  Modern data is customarily of multimodal nature, and analysis tasks typically
require separation into the single components. Although a highly ill-posed
problem, the morphological difference of these components sometimes allow a
very precise separation such as, for instance, in neurobiological imaging a
separation into spines (pointlike structures) and dendrites (curvilinear
structures). Recently, applied harmonic analysis introduced powerful
methodologies to achieve this task, exploiting specifically designed
representation systems in which the components are sparsely representable,
combined with either performing $\ell_1$ minimization or thresholding on the
combined dictionary.
  In this paper we provide a thorough theoretical study of the separation of a
distributional model situation of point- and curvilinear singularities
exploiting a surprisingly simple single-pass alternating thresholding method
applied to the two complementary frames: wavelets and curvelets. Utilizing the
fact that the coefficients are clustered geometrically, thereby exhibiting
clustered/geometric sparsity in the chosen frames, we prove that at
sufficiently fine scales arbitrarily precise separation is possible. Even more
surprising, it turns out that the thresholding index sets converge to the
wavefront sets of the point- and curvilinear singularities in phase space and
that those wavefront sets are perfectly separated by the thresholding
procedure. Main ingredients of our analysis are the novel notion of cluster
coherence and clustered/geometric sparsity as well as a microlocal analysis
viewpoint.
