Multiresolution Signal Decomposition Transforms, Subbands, and Wavelets phần 6 potx

260 CHAPTER 3. THEORY OF SUBBAND DECOMPOSITION Figure 3.78: Subbands corresponding to nodes in Fig. 3.77. 3.10. SUMMARY 261 Figure 3.79: Four-subband directional filter bank. Chapter 6, we will show this dyadic tree to be a precursor of the orthonormal wavelet transform. Using the AC matrix and the polyphase decomposition, we were able to for- mulate general conditions for PR in the M-band filter structure. This led to a general time-domain formulation of the analysis-synthesis subband system that unifies critically sampled block transforms, LOTs, and critically sampled subband filter banks. The paraunitary filter bank provided an elegant solution in terms of the polyphase matrix. The focus on the two-dimensional subband filter bank was the subsampling or decimation lattice. We showed how the ID results could be generalized to 2D, but in a nontrivial way. 262 CHAPTER 3. THEORY OF SUBBAND DECOMPOSITION Figure 3.80: Subbands corresponding to nodes (5)-(10) in Fig. 3.79. References E. H. Adelson, E. Simoncelli, and R. Hingorani, "Orthogonal Pyramid Trans- forms for Image Coding," Proc. SPIE Visual Communication and Image Process- ing," pp. 50-58, 1987. A. N. Akansu, and Y. Liu, "On Signal Decomposition Techniques," Optical Engineering, pp. 912-920, July 1991. A. N. Akansu, R. A. Haddad, and H. Caglar, "Perfect Reconstruction Bino- mial QMF-Wavelet Transform," Proc. SPIE Visual Communication and Image Processing, Vol. 1360, pp. 609-618, Oct. 1990. A. N. Akansu, R. A. Haddad, and H. Caglar, "The Binomial QMF-Wavelet Transform for Multiresolution Signal Decomposition," IEEE Trans, on Signal Pro- cessing, Vol. 41. No. 1, pp. 13-20, Jan. 1993. R. Ansari, "Efficient IIR and FIR Fan Filters," IEEE Trans. Circuits and Systems, Vol. CAS-34, pp. 941-945, Aug. 1987. R. Ansari and C. Guillemot, "Exact Reconstruction Filter Banks Using Dia- mond FIR Filters," Proc. BILCON, pp. 1412-1424, July 1990. R. Ansari and C. L. Lau, "Two-Dimensional IIR Filters for Exact Reconstruc- tion in Tree-Structured Subband Decomposition," Electronics Letters, Vol. 23, pp. 3.10. SUMMARY 263 633-634, June, 1987. R. Ansari arid S. H. Lee, "Two-Dimensional Non-rectangular Interpolation, Decimation, and Filter Banks," presented at ICASSP, 1988. R. Ansari and B. Liu, "Efficient Sample Rate Alteration Using Recursive(IIR.) Digital Filters;' IEEE Trans. ASSP, Vol. ASSP-32, pp. 1366-1373, Dec. 1983. R,. Ansari and B, Liu, "A Class of Low-noise Computationally Efficient Recur- sive Digital Filters with Applications to Sampling Rate Alterations," IEEE Trans. ASSP, Vol. ASSP-33, pp. 90 97, Feb. 1985. R. Ansari, C. Guillemot, and J. F. Kaiser, "Wavelet Construction Using La- grange Halfband Filters," IEEE Trans. Circuits and Systems, Vol. CAS-38, pp. 1116-1118, Sept. 1991. M. Antonini, M. Barlaud, P. Mathieu, I. Daubechies, "Image Coding Using Vector Quantization in the Wavelet Transform Domain," Proc. ICASSP, pp. 2297- 2300, 1990. 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. 