Compactly Supported Shearlets are Optimally Sparse

TL;DR

This paper proves that a specific class of compactly supported shearlet frames can achieve nearly optimal sparse approximation of cartoon-like images, addressing a key limitation of previous results that required band-limited generators.

Contribution

It provides the first complete proof that compactly supported shearlet frames can attain (almost) optimal sparsity for cartoon-like images, expanding applicability for practical uses.

Findings

Compactly supported shearlet frames achieve near-optimal sparsity.

The proof applies to shearlet generators with weak moment conditions.

This advances the theoretical understanding of shearlet-based image approximation.

Abstract

Cartoon-like images, i.e., C^2 functions which are smooth apart from a C^2 discontinuity curve, have by now become a standard model for measuring sparse (non-linear) approximation properties of directional representation systems. It was already shown that curvelets, contourlets, as well as shearlets do exhibit (almost) optimally sparse approximation within this model. However, all those results are only applicable to band-limited generators, whereas, in particular, spatially compactly supported generators are of uttermost importance for applications. In this paper, we now present the first complete proof of (almost) optimally sparse approximations of cartoon-like images by using a particular class of directional representation systems, which indeed consists of compactly supported elements. This class will be chosen as a subset of shearlet frames -- not necessarily required to be tight -- with shearlet generators having compact support and satisfying some weak moment conditions.