Shearlets and Optimally Sparse Approximations

Details Shearlets and Optimally Sparse Approximations Shearlets and Optimally Sparse Approximations

2011-06-07

arxiv.org/abs/1106.1325v2

Mathematics

Functional Analysis

in "Shearlets: Multiscale Analysis for Multivariate Data", Birkh\"auser-Springer

Scientific paper

Multivariate functions are typically governed by anisotropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the purpose of compression as well as for an efficient analysis is the provision of optimally sparse approximations of such functions. Recently, cartoon-like images were introduced in 2D and 3D as a suitable model class, and approximation properties were measured by considering the decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date the only representation system, which provide optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this optimality benchmark. This chapter shall serve as an introduction to and a survey about sparse approximations of cartoon-like images by band-limited and also compactly supported shearlet frames as well as a reference for the state-of-the-art of this research field.

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