Image decompositions using bounded variation and generalized homogeneous Besov spaces (original) (raw)

2007, Applied and Computational Harmonic Analysis

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 =Ḃ −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.