Several advanced multiscale systems, such as curvelets and shearlets, were introduced during the last decade to overcome the limitations of wavelets and other traditional representations. In fact, even though wavelets are very efficient to handle signals with point singularities, they are suboptimal when dealing with edges and the other types of distributed discontinuities which typically dominate multidimensional data. Shearlets, by contrast, are specially
designed to combine the power of multiscale analysis with ability to handle directional information efficiently. As a result, they provide optimally efficient representations, in a precise sense, for a large class of 2D/3D data.
In this talk, I will first give a brief overview of the sparse approximation properties of shearlets. Next, I will present and discuss several results illustrating the unique ability of the shearlet transform to provide a precise geometric characterization of singularities.These observations point out to a wide range of problems and applications where the shearlet framework can provide further insight.