Quantization is a process that maps a continous or discrete set of values into approximations that belong to a smaller set. Quantization is a lossy: some information about the original data is lost in the process. The key to a successful quantization is therefore the selection of an error criterion – such as entropy and signal-to-noise ratio – and the development of optimal quantizers for this criterion.

Quantization

A computationally efficient, high quality, vector quantization scheme based on a parametric probability density function (PDF) is proposed. In this scheme, the observations are modeled as i.i.d realizations of a multivariate Gaussian mixture density. The mixture model parameters are efficiently estimated using the expectation maximization (EM) algorithm. A low complexity quantization scheme using transform coding and bit allocation techniques which allows for easy mapping from observation to quantized value is developed for both fixed rate and variable rate systems. An attractive feature of this method is that source encoding using the resultant codebook involves very few searches and its computational complexity is minimal and independent of the rate of the system. Furthermore, the proposed scheme is bit scalable and can switch seamlessly between a memoryless quantizer and a quantizer with memory. The usefulness of the approach is demonstrated for speech coding where Gaussian mixture models are used to model speech line spectral frequencies. The performance of the memoryless quantizer is 1-3 bits better than conventional quantization schemes.

PDF optimized parametric vector quantization of speech line spectral frequencies

Advances in residual vector quantization (RVQ) are surveyed. Definitions of joint encoder optimality and joint decoder optimality are discussed. Design techniques for RVQs with large numbers of stages and generally different encoder and decoder codebooks are elaborated and extended. Fixed-rate RVQs, and variable-rate RVQs that employ entropy coding are examined. Predictive and finite state RVQs designed and integrated into neural-network based source coding structures are revisited. Successive approximation RVQs that achieve embedded and refinable coding are reviewed. A new type of successive approximation RVQ that varies the instantaneous block rate by using different numbers of stages on different blocks is introduced and applied to image waveforms, and a scalar version of the new residual quantizer is applied to image subbands in an embedded wavelet transform coding system.

Advances in residual vector quantization: a review

A fixed-rate shape-gain quantizer for the memoryless Gaussian source is proposed. The shape quantizer is constructed from wrapped spherical codes that map a sphere packing in /spl Ropf//sup k-1/ onto a sphere in /spl Ropf//sup k/, and the gain codebook is a globally optimal scalar quantizer. A wrapped Leech lattice shape quantizer is used to demonstrate a signal-to-quantization-noise ratio within 1 dB of the distortion-rate function for rates above 1 bit per sample, and an improvement over existing techniques of similar complexity. An asymptotic analysis of the tradeoff between gain quantization and shape quantization is also given.

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Gaussian source coding with spherical codes

A training algorithm for the design of lattices for vector quantization is presented. The algorithm uses a steepest descent method to adjust a generator matrix, in the search for a lattice whose Voronoi regions have minimal normalized second moment. The numerical elements of the found generator matrices are interpreted and translated into exact values. Experiments show that the algorithm is stable, in the sense that several independent runs reach equivalent lattices. The obtained lattices reach as low second moments as the best previously reported lattices, or even lower. Specifically, we report lattices in nine and ten dimensions with normalized second moments of 0.0716 and 0.0708, respectively, and nonlattice tessellations in seven and nine dimensions with 0.0727 and 0.0711, which improves on previously known values. The new nine- and ten-dimensional lattices suggest that Conway and Sloane's (1993) conjecture on the duality between the optimal lattices for packing and quantization might be false. A discussion of the application of lattices in vector quantizer design for various sources, uniform and nonuniform, is included.

Optimization of lattices for quantization

A two-stage vector quantizer is introduced that uses an unstructured first-stage codebook and a second-stage lattice codebook. Joint optimum two-stage encoding is accomplished by exhaustive search of the parent codebook of the two-stage product code. Due to the relative ease of lattice vector quantization, optimum encoding is feasible for moderate-to-large encoding rates and vector dimensions, provided the first-stage codebook size is kept reasonable. For memoryless Gaussian and Laplacian sources, encoding rates of 2 to 3 b/sample, and vector dimensions of 8 to 35 the signal-to-noise ratio performance is comparable or superior to equivalent-delay encoding results previously reported. For Gaussian sources with memory, the effectiveness of the encoding method is dependent on the feasibility of using a large enough first-stage vector quantizer codebook to exploit most of the source memory. >

Two-stage vector quantization-lattice vector quantization

Methods are presented for intraframe encoding of line spectrum pairs. A two-stage vector quantizer-lattice vector quantizer (VQ-LVQ) has a 2-3 b/vector performance advantage over the split vector quantizer, and can save 1-3 b/vector, compared to multistage vector quantization, depending on the number of stages and the number of survivors retained at each stage. The vector quantizer-lattice vector quantizer has a moderate computational complexity. Among three trellis codes considered, trellis-coded vector quantization with nonlinear prediction provides line spectrum pair encoding performance similar to the split vector quantizer. Hybrid transformations are developed for encoding the reflection coefficients using vector quantization, and offer a small encoding advantage over the commonly used arcsine transformation and log area ratios.

Jianping Pan

Papers

Two-stage vector quantization-lattice vector quantization

Vector quantization of speech line spectrum pair parameters and reflection coefficients