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ABSTRACT: Studies the asymptotic characteristics of uniform scalar
quantizers that are optimal with respect to mean-squared error (MSE).
When a symmetric source density with infinite support is sufficiently
well behaved, the optimal step size Δ<sub>N</sub> for symmetric
uniform scalar quantization decreases as 2σN<sup>-1</sup>V¯
<sup>-1</sup>(1/6N<sup>2</sup>), where N is the number of quantization
levels, σ<sup>2</sup> is the source variance and V¯<sup>-1
</sup>(·) is the inverse of V¯(y)=y<sup>-1</sup> ∫<sub>y
</sub><sup>∞</sup> P(σ<sup>-1</sup>X>x) dx. Equivalently,
the optimal support length NΔ<sub>N</sub> increases as
2σV¯<sup>-1</sup>(1/6N<sup>2</sup>). Granular distortion is
asymptotically well approximated by Δ<sub>N</sub><sup>2</sup>/12,
and the ratio of overload to granular distortion converges to a function
of the limit
τ≡lim<sub>y→∞</sub>y<sup>-1</sup>E[X|X>y],
provided, as usually happens, that τ exists. When it does, its value
is related to the number of finite moments of the source density, an
asymptotic formula for the overall distortion D<sub>N</sub> is obtained,
and τ=1 is both necessary and sufficient for the overall distortion
to be asymptotically well approximated by
Δ<sub>N</sub><sup>2</sup>/12. Applying these results to the class
of two-sided densities of the form b|x|<sup>β</sup>e(-α|x|
<sup>α</sup>), which includes Gaussian, Laplacian, Gamma, and
generalized Gaussian, it is found that τ=1, that Δ<sub>N</sub>
decreases as (ln N)<sup>1</sup>α//N, that D<sub>N</sub> is
asymptotically well approximated by Δ<sub>N</sub><sup>2</sup>/12
and decreases as (ln N)<sup>2</sup>α//N<sup>2</sup>, and that more
accurate approximations to Δ<sub>N</sub> are possible. The results
also apply to densities with one-sided infinite support, such as
Rayleigh and Weibull, and to densities whose tails are asymptotically
similar to those previously mentioned
IEEE Transactions on Information Theory 04/2001; · 3.01 Impact Factor
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ABSTRACT: This paper introduces methods for reducing the table storage required for encoding and decoding with unstructured vector quantization (UVQ) or tree-structured vector quantization (TSVQ). Specifically, a low-storage secondary quantizer is used to compress the code vectors (and test vectors) of the primary quantizer. The relative advantages of uniform and nonuniform secondary quantization are investigated. A Linde-Buzo-Gray (LBG) like algorithm that optimizes the primary UVQ codebook for a given secondary codebook and another that jointly optimizes both primary and secondary codebooks are presented. In comparison to conventional methods, it is found that significant storage reduction is possible (typically a factor of two to three) with little loss of signal-to-noise ratio (SNR). Moreover, when reducing dimension is considered as another method of reducing storage, it is found that the best strategy is a combination of both. The method of secondary quantization is also applied to TSVQ to reduce the table storage required for both encoding and decoding. It is shown that by exploiting the correlation among the test vectors in the tree, both encoder and decoder storage can be significantly reduced with little loss of SNR--by a factor of about four (or two) relative to the conventional method of storing test vectors (or test hyperplanes).
IEEE Transactions on Image Processing 02/1998; 7(4):477-95. · 3.04 Impact Factor
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ABSTRACT: This paper studies the asymptotic characteristics of optimal
fixed-rate uniform scalar quantizers for source densities with infinite
support. Asymptotic formulas for the optimal step size and the resulting
minimum mean-squared error are given. It is found that the asymptotic
significance of overload distortion is directly related to the limit of
y<sup>-1</sup>E[X|X>y] as y tends to infinity
Information Theory. 1997. Proceedings., 1997 IEEE International Symposium on; 08/1997
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ABSTRACT: Methods are presented for reducing the table storage required when encoding and decoding with tree-structured vector quantization (TSVQ). The latter is a technique that requires many fewer arithmetic operations than unstructured vector quantization but at least as much storage. The new methods for reducing storage integrate a secondary quantizer into the design of TSVQ, so as to produce a tree structure that can be efficiently stored. Two of the techniques make use of the hierarchical nature of TSVQ. It is shown that, at the expense of a decrease in signal-to-quantization-noise ratio of 0.3 dB or less, encoder storage can be reduced by a factor of about ten and decoder storage can be reduced by a factor of about five. Comparisons are made with the method of codebook sharing.
Proceedings - ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing 5:602-605vol.5.
