T. Linder and K. Zeger. Asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization. IEEE Trans. Information Theory, pp. 575-579, March 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....idea of dithering in lattice quantization. The idea has already been introduced by Ziv [13] as a means of universal quantization. Interesting results on the rate distortion efficiency of dithered lattice quantizers have already been obtained by Zamir and Feder [14] 17] and by Linder and Zeger [18]. In this paper our major concern is the analysis of the lattice quantization error for dithered and undithered cases. The only overlap between our work and the literature that we are aware of is Theorem 5. This was also reported by Zamir and Feder as a small part of their recent paper [16] Even ....

....moment of a lattice quantizer Q(P 0 ; V) denoted by oe D (P 0 ; V) is defined as D (P 0 ; V) T r(GD (P 0 ; V) 1 2=D kek de (4:5) where GD (P 0 ; V) is as in (2. 6) The quantity oe D (P 0 ; V) also comes out of high bit rate analysis of lattice quantizers [19] 25] [18]. It is proven in [18] that for an undithered lattice quantizer, as the unit volume, jdetV j of a quantizer Q(P 0 ; V) goes to 0, the normalized mean square error approaches the limit oe D (P 0 ; V) The name dimensionless second moment is used in [19] The following fact is on the performance ....

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T. Linder and K. Zeger, "Asymptotic entropy-constrained performance of tessellating and universal randomized lattice quantization", IEEE Trans. Information Theory, vol. IT-40, pp575-579, Mar. 1994.

....through rotation, reflection, and translation. 3 When a tessellation is found that consists of bodies with a low G , the codebook is formed as the intersection of the centroids and D . The desired rate determines the scaling of the tessellation. The structure is called a tessellating quantizer [8]. In previous studies of tessellating quantizers, most attention has been devoted to lattice quantizers, which constitute an important subset of all tessellation quantizers. Lattices are defined in the next section. C. Lattices A lattice is a popular special case of a tessellation. 4 The ....

....if Gersho s conjecture (see Section I B) is true, then a tessellating quantizer is asymptotically optimal when the rate tends to infinity. The optimality does not require the source density to be uniform or even smooth, only that the differential entropy is finite, as proved by Linder and Zeger [8]. It is worth mentioning that a tessellating quantizer with entropy coding performs closer to the ratedistortion bound than the optimal fixed rate vector quantizer. The argument behind this statement is the following: The optimal fixedrate vector quantizer is inferior to (has higher average rate ....

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T. Linder and K. Zeger, "Asymptotic entropy-constrained performance of tessellating and universal randomized lattice quantization," IEEE Trans. Inform. Theory, vol. 40, no. 2, pp. 575--579, Mar. 1994.

....is like Bennett s integral in that f (1) x) and consequently #(x) can be arbitrary. On the other hand, it is like Zador s result (or Gersho s generalization of Bennett s integral [193] in that, in essence, it is assumed that the quantizers have optimal cell shapes. In 1994 Linder and Zeger [326] rigorously derived the asymptotic distortion of quantizers generated by tessellations by showing that the quantizer q # formed by tessellating with some basic cell shape S scaled by a positive number # has average (narrow sense) rth power distortion D# satisfying lim ##0 D# # r vol(S) r k ....

T. Linder and K. Zeger, "Asymptotic Entropy-Constrained Performance of Tessellating and Universal Randomized Lattice Quantization," IEEE Trans. Inform. Theory, vol. 40, pp. 575-- 579, Mar. 1994.

....support limit grows, the fact (7) that D N decreases to zero is sufficient to guarantee that granular distortion D N gran is asymptotically well approximated by D N 2 12. This is demonstrated by the following theorem, whose proof, given in Section V, is based on a result of Linder and Zeger [17] that implies that for uniform scalar quantizers with infinitely many levels, D D 2 1 12 as D0 . Theorem 6: For any source density whatsoever lim N D N gran D N 2 12 = 1. It follows that the asymptotic behavior of D N gran derives directly from that of D N , which may be found from ....

