Minimization of a detail-preserving regularization functional for impulse noise removal

Recently, a powerful two-phase method for restoring images corrupted with high level impulse noise has been developed. The main drawback of the method is the computational efficiency of the second phase which requires the minimization of a non-smooth objective functional. However, it was pointed out in (Chan et al. in Proc. ICIP 2005, pp.125-128) that the non-smooth data-fitting term in the functional can be deleted since the restoration in the second phase is applied to noisy pixels only. In this paper, we study the analytic properties of the resulting new functional F. We show that F, which is defined in terms of edge-preserving potential functions \phi_\alpha, inherites many nice properties from \phi_\alpha , including the first and second order Lipschitz continuity, strong convexity, and positive definiteness of its Hessian. Moreover, we use these results to establish the convergence of optimization methods applied to F. In particular, we prove the global convergence of some conjugate gradient-type methods and of a recently proposed low complexity quasi-Newton algorithm. Numerical experiments are given to illustrate the convergence and efficiency of the two methods.