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Theory and design of two-dimensional filter Banks: A review

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Abstract

There has been considerable interest in the design of multidimensional (MD) filter banks. MD filter banks find application in subband coding of images and video data. MD filter banks can be designed by cascading one-dimensional (1D) filter banks in the form of a tree structure. In this case, the individual analysis and synthesis filters are separable and the filter bank is called a separable filter bank. MD filter banks with nonseparable filters offer more flexibility and usually provide better performance. Nonetheless, their design is considerably more difficult than separable filter banks. The purpose of this paper is to provide an overview of developments in this field on the design techniques for MD filter banks, mostly two-dimensional (2D) filter banks. In some image coding applications, the 2D two-channel filter banks are of great importance, particularly the filter bank with diamond-shaped filters. We will present several design techniques for the 2D two-channel nonseparable filter banks. As the design of MD filters are not as tractable as that of 1D filters, we seek design techniques that do not involve direct optimization of MD filters. To facilitate this, transformations that turn a separable MD filter bank into a nonseparable one are developed. Also, transformations of 1D filter banks to MD filter banks are investigated. We will review some designs of MD filter banks using transformations. In the context of 1D filter bank design, the cosine modulated filter bank (CMFB) is well-known for its design and implementation efficiency. All the analysis filters are cosine modulated versions of a prototype filter. The design cost of the filter bank is equivalent to that of the prototype and the implementation complexity is comparable to that of the prototype plus a low-complexity matrix. The success with 1D CMFB motivate the generalization to the 2D case. We will construct the 2D CMFB by following a very close analogy of 1D case. It is well-known that the 1D lossless systems can be characterized by state space description. In 1D, the connection between the losslessness of a transfer matrix and the unitariness of the realization matrix is well-established. We will present the developments on the study of 2D lossless systems. As in 1D case, the 2D FIR lossless systems can be characterized in terms of state space realizations. We will review this, and then address the factorizability of 2D FIR lossless systems by using the properties of state space realizations.

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References

  1. R. Ansari and C. Guillemot, “Exact reconstruction filter banks using diamond FIR filters,”Proc. Int. Conf. on New Trends in Comm. Control, and Signal Proc., Turkey, July 1990.

  2. R. Ansari and C. L. Lau, “Two-dimensional IIR filters for exact reconstruction in tree-structured subband decomposition,”Electronic Letter, vol. 23, June 1987, pp. 633–634.

    Google Scholar 

  3. R. H. Bamberger and M. J. T. Smith, “A filter bank for the directional decomposition of images: theory and design,”IEEE Trans. on Signal Processing, vol. SP-40, no. 4, April 1992, pp. 882–893.

    Google Scholar 

  4. S. Basu, H. Choi, and C. Chiang, “On non-separable multidimensional perfect reconstruction filter banks,”Proc. of the 27th Annual Asilomar Conference on Signals, Systems and Computers, 1993, pp. 45–49.

  5. N. K. Bose,Applied Multidimensional Systems Theory, Van Nostrand Reinhold, 1982.

  6. T. Chen and P. P. Vaidyanathan, “The role of integer matrices in multidimensional multirate systems,”IEEE Trans. on Signal Processing, vol. SP-41, March 1993.

  7. T. Chen and P. P. Vaidyanathan, “Recent developments in multidimensional multirate systems,”IEEE Trans. on Circuits And Systems For Video Technology, vol. 3, no. 2, April 1993, pp. 116–137.

    Google Scholar 

  8. T. Chen and P. P. Vaidyanathan, “Consideration in multidimensional filter bank design,”Proc. International Symposium on Circuits and Systems, May 1993.

  9. T. Chen and P. P. Vaidyanathan, “Multidimensional multirate filters and filter banks derived from one dimensional filters,”IEEE Trans. on Signal Processing, vol. SP-41, May 1993.

  10. P. L. Chu, “Quadrature mirror filter design for an arbitrary number of equal bandwidth channels,”IEEE Trans. on Acoustic, Speech and Signal Processing, vol. 33, Feb. 1985, pp. 203–218.

    Google Scholar 

  11. A. Cohen and I. Daubechies, “Nonseparable bidimensional wavelet bases,” Preprint, 1993.

  12. R. E. Crochiere, S. A. Waber, and J. L. Flanagan, “Digital coding of speech in subbands,”Bell Sys. Tech. Jour., vol. 55, Oct. 1976, pp. 1069–1085.

    Google Scholar 

  13. R. E. Crochiere and L. R. Rabiner,Multirate Digital Signal Processing, Englewood Cliffs: Prentice Hall, 1983.

    Google Scholar 

  14. Z. Doganata, P.P. Vaidyanathan, and T. Q. Nguyen, “General synthesis procedures for FIR lossless transfer matrices, for perfect-reconstruction multirate filter bank applications,”IEEE Trans. on Acoustic, Speech and Signal Processing, vol. ASSP-36, Oct. 1988, pp. 1561–1574.

