V.R. Algazi, \Useful approximations to optimum quantization," IEEE Trans. Commun. Tech., vol. COM-14, pp. 297-301, June 1966.  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, ....

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V.R. Algazi, \Useful approximations to optimum quantization," IEEE Trans. Commun. Tech., vol. COM-14, pp. 297-301, June 1966.

.... QUANTIZATION 11 approximations and the approximation of r(D)or#(R) and the characterizations of properties of optimal high resolution quantization for both fixed and variable rate quantization for squared error and other error moments appeared during the 1960 s, e.g. 497] 498] 55] 467] [8]. An excellent summary of the early work is contained in a 1970 paper by Elias [143] We close this section with an important practical observation. The current JPEG and related standards can be viewed as a combination of transform coding and variablelength quantization. It is worth pointing out ....

....by using a Taylor series expansion of the source density. For example, Lloyd [330] used this approach to show that, ignoring overload distortion, the approximation error in the PanterDite formula is o(1 N 2 ) which means that it tends to zero, even when multiplied by N 2 . Roe [443] Algazi [8] and Wood [539] also used Taylor series. Overload distortion was first explicitly considered in the work of Shtein (1959) 471] who optimized the cell size of uniform scalar quantization using an explicit formula for the overload distortion (as well as # 2 12 for the granular distortion) and ....

V. R. Algazi, "Useful approximation to optimum quantization," IEEE Trans. Comm., vol. 14, pp. 297--301, June 1966.

....classes of VQs. This dissertation focuses on the distortion rate performance of a particular class of vector quantizers, namely lattice based quantization. Lattice quantization has at its core an infinite lattice , which is a countably infinite set of points in k that is closed under addition [6, 10]. A lattice quantizer codebook is given by R , where R k , is the quantizer support. Figures 1.1 and 1.2 show examples of lattice quantizers with hexagonal support. Note that R is bounded for fixed rate quantization but may be unbounded for variable rate quantization. The complement R C ....

....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 codes, Proc. IRE, vol. 40, pp. 1098 1101, Sept. 1952. 12] Y. Linde, A. Buzo, and R.M. Gray, An ....

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V.R. Algazi, "Useful approximations to optimum quantization," IEEE Trans. Commun., vol. COM-14, pp. 297-301, Jun. 1966.

....find their values for a specific N via simple numerical algorithms. This was first done by Max [7] who published the values of D N and D N for the Gaussian density for N = 1 to 36. Similarly, Paez and Glisson [8] published such values for the Laplacian and Gamma densities 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) ....

V.R. Algazi, "Useful approximations to optimum quantization," IEEE Trans. Commun., COM14, pp. 297-301, June 1966.

....is with Department 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 ....

V. R. Algazi, "Useful approximations to optimum quantization," IEEE Trans. Commun. Tech., COM--14, pp. 297--301, 1966.

.... 11 approximations and the approximation of r(D) or ffi (R) and the characterizations of properties of optimal high resolution quantization for both fixed and variable rate quantization for squared error and other error moments appeared during the 1960 s, e.g. 497] 498] 55] 467] [8]. An excellent summary of the early work is contained in a 1970 paper by Elias [143] We close this section with an important practical observation. The current JPEG and related standards can be viewed as a combination of transform coding and variablelength quantization. It is worth pointing out ....

....by using a Taylor series expansion of the source density. For example, Lloyd [330] used this approach to show that, ignoring overload distortion, the approximation error in the PanterDite formula is o(1=N 2 ) which means that it tends to zero, even when multiplied by N 2 . Roe [443] Algazi [8] and Wood [539] also used Taylor series. Overload distortion was first explicitly considered in the work of Shtein (1959) 471] who optimized the cell size of uniform scalar quantization using an explicit formula for the overload distortion (as well as Delta 2 =12 for the granular distortion) ....

V. R. Algazi, "Useful approximation to optimum quantization," IEEE Trans. Comm., vol. 14, pp. 297--301, June 1966.

....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 ....

....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 expressions given in Section 2 are closed form, whereas the most accurate of previous methods are ....

[Article contains additional citation context not shown here]

V.R. Algazi, "Useful approximations to optimum quantization," IEEE Trans. Commun. Tech., vol. COM-14, pp. 297--301, June 1966.

.... of the Bennett style asymptotic approximations and the approximation of r(D) or ffi(R) and the characterizations of properties of optimal high resolution quantization for both fixed and variable rate quantization for squared error and other error moments appeared during the 1960s, e.g. [350, 351, 45, 330, 6]. An excellent summary of the early work is contained in a 1970 paper by Elias [110] We now momentarily leave the discussion of variable rate scalar quantization to discuss one of the first vector quantizers since this early example provided the vehicle for the development of optimal ....

