Asymptotic Analysis of Optimal Uniform Two-Dimensional Quantization for the Laplace Source
Zoran Perić, Zorica Bečanović Nikolić, Dejan D. Drajic · 2004
The asymptotic optimal quantization problem, even for the simplest case- uniform scalar quantization, is very actual nowadays, [1]. The importance of using the rectangular cells and the optimal density (number) of points for product quantization and Gaussian source is considered in [2-3]. In [4] the granular gain (due to cell shape, being 1.53 dB at the maximum) as well as the boundary gain (due to the increase of the dimensions number) was defined showing that the boundary gain dominates at higher dimensions. In [5] the uniform cubic quantization (only the boundary gain) is considered for 8 and 16 dimensions. In this letter, quantizers are designed and analysed under additional constraint – each scalar quantizer is a uniform one. The optimization of two-dimensional Laplace source quantization is analysed and the existence of a single minimum, depending on the number of points for various levels, is proven. The gain over the optimum uniform scalar quantizer [5] is about (2.8-6.8) dB for rates from 4 to 8 bits per sample (see Fig.1). The resulting gain (obtained using rectangule cells) can even be compared to boundary gain in highdimensional space. Description and optimization The 2-D (two-dimensional) probability density function for independent identically distributed Laplace random variables (source) with the zero mean and the unity variance is given as ( ) ()212 2 1 xxef +−=x, (1) x is the source vector with elements x1 and x2. To simplify the vector quantizer, the Helmert transformation is applied on the source vector giving contours with constant probability densities. The transformation is defined as: ()21 2