Directional Wavelet Bases Constructions with Dyadic Quincunx Subsampling

  • Rujie YinEmail author
  • Ingrid Daubechies


We construct directional wavelet systems that will enable building efficient signal representation schemes with good direction selectivity. In particular, we focus on wavelet bases with dyadic quincunx subsampling. In our previous work (Yin, in: Proceedings of the 2015 international conference on sampling theory and applications (SampTA), 2015), we show that the supports of orthonormal wavelets in our framework are discontinuous in the frequency domain, yet this irregularity constraint can be avoided in frames, even with redundancy factor <2. In this paper, we focus on the extension of orthonormal wavelets to biorthogonal wavelets and show that the same obstruction of regularity as in orthonormal schemes exists in biorthogonal schemes. In addition, we provide a numerical algorithm for biorthogonal wavelets construction where the dual wavelets can be optimized, though at the cost of deteriorating the primal wavelets due to the intrinsic irregularity of biorthogonal schemes.


MRA Directional wavelet bases Perfect reconstruction Dyadic quincunx downsampling Optimization 

Mathematics Subject Classification




This work is support by the NSF Grant 1516988.


  1. 1.
    Bamberger, R.H., Smith, M.J.T.: A filter bank for the directional decomposition of images: theory and design. IEEE Trans. Signal Process. 40(4), 882–893 (1992)CrossRefGoogle Scholar
  2. 2.
    Blanchard, J., Krishtal, I.: Matricial filters and crystallographic composite dilation wavelets. Math. Comput. 81(278), 905–922 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Candes, E., Demanet, L., Donoho, D., Ying, L.: Fast discrete curvelet transforms. Multiscale Model. Simul. 5(3), 861–899 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cohen, A., Daubechies, I., Feauveau, J.-C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45(5), 485–560 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cohen, A., Schlenker, J.-M.: Compactly supported bidimensional wavelet bases with hexagonal symmetry. Constr. Approx. 9(2–3), 209–236 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Do, M.N., Vetterli, M.: The contourlet transform: an efficient directional multiresolution image representation. IEEE Trans. Image Process. 14(12), 2091–2106 (2005)CrossRefGoogle Scholar
  7. 7.
    Durand, S.: M-band filtering and nonredundant directional wavelets. Appl. Comput. Harmon. Anal. 22(1), 124–139 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Easley, G.R., Labate, D.: Critically sampled composite wavelets. In: Proceedings of the 2009 Conference Record of the Forty-Third Asilomar Conference on Signals, Systems and Computers, pp. 447–451. IEEE (2009)Google Scholar
  9. 9.
    Easley, G., Labate, D., Lim, W.-Q.: Sparse directional image representations using the discrete shearlet transform. Appl. Comput. Harmon. Anal. 25(1), 25–46 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Goossens, B, Aelterman, J, Luong, H., Pižurica, A., Philips, W.: Design of a tight frame of 2d shearlets based on a fast non-iterative analysis and synthesis algorithm. SPIE Optics and Photonics, pp. 81381Q (2011)Google Scholar
  11. 11.
    Han, B., Zhao, Z., Zhuang, X.: Directional tensor product complex tight framelets with low redundancy. Appl. Comput. Harmon. Anal. 41(2), 603–637 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kutyniok, G., Lim, W.-Q., Zhuang, X.: Digital shearlet transforms. In: Shearlets, pp. 239–282. Springer, Berlin (2012)Google Scholar
  13. 13.
    Nguyen, T.T., Oraintara, S.: Multiresolution direction filterbanks: theory, design, and applications. IEEE Trans. Signal Process. 53(10), 3895–3905 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sauer, T.: Shearlet multiresolution and multiple refinement. In: Kutyniok, G. (ed.) Shearlets. Multiscale Analysis for Multivariate Data, pp. 199–237. Applied and Numerical Harmonic Analysis, Birkhäuser (2012)Google Scholar
  15. 15.
    Selesnick, I.W., Baraniuk, R.G., Kingsbury, N.C.: The dual-tree complex wavelet transform. IEEE Signal Process. Mag. 22(6), 123–151 (2005)CrossRefGoogle Scholar
  16. 16.
    Yin, R.: Construction of orthonormal directional wavelets based on quincunx dilation subsampling. In: Proceedings of the 2015 International Conference on Sampling Theory and Applications (SampTA), May 2015, pp. 292–296 (2015)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA

Personalised recommendations