ASYMPTOTICALLY OPTIMAL BLOCK QUANTIZATION
ASYMPTOTICALLY OPTIMAL BLOCK QUANTIZATION
复制标题
DOI：
10.1109/tit.1979.1056067
复制
发表时间：
1979-01-01
影响因子：
2.5
通讯作者：
GERSHO, A
中科院分区：
文献类型：
--
作者：
GERSHO, A
1948年W. R.班尼特使用压扩模型进行非均匀量化，并提出了公式D\：= \：\frac{1}{12 N ^{2}} \：\int \：p（x）[N（x）]^{-2} \dx表示均方量化误差，其中为电平数，（x）为输入的概率密度，（x）为压缩曲线的斜率。该公式是一个近似的基础上的假设，即水平的数量是大的，过载失真是可以忽略不计的，是一个有用的工具，分析研究的量化。本文将班尼特公式推广到随机变量向量量化的块量化。该方法是再次基于渐近的情况下，量化的输出矢量的数量，是非常大的。使用由此产生的启发式公式，进行优化，导致一个表达式的最小量化噪声可达到的任何块量化器的给定块大小。结果与Zador的结果是一致的，并专门为已知的结果为一维和二维的情况下，无限块长度的情况下。同样的启发式方法也给出了多维量化的Elias界的另一种推导。我们的方法导致一个严格的方法获得块量化器的最小失真的上限。特别是，我们给出了一个严格的上限，实际上可能是准确的。还探讨了由块“压缩器”映射表示块量化器的想法，然后是均匀分布的随机向量的最佳量化器。用这种块压扩模型来表示最佳量化器并不总是可能的。
In 1948 W. R. Bennett used a companding model for nonuniform quantization and proposed the formulaD \: = \: \frac{1}{12N^{2}} \: \int \: p(x) [ É(x) ]^{-2} \dxfor the mean-square quantizing error whereis the number of levels,(x) is the probability density of the input, and(x) is the slope of the compressor curve. The formula, an approximation based on the assumption that the number of levels is large and overload distortion is negligible, is a useful tool for analytical studies of quantization. This paper gives a heuristic argument generalizing Bennett's formula to block quantization where a vector of random variables is quantized. The approach is again based on the asymptotic situation where, the number of quantized output vectors, is very large. Using the resulting heuristic formula, an optimization is performed leading to an expression for the minimum quantizing noise attainable for any block quantizer of a given block size. The results are consistent with Zador's results and specialize to known results for the one- and two-dimensional cases and for the case of infinite block length. The same heuristic approach also gives an alternate derivation of a bound of Elias for multidimensional quantization. Our approach leads to a rigorous method for obtaining upper bounds on the minimum distortion for block quantizers. In particular, forwe give a tight upper bound that may in fact be exact. The idea of representing a block quantizer by a block "compressor" mapping followed with an optimal quantizer for uniformly distributed random vectors is also explored. It is not always possible to represent an optimal quantizer with this block companding model.