\setheader{{\it CHAPTER \thechapter}}{}{}{\ctsimheadtitle}{}{{\it CHAPTER \thechapter}}
\ctsimfooter
-\section{Overview}\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
\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 of one or more phantom elements.
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)
+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.
0.101 in each direction.
\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
simple. The geometry is fixed by the manufacturer during the
being a very flexible simulator, gives tremendous options in
setting up the geometry for a scan.
+\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
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 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}
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}\index{Scanner!Parallel}
\subsection{Divergent Geometries}\label{geometrydivergent}\index{Equilinear geometry}\index{Equiangular geometry}
\index{Scanner!Equilinear}\index{Scanner!Equiangular}
-\subsubsection{Overview}
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.
length} decreases the size of the detector array.
For equiangular geometry, the detectors are equally spaced around a arc
-covering an angular distance of
-\latexonly{$2\,\alpha$.}\latexignore{\emph{2 \alpha}.}
+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.
\latexonly{$\alpha$}\latexignore{\emph{alpha}} and the \emph{focal
length}. This length,
\latexonly{$d_l$,}\latexignore{Dl,} is calculated as
-\latexonly{\begin{equation} d_l = 4\,f \tan (\alpha / 2)\end{equation}}
-\latexignore{\\\centerline{\emph{4 x F x tan(\alpha/2)}}}
+\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}\label{equilinearfig}
+\begin{figure}
\centerline{\image{10cm;0cm}{equilinear.eps}}
\latexonly{\caption{\label{equilinearfig} Equilinear geometry}}
\end{figure}
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 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 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 cluster of 16 computers with excellent
+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}
Herman\cite{HERMAN80}. They are:
\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{twocollist}
+\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