Direction sensitive analysis of higher order jump discontinuities along circles on the sphere

Abstract

In recent years, scale-discretized directional wavelets and second-generation curvelets have been introduced on the unit sphere, yielding directional and localized polynomial frames for band-limited signals. In this paper, we show that these functions are able to detect the positions and orientations of all higher order jump discontinuities which lie along circles on the 2-sphere. Specifically, we prove upper and lower estimates for the magnitude of the corresponding inner products when the analysis function is concentrated in the neighborhood of such a singularity. Although similar results already exist in certain two-dimensional settings, this paper is the first one to consider frames and signals that are given on the 2-sphere. As a side product of our investigations, we also develop a new localization bound as well as an explicit formula for the auto-correlation function for second-generation curvelets.