Shearlets and Optimally Sparse Approximations
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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 and 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.
Key wordsAnisotropic features Band-limited shearlets Cartoon-like images Compactly supported shearlets Linear and nonlinear approximations Multidimensional data Sparse approximations
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The first author acknowledges support by the Einstein Foundation Berlin and by Forschungsgemeinschaft (DFG) Grant KU 1446/14, and the first and third authors acknowledge support by DFG Grant SPP-1324 KU 1446/13.
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