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Design of Optimal Quincunx Filter Banks for Image Coding

Abstract

Two new optimization-based methods are proposed for the design of high-performance quincunx filter banks for the application of image coding. These new techniques are used to build linear-phase finite-length-impulse-response (FIR) perfect-reconstruction (PR) systems with high coding gain, good frequency selectivity, and certain prescribed vanishing-moment properties. A parametrization of quincunx filter banks based on the lifting framework is employed to structurally impose the PR and linear-phase conditions. Then, the coding gain is maximized subject to a set of constraints on vanishing moments and frequency selectivity. Examples of filter banks designed using the newly proposed methods are presented and shown to be highly effective for image coding. In particular, our new optimal designs are shown to outperform three previously proposed quincunx filter banks in 72% to 95% of our experimental test cases. Moreover, in some limited cases, our optimal designs are even able to outperform the well-known (separable) 9/7 filter bank (from the JPEG-2000 standard).

References

  1. 1.

    ISO/IEC 15444-1 Information technology—JPEG 2000 image coding system—Part 1: Core coding system, 2000

  2. 2.

    Tay DBH, Kingsbury NG: Flexible design of multidimensional perfect reconstruction FIR 2-band filters using transformations of variables. IEEE Transactions on Image Processing 1993,2(4):466–480. 10.1109/83.242356

    Google Scholar 

  3. 3.

    Vaidyanathan PP: Multirate Systems and Filter Banks. Prentice Hall, Upper Saddle River, NJ, USA; 1993.

    Google Scholar 

  4. 4.

    Chen T, Vaidyanathan PP: Multidimensional multirate filters and filter banks derived from one-dimensional filters. IEEE Transactions on Signal Processing 1993,41(5):1749–1765. 10.1109/78.215297

    MATH  Google Scholar 

  5. 5.

    Shpairo JM: Adaptive McClellan transformations for quincunx filter banks. IEEE Transactions on Signal Processing 1994,42(3):642–648. 10.1109/78.277856

    Google Scholar 

  6. 6.

    Kalker TACM, Shah IA: Group theoretic approach to multidimensional filter banks: theory and applications. IEEE Transactions on Signal Processing 1996,44(6):1392–1405. 10.1109/78.506606

    Google Scholar 

  7. 7.

    McClellan JH: The design of two-dimensional digital filters by transformation. Proceedings of the 7th Annual Princeton Conference on Information Sciences and Systems, March 1973, Princeton, NJ, USA 247–251.

    Google Scholar 

  8. 8.

    Phoong S-M, Kim CW, Vaidyanathan PP, Ansari R: New class of two-channel biorthogonal filter banks and wavelet bases. IEEE Transactions on Signal Processing 1995,43(3):649–665. 10.1109/78.370620

    Google Scholar 

  9. 9.

    Gouze A, Antonini M, Barlaud M: Quincunx lifting scheme for lossy image compression. Proceedings of the IEEE International Conference on Image Processing (ICIP '00), September 2000, Vancouver, BC, Canada 1: 665–668.

    Google Scholar 

  10. 10.

    Chan SC, Pun KS, Ho KL: On the design and implementation of a class of multiplierless two-channel 1D and 2D nonseparable PR FIR filterbanks. Proceedings of the IEEE International Conference on Image Processing (ICIP '01), October 2001, Thessaloniki, Greece 2: 241–244.

    Google Scholar 

  11. 11.

    Pun KSC, Nguyen TQ: A novel and efficient design of multidimensional PR two-channel filter banks with hourglass-shaped passband support. IEEE Signal Processing Letters 2004,11(3):345–348. 10.1109/LSP.2003.822602

    Google Scholar 

  12. 12.

    Karlsson G, Vetterli M: Theory of two-dimensional multirate filter banks. IEEE Transactions on Acoustics, Speech, and Signal Processing 1990,38(6):925–937. 10.1109/29.56054

    Google Scholar 

  13. 13.

    Viscito E, Allebach JP: The analysis and design of multidimensional FIR perfect reconstruction filter banks for arbitrary sampling lattices. IEEE Transactions on Circuits and Systems 1991,38(1):29–41. 10.1109/31.101301

    Google Scholar 

  14. 14.

    Tran TD, de Queiroz RL, Nguyen TQ: Linear-phase perfect reconstruction filter bank: lattice structure, design, and application in image coding. IEEE Transactions on Signal Processing 2000,48(1):133–147. 10.1109/78.815484

    Google Scholar 

  15. 15.

