Multiple description lattice vector quantization

We consider the problem of designing a lattice-based multiple description vector quantizer for a two-channel diversity system. The design of such a quantizer can be reduced to the problem of assigning pair labels to points of a vector quantizer codebook. A general labeling procedure based on the structure of the lattice is presented, along with detailed results for the hexagonal lattice: algorithms, asymptotic performance, and numerical simulations. Asymptotically, when compared with the lattice Z, the resulting quantizer achieves the standard second-moment gain of the hexagonal lattice for the central distortion, and, surprisingly, achieves the two-dimensional sphere gain for the side distortion.

[1]  N. Sloane,et al.  On the Existence of Similar Sublattices , 1999, Canadian Journal of Mathematics.

[2]  Tamás Linder,et al.  High-Resolution Source Coding for Non-Difference Distortion Measures: Multidimensional Companding , 1999, IEEE Trans. Inf. Theory.

[3]  A. S. Solodovnikov,et al.  Hypercomplex Numbers: An Elementary Introduction to Algebras , 1989 .

[4]  Tamás Linder,et al.  High-Resolution Source Coding for Non-Difference Distortion Measures: The Rate-Distortion Function , 1997, IEEE Trans. Inf. Theory.

[5]  J. P. Ward.  Quaternions and Cayley Numbers: Algebra and Applications , 1997 .

[6]  Vivek K. Goyal,et al.  Multiple description lattice vector quantization: variations and extensions , 2000, Proceedings DCC 2000. Data Compression Conference.

[7]  Jaroslaw Domaszewicz,et al.  Design of entropy-constrained multiple-description scalar quantizers , 1994, IEEE Trans. Inf. Theory.

[8]  W. R. Bennett,et al.  Spectra of quantized signals , 1948, Bell Syst. Tech. J..

[9]  Radu V. Balan,et al.  The analysis and design of windowed Fourier frame based multiple description source coding schemes , 2000, IEEE Trans. Inf. Theory.

[10]  Rudolf Ahlswede,et al.  The rate-distortion region for multiple descriptions without excess rate , 1985, IEEE Trans. Inf. Theory.

[11]  Robert M. Gray,et al.  An Algorithm for Vector Quantizer Design , 1980, IEEE Trans. Commun..

[12]  Harish Viswanathan,et al.  On the whiteness of high-resolution quantization errors , 2000, IEEE Trans. Inf. Theory.

[13]  Tamás Linder,et al.  The multiple description rate region for high resolution source coding , 1998, Proceedings DCC '98 Data Compression Conference (Cat. No.98TB100225).

[14]  Michael T. Orchard,et al.  Multiple description image coding for noisy channels by pairing transform coefficients , 1997, Proceedings of First Signal Processing Society Workshop on Multimedia Signal Processing.

[15]  Meir Feder,et al.  On universal quantization by randomized uniform/lattice quantizers , 1992, IEEE Trans. Inf. Theory.

[16]  Vinay A. Vaishampayan,et al.  Design of multiple description scalar quantizers , 1993, IEEE Trans. Inf. Theory.

[17]  Meir Feder,et al.  On lattice quantization noise , 1996, IEEE Trans. Inf. Theory.

[18]  David J. Sakrison,et al.  The rate distortion function of a Gaussian process with a weighted square error criterion (Corresp.) , 1968, IEEE Trans. Inf. Theory.

[19]  H. Kuhn.  The Hungarian method for the assignment problem , 1955 .

[20]  N. J. A. Sloane,et al.  Design of asymmetric multiple description lattice vector quantizers , 2000, Proceedings DCC 2000. Data Compression Conference.

[21]  L. Ozarow,et al.  On a source-coding problem with two channels and three receivers , 1980, The Bell System Technical Journal.

[22]  Tamás Linder,et al.  On the asymptotic tightness of the Shannon lower bound , 1994, IEEE Trans. Inf. Theory.

[23]  Ram Zamir.  Gaussian codes and Shannon bounds for multiple descriptions , 1999, IEEE Trans. Inf. Theory.

[24]  N. Jayant.  Subsampling of a DPCM speech channel to provide two “self-contained” half-rate channels , 1981, The Bell System Technical Journal.

[25]  Henrique S. Malvar,et al.  Signal processing with lapped transforms , 1992 .

[26]  Jacob Ziv,et al.  On universal quantization , 1985, IEEE Trans. Inf. Theory.

[27]  H. S. Witsenhausen,et al.  B.S.T.J. brief: On source networks with minimal breakdown degradation , 1980, The Bell System Technical Journal.

[28]  N. J. A. Sloane,et al.  Fast quantizing and decoding and algorithms for lattice quantizers and codes , 1982, IEEE Trans. Inf. Theory.

[29]  Tamás Linder,et al.  Asymptotic entropy-constrained performance of tessellating and universal randomized lattice quantization , 1994, IEEE Trans. Inf. Theory.

[30]  Vivek K Goyal,et al.  Multiple description transform coding: robustness to erasures using tight frame expansions , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[31]  Michael T. Orchard,et al.  Multiple description coding using trellis coded quantization , 2000, Proceedings 2000 International Conference on Image Processing (Cat. No.00CH37101).

[32]  Mike E Keating.  A First Course in Module Theory , 1998 .

