Cartoon-like images are a well-studied class of functions that reasonably approximate natural images. They are typically defined to be functions in R2 (Hölder-) continuous onto two areas separated by an α-Hölder continuous boundary curve.
Well-known results show that both curvelets and shearlets optimally approximate cartoon functions whenever α equals 2. These results have usually been proven for each system separately. Recently, the α-molecules framework has been developed to include all known anisotropic frame constructions based on parabolic scaling, and this framework has unified sparse approximation results for the cartoon-like images.
In this talk we will introduce the concept of α-molecules and their advantages. The main result states that we can identify classes of representation systems which share the same nearly-optimal sparse approximation behavior for cartoon-like images. This is joint work with Philipp Grohs, Gitta Kutyniok and Martin Schöfer.