G.M. Roe, \Quantizing for minimum distortion," IEEE Trans. Inform. Thy., vol. IT-10, pp. 384-385, Oct. 1964. 18  Home/Search   Document Not in Database   Summary   Related Articles   Check

....formula [4] and in Bennett s integral [5, 7, 8] and, nally, they are useful in quantizer design. For one thing, accurate estimates of the optimal support region are needed when designing nearly optimal quantizers directly from the distribution function corresponding to the optimal point density [4, 8, 9, 10, 11, 12, 13] or, equivalently, when nding the asymptotically best compressor function for a compander implementation. Secondly, Lloyd Max 2 style algorithms for designing optimal quantizers [8, 14] begin with an estimate of the support the better the estimate, the more rapidly the algorithm converges. For ....

....optimal support by a factor that depends on the probability density but not N . For example this factor is p 3=2 for the Gaussian density. Finally, Section 5 summarizes. A number of the previous key parameter nding methods are based on similar approaches: optimal companding strategies [11, 12, 15], minimization of distortion expressions like that mentioned above for the di erentiation method [12] and the equi distortion property [18] A di erent approach, curve tting, was used in [16] But none are the same as those presented in this paper. In addition to being generally more accurate, ....

G.M. Roe, \Quantizing for minimum distortion," IEEE Trans. Inform. Thy., vol. IT-10, pp. 384-385, Oct. 1964. 18

....to 6R will vary with the source and quantizer being considered. The Panter Dite formula for #(R) can also be derived directly from Bennett s integral using variational methods, as did Lloyd (1957) 330] Smith (1957) 474] and, much later without apparent knowledge of earlier work, Roe (1964) [443]. It can also be derived without using variational methods by application of Holder s inequality to Bennett s integral [222] with the additional benefit of demonstrating that the claimed minimum is indeed global. Though not known at the time, it turns out that for a Gaussian source with ....

....error has long given it a central role. GRAY AND NEUHOFF: QUANTIZATION 29 Intuitively, the average squared error is the average energy or power in the quantization noise. The most common extension of distortion measures for scalars is the rth power distortion, d(x, y) x y r . For example, Roe [443] generalized Max s formulation to distortion measures of this form. Gish and Pierce [204] considered a more general distortion measure of the form d(x, y) L(x y) where L is a monotone increasing function of the magnitude of its argument and L(0) 0 with the added property that M(v) # 1 v Z ....

[Article contains additional citation context not shown here]

G. M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Theory, vol. 10, pp. 384--385, Oct. 1964.

....Therefore, the complexity of such a VQ is O(2 kR ) which increases exponentially in dimension and rate. In order to achieve performance close to that of optimal VQs under reasonable complexity constraints, one must consider structured vector quantizers. There are many types of structured VQs [9], such as product, multistage, tree structured, trellis, lattice, and hierarchical, many of which have been shown to achieve good performance with moderate complexity. Motivation for Lattice Based Quantization We now discuss several reasons for considering lattice based quantization, which has ....

....Lattices, and Groups, 2nd ed. New York: Springer Verlag, 1993. 7] T.R. Fischer, A pyramid vector quantizer, IEEE Trans. Inform. Theory, vol. IT 32, pp. 568 583, July 1986. 8] A. Gersho, Asymptotically optimal block quantization, IEEE Trans. Inform. Theory, vol. IT 25, pp. 373 380, July 1979. [9] A. Gersho and R. Gray, Vector Quantization and Signal Compression. New York: Springer, 1991. 10] J.D. Gibson and K. Sayood, Lattice quantization, Advances in Electronics and Electron Physics, vol. 72, pp. 259 330, 1988. 11] D.A. Huffman, A method for the construction of minimum redundancy ....

[Article contains additional citation context not shown here]

G.M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Thy., vol. IT-10, pp. 384-385, Oct. 1964.

....for N = 1 to 32. Algazi [9] gave an approximate expression for the MSE as function of the step size and numerically minimized it to find estimates of D N and D N . The values found for the Gaussian density via this approach resemble those given by Max. For symmetric source densities, Roe [10] presented (without derivation) a nonlinear equation whose numerical solution provided close approximations to D N for N = 4 and 36. Bucklew and Gallagher [11] obtained a similar equation for the Gaussian density. Apart from these, there exists an oft cited rule of thumb [c.f. 12, p. 125] for ....

