\setfooter{\thepage}{}{}{}{}{\thepage}%
\section{Overview}\label{conceptoverview}\index{Concepts,Overview}%
-The operation of \ctsim begins with the phantom object. A phantom
+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
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
+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}%
published phantoms of Herman and Shepp-Logan. \ctsim\ also reads text
files of user-defined phantoms.
-The types of phantom elements and their definitions are taken from
-Herman's 1980 book\cite{HERMAN80}.
+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}
Each line in the text file describes an element of the
\begin{verbatim}
element-type cx cy dx dy r a
\end{verbatim}
-The first entry defines the type of the element, one
-of {\tt rectangle}, {\tt ellipse}, {\tt triangle}, {\tt sector}, or {\tt segment}.
-{\tt cx}, {\tt cy}, {\tt dx} and {\tt dy} have different meanings depending on the element type.
+The first entry defines the type of the element, one of
+\rtfsp\texttt{rectangle}, \texttt{}, \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.
-{\tt r} is the rotation applied to the object in degrees counterclockwise,
-and {\tt a} is the X-ray attenuation coefficient of the object.
-Where objects overlap, the attenuations of the overlapped objects are summed.
+\rtfsp\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.
\subsection{Phantom Elements}\label{phantomelements}\index{Concepts,Phantoms,Elements}
as a segment.
\subsection{Phantom Size}
-Also note that the overall dimensions of the phantom are increased by
-1\% above the specified sizes to avoid clipping due to round-off
-errors from polygonal sampling. 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.
+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}%
\subsection{Dimensions}
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
+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\%.
-The other important geometry variables for scanning objects 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.
+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.
\subsubsection{Phantom Diameter}
\begin{figure}
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,
+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)}$$\\}
\subsubsection{View Diameter}
The \emph{view diameter} is the area that is being processed during scanning of phantoms as
well as during rasterization of phantoms. By default, the \emph{view diameter}
-\rtfsp is set equal to the \emph{phantom diameter}. It may be useful, especially for
+\rtfsp is set equal to the \emph{phantom diameter}. It may be useful, especially for
experimental reasons, to process an area larger (and maybe even smaller) than
the phantom. Thus, during rasterization or during projections, \ctsim\ will
ask for a \emph{view ratio},
The \emph{view diameter} is then set as
\latexonly{$$v_d = p_d v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
-By using a
+By using a
\latexonly{$v_r$}\latexignore{\emph{VR}}
less than 1, \ctsim\ will allow
-for a \emph{view diameter} less than
+for a \emph{view diameter} less than
\emph{phantom diameter}.
This will lead to significant artifacts. Physically, this would
be impossible and is analagous to inserting an object into the CT
calculated as
\latexonly{$$f = (v_d / 2) f_r$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$}
-For parallel geometry scanning, the focal length doesn't matter. However,
+For parallel geometry scanning, the focal length doesn't matter. However,
divergent geometry scanning (equilinear and equiangular), the \emph{focal
-length ratio} should be set at \texttt{2} or more to avoid artifacts.
+length ratio} should be set at \texttt{2} or more to avoid artifacts.
+
-
\subsection{Parallel Geometry}\label{geometryparallel}\index{Concepts,Scanner,Geometries,Parallel}
As mentioned above, the focal length is not used in this simple
-geometry. The detector array is set to
-be the same size as the \emph{scan diameter}.
-For optimal scanning in this geometry, the \emph{scan diameter} should
-be equal to the \emph{phantom diameter}. This is accomplished by using
-the default values of \texttt{1} for the \emph{view diameter ratio} and
-the \emph{scan diameter ratio}. If values of less than \texttt{1} are
-used for these two variables, significant distortions will occur.
+geometry. The detector array is set to be the same size as the
+\emph{scan diameter}. For optimal scanning in this geometry, the
+\emph{scan diameter} should be equal to the \emph{phantom
+diameter}. This is accomplished by using the default values of
+\texttt{1} for the \emph{view ratio} and the \emph{scan ratio}. If
+values of less than \texttt{1} are used for these two variables,
+significant distortions will occur.
\subsection{Divergent Geometries}\label{geometrydivergent}\index{Concepts,Scanner,Geometries,Divergent}
\subsubsection{Fan Beam Angle}
-For these divergent beam geometries, the \emph{fan beam angle} needs
+For these divergent beam geometries, the \emph{fan beam angle} needs
to be calculated. For real-world CT scanners, this is fixed at the
time of manufacture. \ctsim, however, calculates the \emph{fan beam angle},
-$\alpha$ from the diameter of the \emph{scan diameter} and the \emph{focal length}
+$\alpha$ from the \emph{scan diameter} and the \emph{focal length}
\latexignore{\\$$\emph{alpha = 2 x asin ( (Sd / 2) / f)}$$\\}
\latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1} ((s_d / 2) / f)\end{equation}}
This is illustrated in figure 2.3.
