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 (Herman
+Shepp-Logan) can be constructed. In fact, \ctsim\ provides a shortcut to construct those published phantoms. \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
+with the center of the ellipse at cx and 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
+cx and cy to define the position of the center of the rectangle with respect
to the origin. dx and 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 cx,cy, with a base
+half-width of dx and a height of 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.
+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
absolute values.
\subsubsection{Phantom Diameter}
+\begin{figure}
+$$\image{5cm;0cm}{scangeometry.eps}$$
+\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}}
square. The diameter of this circle is also the diameter of the boundary
square
\latexonly{$P_d$.}\latexignore{\emph{Pd}.}
-These relationships are diagrammed in figure 1.
+These relationships are diagrammed in figure 2.1.
\subsubsection{View Diameter}
The \emph{view diameter} is the area that is being processed during scanning of phantoms as
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}$$\\}
-By default and for all ordinary scanning, the \emph{scan diameter ratio} is to \texttt{1}. If the \emph{scan diameter ratio} is less than \texttt{1}, you
-can plan on significant artifacts.
+is the diameter over which x-rays are collected and is defined as
+\latexonly{$$S_d = V_d S_{R}$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\}
+By default and for all ordinary scanning, the \emph{scan ratio} is to
+\texttt{1}. If the \emph{scan ratio} is less than \texttt{1},
+you can expect significant artifacts.
\subsubsection{Focal Length}
The \emph{focal length},
\latexonly{$F_l$,}\latexignore{\emph{Fl},}
is the distance of the X-ray source to the center of
the phantom. The focal length is set as a ratio,
-\latexonly{$F_{lR}$,}\latexignore{\emph{FlR},}
+\latexonly{$F_{lR}$,}\latexignore{\emph{FllR},}
of the view radius. Focal length is
calculated as
-\latexonly{$$F_l = F_{lR} (V_d / 2)$$}\latexignore{\\$$\emph{Fl = FlR x (Vd / 2)}$$}
+\latexonly{$$F_l = (V_d / 2) F_R$$}\latexignore{\\$$\emph{Fl = (Vd / 2) x FlR}$$}
For parallel geometry scanning, the focal length doesn't matter. However,
divergent geometry scanning (equilinear and equiangular), the \emph{focal
length ratio} should be set at \texttt{2} or more to avoid artifacts.
\subsection{Parallel Geometry}\label{geometryparallel}\index{Concepts,Scanner,Geometries,Parallel}
-\begin{figure}
-\image{10cm;0cm}{ctsimfig1.eps}
-\caption{Geometry used for a 1st generation, parallel beam CT scanner}\label{fistgenfig}
-\end{figure}
As mentioned above, the focal length is not used in this simple
geometry. The detector array is set to
In the equilinear mode, a single
source produces a fan beam which is read by a linear array of detectors. If
the detectors occupy an arc of a circle, then the geometry is equiangular.
-See figure 2.
+See figure 2.2.
\begin{figure}
\image{10cm;0cm}{ctsimfig2.eps}
\caption{Equilinear and equiangular geometries.}
\end{figure}
\subsubsection{Fan Beam Angle}
-For these divergent beam geometries, the angle of the fan beam needs
+For these divergent beam geometries, the \emph{fan beam angle} needs
to be calculated. For real-world CT scanners, this is fixed at the
-time of manufacture. \ctsim\, however, calculates the fan beam angle,
+time of manufacture. \ctsim, however, calculates the \emph{fan beam angle},
\latexonly{$\alpha$,}\latexignore{\emph{alpha},}
from the diameter of the \emph{scan diameter} and the \emph{focal length}
\latexignore{\\$$\emph{alpha = 2 x asin ( (Sd / 2) / F_l)}$$\\}
\latexonly{$$\alpha = 2 \sin^{-1} ((S_d / 2) / F_l)$$}
-This is illustrated in figure 3.
+This is illustrated in figure 2.3.
\begin{figure}
\image{10cm;0cm}{alphacalc.eps}
\caption{Calculation of $\alpha$}
\end{figure}
-If this quantity is less than or equal to zero, then at least for some
-projections the source is inside the phantom. Perhaps a figure will help at
-this point. Consider first the case where $f_{vR} = f_{lR} =1 $, figure 3. The
-square in the figure bounds the phantom and has sides $l_p$. For this case
-then,
-\latexonly{$$f_l=\sqrt{2}l_p/2 = l_p/\sqrt{2}$$,
-$$f_v = \sqrt{2}l_p$$,
-and
-$$d_{hs} = {l_p}/{2}$$.
-Then
-$$\mathrm{dFocalPastPhm} = ({l_p}/{2}) (\sqrt{2}-1)$$
-}
-\begin{figure}
-\image{5cm;0cm}{ctsimfig3.eps}
-\caption{Equilinear and equiangluar geometry when focal length ratio =
-field of view ratio = 1.}
-\end{figure}
-The angle $\alpha$ is now defined as shown in figure 3, and the detector
-length is adjusted to subtend the angle $2\alpha$ as shown. Note that the
-size of the detector array may have changed and the field of view is not
-used.
-For a circular array of detectors, the detectors are spaced around a
-circle covering an angular distance of $2\alpha$. The dotted circle in
-figure 3 indicates the positions of the detectors in this case. Note that
-detectors at the ends of the range would not be illuminated by the source.
-
-Now, consider increasing the focal length ratio to two leaving the
+Empiric testing with \ctsim\ shows that for very large \emph{fan beam angles},
+greater than approximately
+\latexonly{$120^{\circ}$,}\latexignore{120 degrees,}
+there are significant artifacts. The primary way to manage the
+\emph{fan beam angle} is by varying the \emph{focal length} since the
+\emph{scan diameter} by the size of the phantom.
+
+\subsubsection{Detector Array Size}
+In general, you do not need to be concerned with the detector array
+size. It is automatically calculated by \ctsim. The size of the
+detector array depends upon the \emph{focal length} and the
+\emph{scan diameter}. In general, increasing the \emph{focal length}
+decreases the size of the detector array and increasing the \emph{scan
+diameter} increases the detector array size.
+
+For equiangular geometry, the detectors are spaced around a
+circle covering an angular distance of
+\latexonly{$\alpha$.}\latexignore{\emph{alpha}.}
+The dotted circle in
+figure 2.4 indicates the positions of the detectors in this case.
+
+For equilinear geometry, the detectors are space along a straight
+line. The length of the line depends upon
+\latexonly{$\alpha$}\latexignore{\emph{alpha}}
+and the \emph{focal length}. It is calculated as
+\latexonly{$$\mathrm{detLengh} = 4\,F_l \tan (\alpha / 2)$$}
+\latexignore{\\$$\emph{detLength} = 4 x Fl x tan(alpha/2)$$\\}
+An example of the this geometry is in figure 2.5.
+
+\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 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}
\caption{Equilinear and equiangluar geometry when focal length ratio = 4.}
\end{figure}
+
+
\section{Reconstruction}\label{conceptreconstruction}\index{Concepts,Reconstruction}%
\subsection{Overview}
\subsection{Direct Inverse Fourier}
projects / ctsim.git / blobdiff