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

Quantization, the process of approximating continuous-amplitude signals by digital (discrete-amplitude) signals, is an important aspect of data compression or coding, the field concerned with the reduction of the number of bits necessary to transmit or store analog data, subject to a distortion or fidelity criterion. The independent quantization of each signal value or parameter is termed scalar quantization, while the joint quantization of a block of parameters is termed block or vector quantization. This tutorial review presents the basic concepts employed in vector quantization and gives a realistic assessment of its benefits and costs when compared to scalar quantization. Vector quantization is presented as a process of redundancy removal that makes effective use of four interrelated properties of vector parameters: linear dependency (correlation), nonlinear dependency, shape of the probability density function (pdf), and vector dimensionality itself. In contrast, scalar quantization can utilize effectively only linear dependency and pdf shape. The basic concepts are illustrated by means of simple examples and the theoretical limits of vector quantizer performance are reviewed, based on results from rate-distortion theory. Practical issues relating to quantizer design, implementation, and performance in actual applications are explored. While many of the methods presented are quite general and can be used for the coding of arbitrary signals, this paper focuses primarily on the coding of speech signals and parameters.

Vector quantization in speech coding

This paper proposes a complete practical methodology for minimizing additive distortion in steganography with general (nonbinary) embedding operation. Let every possible value of every stego element be assigned a scalar expressing the distortion of an embedding change done by replacing the cover element by this value. The total distortion is assumed to be a sum of per-element distortions. Both the payload-limited sender (minimizing the total distortion while embedding a fixed payload) and the distortion-limited sender (maximizing the payload while introducing a fixed total distortion) are considered. Without any loss of performance, the nonbinary case is decomposed into several binary cases by replacing individual bits in cover elements. The binary case is approached using a novel syndrome-coding scheme based on dual convolutional codes equipped with the Viterbi algorithm. This fast and very versatile solution achieves state-of-the-art results in steganographic applications while having linear time and space complexity w.r.t. the number of cover elements. We report extensive experimental results for a large set of relative payloads and for different distortion profiles, including the wet paper channel. Practical merit of this approach is validated by constructing and testing adaptive embedding schemes for digital images in raster and transform domains. Most current coding schemes used in steganography (matrix embedding, wet paper codes, etc.) and many new ones can be implemented using this framework.

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Minimizing Additive Distortion in Steganography Using Syndrome-Trellis Codes

Trellis-coded quantization (TCQ) is developed and applied to the encoding of memoryless and Gauss-Markov sources. The theoretical justification for the approach is alphabet-constrained rate distortion theory, which is a dual to the channel capacity argument that motivates trellis-coded modulation (TCM). The authors adopt the notions of signal set expansion, set partitioning, and branch labeling of TCM, but modify the techniques to account for the source distribution, to design TCQ coders of low complexity with excellent mean-squared-error (MSE) performance. For a memoryless uniform source, TCQ provides an MSE within 0.21 dB of the distortion-rate bound at all positive (integral) rates. The performance is superior to that promised by the coefficient of quantization for all of the best lattices known in dimensions 24 or less. For a memoryless Gaussian source, the TCQ performance at rates of 0.5, 1, and 2 b/sample is superior to all previous results the authors found in the literature. The encoding complexity of TCQ is very modest. TCQ is incorporated into a predictive coding structure for the encoding of Gauss-Markov sources. Simulation results for first-, second-, and third-order Gauss-Markov sources are presented. >

Trellis coded quantization of memoryless and Gauss-Markov sources

The past decade has witnessed substantial progress towards the application of low-rate speech coders to civilian and military communications as well as computer-related voice applications. Central to this progress has been the development of new speech coders capable of producing high-quality speech at low data rates. Most of these coders incorporate mechanisms to: represent the spectral properties of speech, provide for speech waveform matching, and "optimize" the coder's performance for the human ear. A number of these coders have already been adopted in national and international cellular telephony standards. The objective of this paper is to provide a tutorial overview of speech coding methodologies with emphasis on those algorithms that are part of the recent low-rate standards for cellular communications. Although the emphasis is on the new low-rate coders, we attempt to provide a comprehensive survey by covering some of the traditional methodologies as well. We feel that this approach will not only point out key references but will also provide valuable background to the beginner. The paper starts with a historical perspective and continues with a brief discussion on the speech properties and performance measures. We then proceed with descriptions of waveform coders, sinusoidal transform coders, linear predictive vocoders, and analysis-by-synthesis linear predictive coders. Finally, we present concluding remarks followed by a discussion of opportunities for future research. >

Speech coding: a tutorial review

For memoryless discrete-time sources and bounded single-letter distortion measures, we derive a bound on the average per-letter distortion achievable by a trellis source code of fixed constraint length. For any fixed code rate greater than R(D^{\ast}) , the rate-distortion function at D^{\ast} , this bound decreases toward D^{\ast} exponentially with constraint length.

J. Omura

Papers

Trellis Encoding of memoryless discrete-time sources with a fidelity criterion