Reconstruction from Uniformly Attenuated SPECT Projection Data Using the DBH Method Page: 4 of 21
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4
p'(9,s)= a Jf(s6+tV-)e-'dt = [6-OVf](sO+tV )e'dt. (3)
Notice that by using the chain rule, we obtain
[.5](x-sinO,y+cosO)= -cos[ ](x-sinO,y+cosO) -sinO[-](x-sin0,y+cosO) (4)
Jy
= -z[6- Vf](x - z sinO, y + zcos B).
By changing the variables r = t- r -6 and using (4), we can rewrite f(x,y) as
f(x,y)= - { j[6- Vf](s0+toI)e-(tro 'dt}_ d9
- c / 2 --o
-~ Je - Jiiof](x - isi 9 , y + iCos 9)d~di
pr12
-[f](x-isin0,y+ cosO)d~dr
1 1e '[f(x -i,y) -f(x+z,y)] d
- cosh(pr)f(x - zy) dz.
This represents f(x,y) as the hyperbolic-cosine-weighted Hilbert transform of f(x,y) along horizontal
lines. It follows that the reconstruction of f(x,y) reduces to the inversion of (5). For p = 0, this is called the
inversion of the Hilbert transform [33]. For p : 0 , several formulas have recently been reported in [28, 30]
to obtain f(x,y) from (5). In this paper, we refer to the DBP operation followed by the inversion of a finite
weighted Hilbert transform as the DBH method for the exponential Radon transform. Note that the term
"finite" refers to the limits of the last integration in (5) as being finite.
Due to the uniform attenuation, the derivative with respect to s is, in general, no longer odd, i.e.,
p'(9,?-0) # -p'(9+z,-F -6). However, the weighted backprojection of the derivative over 360 is still zero.
This property can be seen from (5) by replacing (-f/2,f/2), the interval of integration, with (0,2z). We
also prove for a point source in Appendix A that the weighted backprojection of the derivative over 3600 is
zero. Due to the superposition property of a linear algorithm, the weighted backprojection
for any arbitrary object is zero.
B. DBP operation for fan-beam data
A typical fan-beam data acquisition geometry with a circular focal-point trajectory is shown in Fig. 2,
where each projection ray is represented by (#,o). One particular projection ray is the arrowed line from the
focal point S for the angle 6 with the ray angle a. In this paper, the fan-beam uniformly attenuated
projection of the function f(x,y) is defined as
[Df]($, a)= f f(S+ rd(,6, ))e"d, (6)
where Df is the projection operator for the uniformly attenuated fan-beam projection data, a E [-m, ajnI,
and a(,8,o) is a unit vector in R2 representing the direction from the focal point to the collimation hole, as
shown in Fig. 2. Here, 6,m E (0, z/2) denotes the maximum angle subtended by the fan-beam. Let R be the
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Huang, Qiu; You, Jiangsheng; Zeng, Gengsheng L. & Gullberg, Grant T. Reconstruction from Uniformly Attenuated SPECT Projection Data Using the DBH Method, article, March 20, 2008; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc901800/m1/4/: accessed December 16, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.