40, pp. 882-893, April 1992. T. P. Barnwell III, "Subband Coder Design Incorporating Recursive Quadra- ture Filters and Optimum ADPCM Coders," IEEE Trans. ASSP, Vol. ASSP-30, pp. 751-765, Oct. 1982. V. Belevitch, Classical Network Theory. Holden-Day, 1968. M. G. Bellanger, "Computation Rate and Storage Estimation in Multirate Digital Filtering with Half-band Filters," IEEE Trans. ASSP, Vol. ASSP-25. pp. 344-346, Aug. 1977. M. G. Bellanger and J. L. Daguet, "TDM-FDM Transmultiplexer: Digital Poly- phase and FFT," IEEE Trans. Communications, Vol. COM-22, pp. 1199-1204, Sept. 1974. M. G. Bellanger, J. L. Daguet, and G. P. Lepagnol, "Interpolation, Extrapola- tion, and Reduction of Computation Speed in Digital Filters," IEEE Trans. ASSP. Vol. ASSP-22,, pp. 231-235, Aug. 1974. M. G. Bellanger, G. Bonnerot, and M. Coudreuse, "Digital Filtering by Poly- phase Network: Application to Sample-rate Alteration and Filter Banks," IEEE Trans. ASSP, Vol. ASSP-24, pp. 109-114, April 1976. P. J. Burt and E. H. Adelson, "The Laplacian Pyramid as a Compact Image Code," IEEE Trans. Comm., pp. 532-540, April 1983. 264 CHAPTER 3. THEORY OF SUBBAND DECOMPOSITION T. Caelli and M. Hubner, "Coding Images in the Frequency Domain: Filter Design and Energy Processing Characteristics of the Human Visual System," IEEE Trans, on Systems. Man and Cybernetics, pp. 1018 1021, May 1980. B. Chitprasert arid K. R. Rao, "Discrete Cosine Transform Filtering/' Proc. IEEE ICASSP, pp. 1281 1284, 1990. L. Chen, T. Q. Nguyen, and K-P Chan. "Symmetric extension methods for M- channel linear-phase perfect reconstruction filter banks", IEEE Trans, on Signal Processing, Vol. 13, No. 12, pp. 2505-2511, Dec. 1995. P. L. Chu, "Quadrature Mirror Filter Design for an Arbitrary Number of Equal Bandwidth Channels," IEEE Trans. ASSP, Vol. ASSP-33, pp. 203-218, Feb. 1985. A. Cohen, I. Daubechies, and J. C. Feauveau, "Biorthogonal Bases of Com- pactly Supported Wavelets," Technical Memo., #11217-900529-07, AT&T Bell Labs., Murray Hill. R. E. Crochiere and L. R. Rabiner, "Optimum FIR Digital Filter Implemen- tations for Decimation, Interpolation, and Narrow-band Filtering," IEEE Trans. ASSP, Vol. ASSP-23, pp. 444-456, Oct. 1975. R. E, Crochiere and L. R. Rabiner, "Further Considerations in the Design of Decimators and Interpolators," IEEE Trans. ASSP, Vol. ASSP-24, pp. 296-311, Aug. 1976. R. E. Crochiere and L. R. Rabiner, "Interpolation and Decimation of Digital Signals-A Tutorial Review," Proc. IEEE, Vol. 69, pp. 300-331, March 1981. R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing. Pren- tice-Hall, 1983. R. E. Crochiere, S. A.Weber, and J. L. Flanagan, "Digital Coding of Speech Subbands," Bell Syst. Tech. J., Vol. 55, pp. 1069-1085, 1976. A. Croisier, D. Esteban, and C. Galand, "Perfect Channel Splitting by use of Interpolation/Decimation/Tree Decomposition Techniques," Int'l Conf. on Infor- mation Sciences and Systems, Patras, 1976. A. W. Crooke and J. W. Craig, "Digital Filters for Sample-Rate Reduction." IEEE Trans. Audio and Electroacous., Vol. AU-20, pp. 308-315, Oct. 1972. P. Delsarte, B. Macq, and D. T.M. Slock, "Signal-Adapted Multiresolution Transform for Image Coding," IEEE Trans. Information Theory, Vol. 38, pp. 897 904, March 1992. R. L. de Queiroz, T. Q. Nguyen, and K. R. Rao, "The GenLOT: generalized linear-phase lapped orthogonal transform", IEEE Trans, on Signal Processing, Vol. 44, No. 3, pp. 497-507, March 1996. 