-\chapter{Concepts}\index{Concepts}%
-\setheader{{\it CHAPTER \thechapter}}{}{}{\ctsimheadtitle}{}{{\it CHAPTER \thechapter}}%
-\ctsimfooter%
+\chapter{Concepts}
+\setheader{{\it CHAPTER \thechapter}}{}{}{\ctsimheadtitle}{}{{\it CHAPTER \thechapter}}
+\ctsimfooter
-\section{Overview}\label{conceptoverview}\index{Conceptual Overview}%
+\section{Conceptual Overview}\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
+simulated. This projection data can be reconstructed using various
user-controlled algorithms producing an image of the phantom
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 primary objects that we need to be
-concerned with are the \emph{phantom} and the \emph{scanner}.
+concerned with are the \helprefn{phantom}{conceptphantom} and the
+\helprefn{scanner}{conceptscanner}.
\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.
+scanned. A phantom is composed of one or more phantom elements.
These elements are simple geometric shapes, specifically,
rectangles, triangles, ellipses, sectors and segments. With these
elements, the standard phantoms used in the CT literature can be
user-defined phantoms.
The types of phantom elements and their definitions are taken with
-permission from G.T. Herman's 1980 book\cite{HERMAN80}.
+permission from G.T. Herman's publication\cite{HERMAN80}.
-\subsection{Phantom File}\label{phantomfile}\index{Phantom file syntax}
+\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{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.
+\texttt{rectangle}, \texttt{ellipse}, \texttt{triangle},
+\texttt{sector}, or \texttt{segment}.
-\rtfsp\texttt{r} is the rotation applied to the object in degrees
-counterclockwise, and \texttt{a} is the X-ray attenuation
+For all phantom elements, \texttt{r} is the rotation applied to the object in degrees
+counterclockwise and \texttt{a} is the X-ray attenuation
coefficient of the object. Where objects overlap, the attenuations
of the overlapped objects are summed.
+As opposed to the \texttt{r} and \texttt{a} fields, the \texttt{cx},
+\texttt{cy}, \texttt{dx} and \texttt{dy} fields have different
+meanings depending on the element type.
+
-\subsection{Phantom Elements}\label{phantomelements}\index{Phantom elements}
+
+\subsection{Phantom Elements}\label{phantomelements}\index{Phantom!Elements}
\subsubsection{ellipse}
Ellipses use \texttt{dx} and \texttt{dy} to define the semi-major and
-semi-minor axis lengths, with the center of the ellipse at \texttt{cx}
-and \texttt{cy}. Of note, the commonly used phantom described by
+semi-minor axis lengths with the center of the ellipse at \texttt{(cx,cy)}.
+Of note, the commonly used phantom described by
Shepp and Logan\cite{SHEPP74} uses only ellipses.
\subsubsection{rectangle}
-Rectangles use \texttt{cx} and \texttt{cy} to define the position of
+Rectangles use \texttt{(cx,cy)} to define the position of
the center of the rectangle with respect to the origin. \texttt{dx}
and \texttt{dy} are the half-width and half-height of the rectangle.
\subsubsection{triangle}
-Triangles are drawn with the center of the base at \texttt{(cx,cy)},
-with a base half-width of \texttt{dx} and a height of \texttt{dy}.
+Triangles are drawn with the center of the base at \texttt{(cx,cy)}
+and a base half-width of \texttt{dx} and a height of \texttt{dy}.
Rotations are then applied about the center of the base.
\subsubsection{segment}
below the x-axis. The sector is then rotated and translated the
same as a segment.
-\subsection{Phantom Size}\index{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.
+defined as a rectangle of size 0.1 by 0.1, the phantom size is
+0.101 in each direction.
-\section{Scanner}\label{conceptscanner}\index{Scanner concepts}%
-\subsection{Dimensions}
+\section{Scanner}\label{conceptscanner}\index{Scanner!Concepts}%
Understanding the scanning geometry is the most complicated aspect of
using \ctsim. For real-world CT simulators, this is actually quite
simple. The geometry is fixed by the manufacturer during the
-construction of the scanner and can not be changed. Conversely,
-real-world CT scanners can only take objects up to a fixed size.
+construction of the scanner and can not be changed. \ctsim,
+being a very flexible simulator, gives tremendous options in
+setting up the geometry for a scan.
-\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 with the size of
+\subsection{Dimensions}
+The geometry for a scan 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\%.
+above, the phantom dimensions are padded by 1\%.
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
+the \emph{view diameter}, \emph{scan diameter}, \emph{focal
+length}, and \emph{center-detector length}. These variables are input into \ctsim\ in terms of
ratios rather than absolute values.
