\setfooter{\thepage}{}{}{}{}{\thepage}%
\section{Overview}\label{conceptoverview}\index{Concepts,Overview}%
-In \ctsim, a phantom object, or a geometrical description of the object
-of a CT study is constructed and an image can be created. Then a
-scanner geometry can be specified, and the projection data simulated.
-Finally that projection data can be reconstructed using various user
-controlled algorithms producing an image of the phantom or study object.
+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
+image of the phantom object. This reconstruction can then be
+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 'phantom' and
-the 'scanner'.
+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}%
\subsection{Overview}\label{phantomoverview}\index{Concepts,Phantoms,Overview}%
describe the object being 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 the standard phantoms used in the CT literature (the Herman
-and the Shepp-Logan) can be constructed. In fact
-\ctsim\ provides a shortcut to construct those phantoms for you. It also
-allows you to write a file in which the composition of your own phantom is
-described.
+With these elements, standard phantoms used in the CT literature can
+be constructed. In fact, \ctsim\ provides a shortcut to load the
+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 from
+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
phantom. Each line contains seven entries, in the following form:
\begin{verbatim}
-item cx cy dx dy r a
+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}.
Where objects overlap, the attenuations of the overlapped objects are summed.
-
\subsection{Phantom Elements}\label{phantomelements}\index{Concepts,Phantoms,Elements}
\subsubsection{ellipse}
-Ellipses use dx and dy to define the semi-major and semi-minor axis lengths,
-with the centre of the ellipse at cx and cy. Of note, the commonly used
-phantom described by Shepp and Logan\cite{SHEPP74} uses only ellipses.
+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
+Shepp and Logan\cite{SHEPP74} uses only ellipses.
\subsubsection{rectangle}
-Rectangles use
-cx and cy to define the position of the centre of the rectangle with respect
-to the origin. dx and dy are the half-width and half-height of the
-rectangle.
+Rectangles use \texttt{cx} and \texttt{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 centre of the base at cx,cy, with a base
-width of 2*dx in x direction, and a height of dy. Rotations are then
-applied about the origin.
-
-\subsubsection{sector}
-It appears that dx and dy
-define the end points of a radius of the sector, from which the radius and
-the angle of the two arms of the sector are calculated. But then
-orientation and centering of the sector don't make much sense yet.
+Triangles are drawn with the center of the base at \texttt{(cx,cy)},
+with a base half-width of \texttt{dx} and a height of \texttt{dy}.
+Rotations are then applied about the center of the base.
\subsubsection{segment}
-Segments are the segments of a circle between a chord and the
-perimeter of the circle. This also isn't clear to me, but it appears that
-perhaps the distance from chord to circle perimeter, and circle radius is
-defined by dx and dy. Chord is always horizontal through the origin, then
-translated and then rotated (???).
+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}.
+
+\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}
-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.
-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.
+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.
\section{Scanner}\label{conceptscanner}\index{Concepts,Scanner}%
-\subsection{Sizes}
-Understanding the scanning geometry is the most complicated aspect
-of using \ctsim. For our 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.
+\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 is setting up the geometry for a scan.
+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
$$\image{5cm;0cm}{scangeometry.eps}$$
\caption{Phantom Geometry}
\end{figure}
-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}}
+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},}
+\latexonly{$p_d$,}\latexignore{\emph{Pd},}
is then
\latexignore{\\$$\emph{Pl x sqrt(2)}$$\\}
-\latexonly{$$P_d = P_l \sqrt{2}$$}
+\latexonly{$$p_d = p_l \sqrt{2}$$}
CT scanners actually collect projections around a circle rather than a
square. The diameter of this circle is also the diameter of the boundary
square
-\latexonly{$P_d$.}\latexignore{\emph{Pd}.}
+\latexonly{$p_d$.}\latexignore{\emph{Pd}.}
These relationships are diagrammed in figure 2.1.
\subsubsection{View Diameter}
is set equal to the \emph{phantom diameter}. It may be useful, especially for
experimental reasons, to process an area larger (and maybe even smaller) than
the phantom. Thus, during rasterization or during projections, \ctsim\ will
-ask for a \emph{view diameter ratio},
-\latexonly{$V_{dR}$.}\latexignore{\emph{VdR}.}
+ask for a \emph{view ratio},
+\latexonly{$v_{R}$.}\latexignore{\emph{VR}.}
The \emph{view diameter} is then set as
-\latexonly{$$V_d = P_d V_{dR}$$}\latexignore{\\$$\emph{Vd = Pd x VdR}$$}
+\latexonly{$$v_d = p_d v_{R}$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
By using a
-\latexonly{$V_{dR}$}\latexignore{\emph{VdR}}
+\latexonly{$v_{R}$}\latexignore{\emph{VR}}
less than 1, \ctsim\ will allow
for a \emph{view diameter} less than
\emph{phantom diameter}.
