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<head><title>Divergent Geometries</title></head>
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<A HREF="ctsim_contents.html">Contents</A> <A HREF="ctsim15.html#conceptscanner">Up</A> <A HREF="ctsim22.html#geometryparallel"><<</A> <A HREF="ctsim24.html#topic16">>></A> </CENTER><HR>
<H3>Divergent Geometries</H3>
For both equilinear (second generation) and equiangular
(third, fourth, and fifth generation) geometries,
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.
<CENTER><img src="divergent.gif"></CENTER>
<P>
<A HREF="ctsim24.html#topic16"><B>Fan Beam Angle</B></A><BR>
For these divergent beam geometries, the <EM>fan beam angle</EM>
needs to be calculated. For real-world CT scanners, this is fixed
at the time of manufacture. <TT>CTSim</TT>, however, calculates the
<EM>fan beam angle</EM>, alpha, from the <EM>scan diameter</EM> and
the <EM>focal length</EM> as
<CENTER><EM>alpha = 2 x asin (
(Sd / 2) / f)</EM></CENTER>
<CENTER><img src="alphacalc.gif"></CENTER>
<P>
Empiric testing with <TT>CTSim</TT> shows that for very large <EM>fan beam angles</EM>,
greater than approximately
120 degrees,
there are significant artifacts. The primary way to manage the
<EM>fan beam angle</EM> is by varying the <EM>focal length</EM> since the
<EM>scan diameter</EM> is usually fixed at the size of the phantom.<P>
To illustrate, the <EM>scan diameter</EM> can be defined as
<BR>
<CENTER><EM>Sd = Sr x Vr x Pd</EM></CENTER><BR>
<P>
Further, the <EM>focal length</EM> can be defined as
<BR>
<CENTER><EM>F = FR x (VR x Pd)<CENTER></CENTER><BR>
</EM></CENTER><P>
Substituting these equations into the above
equation, We have,
<BR>
<CENTER><EM>alpha = 2 sin (Sr / Fr)</EM></CENTER><BR>
<P>
Since in normal scanning s_r = 1, alpha depends only upon the
<EM>focal length ratio</EM> in normal scanning.<P>
<A HREF="ctsim25.html#topic17"><B>Detector Array Size</B></A><BR>
In general, you do not need to be concerned with the detector
array size -- it is automatically calculated by <TT>CTSim</TT>. For the
particularly interested, this section explains how the detector
array size is calculated.<P>
For parallel geometry, the detector length is simply the scan
diameter.<P>
For divergent beam geometries, the size of the detector array also
depends upon the <EM>focal length</EM>: increasing the <EM>focal
length</EM> decreases the size of the detector array.<P>
For equiangular geometry, the detectors are equally spaced around a arc
covering an angular distance of alpha as viewed from the source. When
viewed from the center of the scanning, the angular distance is
<BR>
<EM>pi + alpha - 2 x acos ((Sd / 2) / C))</EM><BR>
The dotted circle
indicates the positions of the detectors in this case.<P>
<CENTER><img src="equiangular.gif"></CENTER>
<P>
For equilinear geometry, the detectors are equally spaced along a straight
line. The detector length depends upon
<EM>alpha</EM> and the <EM>focal
length</EM>. This length,
Dl, is calculated as
<BR>
<CENTER><EM>2 x (F + C) x tan(alpha/2)</EM></CENTER>
<CENTER><img src="equilinear.gif"></CENTER>
<P>
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