Smooth affine shear tight frames with MRA structure

TL;DR

This paper introduces a comprehensive framework for affine shear tight frames with multiresolution analysis structure, including smooth generators and filter banks, advancing shearlet theory and practical implementation.

Contribution

It provides a complete characterization of affine shear tight frames, introduces MRA-structured shear tight frames, and offers methods for their construction with smooth generators and filter banks.

Findings

Complete characterization of affine shear tight frames

Construction of smooth affine shear tight frames with MRA structure

Efficient implementation via filter banks similar to wavelet transforms

Abstract

Finding efficient representations is one of the most challenging and heavily sought problems in mathematics. Representation using shearlets recently receives a lot of attention due to their desirable properties in both theory and applications. Using the framework of frequency-based affine systems, in this paper we introduce and systematically study affine shear tight frames which include all known shearlet tight frames as special cases. Our results in this paper will resolve several key questions on shearlets. We provide a complete characterization for an affine shear tight frame and then use it to obtain smooth affine shear tight frames with all their generators in the Schwarz class. Though multiresolution analysis (MRA) is the foundation and key feature of wavelet analysis for fast numerical implementation of a wavelet transform, all the known shearlets so far do not possess any MRA structure and filter banks. In order to study affine shear tight frames with MRA structure, we introduce the notion of a sequence of affine shear tight frames and then we provide a complete characterization for it. Based on our characterizations, we present two different approaches, i.e., non-stationary and quasi-stationary, for the construction of sequences of affine shear tight frames with MRA structure such that all their generators are smooth (in the Schwarz class) and they have underlying filter banks. Consequently, their associated transforms can be efficiently implemented using filter banks similarly as a fast wavelet transform does.