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Quantization
 IEEE TRANS. INFORM. THEORY
, 1998
"... 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 analogtodigital conversion was first recognized during the early development of pulsecode modula ..."
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Cited by 700 (12 self)
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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 analogtodigital 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 highresolution 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 analogtodigital 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.
Voronoi diagrams  a survey of a fundamental geometric data structure
 ACM COMPUTING SURVEYS
, 1991
"... This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. ..."
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Cited by 621 (5 self)
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This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
On Universal Quantization by Randomized Uniform/Lattice Quantizers
 IEEE Trans. Inform. Theory
, 1992
"... Uniform quantization with dither, or lattice quantization with dither in the vector case, followed by a universal lossless source encoder (entropy coder), is a simple procedure for universal coding with distortion of a source that may take continuously many values. The rate of this universal codi ..."
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Cited by 50 (15 self)
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Uniform quantization with dither, or lattice quantization with dither in the vector case, followed by a universal lossless source encoder (entropy coder), is a simple procedure for universal coding with distortion of a source that may take continuously many values. The rate of this universal coding scheme is examined, and we derive a general expression for it. An upper bound for the redundancy of this scheme, defined as the difference between its rate and the minimal possible rate, given by the rate distortion function of the source, is derived. This bound holds for all distortion levels. Furthermore, we present a composite upper bound on the redundancy as a function of the quantizer resolution which leads to a tighter bound in the high rate (low distortion) case. Key Words: Uniform and Lattice Quantization, Randomized Quantization, Universal Coding, RateDistortion Performance Meir Feder was also supported by The Andrew W. Mellon Foundation, Woods Hole Oceanographic Institu...
New Trellis Codes Based on Lattices and Cosets
, 1987
"... A new technique is proposed for constructing trellis codes, which provides an alternative to Ungerboeck’s method of “set partitioning.” The new codes use a signal constellation consisting of points,from an ndimensional lattice A, with an equal number of hints from each coset of a sublattice A’. On ..."
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Cited by 38 (7 self)
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A new technique is proposed for constructing trellis codes, which provides an alternative to Ungerboeck’s method of “set partitioning.” The new codes use a signal constellation consisting of points,from an ndimensional lattice A, with an equal number of hints from each coset of a sublattice A’. One part of the input stream drives a generalized convolutional code whose outputs are co&s of A’, while the other part selects points from these cosets. Several of the new codes are better than those previously known.
Soft decoding techniques for codes and lattices, including the Golay code and the Leech lattice
 IEEE Trans. Inform. Theory
, 1986
"... AbstrtiTwo kinds of a&orithms are considered. 1) ff 59 is a binary code of length n, a “soft decision ” decodhg afgorithm for Q changes ao arbitrary point of R ” into a nearest codeword (nearest in Euclideao distance). 2) Similarly, a deco&g afgorithm for a lattice A in R ” changes an arbit ..."
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Cited by 33 (3 self)
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AbstrtiTwo kinds of a&orithms are considered. 1) ff 59 is a binary code of length n, a “soft decision ” decodhg afgorithm for Q changes ao arbitrary point of R ” into a nearest codeword (nearest in Euclideao distance). 2) Similarly, a deco&g afgorithm for a lattice A in R ” changes an arbitraq point of R ” into a closest lattice point. Some general methods are given for constructing such algorithnq and arc used to obtain new and faster decoding algorithms for the C&set lattice E,, the Cofay code and the Leech lattice. L I.
Asymptotic Performance of Vector Quantizers with a Perceptual Distortion Measure
 in Proc. IEEE Int. Symp. on Information Theory, p. 55
, 1997
"... Gersho's bounds on the asymptotic performance of vector quantizers are valid for vector distortions which are powers of the Euclidean norm. Yamada, Tazaki and Gray generalized the results to distortion measures that are increasing functions of the norm of their argument. In both cases, the dist ..."
