Edge detection with polynomial frames on the sphere

TL;DR

This paper extends polynomial frame methods on the sphere to detect discontinuities along smooth curves, providing asymptotic estimates for frame coefficients near such singularities.

Contribution

It generalizes previous work from circular edges to arbitrary smooth curves on the sphere, with new asymptotic estimates for frame coefficients.

Findings

Established upper and lower bounds for frame coefficients near smooth curve discontinuities.

Demonstrated uniform asymptotic estimates in neighborhoods of the curves.

Extended the applicability of polynomial frame techniques for edge detection on the sphere.

Abstract

In a recent article, we have shown that a variety of localized polynomial frames, including isotropic as well as directional systems, are suitable for detecting jump discontinuities along circles on the sphere. More precisely, such edges can be identified in terms of their position and orientation by the asymptotic decay of the frame coefficients in an arbitrary small neighborhood. In this paper, we will extend these results to discontinuities which lie along general smooth curves. In particular, we prove upper and lower estimates for the frame coefficients when the analysis function is concentrated in the vicinity of such a singularity. The estimates are given in an asymptotic sense, with respect to some dilation parameter, and they hold uniformly in a neighborhood of the smooth curve segment under consideration.