Entropy-constrained code-excited linear predictive (EC-CELP) quantization

Majid Foodeei 1;2 and Eric Dubois 2;1 Abstract | In this contribution an entropy-constrained code-excited linear predictive quantizer (EC-CELP) is introduced. EC-CELP addresses the problems associated with high quality, low delay, low complexity, low bit rate source coding of sources with memory. The motivation comes from the applications in motion-compensated time varying image compression. As well, there is an information theoretic interest in instrumentable coding schemes with performance near the rate-distortion bound having minimal delay and computational cost. EC-CELP implicitly combines the advantages of VQ, predictive coding (PC), and analysis-by-synthesis with the merits of the entropy-constrained codebook design. Simulation results show the clear advantages of EC-CELP over alternative schemes. EC-CELP quantization { Recently, it has been shown that a CELP coder followed by an entropy coder operating on a highly correlated Gauss-Markov source, has an excellent ratedistortion performance in the low bit-rate region [1]. To establish and extend those results to an entropy constrained context, and to obtain locally optimum variable-length predictive block coding with respect to a delity criterion, an extension of the entropy constrained VQ (ECVQ) [2] algorithm based on a Lagrangian formulation is introduced. For correlated sources, the CELP coder linear memory removal is e ective and a better utilization of the VQ space lling feature is possible. The CELP analysis-by-synthesis feature ensures that the coder con guration does not sacri ce the inter-block memory removal as is the case for predictive VQ (PVQ). The entropy constrained strategy allows for a design that is more suitable to the combination of CELP and variable-length coding. In the EC-CELP encoder, the input vector S is encoded by the CELP encoder ( ) which generates the output index I 2 I. This index is the input to the entropy encoder ( ) which results in the output C 2 C (C = fCIgI2I). The inverse of the entropy coder at the decoder ( 1) generates the output I. The CELP decoder ( ) reconstructs the signal Ŝ from this index. The CELP encoder has an analysis-by-synthesis structure and uses an exhaustive search through the excitation signal codebook. The Zero-State-Response (ZSR) and Zero-Input-Response (ZIR) lters are separated to reduce the complexity and better formulation of EC design algorithm. The EC-CELP algorithm, as in the case of ECVQ, is an iterative descent algorithm which nds the convex hull of the N -th order operational rate-distortion function (RDF). The lower bound to the operational RDF is the N -th order RDF, which as N goes to in nity becomes RDF (for squared error). Starting from the initial coder, the iterative descent algorithm repeatedly updates the mappings ( (t); (t); (t)) until some stopping criterion for the convergence of the functional is met. The three basic steps are to update one mapping while keeping the other two xed. In updating the mapping (t+1), the centroid rule amounts to an EC \closed loop" CELP design. 1 This research was supported in part by a grant from the Canadian Institute for Telecommunications Research (CITR) under the NCE program of the Government of Canada. 1Electrical Engineering, McGill University, 3480 University Street, Montr eal, PQ, CANADA H3A2A7 2 INRS-T el ecommunications, Universit e du Qu ebec, 16 Place du Commerce, Verdun, PQ, CANADA H3E1H6 E-mail: foodeei, [email protected] Results and Conclusion { The performance of EC-CELP quantizer for the highly correlated Gauss-Markov source was compared with alternative coding con gurations of entropy constrained DPCM coder (EC-DPCM) [3], ECVQ [2], and entropy constrained block transform quantization (EC-BTQ) [4]. The performance of EC-CELP was the closest to the RDF with low delay (small block length) and computational cost (small codebook size). The results shown in Fig. 1, are based on a training sequence of 100,000 samples from Gauss-Markov models with correlation coe cients a = 0:9 and 0:98 (using predictor coe cient equal to a). Initial codebook sizes of M = 256; 1024 for block lengths N = 3; 8 (respectively) are used. ECVQ results are simulated and EC-BTQ results are taken from the available results in [4]. The EC-CELP performance gap over ECVQ is higher for a = 0:98. Since the codebook size for rates below one bps becomes relatively small (much smaller than the initial codebook size M), the CELP coder complexity is also low. More importantly, for highly correlated signals, the EC-CELP is the only coder which can achieve close to RDF with practical block length N = 3 and hence low delay. Although for a = 0:98 results for EC-BTQ were not available, it is safe to predict that like ECVQ for block size N = 8, only a fraction of possible memory gain can be obtained using EC-BTQ. The journal paper of this work will include the detail algorithm and additional results. References [1] M. Foodeei and E. Dubois, \Rate-distortion performance of source coders in the low bit-rate region for highly correlated Gauss-Markov source," in GlobCOM Conf., Comm. Theory Mini-Conf., pp. 123{127, Dec. 1993. [2] P. A. Chou, T. Lookabaugh, and R. M. Gray, \Entropyconstrained vector quantization," IEEE Trans. Acoust. Speech Signal Process., pp. 31{42, Jan. 1989. [3] N. Farvardin and J. W. Modestino, \Rate-distortion performance of DPCM schemes for autoregressive sources," IEEE Trans. Inf. Theory, vol. 31, pp. 402{418, May 1985. [4] N. Farvardin and F. Y. Lin, \Performance of entropyconstrained block transform quantizers," IEEE Trans. Inf. Theory, vol. 37, pp. 1433{1439, Sept. 1991.