On source coding with side-information-dependent distortion measures

High-resolution bounds in lossy coding of a real memoryless source are considered when side information is present. Let X be a “smooth” source and let Y be the side information. First we treat the case when both the encoder and the decoder have access to Y and we establish an asymptotically tight (high-resolution) formula for the conditional rate-distortion function R_X|Y(D) for a class of locally quadratic distortion measures which may be functions of the side information. We then consider the case when only the decoder has access to the side information (i.e., the “Wyner-Ziv problem”). For side-information-dependent distortion measures, we give an explicit formula which tightly approximates the Wyner-Ziv rate-distortion function R^WZ(D) for small D under some assumptions on the joint distribution of X and Y. These results demonstrate that for side-information-dependent distortion measures the rate loss R^WZ(D)-R_X|Y(D) can be bounded away from zero in the limit of small D. This contrasts the case of distortion measures which do not depend on the side information where the rate loss vanishes as D→0