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Construction of parametric biorthogonal wavelet filter banks with two parameters for image coding

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Abstract

We had presented a simple technique, which is based on the theory of Diophantine equation, for parametrization of popular biorthogonal wavelet filter banks (BWFBs) having the linear phase and arbitrary multiplicity of vanishing moments (VMs), and constructed a type of parametric BWFBs with one free parameter [15]. Here we generalize this technique to the case of two parameters, and construct a type of parametric BWFBs with two free parameters. The closed-form parameter expressions of the BWFBs are derived, with which any two-parameter family of BWFBs having preassigned VMs can be constructed, and six families, i.e., 9/11, 10/10, 13/11, 10/14, 17/11, and 10/18 families, are considered here. Two parameters provide two degrees of freedom to optimize the resulting BWFBs with respect to other criteria. In particular, in each family, three specific rational-coefficient BWFBs with attractive features are obtained by adjusting the parameters: the first is not only very close to a quadrature mirror filter (QMF) bank, but has optimum coding gain; the second possesses characteristics that are close to the irrational BWFB with maximum VMs by Cohen et al.; and the last which has binary coefficients can realize a multiplication-free discrete wavelet transform. In addition, two BWFBs are systematically verified to exhibit performance competitive to several state-of-the-art BWFBs for image compression, and yet require lower computational costs.

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References

  1. Cohen, A., Daubechies, I., Feauveau, J.C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45, 485–560 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Vetterli, M., Herley, C.: Wavelets and filter banks: theory and design. IEEE Trans. Image Process. 40(9), 2207–2232 (1992)

    MATH  Google Scholar 

  3. Ansari, R., Guillemot, C., Kaiser, J.F.: Wavelet construction using lagrange halfband filters. IEEE Trans. Circ. Syst. 38(9), 1116–1118 (1991)

    Article  Google Scholar 

  4. Villasenor, J.D., Belzer, B., Liao, J.: Wavelet filter evaluation for image compression. IEEE Trans. Image Process. 4(8), 1053–1060 (1995)

    Article  Google Scholar 

  5. Sweldens, W.: The lifting scheme: a custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3, 186–200 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Tsai, M.J., Villasenor, J.D., Chen, F.: Stack-run image coding. IEEE Trans. Circ. Syst. Video Tech. 6(5), 519–521 (1996)

    Article  Google Scholar 

  7. Cooklev, T., Nishihara, A., Sablatash, M.: Regular orthonormal and biorthogonal wavelet filters. Signal Process. 57, 121–137 (1997)

    Article  MATH  Google Scholar 

  8. Wei, D., Tian, J. Jr., Wells, R.O., Burrus, C.S.: A new class of biorthogonal wavelet systems for image transform coding. IEEE Trans. Image Process. 7(7), 1000–1013 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Tay, D.B.H.: Rationalizing the coefficients of popular biorthogonal wavelet filters. IEEE Trans. Circ. Syst. Video Tech. 10(6), 998–1005 (2000)

    Article  Google Scholar 

  10. He, T.-X.: Biorthogonal wavelets with certain regularities. Appl. Comput. Harmon. Anal. 11, 227–242 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Winger, L.L., Venetsanopoulos, A.N.: Biorthogonal nearly coiflet wavelets for image compression. Signal Process. Image Commu. 16, 859–869 (2001)

    Article  Google Scholar 

  12. Averbuch, A.Z., Zheludev, V.A.: Construction of biorthogonal discrete wavelet transforms using interpolatory splines. Appl. Comput. Harmon. Anal. 12, 25–56 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kotteri, K.A., Bell, A.E., Carletta, J.E.: Design of multiplierless, high-performance, wavelet filter banks with image compression applications. IEEE Trans. Circ. Syst.–I: Regular Papers 51(3), 483–494 (2004)

    Article  Google Scholar 

  14. Liu, Z., Zheng, N., Liu, Y., Wetering, H. van de : Optimization design of biorthogonal wavelets for embedded image coding. IEICE Trans. Info. Syst. E 90(D(2), 569–578 (2007)

    Article  Google Scholar 

  15. Liu, Z., Zheng, N.: Parametrization construction of biorthogonal wavelet filter banks for image coding. Signal Image Video Process. 1(1), 63–76 (2007)

    Article  Google Scholar 

  16. Liu, Z., Zheng, N.: Parametrization construction of integer wavelet transforms for embedded image coding. Int. J. Comput. Math. 84(9), 1339–1352 (2007)

    Article  MATH  Google Scholar 

  17. Daubechies, I., Sweldens, W.: Factoring wavelet transforms into lifting steps. J. Fourier Anal. Appl. 4(3), 247–269 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. ISO/IEC 15444-1: Information Technology—JPEG 2000 Image Coding System: Core Coding System, 2nd edn. (2004)

  19. ISO/IEC 15444-2: Information Technology—JPEG 2000 Image Coding System: Extensions (2004)

  20. Katto, J., Yasuda, Y.: Performance evaluation of subband coding and optimization of its filter coefficients. In: Proceeding of SPIE Symposium on Visual Comm. and Image Processing, vol. 1605, pp. 95–106 (1991)

  21. Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. Constructive Approx. 5, 49–68 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Strang, G., Nguyen, T.: Wavelets and Filter Banks, pp. 214–218. Wellesley-Cambridge Press, Wellesley (1996)

    Google Scholar 

  23. Calderbank, A.R., Daubechies, I., Sweldens, W., Yeo, B.-L.: Wavelet transforms that map integers to integers. Appl. Comput. Harmon. Anal. 5, 332–369 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mallat, S.G.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Patt. Anal. Mach. Intell. 11(7), 674–693 (1989)

    Article  MATH  Google Scholar 

  25. Woods, J.W., Naveen, T.: A filter based bit allocation scheme for subband compression of HDTV. IEEE Trans. Image Process. 1(3), 436–440 (1992)

    Article  Google Scholar 

  26. Said, A., Pearlman, W.A.: A new, fast, efficient image codec based on set partitioning in hierarchical trees. IEEE Trans. Circ. Syst. Video Tech. 6(3), 243–250 (1996)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Changshu Institute of Technology, 215500, Changshu, China

    Zaide Liu

  2. Department of Mechanical Engineering, Lanzhou Polytechnic College, 730050, Lanzhou, China

    Chengxiu Gao

Authors
  1. Zaide Liu
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  2. Chengxiu Gao
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Correspondence to Zaide Liu.

Additional information

This work was supported by the Natural Science Foundation of Jiangsu province, China under Grant 07KJD520005.

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Liu, Z., Gao, C. Construction of parametric biorthogonal wavelet filter banks with two parameters for image coding. SIViP 2, 195–206 (2008). https://doi.org/10.1007/s11760-008-0050-y

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  • DOI: https://doi.org/10.1007/s11760-008-0050-y

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