V. A. Vaishampayan and J. Domaszewicz, "Design of entropyconstrained multiple-description scalar quantizers," IEEE Transactions on Information Theory, vol. 40, no. 1, pp. 245-- 250, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....13 15 17 21 16 18 20 23 19 22 Delta The indices i and j are transmitted over the two channels, and a side description is produced from each. The central decoder uses the inverse function a m ( Delta) to deduce l from i and j. At high resolution conditions, when entropy coding is applied, [18], the quantizer becomes a uniform and unbounded scalar quantizer with step size Delta, paralleling the well known Gish and Pierce result in the single description case [10] The side quantizers are defined in this case by the rows and the columns of an infinite matrix. Q 0 ( Delta) am ....

....scheme exceeds the optimum by at most 0:5 P 3 log 2eGKi bits per sample for high resolution conditions, with some possible improvement using refinement time sharing or dependent dithering. In contrast, the ECD PMDSQ is based on a more complex building block the periodic MDSQ. Unlike in [18], the analysis of the ECD PMDSQ is not asymptotic, and holds for any resolution. We have shown that this scheme exceeds the optimum by only log( 12 ) bits per sample at high resolution and by at most log( 12 ) 2D(X;X at lower resolutions. These two results resemble the results of the ECDQ ....

V. A. Vaishampayan, Design of entropy-constrained multiple-description scalar quantizers, IEEE Trans. Information Theory IT-40 (Jan. 1994), 245--251.

....[3] to the MD case and showed that for a Gaussian source the outer bounds are asymptotically tight. C. Code Constructions Several efforts have also been made to design practical MD coding systems. In [27] a design procedure for the construction of fixed rate scalar quantizers was presented. In [29], that design procedure was extended to the entropy constrained case. It is shown in [28] that at high rates, for the case of balanced descriptions (R 1 = R 2 = R) and Gaussian sources, the distortion product D 0 D 1 of the entropy constrained MD scalar quantizer takes the form: 4 ( At ....

V.A. Vaishampayan and J. Domaszewicz. Design of entropyconstrained multiple-description scalar quantizers. IEEE Trans. Information Theory, 40(1):245--250, January 1994.

....previous results given in [1] I. Introduction Multiple description coding achieves more graceful degradation in reconstruction performance than single description coding in the event of channel failure. Multiple description scalar quantization (MDSQ) 2] and entropyconstrained MDSQ (ECMDSQ) [3] achieve the multiple description property in practical source coding systems. An asymptotic analysis for MDSQ ECMDSQ is given in [1] In this paper, we introduce a new and straightforward asymptotic analysis of the multiple description scalar quantizer (MDSQ) The analysis provides insight into ....

....This analysis is applied to optimizing a two stage MDSQ in Section V. Section VI, the granular distortions of level constrained MDSQ and ECMDSQ are derived, and the paper is summarized in Section VII. II. MDSQ and ECMDSQ The design and analysis of MDSQ and ECMDSQ were given in a series of papers [2, 3, 1] by Vaishampayan et al. We briefly review the results related to our work and then consider an assumption in the derivation of these results and its implications. The basic idea of MDSQ is to create two coarse side quantizers which produce acceptable side distortions when used alone; the two ....

V.A. Vaishampayan, "Design of entropy-constrained multiple description scalar quantizers," IEEE Trans. IT, vol. 40, no. 1, pp. 245--250, Jan 1994.

....quantizer cells with finite diagonals in the index assignment matrix, as well as a method to approximate optimal cell sizes. Based on these insights, a Universal Multiple Description Scalar Quantizer (UMDSQ) is proposed which can achieve nearly the same performance as the fully optimized ECMDSQ [1], at much lower design complexity. The design requires selection of only two parameters, and the resulting UMDSQ can provide a continuum of trade o# points between the central and side distortions as the two parameters are varied. 1 Introduction This paper introduces a new high rate analysis ....

