Rate-distortion methods for: Image and video compression

IEEE Signal Processing Magazine

Volumn 15, Issue 6, 1998, Pages 23-50

  Ortega, Antonio ^a   Ramchandran, Kannan ^a  

^a University of Illinois at Urbana Champaign   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DECODING; IMAGE CODING; IMAGE COMMUNICATION SYSTEMS; INFORMATION THEORY; OPTIMIZATION; SIGNAL DISTORTION; STANDARDS;

MOTION PICTURE EXPERTS GROUP (MPEG) STANDARDS; RATE-DISTORTION (R-D) OPTIMIZATION TECHNIQUES; SHANNON THEORY;

IMAGE COMPRESSION;

EID: 0032202963     PISSN: 10535888     EISSN: None     Source Type: Journal    
DOI: 10.1109/79.733495     Document Type: Article

References (125)

1. G.M. Schuster and A.K. Katsaggelos, Rate-Distortion Based Video Compression, Optimal Video Frame Compression and Object Boundary Encoding. Kluwer Academic Publishers, 1997.
2. G. Sullivan and T. Wiegand, "Rate-distortion optimization for video compression," this issue, pp. 74-90.
3. C.E. Shannon, "A mathematical theory of communication, " Bell Syst. Tech. Journal, vol. 27, pp. 379-423, 1948.
4. C.E. Shannon, "Coding theorems for a discrete source with a fidelity criterion," in IRE National Convention Record, Part 4, pp. 142-163, 1959.
5. W. Pennebaker and J. Mitchell, JPEG Still Image Data Compression Standard. Van Nostrand Reinhold, 1994.
6. D. Lee, "New work item proposal: JPEG 2000 image coding system." ISO/TEC JTC1/SC29/WG1 N390, 1996.
7. Testing Ad Hoc Group, "JPEG2000 testing results." ISO/IEC JTC1J/SC29/WG1/N705, Nov. 1997.
8. N. Jayant, J. Johnston, and R. Safranek, "Signal compression based on models of human perception," Proc of the IEEE, Oct. 1993.
9. T. Berger, Rate-Distortion Theory. A Mathematical Theory for Data Compression. Prentice-Hall, 1971.
10. R.M. Gray, Source Coding Theory. Kluwer Academic Publishers, 1990.
11. W. Bennett, "Spectra of quantized signals," Bell Sys. Tech. J., vol. 27, pp. 446-472, Jul. 1948.
12. P. Zador, Development and Evaluation of Procedures for Quantizing Multivariate Distributions. PhD thesis, Stanford University, Stanford, CA, 1964.
13. A. Gersho, "Asymptotically optimal block quantization," IEEE Trans. on Info. Th., vol. IT-23, pp. 373-380, Jul. 1979.
14. D.L. Neuhoff, "The other asymptotic theory of source coding," in Joint IEEE/DLMACS Workshop on Coding and Quantization, (Rutgers University, Piscataway, NJ), Oct 1992. Also available as ftp://ftp.eecs.umich.cdu/pcople/neuhoff/OthAsymptThy.ps.
15. T.M. Cover and J.A. Thomas, Elements of Information Theory. New York: Wiley, 1991.
16. N. Farvardin and J.W. Modestino, "Optimum quantizer performance for a class of non-Gaussian memoryless sources, IEEE Trans. on Info. Th., vol. IT-30, pp. 485-497, May 1984.
17. P.A. Chou, T. Lookabaugh, and R.M. Gray, "Entropy-constrained vector quantization," IEEE Trans. ASSP,vol. 37, pp. 31-42, Jan. 1989.
18. M. Effros, "Optimal modeling for complex system design," this issue, pp. 51-73.
19. P.H. Westerink, J. Biemond, and D.E. Boekee, "An optimal bit allocation algorithm for sub-band coding," Proc. of ICASSP, pp. 757-760, 1988.
20. J.M. Shapiro, "Embedded image coding using zerotrees of wavelet coefficients," IEEE Trans. on Signal Proc., vol. 41, pp. 3445-3462, Dec. 1993.
21. A. Said and W. A. Pearlman, "A new fast and efficient image coder based on set partitioning in hierarchical trees," IEEE Trans. Circuits and Systems for Video Technology, pp. 243-250, June 1996.
