D. H. Lee and D. L. Neuho#, "Asymptotic distribution of the errors in scalar and vector quantizers," IEEE Trans. Inform. Theory, vol. 42, pp. 446-460, Mar.1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....asymptotically optimal performance by taking advantage of the following two observations: 1. CS is optimal for the MSE criterion (w(x) 1) The base and enhancement layer rates in (6) reduce to, blw(x) 1 : h(X) log(Ab) Relw(x) l : h(Z) log(Ae) log(Ab) log(Ae) For MSE, K(z) fz(z) [9], and distortion can be rewritten as i 2 Dcs Iw(x) l 12 Ae i 2(n(X) Rb R) 5ns(Rb 4 Re)lw(x) l. 7) 2. For an optimally companded ECSQ, the WMSE of the original signal equals MSE of the compressed signal. For the optimal compressor function, 2) reduces to D = A2 12, which equals the ....

D. H. Lee and D. L. Neuhoff, "Asymptotic distribution of the errors in scalar and vector quantizers," IEEE Trans. Inform. Theory, vol. 42, pp. 446-60, March 1996.

....and hence, the best quantizer at the enhancement layer is also uniform. The compression function is given by c(x) ce(x ) 1. The base and enhancement layer rates in (6) reduce to, nlw(x = x) log( nelw(x = z) logCXe) logCX) logCXe) For MSE, 5) reduces to K(z) fz(z) [8], and distortion can be rewritten as D Iw(x) 1 12 1 : 8(Rb 2. For an optimally companded ECSQ, the WMSE of the original signal equals MSE of the compressed signal. For the optimal compressor function, Bennett s integral (1) reduces to D = A2 12 (using (2) which equals the MSE (in ....

D.H. Lee and D. L. Neuhoff, "Asymptotic distribution of the errors in scalar and vector quantizers," IEEE Trans. Inform. Theory, vol. 42, pp. 446-60, March 1996.

....schemes incorporating uniform, lattice or linear trellis quantizers. In light of this model and the wide use of lattices in signal coding, it is interesting to characterize the statistical properties of a random vector which is uniformly distributed over the basic cell of a lattice; see e.g. [12]. Thus, we analyze in this paper the spectral properties and the divergence from Gaussianity of this random vector, referred to as lattice quantization noise. We mainly focus on optimal lattice quantizers, i.e. lattice quantizers that minimize the power of the quantization noise, and on their ....

D.H. Lee and D.L. Neuhoff. Asymptotic distribution of the errors in scalar and vector quantizers. IEEE Trans. Information Theory, submitted.

....error is white in terms of the joint characteristic functions of pairs of samples, two dimensional analogs of Widrow s [529] condition. Zador [562] found high resolution expressions for the characteristic function of the error produced by randomly chosen vector quantizers. Lee and Neuho# [312], 379] found high resolution expressions for the density of the error produced by fairly general (deterministic) scalar and vector quantizers in terms of their point density and their shape profile, which is a function that conveys more cell shape information than the inertial profile. As a side ....

.... s integral on the second stage can be found in [311] 309] In order to apply Bennett s integral, it was necessary to find the form of the probability density of the quantization error produced by the first stage. This motivated the asymptotic error density analysis of vector quantization in [312], 379] Multistage quantizers have been improved in a number of ways. More sophisticated (than greedy) encoding algorithms can take advantage of the direct sum nature of the codebook to make optimal or nearly optimal searches, though with some (and sometimes a great deal of) increased ....

D. H. Lee and D. L. Neuho#, "Asymptotic Distribution of the Errors in Scalar and Vector Quantizers," IEEE Trans. Inform. Theory, vol. 42, pp. 446--460, March 1996.

....error is white in terms of the joint characteristic functions of pairs of samples, two dimensional analogs of Widrow s [529] condition. Zador [562] found high resolution expressions for the characteristic function of the error produced by randomly chosen vector quantizers. Lee and Neuhoff [312], 379] found high resolution expressions for the density of the error produced by fairly general (deterministic) scalar and vector quantizers in terms of their point density and their shape profile, which is a function that conveys more cell shape information than the inertial profile. As a side ....

