Object dependency of resolution in reconstruction algorithms with interiteration filtering applied to PET data

In this paper, we study the resolution properties of those algorithms where a filtering step is applied after every iteration. As concrete examples we take filtered preconditioned gradient descent algorithms for the Poisson log likelihood for PET emission data. For nonlinear estimators, resolution can be characterized in terms of the linearized local impulse response (LLIR). We provide analytic approximations for the LLIR for the class of algorithms mentioned above. Our expressions clearly show that when interiteration filtering (with linear filters) is used, the resolution properties are, in most cases, spatially varying, object dependent and asymmetric. These nonuniformities are solely due to the interaction between the filtering step and the Poisson noise model. This situation is similar to penalized likelihood reconstructions as studied previously in the literature. In contrast, nonregularized and postfiltered maximum-likelihood expectation maximization (MLEM) produce images with nearly "perfect" uniform resolution when convergence is reached. We use the analytic expressions for the LLIR to propose three different approaches to obtain nearly object independent and uniform resolution. Two of them are based on calculating filter coefficients on a pixel basis, whereas the third one chooses an appropriate preconditioner. These three approaches are tested on simulated data for the filtered MLEM algorithm or the filtered separable paraboloidal surrogates algorithm. The evaluation confirms that images obtained using our proposed regularization methods have nearly object independent and uniform resolution.