Dual Norm Based Iterative Methods for Image Restoration

A convergent iterative regularization procedure based on the square of a dual norm is introduced for image restoration models with general (quadratic or non-quadratic) convex fidelity terms. Iterative regularization methods have been previously employed for image deblurring or denoising in the presence of Gaussian noise, which use L2 (Tadmor et al. in Multiscale Model. Simul. 2:554–579, 2004; Osher et al. in Multiscale Model. Simul. 4:460–489, 2005; Tadmor et al. in Commun. Math. Sci. 6(2):281–307, 2008), and L1 (He et al. in J. Math. Imaging Vis. 26:167–184, 2005) data fidelity terms, with rigorous convergence results. Recently, Iusem and Resmerita (Set-Valued Var. Anal. 18(1):109–120, 2010) proposed a proximal point method using inexact Bregman distance for minimizing a convex function defined on a non-reflexive Banach space (e.g. BV(Ω)), which is the dual of a separable Banach space. Based on this method, we investigate several approaches for image restoration such as image deblurring in the presence of noise or image deblurring via (cartoon+texture) decomposition. We show that the resulting proximal point algorithms approximate stably a true image. For image denoising-deblurring we consider Gaussian, Laplace, and Poisson noise models with the corresponding convex fidelity terms as in the Bayesian approach. We test the behavior of proposed algorithms on synthetic and real images in several numerical experiments and compare the results with other state-of-the-art iterative procedures based on the total variation penalization as well as the corresponding existing one-step gradient descent implementations. The numerical experiments indicate that the iterative procedure yields high quality reconstructions and superior results to those obtained by one-step standard gradient descent, with faster computational time.