[33]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[34]  Albert Wang,et al.  Multiple description decoding of overcomplete expansions using projections onto convex sets , 1999, Proceedings DCC'99 Data Compression Conference (Cat. No. PR00096).

[35]  Vinay A. Vaishampayan,et al.  Asymptotic performance of multiple description transform codes , 1997, IEEE Trans. Inf. Theory.

[36]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[37]  J. Humphreys.  Introduction to Lie Algebras and Representation Theory , 1973 .

[38]  R. Sundaram.  A First Course in Optimization Theory , 1996 .

[39]  Takehiro Moriya.  Two-Channel Conjugate Vector Quantizer for Noisy Channel Speech Coding , 1992, IEEE J. Sel. Areas Commun..

[40]  Toby Berger,et al.  New results in binary multiple descriptions , 1987, IEEE Trans. Inf. Theory.

[41]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[42]  Toby Berger,et al.  Rate distortion theory : a mathematical basis for data compression , 1971 .

[43]  Michael Fleming,et al.  Generalized multiple description vector quantization , 1999, Proceedings DCC'99 Data Compression Conference (Cat. No. PR00096).

[44]  Toby Berger,et al.  Lossy Source Coding , 1998, IEEE Trans. Inf. Theory.

[45]  Vivek K. Goyal,et al.  Multiple description perceptual audio coding with correlating transforms , 2000, IEEE Trans. Speech Audio Process..

[46]  Tanya Y. Berger-Wolf,et al.  Optimal Index Assignment for Multichannel Communication (soda 1999, to Appear) , 1998 .

[47]  Abbas El Gamal,et al.  Achievable rates for multiple descriptions , 1982, IEEE Trans. Inf. Theory.

[48]  Vivek K. Goyal,et al.  Quantized Overcomplete Expansions in IRN: Analysis, Synthesis, and Algorithms , 1998, IEEE Trans. Inf. Theory.

[49]  A. Odlyzko,et al.  Lattice points in high-dimensional spheres , 1990 .

[50]  Aaron D. Wyner,et al.  Coding Theorems for a Discrete Source With a Fidelity CriterionInstitute of Radio Engineers, International Convention Record, vol. 7, 1959. , 1993 .

[51]  R. Gray.  Source Coding Theory , 1989 .

[52]  Henry Stark,et al.  Probability, Random Processes, and Estimation Theory for Engineers , 1995 .

[53]  D. West.  Introduction to Graph Theory , 1995 .

[54]  Hamid Jafarkhani,et al.  Multiple description trellis-coded quantization , 1999, IEEE Trans. Commun..

[55]  Aaron D. Wyner,et al.  The rate-distortion function for source coding with side information at the decoder , 1976, IEEE Trans. Inf. Theory.

[56]  Allen Gersho,et al.  Vector quantization and signal compression , 1991, The Kluwer international series in engineering and computer science.

[57]  A. Wyner,et al.  Source coding for multiple descriptions , 1980, The Bell System Technical Journal.

[58]  Vinay A. Vaishampayan,et al.  Asymptotic Analysis of Multiple Description Quantizers , 1998, IEEE Trans. Inf. Theory.

[59]  Toby Berger,et al.  Multiple description source coding with no excess marginal rate , 1995, IEEE Trans. Inf. Theory.

[60]  Jack K. Wolf,et al.  Noiseless coding of correlated information sources , 1973, IEEE Trans. Inf. Theory.

[61]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[62]  John C. Kieffer,et al.  A survey of the theory of source coding with a fidelity criterion , 1993, IEEE Trans. Inf. Theory.

[63]  Allen Gersho,et al.  Asymptotically optimal block quantization , 1979, IEEE Trans. Inf. Theory.

[64]  A. Calderbank,et al.  On reducing granular distortion in multiple description quantization , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[65]  R. Gallager.  Information Theory and Reliable Communication , 1968 .

[66]  N. J. A. Sloane,et al.  On sublattices of the hexagonal lattice , 1997, Discret. Math..

[67]  A. Odlyzko,et al.  Lattice points and the volume/area ratio of convex bodies , 1973 .

[68]  Robert M. Gray,et al.  Asymptotic Performance of Vector Quantizers with a Perceptual Distortion Measure , 1997, IEEE Trans. Inf. Theory.

[69]  William E. Baylis,et al.  The Pauli algebra approach to special relativity , 1989 .

[70]  Wolfgang Ebeling,et al.  Lattices and Codes: A Course Partially Based on Lectures by Friedrich Hirzebruch , 1994 .

[71]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

[72]  Bhaskar D. Rao,et al.  Theoretical analysis of the high-rate vector quantization of LPC parameters , 1995, IEEE Trans. Speech Audio Process..

[73]  Vivek K. Goyal,et al.  Quantized frame expansions as source-channel codes for erasure channels , 1999, Proceedings DCC'99 Data Compression Conference (Cat. No. PR00096).

[74]  Philip A. Chou,et al.  Entropy-constrained vector quantization , 1989, IEEE Trans. Acoust. Speech Signal Process..

[75]  F. Fricker.  Einführung in die Gitterpunktlehre , 1982 .

[76]  S. Weintraub,et al.  Algebra: An Approach via Module Theory , 1992 .