Higher order analysis of the geometry of singularities using the Taylorlet transform

TL;DR

The paper introduces the Taylorlet transform, an extension of shearlet analysis that uses higher order shears and vanishing moments to detect detailed geometric features of singularities such as position, orientation, and curvature.

Contribution

It develops the Taylorlet transform with higher order shears and vanishing moment conditions, enabling more robust detection of geometric singularity features.

Findings

The Taylorlet transform exhibits decay rates depending on higher order shearing variables.

Construction of analyzing functions fulfilling multiple vanishing moment conditions.

Enhanced detection of singularity geometry compared to traditional methods.

Abstract

We consider an extension of the continuous shearlet transform which additionally uses higher order shears. This extension, called the Taylorlet transform, allows for a detection of the position, the orientation, the curvature and other higher order geometric information of singularities. Employing the novel vanishing moment conditions of higher order, $\int_\mathbb{R} g(t^k)t^m dt=0$, on the analyzing function, we can show that the Taylorlet transform exhibits different decay rates for decreasing scales depending on the choice of the higher order shearing variables. This enables a more robust detection of the geometric information of singularities. Furthermore, we present a construction that yields analyzing functions which fulfill vanishing moment conditions of different orders simultaneously.