# 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 [51]. 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 [51]. 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

precise.

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. (https://digital.library.unt.edu/ark:/67531/metadc901800/m1/13/: accessed March 19, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.