Shearlets and Optimally Sparse Approximations

TL;DR

This paper surveys the use of shearlet systems for achieving optimally sparse approximations of cartoon-like images, highlighting their unique ability to efficiently represent anisotropic features in 2D and 3D.

Contribution

It provides an overview of shearlet frames, including band-limited and compactly supported types, and discusses their optimal approximation properties for cartoon-like images.

Findings

Shearlet systems provide optimally sparse approximations in 2D and 3D.

Compactly supported shearlet frames satisfy optimality benchmarks.

Shearlets outperform other directional systems in representing anisotropic features.

Abstract

Multivariate functions are typically governed by anisotropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the purpose of compression as well as for an efficient analysis is the provision of optimally sparse approximations of such functions. Recently, cartoon-like images were introduced in 2D and 3D as a suitable model class, and approximation properties were measured by considering the decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date the only representation system, which provide optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this optimality benchmark. This chapter shall serve as an introduction to and a survey about sparse approximations of cartoon-like images by band-limited and also compactly supported shearlet frames as well as a reference for the state-of-the-art of this research field.