On Wiener-type filters in SPECT

For 2D data with Poisson noise we give explicit formulae for the optimal space-invariant Wiener-type filter with some a priori geometric restrictions on the window function. We show that, under some natural geometric condition, this restrictedly optimal Wiener-type filter admits a very efficient approximation by an approximately optimal filter with an unknown object power spectrum. Generalizations to the case of some more general noise model are also given. Proceeding from these results we (a) explain, in particular, an efficiency of some well-known '1D' approximately optimal space-invariant Wiener-type filtering scheme with an unknown object power spectrum in single-photon-emission-computed tomography (SPECT) and positron emission tomography (PET) imaging based on the classical filtered back-projection (FBP) algorithm or its iterative use and (b) also propose an efficient 2D approximately optimal space-invariant Wiener-type filter with an unknown object power spectrum for SPECT imaging based on the generalized FBP algorithm (implementing the explicit formula for the nonuniform attenuation correction) and/or the classical FBP algorithm (used iteratively). An efficient space-variant version of the latter 2D filter is also announced. Numerical examples illustrating the aforementioned results in the framework of simulated SPECT imaging are given.