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In this paper, we present a theoretical analysis of separating imagesconsisting of pointlike and $C^{ \beta}$-curvelike structures, where $\beta \in(1,2] $. Our approach is based on $l_1$-minimization, in which the sparsity ofthe desired solution is exploited by two sparse representation systems. It iswell known that for such components wavelets provide an optimally sparserepresentation for point singularities, whereas $\alpha$-shearlet type with$\alpha$=$\frac{2}{\beta}$ might be best adapted to the $C^{\beta}$-curvilinearsingularities. In our analysis, we first propose a reconstruction frameworkwith a theoretical guarantee on convergence, which is extended to use generalframes 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 thereconstructions. In addition, we show that it is possible to separate these twocomponents as long as $\alpha <2$, i.e., bandlimited $\alpha$-shearlets whichrange from wavelet to shearlet type do not coincide with wavelets in the senseof isotropic fashion.

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