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Achieving Rate-Optimized Bit Allocation In Video Codecs - Qianfu Jiang And (Correct)
....function of oe e . We need to make certain approximation to obtain a simpler formulation. 1) High rate case If the quantization step size Q is small so that the quantized output is a close approximation of the original input e, the quantizer output entropy can be approximated as below [5]: HQ = h(e) 0 log Q (10) where h(e) is the differential entropy of the input. For a Laplacian random variable e with variance oe , h(e) log 2eoe) 11) Then HQ = log 2eoe (12) 2) Low rate case In this case the quantization step size Q is large. The high rate approximation of ....
R.M. Gray, "Source Coding Theory", Kluwer Academic Publishers, 1990.
Parameter Estimation for Multi-dimensional Filtered Poisson.. - Hero (2000) (Correct)
....where d max is the the a priori variance of V in the scalar case, and the sum of the components variances in the vector case. Thus, de ning the inverse R 1 ( we have the lower bound d = MSE minfd max ; R 1 (C)g (6) The function R 1 (C) is often called the distortion rate function [11]. As previously mentioned, the right hand side of this lower bound is often dicult to compute and this bound is seldom useful. When there exists a natural decomposition of the channel C into a cascade of two independent channels C 1 SNYDER SYMPOSIUM, 2000 7 and C 2 a general and computable MSE ....
R. M. Gray, Source Coding Theory, Kluwer Academic, Norwell MA, 1990.
On Optimal Entropy-Constrained Scalar Quantization - György (2000) (Correct)
....of identifying, for a xed n, the quantizers which have minimal distortion in this family, and then nding the optimal choice of n. Proof of Theorem 3. 1 Without loss of generality we will assume that (0) 0 (otherwise we can replace (x) by b (x) x) 0) Let be the Gish Pierce function [10, 11] de ned by (p) 8 : p) p if p 0 0 if p = 0 where (p) 2 R p=2 0 (x) dx. Notice that (p) Ef (pY )g for all p 0, where Y is a random variable that is uniformly distributed over the interval (0; 1=2) Then the strict convexity of (e t ) implies that for all t 1 ....
R. M. Gray, Source Coding Theory. Boston: Kluwer, 1990.
Computationally Inspired Biotechnologies: Improved DNA.. - Reif, LaBean (2000) (1 citation) (Correct)
....a universal base pairing nucleotide like inosine to generate diverse populations of prefixes. 3 Adapting to Biotechnology VQ Methods Used in Computer Science 3. 1 VQ Coding Methods Used in Computer Science We next consider information theoretic Vector Quantization (VQ) Coding methods (see Gray [G90] Gersho, Gallager, and Gray [GGG91] used in computer science for compressing data (such as speech and images) within bounded error. Again let V = B n be the set of all possible n vectors over domain B of consecutive integers, and consider a database of vectors in V . VQ methods (which are also ....
....) Note that in contrast to Error Correcting codes, the VQ coding induces errors, which are bounded by the choice of the clusters and can be tuned by setting the parameter m. For certain statistical source models for the data, for example memoryless or finite state stationary processes (see Gray [G90] p 44) the resulting datarate distortion of VQ coding has been shown to be asymptotically optimal. However, natural data sources such as speech, images, and natural DNA can not be well modeled as memoryless or finite state stationary processes (this seems to be related to the fact that speech ....
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Gray, R. M., "Source Coding Theory", Klewer Academic Publishers, Boston, (1990).
Multiple Description Vector Quantization with Lattice .. - Vaishampayan.. (2001) (7 citations) (Correct)
....quantizer bin has identical volume and that p L (x) is approximately constant over Voronoi regions of the sublattice V ( 0 ) The second assumption is valid in the limit as the Voronoi regions become small and is standard in asymptotic quantization theory. The rate R 0 = H(Q(X) is given by [18] R 0 = Gamma(1=L) X Z V ( p L (x)dx log 2 Z V ( p L (x)dx Gamma(1=L) X Z V ( p L (x)dx log 2 p L ( h(p) Gamma (1=L)log 2 ( 16) For R, we evaluate the entropy H(ff 1 (Q(X) and then use the approximation that p L (x) is roughly constant over each Voronoi region of 0 ....
R. M. Gray, "Source Coding Theory", Kluwer Academic Publishers, 1990.
Optimal Entropy-Constrained Scalar Quantization of a Uniform.. - György, Linder (2000) (Correct)
....N . This fact suggests that an ECSQ which achieves the lower convex hull of D h (R) is the exception rather than the rule. 4 Proofs Proof of Theorem 1 Without loss of generality we will assume that (0) 0 (otherwise we can replace (x) by b (x) x) 0) Let be the Gish Pierce function [9, 10] de ned by (p) 8 : p) p if p 0 0 if p = 0 where (p) 2 R p=2 0 (x) dx. Notice that (p) E[ pY ) for all p 0, where Y is a random variable that is uniformly distributed over the interval (0; 1=2) Then the strict convexity of 7 (e t ) implies that for all t 1 ; t 2 ....
R. M. Gray, Source Coding Theory. Boston: Kluwer, 1990.
Signal Processing for Everyone - Strang (Correct)
....the center of the bin for the reconstruction step. A vector quantization has M dimensional bins for packets of M coefficients at a time. This transform coding is of critical importance to the whole compression process, It is a highly developed form of roundoff, and we mention two basic references [6, 8]. I believe that quantization should be studied and applied in a wide range of numerical analysis and scientific computing. The combination of linear transform and nonlinear compression is fundamental. The transform is a change to a better basis a more efficient representation of the signal. ....
R.M. Gray, Source Coding Theory, Kluwer, 1990.
