Wavelet Frames Generated by Spline Based p-Filter Banks

  • Amir Z. Averbuch
  • Pekka Neittaanmaki
  • Valery A. Zheludev


This chapter presents a design scheme to generate tight and so-called semi-tight frames in the space of discrete-time periodic signals. The frames originate from three- and four-channel perfect reconstruction periodic filter banks. The filter banks are derived from interpolating and quasi-interpolating polynomial splines and from discrete splines. Each filter bank comprises one linear phase low-pass filter (in most cases interpolating) and one high-pass filter, whose magnitude’s response mirrors that of a low-pass filter. In addition, these filter banks comprise one or two band-pass filters. In the semi-tight frames case, all the filters have linear phase and (anti)symmetric impulse response, while in the tight frame case, some of band-pass filters are slightly asymmetric. The design scheme enables to design framelets with any number of LDVMs.


  1. 1.
    A. Averbuch, V. Zheludev, T. Cohen, Interpolatory frames in signal space. IEEE Trans. Signal Process. 54(6), 2126–2139 (2006)CrossRefGoogle Scholar
  2. 2.
    L.M. Bregman, The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7(3), 200–217 (1967)CrossRefGoogle Scholar
  3. 3.
    J. Cai, S. Osher, and Z. Shen. Split Bregman methods and frame based image restoration. Multiscale Model. Simul., 8(2):337–369, 2009/10.Google Scholar
  4. 4.
    J. Cai, Z. Shen, Framelet based deconvolution. J. Comput. Math. 28(3), 289–308 (2010)MATHMathSciNetGoogle Scholar
  5. 5.
    R.H. Chan, S.D. Riemenschneider, L. Shen, Z. Shen, High-resolution image reconstruction with displacement errors: a framelet approach. Internat. J. Imaging Systems Tech. 14(3), 91–104 (2004)CrossRefMathSciNetGoogle Scholar
  6. 6.
    R.H. Chan, S.D. Riemenschneider, L. Shen, Z. Shen, Tight frame: An efficient way for high-resolution image reconstruction. Appl. Comput. Harmon. Anal. 17(1), 91–115 (2004)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    C.K. Chui, W. He, Compactly supported tight frames associated with refinable functions. Appl. Comput. Harmon. Anal. 8(3), 293–319 (2000)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Z. Cvetković, M. Vetterli, Oversampled filter banks. IEEE Trans. Signal Process. 46(5), 1245–1255 (1998)CrossRefGoogle Scholar
  9. 9.
    B. Dong, H. Ji, J. Li, Z. Shen, Y. Xu, Wavelet frame based blind image inpainting. Appl. Comput. Harmon. Anal. 32(2), 268–279 (2012)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets. Appl. Comput. Harmon. Anal. 22(1), 78–104 (2007)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    D. Gabor, Theory of communications. J. Inst. Electr. Eng. 93, 429–457 (1946)Google Scholar
  12. 12.
    V.K. Goyal, J. Kovacevic, J.A. Kelner, Quantized frame expansions with erasures. Appl. Comput. Harmon. Anal. 10(3), 203–233 (2001)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    V.K. Goyal, M. Vetterli, N.T. Thao, Quantized overcomplete expansions in \(\mathbb{R}^N\): Analysis, synthesis and algorithms. IEEE Trans. Inform. Theory 44(1), 16–31 (1998)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    J. Kovacevic, P.L. Dragotti, V.K. Goyal, Filter bank frame expansions with erasures. IEEE Trans. Inform. Theory 48(6), 1439–1450 (2002)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    A.P. Petukhov, Symmetric framelets. Constr. Approx. 19(2), 309–328 (2003)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    G. Polya, G. Szegö, Aufgaben and Lehrsätze aus der Analysis, vol. II (Springer, Berlin, 1971)CrossRefMATHGoogle Scholar
  17. 17.
    L. Shen, M. Papadakis, I.A. Kakadiaris, I. Konstantinidis, D. Kouri, D. Hoffman, Image denoising using a tight frame. IEEE Trans. Image Process. 15(5), 1254–1263 (2006)CrossRefGoogle Scholar
  18. 18.
    Z. Shen. Wavelet frames and image restorations. In R. Bhatia, editor, Proceedings of the International Congress of Mathematicians, Vol. IV, pages 2834–2863, New Delhi, 2010. Hindustan Book Agency.Google Scholar
  19. 19.
    W. Yin, S. Osher, D. Goldfarb, J. Darbon, Bregman iterative algorithms for \(l_1\)-minimization with applications to compressed sensing. SIAM J. Imaging Sci. 1(1), 143–168 (2008)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    V. Zheludev, V. N. Malozemov, and A. B. Pevnyi. Filter banks and frames in the discrete periodic case. In N. N. Uraltseva, editor, Proceedings of the St. Petersburg Mathematical Society, Vol. XIV, volume 228 of Amer. Math. Soc. Transl., Ser. 2, pages 1–11, 2009.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Amir Z. Averbuch
    • 1
  • Pekka Neittaanmaki
    • 2
  • Valery A. Zheludev
    • 1
  1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Mathematical Information TechnologyUniversity of JyväskyläJyväskyläFinland

Personalised recommendations