3 10. SUMMARY 265 D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal Process- ing. Prentice-Hall, 1984, D. Esteban and C. Galand, "Application of Quadrature Mirror Filters to Split- band Voice Coding Schemes," Proc. ICASSP, pp. 191-195, 1977. A. Fettweis, J. A. Nossek, and K. Meerkotter, "Reconstruction of Signals after Filtering and Sampling Rate Reduction," IEEE Trans. ASSP, Vol. ASSP-33, pp. 893 902, Aug. 1985. S. W. Foo and L. F. Turner, "Design of Nonrecursive Quadrature Mirror Fil- ters," IEE Proc., Vol. 129, part G, pp. 61-67, June 1982. R. Forchheimer and T. Kronander, "Image Coding—From Waveforms to An- imation," IEEE Trans. ASSP, Vol. ASSP-37, pp. 2008 2023, Dec. 1989. D. Gabor, "Theory of Communications," Proc. IEE, pp. 429-461, 1946. C. R. Galand and H. J. Nussbaumer, "New Quadrature Mirror Filter Struc- tures," IEEE Trans. ASSP, Vol. 32, pp. 522-531, June 1984. L. Gazsi, "Explicit Formulas for Lattice Wave Digital Filters," IEEE Trans. Circuits and Systems, Vol. CAS-32, pp. 68-88, Jan. 1985. H. Gharavi and A. Tabatabai, "Subband Coding of Digital Image Using Two- Dimensional Quadrature Mirror Filtering," Proc. SPIE Visual Communication and Image Processing, pp. 51-61, 1986. H. Gharavi and A. Tabatabai, "Sub-band Coding of Monochrome and Color Images," IEEE Trans, on Circuits and Systems, Vol. CAS-35, pp. 207-214, Feb. 1988. R. C. Gonzales and P. Wintz, Digital Image Processing. 2nd ed. Addison- Wesley, 1987. D. J. Goodman and M. J. Carey, "Nine Digital Filters for Decimation and Interpolation," IEEE Trans. ASSP, Vol. ASSP-25, pp. 121-126, Apr. 1977. R. A. Gopinath and C. S. Burrus, "On Upsampling, Downsampling and Ratio- nal Sampling Rate Filter Banks," Tech. Rep., CML TR-91-25, Rice Univ., Nov. 1991. R. A. Haddad arid T. W. Parsons, Digital Signal Processing: Theory, Applica- tions and Hardware. Computer Science Press, 1991. R. A. Haddad and A. N. Akansu, "A Class of Fast Gaussian Binomial Filters for Speech and Image Processing," IEEE Trans, on Signal Processing, Vol. 39, pp. 723-727, March 1991. J. H. Husoy, Subband Coding of Still Images and Video. Ph.D. Thesis, Norwe- gian Institute of Technology, 1991. 266 CHAPTER 3. THEORY OF SUBBAND DECOMPOSITION A. Ikonomopoulos and M. Kunt, "High Compression Image Coding via Direc- tional Filtering," Signal Processing, Vol. 8, pp. 179-203, 1985. V. K. Jain and R. E. Crochiere, "Quadrature Mirror Filter Design in the Time Domain,'' IEEE Trans. ASSP, Vol. ASSP-32, pp. 353-361, April 1984. J. D. Johnston, "A Filter Family Designed for Use in Quadrature Mirror Filter Banks," Proc. ICASSP, pp. 291-294, 1980. G. Karlsson and M. Vetterli, "Theory of Two-Dimensional Multirate Filter Banks," IEEE Trans. ASSP, Vol. 38, pp. 925-937, June 1990. C. W. Kim and R. Ansari, "FIR/IIR Exact Reconstruction Filter Banks with Applications to Subband Coding of Images," Proc. Midwest CAS Symp 1991. R. D. Koilpillai and P. P. Vaidyanathan, "Cosine modulated FIR filter banks satisfying perfect reconstruction", IEEE Trans, on Signal Processing, Vol. 40, No. 4, pp. 770-783, Apr., 1992. T. Kronander, Some Aspects of Perception Based Image Coding. Ph.D. Thesis, Linkoping University, 1989. T. Kronander, "A New Approach to Recursive Mirror Filters with a Special Application in Subband Coding of Images," IEEE Trans. ASSP, Vol. 36, pp. 1496 1500, Sept. 1988. M. Kunt, A. Ikonomopoulos, and M. Kocher, "Second Generation Image Cod- ing Techniques," Proc. IEEE, Vol. 73, pp. 549-574, April 1985. H. S. Malvar, "The LOT: A Link Between Block Transform Coding and Mul- tirate Filter Banks," Proc. IEEE ISCAS, pp. 781-784, 1988. H. S. Malvar, "Modulated QMF Filter Banks with Perfect Reconstruction," Electronics Letters, Vol. 26, pp. 906-907, June 1990. H. S. Malvar, "Lapped Transforms for Efficient Transform/Subband Coding," IEEE Trans. ASSP, Vol. 38, pp. 969-978, June 1990. H. S. Malvar, "Efficient Signal Coding with Hierarchical Lapped Transforms," Proc. ICASSP, pp. 1519-1522, 1990. H. S. Malvar, Signal Processing with Lapped Transforms. Artech House, 1992. H. S. Malvar, "Extended lapped transforms: properties, applications and fast algorithms", IEEE Trans, on Signal Processing, Vol. 40, No. 11, pp. 2703-2714, Nov. 1992. H. S. Malvar and D. H. Staelin, "Reduction of Blocking Effects in Image Coding with a Lapped Orthogonal Transform," Proc. IEEE ICASSP, pp. 781-784, April 1988. 3.JO. SUMMARY 267 H. S. Malvar and D. H. Staelin, "The LOT: Transform Coding without Block- ing Effects,' 1 IEEE Trans. ASSP, Vol.37, pp. 553-559, Apr. 1989. D. F. Marshall, W. K. Jenkins, and J. J. Murphy, "The Use of Orthogonal Transforms for Improving Performance of Adaptive Filters," IEEE Trans. Circuits and Systems, Vol. 36, pp. 474-484, April 1989. J. Masson arid Z. Picel, "Flexible Design of Computationally Efficient Nearly Perfect QMF Filter Banks," Proc. IEEE ICASSP, pp. 541-544, 1985. P. C. Millar, "Recursive Quadrature Mirror Filters-Criteria Specification and Design Method, 1 ' IEEE Trans. ASSP, Vol. ASSP-33, pp. 413-420, 1985. F. Miritzer, "On Half-band, Third-band and Nth-band FIR Filters and Their Design," IEEE Trans, on ASSP, Vol. ASSP-30, pp. 734-738, Oct. 1982. F. Miritzer, "Filters for Distortion-free Two-band Multirate Filter Banks," IEEE Trans. ASSP, Vol.33, pp. 626-630, June, 1985. F. Miritzer and B. Liu, "The Design of Optimal Multirate Bandpass and Band- stop Filters," IEEE Trans. ASSP, Vol. ASSP-26, pp. 534-543, Dec. 1978. F. Mintzer and B. Liu, "Aliasing Error in the Design of Multirate Filters," IEEE Trans. IEEE, Vol. ASSP-26, pp. 76-88, Feb. 1978. T. Miyawaki and C. W. Barnes, "Multirate Recursive Digital Filters: A General Approach and Block Structures," IEEE Trans. ASSP, Vol. ASSP-31, pp. 1148 1154, Oct. 1983. K. Nayebi, T. P. Barnwell III, and M. J. T. Smith, "The Time Domain Anal- ysis and Design of Exactly Reconstructing FIR Analysis/Synthesis Filter Banks," Proc. IEEE ICASSP, pp. 1735-1738, April 1990. K. Nayebi, T. P. Barnwell III, and M. J. T. Smith, "Time Domain Filter Bank Analysis: A New Design Theory," IEEE Trans. Signal Processing, Vol. 40, No. 6, pp. 1412-1429, June 1992. K. Nayebi, T. P. Barnwell III, and M. J. T. Smith, "Nonuniform Filter Banks: A Reconstruction and Design Theory," IEEE Trans. Signal Processing, Vol. 41, No. 3, pp. 1114-1127, March 1993. K. Nayebi, T. P. Barnwell III, and M. J. T. Smith, "The Design of Nonuniform Band Filter Banks," Proc. IEEE ICASSP, pp. 1781-1784, 1991. T. Q. Nguyen and R. D. Koilpillai, "Theory and Design of Arbitrary Length Cosine Modulated Filter Banks and Wavelets Satisfying Perfect Reconstruction", IEEE Trans, on Signal Processing, Vol. 44, No.3, pp. 473-483, March 1996. T. Q. Nguyen and P. P. Vaidyanathan, "Two-channel Perfect-reconstruction FIR QMF Structures Which Yield Linear-phase Analysis and Synthesis Filters," IEEE Trans. ASSP, Vol. ASSP-37, No. 5, pp. 676-690, May, 1989. 268 CHAPTER 3. THEORY OF SUBBAND DECOMPOSITION T, Q. Nguyen and P. P. Vaidyanathan, "Structure for M-channel Perfect Re- construction FIR QMF Banks which Yield Linear-Phase Analysis and Synthesis Filters," IEEE Trans. ASSP, Vol. 38, pp. 433 446, March 1990. H. J. Nussbaumer, "Pseudo-QMF Filter Bank," IBM Tech. Disclosure Bull., Vol. 24, pp. 3081-3087, Nov. 1981. H. J. Nussbaumer and M. Vetterli, "Computationally Efficient QMF Filter Banks," Proc. IEEE ICASSP, pp. 11.3.1 -11.3.4, 1984. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd Edi- tion, pp. 119 123. McGrawHill, 1991. S. R. Pillai, W. Robertson, W. Phillips, "Subband Filters Using All-pass Struc- tures," Proc. IEEE ICASSP, pp. 1641-1644, 1991. J. P. Princen and A. B. Bradley, "Analysis/Synthesis Filter Bank Design Based on Time Domain Aliasing Cancellation," IEEE Trans. ASSP, Vol. ASSP-34, pp. 1153-1161, Oct. 1986. J. P. Princen, A. W. Johnson, and A. B. Bradley, "Sub-band/Transform Cod- ing Using Filter Bank Designs Based on Time Domain Aliasing Cancellation." Proc. IEEE ICASSP, pp. 2161-2164, April 1987. T. A. Ramstad, "Digital Methods for Conversion between Arbitrary Sampling Frequencies," IEEE Trans. ASSP, Vol. ASSP-32, pp. 577-591, June 1984. T. A. Ramstad, "Analysis-Synthesis Filter Banks with Critical Sampling," Proc. Int'l Conf. Digital Signal Processing, pp. 130-134, Sept. 1984. T. A. Ramstad, "IIR Filter Bank for Subband Coding of Images," Proc. ISCAS, pp. 827-830, 1988. T. A. Ramstad and J. P. Tanem, "Cosine Modulated Analysis-Synthesis Filter Bank with Critical Sampling and Perfect Reconstruction, Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, pp. 1789-1792, May 1991. T. A. Ramstad, S. O. Aase and J. H. Husoy, Subband Compression of Images: Principles and Examples. Elsevier, 1995. A. Rosenfeld, Ed., Multiresolution Techniques in Computer Vision. Springer- Verlag, New York, 1984. J. H. Rothweiler, "Polyphase Quadrature Filters—A New Subband Coding Technique," Proc. IEEE ICASSP, pp. 1280-1283, 1983. R. W. Schafer and L. R. Rabiner, "A Digital Signal Processing Approach to Interpolation," Proc, IEEE, Vol. 61, pp. 692-702, June 1973. R. R. Shively, "On Multistage FIR Filters with Decimation," IEEE Trans. ASSP, Vol. ASSP-23, pp. 353-357, Aug. 1975. 3.10. SUMMARY 269 E. Simoncelli, Orthogonal Sub-band Image Transforms. M. S. Thesis, Mas- sachusetts Institute of Technology, May 1988. M. J. T. Smith and T. P. Barnwell, "A Procedure for Designing Exact Re- construction Filter Banks for Tree-Structured Sub-band Coders," Proc. IEEE ICASSP, pp. 27.1.1-27.1.4, 1984. M. J. T. Smith and T. P. Barnwell, "Exact Reconstruction Techniques for Tree-Structured Subband Coders," IEEE Trans. ASSP, pp. 434-441, 1986. M. J. T. Smith and T. P. Barnwell, "A New Filter Bank Theory for Time- Frequency Representation," IEEE Trans. ASSP, Vol. ASSP-35, No.3, pp. 314 327, March, 1987. M. J. T. Smith and S. L. Eddins, "Analysis/Synthesis Techniques for Subband Image Coding," IEEE Trans. ASSP, Vol. ASSP-38, pp. 1446-1456, Aug. 1990. A. K. Soman, P. P. Vaidyanathari, and T. Q. Nguyen, "Linear Phase Parau- nitary Filter Banks: Theory, Factorization, and Design", IEEE Trans, on Signal Processing, Vol. 41, No. 12, pp. 3480-3496, Dec. 1993. N. Uzun and R. A. Haddad, "Modeling and Analysis of