X-Git-Url: http://git.kpe.io/?a=blobdiff_plain;f=doc%2Fctsim-concepts.tex;fp=doc%2Fctsim-concepts.tex;h=417c47a90087bffcbcfe86c0872408f83d234e1a;hb=676f8753e1b3edd337240391855f34dde1af24fa;hp=c7c3ffd39350bd3a66b965e2c8aa25e2b5dcc8c9;hpb=7c695eef645b25a1df6a95c4fc72f6b2994a3630;p=ctsim.git diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex index c7c3ffd..417c47a 100644 --- a/doc/ctsim-concepts.tex +++ b/doc/ctsim-concepts.tex @@ -8,13 +8,13 @@ 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. +object. These reconstructions can be visually and 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}. +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 primary objects that we need to be +concerned with are the \emph{phantom} and the \emph{scanner}. \section{Phantoms}\label{conceptphantom} \subsection{Overview}\label{phantomoverview}\index{Phantom Overview}% @@ -23,7 +23,7 @@ object, but the two objects we need to be concerned with are the 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 +elements, the 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 @@ -70,23 +70,23 @@ 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}. +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. +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 drawn between those two points and lies +below the x-axis. The sector is then rotated and translated the +same as a segment. \subsection{Phantom Size}\index{Phantom size} The overall dimensions of the phantom are increased by 1\% above the @@ -106,13 +106,13 @@ 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\%. +In general, the geometry for a scan all starts with 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, the \emph{phantom diameter}. Remember, as mentioned +above, the phantom dimensions are also padded by 1\%. The other important geometry variables for scanning phantoms are the \emph{view diameter}, \emph{scan diameter}, and \emph{focal @@ -129,12 +129,12 @@ 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}$$} CT scanners actually collect projections around a +\latexonly{$p_d$,}\latexignore{\emph{Pd},} is then +\latexignore{\\\centerline{\emph{Pl x sqrt(2)}}\\} +\latexonly{\begin{equation}p_d = p_l \sqrt{2}\end{equation}} +CT scanners actually collect projections around a circle rather than a square. The diameter of this circle is also -the diameter of the boundary square -\latexonly{$p_d$. These +the diameter of the boundary square \latexonly{$p_d$. These relationships are diagrammed in figure~\ref{phantomgeomfig}.} \latexignore{emph{Pd}.} @@ -147,8 +147,9 @@ experimental reasons, to process an area larger (and maybe even smaller) than the phantom. Thus, during rasterization or during projections, \ctsim\ will ask for a \emph{view ratio}, \latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.} The \emph{view -diameter} is then calculated as \latexonly{$$v_d = p_d -v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$} +diameter} is then calculated as +\latexonly{\begin{equation}v_d = p_dv_r\end{equation}} +\latexignore{\\\centerline{\emph{Vd = Pd x VR}}\\} By using a \latexonly{$v_r$}\latexignore{\emph{VR}} @@ -156,7 +157,7 @@ less than 1, \ctsim\ will allow for a \emph{view diameter} less than \emph{phantom diameter}. This will lead to significant artifacts. Physically, this would -be impossible and is analagous to inserting an object into the CT +be impossible and is analogous to inserting an object into the CT scanner that is larger than the scanner itself! \subsubsection{Scan Diameter}\index{Scan diameter} @@ -166,8 +167,10 @@ larger or smaller than the \emph{view diameter}. Thus, the concept of \emph{scan ratio}, \latexonly{$s_r$,}\latexignore{\emph{SR},} is arises. The scan diameter \latexonly{$s_d$}\latexignore{\emph{Sd}} is the diameter over -which x-rays are collected and is defined as \latexonly{$$s_d = -v_d s_r$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\} By default and +which x-rays are collected and is defined as +\latexonly{\begin{equation}s_d =v_d s_r\end{equation}} +\latexignore{\\\centerline{\emph{Sd = Vd x SR}}\\} +By default and for all ordinary scanning, the \emph{scan ratio} is to \texttt{1}. If the \emph{scan ratio} is less than \texttt{1}, you can expect significant artifacts. @@ -180,7 +183,8 @@ the phantom. The focal length is set as a ratio, \latexonly{$f_r$,}\latexignore{\emph{FR},} of the view radius. Focal length is calculated as -\latexonly{$$f = (v_d / 