    Sweldens W: The lifting scheme: a custom-design construction of biorthogonal wavelets. Applied and Computational Harmonic Analysis 1996,3(2):186–200. 10.1006/acha.1996.0015

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Daubechies I, Sweldens W: Factoring wavelet transforms into lifting steps. Journal of Fourier Analysis and Applications 1998,4(3):247–268. 10.1007/BF02476026

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Cooklev T, Nishihara A, Yoshida T, Sablatash M: Multidimensional two-channel linear phase FIR filter banks and wavelet bases with vanishing moments. Multidimensional Systems and Signal Processing 1998,9(1):39–76. 10.1023/A:1008235605754

    MATH  Google Scholar 

  18. 18.

    Kovačević J, Sweldens W: Wavelet families of increasing order in arbitrary dimensions. IEEE Transactions on Image Processing 2000,9(3):480–496. 10.1109/83.826784

    MATH  Google Scholar 

  19. 19.

    Zhou J, Do MN, Kovačević J: Multidimensional orthogonal filter bank characterization and design using the Cayley transform. IEEE Transactions on Image Processing 2005,14(6):760–769.

    Google Scholar 

  20. 20.

    Zhou J, Do MN, Kovačević J: Special paraunitary matrices, Cayley transform, and multidimensional orthogonal filter banks. IEEE Transactions on Image Processing 2006,15(2):511–519.

    MathSciNet  Google Scholar 

  21. 21.

    Feilner M, Van De Ville D, Unser M: An orthogonal family of quincunx wavelets with continuously adjustable order. IEEE Transactions on Image Processing 2005,14(4):499–510.

    MathSciNet  Google Scholar 

  22. 22.

    Van De Ville D, Blu T, Unser M: Isotropic polyharmonic B-splines: scaling functions and wavelets. IEEE Transactions on Image Processing 2005,14(11):1798–1813.

    MathSciNet  Google Scholar 

  23. 23.

    Kovačević J, Vetterli M:Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for. IEEE Transactions on Information Theory 1992,38(2, part 2):533–555. 10.1109/18.119722

    MathSciNet  Google Scholar 

  24. 24.

    Nguyen TT, Oraintara S: Multiresolution direction filterbanks: theory, design, and applications. IEEE Transactions on Signal Processing 2005,53(10):3895–3905.

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Calderbank AR, Daubechies I, Sweldens W, Yeo B-L: Wavelet transforms that map integers to integers. Applied and Computational Harmonic Analysis 1998,5(3):332–369. 10.1006/acha.1997.0238

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Chen Y: Design and application of quincunx filter banks, M.S. thesis. Department of Electrical and Computing Engineering, University of Victoria, Victoria, BC, Canada; 2006.

    Google Scholar 

  27. 27.

    Katto J, Yasuda Y: Performance evaluation of subband coding and optimization of its filter coefficients. Visual Communications and Image Processing (VCIP '91), November 1991, Boston, Mass, USA, Proceedings of SPIE 1605: 95–106.

    Google Scholar 

  28. 28.

    Vetterli M, Kovačević J, Legall DJ: Perfect reconstruction filter banks for HDTV representation and coding. Signal Processing: Image Communication 1990,2(3):349–363. 10.1016/0923-5965(90)90011-6

    Google Scholar 

  29. 29.

    Lobo MS, Vandenberghe L, Boyd S, Lebret H: Applications of second-order cone programming. Linear Algebra and Its Applications 1998,284(1–3):193–228.

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Sturm JF: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software 1999,11(1):625–653. 10.1080/10556789908805766

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Michael Adams', August 2006 https://doi.org/www.ece.uvic.ca/~mdadams

  32. 32.

    Adams MD: ELEC 545 project: a wavelet-based lossy/lossless image compression system. Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC, Canada, April 1999

    Google Scholar 

  33. 33.

    JPEG-2000 test images ISO/IEC JTC 1/SC 29/WG 1 N 545, July 1997

  34. 34.

    SAIC and University of Arizona, "JPEG-2000 VM 0 software", ISO/IEC JTC 1/SC 29/WG 1 N 840, May 1998

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Correspondence to Yi Chen.

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Chen, Y., Adams, M.D. & Lu, W. Design of Optimal Quincunx Filter Banks for Image Coding. EURASIP J. Adv. Signal Process. 2007, 083858 (2006). https://doi.org/10.1155/2007/83858

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Keywords

  • Information Technology
  • Optimal Design
  • Experimental Test
  • Quantum Information
  • Limited Case