Edge detection with trigonometric polynomial shearlets

TL;DR

This paper demonstrates that specific trigonometric polynomial shearlets can effectively detect boundary singularities in periodic functions, supported by theoretical estimates and numerical examples.

Contribution

It introduces a new class of shearlets based on de la Vallée Poussin wavelets for boundary detection in periodic functions, with theoretical bounds and numerical validation.

Findings

Shearlets detect boundary singularities effectively.

Theoretical bounds for shearlet inner products are established.

Numerical examples confirm the theoretical results.

Abstract

In this paper we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vall\'{e}e Poussin type wavelets are able to detect singularities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of corresponding inner products. In the proof we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results.