Paper Info
Title
Parabolic Molecules.
Abstract
Anisotropic decompositions using representation systems based on parabolic scaling such as curvelets or shearlets have recently attracted significant 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 , 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 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.
Year
DOI
Venue
2014
https://doi.org/10.1007/s10208-013-9170-z
Foundations of Computational Mathematics
Keywords
DocType
Volume
Curvelets,Nonlinear approximation,Parabolic scaling,Shearlets,Smoothness spaces,Sparsity equivalence,41AXX,41A25,53B,22E
Journal
14
Issue
ISSN
Citations 
2
1615-3375
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Philipp Grohs116219.49
Gitta Kutyniok232534.77