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Cartoon Approximation with $\alpha$-Curvelets

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

(Report number 2014-07)

Abstract
It is well-known that curvelets provide optimal approximations for so-called cartoon images which are defined as piecewise \(C^2\)-functions, separated by a \(C^2\) singularity curve. In this paper, we consider the more general case of piecewise \(C^\beta\)-functions, separated by a \(C^\beta\) singularity curve for \(\beta \in (1,2]\). We first prove a benchmark result for the possibly achievable best \(N\)-term approximation rate for this more general signal model. Then we introduce what we call \(\alpha\)-curvelets, which are systems that interpolate between wavelet systems on the one hand (\(\alpha = 1\)) and curvelet systems on the other hand (\(\alpha = \frac12\)). Our main result states that those frames achieve this optimal rate for \(\alpha = \frac{1}{\beta}\), up to \(\log\)-factors.

Keywords: Curvelet, Nonlinear Approximation, Sparsity, Cartoon Images

@Techreport{GKKS14_557,
  author = {P. Grohs and S. Keiper and G. Kutyniok and M. Schaefer},
  title = {Cartoon Approximation with $\alpha$-Curvelets},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-07},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-07.pdf },
  year = {2014}
}

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