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.
+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 objects we need to be concerned with are the
-\emph{phantom} and the \emph{scanner}.
+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}.
\section{Phantoms}\label{conceptphantom}
\subsection{Overview}\label{phantomoverview}\index{Phantom Overview}%
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
+elements, the 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
\subsubsection{segment}
Segments are complex. They are the portion of an circle between a
-chord and the perimeter of the circle. \texttt{dy} sets the radius of
-the circle. Segments start with the center of the chord located at
-\texttt{(0,0)} and the chord horizontal. The half-width of the chord
-is set by \texttt{dx}. The portion of an circle lying below the chord
-is then added. The imaginary center of this circle is located at
-\texttt{(0,-dy)}. The segment is then rotated by \texttt{r} and then
-translated by \texttt{cx,cy}.
+chord and the perimeter of the circle. \texttt{dy} sets the
+radius of the circle. Segments start with the center of the chord
+located at \texttt{(0,0)} and the chord horizontal. The half-width
+of the chord is set by \texttt{dx}. The portion of an circle
+lying below the chord is then added. The imaginary center of this
+circle is located at \texttt{(0,-dy)}. The segment is then rotated
+by \texttt{r} and then translated by \texttt{(cx,cy)}.
\subsubsection{sector}
-Sectors are the like a ``pie slice'' from a circle. The radius of the
-circle is set by \texttt{dy}. Sectors are
-defined similarly to segments. In this case, though, a chord is not
-drawn. Instead, the lines are drawn from the origin of the circle
-\texttt{(0,-dy)} to the points \texttt{(-dx,0)} and \texttt{(dx,0)}.
-The perimeter of the circle is then draw between those two points
-below the x-axis. The sector is then rotated and translated the same
-as a segment.
+Sectors are the like a ``pie slice'' from a circle. The radius of
+the circle is set by \texttt{dy}. Sectors are defined similarly to
+segments. In this case, though, a chord is not drawn. Instead,
+the lines are drawn from the origin of the circle \texttt{(0,-dy)}
+to the points \texttt{(-dx,0)} and \texttt{(dx,0)}. The perimeter
+of the circle is then drawn between those two points and lies
+below the x-axis. The sector is then rotated and translated the
+same as a segment.
\subsection{Phantom Size}\index{Phantom size}
The overall dimensions of the phantom are increased by 1\% above the
\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 from 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, or the
-\emph{phantom diameter}. Remember, as mentioned above, the
-phantom dimensions are also padded by 1\%.
+In general, the geometry for a scan all 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\%.
The other important geometry variables for scanning phantoms are
the \emph{view diameter}, \emph{scan diameter}, and \emph{focal
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}$$} CT scanners actually collect projections around a
+\latexonly{$p_d$,}\latexignore{\emph{Pd},} is then
+\latexignore{\\\centerline{\emph{Pl x sqrt(2)}}\\}
+\latexonly{\begin{equation}p_d = p_l \sqrt{2}\end{equation}}
+CT scanners actually collect projections around a
circle rather than a square. The diameter of this circle is also
-the diameter of the boundary square
-\latexonly{$p_d$. These
+the diameter of the boundary square \latexonly{$p_d$. These
relationships are diagrammed in figure~\ref{phantomgeomfig}.}
\latexignore{emph{Pd}.}
smaller) than the phantom. Thus, during rasterization or during
projections, \ctsim\ will ask for a \emph{view ratio},
\latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.} The \emph{view
-diameter} is then calculated as \latexonly{$$v_d = p_d
-v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
+diameter} is then calculated as
+\latexonly{\begin{equation}v_d = p_dv_r\end{equation}}
+\latexignore{\\\centerline{\emph{Vd = Pd x VR}}\\}
By using a
\latexonly{$v_r$}\latexignore{\emph{VR}}
for a \emph{view diameter} less than
\emph{phantom diameter}.
This will lead to significant artifacts. Physically, this would
-be impossible and is analagous to inserting an object into the CT
+be impossible and is analogous to inserting an object into the CT
scanner that is larger than the scanner itself!
\subsubsection{Scan Diameter}\index{Scan diameter}
of \emph{scan ratio}, \latexonly{$s_r$,}\latexignore{\emph{SR},}
is arises. The scan diameter
\latexonly{$s_d$}\latexignore{\emph{Sd}} is the diameter over
-which x-rays are collected and is defined as \latexonly{$$s_d =
-v_d s_r$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\} By default and
+which x-rays are collected and is defined as
+\latexonly{\begin{equation}s_d =v_d s_r\end{equation}}
+\latexignore{\\\centerline{\emph{Sd = Vd x SR}}\\}
+By default and
for all ordinary scanning, the \emph{scan ratio} is to \texttt{1}.
