## Compactly Supported Shearlets: Construction, Optimally Sparse Approximation, and Applications

- March 3, 2010
- 2:20 p.m.
- LeConte 312

## Abstract

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 spatial dimensions and even there are yet far from being understood.

Three 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. One main idea is to parameterize directions by slope through shear matrices rather than angle, which greatly supports the treating of the digital setting. However, previous constructions excluded the possibility of compactly supported shearlets, which are essential for superior spatial localization.

In this talk, we will discuss a novel construction method for shearlet frames generated by compactly supported shearlets, and provide a detailed analysis of the achievable frame bounds. Next, we will prove that a large class of shearlet frames generated by compactly supported shearlets indeed even provide optimally sparse approximations of cartoon-like images such as the very special class of shearlet frames generated by band-limited shearlets previously studied. Finally, we will discuss some applications of these systems to neurobiological imaging.

This is joint work with Wang-Q Lim (University of Osnabrueck).