Shearlets on Bounded Domains

TL;DR

This paper extends shearlet systems to bounded domains, demonstrating they can efficiently approximate cartoon-like images on such domains, which is relevant for applications like image processing and PDE solvers.

Contribution

It introduces a new model for images on bounded domains and proves shearlet frames provide near-optimal sparse approximations in this setting.

Findings

Shearlet frames can be adapted to bounded domains.

Shearlets provide near-optimal sparse approximations for cartoon-like images.

The approach is relevant for practical applications like image processing.

Abstract

Shearlet systems have so far been only considered as a means to analyze $L^2$-functions defined on $\R^2$, which exhibit curvilinear singularities. However, in applications such as image processing or numerical solvers of partial differential equations the function to be analyzed or efficiently encoded is typically defined on a non-rectangular shaped bounded domain. Motivated by these applications, in this paper, we first introduce a novel model for cartoon-like images defined on a bounded domain. We then prove that compactly supported shearlet frames satisfying some weak decay and smoothness conditions, when orthogonally projected onto the bounded domain, do provide (almost) optimally sparse approximations of elements belonging to this model class.