-\chapter{Concepts}\index{Concepts}%
-\setheader{{\it CHAPTER \thechapter}}{}{}{}{}{{\it CHAPTER \thechapter}}%
-\ctsimfooter%
+\chapter{Concepts}
+\setheader{{\it CHAPTER \thechapter}}{}{}{\ctsimheadtitle}{}{{\it CHAPTER \thechapter}}
+\ctsimfooter
-\section{Overview}\label{conceptoverview}\index{Concepts,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
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 \helprefn{phantom}{conceptphantom} and the
+\helprefn{scanner}{conceptscanner}.
-\section{Phantoms}\label{conceptphantom}\index{Concepts,Phantoms}%
-\subsection{Overview}\label{phantomoverview}\index{Concepts,Phantoms,Overview}%
+\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, 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
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}
+\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{}, \texttt{triangle},
-\rtfsp\texttt{sector}, or \texttt{segment}. \texttt{cx},
-\rtfsp\texttt{cy}, \texttt{dx} and \texttt{dy} have different
+\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.
-\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.
-\subsection{Phantom Elements}\label{phantomelements}\index{Concepts,Phantoms,Elements}
+\subsection{Phantom Elements}\label{phantomelements}\index{Phantom!Elements}
\subsubsection{ellipse}
Ellipses use \texttt{dx} and \texttt{dy} to define the semi-major and
\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.
-
-\subsection{Phantom Size}
+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
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{Concepts,Scanner}%
+\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
-construction of the scanner and can not be changed. Conversely,
-real-world CT scanners can only take objects up to a fixed size.
-
-\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\%.
-
-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}
+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.
+
+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
+length}. These variables are input into \ctsim\ in terms of
+ratios rather than absolute values.
+
+\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 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 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}
-$$\image{5cm;0cm}{scangeometry.eps}$$
-\caption{Phantom Geometry}
+\centerline{\image{5cm;0cm}{scangeometry.eps}}
+\latexonly{\caption{\label{phantomgeomfig} Phantom Geometry}}
\end{figure}
-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},}
-\rtfsp is then
-\latexignore{\\$$\emph{Pl x sqrt(2)}$$\\}
-\latexonly{$$p_d = p_l \sqrt{2}$$}
-CT scanners actually collect projections around a circle rather than a
-square. The diameter of this circle is also the diameter of the boundary
-square
-\latexonly{$p_d$.}\latexignore{\rtfsp\emph{Pd}.}
-These relationships are diagrammed in figure 2.1.
-
-\subsubsection{View Diameter}
+
+\subsubsection{View Diameter}\index{View diameter}
The \emph{view diameter} is the area that is being processed
during scanning of phantoms as well as during rasterization of
-phantoms. By default, the \emph{view diameter} \rtfsp is set equal
+phantoms. By default, the \emph{view diameter} is set equal
to the \emph{phantom diameter}. It may be useful, especially for
experimental reasons, to process an area larger (and maybe even
smaller) than the phantom. Thus, during rasterization or during
projections, \ctsim\ will ask for a \emph{view ratio},
\latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.} The \emph{view
-diameter} is then calculated as \latexonly{$$v_d = p_d
-v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
+diameter} is then calculated as
+\latexonly{\begin{equation}v_d = p_dv_r\end{equation}}
+\latexignore{\\\centerline{\emph{Vd = Pd x VR}}\\}
By using a
\latexonly{$v_r$}\latexignore{\emph{VR}}
for a \emph{view diameter} less than
\emph{phantom diameter}.
This will lead to significant artifacts. Physically, this would
-be impossible and is analagous to inserting an object into the CT
+be impossible and is analogous to inserting an object into the CT
scanner that is larger than the scanner itself!
-\subsubsection{Scan Diameter}
+\subsubsection{Scan Diameter}\index{Scan diameter}
By default, the entire \emph{view diameter} is scanned. For
experimental purposes, it may be desirable to scan an area either
larger or smaller than the \emph{view diameter}. Thus, the concept
of \emph{scan ratio}, \latexonly{$s_r$,}\latexignore{\emph{SR},}
-is born. The scan diameter
-\latexonly{$s_d$}\latexignore{\emph{Sd}} is the diameter over
-which x-rays are collected and is defined as \latexonly{$$s_d =
-v_d s_r$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\} By default and
+is arises. The scan diameter,
+\latexonly{$s_d$,}\latexignore{\emph{Sd},} is the diameter over
+which x-rays are collected and is defined as
+\latexonly{\begin{equation}s_d =v_d s_r\end{equation}}
+\latexignore{\\\centerline{\emph{Sd = Vd x SR}}\\}
+By default and
for all ordinary scanning, the \emph{scan ratio} is to \texttt{1}.
