R. M. Gray and D. L. Neuho#, "Quantization," IEEE Transactions on Information Theory, vol. 44, pp. 2325--2383, October 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....later. The objective is to design optimal dynamic quantizers, Q : Q t (X) t = 0, 1, 2, and encoder decoder pairs under some stability criteria. Before introducing these criteria, we first recall the notions of a quantizer, and support width of a random variable. Definition II.1. [10] A quantizer Q(x) is a function that maps a large, possibly infinite, set (where a variable x takes values) into a smaller finite set (where the quantized values lie) and is characterized by a set of thresholds or bin edges that partition the input space into quantization bins and a set of ....

....is a single multidimensional system (Fig. 1) To design such a scheme with digital noiseless channels, one needs to investigate vector quantization, which is a slight generalization of scalar quantization, where the quantization bins and reconstruction values belong to higher dimensions (see [10]) Vector quantization has a higher degree of freedom over scalar quantization of vector components, since for the quantization the components can have a joint density which does not have to be of product form. For the implementation, since the analytic derivations for the optimal quantizer design ....

R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. Inf. Theory, vol. 44, pp. 2325- 2384, October 1998.

....Point density functions also are useful in analyzing the distortion of quantizers. For example, Bennett s integral gives the average distortion in the high resolution case for a nonuniform quantizer in terms of a point density function, source distribution, and size of the quantizer codebook (see [5] for more details) For uniform quantizers, the computation of a point density function is trivial. For nonuniform quantizers however, point density functions are not always guaranteed to exist, and when they do, their computation can be difficult. Point density functions depend on the quantizer ....

R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Transactions on Information Theory, vol. 44. no. 6, pp. 2355-2383, October 1998.

.... Q (x) Q (x) 2 ; 18) where the first inequality follows from (15) and the second one follows from Theorem 1 (inequality (3) 2 B Sliding Block and Trellis Source Codes A class to which Theorem 1 can be directly applied is that of sliding block codes (cf. e.g. [7]) Assume here an arbitrary distortion measure. A finite constraint length, timeinvariant encoder with constraint length l, memory l M , and delay l d = l Gamma l M Gamma 1 is a 11 mapping f : X f1; 2; Mg yielding the channel symbols fy i g defined by y i = f(x i l d i Gammal M ....

....task. However, any sufficiently smoothly parametrizable subset of this class can be dealt with using appropriate grids, similarly as was done in the proof of Corollary 2. Another important related family of source codes that are covered by Theorem 1 consists of the block trellis codes (cf. e.g. [7, 8]) In particular, note that a block trellis coding scheme of constraint length K and search depth L, which consists of a constraint length K sliding block decoder g and a depth L trellis search encoder matched to g, is a member of L Gamma1 K . C Adaptive Differential Pulse Code Modulation We ....

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R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. Inform. Theory, vol. 44, no. 6, pp. 2325-2383, Oct. 1998.

....encoder indices in a way that possible bit errors create a lower level of distortion in the reconstructed data. One usual advantage of the index assignment is that it does not degrade the performance during the clean channel conditions. For a review of di#erent index assignment techniques refer to [11]. In Channel Optimized Quantization, the quantization levels are designed to optimize the performance of the system in the presence of channel noise. Two classic works in this area are those of Kurtenbach and Wintz [12] on scalar quantization over noisy channels and Chang and Donaldson [13] on ....

....as the works of Dunham and Gray [16] and Ayanoglu and Gray [17] on joint source channel trellis coding. Examples of more recent works in this class are present in [18] 20] For a more comprehensive review of the techniques for channel optimized quantization, the interested reader is referred to [11], 20] 21] More recently in this venue, exploiting the residual redundancy [22] in the output of the source coders for improved reconstruction over noisy channels has found increasing attention [22] 41] This redundancy is due to the suboptimal source coding which is caused by, e.g. a ....

