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$\alpha$-Molecules: Curvelets, Shearlets, Ridgelets, and Beyond

by P. Grohs and S. Keiper and G. Kutyniok and M. Schaefer

(Report number 2013-24)

Abstract
The novel framework of parabolic molecules provides for the first time a unifying framework for (sparse) approximation properties of directional representation systems by, in particular, including curvelets and shearlets. However, the considered common bracket is parabolic scaling, which excludes systems such as ridgelets and wavelets. In this paper, we therefore provide a generalization of this framework, which we coin α-molecules, by introducing an additional parameter α, which specifies the extent of anisotropy in the scaling. We show that, for instance, both ridgelets and wavelets are in fact α-molecules. As an application of the concept, we then analyze the sparse approximation behavior of α-molecules. Utilizing the idea of sparsity equivalence, it is possible to identify large classes of α-molecules providing the same sparse approximation behavior.

Keywords: Ridgelets, Wavelets, Curvelets, Nonlinear Approximation, Anisotropic Scaling, Shearlets, Sparsity Equivalence, Parabolic Molecules, Cartoon Images

@Techreport{GKKS13_521,
  author = {P. Grohs and S. Keiper and G. Kutyniok and M. Schaefer},
  title = {$\alpha$-Molecules: Curvelets, Shearlets, Ridgelets, and Beyond},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-24},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-24.pdf },
  year = {2013}
}

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