Quantization Errors in Two Channel Subband Filter Structures," Proc. SPIE Visual Comm. and Image Processing, Nov. 1992. P. P. Vaidyanathan, "Quadrature Mirror Filter Banks, M-Band Extensions and Perfect Reconstruction Techniques," IEEE ASSP Magazine, pp. 4 20, July 1987. P. P. Vaidyanathan, "Theory and Design of M-channel Maximally Decimated Quadrature Mirror Filters with Arbitrary M, Having the Perfect Reconstruction Property," IEEE Trans. ASSP, pp. 476-492, April 1987. P. P. Vaidyanathan, Multirate Systems and Filterbanks. Prentice-Hall, 1993. P. P. Vaidyanathan, "Multirate Digital Filters, Filter Banks, Polyphase Net- works, and Applications: A Tutorial," Proc. IEEE, Vol. 78, pp. 56-93, Jan. 1990. P. P. Vaidyanathan, and Z. Doganata, "The Role of Lossless Systems in Modern Digital Signal Processing," IEEE Trans. Education, Special Issue on Circuits and Systems, Vol. 32, No. 3, pp. 181-197, Aug. 1989. P. P. Vaidyanathan, and P. Q. Hoang, "Lattice Structures for Optimal Design and Robust Implementation of Two-band Perfect Reconstruction QMF Banks." IEEE Trans. ASSP, Vol. ASSP-36, No.l, pp. 81-94, Jan. 1988. P. P. Vaidyanathan, T. Q. Nguyen, Z. Doganata, and T. Saramaki, "Improved Technique for Design of Perfect Reconstruction FIR QMF Banks with Lossless Polyhase Matrices," IEEE Trans. ASSP, pp. 1042-1056, July 1989. [...]... -0.023900270 561 13145 -0.075940 963 79188282 CHAPTER 4 FILTER BANK FAMILIES: DESIGN 16 TAP 0.02193598203004352 0.00157 861 649 766 3704 -0. 060 25449102875281 -0.01189 065 962 05391 0.13753791 563 662 5 0.057454500 563 90939 -0.32 167 02 961 65893 -0.528720271545339 -0.29577 967 4500919 0.0002043110845170894 0.029 066 997894 467 96 -0.035334 860 887081 46 -0.0 068 21045322743358 0.0 260 667 8 468 264 118 0.001033 363 4919441 26 -0.01435930957477529... 0.001050 167 24 TAP(B) 0.4731289 0.1 160 355 -0.09829783 -0.02 561 533 0.044239 76 0.003891522 -0.01901993 0.0014 464 61 0.0 064 85879 -0.001373 861 -0.001392911 0.00038330 96 12 TAP(B) 0.4807 962 0.09808522 -0.0913825 -0.00758 164 0.02745539 -0.0 064 43977 16 TAP(B) 0.4773 469 0.1 067 987 -0.09530234 -0.0 161 1 869 0.035 968 53 -0.0019209 36 -0.009972252 0.002898 163 24 TAP(C) 0. 468 6479 0.12 464 52 -0.09987885 -0.03 464 143 0.05088 162 ... -0.00009 961 78 168 7347404 -0.008795047132402801 0.000708779549084502 0.012204201 560 35413 -0.001 762 6393147953 36 -0.01558455903573820 0.00408285 567 5 060 479 0.01 765 222024089335 -0.003835219782884901 -0.0 167 4 761 38847 368 8 0.018239 062 10 869 841 0.005781735813341397 -0.0 469 267 409090 767 5 0.05725005445073179 0.354522945953839 0.504811839124518 0. 264 955 363 281817 -0.08329095 161 140 063 -0.1391087475849 26 0.033140 360 8 065 9188... 0.09035938422033127 -0.01 468 791729134721 -0. 061 033358 867 07139 0.0 066 061 2 263 8753900 0.0405155508803 568 5 -0.00 263 1418173 168 537 -0.025925804 761 49722 0.0009319532350192227 0.0153 563 89599 161 69 -0.00011 968 3 269 33 261 84 -0.01057032258472372 Table 4.5: The 8-, 16- , and 32-tap PR-CQF coefficients with 40 dB stopband attenuation [M.J.T Smith and T.P Barnwell, ©19 86, IEEE) 4 .6 LEGALL-TABATABAI PR FILTER BANK 291 Figure 4 .6: (a)... -0.09987885 -0.03 464 143 0.05088 162 0.0100 462 1 -0.02755195 -0.00 065 0 466 9 0.01354012 -0.002273145 -0.005182978 0.002329 266 287 16 TAP(C) 0.4721122 0.117 866 6 -0.0992955 -0.0 262 7 56 0 067 8 4 464 0.00199115 -0.02048751 0.0 065 2 566 6 24 TAP(D) 0. 