If the \emph{scan ratio} is less than \texttt{1}, you can expect
significant artifacts.
\latexonly{$f_r$,}\latexignore{\emph{FR},}
of the view radius. Focal length is
calculated as
-\latexonly{$$f = (v_d / 2) f_r$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$}
+\latexonly{\begin{equation}f = (v_d / 2) f_r\end{equation}}
+\latexignore{\\\centerline{\emph{F = (Vd / 2) x FR}}}
For parallel geometry scanning, the focal length doesn't matter.
However, divergent geometry scanning (equilinear and equiangular),
\subsection{Parallel Geometry}\label{geometryparallel}\index{Parallel Geometry}
-As mentioned above, the focal length is not used in this simple
+The simplest geometry, parallel, was used in \mbox{$1^{st}$} generation
+scanners. As mentioned above, the focal length is not used in this simple
geometry. The detector array is set to be the same size as the
\emph{scan diameter}. For optimal scanning in this geometry, the
\emph{scan diameter} should be equal to the \emph{phantom
In the equilinear mode, a single
source produces a fan beam which is read by a linear array of detectors. If
the detectors occupy an arc of a circle, then the geometry is equiangular.
-\latexonly{See figure~\ref{divergentfig}.}
+\latexonly{The configurations are shown in figure~\ref{divergentfig}.}
\begin{figure}
\image{10cm;0cm}{divergent.eps}
\caption{\label{divergentfig} Equilinear and equiangular geometries.}
\emph{scan diameter} by the size of the phantom.
To illustrate, the \emph{scan diameter} can be defined as
-\latexonly{$$s_d = s_r v_r p_d$$}\latexignore{\\$$Sd = Sr x Vr x Pd$$\\}
+\latexonly{\begin{equation}s_d = s_r v_r p_d\end{equation}}
+\latexignore{\\\centerline{\emph{Sd = Sr x Vr x Pd}}\\}
-Further, $f$ can be defined as \latexonly{$$f = f_r (v_r p_d /
-2)$$}\latexignore{\\$$F = FR x (VR x Pd)$$\\}
+Further, $f$ can be defined as
+\latexonly{\[f = f_r (v_r p_d / 2)\]}
+\latexignore{\\\centerline{\emph{F = FR x (VR x Pd)$$\\}}}
Substituting these equations into \latexignore{the above
equation,}\latexonly{equation~\ref{alphacalc},} We have,
\latexonly{
\begin{eqnarray}
-\alpha &= 2\,\sin^{-1} \frac{s_r v_r p_d / 2}{f_r v_r (p_d / 2)} \nonumber \\
-&= 2\,\sin^{-1} (s_r / f_r)
+\alpha &=& 2\,\sin^{-1} \frac{\displaystyle s_r v_r p_d / 2}{\displaystyle f_r v_r (p_d / 2)} \nonumber \\
+&=& 2\,\sin^{-1} (s_r / f_r)
\end{eqnarray}
-} \latexignore{\\$$\alpha = 2 sin (Sr / Fr$$\\}
+} \latexignore{\\\centerline{\emph{\alpha = 2 sin (Sr / Fr)}}\\}
Since in normal scanning $s_r$ = 1, $\alpha$ depends only upon the
-\emph{focal length ratio}.
+\emph{focal length ratio} in normal scanning.
\subsubsection{Detector Array Size}
In general, you do not need to be concerned with the detector
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{$4\,f \tan (\alpha / 2)$}
-\latexignore{\emph{4 x F x tan(\alpha/2)}}
+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}
\latexonly{This geometry is shown in figure~\ref{equilinearfig}.}
-\subsubsection{Examples of Geometry Settings}
-
-
\section{Reconstruction}\label{conceptreconstruction}\index{Reconstruction Overview}%
-\subsection{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
\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
+processors. This has been testing in a 16-CPU cluster with excellent
results.
\subsubsection{Filter projections}
-The projections for a single view have their frequency data multipled by
+The first step in filtered backprojection reconstructions is the filtering
+of each projection. The projections for a each view have their frequency data multipled by
a filter of $|w|$. \ctsim\ permits four different ways to accomplish this
filtering.
\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}
-}
+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}
+
+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
+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.
+
+\latexignore{These formulas are shown in the print documentation of \ctsim.}
+%
+%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{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{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}}
projects / ctsim.git / blobdiff