Optimally sparse approximations of 3D functions by compactly supported shearlet frames

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 3D 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 C β-smooth functions with discontinuities on a piecewise C α-smooth surface. We introduce a pyramid-adapted, hybrid shearlet system for the 3D setting and construct frames for L 2(ℝ3) 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 nonlinear N-term approximations. © 2012 Society for Industrial and Applied Mathematics.