Eitan Tadmor: Images, PDEs and hierarchical constructions in critical regularity spaces

Coffee:15:30 - 16:00

Abstract: Edges are noticeable features in images which can be extracted from noisy data using different variational models. The analysis of such variational models leads to the question of representing general images as the divergence of uniformly bonded vector fields. 
We construct uniformly bounded solutions of div(U)=f for general f’s in the critical regularity space \(L^d(R^d)\). The study of this equation and related problems was motivated by recent results of Bourgain & Brezis. The intriguing 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=∑_j▒u_j  which we introduced earlier in the context of image processing. The \(u_j's\) are constructed recursively as proper minimizers, yielding a multi-scale decomposition of solutions/images U.