The history of the theory and practice of quantization dates to 1948, although similar ideas had appeared in the literature as long ago as 1898. The fundamental role of quantization in modulation and analog-to-digital conversion was first recognized during the early development of pulsecode modulation systems, especially in the 1948 paper of Oliver, Pierce, and Shannon. Also in 1948, Bennett published the first high-resolution analysis of quantization and an exact analysis of quantization noise for Gaussian processes, and Shannon published the beginnings of rate distortion theory, which would provide a theory for quantization as analog-to-digital conversion and as data compression. Beginning with these three papers of fifty years ago, we trace the history of quantization from its origins through this decade, and we survey the fundamentals of the theory and many of the popular and promising techniques for quantization.

this paper. Let us place it within the neural network perspective, and particularly that of learning. The area of neural networks has greatly benefited from its unique position at the crossroads of several diverse scientific and engineering disciplines including statistics and probability theory, physics, biology, control and signal processing, information theory, complexity theory, and psychology (see [45]). Neural networks have provided a fertile soil for the infusion (and occasionally confusion) of ideas, as well as a meeting ground for comparing viewpoints, sharing tools, and renovating approaches. It is within the ill-defined boundaries of the field of neural networks that researchers in traditionally distant fields have come to the realization that they have been attacking fundamentally similar optimization problems.

this article. Some further techniques which go somewhat beyond this model will also be discussed. [Begin Sidebar Inset Article #1] A History of Existing Visual Coding Standards

Long-term memory prediction extends the spatial displacement vector utilized in hybrid video coding by a variable time delay permitting the use of more than one reference frame for motion compensation. This extension provides improved rate-distortion performance. However, motion compensation in combination with transmission errors leads to temporal error propagation that occurs when the reference frames at encoder and decoder dier. In this paper, we present a framework that incorporates an error estimate into rate-constrained motion estimation and mode decision. Experimental results with a Rayleigh fading channel show that long-term memory motion compensation signicantly outperforms single-frame prediction. 1. INTRODUCTION The eciency of long-term memory motion-compensated prediction (MCP) as an approach to improve coding performance has been demonstrated in [1]. The ITU-T has decided to adopt this feature as Annex U to version 3 of the H.263 standard. In this paper, we show that t...

Vector quantization is a data compression method where a set of data points is encoded by a reduced set of reference vectors, the codebook. We discuss a vector quantization strategy which jointly optimizes distortion errors and the codebook complexity, thereby, determining the size of the codebook. A maximum entropy estimation of the cost function yields an optimal number of reference vectors, their positions and their assignment probabilities. The dependence of the codebook density on the data density for different complexity functions is investigated in the limit of asymptotic quantization levels. How different complexity measures influence the efficiency of vector quantizers is studied for the task of image compression, i.e., we quantize the wavelet coefficients of gray level images and measure the reconstruction error. Our approach establishes a unifying framework for different quantization methods like K-means clustering and its fuzzy version, entropy constrained vector quantizati...

Subband and wavelet decompositions are powerful tools in image coding, because of their decorrelating effects on image pixels, the concentration of energy in a few coefficients, their multirate/multiresolution framework, and their frequency splitting which allows for efficient coding matched to the statistics of each frequency band and to the characteristics of the human visual system. Vector quantization provides a means of converting the decomposed signal into bits in a manner that takes advantage of remaining inter- and intra-band correlation as well as of the more flexible partitions of higher dimensional vector spaces. Since 1988 a growing body of research has examined the use of vector quantization for subband/wavelet transform coefficients. We present a survey of these methods. 1 Introduction Image compression maps an original image into a bit stream suitable for communication over or storage in a digital medium. The number of bits required to represent the coded image should b...

Abstract-A two-stage code is a block code in which each block of data is coded in two stages: the first stage codes the identity of a block code among a collection of codes, and the second stage codes the data using the identified code. The collection of codes may he noiseless codes, fixed-rate quantizers, or variable-rate quantizers. We take a vector quantization approach to two-stage coding, in which the first stage code can be regarded as a vector quantizer that “quantizes ” the input data of length n to one of a fixed collection of block codes. We apply the generalized Lloyd algorithm to the first-stage quantizer, using induced measures of rate and distortion, to design locally opti-mal two-stage, codes. On a source of medical images, two-stage variahle-rate vector quantizers designed in this way outperform standard (one-stage) fixed-rate vector quantizers by over 9 dB. The tail of the operational distortion-rate function of the first-stage quantizer determines the optimal rate of convergence of the redundancy of a universal sequence of two-stage codes. We show that there exist two-stage universal noiseless codes, fixed-rate quantizers, and variable-rate quantizers whose per-letter rate and distortion redundancies converge to zero as (k/2)n- ’ logn, when the universe of sources has finite dimension k. This extends the achievability part of Rissanen’s theorem from universal noiseless codes to universal quantizers. Further, we show that the redundancies converge as O(n-’) when the universe of sources is countable, and as O(r~-l+‘) when the universe of sources is infinite-dimensional, under appropriate conditions. Index Terms-Two-stage, adaptive, compression, minimum de-scription length, clustering. I.

ABSTRACT This paper presents an overview of wavelet-based image coding. We develop the basics of image coding with a discussion of vector quantization. We motivate the use of transform coding in practical settings,and describe the properties of various decorrelating transforms. We motivate the use of the wavelet transform in coding using rate-distortion considerations as well as approximation-theoretic considerations. Finally,we give an overview of current coders in the literature. 1

Classification and compression play important roles in communicating digital information. Their combination is useful in many applications, including the detection of abnormalities in compressed medical images. In view of the similarities of compression and low-level classification, it is not surprising that there are many similar methods for their design. Because some of these methods are useful for designing vector quantizers, it seems natural that vector quantization (VQ) is explored for the combined goal. We investigate several VQ-based algorithms that seek to minimize both the distortion of compressed images and errors in classifying their pixel blocks. These algorithms are investigated with both full search and tree-structured codes. We emphasize a nonparametric technique that minimizes both error measures simultaneously by incorporating a Bayes risk component into the distortion measure used for design and encoding. We introduce a tree-structured posterior estimator to produce t...

Over the last one and a half decades, digital video compression technologies have become an integral part of the way we create, communicate, and consume visual information. In this paper, techniques for video compression are reviewed, starting from basic concepts. The rate-distortion performance of modern video compression schemes is the result of an interaction between motion representation techniques, intra-picture prediction techniques, waveform coding of differences, and waveform coding of various refreshed regions. The paper starts with an explanation of the basic concepts of video codec design and then explains how these various features have been integrated into international standards, up to and including the most recent such standard, known as H.264/AVC.