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 β controlling classical smoothness and one parameter α controlling anisotropic smoothness. The class then consists of piecewise Cβ -smooth functions with discontinuities on a piecewise Cα-smooth surface. We introduce a pyramid-adapted, hybrid shearlet system for the three-dimensional setting and construct frames for L(R) with this particular shearlet structure. For the smoothness range 1 < α ≤ β ≤ 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.