....2 p(x)dx L 4 N 11 N Lp(L) 2 L 2 y i 2 p(x)dx S i =1 N , which is strictly positive for all L 0 and N 1. Therefore, D N (L) is a strictly convex function of L , and L N is unique for all N 1. The integral equation (51) is similar to those appearing in [7,10,11]. Solving it would yield the optimal support limit L N and, as a result, the optimal step size D N . Although the problem is quite 19 susceptible to machine computation, it is very difficult, if not impossible, to find closed form expressions for L N , due to the nonlinearity of (51) This is ....

G.M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Thy., pp. 384-385, IT10, Oct. 1964.

....of Electrical Engineering Technion and Hewlett Packard Laboratories Israel, Technion City, Haifa 32000, Israel. 1 Introduction Companding is a very simple and well known technique for non uniform scalar quantization using a high resolution uniform quantizer (A D converter) 1] 9] 11] [10], 6] The idea is to apply a memoryless non linearity (compressor) before the uniform quantizer and the inverse non linearlity (expander) after, so as to obtain an overall effect of non uniform quantization in a simple manner. Theoretically, it is known [5] 6] that for a given input ....

G. M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Theory, vol. IT--10, pp. 384--385.

....to 6R will vary with the source and quantizer being considered. The Panter Dite formula for ffi (R) can also be derived directly from Bennett s integral using variational methods, as did Lloyd (1957) 330] Smith (1957) 474] and, much later without apparent knowledge of earlier work, Roe (1964) [443]. It can also be derived without using variational methods by application of Holder s inequality to Bennett s integral [222] with the additional benefit of demonstrating that the claimed minimum is indeed global. Though not known at the time, it turns out that for a Gaussian source with ....

....long given it a central role. GRAY AND NEUHOFF: QUANTIZATION 29 Intuitively, the average squared error is the average energy or power in the quantization noise. The most common extension of distortion measures for scalars is the rth power distortion, d(x; y) jx Gamma yj r . For example, Roe [443] generalized Max s formulation to distortion measures of this form. Gish and Pierce [204] considered a more general distortion measure of the form d(x; y) L(x Gamma y) where L is a monotone increasing function of the magnitude of its argument and L(0) 0 with the added property that M (v) j ....

[Article contains additional citation context not shown here]

G. M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Theory, vol. 10, pp. 384--385, Oct. 1964.

....described in Part I, the key parameters are the endpoints of the support region of an optimal quantizer. Numerical results for the differentiation method demonstrate its accuracy. A number of the previous key parameter finding methods are based on similar approaches: optimal companding strategies [4, 5, 6], minimizing distortion expressions like that mentioned above for the differentiation method [5] and the equi distortion property [7] A different approach, curve fitting, was used in [8] But none are the same as those presented in this paper. In addition to being generally more accurate, the ....

G.M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Thy., vol. IT-10, pp. 384--385, Oct. 1964.

.... with variance oe 2 , then [146] ffi(R) 1 12 6 p 3oe 2 2 Gamma2R (9) ffi(R) can also be derived directly from Bennett s integral using variational methods (as did Lloyd (1957) 237] and Smith (1957) 334] and, much later without apparent knowledge of these early works, by Roe (1964) [311]) It can also be derived without using variational methods by application of Holder s inequality to Bennett s integral [168] demonstrating that the minimum is global. Though not known at the time, it turns out that for a Gaussian source with independent and identical (IID) samples, the ....

....tractability of squared error has long given it a central role. Intuitively, the average squared error is the average energy or power in the quantization noise. The most common extension of distortion measures for scalars is the rth power distortion, d(x; y) jx Gamma yj r . For example, Roe [311] generalized Max s formulation to distortion measures of this form. Gish and Pierce [155] considered a more general distortion measure of the form d(x; y) L(x Gamma y) where L is a monotone increasing function of the magnitude of its argument and L(0) 0 with the added property that M(v) j ....

[Article contains additional citation context not shown here]

G.M. Roe, "Quantizing for minimum distortion," IEEE Trans. on Information Theory, Vol. 10, pp. 384--385, 1964.

....formula [5] and in Bennett s integral [6, 8, 9] and, finally, they are useful in quantizer design. For one thing, accurate estimates of the optimal support region are needed when designing nearly optimal quantizers directly from the distribution function corresponding to the optimal point density [5, 9, 10, 11, 12, 13, 14] or, equivalently, when finding the asymptotically best compressor function for a compander implementation. Secondly, Lloyd Max style algorithms for designing optimal quantizers [9, 15] begin with an estimate of the support the better the estimate, the more rapidly the algorithm converges. For ....

