Reconstruction from Uniformly Attenuated SPECT Projection Data Using the DBH Method Page: 3 of 21
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3
II. DBH METHOD FOR INVERSE OF THE EXPONENTIAL RADON TRANSFORM
For a transaxial slice, let f(x, y) represent the distribution of a radiopharmaceutical in body tissues, which
is assumed to be a smooth and compactly supported function ofR2. The SPECT image reconstruction
estimates f(x, y) from the detected photon counts. We denote r= (x, y) and D ={(x, y) E 2y2 y }.
We assume f(x,y) 0 outside of D and the attenuation of the body tissues is uniform inside the elliptical
grey region in Fig. 1. For example, these assumptions can be met in brain SPECT after compensating for
the attenuation of the skull [40]. We denote 0 = (cos 9, sin 9) and ' = (- sin 9, cos 9). Let p >o be the constant
attenuation coefficient and Las be the distance between the point C on the boundary of the elliptical
attenuator and the s-axis in Fig. 1. All photons detected in the collimator hole at (9, s) are proportional to
f f(sO+tO')e- , -L0,,-t4dt; here the integral is carried out along the line PC. The factor e-40- can be estimated
if the boundary of the uniform attenuation region is known. After multiplying the measured projections by
e~'1', the modified projections can be expressed as the eRT
[R f](,s)= Jf(s B+t e'dt . (1)
For simplicity, we use p(9, s) = [Rf](9, s) throughout this paper. The reconstruction process estimates
f(x, y) from p(O, s).
Y
Collimation hole
Convex uniform attenuator Isotope distrbution:f(xy)
Fig. 1: Illustration of projection coordinates corresponding to a parallel-beam scanning geometry for SPECT.
The DBH method involves calculating the derivative of the projection data, backprojection, and the
inversion of the finite weighted Hilbert transform. The first two steps, the operation of derivative and
backprojection, are also named as the DBP operation in [37] for the reconstruction of CT data acquired over
less than 3600. According to [28], the DBP operation of the eRT is the one-dimensional (1D) weighted
Hilbert transform of the original image along certain lines. Through coordinate transformations, we obtain a
similar relation for the uniformly attenuated fan-beam projection data. Based on work in [30], an exact
reconstruction can be achieved in a well-defined subset of the support of f(x,y) from the DBP of truncated
projections.
A. DBP operation and the weighted Hilbert transform
We define an intermediate function f(x,y) as
1/2
f(x, y) = J e-''u p'(0, - 6)d, (2)
2 -zi2
where p'(9,s) is the partial derivative of p(9, s) with respect to the variable s. Equation (2) is usually called
the operation of derivative and backprojection (DBP). The DBP operation yields an intermediate function
f(x,y), which is different from the desired function f(x,y) . Let= (a /ax, a /ay), then from (1) we have
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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/m1/3/: accessed October 23, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.