Reconstruction from Uniformly Attenuated SPECT Projection Data Using the DBH Method Page: 2 of 21
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explicit formulas. The developments of analytical methods have led to new insights into the mathematics of
computerized tomography in SPECT. These insights have spurred the development of new reconstruction
algorithms for various collimator geometries and detector orbit configurations with both uniform [13-15]
and non-uniform  attenuation correction. Analytical algorithms also provide theoretical consistency
conditions. These conditions are extremely important in the evaluation and understanding of the
performance of iterative reconstruction algorithms and for determining if they may provide accurate
solutions. Beyond the theoretical aspects there are practical reasons for developing analytical reconstruction
algorithms. Analytical methods provide: a) a quick reconstruction that allows evaluation of subject
positioning and determination of whether or not an experiment needs to be redone, b) a method for
analytical evaluation of various factors that affect resolution and signal-to-noise, which are difficult to do
with a non-linear iterative algorithm, c) a means by which to compare and to evaluate the development of
iterative reconstruction methods, and d) a method useful in quality control; for example, the evaluation of
whether or not geometrical calibrations, such as the center of rotation, are measured accurately.
There is considerable interest in developing algorithms that are able to reconstruct data acquired for less
than 3600 because of potential clinical applications. The data from a half-scan acquisition is sufficient for an
exact reconstruction for the attenuation-free case . However, analytical methods [14, 15] developed
earlier for uniform attenuation required the projection data to be available over a full-scan. The same
requirement was necessary for other analytical reconstruction methods such as the Fourier methods in [17-
22], the integral geometry method in , and the Cormack-type inversion methods in [24, 25]. Recently
progress has been made in the reconstruction of attenuated half-scan projection data [26-30]. The
uniqueness of analytical reconstruction solutions when the scanning range is an open subset of [0, 3600)
was proven in [31, 32], and stable reconstructions for half-scan data were later proposed for iterative
algorithms in [26, 27] and an analytical algorithm in . The method in  was an explicit formula of
the convolution-backprojection type and was designed to reconstruct images from non-truncated
measurements in the parallel-beam geometry. Some progress has been made in developing algorithms
which are capable of dealing with fan-beam data and truncated measurements [29, 30]. The method in 
performs least-squares fitting to find a kernel function numerically and the algorithm in  utilizes the
calculation of a power series expansion. The algorithms in [28-30] can be implemented in a common
procedure of derivative, backprojection, and Hilbert transform (DBH). In the DBH method, one of the key
steps is the inversion of the finite Hilbert transform, which is an old research topic surveyed in . The
recent works [34-36] show progress on that subject, aiming at an exact image reconstruction from data
acquisitions of less than 3600.
The DBH method has been used in x-ray CT to reconstruct a region-of-interest (ROI) from truncated data
[37-39]. Recently, progress has been made [28, 30] in applying the DBH method in the reconstruction of
truncated uniformly attenuated SPECT data either obtained by half-scan (parallel geometry) or short-scan
(fan-beam) acquisitions. This paper further explores the DBH method for the analytical reconstruction of
exponential Radon transform (eRT) data acquisitions of less than 3600 in both parallel- and fan-beam
acquisition geometries and develops a stable method to obtain a numerical kernel for the inverse of the
finite weighted Hilbert transform. We demonstrate that the DBH method is effective in reconstructing an
ROI from truncated half-scan (parallel-beam) or short-scan (fan-beam) uniformly attenuated data.
The paper is organized in five sections. In Section I, we motivate the problem under consideration. In
Section II, the DBH method is developed for the 2D parallel-beam and fan-beam geometries. Section III
develops the numerical implementation of the finite weighted Hilbert transform. Section IV gives
implementation details and numerical results demonstrating the effectiveness of the DBH method. In
Section V, we offer our conclusions and directions for further work.
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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/m1/2/: accessed August 20, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.