A globally convergent algorithm for a class of gradient compounded non-Lipschitz models applied to non-additive noise removal

Non-Lipschitz regularization has got much attention in image restoration with additive noise removal recently, which can preserve neat edges in the restored image. In this paper, we consider a class of minimization problems with gradient compounded non-Lipschitz regularization applied to non-additive noise removal, with Poisson and multiplicative one as examples. The existence of a solution of the general model is discussed. We also extend the recent iterative support shrinkage strategy to give an algorithm to minimize it, where the subproblem at each iteration is allowed to be solved inexactly. Moreover, this paper is the first one to give the subdifferential of the gradient compounded non-Lipschitz regularization term, based on which we are able to establish the global convergence of the iterative sequence to a stationary point of the original objective function. This is, to our best knowledge, stronger than all the convergence results for gradient compounded non-Lipschitz minimization problems in the current published literature. Numerical experiments show that our proposed method performs well.