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 \chapter{Concepts}\index{Concepts}%
 \setheader{{\it CHAPTER \thechapter}}{}{}{}{}{{\it CHAPTER \thechapter}}%
-\setfooter{\thepage}{}{}{}{}{\thepage}%
+\setfooter{\thepage}{}{}{}{\small Version 0.2}{\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 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.
+
+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\cite{HERMAN80} and
+Shepp-Logan\cite{SHEPP74}. \ctsim\ also reads text files of
+user-defined phantoms.
+
+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}
+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
+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
+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}
+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.
 
 \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 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}
+\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},}
+\rtfsp is then
+\latexignore{\\$$\emph{Pl x sqrt(2)}$$\\}
+\latexonly{$$p_d = p_l \sqrt{2}$$}
+CT scanners actually collect projections around a circle rather than a
+square. The diameter of this circle is also the diameter of the boundary
+square
+\latexonly{$p_d$.}\latexignore{\rtfsp\emph{Pd}.}
+These relationships are diagrammed in figure 2.1.
+
+\subsubsection{View Diameter}
+The \emph{view diameter} is the area that is being processed
+during scanning of phantoms as well as during rasterization of
+phantoms. By default, the \emph{view diameter} \rtfsp is set equal
+to the \emph{phantom diameter}. It may be useful, especially for
+experimental reasons, to process an area larger (and maybe even
+smaller) than the phantom. Thus, during rasterization or during
+projections, \ctsim\ will ask for a \emph{view ratio},
+\latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.} The \emph{view
+diameter} is then calculated as \latexonly{$$v_d = p_d
+v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$}
+
+By using a
+\latexonly{$v_r$}\latexignore{\emph{VR}}
+less than 1, \ctsim\ will allow
+for a \emph{view diameter} less than
+\emph{phantom diameter}.
+This will lead to significant artifacts. Physically, this would
+be impossible and is analagous to inserting an object into the CT
+scanner that is larger than the scanner itself!
+
+\subsubsection{Scan Diameter}
+By default, the entire \emph{view diameter} is scanned. For
+experimental purposes, it may be desirable to scan an area either
+larger or smaller than the \emph{view diameter}. Thus, the concept
+of \emph{scan ratio}, \latexonly{$s_r$,}\latexignore{\emph{SR},}
+is born. The scan diameter
+\latexonly{$s_d$}\latexignore{\emph{Sd}} is the diameter over
+which x-rays are collected and is defined as \latexonly{$$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. Moreover, a value of less than \texttt{1},
+though it can be given to \ctsim, is physically impossible and it
+analagous to have having the x-ray source with the \emph{view
+diameter}.
+
+
+\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 ratio} and the \emph{scan 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}, $\alpha$, from the \emph{scan diameter} and
+the \emph{focal length} \latexignore{\\$$\emph{alpha = 2 x asin (
+(Sd / 2) / f)}$$\\}
+\latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1}
+((s_d / 2) / f)\end{equation}} This is illustrated in figure 2.3.
+\begin{figure}
+\image{10cm;0cm}{alphacalc.eps}
+\caption{Calculation of $\alpha$}
+\end{figure}
+
+
+Empiric testing with \ctsim\ shows that for very large \emph{fan beam angles},
+greater than approximately
+\latexonly{$120^\circ$,}\latexignore{120 degrees,}
+there are significant artifacts. The primary way to manage the
+\emph{fan beam angle} is by varying the \emph{focal length} since the
+\emph{scan diameter} by the size of the phantom.
+
+To illustrate, the \emph{scan diameter} can be defined as
+\latexonly{$$s_d = s_r v_r p_d$$}\latexignore{\\$$Sd = Sr x Vr x Pd$$\\}
+
+Further, $f$ can be defined as \latexonly{$$f = f_r (v_r p_d /
+2)$$}\latexignore{\\$$F = FR x (VR x Pd)$$\\}
+
+Substituting these equations into \latexignore{the above
+equation,}\latexonly{equation~\ref{alphacalc},} We have,
+\latexonly{
+\begin{eqnarray}
+\alpha &= 2\,\sin^{-1} \frac{s_r v_r p_d / 2}{f_r v_r (p_d / 2)} \nonumber \\
+&= 2\,\sin^{-1} (s_r / f_r)
+\end{eqnarray}
+} \latexignore{\\$$\alpha = 2 sin (Sr / Fr$$\\}
+
+Since in normal scanning $s_r$ = 1, $\alpha$ 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. For those
+interested, this section explains how the detector array size is
+calculated.
+
+For parallel geometry, the detector length is equal to the scan
+diameter.
+
+For divergent beam geometries, the size of the detector array also
+depends upon the \emph{focal length}. Increasing the \emph{focal
+length} decreases the size of the detector array while 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{$2\,\alpha$.}\latexignore{\emph{2 \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{$4\,f \tan (\alpha / 2)$}
+\latexignore{\emph{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}
+
 
 \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.
+
+\subsubsection{Multiple Computer Processing}
+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 reducing the higher
+frequencies. This is performed 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.