Reconstruction from Uniformly Attenuated SPECT Projection Data Using the DBH Method Page: 16 of 21
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Using (27) could probably help reduce the numerical errors in calculating the partial derivative with respect
top, which is usually under-sampled in SPECT imaging.
V. DISCUSSION AND CONCLUSION
The DBH method was applied to the inversion of the eRT for both parallel- and fan-beam data
acquisition geometries for SPECT. In the DBH method, the backprojection of the differentiated projections
was obtained first and an inverse of the finite weighted Hilbert transform was performed to restore the
image. Computer simulations show that the DBH algorithm produces a reconstruction of uniformly
attenuated SPECT data when the attenuation coefficient is 0.15cm". The effectiveness of our method was
demonstrated with the reconstruction of parallel-beam half-scan and fan-beam short-scan truncated
projections. However, issues such as sampling resolution, noise effects, and clinical efficacy have to be
We developed a matrix inversion method to numerically generate the kernel for the inverse of the finite
weighted Hilbert transform. Our numerical method consists of forming a small nonsingular matrix
(256x256 in our computer simulations) by numerically evaluating definite integrals in (23), which do not
contain any essential singularities. The singularity at s = t in (23) was transformed into a removable
singularity by adding an integral term equal to zero. This matrix does not have any zero eigenvalues. The
eigenvalues depend on the attenuation coefficient. For attenuation coefficients in the range of most clinical
nuclear medicine applications, our computer simulations indicate that the inversion of the matrix is stable
and insensitive to computer numerical errors.
The most important aspect of this work is the development of a stable method to obtain a numerical
kernel of the inverse of the finite weighted Hilbert transform. A new result in this paper is the application of
the DBH algorithm with this new kernel to parallel-beam half-scan, fan-beam short-scan, and truncated
data. This method requires H,(s) to be known in the interval (-q, q), whereas in  H,(s) is required in
the interval (-3q, 3q). In  a power series expansion method was used to solve for the inverse to the
integral equation in (19) starting with (20). Our proposed matrix inversion method started instead with (21)
to derive an inversion procedure. In  a different matrix inversion method was developed, where the
equation in (5) of this paper was first discretized and transformed into a system of linear equations, a matrix
inversion method was used to solve this system. Computer simulations showed that the method developed
in this paper is more stable than the one in .
The analytical reconstruction algorithm developed in this paper for the correction of uniform attenuation
is computationally more efficient than an iterative algorithm and may have potential applications in clinical
SPECT. A possible application is in brain imaging after the data have been compensated for the attenuation
of the bone. The two dimensional (2D) algorithm could also be extended to the development of a fully three
dimensional (3D) analytical reconstruction algorithm with uniform attenuation correction for breast
imaging with rotating slant hole collimation . This would be an important extension of the 2D
algorithm to fully 3D tomographic reconstruction. It is also possible that the DBH reconstruction method
could be applied to other more complicated geometries such as variable focal-length fan-beam geometries.
The DBH method is a new direction in the development of analytical reconstruction algorithms for
tomographic imaging. It provides important insight into understanding the limitations and possibilities for
reconstructing truncated projections with uniform attenuation correction. The development of this method
is actively being carried out in CT [37-39] and has potential for SPECT [28-30] with uniform attenuation.
There is potential of extending our algorithm to fully 3D applications and to the reconstruction of cone-
beam data with correction for uniform attenuation . A more difficult problem, which is still unsolved, is
the development of an analytical algorithm for reconstructing an ROI from truncated projections with a non
uniform attenuation distribution.
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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/m1/16/: accessed June 17, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.