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Eitan Tadmor : Multi-scale construction of solutions to problems with critical regularity



Edges are noticeable features in images which can be extracted from noisy data using different variational models. The analysis of such models leads to the question of representing general L^2-data as the divergence of uniformly bounded vector fields.
We use a multi-scale approach to construct uniformly bounded solutions of div(U)=f for general f’s in the critical regularity space L^2(T^2). The study of this equation and related problems was motivated by results of Bourgain & Brezis. The intriguing critical aspect here is that although the problems are linear, construction of their solution is not. These constructions are special cases of a rather general framework for solving linear equations in critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical representations $U = \sum_j u_j$ which we introduced earlier in the context of image processing, yielding a multi-scale decomposition of “images” U.

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