    Google Scholar 

  15. D. E. Dudgeon and R. M. Mersereau,Multidimensional Digital Signal Processing, Englewood Cliffs: Prentice Hall, 1984.

    Google Scholar 

  16. R. Eising, “Realization and stabilization of 2-D systems,”IEEE Trans. on Automatic Control, vol. 23, Oct. 1978, pp. 793–799.

    Google Scholar 

  17. B. L. Evans, R. H. Bamberger, and J. H. McClellan, “Rules for multidimensional multirate structures,”IEEE Trans. on Signal Processing, vol. SP-42, April 1994, pp. 762–771.

    Google Scholar 

  18. R. A. Gopinath and C. S. Burrus, “On upsampling, downsampling, and rational sampling rate filter banks,”IEEE Trans. on Signal Processing, vol. SP-42, April 1994, pp. 812–824.

    Google Scholar 

  19. M. Ikehara, “Cosine-modulated 2 dimensional FIR filter banks satisfying perfect reconstruction,”Proc. International Conf. on Acoustic, Speech, and Signal Processing, vol. III, April 1994, pp. 137–140.

    Google Scholar 

  20. M. Ikehara, “Modulated 2 dimensional perfect reconstruction FIR filter banks with permissible passbands,”Proc. International Conf. on Acoustic, Speech, and Signal Processing, May 1995, pp. 1468–1471.

  21. N. S. Jayant and P. Noll,Digital Coding of Waveforms, Englewood Cliffs: Prentice Hall, 1984.

    Google Scholar 

  22. A. A. C. M. Kalker, “Commutativity of up/down sampling,”Electron. Lett., vol. 28, no. 6, March 1992, pp. 567–569.

    Google Scholar 

  23. G. Karlsson and M. Vetterli, “Theory of two-dimensional multirate filter banks,”IEEE Trans. on Acoustic, Speech and Signal Processing, vol. SP-38, June 1990, pp. 925–937.

    Google Scholar 

  24. C. W. Kim and R. Ansari, “FIR/IIR exact reconstruction filter banks with applications to subband coding of images,” Midwest CAS Symposium, May 1991.

  25. C. W. Kim and R. Ansari, “Subband decomposition procedure for quincunx sampling grids,”Proc. SPIE Visual Communications and Image Processing, Boston, Nov. 1991.

  26. R. D. Koilpillai and P. P. Vaidyanathan, “Cosine-modulated FIR filter banks satisfying perfect reconstruction,”IEEE Trans. on Signal Processing, vol. 40, April 1992, pp. 770–783.

    Google Scholar 

  27. J. Kovacevic and M. Vetterli, “The commutativity of up/downsampling in two dimensions,”IEEE Trans. on Information Theory, vol. 37, no. 4, May 1991, pp. 695–698.

    Google Scholar 

  28. J. Kovacevic and M. Vetterli, “Non-separable multidimensional perfect reconstruction filter banks and wavelet bases forR n,”IEEE Trans. on Information Theory, vol. 38, no. 2, March 1992, pp. 533–555.

    Google Scholar 

  29. J. Kovacevic, “Local cosine bases in two dimensions,”Proc. International Conf. on Acoustic, Speech, and Signal Processing, vol. IV, May 1995, pp. 2125–2128.

    Google Scholar 

  30. M. Kunt, A. Ikonomopoulos and M. Kocher, “Second generation image coding techniques,”Proc. IEEE, vol. 73, April 1985, pp. 549–574.

    Google Scholar 

  31. Y. Lin and P. P. Vaidyanathan, “Theory and design of two-dimensional cosine modulated filter banks,” Tech. report, California Institute of Technology, Pasadena, CA, March 1995.

    Google Scholar 

  32. Y. Lin and P. P. Vaidyanathan, “Two-dimensional paraunitary cosine modulated perfect reconstruction filter banks,”Proc. International Symposium on Circuits and Systems, April 1995, pp. 752–755.

  33. Y. Lin and P. P. Vaidyanathan, “On the sampling of two-dimensional bandpass signal,” In preparation.

  34. V. C. Liu and P. P. Vaidyanathan, “On factorization of a subclass of 2D digital FIR lossless matrices for 2D QMF bank applications,”IEEE Trans. on Circuits and Systems, vol. 37, no. 6, June 1990, pp. 852–854.

    Google Scholar 

  35. H. S. Malvar,Signal Processing with Lapped Transforms, Norwood, MA: Artech House, 1992.

    Google Scholar 

  36. J. H. McClellan, “The design of two-dimensional digital filters by transformations,”Proc. Seventh Annual Princeton Conf. Information Sciences and Systems, 1973, pp. 247–251.

