}\latexignore{\emph{Pd},} \rtfsp is then
+\latexignore{\\$$\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
@@ -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!
 
-\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},}
-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
@@ -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.
 
-\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
@@ -191,7 +190,7 @@ 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
@@ -203,7 +202,7 @@ 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}
+\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,
@@ -211,10 +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.
-See figure 2.2.
+\latexonly{See figure~\ref{divergentfig}.}
 \begin{figure}
 \image{10cm;0cm}{divergent.eps}
-\caption{Equilinear and equiangular geometries.}
+\caption{\label{divergentfig} Equilinear and equiangular geometries.}
 \end{figure}
 
 
@@ -226,10 +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}
-((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}
-\caption{Calculation of $\alpha$}
+\caption{\label{alphacalcfig} Calculation of $\alpha$}
 \end{figure}
 
 
@@ -260,9 +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
-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.
@@ -276,10 +276,11 @@ 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}\label{equiangularfig}
-\image{10cm;0cm}{equiangular.eps} \caption{Equiangular geometry}
+\begin{figure}
+\image{10cm;0cm}{equiangular.eps}
+\caption{\label{equiangularfig}Equiangular geometry}
 \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
@@ -288,15 +289,15 @@ 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}
-\caption{Equilinear geometry}
+\caption{\label{equilinearfig} Equilinear geometry}
 \end{figure}
-This geometry is shown in figure~2.5.
+\latexonly{This geometry is shown in figure~\ref{equilinearfig}.}
 
 
 \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
@@ -305,7 +306,7 @@ accurate as filtered backprojection. The difference is due primarily
 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.
@@ -336,3 +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.
+
+\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}
+}