Shearlets and Optimally Sparse Approximations∗

Abstract: Multivariate functions are typically governed by aniso-tropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the pur-pose of compression as well as for an efficient analysis is the pro-vision of optimally sparse approximations of such functions. Re-cently, cartoon-like images were introduced in 2D and 3D as a suit-able model class, and approximation properties were measured by considering the decay rate of the L2 error of the best N-term approx-imation. Shearlet systems are to date the only representation sys-tem, which provide optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other di-rectional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this opti-mality benchmark. This chapter shall serve as an introduction to and a survey about sparse approximations of cartoon-like images by band-limited and also compactly supported shearlet frames as well as a reference for the state-of-the-art of this research field.