....with finite diagonals in the index assignment matrix, as well as methods to approximate optimal cell sizes. Based on these insights, a class of multiple description scalar quantizers is proposed which are universal in nature and can achieve almost the same performance as the fully optimized ECMDSQ [1], at lower complexity. The system considered in this paper is a quantizer followed by an entropy coder, which is usually the case in practice. The motivation to consider a new approach to the design and high rate analysis of ECMDSQ is to resolve some di#culties in its use in practical ....

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V.A. Vaishampayan, "Design of entropy-constrained multiple description scalar quantizers," IEEE Trans. IT, vol. 40, no. 1, pp. 245--250, Jan 1994.

....locally minimize the average side distortion under constraints on the side rates. This approach constrasts with the previous entropy constrained multiple description quantization schemes in which the IA was considered xed and the side rates depended on the boundaries of the quantization cells [12]. We propose an IA design method based on a local search algorithm with a large scale neighborhood. This algorithm uses entropy constraints and allows to design multiple description coders with di erent side rates and side distortions without modifying the central codebook. It is a combinatorial ....

....between two indices sharing a description is minimized. The asymptotic soundness of this min max criterion is proved. In [10] an iterative algorithm similar to the generalized Lloyd algorithm (GLA) is described and applied to the optimization of multiple description vector quantizers. In [12], a similar technique is presented for entropy constrained multiple description scalar quantizers. In both cases, the index assignment problem is not treated explicitly. Other contributions from Vaishampayan, Sloane et al. 13] 3] describe index assignment methods for lattice vector quantizers. A ....

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V. A. Vaishampayan and J. Domaszewicz. Design of entropy-constrained multiple description scalar quantizers. IEEE Trans. Inform. Theory, 40(1):245-250, January 1994.

....annealing for multiple description vector quantizers. 5.2.1 Training approach Practical design of multiple description quantizers was first investigated by Vaishampayan in [96] where a design method for two description scalar quantizers is presented. This work has been generalized in [97] to entropyconstrained scalar quantizers. It is based on the traditional Lloyd approach in which each iteration decreases a weighted sum of the rates and distortions. This locally optimal technique is also used in [33] for the design of multiple description vector quantizers on more than two ....

....of the rates and distortions. This locally optimal technique is also used in [33] for the design of multiple description vector quantizers on more than two channels. We now describe a GLA like design algorithm for a multiple description vector quantizer. The design algorithms of Vaishampayan [96] [97] are specialized to the scalar (k = 1) case and make use of initial index assignments described in the next subsection, while the algorithms from Fleming et al. 33] are generalized to m 2. So we will take a middle approach and give the local optimization rules for an entropy constrained MDVQ on ....

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V. A. Vaishampayan and J. Domaszewicz. Design of entropyconstrained multiple description scalar quantizers. IEEE Trans. Inform. Theory, 40(1):245--250, January 1994.

....and the quantizers refine each other in a symmetric fashion. For a given number of side levels n, the central distortion is smaller at the cost of higher side distortions than for an MDSQ as in Fig. 3(a) The optimal design for fixed rate MDSQ in [33] was extended to entropy constrained MDSQ in [34]. One of the most satising aspects of the theory of MDSQ is that at high rates, all the decay exponent trade offs discussed in Section II B can be obtained. Furthermore, the factor by which the distortion product DoDi exceeds the bounds of Theorem I is approximately constant [35] For R R1 R2 and ....

V. A. Vaishampayan and J. Domaszewicz. Design of entropy-constrained multiple-description scalar quantizers. IEEE Trans. Inform. Th., 40(1):24550, January 1994.

....descriptions can be decoded and the pictures can be remultiplexed to reconstruct a video sequence whose frame rate is essentially proportional to the number of descriptions received. More sophisticated forms of multiple description coding have been investigated over the years; some highlights are [25, 26, 27, 6]. For an overview see [7] A particularly ecient and practical system is based on layered audio or video coding [18, 10] Reed Solomon coding [28] priority encoded transmission [1] and optimized bit allocation [4, 19, 11, 12] In such a system the audio and or video signal is partitioned into ....

