\setheader{{\it CHAPTER \thechapter}}{}{}{\ctsimheadtitle}{}{{\it CHAPTER \thechapter}}
\ctsimfooter
-\section{Overview}\index{Conceptual Overview}
+\section{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
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}
Each line in the text file describes an element of the
\end{verbatim}
The first entry defines the type of the element, either
\texttt{rectangle}, \texttt{ellipse}, \texttt{triangle},
-\texttt{sector}, or \texttt{segment}. \texttt{cx},
-\texttt{cy}, \texttt{dx} and \texttt{dy} have different
-meanings depending on the element type.
+\texttt{sector}, or \texttt{segment}.
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}
\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}
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
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,
+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{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
\subsection{Parallel Geometry}\label{geometryparallel}\index{Parallel geometry}\index{Scanner!Parallel}
-The simplest geometry, parallel, was used in \mbox{$1^{st}$} generation
+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
\subsection{Divergent Geometries}\label{geometrydivergent}\index{Equilinear geometry}\index{Equiangular geometry}
\index{Scanner!Equilinear}\index{Scanner!Equiangular}
\subsubsection{Overview}
-Next consider the case of equilinear (second generation) and equiangular
-(third, fourth, and fifth generation) geometries. In these cases,
+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
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}:
+the \emph{focal length} as
\latexignore{\centerline{\emph{alpha = 2 x asin (
(Sd / 2) / f)}}}
\latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1}
\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{\begin{equation} = f_r (v_r p_d / 2)\end{equation}}
+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
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.
+depends upon the \emph{focal length}: increasing the \emph{focal
+length} decreases the size of the detector array.
-For equiangular geometry, the detectors are spaced around a circle
+For equiangular geometry, the detectors are equally spaced around a arc
covering an angular distance of
\latexonly{$2\,\alpha$.}\latexignore{\emph{2 \alpha}.}
The dotted circle
\latexonly{\caption{\label{equiangularfig}Equiangular geometry}}
\end{figure}
-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}}
+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{This geometry is shown in figure~\ref{equilinearfig}.}
\begin{figure}\label{equilinearfig}
\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
planned for a future release. This method does not give results as
accurate as filtered backprojection. This is due primarily
-because interpolation occurs in the frequency domain rather than the
+to interpolation occurring in the frequency domain rather than the
spatial domain.
\subsection{Filtered Backprojection}\index{Filtered backprojection}
The technique is comprised of two sequential steps:
-filtering projections followed backprojecting the filtered projections. Though
+filtering projections followed by backprojecting the filtered projections. Though
these two steps are sequential, each view position can be processed independently.
-\subsubsection{Multiple Computer Processing}
+\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 16-CPU cluster with excellent
+processors. This has been testing in a cluster of 16 computers with excellent
results.
\subsubsection{Filter projections}
projects / ctsim.git / blobdiff