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 article focuses on the compressed representations of the pictures

This paper places frames in a new setting, where some of the elements are deleted. Since proper subsets of fi'ames are sometimes them- selves frames, a quantized frame expansion can be a useful representation even when some transform coefficients are lost in transmission. This yields robustness to losses in packet networks such as the Internet

Generalized multiple description coding (GMDC) is source coding for multiple channels such that a decoder which receives an arbitrary subset of the channels may produce a useful reconstruction. This paper reports on applications of two recently proposed methods for GMDC to image coding. The first produces statistically correlated streams such that lost streams can be estimated from the received data. The second uses quantized frame expansions and hence is conceptually similar to block channel coding, except it is done prior to quantization. 1

We consider the problem of coding images for transmission over error-prone channels. The impairments we target are transient channel shutdowns, as would occur in a packet network when a packet is lost, or in a wireless system during a deep fade: when data is delivered it is assumed to be error-free, but some of the data may never reach the receiver. The proposed algorithms are based on a combination of multiple description scalar quantizers with techniques successfully applied to the construction of some of the most ecient subband coders. A given image is encoded into multiple independent packets of roughly equal length. When packets are lost, the quality of the approximation computed at the receiver depends only on the number of packets received, but does not depend on exactly which packets are actually received. When compared with previously reported results on the performance of robust image coders based on multiple descriptions, on standard test images, our coders attain s...

The objective of multiple description coding (MDC) is to encode a source into two (or more) bitstreams supporting two quality levels of decoding. A high-quality reconstruction should be decodable from the two bitstreams together, while lower, but still acceptable, quality reconstructions should be decodable from either of the two individual bitstreams. This paper describes techniques for meeting MDC objectives in the framework of standard transform-based image coding through the design of pairwise transforms.

In this paper, we present an ecient Multiple Description Coding (MDC) technique to achieve robust communication over unreliable channels such as a lossy packet network. We first model such unreliable channels as erasure channels and then we present a MDC system using polyphase transform and selective quantization to recover channel erasures. Different from previous MDC work, our system explicitly separates description generation and redundancy addition which greatly reduces the implementation complexity specially for systems with more than two descriptions. Our system also realizes a Balanced Multiple Description Coding (BMDC) framework which can generate descriptions of statistically equal rate and importance. This property is well matched to communication systems with no priority mechanisms for data delivery, such as today's Internet.

Abstract—Multiple description (MD) coding is source coding in which several descriptions of the source are produced such that various reconstruction qualities are obtained from different subsets of the descriptions. Unlike multiresolution or layered source coding, there is no hierarchy of descriptions; thus, MD coding is suitable for packet erasure channels or networks without priority provisions. Generalizing work by Orchard, Wang, Vaishampayan, and Reibman, a transform-based approach is developed for producing descriptions of an-tuple source,. The descriptions are sets of transform coefficients, and the transform coefficients of different descriptions are correlated so that missing coefficients can be estimated. Several transform optimization results are presented for memoryless Gaussian sources, including a complete solution of the aP, aPcase with arbitrary weighting of the descriptions. The technique is effective only when independent components of the source have differing variances. Numerical studies show that this method performs well at low redundancies, as compared to uniform MD scalar quantization. Index Terms—Erasure channels, integer-to-integer transforms, packet networks, robust source coding.

We propose the use of a tight frame expansion for generalized multiple description coding. This method is conceptually very similar to using a block channel code, except it is done on continuousvalued data prior to quantization. A sketch of the distortion analysis is given. I. Introduction Recently the problem of transmitting data over heterogenous networks has received considerable attention. A typical scenario might require data to move from a fiber link to a wireless link, which necessitates dropping packets to accommodate the lower capacity of the latter. If the network is able to provide preferential treatment to some packets, then the use of a multiresolution or layered source coding system is the obvious solution. But what if the network will not look inside packets and discriminate? Then packets will be dropped at random, and it is not clear how the source (or source-channel) coding should be designed. If packet retransmission is not an option (e.g., due to a delay constraint...

Quantized frame expansions are proposed as a method for generalized multiple description coding, where each quantized coe cient is a description. Whereas previous investigations have revealed the robustness of frame expansions to additive noise and quantization, this represents a new application of frame expansions. The performance of a system based on quantized frame expansions is compared to that of a system with a conventional block channel code. The new system performs well when the number of lost descriptions (erasures on an erasure channel) is hard to predict. 1