# 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 May 24, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.