.... s integral on the second stage can be found in [311] 309] In order to apply Bennett s integral, it was necessary to find the form of the probability density of the quantization error produced by the first stage. This motivated the asymptotic error density analysis of vector quantization in [312], 379] Multistage quantizers have been improved in a number of ways. More sophisticated (than greedy) encoding algorithms can take advantage of the direct sum nature of the codebook to make optimal or nearly optimal searches, though with some (and sometimes a great deal of) increased ....

D. H. Lee and D. L. Neuhoff, "Asymptotic Distribution of the Errors in Scalar and Vector Quantizers," IEEE Trans. Inform. Theory, vol. 42, pp. 446--460, March 1996.

....integral on the second stage can be found in [224, 222] In order to apply Bennett s integral, it was necessary to find the form of the probability density of the quantization error produced by the first stage. This motivated the asymptotic error density analysis of vector quantization in [225, 271]. An interesting sidelight of this work is that it turns out that one can learn much about the point density and cell shapes of a high resolution quantizer from the histogram of the lengths of the quantization errors. Multistage quantizes have been improved in a number of ways. More ....

D.H. Lee and D.L. Neuhoff, "Asymptotic Distribution of the Errors in Scalar and Vector Quantizers," IEEE Trans. Information Theory, Vol. 42, pp. 446-460, March 1996.

....for a survey of more recent work. High rate quantization theory had humble beginnings in Bennett s paper for scalar quantizers [10] developed slowly through the years through the efforts of many [13 21,9,22 29] and is not so well known. Recent results have significantly expanded its usefulness [11,30 32], and it is still developing. 3) Underlying Principles: Rate distortion theory is a deep and elegant theory based on the law of large numbers. High rate quantization theory is a simpler, less elegant theory based on geometric characterizations and integral approximations over small cells. 4) ....

....that VQ s with moderate dimension were both good and feasible. 8 D. L. NEUHOFF Recently, in high rate theory there has been developed an asymptotic formula for the density of the error that results from a VQ with many mostly small cells, with neighboring cells having similar sizes and shapes [30]. Taking the second moment of this density yields, of course, Bennett s integral. The form of the density depends intimately on the point density and the cell shapes. The original motivation for finding the asymptotic error density was a high rate analysis of two stage VQ [32] where a second ....

D.H. Lee and D.L. Neuhoff, "The asymptotic distribution of the error in scalar and vector quantizers," 1990 IEEE Int'l Symp. Inform. Theory, San Diego, Jan. 1990.

....shape profile, shape function, inertial profile. 1 I. INTRODUCTION When a k dimensional vector quantizer with quantization rule Q is applied to a random vector X = X 1 . X k ) the resulting quantization error is the k dimensional random vector U = XQ (X) In a recent paper, Lee and Neuhoff [1] found an approximate formula for distribution of the length of this error, suitably normalized, that applies when the number of quantization points, N, is large. Specifically, they considered, the normalized error length W N D = N 1 k X Q(X) and showed that when N is large, the ....

....probability density, l(x) is the quantizer point density, which indicates the sizes of the cells in the vicinity of x, and g(x,r) is the quantizer shape profile, which reflects the shapes of the cells in the vicinity of x. Precise definitions of l(x) and g(x,r) will be given later. The result in [1] was derived using informal approximation arguments such as F f(x) dx vol(F) f(x ) for any point x in F. The purpose of the present paper is to put this result on a firm foundation by giving careful definitions of point density and shape profile and by finding precise conditions under ....

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D. H. Lee and D.L. Neuhoff, "Asymptotic Distribution of the Errors in Scalar and Vector Quantizers," to appear in IEEE Trans. Inform. Theory.

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D. H. Lee and D. L. Neuho#, "Asymptotic distribution of the errors in scalar and vector quantizers," IEEE Trans. Inform. Theory, vol. 42, pp. 446-460, Mar.1996.

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D. H. Lee and D. L. Neuhoff, "Asymptotic distribution of the errors in scalar and vector quantizers," IEEE Trans. Inform. Theory, vol. 42, pp. 446-460, Mar.1996.

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