Optimally Sparse Representations of Cartoon-Like Cylindrical Data
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.
Frames, Image processing, Parseval frames, Shearlets, Sparsity, Wavelets
Easley, Glenn R., Kanghui Guo, Demetrio Labate, and Basanta R. Pahari. "Optimally sparse representations of cartoon-like cylindrical data." The Journal of Geometric Analysis (2020): 1-21.
Journal of Geometric Analysis