Empiric testing with \ctsim\ shows that for very large \emph{fan beam angles},
-greater than approximately
+greater than approximately
\latexonly{$120^\circ$,}\latexignore{120 degrees,}
there are significant artifacts. The primary way to manage the
\emph{fan beam angle} is by varying the \emph{focal length} since the
\emph{scan diameter} by the size of the phantom.
To illustrate, the \emph{scan diameter} can be defined as
-\latexonly{$$s_d = v_r s_r p_d$$}\latexignore{\\$$Sd = Vr x Sr x Pd$$\\}
+\latexonly{$$s_d = s_r v_r p_d$$}\latexignore{\\$$Sd = Sr x Vr x Pd$$\\}
-If $v_r = 1$ and $s_r = 1$, then $s_d = p_d$. Further, $f = f_r v_r (p_d / 2)$
-Plugging these equations into
+Further, $f$ can be defined as
+\latexonly{$$f = f_r (v_r p_d / 2)$$}
+Plugging these equations into
\latexignore{the above equation,}\latexonly{equation~\ref{alphacalc},}
We have,
\latexonly{
\begin{eqnarray}
-\alpha &= 2\,\sin^{-1} \frac{p_d / 2}{f_r (p_d / 2)} \nonumber \\
-&= 2\,\sin^{-1} (1 / f_r)
+\alpha &= 2\,\sin^{-1} \frac{s_r v_r p_d / 2}{f_r v_r (p_d / 2)} \nonumber \\
+&= 2\,\sin^{-1} (s_r / f_r)
\end{eqnarray}
}
-Thus, $\alpha$ depends only upon the \emph{focal length ratio}.
+Since in normal scanning $s_r = 1$, $\alpha$ depends only upon the \emph{focal length ratio}.
\subsubsection{Detector Array Size}
In general, you do not need to be concerned with the detector array
-size. It is automatically calculated by \ctsim. The size of the
-detector array depends upon the \emph{focal length} and the
-\emph{scan diameter}. In general, increasing the \emph{focal length}
-decreases the size of the detector array and increasing the \emph{scan
+size. It is automatically calculated by \ctsim.
+
+For parallel geometry, the detector length is equal to the scan
+diameter.
+
+For divergent beam geometrys, the size of the
+detector array also depends upon the \emph{focal length}.
+Increasing the \emph{focal length}
+decreases the size of the detector array while increasing the \emph{scan
diameter} increases the detector array size.
For equiangular geometry, the detectors are spaced around a
-circle covering an angular distance of
+circle covering an angular distance of
\latexonly{$\alpha$.}\latexignore{\emph{alpha}.}
The dotted circle in
\begin{figure}
\image{10cm;0cm}{equiangular.eps}
\caption{Equiangluar geometry}
\end{figure}
-figure 2.4 indicates the positions of the detectors in this case.
+figure 2.4 indicates the positions of the detectors in this case.
For equilinear geometry, the detectors are space along a straight
-line. The length of the line depends upon
+line. The length of the line depends upon
\latexonly{$\alpha$}\latexignore{\emph{alpha}}
and the \emph{focal length}. It is calculated as
\latexonly{$$\mathrm{detLengh} = 4\,f \tan (\alpha / 2)$$}
\subsubsection{Examples of Geometry Settings}
-Consider increasing the focal length ratio to two leaving the
-field of view ratio as 1, as in Figure 4. Now the detectors array is
-denser, and the real field of view is closer to that specified, but note
-again that the field of view is not used. Instead, the focal length is
-used to give a distance from the center of the phantom to the source, and
-the detector array is adjusted to give an angular coverage to include the
-whole phantom.
-
\section{Reconstruction}\label{conceptreconstruction}\index{Concepts,Reconstruction}%
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.
-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 results.
+
+\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
+results.
\subsubsection{Filter projections}
The projections for a single view have their frequency data multipled by
a filter of $|w|$. \ctsim\ permits four different ways to accomplish this
-filtering. Two of the methods use convolution of the projection data with the
+filtering.
+
+Two of the methods use convolution of the projection data with the
inverse Fourier transform of $|w|$. The other two methods perform an Fourier
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 multiple
-filters for this purpose.
+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
+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. In general, the trade-off is between quality and execution
+time.
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