Optimally sparse approximations of 3D functions by compactly supported shearlet frames
Abstract
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<\alpha =< \beta =< 2 we show that pyramid-adapted shearlet systems provide a nearly optimally sparse approximation rate within the generalized cartoon-like image model class measured by means of non-linear N-term approximations.
Cite
@article{arxiv.1109.5993,
title = {Optimally sparse approximations of 3D functions by compactly supported shearlet frames},
author = {Gitta Kutyniok and Jakob Lemvig and Wang-Q Lim},
journal= {arXiv preprint arXiv:1109.5993},
year = {2012}
}
Comments
56 pages, 6 figures