R. M. Gray and D. L. Neuho#, "Quantization,"IEEE Trans. Inform. Theory, vol. 44, No. 6, Oct. 1998.

....the literature. These methods include optimized rate allocation, unequal error protection, optimized index assignment, channel optimized quantization, and more recently, exploiting the source residual redundancies. For a comprehensive review of these techniques the interested reader is referred to [2] [7] The work presented in this manuscript falls into the category of joint source channel coders which use the residual redundancy [8] in the output of the source coder for improved reconstruction over noisy channels. This redundancy is due to the suboptimal source coding which is caused by, ....

R. M. Gray and D. L. Neuho#, "Quantization,"IEEE Trans. Inform. Theory, vol. 44, No. 6, Oct. 1998.

....for coverage control In this paper we design coordination algorithms implementable by a multi vehicle network with limited sensing and communication capabilities. Our approach is related to the classic Lloyd algorithm from quantization theory; see [13] for a reprint of the original report and [14] for a historical overview. We present Lloyd descent algorithms that take into careful consideration all constraints on the mobile sensing network. In particular, we design coverage algorithms that are adaptive, distributed, asynchronous, and verifiably asymptotically correct: Adaptive: Our ....

....guarantees exponential convergence is that the Hessian of be positive definite at C. This property is known to be an open problem, see [12] Note that this gradient descent is not guaranteed to find the global minimum. For example, in the vector quantization and signal processing literature [14], it is known that for bimodal distribution density functions, the solution to the gradient flow reaches local minima where the number of generators allocated to the two region of maxima are not optimally partitioned. B. A family of discrete time Lloyd algorithms Let us consider the following ....

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R. M. Gray and D. L. Neuho#, "Quantization," IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2325--2383, 1998, Commemorative Issue 1948-1998.

....their corresponding standard deviation, which can be computed during training. Thus, USQ offers the advantages of low complexity encoding, simple design (no training is required) and easy performance trade off. Additionally, given a desired rate, the stepsize for a USQ can be optimally determined [20]. The USQ indices are lossless encoded with a Huffman entropy coder. To achieve lower bitrates than that achievable by Huffman coding, a bitmap can be transmitted to the decoder to indicate the position of the non zero coefficients in every frame, and the non zero coefficients can be entropy coded ....

R. M. Gray and D. L. Neuhoff, "Quantization," IEEE transactions on information theory, vol. 44, pp. 2325--2383, October 1998.

....steganography, additive noise 1. DATA HIDING AS ADDITIVE NOISE 1.1. Motivation The motivation to model the steganographic process as the addition of noise arises from a number of factors. In the process of sampling and transmitting signals there are numerous sources of noise such as quantization[1], sensor[2] and channel[3] A number of steganographic hiding schemes have used this as a foundation for noise based data hiding. The goal is to disguise the message as a naturally present noise and add it to the coverimage. While the additive noise framework is especially well suited to schemes ....

....and x c is the pixel value prior to embedding. Generally speaking, f # [n] is the probability that a pixel will be altered by n. In this model it is assumed that the noise acts independently on each pixel. So f # [0] is the probability that, after embedding, a pixel is unchanged. Whereas f # [ 1] is the probability that the pixel is decreased by one. Many times it is more convenient to work with a continuous probability density function, f # (x) rather than the discrete probability mass function. Of course, when digital media is stored, the values must be quantized to a finite number of ....

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R. M. Gray and D. L. Neuho#, "Quantization," IEEE Trans. on Information Theory 44, pp. 2325 -- 2383, Oct. 1998.

....the quantized value and that of the analog signal is called quantization error. Therefore, during the analog signal conversion into digital, the quantization error or quantization noise, being the difference between the input and the output signal should be taken into account. It can be proved [9] that if the analog signal amplitude does not exceed the Full Scale Range (FSR) of the ADC and quantization error is a uniformly random variable, the variance of the quantization error (020 is given by the relation: O 2q (4) 12 From equation (4) we can conclude that when b tends to ....