....lim N L N V 1 1 6N 2 ( lim N V 1 VL N ( V 1 1 6N 2 ( 1. which demonstrates (17) of Theorem 5. 23 D. Proof of Theorem 6 Let Q D N denote a uniform scalar quantizer with N levels and step size D . We use the following lemma, which is a special case of Lemma 1 of [17] regarding the MSE of a uniform scalar quantizer Q D with step size D and infinitely many levels. Lemma 21: For any source density p(x) lim D0 x Q D (x) 2 p(x)dx D 2 = 1 12 . To prove Theorem 6, we first note that D N gran D N 2 = x Q D N N (x) 2 p(x)dx L ....

T. Linder and K. Zeger, "Asymptotic entropy-constrained performance of tessellating and universal randomized lattice quantization," IEEE Trans. Inform. Thy., 40, pp. 575-579, Mar. 1994.

....is like Bennett s integral in that f (1) x) and consequently (x) can be arbitrary. On the other hand, it is like Zador s result (or Gersho s generalization of Bennett s integral [193] in that, in essence, it is assumed that the quantizers have optimal cell shapes. In 1994 Linder and Zeger [326] rigorously derived the asymptotic distortion of quantizers generated by tessellations by showing that the quantizer q ff formed by tessellating with some basic cell shape S scaled by a positive number ff has average (narrow sense) rth power distortion D ff satisfying lim ff 0 D ff ff r ....

T. Linder and K. Zeger, "Asymptotic Entropy-Constrained Performance of Tessellating and Universal Randomized Lattice Quantization," IEEE Trans. Inform. Theory, vol. 40, pp. 575-- 579, Mar. 1994.

....uniformly distributed on the unit square, the mean squared error of any N point quantizer is bounded from below by M(hexagon) N . This result was independently rederived in a simpler fashion by Newman (1964) 274] The lower bound is asymptotically achievable by a lattice with hexagonal cells; see [231] for a rigorous proof. It follows then that the ratio of ffi 2 (R) to M(hexagon)oe 2 2 Gamma2R tends to one, and also, that Gersho s conjecture holds for dimension two. Zador s thesis (1963) 389] was the next rigorous work. As mentioned earlier, it contains two principal results. For ....

....f (1) x) This is like Bennett s integral in that f (1) x) and consequently (x) can be arbitrary, but like Zador s result (or Gersho s generalization of Bennett s integral [147] in that, in essence, it is assumed that the quantizers have optimal cell shapes. 50 In 1994 Linder and Zeger [231] generalized (36) to quantizers generated by tesselations by showing that the quantizer q ff based formed by tesselating with some basic cell shape S scaled by a postive number ff, has average (narrow sense) rth power distortion D ff satisfying lim ff 0 D ff ff r vol(S) r=k M(S) 1 They ....

T. Linder and K. Zeger, "Asymptotic Entropy-Constrained Performance of Tesselating and Universal Randomized Lattice Quantization, " IEEE Trans. Information Theory, Vol. 40, pp. 575-579, Mar. 1994.

....is subtracted to obtain the reconstructions and . Equation (68) shows that and satisfy the desired distortions, while the entropies in (70) characterize the region of admissible rates. However, since and are not Gaussian, this rate region does not coincide with . Nevertheless, it follows from [13] and [31] that (71) for any satisfying the conditions of Theorem 2, where and are the Gaussian variables associated with , is the normalized second moment of and (assuming both have the same structure) and as . Thus at high resolution, multiterminalECDQ has a combined rate redundancy of bits per ....

T. Linder and K. Zeger, "Asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization," IEEE Trans. Inform. Theory, vol. 40, pp. 575--579, Mar. 1994.

....is in the sense that first, b X Gamma X is independent of X and is distributed as N , and second, H(Q 1 jZ) I(X; X N) 2) where I( Delta; Delta) denotes mutual information 1 . The first property asserts that the distortion is E( b X Gamma X) 2 = EN 2 = D. As shown in [26] 24] and [15], for small D and smooth sources the coding rate (2) is about 1 2 log 2 e=12 0:254 bits higher than the rate distortion function of the source, defined as [2] R(D) inf fU :E(U GammaX ) 2 Dg I(X; U) 3) Furthermore, for every D and all sources H(Q 1 jZ) Gamma R(D) 1 2 log 4 e ....

