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Fast preconditioned conjugate gradient reconstruction for 2D SPECT

To be clinically useful, reconstruction in SPECT should be accomplished rapidly and also result in good image quality. Penalized likelihood methods are desirable in that they can easily incorporate system and noise models as well as prior information into an objective function. The reconstruction, computed by maximizing the objective, requires an iterative method, and this can be slow. One class of optimization methods that has been used for SPECT are gradient methods, wherein at each iteration, a search direction is established and the objective maximized along this direction. Of the many variants of gradient methods, preconditioned conjugate gradient (PCG) methods have shown promise for their speed. The speed for PCG depends on choice of a preconditioner. In this work, we follow the development in [1] for the derivation of fast preconditioners for a penalized weighted least-squares objective, but we add some approximations specific to SPECT.