A comparison of nested primal-dual forward-backward methods for Poisson image deblurring

We consider an inexact version of the popular Fast Iterative Soft-Thresholding Algorithm (FISTA) suited for minimizing the sum of a differentiable convex data fidelity function plus a nondifferentiable convex regularizer whose proximal operator is not computable in closed form. The proposed method is a nested primal-dual forward-backward method inspired by the methodology developed in [10], according to which the proximal-gradient point is approximated by means of a prefixed number of inner primal-dual iterates initialized with an appropriate warm-start strategy. We report some preliminary numerical results on a weighted least squares total-variation based model for Poisson image deblurring, which show the efficiency of the proposed FISTA-like method with respect to other strategies for defining the inner loop associated to the proximal step.