\subsubsection{Scan Diameter}
By default, the entire \emph{view diameter} is scanned. For experimental
purposes, it may be desirable to scan an area either larger or smaller than
-the \emph{view diameter}. Thus, the concept of \emph{scan diameter}
-\latexonly{$S_{dR}$}\latexignore{\emph{SdR}}
+the \emph{view diameter}. Thus, the concept of \emph{scan ratio}
+\latexonly{$s_{R}$}\latexignore{\emph{SR}}
is born. The scan diameter
-\latexonly{$S_d$}\latexignore{\emph{Sd}}
-is defined as
-\latexonly{$$S_d = V_d S_{dR}$$}\latexignore{\\$$\emph{Sd = Vd x SdR}$$\\}
-By default and for all ordinary scanning, the \emph{scan diameter ratio} is to \texttt{1}. If the \emph{scan diameter ratio} is less than \texttt{1}, you
-can plan on significant artifacts.
+\latexonly{$s_d$}\latexignore{\emph{Sd}}
+is the diameter over which x-rays are collected and is defined as
+\latexonly{$$s_d = v_d s_{R}$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\}
+By default and for all ordinary scanning, the \emph{scan ratio} is to
+\texttt{1}. If the \emph{scan ratio} is less than \texttt{1},
+you can expect significant artifacts.
\subsubsection{Focal Length}
The \emph{focal length},
-\latexonly{$F_l$,}\latexignore{\emph{Fl},}
+\latexonly{$f$,}\latexignore{\emph{F},}
is the distance of the X-ray source to the center of
the phantom. The focal length is set as a ratio,
-\latexonly{$F_{lR}$,}\latexignore{\emph{FlR},}
+\latexonly{$f_{R}$,}\latexignore{\emph{FR},}
of the view radius. Focal length is
calculated as
-\latexonly{$$F_l = F_{lR} (V_d / 2)$$}\latexignore{\\$$\emph{Fl = FlR x (Vd / 2)}$$}
+\latexonly{$$f = (v_d / 2) f_R$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$}
For parallel geometry scanning, the focal length doesn't matter. However,
divergent geometry scanning (equilinear and equiangular), the \emph{focal
the detectors occupy an arc of a circle, then the geometry is equiangular.
See figure 2.2.
\begin{figure}
-\image{10cm;0cm}{ctsimfig2.eps}
+\image{10cm;0cm}{divergent.eps}
\caption{Equilinear and equiangular geometries.}
\end{figure}
time of manufacture. \ctsim, however, calculates the \emph{fan beam angle},
\latexonly{$\alpha$,}\latexignore{\emph{alpha},}
from the diameter of the \emph{scan diameter} and the \emph{focal length}
-\latexignore{\\$$\emph{alpha = 2 x asin ( (Sd / 2) / F_l)}$$\\}
-\latexonly{$$\alpha = 2 \sin^{-1} ((S_d / 2) / F_l)$$}
+\latexignore{\\$$\emph{alpha = 2 x asin ( (Sd / 2) / F)}$$\\}
+\latexonly{$$\alpha = 2 \sin^{-1} ((s_d / 2) / f)$$}
This is illustrated in figure 2.3.
\begin{figure}
\image{10cm;0cm}{alphacalc.eps}
circle covering an angular distance of
\latexonly{$\alpha$.}\latexignore{\emph{alpha}.}
The dotted circle in
+\begin{figure}
+\image{10cm;0cm}{equiangular.eps}
+\caption{Equiangluar geometry}
+\end{figure}
figure 2.4 indicates the positions of the detectors in this case.
For equilinear geometry, the detectors are space along a straight
line. The length of the line depends upon
\latexonly{$\alpha$}\latexignore{\emph{alpha}}
and the \emph{focal length}. It is calculated as
-\latexonly{$$\mathrm{detLengh} = 4\,F_l \tan (\alpha / 2)$$}
-\latexignore{\\$$\emph{detLength} = 4 x Fl x tan(alpha/2)$$\\}
+\latexonly{$$\mathrm{detLengh} = 4\,f \tan (\alpha / 2)$$}
+\latexignore{\\$$\emph{detLength} = 4 x F x tan(alpha/2)$$\\}
+\begin{figure}
+\image{10cm;0cm}{equilinear.eps}
+\caption{Equilinear geometry}
+\end{figure}
An example of the this geometry is in figure 2.5.
\subsubsection{Examples of Geometry Settings}
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 centre of the phantom to the source, and
+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.
-\begin{figure}
-\image{10cm;0cm}{ctsimfig4.eps}
-\caption{Equilinear and equiangluar geometry when focal length ratio = 2
-and the field of view ratio = 1.}
-\end{figure}
-\begin{figure}
-\image{10cm;0cm}{ctsimfig5.eps}
-\caption{Equilinear and equiangluar geometry when focal length ratio = 4.}
-\end{figure}
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
-inverse fourier transform of $|w|$. The other two methods perform an fourier
+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.
\subsubsection{Backprojection of filtered projections}
Backprojection is the process of ``smearing'' the filtered projections over
-the reconstructing image.
+the reconstructing image. Various levels of interpolation can be specified.
+In general, the trade-off is between quality and execution time.
projects / ctsim.git / blobdiff