22. Z. Xiong, K. Ramchandran, and M.T. Orchard, "Space-frequency quantization for wavelet image coding," IEEE Trans. on Image Proc., vol. 6, pp. 677-693, May 1997.
23. R.L. Joshi, H. Jafarkhani, J.H. Kasner, T.R. Fischer, N. Farvardin, M.W. Marcellin, and R.H. Bamberger, "Comparison of different methods of clas-sification in subband coding of images," IEEE Tram. on Image Processing, vol. 6, pp. 1473-1486, Nov. 1997.
24. S. LoPresto, K. Ramchandran, and M.T. Orchard, "Image coding based on mixture modeling of wavelet coefficients and a fast estimation-quantization framework," in Data Compression Conference '97, (Snowbird, Utah), pp. 221-230, 1997.
25. Y. Yoo, A. Onega, and B. Yu, "Adaptive quantization of image subbands with efficient overhead rate selection," in Proc. of the Intl. Conf. on Image Proc., ICIP'96, vol. 2, (Lausanne, Switzerland), pp. 361-364, Sept. 1996.
26. C. Chrysafis and A. Ortega, "Efficient context-based entropy coding for lossy wavelet image compression," in Data Compression Conference '97, (Snowbird, Utah), pp. 241-250, 1997.
27. S.P. Lloyd, "Least squares quantization in PCM," IEEE Trans. on Info. Th., vol. IT-28, pp. 127-135, Mar. 1982. Also, Unpublished Bell Labs Technical Report, 1957.
28. J. Max, "Quantizing for minimum distortion," IEEE Trans. on Info. Th., pp. 7-12, Mar. 1960.
29. Y. Linde, A. Buzo, and R.M. Gray, "An algorithm for vector quantizer design," IEEE Trans. on Comm., vol. COM-28, pp. 84-95, Jan. 1980.
30. P.A. Chou, T. Lookabaugh, and R.M. Gray, "Optimal pruning with applications to tree-structured source coding and modeling," IEEE Trans. on Inform. Theory, vol. 35, pp. 299-315, Mar. 1989.
31. A. Gersho and R.M. Gray, Vector Quantization and Signal Compression. Kluwer Academic Publishers, 1992.
32. J. Mitchell, W. Pennebaker, C.E. Fogg, and D.J. LeGall, MPEG Video Compression Standard. New York: Chapman and Hall, 1997.
33. A.N. Netravali and B.G. Haskell, Digital Pictures: Representation, Compression and Standards, New York: Plenum Press, 2nd ed., 1995.
34. R.J. Clarke, Digital Compression of Still Images and Video. Academic Press, 1995.
35. V. Bhaskaran and K. Konstantinides, Image and Video Compression Standards: Algorithms and Architectures. Kluwer Academic Publishers, 1996.
36. K.R. Rao and J.J. Hwang, Techniaue & Standards for Image, Video & Audio Ceding. Prentice Hall, 1996.
37. B.G. Haskell, A. Puri, and A.N. Netravali, Digital Video: An Introduction to MPEG-2. Chapman and Hall, 1997.
38. ITU-T (formerly CCITT), "Video coding for low bitrate communication," ITU-T Recommendation H.263; version 1, Nov. 1995, version 2, Jan. 1998.
39. S. Wu and A. Gersho, "Rate-constrained picture-adaptive quantization for JPEG baseline coders" in Proc. IEEE International Conference Acoustics, Speech and Signal Processing, ICASSI'93, vol. 5, (Minneapolis, MN), pp. 389-392, April 1993.
40. K. Ramchandran and M. Vetterli, "Rate-distortion optimal fast thresholding with complete JPEG/MPEG decoder compatibility," IEEE Trans. on Image Proc., vol. 3, pp. 700-704, Sept. 1994.
41. M. Crouse and K. Ramchandran, "Joint thresholding and quantizer selection for transform image-coding: Entropy-constrained analysis and applications to baseline JPEG," IEEE Transactions on Image Processing, vol. 6, pp. 285-297, February 1997.
42. K. Ramchandran and M. Vetterli, "Best wavelet packet bases in a rate-distortion sense," IEEE Trans. on Image Proc., vol.2, pp. 160-175, Apr. 1993.
43. M.W. Garrett, "Statistical analysis of a long trace of VBR coded video." Ph.D. Thesis, Chapter IV, Columbia University, 1993.