V. A. Vaishampayan and J. Domaszewicz. Design of entropy-constrained multiple description scalar quantizers. IEEE Trans. Information Theory, 40(1):245-250, January 1994.

....3. We can see that for small values of ff (so for an almost memoryless source) the performance of the proposed system is quite poor. For ff = 0 the gap between the two cases is of 6.02dB while in this case an entropy constrained MD Scalar Quantizer performs only 3. 06dB worse than the ideal bounds[11]) On the other hand for highly correlated sources the gap between the optimal filter banks and the MD bounds is reduced in this experiment to 3.10dB. So in this case the filter bank system could effectively compete with Multiple Description Transform codes based on decorrelating the input sequence ....

V.A. Vaishampayan and J. Domaszewicz. "Design of Entropy-Constrained Multiple Description Scalar Quantizers," IEEE Trans. Inform. Theory, 40(1):245-250,1994.

....a real number, to a set of integers (designing a quantizer) second, mapping each integer to the set of components to be transmitted over the channels (index assignment) This paper deals with the second part of the problem. The relevant works include [3] 4] 1] 12] 10] 11] 6] [8], 2] Applications of this problem arise in video and speech communication. To the best of our knowledge, this paper is rst to give lower bounds for the problem, and provide optimal algorithms that achieve the lower bound. In addition, the general case of k channels, when k 2, has never been ....

Vaishampayan, V. A. and J. Domaszewicz, \Design of entropy-constrained multiple-description scalar quantizers ", IEEE Trans. Inform. Theory, 40 (1994), 245{ 250.

....criterion. Zhang and Berger [33] and Witsenhausen, Wolf, Wyner, and Ziv [30] 31] explored whether the achievable rate region is the rate distortion region for other types of information sources. The rst constructive results for two channels with equal rates were presented by Vaishampayan [26] [25]. In [26] Vaishampayan designs Multiple Description Scalar Quantizers (MDSQs) with good asymptotic properties. We show, however, that this solution is not optimal. An MDSQ is a scalar quantizer (mapping of the source to a nite integer point set) that is designed to work in a diversity based ....

....rate distortion tuples of the type (log n; log n; log n; 0; D 1 ; D 2 k 2 ) when the original information source is quantized into m n k integers, where log n is channel rate. This means that in the corresponding matrix the m numbers do not ll the entire matrix. Vaishampayan [26] [25], 27] designed a solution for this case with two channels which is an arrangement of the numbers into a uniform diagonal. In the next section we examine this solution. However, to avoid the boundary e ects, we will rst consider the in nite diagonal in this section. We consider an arrangement ....

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Vaishampayan, V. A. and J. Domaszewicz, \Design of entropy-constrained multiple-description scalar quantizers", IEEE Transactions on Information Theory, 40 (1994), 245-250.

....51 veloped achievable rate regions and lower bounds to performance. The results were extended by El Gamal and Cover (1982) 139] Ahlswede (1985) 6] and Zhang and Berger (1987) 573] In 1993 Vaishampayan et al. used a Lloyd algorithm to actually design fixed rate [508] and entropyconstrained [509] scalar quantizers for the multiple description problem. High resolution quantization ideas were used to evaluate achievable performance in 1998 by Vaishampayan and Batllo [510] and Linder, Zamir, and Zeger [324] An alternative approach to multiple description quantization using transform coding ....

V. A. Vaishampayan, "Design of entropy-constrained multipledescription scalar quantizers," IEEE Trans. Inform. Theory, Vol.40, pp. 245--250, Jan. 1994.

....for the two descriptions and by D 0 ; D 1 and D 2 the average distortion for each decoder. Practical design of multiple description quantizers was rst investigated by Vaishampayan in [4] where a design method for two description scalar quantizers is presented. This work has been generalized in [5] to entropy constrained scalar quantizers. It is based on the traditional Lloyd approach in which each iteration decreases a weighted sum of rates and distortions. This locally optimal technique is also used in [6] for the design of multiple description vector quantizers on more than two channels. ....