Optimal Entropy-Constrained Scalar Quantization of a Uniform.. - György, Linder (2000) (Correct)
....suggests that an ECSQ which achieves the lower convex hull of D h (R) is the exception rather than the rule. 4 Proofs Proof of Theorem 1 Without loss of generality we will assume that ae(0) 0 (otherwise we can replace ae(x) by b ae(x) ae(x) Gamma ae(0) Let Psi be the Gish Pierce function [9, 10] defined by Psi(p) 8 : Phi(p) p if p 0 0 if p = 0 where Phi(p) 2 R p=2 0 ae(x) dx. Notice that Psi(p) E[ae(pY ) for all p 0, where Y is a random variable that is uniformly distributed over the interval (0; 1=2) Then the strict convexity of 7 ae(e t ) implies that ....
R. M. Gray, Source Coding Theory. Boston: Kluwer, 1990.
Rate-Distortion Performance of Source Coders in the Low.. - Foodeei, Dubois (1993) (1 citation) (Correct)
....the Shannon Lower Bound RSLB(D) These results were based on asymptotic approximations. Zador [11] Gersho [12] and others studied the extension of these results to block quantization (ECVQ) and obtained formulations for the asymptotic quantization distortion D with various degrees of generality [13]. The Zador study provides the bounds for asymptotic ECVQ bounds. Gersho conjecture is that optimum high resolution ECVQ has the form of a lattice. Numerical methods were devised for the cases where high resolution approximation was not used [14] 15] 16] 17] The last 3 references used a ....
R. M. Gray, Source Coding Theory. Boston: Kluwer Academic Press, 1990.
The Other Asymptotic Theory Of Lossy Source Coding - Neuhoff (Correct)
....upon in his 1959 paper [2] and generalized by others [3, 4] achieves the first goal by showing that for stationary, ergodic sources (satisfying a moment finiteness condition) ffi (R) D(R) where D(R) is the information theoretic distortion rate function (DRF) 3 , defined as follows (c.f. [3, 5, 6]) D(R) Delta = lim k 1 D k (R) where D k (R) Delta = inf q2Qk (R) 1 k EkX Gamma Y k 2 ; where Y = Y 1 : Y k ) is the output of the test channel q with X as the input, where Q k (R) Delta = Phi q(yjx) k Gamma1 I q (X; Y ) R Psi , is a collection of ....
....log 2 (p k (x)q(yjx) p(y) dx is the information between input and output of the test channel q. In other words, rate distortion theory relates the operational quantity ffi (R) that we wish to know to the information theoretic quantity D(R) The latter can be evaluated analytically (c.f. [5, 6]) or numerically [7] It is well known that, except for degenerate cases, ffi k (R) D k (R) For small k, this bound is usually rather weak, and the kth order distortion rate function cannot be viewed as having much operational significance. There are several well known bounds to D(R) and D k ....
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R.M.Gray, Source Coding Theory, Boston: Kluwer, 1990.
Fractal Dimension for Data Mining - Krishna Kumaraswamy Skkumar (Correct)
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R. M. Gray. Source Coding Theory. Kluwer Academic Press, Boston, 1990.
Fractal Dimension and Vector Quantization (Extended.. - Kumaraswamy.. (2003) (Correct)
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R. M. Gray. Source Coding Theory. Kluwer Academic Press, Boston, 1990.
Fractal Dimension and Vector Quantization [Extended.. - Kumaraswamy.. (2003) (Correct)
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R. M. Gray. Source Coding Theory. Kluwer Academic Press, Boston, 1990.
On Simple Oversampled A/D Conversion in L²(R) - Cvetkovic, Vetterli (2001) (Correct)
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R. M. Gray, Source Coding Theory. Boston, MA: Kluwer, 1990.
Multiple Description Coding: Compression Meets the Network - Goyal (2001) (38 citations) (Correct)
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R.M. Gray, Source Coding Theory. Boston, MA: Kluwer, 1990.
Learning-Theoretic Methods in Vector Quantization - Linder (2001) (1 citation) (Correct)
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R. M. Gray. Source Coding Theory. Kluwer, Boston, 1990. 52
A Note on Zador's Formula - Sloane, Vaishampayan (Correct)
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R. M. Gray, Source Coding Theory, Kluwer, Boston, 1990.
Lossy Source Coding - Berger, Gibson (1998) (11 citations) (Correct)
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R. M. Gray, Source Coding Theory. Norwell, MA: Kluwer, 1990.
Quantizing Using Lattice Intersections - Sloane, Beferull-Lozano (2002) (1 citation) (Correct)
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R. M. Gray, Source Coding Theory, Kluwer, Boston, 1990.
Asymptotic Analysis Of Lattice-Based Quantization - Moo (1998) (2 citations) (Correct)
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R.M. Gray, Source Coding Theory. Boston: Kluwer, 1990.
Information Theory and Its Application to Image Coding - Chee (1995) (Correct)
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Gray, R.M. (1990). Source coding theory. Kluwer Academic Publishers, Boston, MA.
Asymptotically Optimal Fixed-Rate Lattice Quantization for a.. - Moo, Neuhoff (Correct)
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R.M. Gray, Source Coding Theory. Boston: Kluwer, 1990.
Asymptotically Optimal Fixed-Rate Lattice Quantization for a.. - Moo, Neuhoff (Correct)
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R.M. Gray, Source Coding Theory. Boston: Kluwer, 1990.
Asymptotically Optimal Fixed-Rate Lattice Quantization for a.. - Peter Moo (Correct)
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R.M. Gray, Source Coding Theory. Boston: Kluwer, 1990.
Asymptotically Optimal Fixed-Rate Lattice Quantization for a.. - Moo, Neuhoff (Correct)
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R.M. Gray, Source Coding Theory. Boston: Kluwer, 1990.
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