Irregular Shearlet Frames: Geometry and Approximation Properties

TL;DR

This paper introduces a geometric approach to analyze and construct shearlet frames for 2D data, focusing on their approximation properties and necessary geometric conditions for frame formation.

Contribution

It develops a weighted shearlet density based on group representations and provides necessary geometric conditions for shearlet systems to form frames, including new classes of generators.

Findings

Established necessary conditions for shearlet frames based on geometric density

Analyzed approximation properties and homogeneous approximation abilities of shearlet systems

Presented examples like oversampled and co-shearlet systems to demonstrate the approach

Abstract

Recently, shearlet systems were introduced as a means to derive efficient encoding methodologies for anisotropic features in 2-dimensional data with a unified treatment of the continuum and digital setting. However, only very few construction strategies for discrete shearlet systems are known so far. In this paper, we take a geometric approach to this problem. Utilizing the close connection with group representations, we first introduce and analyze an upper and lower weighted shearlet density based on the shearlet group. We then apply this geometric measure to provide necessary conditions on the geometry of the sets of parameters for the associated shearlet systems to form a frame for L^2(\R^2), either when using all possible generators or a large class exhibiting some decay conditions. While introducing such a feasible class of shearlet generators, we analyze approximation properties of the associated shearlet systems, which themselves lead to interesting insights into homogeneous approximation abilities of shearlet frames. We also present examples, such a oversampled shearlet systems and co-shearlet systems, to illustrate the usefulness of our geometric approach to the construction of shearlet frames.