T. Linder and K. Zeger. Asymptotic entropy constrained performance of tesselating and universal randomized lattice quantization. IEEE Trans. Information Theory, IT-40:575--579, March 1994.

....result, 24) is obtained by replacing the Nyquist sampled processes with their equivalent band limited processes, and changing XB to X, as in (C.1) c) D. Low distortion behavior of r(D) The term r(D) which appears e.g. in Theorem 4 is analogous to the resolution measure defined in [7] In [24] it was shown that r(D) vanishes, as D 0, for vector sources with a finite differential entropy. In this appendix we show a similar result for discrete time processes with a finite entropy rate, and in some cases we even characterize the convergence rate. Let us first recall the following lemma ....

T. Linder and K. Zeger. Asymptotic entropy constrained performance of tesselating and universal randomized lattice quantization. IEEE Trans. Information Theory, IT-40:575--579, March 1994.

....suggested by Ziv [6] in a different context. Interesting results on the rate distortion efficiency of y Work supported in parts by NSF grant MIP 92 15785, Tektronix, Inc. and Rockwell International. dithered lattice quantizers have been obtained by Zamir and Feder [7] and by Linder and Zeger [8]. Our emphasis in this paper is on the statistical relationship between the input and the error vectors of a dithered lattice quantizer. 2 Definitions and Preliminaries Let R D and Z D denote the D dimensional Euclidean space of real numbers and the D dimensional space of integers ....

T. Linder and K. Zeger, "Asymptotic entropyconstrained performance of tessellating and universal randomized lattice quantization", IEEE Trans. Inform. Theory, vol. IT-40, pp575-579, Mar. 1994.

....idea of dithering in lattice quantization. The idea has already been introduced by Ziv [13] as a means of universal quantization. Interesting results on the rate distortion efficiency of dithered lattice quantizers have already been obtained by Zamir and Feder [14] 17] and by Linder and Zeger [18]. In this paper our major concern is the analysis of the lattice quantization error for dithered and undithered cases. The only overlap between our work and the literature that we are aware of is Theorem 5. This was also reported by Zamir and Feder as a small part of their recent paper [16] Even ....

....; V) denoted by oe 2 D (P 0 ; V) is defined as oe 2 D (P 0 ; V) 1 DjdetV j 2=D T r(GD (P 0 ; V) 1 DjdetVj 1 2=D Z P0 kek 2 de (4:5) where GD (P 0 ; V) is as in (2. 6) The quantity oe 2 D (P 0 ; V) also comes out of high bit rate analysis of lattice quantizers [19] 25] [18]. It is proven in [18] that for an undithered lattice quantizer, as the unit volume, jdetV j of a quantizer Q(P 0 ; V) goes to 0, the normalized mean square error approaches the limit oe 2 D (P 0 ; V) The name dimensionless second moment is used in [19] The following fact is on the performance ....

[Article contains additional citation context not shown here]

T. Linder and K. Zeger, "Asymptotic entropy-constrained performance of tessellating and universal randomized lattice quantization", IEEE Trans. Information Theory, vol. IT-40, pp575-579, Mar. 1994.

....support length grows, the fact (7) that D N decreases to zero is sufficient to guarantee that granular distortion D N gran is asymptotically well approximated by D N 2 12 . This is demonstrated by the following theorem, whose proof, given in Section V, is based on a result of Linder and Zeger [16] that implies that for uniform scalar quantizers with infinitely many levels, D D 2 1 12 as D 0 . Theorem 6: For any source density whatsoever lim N D N gran D N 2 12 = 1. It follows that the asymptotic behavior of D N gran derives directly from that of D N , which may be ....

....L N V 1 1 6N 2 ( lim N V 1 V L N ( V 1 1 6N 2 ( 1. which demonstrates (17) of Theorem 5. D. Proof of Theorem 6 Let Q D N denote a uniform scalar quantizer with N levels and step size D . We use the following lemma, which is a special case of Theorem 1 of [16] regarding the MSE of a uniform scalar quantizer Q D with step size D and infinitely many levels. 24 Lemma 21: For any source density p(x) lim D 0 x Q D (x) 2 p(x)dx D 2 = 1 12 . To prove Theorem 6, we first note that D N gran D N 2 = x Q D N N (x) 2 ....

