Image Decomposition Using Total Variation and div(BMO)

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. 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 $G = {\rm div}(L^\infty)$ and $F = {\rm div}(BMO)$ (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 L2 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. Solé, and L. Vese, Multiscale Model. Simul., 1 (2003), pp. 349--370], the authors have proposed approximations to the (BV,G) decomposition model, where the $L^\infty$ space has been substituted by Lp, $1 \leq p < \infty$. 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.