A hierarchical dynamic programming approach to fixed-rate, entropy-coded quantization

In quantization of any source with a nonuniform probability density function, the entropy coding of the quantizer output can result in a substantial decrease in bit rate. A straightforward entropy coding scheme faces us with the problem of variable data rate. A solution in a space of dimensionality N is to select an appropriate subset of elements in the N-fold Cartesian product of a scalar quantizer and represent its elements with codewords of the same length. The drawback is that the search/addressing of this scheme can no longer be achieved independently along the one-dimensional subspaces. A reasonable rule is to select the N-fold symbols of the highest probability. For a memoryless source, this is equivalent to selecting the N-fold symbols with the lowest additive self-information. In this case, due to the additivity property of the self-information, the selected subset has a high degree of structure which can be used to substantially decrease the search/addressing complexity. A dynamic programming approach is used to exploit this structure. We build our recursive structure required for the dynamic programming in a hierarchy of levels. This results in several benefits over the conventional trellis-based approaches. Using this structure, we develop efficient rules (based on aggregating the states) to substantially reduce the search/addressing complexities while keeping the degradation in performance negligible