.... 0 ; t 1 ; t 2 ) while t 0 62 T 0 do j (t 0 ) i arg min i=1;2 d(X; y(son i (t 0 ) j d(X; y(son i (t j ) t 0 son i (t 0 ) t j son i (t j ) end while return (index(t 1 ) index(t 2 ) Figure 3: Encoder algorithm this type of formulation can be found for example in [5]. Just as in a classical TSVQ, the minimization of the Lagrangian sum at each node does not lead to a globally optimal encoding, but makes the encoding complexity linear in the bitrate. The quantization cell Q(t i ) of a leaf node t i 2 T i ; i = 1; 2 is de ned as Q(t i ) fX 2 R k j i ....

[Article contains additional citation context not shown here]

V. A. Vaishampayan and J. Domaszewicz. Design of entropy-constrained multiple description scalar quantizers. IEEE Trans. Inform. Theory, 40(1):245-250, January 1994.

....a real number, to a set of integers (designing a quantizer) second, mapping each integer to the set of components to be transmitted over the channels (index assignment) This paper deals with the second part of the problem. The relevant works include [6] 10] 3] 19] 1] 16] 17] 12] [14], 4] Applications of this problem arise in video and speech ( 8] 9] 18] 2] communication. We are given k channels of capacities log n 1 ; log n 2 ; log n k bits. We consider the information to be sent over those channels as a number M with at most lg m bits, 1 m n 1 n 2 Delta ....

Vaishampayan, V. A. and J. Domaszewicz, "Design of entropy-constrained multiple-description scalar quantizers", IEEE Transactions on Information Theory, 40 (1994), 245--250. 9

....approaches. In the first approach, pioneered by Vaishampayan, MD scalar, vector, or trellis quantizers are designed to produce N = 2 descriptions, using a generalized Lloyd like clustering algorithm that minimizes the Lagrangian of the rates and expected distortions R 1 ; R 2 ; D 1 ; D 2 ; D 1;2 [1, 2, 3, 4]. In the second approach, pioneered by Wang, Orchard, and Reibman, MD quantizers are constructed by separately describing (i.e. quantizing and coding) the N coefficients of an N Theta N block linear transform, which has been designed to introduce a controlled amount of correlation between the ....

V. A. Vaishampayan and J. Domaszewicz. Design of entropy-constrained multiple description scalar quantizers. IEEE Trans. Information Theory, 40(1):245--250, January 1994.

....subset is acceptable, and that better quality is obtained by more descriptions. It is assumed in MDC that losses to different descriptions are uncorrelated, and that the probability of losing all the descriptions is small. MDC has been implemented in several ways. In the scalar quantizer approach [6, 52, 67, 68, 69], optimal index assignments are hard to find in real time, and suboptimal approaches, such as A2 index assignment [67] introduce a large overhead in bit rate [76] Instead of putting each pixel in every description, a pair wise correlatingtransform (PCT) 45, 75] approach has been proposed to ....

V. A. Vaishampayan and J. Domaszewicz. Design of entropy constrained multiple description scalar quantizers. IEEE Trans. on Information Theory, 40:245-251, Jan. 1994.

....51 veloped achievable rate regions and lower bounds to performance. The results were extended by El Gamal and Cover (1982) 139] Ahlswede (1985) 6] and Zhang and Berger (1987) 573] In 1993 Vaishampayan et al. used a Lloyd algorithm to actually design fixed rate [508] and entropyconstrained [509] scalar quantizers for the multiple description problem. High resolution quantization ideas were used to evaluate achievable performance in 1998 by Vaishampayan and Batllo [510] and Linder, Zamir, and Zeger [324] An alternative approach to multiple description quantization using transform coding ....

V. A. Vaishampayan, "Design of entropy-constrained multipledescription scalar quantizers," IEEE Trans. Inform. Theory, Vol.40, pp. 245--250, Jan. 1994.

