Classification of Edges Using Compactly Supported Shearlets

TL;DR

This paper demonstrates how compactly supported shearlets can detect, classify, and analyze the geometric features of singularities and boundaries in functions, improving previous methods with uniform estimates and optimal bounds.

Contribution

It introduces a method using compactly supported shearlets for boundary detection and classification, with uniform estimates and novel wavefront set characterizations in 3D.

Findings

Boundary sets can be extracted via continuous shearlet transform.

Smooth and non-smooth boundary components are classified.

Bounds on shearlet estimates are proven to be optimal.

Abstract

We analyze the detection and classification of singularities of functions $f = \chi_B$, where $B \subset \mathbb{R}^d$ and $d = 2,3$. It will be shown how the set $\partial B$ can be extracted by a continuous shearlet transform associated with compactly supported shearlets. Furthermore, if $\partial S$ is a $d-1$ dimensional piecewise smooth manifold with $d=2$ or $3$, we will classify smooth and non-smooth components of $\partial S$. This improves previous results given for shearlet systems with a certain band-limited generator, since the estimates we derive are uniform. Moreover, we will show that our bounds are optimal. Along the way, we also obtain novel results on the characterization of wavefront sets in $3$ dimensions by compactly supported shearlets. Finally, geometric properties of $\partial S$ such as curvature are described in terms of the continuous shearlet transform of $f$.