Reconstruction from Uniformly Attenuated SPECT Projection Data Using the DBH Method Page: 11 of 21
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concentration. These ellipses were described in , and were scaled to be contained in the unit disk D in
our simulations. The phantom is represented by a 256x256 grid. In terms of pixels, the lengths of the major
and minor axes of the most outer ellipse are 184 and 138, respectively. The three detectors around the
phantom indicate the range of the projections being sampled. Projection data were evenly sampled on a grid
of 400x256 over [ir/2, 3/2)x(-1,1), and are shown in Fig. 6(b). All simulated projection data were
analytically generated based on the weighted line integrals of the modified Shepp-Logan phantom with a
uniform attenuation coefficient of 0.15 cm1.
0.1 0.1 cm
10.8 cm -_>
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Fig. 6: (a) The modified Shepp-Logan phantom. The numbers in the figure without dimensions indicate the relative distributions of the radiopharmaceutical
for each of the ellipses. For example, the five small ellipses have a value 0.4. These values were scaled to obtain the total desired counts in the statistical
studies. The detectors around the phantom indicate the range of the sampled projections. (b) The analytically computed noise-free projection data. The view
angle is along the vertical axis and the detector bin is along the horizontal axis.
In the study of the effect of noise with the DBH method, we converted the analytically generated
projection data into Poisson random samples. The total counts were 2x 107, with each bin containing about
200 photons on average. Projection data were first modified to place the detector at the center of rotation, as
explained in Section II. Once the modified projections p(O,s) were obtained, the DBP image f(x,y) was
calculated by (2) on a 256x256 grid, the same size used to sample the original modified Shepp-Logan
phantom. At each fixed view angle, the modified projections were convolved with a discrete band-limited
five-point kernel to yield the derivative p'(0, s). The numerical implementation of the derivative was a mid-
point method. The DBP image was obtained by backprojecting the derivative over 1800 with an exponential
weighting function e-"' . Then the finite inverse Hilbert transform (26) was performed to find f(x,y)
along each row. In other words, using the four steps in Section III, the desired image f(x,y) (corresponding
to h(t = x)) at each fixed y is reconstructed from its weighted Hilbert transform f(x,y) (corresponding to
H, (s) = f(x = s, y)) at the same y .
The reconstructed images from noise-free and noisy projection data are shown in Fig. 7(a). The grayscale
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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. (https://digital.library.unt.edu/ark:/67531/metadc901800/m1/11/?rotate=90: accessed May 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.