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 for the derivation of fast preconditioners for a penalized weighted least-squares objective, but we add some approximations specific to SPECT. In particular, we make use of our knowledge of the nature of attenuation and depth-dependent blur in SPECT in designing the preconditioner. The resulting preconditioners lead to tremendous speedups for penalized weighted least-squares objectives, but are still not practical in that extensive computation is required just to obtain the preconditioner itself. With further approximations, however, the preconditioners become practical to compute and lead to enhanced reconstruction speed relative to using no preconditioner (conjugate gradient) or commonly used diagonal preconditioners. We show results for 2 cases: (1) Poisson noise, attenuation, and depth independent blur, and (2) same, but with depth dependent blur.