Parabolic molecules
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Date
2012-06
Publication Type
Report
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Abstract
Anisotropic decompositions using representation systems based on parabolic scaling such as curvelets or shearlets have recently attracted significantly increased attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. However, it seems quite cumbersome to prove such a canon of results for each system separately, while many of the systems exhibit certain similarities. In this paper, with the introduction of the notion of parabolic molecules, we aim to provide a comprehensive framework which includes customarily employed representation systems based on parabolic scaling such as curvelets and shearlets. It is shown that pairs of parabolic molecules have the fundamental property to be almost orthogonal in a particular sense. This result is then applied to analyze parabolic molecules with respect to their ability to sparsely approximate data governed by anisotropic features. For this, the concept of sparsity equivalence is introduced which is shown to allow the identification of a large class of parabolic molecules providing the same sparse approximation results as curvelets and shearlets. Finally, as another application, smoothness spaces associated with parabolic molecules are introduced providing a general theoretical approach which even leads to novel results for, for instance, compactly supported shearlets.
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published
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Volume
2012-14
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Publisher
Seminar for Applied Mathematics, ETH Zurich
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Subject
Curvelets; Nonlinear Approximation; Parabolic Scaling; Shearlets; Smoothness Spaces; Sparsity Equivalence
Organisational unit
03941 - Grohs, Philipp (ehemalig)
02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics
Notes
Funding
247277 - Automated Urban Parking and Driving (EC)
Related publications and datasets
Is previous version of: https://doi.org/10.3929/ethz-b-000096910