X-Git-Url: http://git.kpe.io/?a=blobdiff_plain;ds=inline;f=doc%2Fctsim-concepts.tex;h=e8de3431d4cf6806ef53901dbdf2f8d411fb2f21;hb=8ab133ae5386023557f5a008ef0d2b081ff288e5;hp=1d2d01bfe27a061aaf2b9d7fdec461f200feec94;hpb=33dd4470441860e1176a737ee4fd1bb80a200746;p=ctsim.git diff --git a/doc/ctsim-concepts.tex b/doc/ctsim-concepts.tex index 1d2d01b..e8de343 100644 --- a/doc/ctsim-concepts.tex +++ b/doc/ctsim-concepts.tex @@ -125,28 +125,28 @@ 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 +\rtfsp is then \latexignore{\\$$\emph{Pl x sqrt(2)}$$\\} \latexonly{$$p_d = p_l \sqrt{2}$$} CT scanners actually collect projections around a circle rather than a square. The diameter of this circle is also the diameter of the boundary square -\latexonly{$p_d$.}\latexignore{\emph{Pd}.} +\latexonly{$p_d$.}\latexignore{\rtfsp\emph{Pd}.} These relationships are diagrammed in figure 2.1. \subsubsection{View Diameter} The \emph{view diameter} is the area that is being processed during scanning of phantoms as well as during rasterization of phantoms. By default, the \emph{view diameter} -is set equal to the \emph{phantom diameter}. It may be useful, especially for +\rtfsp is set equal to the \emph{phantom diameter}. It may be useful, especially for experimental reasons, to process an area larger (and maybe even smaller) than the phantom. Thus, during rasterization or during projections, \ctsim\ will ask for a \emph{view ratio}, -\latexonly{$v_{R}$.}\latexignore{\emph{VR}.} +\latexonly{$v_r$.}\latexignore{\rtfsp \emph{VR}.} The \emph{view diameter} is then set as -\latexonly{$$v_d = p_d v_{R}$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$} +\latexonly{$$v_d = p_d v_r$$}\latexignore{\\$$\emph{Vd = Pd x VR}$$} By using a -\latexonly{$v_{R}$}\latexignore{\emph{VR}} +\latexonly{$v_r$}\latexignore{\emph{VR}} less than 1, \ctsim\ will allow for a \emph{view diameter} less than \emph{phantom diameter}. @@ -158,11 +158,11 @@ scanner that is larger than the scanner itself! By default, the entire \emph{view diameter} is scanned. For experimental purposes, it may be desirable to scan an area either larger or smaller than the \emph{view diameter}. Thus, the concept of \emph{scan ratio} -\latexonly{$s_{R}$}\latexignore{\emph{SR}} +\latexonly{$s_r$}\latexignore{\emph{SR}} is born. The scan diameter \latexonly{$s_d$}\latexignore{\emph{Sd}} is the diameter over which x-rays are collected and is defined as -\latexonly{$$s_d = v_d s_{R}$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\} +\latexonly{$$s_d = v_d s_r$$}\latexignore{\\$$\emph{Sd = Vd x SR}$$\\} By default and for all ordinary scanning, the \emph{scan ratio} is to \texttt{1}. If the \emph{scan ratio} is less than \texttt{1}, you can expect significant artifacts. @@ -172,15 +172,16 @@ The \emph{focal length}, \latexonly{$f$,}\latexignore{\emph{F},} is the distance of the X-ray source to the center of the phantom. The focal length is set as a ratio, -\latexonly{$f_{R}$,}\latexignore{\emph{FR},} +\latexonly{$f_r$,}\latexignore{\emph{FR},} of the view radius. Focal length is calculated as -\latexonly{$$f = (v_d / 2) f_R$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$} +\latexonly{$$f = (v_d / 2) f_r$$}\latexignore{\\$$\emph{F = (Vd / 2) x FR}$$} For parallel geometry scanning, the focal length doesn't matter. However, divergent geometry scanning (equilinear and equiangular), the \emph{focal length ratio} should be set at \texttt{2} or more to avoid artifacts. + \subsection{Parallel Geometry}\label{geometryparallel}\index{Concepts,Scanner,Geometries,Parallel} As mentioned above, the focal length is not used in this simple @@ -192,6 +193,7 @@ the default values of \texttt{1} for the \emph{view diameter ratio} and the \emph{scan diameter ratio}. If values of less than \texttt{1} are used for these two variables, significant distortions will occur. + \subsection{Divergent Geometries}\label{geometrydivergent}\index{Concepts,Scanner,Geometries,Divergent} \subsubsection{Overview} Next consider the case of equilinear (second generation) and equiangular @@ -206,27 +208,44 @@ See figure 2.2. \caption{Equilinear and equiangular geometries.} \end{figure} + \subsubsection{Fan Beam Angle} For these divergent beam geometries, the \emph{fan beam angle} needs to be calculated. For real-world CT scanners, this is fixed at the time of manufacture. \ctsim, however, calculates the \emph{fan beam angle}, -\latexonly{$\alpha$,}\latexignore{\emph{alpha},} -from the diameter of the \emph{scan diameter} and the \emph{focal length} -\latexignore{\\$$\emph{alpha = 2 x asin ( (Sd / 2) / F)}$$\\} -\latexonly{$$\alpha = 2 \sin^{-1} ((s_d / 2) / f)$$} +$\alpha$ from the diameter of the \emph{scan diameter} and the \emph{focal length} +\latexignore{\\$$\emph{alpha = 2 x asin ( (Sd / 2) / f)}$$\\} +\latexonly{\begin{equation}\label{alphacalc}\alpha = 2 \sin^{-1} ((s_d / 2) / f)\end{equation}} This is illustrated in figure 2.3. \begin{figure} \image{10cm;0cm}{alphacalc.eps} \caption{Calculation of $\alpha$} \end{figure} + Empiric testing with \ctsim\ shows that for very large \emph{fan beam angles}, greater than approximately -\latexonly{$120^{\circ}$,}\latexignore{120 degrees,} +\latexonly{$120^\circ$,}\latexignore{120 degrees,} there are significant artifacts. The primary way to manage the \emph{fan beam angle} is by varying the \emph{focal length} since the \emph{scan diameter} by the size of the phantom. +To illustrate, the \emph{scan diameter} can be defined as +\latexonly{$$s_d = v_r s_r p_d$$}\latexignore{\\$$Sd = Vr x Sr x Pd$$\\} + +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 +\latexignore{the above equation,}\latexonly{equation~\ref{alphacalc},} +We have, +\latexonly{ +\begin{eqnarray} +\alpha &= 2\,\sin^{-1} \frac{p_d / 2}{f_r (p_d / 2)} \nonumber \\ +&= 2\,\sin^{-1} (1 / f_r) +\end{eqnarray} +} + +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 @@ -257,6 +276,7 @@ and the \emph{focal length}. It is calculated as \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 @@ -301,4 +321,3 @@ filters for this purpose. 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. -