Optimally sparse representations of 3D data with \(C^2\) surface singularities using Parseval frames of shearlets (Q2904728)

The authors investigate sparse approximations of three-dimensional data with singularities on two-dimensional manifolds. The authors introduce shearlet-like systems with elements that are smooth and have Fourier transforms that are smooth and have compact support. In [the authors, Math. Model. Nat. Phenom. 8, No. 1, 82--105 (2013; Zbl 1266.42076)], the authors show that this type of system constitutes a tight frame for \(L^2(\mathbb{R}^3)\). A cartoon-like image model class consisting of piecewise \(C^2\)-functions of three variables with discontinuities on a \(C^2\)-surface is introduced. Optimal rates of approximation within this model class are derived. It is proved that the shearlet-like tight frames provide nearly optimal sparse approximations of three-dimensional cartoon-like images.NEWLINENEWLINEConcurrently, similar results were obtained by \textit{G. Kutyniok} et al. [SIAM J. Math. Anal. 44, No. 4, 2962--3017 (2012; Zbl 1252.42043)], where nearly optimal rates of approximation within a generalized cartoon-like model class were obtained for shearlet frames with compactly supported generators.