2) f_r$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$} +\latexonly{\begin{equation}f = (v_d / 2) f_r\end{equation}} +\latexignore{\\\centerline{\emph{F = (Vd / 2) x FR}}} For parallel geometry scanning, the focal length doesn't matter. However, divergent geometry scanning (equilinear and equiangular), @@ -192,7 +196,8 @@ source inside of the \emph{view diameter}. \subsection{Parallel Geometry}\label{geometryparallel}\index{Parallel Geometry} -As mentioned above, the focal length is not used in this simple +The simplest geometry, parallel, was used in \mbox{$1^{st}$} generation +scanners. As mentioned above, the focal length is not used in this simple geometry. The detector array is set to be the same size as the \emph{scan diameter}. For optimal scanning in this geometry, the \emph{scan diameter} should be equal to the \emph{phantom @@ -210,7 +215,7 @@ the x-ray beams diverge from a single source to a detector array. In the equilinear mode, a single source produces a fan beam which is read by a linear array of detectors. If the detectors occupy an arc of a circle, then the geometry is equiangular. -\latexonly{See figure~\ref{divergentfig}.} +\latexonly{The configurations are shown in figure~\ref{divergentfig}.} \begin{figure} \image{10cm;0cm}{divergent.eps} \caption{\label{divergentfig} Equilinear and equiangular geometries.} @@ -241,7 +246,8 @@ there are significant artifacts. The primary way to manage the \emph{scan diameter} by the size of the phantom. To illustrate, the \emph{scan diameter} can be defined as -\latexonly{$$s_d = s_r v_r p_d$$}\latexignore{\\$$Sd = Sr x Vr x Pd$$\\} +\latexonly{\begin{equation}s_d = s_r v_r p_d\end{equation}} +\latexignore{\\\centerline{\emph{Sd = Sr x Vr x Pd}}\\} Further, $f$ can be defined as \latexonly{$$f = f_r (v_r p_d / 2)$$}\latexignore{\\$$F = FR x (VR x Pd)$$\\} @@ -253,7 +259,7 @@ equation,}\latexonly{equation~\ref{alphacalc},} We have, \alpha &= 2\,\sin^{-1} \frac{s_r v_r p_d / 2}{f_r v_r (p_d / 2)} \nonumber \\ &= 2\,\sin^{-1} (s_r / f_r) \end{eqnarray} -} \latexignore{\\$$\alpha = 2 sin (Sr / Fr$$\\} +} \latexignore{\\\centerline{\emph{\alpha = 2 sin (Sr / Fr)}}\\} Since in normal scanning $s_r$ = 1, $\alpha$ depends only upon the \emph{focal length ratio}. @@ -285,8 +291,9 @@ figure~\ref{equiangularfig} indicates the positions of the detectors in this cas 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)}} +length}. It is calculated as +\latexonly{\begin{equation}4\,f \tan (\alpha / 2)\end{equation}} +\latexignore{\\\centerline{\emph{4 x F x tan(\alpha/2)}}} \begin{figure}\label{equilinearfig} \image{10cm;0cm}{equilinear.eps} \caption{\label{equilinearfig} Equilinear geometry} @@ -294,11 +301,8 @@ length}. It is calculated as \latexonly{$4\,f \tan (\alpha / 2)$} \latexonly{This geometry is shown in figure~\ref{equilinearfig}.} -\subsubsection{Examples of Geometry Settings} - - \section{Reconstruction}\label{conceptreconstruction}\index{Reconstruction Overview}% -\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 @@ -314,11 +318,12 @@ 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 +processors. This has been testing in a 16-CPU cluster with excellent results. \subsubsection{Filter projections} -The projections for a single view have their frequency data multipled by +The first step in filtered backprojection reconstructions is the filtering +of each projection. The projections for a each view have their frequency data multipled by a filter of $|w|$. \ctsim\ permits four different ways to accomplish this filtering. @@ -342,8 +347,9 @@ interpolation can be specified. Images can be compared statistically. Three measurements can be calculated by \ctsim. They are taken from the standard measurements used by Herman\cite{HERMAN80}. -$d$ is the standard error, $e$ is the maximum error, and -$r$ is the maximum error of a 2 by 2 pixel area. +$d$ is the normalized root mean squared distance measure, +$r$ is the normalized mean absolute distance measure, +and $e$ is the worst case distance measure over a $2\times2$ area. To compare two images, $A$ and $B$, each of which has $n$ columns and $m$ rows, these values are calculated as below. @@ -351,7 +357,8 @@ these values are calculated as below. \latexonly{ \begin{equation} -d = \frac{\sum_{i=0}^{n}{\sum_{j=0}^{m}{(A_{ij} - B_{ij})^2}}}{m n} +d = \sqrt{\frac{\sum_{i=1}^{n}{\sum_{j=1}^{m}{(A_{ij} - B_{ij})^2}}} + {\sum_{i=1}^{n}{\sum_{j=1}^{m}{(A_{ij} - A^{\_})^2}}}} \end{equation} \begin{equation} r = \max(|A_{ij} - B{ij}|)