Shearlets pp 239-282 | Cite as

Digital Shearlet Transforms

  • Gitta KutyniokEmail author
  • Wang-Q Lim
  • Xiaosheng Zhuang
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Over the past years, various representation systems which sparsely approximate functions governed by anisotropic features such as edges in images have been proposed. We exemplarily mention the systems of contourlets, curvelets, and shearlets. Alongside the theoretical development of these systems, algorithmic realizations of the associated transforms were provided. However, one of the most common shortcomings of these frameworks is the lack of providing a unified treatment of the continuum and digital world, i.e., allowing a digital theory to be a natural digitization of the continuum theory. In fact, shearlet systems are the only systems so far which satisfy this property, yet still deliver optimally sparse approximations of cartoon-like images. In this chapter, we provide an introduction to digital shearlet theory with a particular focus on a unified treatment of the continuum and digital realm. In our survey we will present the implementations of two shearlet transforms, one based on band-limited shearlets and the other based on compactly supported shearlets. We will moreover discuss various quantitative measures, which allow an objective comparison with other directional transforms and an objective tuning of parameters. The codes for both presented transforms as well as the framework for quantifying performance are provided in the Matlab toolbox ShearLab.

Key words

Digital shearlet system Fast digital shearlet transform Performance measures Pseudo-polar Fourier transform Pseudo-polar grid ShearLab Software package Tight frames 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The first author would like to thank David Donoho and Morteza Shahram for many inspiring discussions on topics in this area. She also acknowledges support by the Einstein Foundation Berlin and by Forschungsgemeinschaft (DFG) Grant SPP-1324 KU 1446/13 and DFG Grant KU 1446/14. The second author was supported by DFG Grant SPP-1324 KU 1446/13, and the third author was supported by DFG Grant KU 1446/14.


  1. 1.
    A. Averbuch, R. R. Coifman, D. L. Donoho, M. Israeli, and Y. Shkolnisky, A framework for discrete integral transformations I—the pseudo-polar Fourier transform, SIAM J. Sci. Comput. 30 (2008), 764–784.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    D. H. Bailey and P. N. Swarztrauber, The fractional Fourier transform and applications, SIAM Review, 33 (1991), 389–404.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    E. J. Candès, L. Demanet, D. L. Donoho and L. Ying, Fast discrete curvelet transforms, Multiscale Model. Simul. 5 (2006), 861–899.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    E. J. Candès and D. L. Donoho, Ridgelets: a key to higher-dimensional intermittency?, Phil. Trans. R. Soc. Lond. A. 357 (1999), 2495–2509.zbMATHCrossRefGoogle Scholar
  5. 5.
    E. J. Candès and D. L. Donoho, New tight frames of curvelets and optimal representations of objects with C 2 singularities, Comm. Pure Appl. Math. 56 (2004), 219–266.CrossRefGoogle Scholar
  6. 6.
    E. J. Candès and D. L. Donoho, Continuous curvelet transform: I. Resolution of the wavefront set, Appl. Comput. Harmon. Anal. 19 (2005), 162–197.Google Scholar
  7. 7.
    E. J. Candès and D. L. Donoho, Continuous curvelet transform: II. Discretization of frames, Appl. Comput. Harmon. Anal. 19 (2005), 198–222.Google Scholar
  8. 8.
    M. N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Trans. Image Process. 14 (2005), 2091–2106.MathSciNetCrossRefGoogle Scholar
  9. 9.
    D. L. Donoho, Wedgelets: nearly minimax estimation of edges, Ann. Statist. 27 (1999), 859–897.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    D. L. Donoho, G. Kutyniok, M. Shahram, and X. Zhuang, A rational design of a digital shearlet transform, Proceeding of the 9th International Conference on Sampling Theory and Applications, Singapore, 2011.Google Scholar
  11. 11.
    D. L. Donoho, A. Maleki, M. Shahram, V. Stodden, and I. Ur-Rahman, Fifteen years of Reproducible Research in Computational Harmonic Analysis, Comput. Sci. Engr. 11 (2009), 8–18.CrossRefGoogle Scholar
  12. 12.
    G. Easley, D. Labate, and W.-Q Lim, Sparse directional image representations using the discrete shearlet transform, Appl. Comput. Harmon. Anal. 25 (2008), 25–46.Google Scholar
  13. 13.
    B. Han, G. Kutyniok, and Z. Shen, Adaptive multiresolution analysis structures and shearlet systems, SIAM J. Numer. Anal. 49 (2011), 1921–1946.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    E. Hewitt and K.A. Ross, Abstract Harmonic Analysis I, II, Springer-Verlag, Berlin/ Heidelberg/New York, 1963.Google Scholar
  15. 15.
    P. Kittipoom, G. Kutyniok, and W.-Q Lim, Construction of compactly supported shearlet frames. Constr. Approx. 35 (2012), 21–72.Google Scholar
  16. 16.
    G. Kutyniok, M. Shahram, and X. Zhuang, ShearLab: A rational design of a digital parabolic scaling algorithm, preprint.Google Scholar
  17. 17.
    G. Kutyniok and T. Sauer, Adaptive directional subdivision schemes and shearlet multiresolution analysis, SIAM J. Math. Anal. 41 (2009), 1436–1471.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    W.-Q Lim, The Discrete Shearlet Transform: A new directional transform and compactly supported shearlet frames, IEEE Trans. Imag. Proc. 19 (2010), 1166–1180.Google Scholar
  19. 19.
    W.-Q Lim, Nonseparable Shearlet Transforms, preprint.Google Scholar
  20. 20.
    S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. New York: Academic, 1999.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations