Shrink and Stretch Sequential Scalar (S4) Quantizers

A simple backward adaptation method for constructing adaptive scalar quantizers is presented. The method needs no excess memory apart from that used to describe the current state of the quantizer and its complexity is linear in the length of the sequence to be quantized. Furthermore, it is direct and does not go through auxiliary steps such as probability density function (PDF) estimations. The basic idea is that if the current value of the sequence belongs to a certain cell (the cell is "hit"), we shrink that cell by a certain factor (with a certain probability, assuming joint randomness) and stretch all the other cells to fill the remaining space. The probability of shrinking a cell is optimally set to be proportional to 1/length(cell)². In the high resolution limit, the equilibrium of the quantizer is reached when the length of the quantizer cells is proportional to 1/PDF(cell)^1/3 which is the optimal density of a scalar quantizer. This method is shown to converge to the optimal quantizer even for probability density functions for which the Lloyd-Max algorithm converges to a local minimum, e.g., mixed Gaussian with different weights