Dualizable Shearlet Frames and Sparse Approximation

Abstract

Shearlet systems have been introduced as directional representation systems, which provide optimally sparse approximations of a certain model class of functions governed by anisotropic features while allowing faithful numerical realizations by a unified treatment of the continuum and digital realm. They are redundant systems, and their frame properties have been extensively studied. In contrast to certain band-limited shearlets, compactly supported shearlets provide high spatial localization, but do not constitute Parseval frames. Thus reconstruction of a signal from shearlet coefficients requires knowledge of a dual frame. However, no closed and easily computable form of any dual frame is known. In this paper, we introduce the class of dualizable shearlet systems, which consist of compactly supported elements and can be proven to form frames for $L^2(\mathbb{R}^2)$. For each such dualizable shearlet system, we then provide an explicit construction of an associated dual frame, which can be stated in closed form and efficiently computed. We also show that dualizable shearlet frames still provide optimally sparse approximations of anisotropic features.