Harmonic Analysis and Signal Processing Seminar (joint with the Numerical Analysis and Scientific Computing Seminar)
Anisotropic Multiscale Systems on Bounded Domains
Friday, March 3, 2017, 10am, WWH 1302
Many important problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shear layers in solutions of transport dominated equations. While the ability to reliably capture and sparsely represent anisotropic structures is obviously the more important the higher the number of spatial variables is, principal difficulties arise already in two spatial dimensions. Since it was shown that the well-known (isotropic) wavelet systems are not capable of efficiently approximating such anisotropic features, the need arose to introduce appropriate anisotropic representation systems. Among various suggestions, shearlets are the most widely used today. Main reasons for this are their optimal sparse approximation properties within a model situation in combination with their unified treatment of the continuum and digital realm, leading to faithful implementations. However, a current major drawback of this and also other anisotropic systems is the fact that they were designed as systems for L²(ℝ²), whereas applications such as discretization schemes for partial differential equations typically require systems defined on a bounded domain.
In this talk, we will present a novel construction of shearlets on bounded domains, which, in particular, still provide optimal sparse approximations of anisotropic features and constitute a frame with controllable frame bounds. We will then discuss applications to imaging science in combination with sparse regularization methods and to adaptive numerical solution of partial differential equations. This is joint work with P. Grohs, J. Ma, P. Petersen, and M. Raslan.