| This manuscript consists of a study of image segmentation and decomposition models in a variational approach. In the segmentation case, we consider images that are corrupted by additive and multiplicative noise. In the additive case, we decompose the data f into the sum u + w + noise. Here, u is a piecewise-constant component, capturing edges and discontinuities, and it is modeled in a level set approach, while w is a smooth component, capturing the intensity inhomogeneities. The additive noise is removed from the initial data. In the multiplicative case, we consider a piecewise-constant segmentation model of the data corrupted by multiplicative noise, in a multiphase level set approach. The fidelity term is chosen appropriately for such degradation model. Then, we extend this model to piecewise-smooth segmentation, decomposing the data u into the product u · w · noise, where again u is piecewise-constant, while w is smooth. In the image decomposition case, we focus on the modeling of oscillatory components (texture or noise). In general, we decompose a given 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 [Mey01] proposed refinements of the total variation model (L. Rudin, S. Osher and E. Fatemi [ROF92]) that better represent the oscillatory part v: the spaces of generalized functions G = div(Linfinity), F = div(BMO) = BM˙O -1, and E = B&d2;-1infinity,infinity have been proposed to model v, instead of the standard L2 space, while keeping u ∈ BV a function of bounded variation. D. Mumford and B. Gidas [MG01] 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, F, or E. 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. We also generalize Meyer's (BV, E) model and consider the homogenenous Besov spaces B&d2;sp,q , -2 < s < 0, 1 ≤ p, q ≤ infinity, to represent the oscillatory part v. Several theoretical and numerical results will be presented. |
