X-Git-Url: http://git.kpe.io/?p=ctsim.git;a=blobdiff_plain;f=doc%2Fctsim-concepts.tex;h=ddf6bed819ad82cc4516a7f67239494adfef9976;hp=81dbf42b2da366baf1252f0b94369996c0d71fe4;hb=d3fa225aa232e132cc198672c4fc148f96a1ab8c;hpb=eb4b8ecaf864329867c9d68c5911d2a2673d8a04 diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex index 81dbf42..ddf6bed 100644 --- a/doc/ctsim-concepts.tex +++ b/doc/ctsim-concepts.tex @@ -1,14 +1,15 @@ \chapter{Concepts}\index{Concepts}% \setheader{{\it CHAPTER \thechapter}}{}{}{}{}{{\it CHAPTER \thechapter}}% -\setfooter{\thepage}{}{}{}{}{\thepage}% +\setfooter{\thepage}{}{}{}{\small Version 0.2}{\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. +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 @@ -18,14 +19,15 @@ object, but the two objects we need to be concerned with are the \section{Phantoms}\label{conceptphantom}\index{Concepts,Phantoms}% \subsection{Overview}\label{phantomoverview}\index{Concepts,Phantoms,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}. @@ -36,7 +38,7 @@ 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 +The first entry defines the type of the element, either \rtfsp\texttt{rectangle}, \texttt{}, \texttt{triangle}, \rtfsp\texttt{sector}, or \texttt{segment}. \texttt{cx}, \rtfsp\texttt{cy}, \texttt{dx} and \texttt{dy} have different @@ -138,15 +140,16 @@ square 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}$$} +The \emph{view diameter} is the area that is being processed +during scanning of phantoms as well as during rasterization of +phantoms. By default, the \emph{view diameter} \rtfsp is set equal +to the \emph{phantom diameter}. It may be useful, especially for +experimental reasons, to process an area larger (and maybe even +smaller) than the phantom. Thus, during rasterization or during +projections, \ctsim\ will ask for a \emph{view ratio}, +\latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.} The \emph{view +diameter} is then calculated as \latexonly{$$v_d = p_d +v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$} By using a \latexonly{$v_r$}\latexignore{\emph{VR}} @@ -158,17 +161,17 @@ be impossible and is analagous to inserting an object into the CT scanner that is larger than the scanner itself! \subsubsection{Scan Diameter} -By default, the entire \emph{view diameter} is scanned. For experimental -purposes, it may be desirable to scan an area either larger or smaller than -the \emph{view diameter}. Thus, the concept of \emph{scan ratio} -\latexonly{$s_r$}\latexignore{\emph{SR}} +By default, the entire \emph{view diameter} is scanned. For +experimental purposes, it may be desirable to scan an area either +larger or smaller than the \emph{view diameter}. Thus, the concept +of \emph{scan ratio}, \latexonly{$s_r$,}\latexignore{\emph{SR},} is born. The scan diameter -\latexonly{$s_d$}\latexignore{\emph{Sd}} -is the diameter over which x-rays are collected and is defined as -\latexonly{$$s_d = v_d s_r$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\} -By default and for all ordinary scanning, the \emph{scan ratio} is to -\texttt{1}. If the \emph{scan ratio} is less than \texttt{1}, -you can expect significant artifacts. +\latexonly{$s_d$}\latexignore{\emph{Sd}} is the diameter over +which x-rays are collected and is defined as \latexonly{$$s_d = +v_d s_r$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\} By default and +for all ordinary scanning, the \emph{scan ratio} is to \texttt{1}. +If the \emph{scan ratio} is less than \texttt{1}, you can expect +significant artifacts. \subsubsection{Focal Length} The \emph{focal length}, @@ -180,9 +183,13 @@ of the view radius. Focal length is calculated as \latexonly{$$f = (v_d / 2) f_r$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$} -For parallel geometry scanning, the focal length doesn't matter. However, -divergent geometry scanning (equilinear and equiangular), the \emph{focal -length ratio} should be set at \texttt{2} or more to avoid artifacts. +For parallel geometry scanning, the focal length doesn't