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

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, ${\int }_{\mathbb {R}} g(\pm t^{k})t^{m} dt= 0$ for $k,m\in \mathbb {N}$ , k ≥ 1, on the analyzing function $g\in \mathcal {S}(\mathbb {R})$ , 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 faster detection of the geometric information of singularities in terms of the decay rate with respect to the dilation parameter. Furthermore, we present a construction that yields analyzing functions which fulfill vanishing moment conditions of different orders simultaneously.