\setfooter{\thepage}{}{}{}{}{\thepage}%
\section{Overview}\label{conceptoverview}\index{Concepts,Overview}%
+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
+\emph{phantom} and the \emph{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 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}.
+
+\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}
+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}.
+{\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 \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 \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}
-\subsubsection{sector}
+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 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 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{Geometries}
-\subsection{Focal Length}
-\subsection{Field of View}
+\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 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}
+\begin{figure}
+$$\image{5cm;0cm}{scangeometry.eps}$$
+\caption{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},}
+is then
+\latexignore{\\$$\emph{Pl x sqrt(2)}$$\\}
+\latexonly{$$p_d = p_l \sqrt{2}$$}
+CT scanners actually collect projections around a circle rather than a
+square. The diameter of this circle is also the diameter of the boundary
+square
+\latexonly{$p_d$.}\latexignore{\emph{Pd}.}
+These relationships are diagrammed in figure 2.1.
+
+\subsubsection{View Diameter}
+The \emph{view diameter} is the area that is being processed during scanning of phantoms as
+well as during rasterization of phantoms. By default, the \emph{view diameter}
+is set equal to the \emph{phantom diameter}. It may be useful, especially for
+experimental reasons, to process an area larger (and maybe even smaller) than
+the phantom. Thus, during rasterization or during projections, \ctsim\ will
+ask for a \emph{view ratio},
+\latexonly{$v_{R}$.}\latexignore{\emph{VR}.}
+The \emph{view diameter} is then set as
+\latexonly{$$v_d = p_d v_{R}$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
+
+By using a
+\latexonly{$v_{R}$}\latexignore{\emph{VR}}
+less than 1, \ctsim\ will allow
+for a \emph{view diameter} less than
+\emph{phantom diameter}.
+This will lead to significant artifacts. Physically, this would
+be impossible and is analagous to inserting an object into the CT
+scanner that is larger than the scanner itself!
+
+\subsubsection{Scan Diameter}
+By default, the entire \emph{view diameter} is scanned. For experimental
+purposes, it may be desirable to scan an area either larger or smaller than
+the \emph{view diameter}. Thus, the concept of \emph{scan ratio}
+\latexonly{$s_{R}$}\latexignore{\emph{SR}}
+is born. The scan diameter
+\latexonly{$s_d$}\latexignore{\emph{Sd}}
+is the diameter over which x-rays are collected and is defined as
+\latexonly{$$s_d = v_d s_{R}$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\}
+By default and for all ordinary scanning, the \emph{scan ratio} is to
+\texttt{1}. If the \emph{scan ratio} is less than \texttt{1},
+you can expect significant artifacts.
+
+\subsubsection{Focal Length}
+The \emph{focal length},
+\latexonly{$f$,}\latexignore{\emph{F},}
+is the distance of the X-ray source to the center of
+the phantom. The focal length is set as a ratio,
+\latexonly{$f_{R}$,}\latexignore{\emph{FR},}
+of the view radius. Focal length is
+calculated as
+\latexonly{$$f = (v_d / 2) f_R$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$}
+
+For parallel geometry scanning, the focal length doesn't matter. However,
+divergent geometry scanning (equilinear and equiangular), the \emph{focal
+length ratio} should be set at \texttt{2} or more to avoid artifacts.
+
+
+\subsection{Parallel Geometry}\label{geometryparallel}\index{Concepts,Scanner,Geometries,Parallel}
+
+As mentioned above, the focal length is not used in this simple
+geometry. The detector array is set to
+be the same size as the \emph{scan diameter}.
+For optimal scanning in this geometry, the \emph{scan diameter} should
+be equal to the \emph{phantom diameter}. This is accomplished by using
+the default values of \texttt{1} for the \emph{view diameter ratio} and
+the \emph{scan diameter ratio}. If values of less than \texttt{1} are
+used for these two variables, significant distortions will occur.
+
+
+\subsection{Divergent Geometries}\label{geometrydivergent}\index{Concepts,Scanner,Geometries,Divergent}
+\subsubsection{Overview}
+Next consider the case of equilinear (second generation) and equiangular
+(third, fourth, and fifth generation) geometries. In these cases,
+the x-ray beams diverge from a single source to a detector array.
+In the equilinear mode, a single
+source produces a fan beam which is read by a linear array of detectors. If
+the detectors occupy an arc of a circle, then the geometry is equiangular.
+See figure 2.2.
+\begin{figure}
+\image{10cm;0cm}{divergent.eps}
+\caption{Equilinear and equiangular geometries.}
+\end{figure}
+
+
+\subsubsection{Fan Beam Angle}
+For these divergent beam geometries, the \emph{fan beam angle} needs
+to be calculated. For real-world CT scanners, this is fixed at the
+time of manufacture. \ctsim, however, calculates the \emph{fan beam angle},
+\latexonly{$\alpha$,}\latexignore{\emph{alpha},}
+from the diameter of the \emph{scan diameter} and the \emph{focal length}
+\latexignore{\\$$\emph{alpha = 2 x asin ( (Sd / 2) / f)}$$\\}
+\latexonly{$$\alpha = 2 \sin^{-1} ((s_d / 2) / f)$$}
+This is illustrated in figure 2.3.
+\begin{figure}
+\image{10cm;0cm}{alphacalc.eps}
+\caption{Calculation of $\alpha$}
+\end{figure}
+
+
+Empiric testing with \ctsim\ shows that for very large \emph{fan beam angles},
+greater than approximately
+\latexonly{$120^\circ$,}\latexignore{120 degrees,}
+there are significant artifacts. The primary way to manage the
+\emph{fan beam angle} is by varying the \emph{focal length} since the
+\emph{scan diameter} by the size of the phantom.
+
+$$s_d = p_d v_R s_R$$
+If $v_r = 1$ and $s_R = 1$, then $s_d = p_d$. Further, $f = f_R v_R (p_d / 2)$
+Plugging these equations into the above equation,
+$$\alpha = 2\,\sin^{-1} \frac{p_d / 2}{f_R (p_d / 2)}$$
+$$\alpha = 2\,\sin^{-1} (1 / f_R)$$
+
+Thus, $\alpha$ depends only upon the \emph{focal length ratio}.
+
+\subsubsection{Detector Array Size}
+In general, you do not need to be concerned with the detector array
+size. It is automatically calculated by \ctsim. The size of the
+detector array depends upon the \emph{focal length} and the
+\emph{scan diameter}. In general, increasing the \emph{focal length}
+decreases the size of the detector array and increasing the \emph{scan
+diameter} increases the detector array size.
+
+For equiangular geometry, the detectors are spaced around a
+circle covering an angular distance of
+\latexonly{$\alpha$.}\latexignore{\emph{alpha}.}
+The dotted circle in
+\begin{figure}
+\image{10cm;0cm}{equiangular.eps}
+\caption{Equiangluar geometry}
+\end{figure}
+figure 2.4 indicates the positions of the detectors in this case.
+
+For equilinear geometry, the detectors are space along a straight
+line. The length of the line depends upon
+\latexonly{$\alpha$}\latexignore{\emph{alpha}}
+and the \emph{focal length}. It is calculated as
+\latexonly{$$\mathrm{detLengh} = 4\,f \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}
+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 center of the phantom to the source, and
+the detector array is adjusted to give an angular coverage to include the
+whole phantom.
+
+
\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. Various levels of interpolation can be specified.
+In general, the trade-off is between quality and execution time.
projects / ctsim.git / blobdiff