matter. +However, divergent geometry scanning (equilinear and equiangular), +the \emph{focal length ratio} should be set at \texttt{2} or more +to avoid artifacts. Moreover, a value of less than \texttt{1}, +though it can be given to \ctsim, is physically impossible and it +analagous to have having the x-ray source with the \emph{view +diameter}. \subsection{Parallel Geometry}\label{geometryparallel}\index{Concepts,Scanner,Geometries,Parallel} @@ -213,13 +220,14 @@ See figure 2.2. \subsubsection{Fan Beam Angle} -For these divergent beam geometries, the \emph{fan beam angle} needs -to be calculated. For real-world CT scanners, this is fixed at the -time of manufacture. \ctsim, however, calculates the \emph{fan beam angle}, -$\alpha$ from the \emph{scan diameter} and the \emph{focal length} -\latexignore{\\$$\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)}$$\\} +\latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1} +((s_d / 2) / f)\end{equation}} This is illustrated in figure 2.3. \begin{figure} \image{10cm;0cm}{alphacalc.eps} \caption{Calculation of $\alpha$} @@ -236,37 +244,39 @@ there are significant artifacts. The primary way to manage the To illustrate, the \emph{scan diameter} can be defined as \latexonly{$$s_d = s_r v_r p_d$$}\latexignore{\\$$Sd = Sr x Vr x Pd$$\\} -Further, $f$ can be defined as -\latexonly{$$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)$$\\} + +Substituting these equations into \latexignore{the above +equation,}\latexonly{equation~\ref{alphacalc},} We have, \latexonly{ \begin{eqnarray} \alpha &= 2\,\sin^{-1} \frac{s_r v_r p_d / 2}{f_r v_r (p_d / 2)} \nonumber \\ &= 2\,\sin^{-1} (s_r / f_r) \end{eqnarray} -} +} \latexignore{\\$$\alpha = 2 sin (Sr / Fr$$\\} -Since in normal scanning $s_r = 1$, $\alpha$ depends only upon the \emph{focal length ratio}. +Since in normal scanning $s_r$ = 1, $\alpha$ depends only upon the +\emph{focal length ratio}. \subsubsection{Detector Array Size} -In general, you do not need to be concerned with the detector array -size. It is automatically calculated by \ctsim. +In general, you do not need to be concerned with the detector +array size. It is automatically calculated by \ctsim. For those +interested, this section explains how the detector array size is +calculated. For parallel geometry, the detector length is equal to the scan diameter. -For divergent beam geometrys, the size of the -detector array also depends upon the \emph{focal length}. -Increasing the \emph{focal length} -decreases the size of the detector array while increasing the \emph{scan -diameter} increases the detector array size. +For divergent beam geometries, the size of the detector array also +depends upon the \emph{focal length}. Increasing the \emph{focal +length} decreases the size of the detector array while increasing +the \emph{scan diameter} increases the detector array size. -For equiangular geometry, the detectors are spaced around a -circle covering an angular distance of -\latexonly{$\alpha$.}\latexignore{\emph{alpha}.} -The dotted circle in +For equiangular geometry, the detectors are spaced around a circle +covering an angular distance of +\latexonly{$2\,\alpha$.}\latexignore{\emph{2 \alpha}.} The dotted +circle in \begin{figure} \image{10cm;0cm}{equiangular.eps} \caption{Equiangluar geometry} @@ -275,10 +285,9 @@ figure 2.4 indicates the positions of the detectors in this case. For equilinear geometry, the detectors are space along a straight line. The length of the line depends upon -\latexonly{$\alpha$}\latexignore{\emph{alpha}} -and the \emph{focal length}. It is calculated as -\latexonly{$$\mathrm{detLengh} = 4\,f \tan (\alpha / 2)$$} -\latexignore{\\$$\emph{detLength} = 4 x F x tan(alpha/2)$$\\} +\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} \image{10cm;0cm}{equilinear.eps} \caption{Equilinear geometry} @@ -320,12 +329,12 @@ 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 -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} -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.