  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.