44. A.R. Reibman and B.G. Haskell, "Constraints on variable bit-rate video for ATM networks,"IEEE Trans. on CAS for video tech., vol. 2, pp. 361-372, Dec. 1992.
45. S.-W. Wu and A. Gersho, "Rate-constrained optimal block-adaptive coding for digital tape recording of HDTV," IEEE Trans. on Circuits and Sys. for Video Tech., vol. 1, pp. 100-112, Mar. 1991.
46. A. Ortega, K. Ramchandran, and M. Vetterli, "Optimal trellis-based buffered compression and fast approximation," IEEE Trans. on Image Proc., vol. 3, pp. 26-40, Jan. 1994.
47. W. Ding and B. Liu, "Rate control of MPEG video coding and recording by rate-quantization modeling," IEEE Transactions on Circuits and Systems for Video Technology, vol. 6, pp. 12-20, Feb 1996.
48. L.-J. Lin and A. Ortega, "Bit-rate control using piecewise approximated rate-distortion characteristics," IEEE Trans. on Circ. and Sys. for Video Tech., vol. 8, pp. 446-459, Aug. 1998.
49. C.-Y. Hsu, A. Ortega, and A. Reibman, "Joint selection of source and channel rate for VBR video transmission under ATM policing constraints," IEEE J. on Set. Areas in Comm., vol. 15, pp. 1016-1028, Aug. 1997.
50. J.-J. Chen and D.W. Lin, "Optimal bit allocation for coding of video signals over ATM networks" IEEE J. on Sel. Areas in Comm., vol. 15, pp. 1002-1015, Aug. 1997.
51. W. Ding, "Joint encoder and channel rate control of VBR video over ATM networks," IEEE Transactions on Circuits and Systems for Video Technology, vol. 7, pp. 266-278, April 1997.
52. C.-Y. Hsu, A. Ortega, and M. Khansari, "Rate control for robust video transmission over burst-error wireless channels," IEEE J. on Sel. Areas in Comm., 1998. to appear.
53. T. Cover, "Broadcast channels," IEEE Trans. on Inform. Theory, vol. IT-18, pp. 2-14, Jan. 1972.
54. G.M Schuster and A.K. Katsaggelos, "The minimun-average and minimum-maximum criteria in lossy compression," Vistas in Astronomy, vol. 41, no. 3, pp. 427-437, 1997.
55. D.T. Hoang, Fast and Efficient Algorithms for Text and Video Compression. PhD thesis, Brown University, 1997.
56. D.T. Hoang, E.L. Linzer, and J.S. Vitter, "Lexicographic bit allocation for MPEG video," Journal of Visual Communication and Image Representation, vol. 8, Dec. 1997.
57. K.M. Uz, J.M. Shapiro, and M. Czigler, "Optimal bit allocation in the presence of quantizer feedback," in Proc. of ICASSP'93, vol. V, (Minneapolis, MN), pp. 385-388, Apr. 1993.
58. E. Frimout, J. Biemond, and R.L. Lagendijk, "Forward rate control for MPEG recording," in Proc. of SPIE Visual Communications and Image Processing '93, (Cambridge, MA), pp. 184-194, Nov. 1993.
59. J. Karto and M. Ohta, "Mathematical analysis of MPEG compression capability and its application to rate control," in Proc. of ICIP'95, vol. II, (Washington, D.C.), pp. 555-559, 1995
60. H.-M. Hang and J.-J. Chen, "Source model for transform video coder and its application," IEEE Trans. on CAS for Video Tech., vol. 7, pp. 287-311, Apr. 1997.
61. G.M. Schuster and A.K. Katsaggelos, "The min-max approach in video coding," in Proc. of 1997 Intl. Conf. Acoust., Speech and Signal Processing, ICASSP'97, vol. IV, (Munich, Germany), pp. 3105-3108, Apr. 1997.
62. A. Ortega and M. Khansari, "Rate control for video coding over variable bit rate channels with applications to wireless transmission," in Proc. of the 2nd. Intl. Conf on Image Proc., ICIP'95 (Washington, D.C.), Oct. 1995.
63. C.-Y. Hsu, A. Ortega, and M. Khansari, "Rate control for robust video transmission over wireless channels," in Proc. of Visual Comm. and Image Proc., VCIP'97 (San Jose, CA), Feb. 1997.