465 4288 0.1301121 -0.09984422 -0.04089222 0.05402985 0.01547393 -0.03295839 -0.004013781 0.019 763 8 -0.001571418 -0.01 061 4 0.00 469 84 26 Table 4.4: Johnston QMF coefficients... 0.224143 868 0420 0.8 365 163 037378 0.482 962 9131445 6 tap 0.0352 262 935542 -0.0854412721235 -0.1350110232992 0.4598774983 863 0.8 068 9151040 46 0.33 267 05543970 8 tap -0.0105973984294 0.0328830189591 0.0308413834495 -0.1870348133 969 -0.027983 763 8710 0 .63 08807718592 0.7148 465 672 569 0.2303778109845 8 tap 8 tap -0.075 765 7137833 -0.02 963 55292117 047 169 86 968 533 0.8037387521124 0.2978578127957 -0.0992195317257 -0.01 260 3 969 0937... case CHAPTER 4 FILTER BANK FAMILIES: 2 76 n 0 1 2 3 0 1 2 3 4 5 0 1 2 3 4 5 6 7 Mini Phase 4 tap 0.482 962 91314453 0.8 365 163 0373780 0.224143 868 04201 -0.129409522551 26 6 tap 0.33 267 055439701 0.8 068 9151040 469 0.4598774983 863 0 -0.13501102329922 -0.08544127212359 0.0352 262 9355424 8 tap 0.23037781098452 0.7148 465 672 569 1 0 .63 0880771859 26 -0.027983 763 87108 -0.1870348133 969 3 0.03084138344957 0.03288301895913... -0.044524230 0.054812130 0.019472180 -0.034 964 400 -0.007 961 7310 0.022704150 0.002 069 4700 -0.014228990 0.00084 268 330 0.0081819410 -0.001 969 6720 -0.0039715520 0.0022551390 32 TAP(E) 0.45 964 550 0.138 764 20 -0.09 768 3790 -0.051382570 0.055707210 0.0 266 24310 -0.0383 061 30 -0.014 569 000 0.028122590 0.0073798 860 -0.021038230 -0.00 261 20410 0.01 568 0820 -0.000 962 45920 -0.01127 565 0 0.0051232280 Table 4.4 (continued):... two-band filter bank is possible if the linear-phase requirement is relaxed The Smith-Barnwell filters were called conjugate quadrature filters 4.5 SMITH-BARNWELL PR-CQF FAMILY 8 TAP 0.48998080 0. 069 42827 -0.07 065 183 0.00938715 12 TAP(A) 0.48438940 0.088 469 92 -0.08 469 594 -0.0027103 26 0.0188 565 9 -0.00380 969 9 16 TAP(A) 0.4810284 0.09779817 -0.09039223 -0.00 966 63 76 0.02 764 14 -0.0025897 56 -0.0050545 26 0.001050 167 ... THEORY OF SUBBAND DECOMPOSITION L Vandendorpe, "Optimized Quantization for Image Subband Coding," Signal Processing, Image Communication, Vol 4, No 1, pp 65 -80, Nov 1991 M Vetterli and C Her ley, "Wavelets and Filter Banks: Theory and Design," IEEE Trans Signal Processing, Vol 40, No 9, pp 2207-2232, Sept 1992 M Vetterli, "Multi-dimensional Sub-band Coding: Some Theory and Algorithms," Signal Processing, . tap -0.0105973984294 0.0328830189591 0.0308413834495 -0.1870348133 969 -0.027983 763 8710 0 .63 08807718592 0.7148 465 672 569 0.2303778109845 8 tap -0.075 765 7137833 -0.02 963 55292117 0.49 761 865 938 36 0.8037387521124 0.2978578127957 -0.0992195317257 -0.01 260 3 969 0937 0.0322230981272 8. tap 0.33 267 055439701 0.8 068 9151040 469 0.4598774983 863 0 -0.13501102329922 -0.08544127212359 0.0352 262 9355424 8 tap 0.23037781098452 0.7148 465 672 569 1 0 .63 0880771859 26 -0.027983 763 87108 -0.1870348133 969 3 0.03084138344957 0.03288301895913 -0.01059739842942 Non- . tap -0.1294095225512 0.224143 868 0420 0.8 365 163 037378 0.482 962 9131445 6 tap 0.0352 262 935542 -0.0854412721235 -0.1350110232992 0.4598774983 863 0.8 068 9151040 46 0.33 267 05543970 8 tap -0.0105973984294 0.0328830189591 0.0308413834495 -0.1870348133 969 -0.027983 763 8710 0 .63 08807718592 0.7148 465 672 569 0.2303778109845 8