An asymptotic analysis of separating pointlike and $C^β-$ curvelike singularities

Van Tiep Do, Alex Goessmann

Published 2021-07-05Version 1

In this paper, we present a theoretical analysis of separating an image consists of pointlike and $C^{1/ \alpha}$ curvelike structures, $\alpha \in [1,2) $, by using wavelets and a pair of dual $\alpha-$ shearlet types with flexible scaling. Our analyzing method is based on $l_1-$ minimization which is extended to use general frames instead of Parseval frames. It is well known that for such components wavelets provide an optimally sparse representation for point singularities, whereas $\alpha-$shearlet type might be best adapted to the $C^{1/\alpha}$ curvilinear singularities. Our theoretical results show that it is possible to separate these two components as long as $\alpha <2$, i.e., $\alpha-$ shearlets which range from wavelets and shearlets do not coincide with wavelets in sense of isotropic fashion. The aim of this paper is to provide a generalized theoretical guarantee for the successful asymptotic separation by using two general frames. We then transfer this result to derive an asymptotic accuracy of image geometric separation by using radial wavelets and a dual pair of well-localized bandlimited $\alpha-$ shearlet frames.