Optimally Sparse Representations of Cartoon-Like Cylindrical Data

Abstract

Sparse representations of multidimensional data have received a significant attention in the literature due to their applications in problems of data restoration and feature extraction. In this paper, we consider an idealized class C2(Z) ⊂ L2(R3) of 3-dimensional data dominated by surface singularities that are orthogonal to the xy plane. To deal with this type of data, we introduce a new multiscale directional representation called cylindrical shearlets and prove that this new approach achieves superior approximation properties not only with respect to conventional multiscale representations but also with respect to 3-dimensional shearlets and curvelets. Specifically, the N-term approximation fNS obtained by selecting the N largest coefficients of the cylindrical shearlet expansion of a function f∈ C(Z) satisfies the asymptotic estimate ‖f-fNS‖22≤cN-2(lnN)3,asN→∞. This is the optimal decay rate, up the logarithmic factor, outperforming 3d wavelet and 3d shearlet approximations which only yield approximation rates of order N- 1 / 2 and N- 1 (ignoring logarithmic factors), respectively, on the same type of data.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1007/s12220-020-00493-0

Keywords

Frames, Image processing, Parseval frames, Shearlets, Sparsity, Wavelets

Publication Date

1-1-2020

Journal Title

Journal of Geometric Analysis

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