\chapter{Concepts}\index{Concepts}%
-\setheader{{\it CHAPTER \thechapter}}{}{}{}{}{{\it CHAPTER \thechapter}}%
-\setfooter{\thepage}{}{}{}{}{\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.
+\setheader{{\it CHAPTER \thechapter}}{}{}{\ctsimheadtitle}{}{{\it CHAPTER \thechapter}}%
+\ctsimfooter%
+
+\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.
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}.
-\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 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}.
-\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, one of
-\rtfsp\texttt{rectangle}, \texttt{}, \texttt{triangle},
+The first entry defines the type of the element, either
+\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.
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
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
\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}$$}
-CT scanners actually collect projections around a circle rather than a
-square. The diameter of this circle is also the diameter of the boundary
-square
-\latexonly{$p_d$.}\latexignore{\rtfsp\emph{Pd}.}
-These relationships are diagrammed in figure 2.1.
-
-\subsubsection{View Diameter}
-The \emph{view diameter} is the area that is being processed during scanning of phantoms as
-well as during rasterization of phantoms. By default, the \emph{view diameter}
-\rtfsp is set equal to the \emph{phantom diameter}. It may be useful, especially for
-experimental reasons, to process an area larger (and maybe even smaller) than
-the phantom. Thus, during rasterization or during projections, \ctsim\ will
-ask for a \emph{view ratio},
-\latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.}
-The \emph{view diameter} is then set as
-\latexonly{$$v_d = p_d v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
+The phantom diameter is automatically calculated by \ctsim\ from
+the phantom definition. The maximum of the phantom length and
+height is used to define the square that completely surrounds the
+phantom. Let \latexonly{$p_l$}\latexignore{\emph{Pl}} be the width
+and height of this square. The diameter of this boundary box,
+\latexonly{$p_d$,}\latexignore{\emph{Pd},} \rtfsp is then
+\latexignore{\\$$\emph{Pl x sqrt(2)}$$\\} \latexonly{$$p_d = p_l
+\sqrt{2}$$} CT scanners actually collect projections around a
+circle rather than a square. The diameter of this circle is also
+the diameter of the boundary square
+\latexonly{$p_d$. These
+relationships are diagrammed in figure~\ref{phantomgeomfig}.}
+\latexignore{emph{Pd}.}
+
+\subsubsection{View Diameter}\index{View diameter}
+The \emph{view diameter} is the area that is being processed
+during scanning of phantoms as well as during rasterization of
+phantoms. By default, the \emph{view diameter} \rtfsp is set equal
+to the \emph{phantom diameter}. It may be useful, especially for
+experimental reasons, to process an area larger (and maybe even
+smaller) than the phantom. Thus, during rasterization or during
+projections, \ctsim\ will ask for a \emph{view ratio},
+\latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.} The \emph{view
+diameter} is then calculated as \latexonly{$$v_d = p_d
+v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
By using a
\latexonly{$v_r$}\latexignore{\emph{VR}}
be impossible and is analagous to inserting an object into the CT
scanner that is larger than the scanner itself!
-\subsubsection{Scan Diameter}
-By default, the entire \emph{view diameter} is scanned. For experimental
-purposes, it may be desirable to scan an area either larger or smaller than
-the \emph{view diameter}. Thus, the concept of \emph{scan ratio}
-\latexonly{$s_r$}\latexignore{\emph{SR}}
-is born. The scan diameter
-\latexonly{$s_d$}\latexignore{\emph{Sd}}
-is the diameter over which x-rays are collected and is defined as
-\latexonly{$$s_d = v_d s_r$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\}
-By default and for all ordinary scanning, the \emph{scan ratio} is to
-\texttt{1}. If the \emph{scan ratio} is less than \texttt{1},
-you can expect significant artifacts.
-
-\subsubsection{Focal Length}
+\subsubsection{Scan Diameter}\index{Scan diameter}
+By default, the entire \emph{view diameter} is scanned. For
+experimental purposes, it may be desirable to scan an area either
+larger or smaller than the \emph{view diameter}. Thus, the concept
+of \emph{scan ratio}, \latexonly{$s_r$,}\latexignore{\emph{SR},}
+is arises. The scan diameter
+\latexonly{$s_d$}\latexignore{\emph{Sd}} is the diameter over
+which x-rays are collected and is defined as \latexonly{$$s_d =
+v_d s_r$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\} By default and
+for all ordinary scanning, the \emph{scan ratio} is to \texttt{1}.
+If the \emph{scan ratio} is less than \texttt{1}, you can expect
+significant artifacts.
+
+\subsubsection{Focal Length}\index{Focal length}
The \emph{focal length},
\latexonly{$f$,}\latexignore{\emph{F},}
is the distance of the X-ray source to the center of
calculated as
\latexonly{$$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} 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
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,
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}
\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)}$$\\}
+\latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1}
+((s_d / 2) / f)\end{equation}
+ This is illustrated in figure~\ref{alphacalcfig}.}
\begin{figure}
\image{10cm;0cm}{alphacalc.eps}
-\caption{Calculation of $\alpha$}
+\caption{\label{alphacalcfig} Calculation of $\alpha$}
\end{figure}
To illustrate, the \emph{scan diameter} can be defined as
\latexonly{$$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 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 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}
+\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{$$\mathrm{detLengh} = 4\,f \tan (\alpha / 2)$$}
-\latexignore{\\$$\emph{detLength} = 4 x F x tan(alpha/2)$$\\}
-\begin{figure}
+\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}\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{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
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.
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.
+
+\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}
+}
projects / ctsim.git / blobdiff