-\subsubsection{Phantom Diameter}\index{Phantom diameter}
-\begin{figure}
-$$\image{5cm;0cm}{scangeometry.eps}$$
-\caption{\label{phantomgeomfig} Phantom Geometry}
-\end{figure}
+\subsubsection{Phantom Diameter}\index{Phantom!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
+\latexonly{$p_d$,}\latexignore{\emph{Pd},} is given by the
+Pythagorean theorem and is
\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}.}
+CT scanners collect projections around a
+circle rather than a square. The diameter of this circle is
+the diameter of the boundary square \latexonly{$p_d$.}
+\latexignore{\emph{Pd}.}
+\latexonly{These relationships are diagrammed in figure~\ref{phantomgeomfig}.}
+\begin{figure}
+\centerline{\image{8cm;0cm}{scangeometry.eps}}
+\latexonly{\caption{\label{phantomgeomfig} Phantom Geometry}}
+\end{figure}
\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
+phantoms. By default, the \emph{view diameter} 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
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
+arises. The scan diameter,
+\latexonly{$s_d$,}\latexignore{\emph{Sd},} is the diameter over
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}}\\}
\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),
+However, for 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
+physically impossible and it analagous to having the x-ray
source inside of the \emph{view diameter}.
+\subsubsection{Center-Detector Length}\index{Center-Detector length}
+The \emph{center-detector length},
+\latexonly{$c$,}\latexignore{\emph{C},}
+is the distance from the center of
+the phantom to the center of the detector array. The center-detector length is set as a ratio,
+\latexonly{$c_r$,}\latexignore{\emph{CR},}
+of the view radius. The center-detector length is
+calculated as
+\latexonly{\begin{equation}f = (v_d / 2) c_r\end{equation}}
+\latexignore{\\\centerline{\emph{F = (Vd / 2) x CR}}}
+
+For parallel geometry scanning, the center-detector length doesn't matter.
+A value of less than \texttt{1} is physically impossible and it analagous to
+having the detector array inside of the \emph{view diameter}.
-\subsection{Parallel Geometry}\label{geometryparallel}\index{Parallel Geometry}
-The simplest geometry, parallel, was used in \mbox{$1^{st}$} generation
+\subsection{Parallel Geometry}\label{geometryparallel}\index{Parallel geometry}\index{Scanner!Parallel}
+
+The simplest geometry, parallel, was used in first 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
significant distortions will occur.
-\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,
+\subsection{Divergent Geometries}\label{geometrydivergent}\index{Equilinear geometry}\index{Equiangular geometry}
+\index{Scanner!Equilinear}\index{Scanner!Equiangular}
+For both equilinear (second generation) and equiangular
+(third, fourth, and fifth generation) geometries,
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.
-\latexonly{The configurations are shown in figure~\ref{divergentfig}.}
+\latexonly{These configurations are shown in figure~\ref{divergentfig}.}
\begin{figure}
-\image{10cm;0cm}{divergent.eps}
-\caption{\label{divergentfig} Equilinear and equiangular geometries.}
+\centerline{\image{10cm;0cm}{divergent.eps}}
+\latexonly{\caption{\label{divergentfig} Equilinear and equiangular geometries.}}
\end{figure}
-\subsubsection{Fan Beam Angle}
+\subsubsection{Fan Beam Angle}\index{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)}$$\\}
+the \emph{focal length} as
+\latexignore{\centerline{\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}.}
+((s_d / 2) / f)\end{equation}}
+\latexonly{This is illustrated in figure~\ref{alphacalcfig}.}
\begin{figure}
-\image{10cm;0cm}{alphacalc.eps}
-\caption{\label{alphacalcfig} Calculation of $\alpha$}
+\centerline{\image{10cm;0cm}{alphacalc.eps}}
+\latexonly{\caption{\label{alphacalcfig} Calculation of $\alpha$}}
\end{figure}
\latexonly{$120^\circ$,}\latexignore{120 degrees,}
there are significant artifacts. The primary way to manage the
\emph{fan beam angle} is by varying the \emph{focal length} since the
-\emph{scan diameter} by the size of the phantom.
+\emph{scan diameter} is usually fixed at the size of the phantom.
To illustrate, the \emph{scan diameter} can be defined as
\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{\[f = f_r (v_r p_d / 2)\]}
+Further, the \emph{focal length} can be defined as
+\latexonly{\begin{equation} f = f_r (v_r p_d / 2)\end{equation}}
\latexignore{\\\centerline{\emph{F = FR x (VR x Pd)$$\\}}}
Substituting these equations into \latexignore{the above
\subsubsection{Detector Array Size}
In general, you do not need to be concerned with the detector
-array size. It is automatically calculated by \ctsim. For the
+array size -- it is automatically calculated by \ctsim. For the
particularly interested, this section explains how the detector
array size is calculated.
-For parallel geometry, the detector length is equal to the scan
+For parallel geometry, the detector length is simply the scan
diameter.
For divergent beam geometries, the size of the detector array also
-depends upon the \emph{focal length}. Increasing the \emph{focal
-length} decreases the size of the detector array while increasing
-the \emph{scan diameter} increases the detector array size.
-
-For equiangular geometry, the detectors are spaced around a circle
-covering an angular distance of
-\latexonly{$2\,\alpha$.}\latexignore{\emph{2 \alpha}.} The dotted
-circle in
+depends upon the \emph{focal length}: increasing the \emph{focal
+length} decreases the size of the detector array.