64. H. Everett, "Generalized Lagrange multiplier method for solving problems of optimum allocation of resources," Operations Research, vol. 11, pp. 399-417, 1963.
65. Y. Shoham and A. Gersho, "Efficient bit allocation for an arbitrary set of quantizers," IEEE Trans. ASSP, vol. 36, pp. 1445-1453, Sep. 1988.
66. S.-Z. Kiang, R.L. Baker, G.J. Sullivan, and C.-Y. Chiu, "Recursive optimal pruning with applications to tree structured vector quantizers," IEEE Trans. on Image Proc., vol. 1, pp. 162-169, Apr. 1992.
67. D.W. Lin, M -H. Wang, and J.-J. Chen, "Optimal delayed-coding of video sequences subject to a buffer-size constraint," in Proc. of SPIE Visual Communications and Image Processing '93, (Cambridge, MA), Nov. 1993.
68. G.J. Sullivan and B.L. Baker, "Efficient quadtree coding of images and video," IEEE Trans. on Image Proc., vol. 3, pp. 327-331, May 1994.
69. G.M. Schuster and A.K Katsaggelos, "An optimal quadtree-based motion estimation and motion based interpolation scheme for video compression," IEEE Trans. on Image Proc., vol. 7, no. 11, pp. 1505-1523, Nov. 1998.
70. M.L. Fisher, "The Lagrangian relaxation method for solving integer programming problems," Management Science, vol. 27, pp. 1-18, Jan. 1981.
71. J.-J. Chen and D.W. Lin, "Optimal coding of video sequences over ATM networks," in Proc. of the 2nd Intl. Conf. on Image Processing, ICIP-95, vol. I, (Washington, D.C.), pp. 21-24, Oct. 1995.
72. A. Ortega, "Optimal rate allocation under multiple rate constraints," in Proc. of the Data Compression Conference, DCC'96 (Snowbird, UT), pp. 349-358, Apr. 1996.
73. J.-J. Chen and D.W. Lin, "Optimal bit allocation for video coding under multiple constraints," in Proc. IEEE Intl. Conf. on Image Proc., ICIP'96, 1996.
74. B. Bellman, Dynamic Programming. Princeton University Press, 1957.
75. D.P. Bertsekas, Dynamic Programming. Prentice-Hall, 1987.
76. G.D Forney, "The Viterbi algorithm," Proc. of the IEEE, vol. 61, pp. 268-278, March 1973.
77. M.W. Marcellin and T.B. Fischer, "Trellis coded quantization of memoryless and Gauss-Markov sources," IEEE Trans. on Comm., vol. 38, pp. 82-93, Jan. 1990.
78. T.B. Fischer and M. Wang, "Entropy-constrained trellis-coded quantization," IEEE Trans. on Info. Th., vol. 38, pp. 415-426, Mar. 1992.
79. R. Laroia and N. Farvardin, "A structured fixed-rate vector quantizer derived from a variable length scalar quantizer part I: memoryless sources," IEEE Trans. on Inform. Theory, vol. IT-39, pp. 85 1-867, May 1993.
80. K. Ramchandran, A. Ortega, and M. Vetterli, "Bit allocation for dependent quantization with applications to multiresolution and MPEG video coders," IEEE Trans. on Image Proc., vol. 3, pp 533-545, Sept. 1994.
81. A. Ortega and K. Ramchandran, "Forward-adaptive quantization with optimal overhead cost for image and video coding with applications to MPEG video coders," in Proc. of SPIE, Digital Video Compression: Algorithms & Technologies 95, (San Jose, CA), Feb. 1995.
82. T. Wiegand, M. Lightstone, D. Mukherjee, T. Campbell, and S.K. Mitra, "Rate-distortion optimized mode selection for very low bit-rate video coding and the emerging H.263 standard," IEEE Trans. on Circ. and Sys. for Video Tech., vol. 6, pp. 182-190, Apr. 1996.
83. G.M. Schuster and A.K. Katsaggelos, "A video compression scheme with optimal bit allocation between displacement vector field and displaced frame difference," IEEE J. on Sel. Areas in Comm., vol. 15, pp. 1739-1751, Dec. 1997.
