## Compactly supported shearlet frames and optimally sparse approximations of functions in $L^2(\mathbb{R}^3)$ with piecewise $C^\alpha$ singularities

Publication: Research - peer-reviewJournal article – Annual report year: 2011

### Documents

We study efficient and reliable methods of capturing and sparsely representing anisotropic structures in 3D data. As a model class for multidimensional data with anisotropic features, we introduce generalized three-dimensional cartoon-like images. This function class will have two smoothness parameters: one parameter \beta controlling classical smoothness and one parameter \alpha controlling anisotropic smoothness. The class then consists of piecewise C^\beta-smooth functions with discontinuities on a piecewise C^\alpha-smooth surface. We introduce a pyramid-adapted, hybrid shearlet system for the three-dimensional setting and construct frames for L^2(R^3) with this particular shearlet structure. For the smoothness range 1
Original language English Arxiv.com 2011 arXiv:1109.5993v1 E-pub ahead of print

### Keywords

• Sparse approximations, Cartoon-like images, Anisotropic features, Nonlinear approximations, Multi-dimensional data, Shearlets