R.M. Gray, D.L. Neuhoff, "Quantization", IEEE Wansactions on information theory, vol. 44, no. 6 1998, pp. 2325-2388.

....conbribubion is bhe inclusion of bhe case in which bhe rathe equals bhe joinb condibional enbropy of bhe quanbizabion indices given bhe side informabion, bhab is, opbimal quanbizabion for Slepian Wolf coding. This work exbends bhe framework for opbimal quanbizer design for non disbribubed sources [12, 13], especially bhe Lloyd algoribhm. The formulabion and bhe solubion of bhe problem sbudied here were developed independenbly of [9, 11] and while bhere are several similaribies in bhe breabmenb of bhe disborbion, our framework is more general as far as rathe measures is concerned. This greather ....

....case, the index ql minimizing the cost ji(xi, qi) is chosen. In this section we shall discover how similar the distributed quantizer design is to the non distributed one. Observe also the similarity of our definitions with the mod ified distortion measures used in quantization of noisy sources [14, 13], where the distortion between a noise corrupted observation V of an unseen original U and its reconstruction 0 is defined as Eld(U, A fundamental property of the distortion, rate and Lagrangian cost functions (1) is that their expectation is precisely the expected distortion, rate and cost, ....

R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. Inform. Theory, vol. 44, no. 6, pp. 2325 2383, Oct. 1998.

....is with the Electrical Engineering Department and the Institute for Systems Research, University of Maryland, College Park, MD 20742 USA. Publisher Item Identifier S 0018 9448(98)05288 2. of the source. The body of literature on this subject is vast, and we refer the reader to [23] 25] 61] [71], and [128] in this issue. In selecting a model for a communication situation, several factors must be considered. These include the physical and statistical nature of the channel disturbances, the information available to the transmitter, the information available to the receiver, the presence ....

R. Gray and D. Neuhoff, "Quantization," this issue, pp. 2325--2383.

....Department of Electrical Engineering, Southern Methodist University, Dallas, TX 75275 USA. Publisher Item Identifier S 0018 9448(98)06886 2. explain why he would delay further consideration of lossy compression until 10 years later. By 1959, work in scalar quantization and PCM was well underway [196] and differential encoding had received considerable attention [180] 186] 215] Shannon coined the term rate distortion function when he revisited the source coding problem in 1959 [2] The insights and contributions in that paper are stunning. In particular, rate distortion terminology is ....

....local optimality and the encoding stage was still exponentially complex in the product, the possibility of actually using a VQ and testing its performance became possible. We leave further broad discussion of scalar and vector quantization to the excellent paper in this issue by Gray and Neuhoff [196]. However, later when discussing particular lossy source compression techniques, we will identify the role of vector quantization and the type of VQ employed. BERGER AND GIBSON: LOSSY SOURCE CODING 2711 In many applications, it was (and is) necessary to encode several independent memoryless ....

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R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. Inform. Theory, this issue, pp. 2325--2383.

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R.M. Gray and D.L. Neuhoff, "Quantization," IEEE Transactions on Information Theory, Vol. 44, pp. 2325--2384, October 1998.

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R.M. Gray and D.L. Neuhoff, "Quantization," IEEE Transactions on Information Theory, Vol. 44, pp. 2325--2384, October 1998.

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R.M. Gray and D.L. Neuho#, "Quantization," IEEE Transactions on Information Theory, Vol. 44, pp. 2325--2384, October 1998.

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R.M. Gray and D.L. Neuho#, "Quantization," IEEE Transactions on Information Theory, vol. 44, pp. 2325--2384, Oct. 1998. 28

....vector quantization as the rate or codebook size grows asymptotically large, in contrast to the Shannon rate distortion theory results characterizing the tradeo# for fixed rate when the dimension becomes asymptotically large. The history and generality of the results may be found, e.g. in [18]. Bucklew [3] developed further asymptotic properties of high rate quantization, most notably providing a mismatch result that quantified the This work was supported in part by the National Science Foundation under Grant Number 0073050, by the Hewlett Packard Corporation, and by Natural Sciences ....

