Bendlets: A Second-Order Shearlet Transform with Bent Elements

TL;DR

Bendlets are a second-order shearlet system that uses bending to improve the characterization of discontinuities in images, providing enhanced detection of location, orientation, and curvature.

Contribution

This paper introduces bendlets, a novel second-order shearlet system incorporating bending, which improves curvature analysis over existing directional systems.

Findings

Decay rates enable precise localization and curvature detection.

Bendlets outperform traditional shearlets in curvature characterization.

Implementation in ShearLab confirms theoretical results.

Abstract

We introduce bendlets, a shearlet-like system that is based on anisotropic scaling, translation, shearing, and bending of a compactly supported generator. With shearing being linear and bending quadratic in spatial coordinates, bendlets provide what we term a second-order shearlet system. As we show in this article, the decay rates of the associated transform enable the precise characterization of location, orientation and curvature of discontinuities in piecewise constant images. These results yield an improvement over existing directional representation systems where curvature only controls the constant of the decay rate of the transform. We also detail the construction of shearlet systems of arbitrary order. A practical implementation of bendlets is provided as an extension of the ShearLab toolbox, which we use to verify our theoretical classification results.