84. Y. Yoo, A. Ortega, and K. Ramchandran, "A novel hybrid technique for discrete rate-distortion optimization with applications to fast codebook search for scalar vector quantization," in Proc. of ICASSP'96 (Atlanta, GA), Apr. 1996.
85. M. Vetterli and J. Kovacevic, Wavelets and Subband Coding. Prentice Hall, 1995.
86. G. Strang and T. Nguyen, Wavelets and Filter Banks. Wellesley-Cambridge Press, 1996.
87. N.S. Jayant and P. Noll, Digital Coding of Waveforms: Principles and Applications to Speech and Video. Englewood Cliffs, New Jersey: Prentice-Hall, 1984.
88. A.K. Jain, Fundamentals of Digital Image Processing. Englewood Cliffs, NJ: Prentice-Hall, 1989
89. R.L. Joshi, T.R. Fischer, and R.H. Bamberger, "Optimum classification in subband coding of images," in Proc. of ICIP'94 (Austin, TX), pp. 883-887, Nov. 1994.
90. M.C. Chen and A.N. Willson Jr., "Rate-distortion optimal motion estimation algorithms for motion-compensated transform video coding," IEEE Trans. on Circ. and Sys. for Video Tech., vol. 8, pp 147-158, Apr. 1998.
91. D. Tzovaras and M. Strintzis, "Motion and disparity field estimation using rate-distortion optimization," IEEE Trans. on Cric. and Sys. for Video Tech., vol. 8, pp. 171-180, Apr. 1998.
92. D.T. Hoang, P.M. Long, and J.V. Vitter, "Efficient cost measures for motion estimation at low bit rates," IEEE Trans. on Cric. and Sys. for Video Tech., vol. 8, pp. 488-500, Aug. 1998.
93. J. Ribas-Corbera and D.L. Neuhoff, "Optimal bit allocations for lossless video coders: Morion vectors vs difference frames," in Proc. of ICIP'95, vol. 3, pp. 180-183,1995.
94. J. Ribas-Corbera and D.L. Neuhoff, "Optimizing block size in motion-compensated video coding," Journal of Electronic Imaging, vol. 7, pp. 155-165, Jan. 1998.
95. W.C. Chung, F. Kossentini, and M.J.T. Smith, "A new approach to scalable video coding," in Proc. of DCC'95, (Snowbird, UT), pp. 381-390, Apr 1995.
96. G.M. Schuster, G. Melnikov, and A. Katsaggelos, "Operationally optimal vertex-based shape coding," this issue, pp. 91-108.
97. G.M. Schuster and A.K. Katsaggelos, "An optimal boundary encoding scheme in the rate-distortion sense," IEEE Trans. on Image Proc., vol. 7, pp. 13-26, Jan. 1998.
98. G.M. Schuster and A.K. Katsaggelos, "A video compression scheme with optimal bit allocation among segmentation, displacement vector field, and displaced frame difference," IEEE Trans. on Image Proc., vol. 6, pp. 1487-1502, Nov 1997.
99. C. Herley, J. Kovacevic, K. Ramchandran, and M. Vetterli, "Tilings of the time-frequency plane: construction of arbitrary orthogonal bases and fast tiling algorithms," IEEE Trans. on Signal Proc., Special Issue on Wavelets, vol. 41, pp. 3341-59, Dec. 1993.
100. Z. Xiong, K. Ramchandran, C. Herley, and M.T. Orchard, "Flexible time segmentations for time-varying wavelet packets," IEEE Trans. on Signal Proc., vol. 45, pp. 333-345, Feb. 1997.
101. Z. Xiong, M.T. Orchard, and K. Ramchandran, "Wavelet packet image coding using joint space-frequency quantization," IEEE Trans. on Image Proc., pp. 892-898, June 1998.
102. K. Ramchandran, M. Vetterli, and C. Herley, "Wavelets, subband coding and best bases," Proceedings of the IEEE, pp. 541-560, April 1996. Special Issue on Wavelets: Invited Paper.
103. R. Coifman and V. Wickhauser, "Entropy-based algorithms for best basis selection," IEEE Trans. Inform Theory, vol. IT-38, pp. 713-718, Mar. 1992.
104. C.-T. Chen and A. Wong, "A self-governing rate buffer control strategy for pseudoconstant bit rate video coding," IEEE Trans. on Image Proc., vol. 2, pp. 50-59, Jan. 1993.