R.M. Gray and D.L. Neuho#, "Quantization," IEEE Transactions on Information Theory, vol. 44, pp. 2325--2384, Oct. 1998. 28

.... tradeoff between average distortion and rate for k dimensional quantization in the limit of large rate, where rate was measured either by the log of the number of quantization levels or by the Shannon entropy of the quantized vector [18] The history and generality of the results may be found in [10]. Most notably, Bucklew and Wise [2] demonstrated Zador s fixed rate result for rth power distortion measures of the form jjx Gamma yjj r , assuming only that E(jjXjj r ffi ) 1 for some ffi 0. Their result was subsequently simplified and elaborated by Graf and Luschgy [8] Zador s ....

....since Zador requires tail conditions on the marginal densities induced by the pdf. In particular, he requires that the marginal pdfs f i ; i = 1; k each have the property that f i (t) jtj Gammal for jtj c i , where the c i are nonzero, finite constants, and where l 3. As noted in [10], the variable rate results reported by Zador in his PhD thesis [17] and in [19] are incorrect as they are for the fixed rate case and do not include the needed differential entropy term, so it is the results in his Bell Labs Technical Report [18] which are considered here. The conditions of the ....

[Article contains additional citation context not shown here]

R.M. Gray and D.L. Neuhoff, "Quantization," IEEE Trans. Inform. Theory, vol. 44, pp. 2325--2384, October 1998.

.... tradeoff between average distortion and rate for k dimensional quantization in the limit of large rate, where rate was measured either by the log of the number of quantization levels or by the Shannon entropy of the quantized vector [16] The history and generality of the results may be found in [10]. Most notably, Bucklew and Wise [2] demonstrated Zador s fixed rate result for rth power distortion measures of the form jjx Gamma yjj r , assuming only that E(jjXjj r ffi ) 1 for some ffi 0. Their result was subsequently simplified and elaborated by Graf and Luschgy [8] Zador s ....

....since Zador requires tail conditions on the marginal densities induced by the pdf. In particular, he requires that the marginal pdfs f i ; i = 1; k each have the property that f i (t) jtj Gammal for jtj c i , where the c i are nonzero, finite constants, and where l 3. As noted in [10], the variable rate results reported by Zador in his PhD thesis [15] and in [17] are incorrect as they are for the fixed rate case and do not include the needed differential entropy term, so it is the results in his Bell Labs Technical Report [16] which are considered here. The conditions of the ....

[Article contains additional citation context not shown here]

R.M. Gray and D.L. Neuhoff, "Quantization," IEEE Transactions on Information Theory, Vol. 44, pp. 2325--2384, October 1998.

.... tradeo# between average distortion and rate for k dimensional quantization in the limit of large rate, where rate was measured either by the log of the number of quantization levels or by the Shannon entropy of the quantized vector [18] The history and generality of the results may be found in [10]. Most notably, Bucklew and Wise [2] demonstrated Zador s fixed rate result for rth power distortion measures of the form x y r , assuming only that E( X r # ) # for some # 0. Their result was subsequently simplified and elaborated by Graf and Luschgy [8] Zador s ....

....since Zador requires tail conditions on the marginal densities induced by the pdf. In particular, he requires that the marginal pdfs f i ; i = 1, k each have the property that f i (t) # t l for t # c i , where the c i are nonzero, finite constants, and where l 3. As noted in [10], the variable rate results reported by Zador in his PhD thesis [17] and in [19] are incorrect as they are for the fixed rate case and do not include the needed di#erential entropy term, so it is the results in his Bell Labs Technical Report [18] which are considered here. The conditions of the ....

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R.M. Gray and D.L. Neuho#, "Quantization," IEEE Trans. Inform. Theory, vol. 44, pp. 2325--2384, October 1998.

