A UNIFIED ANALYSIS OF EXACT METHODS OF INVERTING THE 2-D EXPONENTIAL RADON-TRANSFORM, WITH IMPLICATIONS FOR NOISE-CONTROL IN SPECT

Exact methods of inverting the two-dimensional (2-D) exponential Radon transform have been proposed by Bellini et al. and by Inouye et al., both of whom worked in the spatial-frequency domain to estimate the 2- D Fourier transform of the unattenuated sinogram; by Hawkins et al., w ho worked with circularly harmonic Bessel transforms; and by Tretiak a nd Metz, who followed filtering of appropriately-modified projections by exponentially-weighted backprojection. With perfect sampling, all f our of these methods are exact in the absence of projection-data noise , but empirical studies have shown that they propagate noise different ly, and no underlying theoretical relationship among the methods has b een evident. In this paper, an analysis of the 2-D Fourier transform o f the modified sinogram reveals that all previously-proposed linear me thods can be interpreted as special cases of a broad class of methods, and that each method in the class can be implemented, in principle, b y any one of four distinct techniques. Moreover, the analysis suggests a new member of the class that is predicted to have noise properties better than those of previously-proposed members.