T. Linder and K. Zeger, "Asymptotic entropy-constrained performance of tessellating and universal randomized lattice quantization," IEEE Trans. Inform. Thy., 40, pp. 575-579, Mar.

....information divergence, and shows that this distance vanishes asymptotically for large (optimal) lattice dimension. A slight generalization of the lattice quantizer is the tessellating quantizer, in which the basic cell P 0 may be rotated, and not only translated, to get the i th cell [8] [14]. For example, an equilateral triangle cell generates a tessellating quantizer which is not a lattice quantizer. Despite their slight generality, tessellating quantizers are not considered in this work since their resulting noise can not be modeled as additive. II Lattice Quantization Noise ....

T. Linder and K. Zeger. Asymptotic entropy constrained performance of tesselating and universal randomized lattice quantization. IEEE Trans. Information Theory, IT-40:575--579, March 1994.

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T. Linder and K. Zeger, "Asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization," IEEE Trans. Inform. Theory, vol. 40, pp 575--579, Mar. 1994.

....b(2, 1) 1 12. In the scalar case it can be argued that a sequence of increasing rate uniform quantizers followed by optimum entropy coders will be asymptotically optimal if the assumptions of Theorem 1, which are more general than those of Gish and Pierce, are met. This follows from a result of [28] which shows that under assumptions equivalent to those of Theorem 1 a sequence of k dimensional lattice vector quantizers QR will have asymptotic (large entropy R) distortion of D(QR ) G(#)2 (2 k) h(f) R) whereG(#) is the normalized moment of inertia of the basic Voronoi cell of the base ....

....raster image intensities or speech samples produced by physical sensors, then f g will be bounded due to the finite dynamic range of real sensors. The special role of uniform scalar quantizers provides an illustration. If both f and g meet the requirements of Theorem 1, then the results of [28] imply that a sequence of uniform quantizers with an optimal entropy code for g will yield the same average squared error when applied to f with a rate increase of H f (S Q ) where SQ is the partition corresponding to the uniform quantizer. If the relative entropy I(f is finite, then H ....

T. Linder and K. Zeger, "Asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization," IEEE Trans. Inform. Theory, vol. 40, pp 575--579, Mar. 1994.

....entropy coded scalar quantization is uniform. More generally, the entropy of a lattice quantizer that encodes a smooth dimensional vector source with squared distortion is given for small by [3] 2) where denotes the normalized second moment of the basic cell of the lattice (see also [13] and [14]) The above implies (by means of the Shannon lower bound [15] that the asymptotic rate redundancy of an entropy coded lattice quantizer above the rate distortion function is bits per dimension [2] 4] In this paper, we will show that analogous results hold for locally quadratic non difference ....

....then is the same positive constant for all and thus the companding scheme is not asymptotically optimal, contrary to the case of an invertible . V. PROOFS Proof of Proposition 2: The entropy of is equal to the entropy of the lattice quantizer output since is invertible. It was proved in [14] using a result of Csisz ar [33] that if a random vector is lattice quantized by the scaled lattice quantizer , and has a density and finite differential entropy , then the quantizer s entropy is given asymptotically by provided is finite for some . Thus by setting and using the identity valid ....

T. Linder and K. Zeger, "Asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization," IEEE Trans. Inform. Theory, vol. 40, pp. 575--579, Mar. 1994.

.... the entropy of a lattice quantizer Q that encodes a smooth k dimensional vector source X with squared distortion D is given for small D by [3] H(Q(X) h(X) Gamma k 2 log(D= kL(P 0 ) 2) where L(P 0 ) denotes the normalized second moment of the basic cell of the lattice (see also [13] and [14]) The above implies (by means of the Shannon lower bound [15] that the asymptotic rate redundancy of an entropy coded lattice quantizer above the rate distortion function is 1 2 log(2 eL(P 0 ) bits per dimension [2, 4] In this paper we will show that analogous results hold for locally ....