105. J. Choi and D. Park, "A stable feedback control of the buffer state using the controlled Lagrange multiplier method," IEEE Trans. on Image Proc., vol. 3, pp. 546-558, Sept. 1994.
106. J. Lee and B.W. Dickinson, "Rate distortion optimized frame-type selection for MPEG coding," IEEE Trans. on Circ. and Sys. for Video Tech., vol. 7, pp. 501-510, June 1997.
107. A. Eleftheriadis and D. Anastassiou, "Constrained and general dynamic rate shaping of compression digital video," in Proc. of ICIP'95, vol. III (Washington, D.C.), pp. 396-399, 1995.
108. A. Ortega and M. Khansari, "Rate control for video coding over variable bit rate channels with applications to wireless transmission," in Proc. of ICIP'95, (Washington, D.C.), Nov 1995.
109. C.-Y. Hsu and A. Ortega, "A Lagrangian optimization approach to rate control for delay-constrained video transmission over burst-error channels," in Proc. of ICASSP'98, (Seattle, WA), May 1998.
110. M. Khansari, A. Jalali, E. Dubois, and V. Mermelstein, "Low bit-rate video transmission over fading channels for wireless microcellular systems," IEEE Trans. on Circ. and Sys. for Video Tech., vol. 6, pp. 1-11, Feb. 1995.
111. S. Lin and J.D.J. Costello, Error Control Coding: fundamentals and Applications. Prentice-Hall, 1984.
112. J. Modestino, D.G. Daut, and A. Vickers, "Combined source-channel coding of images using the block cosine transform," IEEE Trans. on Commun., vol. COM-29, pp. 1261-1274, Sept. 1981.
113. N. Farvardin and V. Vaishampayan, "On the performance and complexity of channel-optimized vector quantizers," IEEE Trans. on Inform. Theory, vol. IT-37, Jan. 1991.
114. F.H. Liu, P. Ho, and V. Cuperman, "Joint source and channel coding using a non-linear receiver," in Proc. of ICC'93, June 1993.
115. W.F. Schreiber, "Considerations in the design of HDTV systems for terrestrial broadcasting," SMPTE Journal, pp. 668-677, Sept. 1991.
116. K. Ramchandran, A. Ortega, K.M. Uz, and M. Vetterli, "Multiresolution broadcast for digital HDTV using joint source-channel coding," IEEE J. on Sel. Areas in Comm., vol. 11, pp. 6-23, Jan. 1993.
117. S. McCanne, M. Vetterli, and V. Jacobson, "Low-complexity video coding for receiver-driven layered multicast," IEEE J. on Sel. Areas in Comm., vol. 15, pp. 983-1001, Aug. 1997.
118. J. Hagenauer, "Rate-compatible punctured convolutional codes (rcpc codes) and their applications," IEEE Trans. on Comm., vol. COM 36, pp. 389-400, Apr. 1988.
119. K. Ramchandran and M. Vetterli, "Multiresolution Joint Source-Channel Coding for Wireless Channels," in Wireless Communications: A Signal Processing Perspective, Editors: V. Poor and G. Wornell. Prentice-Hall, 1998.
120. B. Schafer, "Terrestrial transmission of DTVB signals - the European specification," in International Broadcasting Convention, no. 413, September 1995.
121. H. Kumazawa, M. Kasahara, and T. Namekawa, "A construction of vector quantizers for noisy channels," Electron. Eng. Japan, vol. 67-B, pp. 39-47, 1984.
122. K.A. Zeger and A. Gersho, "Zero redundancy channel coding in vector quantization," IEEE Electron. Letters, pp. 654-655, June 1987.
123. I. Kozintsev and K. Ramchandran, "Robust image transmission over energy-constrained time-varying channels using multiresolution joint source-channel coding," IEEE Trans. on Signal Processing, Special Issue on Wavelets and Filter Banks, vol. 46, pp. 1012-1026, April 1998.
124. M.W. Marcellin and T.R. Fischer, "Joint trellis coded quantization and modulation," IEEE Trans. on Comm., Jan. 1993.
125. I. Kozintsev and K. Ramchandran, "A hybrid compressed-uncompressed framework for wireless image transmission," in Proc. of ICIP'97, (Santa Barbara, California), Oct. 1997.