A General Non-Lipschitz Infimal Convolution Regularized Model: Lower Bound Theory and Algorithm

The convex infimal convolution model proposed in Chambolle and Lions [Numer. Math., 76 (1997), pp. 167--188] is a fundamental model to extract two useful components from a single input image and has various low level vision applications. In many of them, one target component has an (ap-proximately) piecewise constant structure and the other is a smoothly varying function or repeated texture pattern. In this paper, we propose and study a general non-Lipschitz infimal convolution (GnLIC) regularization model, which covers most existing applications in this type. Therein the non-Lipschitz regularization enforces the piecewise constant property of the first component. For this GnLIC model, we prove a lower bound theory for its local minimizers and a local version for its stationary points. Motivated by these, we naturally extend previous works to design an inexact iterative support shrinking algorithm with proximal linearization for our GnLIC model (InISSAPL-GnLIC). Moreover, we establish the sequence convergence property and a sequence lower bound theory for InISSAPL-GnLIC, provided that an inexact subgradient condition generated by a sub -solver holds. The subsolver is constructed by efficient ADMM and a specially designed feasibilization operation. We finally give numerical experiments in two low level vision applications: Retinex and cartoon-texture decomposition. These tests demonstrate that our non-Lipschitz regularization based method can indeed extract the piecewise constant component better than existing approaches, which is consistent with the established lower bound theory.