....by using a Taylor series expansion of the source density. For example, Lloyd [237] used this approach to show that, ignoring overload distortion, the approximation error in the Panter Dite formula is o(1=N 2 ) which means that it tends to zero, even when divided by 1=N 2 . Roe [311] Algazi [6] and Wood [379] also used Taylor series. Overload distortion was first explcitly considered in the work of Shteyn (1959) 333] who optimized the cell size of uniform scalar quantization using an explicit formula for the overload distortion (as well as Delta 2 =12 for the graunular distortion) ....

V.R. Algazi, "Useful approximation to optimum quantization," IEEE Trans. Comm. Tech., Vol. 14, pp. 297--301, 1966.

....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 ....

V.R. Algazi, "Useful approximations to optimum quantization," IEEE Trans. Commun. Tech., vol. COM-14, pp. 297--301, June 1966.

.... 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 ....

V.R. Algazi, "Useful approximations to optimum quantization," IEEE Trans. Commun. , vol. COM-14, pp. 297-301, Jun. 1966.

....here is slightly simpler than the general one described by Bruce. If the decisions levels x[k] are known, the y[k] can be derived by the centroid formula given above. Therefore, to find a K interval quantization of a probability density function p, all we need do is find the K 1 decision levels x[1], x[2] x[K 1] The Algorithm: We will consider one decision level at a time, advancing from left to right (small k to large) accumulating information about the quality of different placements of x[k] Given a placement of x[k] from 0 to 254, we want to find the location of x[k 1] which ....

.... one decision level at a time, advancing from left to right (small k to large) accumulating information about the quality of different placements of x[k] Given a placement of x[k] from 0 to 254, we want to find the location of x[k 1] which minimizes the distortion so far : the sum D[0] D[1] D[2] D[k 1] This is facilitated by accumulating these partial sums of left distortions in a table. We generate a two dimensional table containing, for all k and x , the minimum left distortion possible, given that x[k] x . In addition, we ciq thesis at almond.srv.cs.cmu.edu Page ....

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Algazi, V. R., "Useful Approximations to Optimum Quantization," IEEE Trans. on Communication Technology, Vol. COM-14, No. 3, June 1966, p. 297.

.... 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 ....

V.R. Algazi, "Useful approximations to optimum quantization," IEEE Trans. Commun. , vol. COM-14, pp. 297-301, Jun. 1966.

.... 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 ....

V.R. Algazi, "Useful approximations to optimum quantization," IEEE Trans. Commun. , vol. COM-14, pp. 297-301, Jun. 1966.

.... 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 ....

V.R. Algazi, "Useful approximations to optimum quantization," IEEE Trans. Commun. , vol. COM-14, pp. 297-301, Jun. 1966.

....the definition of l(x) The reason is that the point density is generally intended as a high level model for what the quantizer does or should do. For example, the point density that minimizes expected q th power distortion 4 is l(x) p X (x) 1 (1 q) p X (x) 1 (1 q) dx [14,18]. Let us also note that when using the notion of a point density, there is a tacit assumption that N is large and neighboring cells have similar widths. The main result of this section is that when the quantizer has many points the normalized quantization error V = D N(X Q(X) 2.2) has ....

....of a quantizer is E X Q(X) q = 1 N q E V q = 2 N q 0 v q 1 2 l(x) v p X (x) l(x) dx dv = 1 N q 1 2 q (q 1) p X (x) l(x) q dx . The latter expression is Bennett s integral for scalar quantizers [4] as generalized by Algazi [18] to qth power distortion, giving an asymptotic formula for the distortion of a scalar quantizer with a given point density and with points centered in their cells. 5) p V (v) has the form of a plateau with sloping sides, centered at the origin. That is, p V (v) p(x) l(x) dx ....

V.R. Algazi, "Useful approximations to optimum quantization," IEEE Trans. Commun., vol. COM-14, pp. 297-301, June 1966.

....find their values for a specific N via simple numerical algorithms. This was first done by Max [6] who published the values of D N and D N for the Gaussian density for N = 1 to 36. Similarly, Paez and Glisson [7] published such 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 ....

V.R. Algazi, "Useful approximations to optimum quantization," IEEE Trans. Commun., COM14, pp. 297-301, June 1966.

.... particularly, scalar, product and transform quantizers. Section III presents a rigorous formulation of Bennett s integral, and Section IV presents the proof, with certain details left to the Appendix. For scalar quantizers, Bennett s integral was extended to rth power distortion by Algazi [6]. It was extended to vector quantizers with congruent cells (as in a lattice quantizer) by Gersho [5] Yamada et al. 7] gave a Bennett like lower bound to distortion that applies to all vector quantizers and difference distortion measures. Bucklew [8,9] extended Bennett s integral to companders ....

V.R. Algazi, "Useful approximations to optimum quantization," IEEE Trans. Comm., vol. COM-14, pp. 297-301, June 1966.

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V.R. Algazi, "Useful approximations to optimum quantization, "IEEE Trans. Commun., vol. COM-14, pp. 297-301, June 1966.

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