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. 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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. 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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. 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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. 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[...]... -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