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Cited by 31 (3 self)
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Gersho's bounds on the asymptotic performance of vector quantizers are valid for vector distortions which are powers of the Euclidean norm. Yamada, Tazaki and Gray generalized the results to distortion measures that are increasing functions of the norm of their argument. In both cases, the distortion is uniquely determined by the vector quantization error, i.e., the Euclidean difference between the original vector and the codeword into which it is quantized. We generalize these asymptotic bounds to inputweighted quadratic distortion measures, a class of distortion measure often used for perceptually meaningful distortion. The generalization involves a more rigorous derivation of a fixed rate result of Gardner and Rao and a new result for variable rate codes. We also consider the problem of source mismatch, where the quantizer is designed using a probability density different from the true source density. The resulting asymptotic performance in terms of distortion increase in dB is shown...
Multipledescription vector quantization with lattice codebooks: Design and analysis
 IEEE Trans. Inform. Theory
, 2001
"... Abstract—The problem of designing a multipledescription vector quantizer with lattice codebook 3 is considered. A general solution is given to a labeling problem which plays a crucial role in the design of such quantizers. Numerical performance results are obtained for quantizers based on the latti ..."
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Cited by 29 (1 self)
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Abstract—The problem of designing a multipledescription vector quantizer with lattice codebook 3 is considered. A general solution is given to a labeling problem which plays a crucial role in the design of such quantizers. Numerical performance results are obtained for quantizers based on the lattices 2 and, =1 2 4 8 that make use of this labeling algorithm. The highrate squarederror distortions for this family ofdimensional vector quantizers are then analyzed for a memoryless source with probability density function (pdf) and differential entropy ( ). For any (0 1) and rate pair (),it is shown that the twochannel distortion 0 and the channel 1 (or channel 2) distortion satisfy
Efficient Approximation Algorithms for the Hamming Center Problem
, 1999
"... The Hamming center problem for a set S of k binary strings, each of length n, asks for a binary string of length n that minimizes the maximum Hamming distance between and any string in S. The decision version of this problem is known to be NPcomplete [6]. We provide several approximation algorit ..."
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Cited by 28 (2 self)
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The Hamming center problem for a set S of k binary strings, each of length n, asks for a binary string of length n that minimizes the maximum Hamming distance between and any string in S. The decision version of this problem is known to be NPcomplete [6]. We provide several approximation algorithms for the Hamming center problem. Our main result is a randomized ( 4 3 + ")approximation algorithm running in polynomial time if the Hamming radius of S is at least superlogarithmic in k. Furthermore, we show how to nd in polynomial time a set B of O(log k) strings of length n such that for each string in S there is at least one string in B within Hamming distance not exceeding the radius of S. 1 Introduction Let Z n 2 be the set of all strings of length n over the alphabet f0; 1g. For any 2 Z n 2 we use the notation [i] to refer to the symbol placed at the ith position of , where i = 1; ::; n, and we let [i::j] represent the substring of starting at position i and endin...
Multiple description vector quantization with lattice codebooks: Design and analysis
, 2000
"... The problem of designing a multiple description vector quantizer with lattice codebook Λ is considered. A general solution is given to a labeling problem which plays a crucial role in the design of such quantizers. Numerical performance results are obtained for quantizers based on the lattices A2 an ..."
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Cited by 25 (9 self)
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The problem of designing a multiple description vector quantizer with lattice codebook Λ is considered. A general solution is given to a labeling problem which plays a crucial role in the design of such quantizers. Numerical performance results are obtained for quantizers based on the lattices A2 and Z i, i = 1, 2, 4, 8, that make use of this labeling algorithm. The highrate squarederror distortions for this family of Ldimensional vector quantizers are then analyzed for a memoryless source with probability density function p and differential entropy h(p) < ∞. For any a ∈ (0, 1) and rate pair (R, R), it is shown that the twochannel distortion ¯ d0 and the channel 1 (or channel 2) distortion ¯ ds satisfy and lim ¯d02
The Optimal Lattice Quantizer in Three Dimensions
, 1983
"... The bodycentered cubic lattice is shown to have the smallest mean squared error of any lattice quantizer in three dimensions, assuming that the input to the quantizer has a uniform distribution. ..."
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Cited by 24 (6 self)
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The bodycentered cubic lattice is shown to have the smallest mean squared error of any lattice quantizer in three dimensions, assuming that the input to the quantizer has a uniform distribution.