X-Git-Url: http://git.kpe.io/?a=blobdiff_plain;f=doc%2Fctsim-concepts.tex;h=e76a77f6c54cf4f9221c6661bff3466cea0eb572;hb=82ea0c94394a5a175b260160760155a6686203a1;hp=ddf6bed819ad82cc4516a7f67239494adfef9976;hpb=d3fa225aa232e132cc198672c4fc148f96a1ab8c;p=ctsim.git

diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex
index ddf6bed..e76a77f 100644
--- a/doc/ctsim-concepts.tex
+++ b/doc/ctsim-concepts.tex
@@ -1,29 +1,29 @@
 \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}%
+\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.
+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 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 primary objects that 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
+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
@@ -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
@@ -70,32 +70,32 @@ 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}
-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}
+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 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
@@ -106,40 +106,39 @@ 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\%.
+In general, the geometry for a scan all 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 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}$$}
-CT scanners actually collect projections around a circle rather than a
-square. The diameter of this circle is also the diameter of the boundary
-square
-\latexonly{$p_d$.}\latexignore{\rtfsp\emph{Pd}.}
-These relationships are diagrammed in figure 2.1.
-
-\subsubsection{View Diameter}
+The phantom diameter is automatically calculated by \ctsim\ from
+the phantom definition. The maximum of the phantom length and
+height is used to define the square that completely surrounds the
+phantom. Let \latexonly{$p_l$}\latexignore{\emph{Pl}} be the width
+and height of this square. The diameter of this boundary box,
+\latexonly{$p_d$,}\latexignore{\emph{Pd},} is then
+\latexignore{\\\centerline{\emph{Pl x sqrt(2)}}\\}
+\latexonly{\begin{equation}p_d = p_l \sqrt{2}\end{equation}}
+CT scanners actually collect projections around a
+circle rather than a square. The diameter of this circle is also
+the diameter of the boundary square \latexonly{$p_d$.  These
+relationships are diagrammed in figure~\ref{phantomgeomfig}.}
+\latexignore{emph{Pd}.}
+
+\subsubsection{View Diameter}\index{View diameter}
 The \emph{view diameter} is the area that is being processed
 during scanning of phantoms as well as during rasterization of
 phantoms. By default, the \emph{view diameter} \rtfsp is set equal
@@ -148,8 +147,9 @@ experimental reasons, to process an area larger (and maybe even
 smaller) than the phantom. Thus, during rasterization or during
 projections, \ctsim\ will ask for a \emph{view ratio},
 \latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.} The \emph{view
-diameter} is then calculated as \latexonly{$$v_d = p_d
-v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
+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}}
@@ -157,23 +157,25 @@ less than 1, \ctsim\ will allow
 for a \emph{view diameter} less than
 \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!
 
-\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
+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}
+\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
@@ -181,20 +183,21 @@ the phantom. The focal length is set as a ratio,
 \latexonly{$f_r$,}\latexignore{\emph{FR},}
 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, 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
+The simplest geometry, parallel, was used in \mbox{$1^{st}$} 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
@@ -204,7 +207,7 @@ 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,
@@ -212,10 +215,10 @@ 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{The configurations are shown in 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 +230,11 @@ 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}
 
 
@@ -242,28 +246,30 @@ there are significant artifacts. The primary way to manage the
 \emph{scan diameter} by the size of the phantom.
 
 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$$\\}
+\latexonly{\begin{equation}s_d = s_r v_r p_d\end{equation}}
+\latexignore{\\\centerline{\emph{Sd = Sr x Vr x Pd}}\\}
 
-Further, $f$ can be defined as \latexonly{$$f = f_r (v_r p_d /
-2)$$}\latexignore{\\$$F = FR x (VR x Pd)$$\\}
+Further, $f$ can be defined as
+\latexonly{\[f = f_r (v_r p_d / 2)\]}
+\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{s_r v_r p_d / 2}{f_r v_r (p_d / 2)} \nonumber \\
-&= 2\,\sin^{-1} (s_r / f_r)
+\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{\\$$\alpha = 2 sin (Sr / Fr$$\\}
+} \latexignore{\\\centerline{\emph{\alpha = 2 sin (Sr / Fr)}}\\}
 
 Since in normal scanning $s_r$ = 1, $\alpha$ depends only upon the
-\emph{focal length ratio}.
+\emph{focal length ratio} in normal scanning.
 
 \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.
@@ -279,27 +285,25 @@ 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}
+length}. It is calculated as
+\latexonly{\begin{equation}4\,f \tan (\alpha / 2)\end{equation}}
+\latexignore{\\\centerline{\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}
-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}
 
+\section{Reconstruction}\label{conceptreconstruction}\index{Reconstruction Overview}%
 
-\section{Reconstruction}\label{conceptreconstruction}\index{Concepts,Reconstruction}%
-\subsection{Overview}
 \subsection{Direct Inverse Fourier}
 This method is not currently implemented in \ctsim, however it is
 planned for a future release. This method does not give results as
@@ -307,7 +311,7 @@ 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.
@@ -315,11 +319,12 @@ these two steps are sequential, each view position can be processed individually
 \subsubsection{Multiple Computer Processing}
 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
+processors. This has been testing in a 16-CPU cluster with excellent
 results.
 
 \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
 filtering.
 
@@ -338,3 +343,27 @@ 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}. They are:
+\begin{description}
+\item[$d$] The normalized root mean squared distance measure.
+\item[$r$] The normalized mean absolute distance measure.
+\item[$e$] The worst case distance measure over a $2\times2$ area.
+\end{description}
+
+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 for $p$. 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.}
+\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,2j+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}}