High-Resolution Functional Quantization

Suppose a function of N real source variables X₁ ^N = (X₁, X₂, ..., X_N) is desired at a destination constrained to receive a limited number of bits. If the result of evaluating the function, Y = G(X₁ ^N), can be itself encoded, this is the optimal strategy-the origin of Y becomes irrelevant to the communication problem. We consider two alternative scenarios: distributed quantization, in which each X_i must be separately encoded; and linear transform coding of X₁ ^N . Optimal fixed- and variable-rate scalar quantizers are derived under the conventional assumptions of high-resolution quantization theory, and we find optimal transforms for transform coding. For certain classes of functions, examples demonstrate large improvements over using quantizers designed to minimize distortion of the X_is.