....and for communicating over a lossy packet network. Source codes designed for this channel model are called multiple description source codes. For memoryless sources, rate distortion theoretic results have been presented in [1] and [2] multiple description quantizers have been designed in [3] and [4] and an asymptotic analysis has been presented in [5] For sources with memory, subsampling approaches have been considered in [6] 7] Here we introduce the multiple description transform coder (MDTC) for sources with memory and present an asymptotic analysis for the squared error distortion ....

V.A. Vaishampayan and J. Domaszewicz, "Design of entropy-constrained multiple description scalar quantizers," IEEE Trans. Inform. Th., vol. 40, pp.245-250, January 1994.

....with memory. An important special case of the multiple description problem is the problem of successive refinement of information [9] In [9] a necessary and sufficient condition for a rate distortion problem to be successively refinable is derived. The design of MDSQ s was addressed in [1] and [10], respectively. Vector quantizer design for the multiple description problem was addressed in [11] Applications of multiple description source codes that have been explored so far are to speech transmission over packet switched networks and to communication over correlated Rayleigh fading ....

V. A. Vaishampayan and J. Domaszewicz, "Design of entropy-constrained multiple description scalar quantizers," IEEE Trans. Inform. Th., pp. 245--250, January 1994. 13

....descriptions will result in high coding performance under the MD constraints. 1.4. Related Work The problem of Multiple Descriptions was first studied by El Gamal and Cover [4] in a purely information theoretic framework. The design of MD scalar quantizers has been pioneered by Vaishampayan [17, 18]. An alternative framework for the generation of multiple descriptions was proposed by Wang et al. in [19] consisting of introducing correlations between pairs of transform coefficients of an image, such that if one of the descriptions is lost, the other one can be statistically estimated using ....

....be resolved somehow will arise at the central decoder. ffl We are currently working on proving the conjectured intractability result mentioned in Section 3.2. This and an algorithm for generating index assignments based on a source description, capable of generalizing asymptotically optimal MDSQs [17, 18] (and hopefully improving upon their performance at low rates) are the current focus of our research efforts. ffl This is a still image coder. We intend to extend this construction to coding video signals, and integrate it with our previously developed rate control policies for the multipath ....

V. Vaishampayan and J. Domaszewicz. Design of Entropy-Constrained Multiple Description Scalar Quantizers. IEEE Transactions on Information Theory, 40(1):245--250, January 1994.

....to performance analysis and optimization. 2.2. 1 Review of Multiple Description Scalar Quantization The rst practical 1 results on multiple descriptions were presented by Vaishampayan, where a simple procedure is given to design MD scalar quantizers with some remarkable asymptotic properties [32, 34]. Fig. 2 illustrates the role of a MD quantizer in a two channel diversity system. Channel 1 Channel 2 Receiver 1 Receiver 0 Receiver 2 User Source Scalar Quantizer Index Assignment Multiple Description Scalar Quantizer Figure 2: Basic architecture of a diversity system using a MD ....

....Further work is required to complete the proof of the conjectured intractability result mentioned in Section 2.3. If our conjecture proves true, an approximation algorithm for generating index assignments based on a source description, capable of generalizing asymptotically optimal MD quantizers [32, 34] (and hopefully improving upon their performance at low rates) would be extremely useful. Perhaps the single most important problem yet to be resolved in the context we set up in this work is that of redundancy allocation (in the N 2 case) Recall that we motivated the problem of more than two ....

V. A. Vaishampayan and J. Domaszewicz. Design of Entropy-Constrained Multiple Description Scalar Quantizers. IEEE Trans. Inform. Theory, 40(1):245{ 250, 1994.

....be emphasized that, to this date, Ozarow s result for the gaussian source and squared error metric is the only one presenting a complete characterization of a MD rate distortion region. 1. 2 Multiple Description Scalar Quantization The first practical results on MDs were presented by Vaishampayan [10, 11], where a simple procedure is given for designing MD scalar quantizers with remarkable good asymptotic properties. A MD scalar quantizer Work done at AT T Labs Research, between 6 98 and 9 98. y Beckman Institute and Dept. of Computer Science, University of Illinois at Urbana Champaign. 405 ....

