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


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.


Anisotropic scaling Curvelets Nonlinear approximation Ridgelets Shearlets Sparsity equivalence Wavelets 



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.


  1. 1.
    Berger, T.: Rate-Distortion Theory. Wiley Online Library, New York (1971)Google Scholar
  2. 2.
    Candès, E.J.: Ridgelets theory and applications. PhD thesis, Stanford University (1998)Google Scholar
  3. 3.
    Candès, E.J., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with \(C^2\) singularities. Commun. Pure Appl. Math. 56, 219–266 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)CrossRefMATHGoogle Scholar
  5. 5.
    DeVore, R.: Nonlinear approximation. Acta Numer. 7, 51–150 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Donoho, D.L.: Sparse components of images and optimal atomic decomposition. Constr. Approx. 17, 353–382 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gribonval, R., Nielsen, M.: Non-linear approximation with dictionaries. I. Direct estimates. J. Fourier Anal. Appl. 10, 51–71 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Grohs, P.: Ridgelet-type frame decompositions for sobolev spaces related to linear transport. J. Fourier Anal. Appl. 18(2), 309–325 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Grohs, P., Keiper, S., Kutyniok, G., Schäfer, M.: \(\alpha \)-Molecules: curvelets, shearlets, ridgelets, and beyond. In: Proceedings SPIE, vol. 2013 (2013)Google Scholar
  10. 10.
    Grohs, P., Keiper, S., Kutyniok, G., Schäfer, M.: \(\alpha \)-Molecules. (2013, preprint)Google Scholar
  11. 11.
    Grohs, P., Kutyniok, G.: Parabolic molecules. Found. Comput. Math. 14(2), 299–337 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Guo, K., Labate, D.: Sparse multidimensional representations using anisotropic dilation and shear operators. Wavelets and Splines (Athens, GA, 2005). Nashboro Press 14, 189–201 (2005)Google Scholar
  13. 13.
    Keiper, S.: A flexible shearlet transform—sparse approximation and dictionary learning. Bachelor’s thesis, TU Berlin (2013)Google Scholar
  14. 14.
    Kutyniok, G., Labate, D.: Introduction to shearlets. Shearlets: Multiscale Analysis for Multivariate Data, pp. 1–38. Birkhäuser, Boston (2012)CrossRefGoogle Scholar
  15. 15.
    Kutyniok, G., Lemvig, J., Lim, W.-Q.: Compactly supported shearlet frames and optimally sparse approximations of functions in \(L^2(\mathbb{R}^3)\) with piecewise \(C^\alpha \) singularities. SIAM J. Math. Anal. 44, 2962–3017 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kutyniok, G., Lim, W.-Q.: Compactly supported shearlets are optimally sparse. J. Approx. Theory 163(11), 1564–1589 (2011)MathSciNetCrossRefMATHGoogle Scholar

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

Personalised recommendations