Reconstruction from Uniformly Attenuated SPECT Projection Data Using the DBH Method Page: 13 of 21
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the projections were sampled from E [Z/2, 3x/2]. Therefore, the backprojection of the differentiated
projections are weighted by a factor greater than 1 in the upper half of the image because T.6' <0 and less
than 1 in the lower half of the image because .6' > 0. Thus, the reconstruction in the upper half of the
image is noisier.
For those pixels being generally further from the detector, this effect of the noise being worse can be
observed from the variance images for three different noise levels in Fig. 8. For total counts of 1x 105,
5x 105 and 1 x 106, we performed 1000 simulations without truncated projections with random noise
generated for each realization. The ensemble variances [49, 50] of the reconstructed images were calculated
and are shown in Fig. 8 for the three noise levels. The brightness at each pixel is the value of the variance at
that pixel. The images in Fig. 8 show higher variance in the top-half of each image. Also, it is clear that the
higher the total counts, the lower the ensemble variance. The mean and the standard deviation of the value
for the pixel at the origin (x, y) = (0,0) are tabulated in Table 1. The Tests 1, 2, and 3 refer to the tests
performed for total counts of 1x 105, 5x105 and 1x 106, respectively. When the counts are low, we observe
that the noise causes large deviations. In order to investigate how well the sample mean estimates the
population mean we calculated the 95% confidence interval (K- / J- t, V+o-/J- t) for the sample mean
X with sample standard deviation c . For n =1000 realizations, the degrees of freedom is 999 giving a
critical value of t=t0.025,n-1 =1.962 for the t-distribution from the table on page 485 of . The lower and
upper confidence limits were calculated and are tabulated in the last two columns of Table 1. As can be
seen in Fig. 9, the confidence interval shrinks as the noise decreases, which means the estimation is more
Test Sample Sample _ _
# mean deviation - to 025,n-1 + to.o25,n-
1 0.2821 0.2263 0.268059 0.296141
2 0.2897 0.1045 0.283216 0.296184
3 0.2913 0.0759 0.286591 0.296009
4 0.3050 0.2304 0.290705 0.319295
5 0.2861 0.1056 0.279548 0.292652
6 0.2848 0.0474 0.281859 0.287741
Table 1: Statistics used in the calculation of the 95% confidence interval for the sample means. Tests 1, 2, 3 are for non-truncated data with total counts of
1 X 105, 5 X 105 and 1 X 106, and Tests 4, 5, 6 are for truncated data with total counts of 1 X 105, 5 X 105 and 1 X 106 The limits of the 95% confidence interval in
the last two columns are displayed in Fig. 9. When n =1000 , to.025,n-1 =1.962 .
For the study with truncated data, we considered the detector to sample only the central 156 rays of the
modified Shepp-Logan phantom at each angle as illustrated for the elliptical object in Fig. 4. This small
detector defines the FOV to be within the thick circle with a radius of 78. Recall that the maximum length
of the ellipse of the modified Shepp-Logan phantom was 184; thus, a total of 28 pixels along the major axis
were truncated. As pointed out previously, the derivative and backprojection are local operations; thus, the
values of the intermediate function f(x,y) between the two dashed lines illustrated in Fig. 4 are sufficient
to provide an exact reconstruction of f(x,y) within that region. In this simulation, 400x 156 noise-free and
noisy truncated projection data were generated, and (2) was then used to backproject the derivative of the
truncated data. The inverse (26) of the finite weighted Hilbert transform was applied to reconstruct the
object function f(x,y) between the two dashed lines in Fig. 4.
The reconstructed images are shown in Fig. 10. Three variance images (see Fig. 11) were calculated from
1000 noise realizations with truncation for total counts of 1x105, 5x105 and 1x106, respectively. The 95%
confidence intervals for the mean value of the pixel at the origin (x,y) = (0,0) are tabulated in Table 1. The
Tests 4, 5, and 6 refer to the tests performed for the truncated case for total counts of 1x105,
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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/m1/13/: accessed December 15, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.