Journal of Fourier Analysis and Applications

, Volume 22, Issue 6, pp 1235–1293 | Cite as

Cartoon Approximation with \(\alpha \)-Curvelets

  • Philipp Grohs
  • Sandra Keiper
  • Gitta Kutyniok
  • Martin Schäfer
Article

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 (\(\alpha = \frac{1}{2}\)). Our main result states that those frames achieve this optimal rate for \(\alpha = \frac{1}{\beta }\), up to \(\log \)-factors.

Keywords

Anisotropic scaling Curvelets Nonlinear approximation Ridgelets Shearlets Sparsity equivalence Wavelets 

Notes

Acknowledgments

PG was supported in part by Swiss National Fund (SNF) Grant 146356. SK acknowledges support from the Berlin Mathematical School and the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics. GK was supported in part by the Einstein Foundation Berlin, by the Einstein Center for Mathematics Berlin (ECMath), by Deutsche Forschungsgemeinschaft (DFG) Grant KU 1446/14, by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”, and by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Philipp Grohs
    • 1
    • 2
  • Sandra Keiper
    • 3
  • Gitta Kutyniok
    • 3
  • Martin Schäfer
    • 3
  1. 1.University of ViennaOskar Morgenstern Platz 1ViennaAustria
  2. 2.Seminar for Applied MathematicsETH ZürichZurichSwitzerland
  3. 3.Department of MathematicsTechnische Universitaet BerlinBerlinGermany

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