\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.
+
+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'.
\section{Phantoms}\label{conceptphantom}\index{Concepts,Phantoms}%
\subsection{Overview}\label{phantomoverview}\index{Concepts,Phantoms,Overview}%
-\subsection{Phantom Elements}\label{phantoelements}\index{Concepts,Phantoms,Elements}
+
+\ctsim\ uses geometrical objects to
+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.
+
+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
+\end{verbatim}
+The first entry defines the type of the element, one
+of {\tt rectangle}, {\tt ellipse}, {\tt triangle}, {\tt sector}, or {\tt segment}.
+{\tt cx}, {\tt cy}, {\tt dx} and {\tt dy} have different meanings depending on the element type.
+
+{\tt r} is the rotation applied to the object in degrees counterclockwise,
+and {\tt 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}
+
\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.
+
\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.
+
\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.
+
\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 (???).
+
+\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.
\section{Scanner}\label{conceptscanner}\index{Concepts,Scanner}%
-\subsection{Geometries}
-\subsection{Focal Length}
-\subsection{Field of View}
+\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.
+\ctsim, being a very flexible simulator,
+gives tremendous options is 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 diameter}, \emph{scan diameter}, and \emph{focal length}.
+These variables are all input into \ctsim\ in terms of ratios rather than
+absolute values.
+
+\subsubsection{Phantom Diameter}
+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 then
+\latexignore{\\$$\emph{Pl x sqrt(2)}$$\\}
+\latexonly{$$P_d = P_l \sqrt{2}$$}
+CT scanners actually collect projections around a circle rather than a
+square. The diameter of this circle is also the diameter of the boundary
+square
+\latexonly{$P_d$.}\latexignore{\emph{Pd}.}
+These relationships are diagrammed in figure 1.
+
+\subsubsection{View Diameter}
+The \emph{view diameter} is the area that is being processed during scanning of phantoms as
+well as during rasterization of phantoms. By default, the \emph{view diameter}
+is set equal to the \emph{phantom diameter}. It may be useful, especially for
+experimental reasons, to process an area larger (and maybe even smaller) than
+the phantom. Thus, during rasterization or during projections, \ctsim\ will
+ask for a \emph{view diameter ratio},
+\latexonly{$V_{dR}$.}\latexignore{\emph{VdR}.}
+The \emph{view diameter} is then set as
+\latexonly{$$V_d = P_d V_{dR}$$}\latexignore{\\$$\emph{Vd = Pd x VdR}$$}
+
+By using a
+\latexonly{$V_{dR}$}\latexignore{\emph{VdR}}
+less than 1, \ctsim\ will allow
+for a \emph{view diameter} less than
+\emph{phantom diameter}.
+This will lead to significant artifacts. Physically, this would
+be impossible and is analagous to inserting an object into the CT
+scanner that is larger than the scanner itself!
+
+\subsubsection{Scan Diameter}
+By default, the entire \emph{view diameter} is scanned. For experimental
+purposes, it may be desirable to scan an area either larger or smaller than
+the \emph{view diameter}. Thus, the concept of \emph{scan diameter}
+\latexonly{$S_{dR}$}\latexignore{\emph{SdR}}
+is born. The scan diameter
+\latexonly{$S_d$}\latexignore{\emph{Sd}}
+is defined as
+\latexonly{$$S_d = V_d S_{dR}$$}\latexignore{\\$$\emph{Sd = Vd x SdR}$$\\}
+By default and for all ordinary scanning, the \emph{scan diameter ratio} is to \texttt{1}. If the \emph{scan diameter ratio} is less than \texttt{1}, you
+can plan on significant artifacts.
+
+\subsubsection{Focal Length}
+The \emph{focal length},
+\latexonly{$F_l$,}\latexignore{\emph{Fl},}
+is the distance of the X-ray source to the center of
+the phantom. The focal length is set as a ratio,
+\latexonly{$F_{lR}$,}\latexignore{\emph{FlR},}
+of the view radius. Focal length is
+calculated as
+\latexonly{$$F_l = F_{lR} (V_d / 2)$$}\latexignore{\\$$\emph{Fl = FlR x (Vd / 2)}$$}
+
+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
+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 diameter}. This is accomplished by using
+the default values of \texttt{1} for the \emph{view diameter ratio} and
+the \emph{scan diameter ratio}. If values of less than \texttt{1} are
+used for these two variables, significant distortions will occur.
+
+\subsection{Divergent Geometries}\label{geometrydivergent}\index{Concepts,Scanner,Geometries,Divergent}
+\subsubsection{Overview}
+Next consider the case of equilinear (second generation) and equiangular
+(third, fourth, and fifth generation) geometries. In these cases,
+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
+the detectors occupy an arc of a circle, then the geometry is equiangular.
+See figure 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
+to be calculated. For real-world CT scanners, this is fixed at the
+time of manufacture. \ctsim\, however, calculates the 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)$$}
+This is illustrated in figure 3.
+\begin{figure}
+\image{10cm;0cm}{alphacalc.eps}
+\caption{Calculation of $\alpha$}
+\end{figure}
+
+If this quantity is less than or equal to zero, then at least for some
+projections the source is inside the phantom. Perhaps a figure will help at
+this point. Consider first the case where $f_{vR} = f_{lR} =1 $, figure 3. The
+square in the figure bounds the phantom and has sides $l_p$. For this case
+then,
+\latexonly{$$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
+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
+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}
\section{Reconstruction}\label{conceptreconstruction}\index{Concepts,Reconstruction}%
+\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
+accurate as filtered backprojection. The difference is due primarily
+because interpolation occurs in the frequency domain rather than the
+spatial domain.
+
\subsection{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.
+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 results.
+
+\subsubsection{Filter projections}
+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
+transform of the projection data and multiply that by the $|w|$ filter and
+then perform an inverse fourier transform.
+
+Though multiplying by $|w|$ gives the sharpest reconstructions, in practice, superior results are obtained by mutiplying the $|w|$ filter by
+another filter that attenuates the higher frequencies. \ctsim\ has multiple
+filters for this purpose.
+
+\subsubsection{Backprojection of filtered projections}
+Backprojection is the process of ``smearing'' the filtered projections over
+the reconstructing image.
+
projects / ctsim.git / blobdiff