  37. H. J. Nussbaumer, “Pseudo QMF filter bank,”IBM Tech. Disclosure Bulletin, vol. 24, Nov. 1981, pp. 3081–3087.

    Google Scholar 

  38. S. Phoong, C. W. Kim, P. P. Vaidyanathan, and R. Ansari, “A new class of two-channel biorthogonal filter banks and wavelet bases,”IEEE Trans. on Signal Processing, vol. SP-43, no. 3, March 1995, pp. 649–665.

    Google Scholar 

  39. T. A. Ramstad, “Cosine modulated analysis-synthesis filter bank with critical sampling and perfect reconstruction,”Proc. IEEE Int. Conf. Acoustic, Speech and Signal Processing, Toronto, Canada, May 1991, pp. 1789–1792.

    Google Scholar 

  40. R. P. Roesser, “A discrete state-space model for linear image processing,”IEEE Trans. on Automatic Control, vol. 20, Feb. 1975, pp. 1–10.

    Google Scholar 

  41. J. H. Rothweiler, “Polyphase quadrature filters, a new subband coding technique,”Proc. of the IEEE Int. Conf. on Acoustic, Speech and Signal Processing, April 1973, pp. 1980–1983.

  42. I. A. Shah and A. A. C. Kalker, “Generalized theory of multidimensional M-band filter bank design,”EUSIPCO, 1992, pp. 969–972.

  43. I. A. Shah and A. A. C. Kalker, “Algebraic theory of multidimensional filter banks and their design using transformations,” preprint.

  44. M. J. T. Smith and S. L. Eddins, “Analysis/synthesis techniques for subband image coding,”IEEE Trans. on Acoustic, Speech and Signal Processing, vol. 38, no. 8, 1990, pp. 1446–1456.

    Google Scholar 

  45. D. B. H. Tay and N. G. Kingsbury, “Flexible design of multidimensional perfect reconstruction FIR 2-band filters using transformation of variables,”IEEE Trans. on Image Processing, vol. 2, no. 4, Oct. 1993, pp. 466–480.

    Google Scholar 

  46. P. P. Vaidyanathan, “The discrete-time bounded-real lemma in digital filtering,”IEEE Trans. on Circuits and Systems, vol. 32, Sept. 1985, pp. 918–924.

    Google Scholar 

  47. P. P. Vaidyanathan and T. Q. Nguyen, “A “Trick” for the design of FIR halfband filters,”IEEE Trans. on Circuits and Systems, vol. 34, March 1987, pp. 297–300.

    Google Scholar 

  48. P. P. Vaidyanathan, “Fundamentals of multidimensional multirate digital signal processing,”Sadahana, vol. 15, Nov. 1990, pp. 157–176.

    Google Scholar 

  49. P. P. Vaidyanathan, “New results in multidimensional multirate systems,”Proc. International Symposium on Circuits and Systems, 1991, pp. 468–471.

  50. P. P. Vaidyanathan,Multirate Systems and Filter Banks, Englewood Cliffs: Prentice Hall, 1993.

    Google Scholar 

  51. S. Venkataraman and B. C. Levy, “State space representations of 2D FIR lossless transfer matrices,”IEEE Trans. on Circuits and Systems, vol. 41, Feb. 1994, pp. 117–131.

    Google Scholar 

  52. M. Vetterli, “Multidimensional subband coding: Some theory and algorithms,”Signal Processing, vol. 6, no. 2, Feb. 1984, pp. 97–112.

    Google Scholar 

  53. E. Viscito and J. P. Allebach, “The analysis and design of multidimensional FIR perfect reconstruction filter banks for arbitrary sampling lattices,”IEEE Trans. on Circuits and Systems, vol. CAS-38, no. 1, Jan. 1991, pp. 29–41.

    Google Scholar 

  54. J. W. Woods and S. D. O'Neil, “Subband coding of images,”IEEE Trans. on Accoust. Speech and Signal Proc., vol. 34, Oct. 1986, pp. 1278–1288.

    Google Scholar 

  55. J. W. Woods,Subband Image Coding, Norwell, MA: Kluwer Academic Publishers, Inc., 1991.

    Google Scholar 

  56. P. Yip, and K. R. Rao, “Fast discrete transforms,” inHandbook of Digital Signal Processing, edited by D. F. Elliott, San Diego, CA: Academic Press, 1987.

    Google Scholar 

  57. D. C. Youla, “The synthesis of networks containing lumped and distributed elements,”Proc. Symp. on Generalized Networks, New York: Polytechnic Institute of Brooklyn Press, Apr. 1966.

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, 136-93, California Institute of Technology, 91125, Pasadena, CA

    Yuan-Pei Lin

  2. Department of Electrical Engineering, 136-93, California Institute of Technology, 91125, Pasadena, CA

    P. P. Vaidyanathan

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Lin, YP., Vaidyanathan, P.P. Theory and design of two-dimensional filter Banks: A review. Multidim Syst Sign Process 7, 263–330 (1996). https://doi.org/10.1007/BF01826246

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