Image decomposition using total variation and div(BMO)

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. Meyer [ Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, Univ. Lecture Ser. 22, AMS, Providence, RI, 2001] proposed refinements of the total variation model (Rudin, Osher, and Fatemi [Phys. D, 60 ( 1992), pp. 259 - 268]) that better represent the oscillatory part v: the spaces of generalized functions G = div(L-infinity) and F = div(BMO) ( this last space arises in the study of Navier - Stokes equations; see Koch and Tataru [Adv. Math., 157 ( 2001), pp. 22 - 35]) have been proposed to model v, instead of the standard L-2 space, while keeping u a function of bounded variation. Mumford and Gidas [ Quart. Appl. Math., 59 ( 2001), pp. 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. However, there is no simple solution to obtain in practice such decompositions f = u+v when working with G or F. In earlier works [L. Vese and S. Osher, J. Sci. Comput., 19 (2003), pp. 553 - 572], [ L. A. Vese and S. J. Osher, J. Math. Imaging Vision, 20 ( 2004), pp. 7 - 18], [S. Osher, A. Sole, and L. Vese, Multiscale Model. Simul., 1 ( 2003), pp. 349 - 370], the authors have proposed approximations to the (BV, G) decomposition model, where the L-infinity space has been substituted by L-p, 1 <= p < infinity. In the present paper, we introduce energy minimization models to compute (BV, F) decompositions, and as a by-product we also introduce a simple model to realize the (BV, G) decomposition. In particular, we investigate several methods for the computation of the BMO norm of a function in practice. Theoretical, experimental results and comparisons to validate the proposed new methods are presented.