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R. M. Gray and D. L. Neuho#, "Quantization," IEEE Transactions on Information Theory, vol. 44, pp. 2325--2383, October 1998.

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R. Gray and D.L. Neuho#, "Quantization," IEEE Transactions on Information Theory, vol. 44, pp. 2325 - 2383, October 1998

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R.M. Gray & D.L. Neuhoff, "Quantization", IEEE Trans. Inform. Theory, Vol. 44, No. 6, Oct. 1998.

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R.M. Gray & D.L. Neuhoff, "Quantization", IEEE Transactions on Information Theory, Vol. 4, No. 6, Oct 1998.

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R. M. Gray and D. L. Neuho#, "Quantization," IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2325-2383, Oct. 1998.

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R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. on Information Theory, vol. 44, Oct. 1998.

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R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. Inform. Theory, this issue, pp. 2325--2383.

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R. M. Gray and D. L. Neuho#, "Quantization," IEEE Trans. Inform. Th., vol. 44, no. 6, pp. 2325--2383, Oct. 1998.

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R. M. Gray and D. L. Neuho, "Quantization," IEEE Trans. on Information Theory, v. 44, no. 6, pp. 1-63, 1998.

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R. M. Gray and D. L. Neuho#, "Quantization," IEEE Transactions on Information Theory, vol. 44, pp. 2325--2383, October 1998.

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R.M. Gray and D.L. Neuhoff "Quantization", IEEE Transactions on Information Theory, vol.44, (no.6), IEEE, Oct. 1998.

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R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. Information Theory, 44: 2325-2384, October 1998.

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R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. Inform. Theory, vol. 44, no. 6, pp. 2325--2383, 1998.

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R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Tran. Inform. Theory, vol. 44, no. 6, Oct. 1998.

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R. M. Gray and D. L. Neuho#, "Quantization," IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2325--2383, 1998, Commemorative Issue 1948-1998.

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R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. Inform. Theory, vol. 44, pp. 2325--2383, Oct. 1998.

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R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Transactions on Information Theory, vol. 44(6), pp. 2325--2383, 1998.

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R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. Inform. Theory, vol. 44, pp. 2325--2383, Oct. 1998.

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R. M. Gray and D. L. Neuho#, "Quantization," IEEE Transactions on Information Theory, vol. 44, pp. 2325--2383, October 1998.

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R.M. Gray and D.L. Neuhoff, "Quantization," IEEE Trans. Inform. Theory, vol. 44, pp. 2325-2383, Oct. 1998.

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R.M. Gray and D.L. Neuho#, "Quantization," IEEE Trans. on Information Theory, vol. 44, Oct. 1998.

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R. Gray and D. Neuhoff, "Quantization," IEEE Trans. Inform. Theory, vol. 44, pp. 2325--2383, Nov. 1998.

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R. M. Gray and D. L. Neuho#, "Quantization,"IEEE Trans. Inform. Theory, vol. 44, No. 6, Oct. 1998.

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R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. on Information Theory, vol. 44, Oct. 1998.

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R. M. Gray and D. L. Neuho#, "Quantization," IEEE Trans. Inform. Theory, (Special Commemorative Issue), vol. IT-44, pp. 2325--2383, Oct. 1998.

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R.M. Gray and D.L. Neuhoff, "Quantization," IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 1--63, October 1998.

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R. M. Gray and D. L. Neuhoff, "Quantization," IEEE Trans. Inform. Theory, (Special Commemorative Issue), vol. IT-44, pp. 2325--2383, Oct. 1998.

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R. Gray and D. Neuho#, "Quantization," IEEE Transactions on Information Theory, vol. 44, pp. 2325--2383, Oct. 1998.

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R.M. Gray and D.L. Neuhoff, "Quantization," IEEE Trans. Inform. Theory, vol. 44, pp. 2325-2383, Oct. 1998.

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R. M. Gray and D. L. Neuho#, "Quantization," IEEE Trans. Inf. Theory 44, pp. 2325--2384, 1998.

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