Parabolic Molecules

TL;DR

This paper introduces the concept of parabolic molecules to unify and analyze various anisotropic representation systems like curvelets and shearlets, demonstrating their near orthogonality and sparse approximation capabilities for data with anisotropic features.

Contribution

It provides a comprehensive framework for parabolic molecules, establishing their near orthogonality, sparse approximation properties, and introducing sparsity equivalence among different systems.

Findings

Parabolic molecules are almost orthogonal in a specific sense.

They can sparsely approximate data with anisotropic features.

The framework applies to systems like curvelets and shearlets, including compactly supported variants.

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 {\em 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 {\em 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.