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. (1979) and by Inouye et al. (1989), both of whom worked in the spatial-frequency domain to estimate the 2-D Fourier transform of the unattenuated sinogram; by Hawkins et al. (1988), who worked with circularly harmonic Bessel transforms; and by Tretiak and Metz (1980), who followed filtering of appropriately-modified projections by exponentially-weighted backprojection. With perfect sampling, all four of these methods are exact in the absence of projection-data noise, but empirical studies have shown that they propagate noise differently, and no underlying theoretical relationship among the methods has been evident. Here, an analysis of the 2-D Fourier transform of the modified sinogram reveals that all previously-proposed linear methods can be interpreted as special cases of a broad class of methods, and that each method in the class can be implemented, in principle, by 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.