Primal-Dual Algorithms for Total Variation Based Image Restoration under Poisson Noise Dedicated to Professor Lin Qun on the Occasion of his 80th Birthday

We consider the problem of restoring images corrupted by Poisson noise. Under the framework of maximum a posteriori estimator, the problem can be converted into a minimization problem where the objective function is composed of a Kullback-Leibler (KL)-divergence term for the Poisson noise and a total variation (TV) regularization term. Due to the logarithm function in the KL-divergence term, the non-differentiability of TV term and the positivity constraint on the images, it is not easy to design stable and efficiency algorithm for the problem. Recently, many researchers proposed to solve the problem by Alternating Direction Method of Multipliers (ADMM). Since the approach introduces some auxiliary variables and requires the solution of some linear systems, the iterative procedure can be complicated. Here we formulate the problem as two new constrained minimax problems and solved them by Chambolle-Pock’s first order primal-dual approach. The convergence of our approach is guaranteed by their theory. Comparing with ADMM approaches, our approach requires about half of the auxiliary variables and is matrix-inversion free. Numerical results show that our proposed algorithms are efficient and outperform the ADMM approach.