# Reconstruction from Uniformly Attenuated SPECT Projection Data Using the DBH Method Page: 5 of 21

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radius of the circular focal point trajectory. The coordinates in parallel- and fan-beam geometries are related

by

s=Rsina, 9=a+/3. (7)

The distance between the focal point and the s-axis is R cos o . Thus, the definitions of (1) and (6) can be

related by the following equation

p(o-+ ,8,Rsino-) = e-R cosa[Duf]($, C) = g($,C) (8)

We will use g(6, a) for the modified projection of (8). By the chain rule, we have -(/(#,6)= - (,s) and

a6 (#,6) = (,s)+Rcos (,s). (9)

Then, we obtain

ap 1 a a

as (0, s) = [(a -- )g](3,o). (10)

aJs Rcoso d3Jc 3J/

Let r=JHJ and ?=(x,y)=r(cos p,sin(), and K(r,p,3) denotes the length of SP, and 7'(r, , p) the angle

between SO and SP. It is straightforward to derive the following geometric identities [41]:

K(r,p,,$)= r2 + R2 +2rRsin(- p) , (11)

a'(r, p, p) = arctan r cos(p - () (12)

R + rsin(# -(o)

(6, c)= (sin(g + C), - cos(# + o)). (13)

For a given F, 9, and 6 the following differential relationship is satisfied [42]:

d9 = d(3+ C'(r, p, ,8)) = Rcos a'(r,p,#) d,8.

K(r, (o,,$)

Focal POmtl S

Fig. 2: Illustration of a typical fan-beam acquisition geometry. Each projection ray is uniquely determined by (#8, a). Points 0 and S represent the

coordinate origin and focal point, respectively. P is a point at which the value needs to be reconstructed, a denotes the angle between OS and the

projection ray, CT' denotes the angle between OS and PS, #8 denotes the angle between OS and the y-axis, and R is the radius of the circular orbit.

For simplicity, only the fan-beam short-scan reconstruction will be discussed in this subsection. The short-

scan is the scanning range of [-0.5ff - CTm,, 0.5ff+ C.], which is 1800 plus the fan angle. We assume R sinT, =1

so that the inner disk in Fig. 3 is the unit disk D, i.e., the projection is not truncated. In Fig. 3 fory E (-1, 1) ,

#y= arcsin(y/R). After transforming the integration over 9 in (2) to the integration over p3 in the fan-

beam geometry, (2) becomes:

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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/m1/5/: accessed September 25, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.