[IEEE Trans. on Information Theory, November 2000, pp. 2697-2704]
On Source Coding with Side Information Dependent
Distortion Measures
Tamás Linder, Ram Zamir, and Kenneth Zeger
Abstract
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 {\em 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 $R^{\rm WZ}(D)$ 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 $R^{\rm WZ}(D)-R_{X|Y}(D)$ 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 $D\to 0$.