X-Git-Url: http://git.kpe.io/?a=blobdiff_plain;f=doc%2Fctsim-concepts.tex;h=3a01615190f08910ee6a0a81ae9201ac403d35c4;hb=47601e0ab94ccdc360824178cf068a05bcbdb0eb;hp=ddf6bed819ad82cc4516a7f67239494adfef9976;hpb=d3fa225aa232e132cc198672c4fc148f96a1ab8c;p=ctsim.git

diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex
index ddf6bed..3a01615 100644
--- a/doc/ctsim-concepts.tex
+++ b/doc/ctsim-concepts.tex
@@ -1,6 +1,6 @@
 \chapter{Concepts}\index{Concepts}%
-\setheader{{\it CHAPTER \thechapter}}{}{}{}{}{{\it CHAPTER \thechapter}}%
-\setfooter{\thepage}{}{}{}{\small Version 0.2}{\thepage}%
+\setheader{{\it CHAPTER \thechapter}}{}{}{\ctsimheadtitle}{}{{\it CHAPTER \thechapter}}%
+\ctsimfooter%
 
 \section{Overview}\label{conceptoverview}\index{Concepts,Overview}%
 The operation of \ctsim\ begins with the phantom object.  A
@@ -39,7 +39,7 @@ phantom.  Each line contains seven entries, in the following form:
 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.
@@ -114,30 +114,29 @@ 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}
 \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}$$}
-CT scanners actually collect projections around a circle rather than a
-square. The diameter of this circle is also the diameter of the boundary
-square
-\latexonly{$p_d$.}\latexignore{\rtfsp\emph{Pd}.}
-These relationships are diagrammed in figure 2.1.
+The phantom diameter is automatically calculated by \ctsim\ from
+the phantom definition. The maximum of the phantom length and
+height is used to define the square that completely surrounds the
+phantom. Let \latexonly{$p_l$}\latexignore{\emph{Pl}} be the width
+and height of this square. The diameter of this boundary box,
+\latexonly{$p_d$,}\latexignore{\emph{Pd},} \rtfsp is then
+\latexignore{\\$$\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}
 The \emph{view diameter} is the area that is being processed
@@ -165,7 +164,7 @@ 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
@@ -186,10 +185,9 @@ 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
-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}
@@ -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.
-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}
 
 
@@ -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}
-((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}
 
 
@@ -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
-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.
@@ -279,20 +278,20 @@ covering an angular distance of
 circle in
 \begin{figure}
 \image{10cm;0cm}{equiangular.eps}
-\caption{Equiangluar geometry}
+\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
 \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}
-\caption{Equilinear geometry}
+\caption{\label{equilinearfig} Equilinear geometry}
 \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}
@@ -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.
+
+\section{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}
+}