Continuous Shearlet Tight Frames

TL;DR

This paper introduces a general method for constructing continuous tight frames for 2D square-integrable functions using shearlet transforms, enabling flexible and optimal directional data approximation.

Contribution

It provides a new, broad construction of shearlet-based tight frames from smooth functions, extending previous bandlimited approaches with improved approximation properties.

Findings

Constructs continuous tight frames from smooth functions with anisotropic moments

Frames achieve similar approximation properties as previous shearlet systems

Representation formulas are shown to be optimal for shearlet transforms

Abstract

Based on the shearlet transform we present a general construction of continuous tight frames for $L^2(\mathbb{R}^2)$ from any sufficiently smooth function with anisotropic moments. This includes for example compactly supported systems, piecewise polynomial systems, or both. From our earlier results it follows that these systems enjoy the same desirable approximation properties for directional data as the previous bandlimited and very specific constructions due to Kutyniok and Labate. We also show that the representation formulas we derive are in a sense optimal for the shearlet transform.