Construction of Compactly Supported Shearlet Frames

TL;DR

This paper develops compactly supported shearlet frames that are practically useful and theoretically sound, providing optimal sparse approximations of cartoon-like images with explicit frame bounds.

Contribution

It introduces a method to construct compactly supported cone-adapted shearlet frames with proven frame bounds and optimal approximation properties.

Findings

Constructed shearlet frames with explicit frame bounds

Proved these frames provide optimally sparse image approximations

Derived sufficient conditions for shearlet systems to form frames

Abstract

Shearlet tight frames have been extensively studied during the last years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital setting. However, these studies only concerned shearlet tight frames generated by a band-limited shearlet, whereas for practical purposes compact support in spatial domain is crucial. In this paper, we focus on cone-adapted shearlet systems which -- accounting for stability questions -- are associated with a general irregular set of parameters. We first derive sufficient conditions for such cone-adapted irregular shearlet systems to form a frame and provide explicit estimates for their frame bounds. Secondly, exploring these results and using specifically designed wavelet scaling functions and filters, we construct a family of cone-adapted shearlet frames consisting of compactly supported shearlets. For this family, we derive estimates for the ratio of their frame bounds and prove that they provide optimally sparse approximations of cartoon-like images.