G.M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Thy., vol. IT-10, pp. 384--385, Oct. 1964.

.... work on the optimal support of uniform scalar quantization, which is lattice quantization in one dimension, has included numerical optimization [3, 4] and curve fitting [5] 8] Nonlinear equations that can be solved numerically to give good approximations to aN and DN have also been developed [6, 9, 10]. The most comprehensive work to date is that of Hui and Neuhoff [11] who found asymptotic formulas for aN and DN for a large class of sources. For multidimensional lattice quantization, determination of aN has been performed largely by experimentation [12] 17] Jeong and Gibson [18] developed a ....

G.M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Thy., vol. IT10, pp. 384-385, Oct. 1964.

.... work on the optimal support of uniform scalar quantization, which is lattice quantization in one dimension, has included numerical optimization [3, 4] and curve fitting [5] 8] Nonlinear equations that can be solved numerically to give good approximations to aN and DN have also been developed [6, 9, 10]. The most comprehensive work to date is that of Hui and Neuhoff [11] who found asymptotic formulas for aN and DN for a large class of sources. For multidimensional lattice quantization, determination of aN has been performed largely by experimentation [12] 17] Jeong and Gibson [18] developed a ....

G.M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Thy., vol. IT10, pp. 384-385, Oct. 1964.

.... work on the optimal support of uniform scalar quantization, which is lattice quantization in one dimension, has included numerical optimization [3, 4] and curve fitting [5] 8] Nonlinear equations that can be solved numerically to give good approximations to aN and DN have also been developed [6, 9, 10]. The most comprehensive work to date is that of Hui and Neuhoff [11] who found asymptotic formulas for aN and DN for a large class of sources. For multidimensional lattice quantization, determination of aN has been performed largely by experimentation [12] 17] Jeong and Gibson [18] developed a ....

G.M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Thy., vol. IT10, pp. 384-385, Oct. 1964.

.... work on the optimal support of uniform scalar quantization, which is lattice quantization in one dimension, has included numerical optimization [3, 4] and curve fitting [5] 8] Nonlinear equations that can be solved numerically to give good approximations to aN and DN have also been developed [6, 9, 10]. The most comprehensive work to date is that of Hui and Neuhoff [11] who found asymptotic formulas for aN and DN for a large class of sources. For multidimensional lattice quantization, determination of aN has been performed largely by experimentation [12] 17] Jeong and Gibson [18] developed a ....

G.M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Thy., vol. IT10, pp. 384-385, Oct. 1964.

....values for the Laplacian and Gamma densities for N = 1 to 32. Algazi [8] gave an approximate formula for D N and numerically minimized it to find estimates of D N and D N . The values found for the Gaussian density via this approach resemble those given by Max. For symmetric source densities, Roe [9] presented (without derivation) a nonlinear equation whose numerical solution provided close approximations to D N for N = 4 and 36. Bucklew and Gallagher [10] obtained a similar equation for the Gaussian density. Apart from these, there exists an oft cited rule of thumb [c.f. 11, p. 125] for ....

....L 4 N 1 1 N L p(L) 2 L 2 y i 2 p(x)dx S i =1 N , which is strictly positive for all L 0 and N 1. Therefore, D N (L) is a strictly convex function of L , and L N is unique for all N 1. The integral equation (51) is similar to those appearing in [6,9,10]. Solving it would yield the optimal support length L N and, as a result, the optimal step size D N . Although the problem is quite susceptible to machine computation, it is very difficult, if not impossible, to find closed form expressions for L N , due to the nonlinearity of (51) This is ....

G.M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Thy., pp. 384-385, IT10, Oct. 1964.

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Roe, G. M., "Quantizing for Minimum Distortion," IEEE Trans. on Information Theory, (Correspondence), Vol. IT10, Oct. 1964, p. 384.

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G.M. Roe, "Quantizing for minimum distortion," IEEE Trans. on Inform. Theory, vol. IT-10, pp. 384-385, Oct. 1964. 27

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G.M. Roe, "Quantizing for minimum distortion," IEEE Trans. Inform. Theory, vol. IT-10, pp. 384- 385, Oct. 1964.

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