+
+For equiangular geometry, the detectors are equally spaced around a arc
+covering an angular distance of $\alpha$ as viewed from the source. When
+viewed from the center of the scanning, the angular distance is
+\latexonly{$$\pi + \alpha - 2 \, \cos^{-1} \Big( \frac{s_d / 2}{c} \Big)$$}
+\latexignore{\\\emph{pi + \alpha - 2 x acos ((Sd / 2) / C))}\\}
+The dotted circle
+\latexonly{in figure~\ref{equiangularfig}}
+indicates the positions of the detectors in this case.
+
\begin{figure}
-\image{10cm;0cm}{equiangular.eps}
-\caption{\label{equiangularfig}Equiangular geometry}
+\centerline{\image{10cm;0cm}{equiangular.eps}}
+\latexonly{\caption{\label{equiangularfig}Equiangular geometry}}
\end{figure}
-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
+For equilinear geometry, the detectors are equally spaced along a straight
+line. The detector length depends upon
\latexonly{$\alpha$}\latexignore{\emph{alpha}} and the \emph{focal
-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{\label{equilinearfig} Equilinear geometry}
-\end{figure}
+length}. This length,
+\latexonly{$d_l$,}\latexignore{Dl,} is calculated as
+\latexonly{\begin{equation} d_l = 2\,(f + c) \tan (\alpha / 2)\end{equation}}
+\latexignore{\\\centerline{\emph{2 x (F + C) x tan(\alpha/2)}}}
\latexonly{This geometry is shown in figure~\ref{equilinearfig}.}
+\begin{figure}
+\centerline{\image{10cm;0cm}{equilinear.eps}}
+\latexonly{\caption{\label{equilinearfig} Equilinear geometry}}
+\end{figure}
-\section{Reconstruction}\label{conceptreconstruction}\index{Reconstruction Overview}%
+\section{Reconstruction}\label{conceptreconstruction}\index{Reconstruction overview}%
\subsection{Direct Inverse Fourier}
-This method is not currently implemented in \ctsim, however it is
+This method is not currently implemented in \ctsim; however, it is
planned for a future release. This method does not give results as
-accurate as filtered backprojection. The difference is due primarily
-because interpolation occurs in the frequency domain rather than the
+accurate as filtered backprojection. This is due primarily
+to interpolation occurring in the frequency domain rather than the
spatial domain.
-\subsection{Filtered Backprojection}\index{Filtered backprojection}
+\subsection{Filtered Backprojection}\index{Filtered backprojection}\index{Symmetric multiprocessing}\index{SMP}
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.
-
-\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 excellent
+filtering projections followed by backprojecting the filtered projections. Though
+these two steps are sequential, each view position can be processed independently.
+
+\subsubsection{Parallel Computer Processing}\index{Parallel processing}
+Since each view can be processed independently, filtered backprojection is amendable to
+parallel processing. Indeed, this has been used in commercial scanners to speed reconstruction.
+This parallelism is exploited both in the \ctsim\ graphical shell and
+in the \helpref{LAM}{ctsimtextlam} version of \ctsimtext. \ctsim\ can distribute it's workload
+amongst multiple processors working in parallel.
+
+The graphical shell will automatically take advantage of multiple CPU's when
+running on a \emph{Symmetric Multiprocessing}
+computer. Dual-CPU computers are commonly available which provide a near doubling
+in reconstruction speeds. \ctsim, though, has no limits on the number of CPU's
+that can be used in parallel. The \emph{LAM} version
+of \ctsimtext\ is designed to work in a cluster of computers.
+This has been testing with a cluster of 16 computers in a
+\urlref{Beowulf-class}{http://www.beowulf.org} cluster with excellent
results.
\subsubsection{Filter projections}
projections over the reconstructing image. Various levels of
interpolation can be specified.
-\section{Image Comparison}\index{Image comparison}
+\section{Image Comparison}\label{conceptimagecompare}\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}
+
+\begin{itemize}\itemsep=0pt
+\item[]\textbf{$d$}\quad The normalized root mean squared distance measure.
+\item[]\textbf{$r$}\quad The normalized mean absolute distance measure.
+\item[]\textbf{$e$}\quad The worst case distance measure over a \latexonly{$2\times2$}\latexignore{\emph{2 x 2}} pixel area.
+\end{itemize}
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
+image, and $\bar{p}$ denotes the average pixel value of $p$. 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.
%Tex2RTF can not handle the any subscripts or superscripts for the inner summation unless
% have a space character before the \sum
\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{\[\label{requation}r = \frac{ \displaystyle \sum_{i=1}^{n}{ \sum_{j=1}^{m}{|p_{i,j} - r_{i,j}|}}}{ \displaystyle \sum_{i=1}^{n}{ \sum_{j=1}^{m}{|p_{i,j}|}}}\]}
+\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{\[\label{bigpequation}P_{k,l} = \textstyle \frac{1}{4} (p_{2k,2l} + p_{2k+1,2l} + p_{2k,2l+l} + p_{2k+1,2l+1})\]}
+\latexonly{\begin{equation}\label{bigpequation}P_{k,l} = \textstyle \frac{1}{4} (p_{2k,2l} + p_{2k+1,2l} + p_{2k,2l+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}}
+\begin{comment}
+\end{comment}
projects / ctsim.git / blobdiff