(Φ,Φ*) Image Decomposition Models and Minimization Algorithms

Abstract

We propose in this paper minimization algorithms for image restoration using dual functionals and dual norms. In order to extract a clean image u from a degraded version f=Ku+n (where f is the observation, K is a blurring operator and n represents additive noise), we impose a standard regularization penalty Φ(u)=∫ φ(|Du|)dx< ∞ on u, where φ is positive, increasing and has at most linear growth at infinity. However, on the residual f?Ku we impose a dual penalty Φ*(f-Ku)< ∞, instead of the more standard $\|f-\mathit{Ku}\|^{2}_{L^{2}}$ fidelity term. In particular, when φ is convex, homogeneous of degree one, and with linear growth (for instance the total variation of u), we recover the (BV,BV *) decomposition of the data f, as suggested by Y. Meyer (Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, vol. 22, Am. Math. Soc., Providence, 2001). Practical minimization methods are presented, together with theoretical, experimental results and comparisons to illustrate the validity of the proposed models. Moreover, we also show that by a slight modification of the associated Euler-Lagrange equations, we obtain well-behaved approximations and improved results.