Sparsity regularization for image reconstruction with Poisson data

This work investigates three penalized-likelihood expectation maximization (EM) algorithms for image reconstruction with Poisson data where the images are known a priori to be sparse in the space domain. The penalty functions considered are the 1 norm, the 0 “norm,” and a penalty function based on the sum of logarithms of pixel values, R(x) = ∑np j=1 log (xj δ + 1 ) . Our results show that the 1 penalized algorithm reconstructs scaled versions of the maximum-likelihood (ML) solution, which does not improve the sparsity over the traditional ML estimate. Due to the singularity of the Poisson log-likelihood at zero, the 0 penalized EM algorithm is equivalent to the maximum-likelihood EM algorithm. We demonstrate that the penalty based on the sum of logarithms produces sparser images than the ML solution. We evaluated these algorithms using experimental data from a position-sensitive Compton-imaging detector, where the spatial distribution of photon-emitters is known to be sparse.