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

2011/09/27 by Kutyniok, Gitta, Lemvig, Jakob, Lim, Wang-Q
#41A30 #42C40 (Primary) 42C15 #94A08 (Secondary) #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Information Theory (cs.IT) #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1109.5993

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 βcontrolling classical smoothness and one parameter αcontrolling 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 three-dimensional setting and construct frames for L²(R³) with this particular shearlet structure. For the smoothness range 1

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