An asymptotic analysis of separating pointlike and $C^{\beta}$-curvelike singularities

TL;DR

This paper provides a theoretical analysis of separating pointlike and curvilinear singularities in images using $l_1$-minimization with wavelets and shearlets, establishing convergence guarantees and asymptotic accuracy.

Contribution

It introduces a reconstruction framework with theoretical guarantees, extending to general frames and constructing dual bandlimited shearlets for improved separation.

Findings

Proves convergence of the separation method.

Derives asymptotic accuracy of reconstructions.

Shows shearlets with $eta eq 2$ are distinct from wavelets.

Abstract

In this paper, we present a theoretical analysis of separating images consisting of pointlike and $C^{ \beta}$-curvelike structures, where $\beta \in (1,2] $. Our approach is based on $l_1$-minimization, in which the sparsity of the desired solution is exploited by two sparse representation systems. It is well known that for such components wavelets provide an optimally sparse representation for point singularities, whereas $\alpha$-shearlet type with $\alpha$=$\frac{2}{\beta}$ might be best adapted to the $C^{\beta}$-curvilinear singularities. In our analysis, we first propose a reconstruction framework with a theoretical guarantee on convergence, which is extended to use general frames instead of Parseval frames. We then construct a dual pair of bandlimited $\alpha$-shearlets which possesses a good time and frequency localization. Finally, we apply the result to derive an asymptotic accuracy of the reconstructions. In addition, we show that it is possible to separate these two components as long as $\alpha <2$, i.e., bandlimited $\alpha$-shearlets which range from wavelet to shearlet type do not coincide with wavelets in the sense of isotropic fashion.