Fixed point algorithm in PET reconstruction

This paper presents a class of iterative reconstruction algorithms based on Amari´s α-divergence, which is parameterized by α, for positron emission tomography. We use the α-divergence to model the discrepancy between projections and estimates. Then a multiplicative update rule is developed to minimize it. Instead of directly optimizing the model, we solve the corresponding Kuhn-Tucker conditions as a non-linear system of equations by a fixed point algorithm. Three well-known algorithms (ML-EM, Anderson´s algorithm and Liu´s algorithm) are special examples in our method. Except for the ML-EM, the other two have not provided the rigorous proofs of convergence. Although we do not prove convergence for all the proposed algorithms, too, the case of Anderson is provided. The experiments were performed on both simulated phantom and real PET data to study the interesting and useful behavior of the method in cases where different parameters (α) were used.