V. A. Vaishampayan and J. Domaszewicz. Design of Entropy-Constrained Multiple Description Scalar Quantizers. IEEE Trans. Inform. Theory, 40(1):245-- 250, 1994.

....is a good figure of merit for high rates. For a gaussian source and m = 2, it was shown in [1] that there is a gap of 8.69dB between the distortion product of the MD scalar quantizer and the MD rate distortion bound; this gap is reduced to 3. 06dB when the entropy constrained MD scalar quantizer [13] is considered. 1.3 Multiple Description Vector Quantization For the squared error distortion measure, it is known from classical quantization theory [8] that the error of a scalar quantizer with cubic Voronoi regions is 1:53dB higher 1 However, the sub exponential terms in the MSE are not the ....

V. A. Vaishampayan and J. Domaszewicz. Design of Entropy-Constrained Multiple Description Scalar Quantizers. IEEE Trans. Inform. Theory, 40(1):245--250, 1994.

....bounds on the value of D 0 which, when evaluated numerically for an unbiased coin and Hamming metric, differ by a very small amount. 2.2 Multiple Description Quantization Theory 2.2. 1 Scalar Quantization The first practical 1 results on multiple descriptions were presented by Vaishampayan [35, 38], where a simple procedure is given to design MD scalar quantizers with some remarkable asymptotic properties. Fig. 1 illustrates the role of a MD quantizer in a two channel diversity system. A MD quantizer consists of two main components: a scalar quantizer (that maps continuous valued random ....

....work is also required to complete the proof of the conjectured intractability result mentioned in Section 4. If our conjecture proves true, an approximation algorithm for generating index assignments based on a source description, capable of generalizing asymptotically optimal MD quantizers [35, 38] (and hopefully improving upon their performance at low rates) would be extremely useful. Our current research efforts are focused on the problem of applying the tools presented in this paper to the development of a complete system for image video transmission over the Internet. Acknowledgements. ....

V. A. Vaishampayan and J. Domaszewicz. Design of Entropy-Constrained Multiple Description Scalar Quantizers. IEEE Trans. Inform. Theory, 40(1):245-- 250, 1994.

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V. A. Vaishampayan and J. Domaszewicz, "Design of entropyconstrained multiple-description scalar quantizers," IEEE Transactions on Information Theory, vol. 40, no. 1, pp. 245-- 250, 1994.

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V.A. Vaishampayan and J. Domaszewicz. Design of entropyconstrained multiple-description scalar quantizers. IEEE Trans. Information Theory, 40(1):245--250, January 1994.

No context found.

V.A. Vaishampayan and J. Domaszewicz. Design of entropyconstrained multiple-description scalar quantizers. IEEE Trans. Information Theory, 40(1):245--250, January 1994.

No context found.

V.A. Vaishampayan and J. Domaszewicz. "Design of Entropy-Constrained Multiple Description Scalar Quantizers," IEEE Trans. Inform. Theory, 40(1):245-250,1994.

No context found.

V. A. Vaishampayan, J. Domaszewicz, Design of entropy-constrained multiple-description scalar quantizers, IEEE Trans. Inform. Theory, vol. 40, pp. 245-250, Jan. 1994.

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V. A. Vaishampayan and J. Domaszewicz, "Design of entropy constrained multiple description scalar quantizers," IEEE Trans. Info. Theory, vol. 40, pp. 245-251, Jan. 1994.

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V.A. Vaishampayan and J. Domaszewicz, "Design of entropy-constrained multiple-description scalar quantizers," IEEE Trans. Inform. Theory, vol. 40, pp. 245-250, Jan. 1994.

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V.A. Vaishampayan and J. Domaszewicz. Design of entropy-constrained multiple-description scalar quantizers. IEEE Trans. Information Theory, 40(1):245--250, January 1994.

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