If the \emph{scan ratio} is less than \texttt{1}, you can expect
significant artifacts.
-\subsubsection{Focal Length}
+\subsubsection{Focal Length}\index{Focal length}
The \emph{focal length},
\latexonly{$f$,}\latexignore{\emph{F},}
is the distance of the X-ray source to the center of
\latexonly{$f_r$,}\latexignore{\emph{FR},}
of the view radius. Focal length is
calculated as
-\latexonly{$$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),
source inside of the \emph{view diameter}.
-\subsection{Parallel Geometry}\label{geometryparallel}\index{Concepts,Scanner,Geometries,Parallel}
+\subsection{Parallel Geometry}\label{geometryparallel}\index{Parallel geometry}\index{Scanner!Parallel}
-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
significant distortions will occur.
-\subsection{Divergent Geometries}\label{geometrydivergent}\index{Concepts,Scanner,Geometries,Divergent}
+\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,
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.
-See figure 2.2.
+\latexonly{These configurations are shown in figure~\ref{divergentfig}.}
\begin{figure}
-\image{10cm;0cm}{divergent.eps}
-\caption{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)}$$\\}
+the \emph{focal length}:
+\latexignore{\centerline{\emph{alpha = 2 x asin (
+(Sd / 2) / f)}}}
\latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1}
-((s_d / 2) / f)\end{equation}} This is illustrated in figure 2.3.
+((s_d / 2) / f)\end{equation}}
+\latexonly{This is illustrated in figure~\ref{alphacalcfig}.}
\begin{figure}
-\image{10cm;0cm}{alphacalc.eps}
-\caption{Calculation of $\alpha$}
+\centerline{\image{10cm;0cm}{alphacalc.eps}}
+\latexonly{\caption{\label{alphacalcfig} Calculation of $\alpha$}}
\end{figure}
\emph{scan diameter} by the size of the phantom.
To illustrate, the \emph{scan diameter} can be defined as
-\latexonly{$$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)$$\\}
+Further, $f$ can be defined as
+\latexonly{\begin{equation} = f_r (v_r p_d / 2)\end{equation}}
+\latexignore{\\\centerline{\emph{F = FR x (VR x Pd)$$\\}}}
Substituting these equations into \latexignore{the above
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
-array size. It is automatically calculated by \ctsim. For those
-interested, this section explains how the detector array size is
-calculated.
+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
diameter.
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
-\begin{figure}\label{equiangularfig}
-\image{10cm;0cm}{equiangular.eps} \caption{Equiangular geometry}
+\latexonly{$2\,\alpha$.}\latexignore{\emph{2 \alpha}.}
+The dotted circle
+\latexonly{in figure~\ref{equiangularfig}}
+indicates the positions of the detectors in this case.
+
+\begin{figure}
+\centerline{\image{10cm;0cm}{equiangular.eps}}
+\latexonly{\caption{\label{equiangularfig}Equiangular geometry}}
\end{figure}
-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
\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)}}}
+\latexonly{This geometry is shown in figure~\ref{equilinearfig}.}
\begin{figure}\label{equilinearfig}
-\image{10cm;0cm}{equilinear.eps}
-\caption{Equilinear geometry}
+\centerline{\image{10cm;0cm}{equilinear.eps}}
+\latexonly{\caption{\label{equilinearfig} Equilinear geometry}}
\end{figure}
-This geometry is shown in figure~2.5.
-\subsubsection{Examples of Geometry Settings}
+\section{Reconstruction}\label{conceptreconstruction}\index{Reconstruction Overview}%
-
-\section{Reconstruction}\label{conceptreconstruction}\index{Concepts,Reconstruction}%
-\subsection{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
+accurate as filtered backprojection. This is due primarily
because interpolation occurs in the frequency domain rather than the
spatial domain.
-\subsection{Filtered Backprojection}
+\subsection{Filtered Backprojection}\index{Filtered backprojection}
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.
+filtering projections followed backprojecting the filtered projections. Though
+these two steps are sequential, each view position can be processed independently.
\subsubsection{Multiple Computer 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 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.
Backprojection is the process of ``smearing'' the filtered
projections over the reconstructing image. Various levels of
interpolation can be specified.
+
+\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{twocollist}
+\twocolitem{\textbf{$d$}}{The normalized root mean squared distance measure.}
+\twocolitem{\textbf{$r$}}{The normalized mean absolute distance measure.}
+\twocolitem{\textbf{$e$}}{The worst case distance measure over a \latexonly{$2\times2$}\latexignore{\emph{2 x 2}} pixel area.}
+\end{twocollist}
+
+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 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.
+
+\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{\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{\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