Image decompositions using bounded variation and generalized homogeneous Besov spaces

Abstract

This paper is devoted to the decomposition of an image f into u + v , with u a piecewise-smooth or “cartoon” component, and v an oscillatory component (texture or noise), in a variational approach. Y. Meyer [Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, vol. 22, Amer. Math. Soc., Providence, RI, 2001] proposed refinements of the total variation model [L. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992) 259–268] that better represent the oscillatory part v: the weaker spaces of generalized functions G = div ( L ∞ ) , F = div ( BMO ) , and E = B ˙ ∞ , ∞ −1 have been proposed to model v, instead of the standard L 2 space, while keeping u ∈ BV , a function of bounded variation. Such new models separate better geometric structures from oscillatory structures, but it is difficult to realize them in practice. D. Mumford and B. Gidas [D. Mumford, B. Gidas, Stochastic models for generic images, Quart. Appl. Math. 59 (1) (2001) 85–111] also show that natural images can be seen as samples of scale invariant probability distributions that are supported on distributions only, and not on sets of functions. In this paper, we consider and generalize Meyer's ( BV , E ) model, using the homogeneous Besov spaces B ˙ p , q α , − 2 < α < 0 , 1 ⩽ p , q ⩽ ∞ , to represent the oscillatory part v. Theoretical, experimental results and comparisons to validate the proposed methods are presented.