International Conference on Multivariate Approximation
September 24-27, 2011

Latex rendering failed!

Shearlets and sparse approximation

Gitta Kutyniok
Many important problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds. While the ability to reliably capture and sparsely represent anisotropic structures is obviously the more important the higher the number of spatial variables is, the principal difficulties arise already in two and three spatial dimensions and even there are yet far from being understood. Five years ago, shearlets were introduced as a means to sparsely encode anisotropic singularities of 2D data in an optimal way, while -- in contrast to previously introduced directional representation systems -- providing a unified treatment of the continuous and digital world. In this talk, we will first give a general introduction to the theory of shearlets. Then some very recent results on the construction of compactly supported shearlet in 2D and 3D will be highlighted, in particular, showing that these shearlet frames provide optimally sparse approximations of anisotropic features. Finally, we will discuss applications of shearlet decompositions such as denoising and data separation.