Advertisement

Parabolic Molecules: Curvelets, Shearlets, and Beyond

Conference paper
First Online:
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 83)

Abstract

Anisotropic representation systems such as curvelets and shearlets have had a significant impact on applied mathematics in the last decade. The main reason for their success is their superior ability to optimally resolve anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shock fronts in solutions of transport dominated equations. By now, a large variety of such anisotropic systems have been introduced, for instance, second-generation curvelets, bandlimited shearlets, and compactly supported shearlets, all based on a parabolic dilation operation. These systems share similar approximation properties, which are usually proven on a case-by-case basis for each different construction. The novel concept of parabolic molecules, which was recently introduced by two of the authors, allows for a unified framework encompassing all known anisotropic frame constructions based on parabolic scaling. The main result essentially states that all such systems share similar approximation properties. One main consequence is that at once all the desirable approximation properties of one system within this framework can be deduced virtually for any other system based on parabolic scaling. This paper motivates and surveys recent results in this direction.

Keywords

Curvelets Nonlinear approximation Parabolic scaling Shearlets 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Andersson, F., de Hoop, M., Smith, H., Uhlmann, G.: A multi-scale approach to hyperbolic evolution equations with limited smoothness. Comm. PDE 33, 988–1017 (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Borup, L., Nielsen, M.: Frame decompositions of decomposition spaces. J. Fourier Anal. Appl. 13, 39–70 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Candès, E.J., Demanet, L.: The curvelet representation of wave propagators is optimally sparse. Comm. Pure Appl. Math. 58, 1472–1528 (2002)CrossRefGoogle Scholar
  4. 4.
    Candès, E.J., Donoho, D.L.: Ridgelets: a key to higher-dimensional intermittency? Phil. Trans. R. Soc. Lond. A. 357, 2495–2509 (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Candès, E.J., Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with \(C^2\) singularities. Comm. Pure Appl. Math. 56, 219–266 (2004)CrossRefGoogle Scholar
  6. 6.
    Candès, E.J., Donoho, D.L.: Continuous curvelet transform: I. Resolution of the wavefront set. Appl. Comput. Harmon. Anal. 19, 162–197 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Candès, E.J., Donoho, D.L.: Continuous curvelet transform: II. Discretization and frames. Appl. Comput. Harmon. Anal. 19, 198–222 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Casazza, P.G., Kutyniok, G. (eds.): Finite Frames: Theory and Applications. Birkhäuser, Boston (2012)Google Scholar
  9. 9.
    Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Bostan (2003)Google Scholar
  10. 10.
    DeVore, R.A.: Nonlinear approximation. Acta Numerica 7, 51–150 (1998)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Donoho, D.L.: Sparse components of images and optimal atomic decomposition. Constr. Approx. 17, 353–382 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Flinth, A.: 3D parabolic molecules. Bachelor’s thesis, Technische Universität Berlin (2013)Google Scholar
  13. 13.
    Gribonval, R., Nielsen, M.: Nonlinear approximation with dictionaries. I. Direct estimates. J. Fourier Anal. Appl. 10, 51–71 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Grohs, P.: Continuous shearlet frames and resolution of the wavefront set. Monatsh. Math. 164, 393–426 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Grohs, P.: Bandlimited shearlet frames with nice duals. J. Comput. Appl. Math. 244, 139–151 (2013)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Grohs, P.: Intrinsic localization of anisotropic frames. Appl. Comput. Harmon. Anal. 35, 264–283 (2013)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Grohs P., Kutyniok, G.: Parabolic molecules. Found. Comput. Math. 14(2), 299–337 (2014)Google Scholar
  18. 18.
    Grohs, P., Keiper, S., Kutyniok, G., Schäfer, M.: \(\alpha \)-Molecules. Preprint (2013)Google Scholar
  19. 19.
    Guo, K., Kutyniok, G., Labate, D.: Sparse multidimensional representations using anisotropic dilation and shear operators. Wavelets and Splines, pp. 189–201 (2005). Nashboro Press, Athens (2006)Google Scholar
  20. 20.
    Guo, K., Labate, D.: Optimally sparse multidimensional representation using shearlets. SIAM J. Math Anal. 39, 298–318 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Guo, K., Labate, D.: Representation of Fourier integral operators using shearlets. J. Fourier Anal. Appl. 14, 327–371 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Guo, K., Labate, D.: The construction of smooth parseval frames of shearlets. Math. Model Nat. Phenom. 8, 82–105 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Keiper, S.: A flexible shearlet transform—sparse approximation and dictionary learning. Bachelor’s thesis, Technische Universität Berlin (2012)Google Scholar
  24. 24.
    Kittipoom, P., Kutyniok, G., Lim, W.-Q.: Construction of compactly supported shearlet. Constr. Approx. 35, 21–72 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Kutyniok, G., Labate, D.: Resolution of the wavefront set using continuous shearlets. Trans. Amer. Math. Soc. 361, 2719–2754 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Kutyniok, G., Lemvig, J.: Optimally sparse approximations of 3D functions by compactly supported shearlet frames. SIAM J. Math. Anal. 44, 2962–3017 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kutyniok, G., Lim, W.-Q.: Compactly supported shearlets are optimally sparse. J. Approx. Theor. 163, 1564–1589 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Kutyniok, G., Labate, D. (eds.): Shearlets: Multiscale Analysis for Multivariate Data. Birkhäuser, Boston (2012)Google Scholar
  29. 29.
    Smith, H.: A parametrix construction for wave equations with \(C^{1,1}\)-coefficients. Ann. Inst. Fourier 48, 797–835 (1998)CrossRefzbMATHGoogle Scholar
  30. 30.
    Wojtaszczyk, P.: A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Seminar for Applied MathematicsETH ZürichZürichSwitzerland
  2. 2.Department of MathematicsTechnische Universität BerlinBerlinGermany

Personalised recommendations

Cite paper

Buy options