....all k and thus the companding scheme is not asymptotically optimal, contrary to the case of an invertible r(x) 5 Proofs Proof of Proposition 2 The entropy of Q ff;F (X) G(Q ff (F (X) is equal to the entropy of the lattice quantizer output Q ff (F (X) since G is invertible. It was proved in [14] using a result of Csisz ar [33] that if a random vector Y is lattice quantized by the scaled lattice quantizer Q ff , and Y has a density and finite differential entropy h(Y ) then the quantizer s entropy is given asymptotically by lim ff 0 [ H(Q ff (Y ) k log ff ] h(Y ) Gamma log V (P 0 ....

T. Linder and K. Zeger, "Asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization," IEEE Trans. Inform. Theory, vol. 40, pp. 575--579, Mar. 1994.

....denotes base 2 logarithm, and means that the difference between the corresponding quantities goes to zero as D 0. More generally, the entropy of a lattice quantizer Q n that encodes a smooth source X 2 R n with MSE level D (so that the per dimension distortion is D=n) is given for small D by [5, 6, 7] H(Q n (X) h(X) Gamma n 2 log(D= nG n ) 4) where G n denotes the normalized second moment of the lattice. Also, Shannon s ratedistortion function [3] R(D) inf fI(X; Y ) E[d(X; Y ) Dg ; 5) characterizing the minimum achievable rate at distortion level D by any (multi dimensional) ....

....noise channel (see [4, 8] X Y = X N; where for the MSE case N is Gaussian with variance D. In this regard it was demonstrated in [6, 18] that entropy coded randomized (dithered) uniform lattice quantization (ECDQ) simulates (in the rate distortion sense) an additive noise test channel. In [7] it was shown that at high resolution the randomization of ECDQ is not necessary, and its redundancy above the rate distortion function is asymptotically 1 2 log(2 eG n ) bit per dimension. For a non difference distortion measure, the additive noise N in the the asymptotically optimal test ....

T. Linder and K. Zeger, "Asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization," IEEE Trans. Inform. Theory, vol. 40, pp. 575--579, Mar. 1994.

....uniform. More generally, the entropy of a lattice quantizer Q that encodes a smooth k dimensional vector source X with squared distortion D is given for small D by [3] H(Q(X) h(X) Gamma k 2 log(D= nG k ) 2) where G k denotes the normalized second moment of the lattice (see also [13] and [14]) The above implies (by means of the Shannon lower bound [15] that the asymptotic rate redundancy of an entropy coded lattice quantizer above the rate distortion function is 1 2 log(2 eG n ) bit per dimension [2, 4] In this paper we will show that analogous results hold for locally quadratic ....

....all k and thus the companding scheme is not asymptotically optimal, contrary to the case of an invertible r(x) 5 Proofs Proof of Proposition 2 The entropy of Q ff;F (X) G(Q ff (F (X) is equal to the entropy of the lattice quantizer output Q ff (F (X) since G is invertible. It was proved in [14] using a result of Csisz ar [33] that if a random vector Y is lattice quantized by the scaled lattice quantizer Q ff , and Y has a density and finite differential entropy h(Y ) then the quantizer s entropy is given asymptotically by lim ff 0 [ H(Q ff (Y ) k log ff ] h(Y ) Gamma log V (P 0 ....

T. Linder and K. Zeger, "Asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization," IEEE Trans. Inform. Theory, vol. 40, pp. 575--579, Mar. 1994.

....Their only condition on the distribution of the source vector X was that EkXk r ffl 1 for some ffl 0. Zamir and Feder [18] determined the precise asymptotics of the entropy rate of randomized lattice quantizers for all sources with densities. With the same conditions Linder and Zeger [12] gave the exact asymptotics of the entropy rate of tessellating vector quantizers of small rth power distortion. The Shannon lower bound is an extremely useful tool to relate these asymptotic distortion and rate formulas to the theoretical rate distortion limits. However, the existing results are ....

T. Linder, , and K. Zeger. Asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization. to appear in IEEE Trans. Inform. Theory, 1994.

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T. Linder and K. Zeger. Asymptotic entropy constrained performance of tessellating and universal randomized lattice quantization. IEEE Trans. Information Theory, pp. 575-579, March 1994.

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