Dominance--A relation between distortion measures

A relation between distortion measures for source coding, called dominance, is defined so that if
d_{1}
and
d_{2}
are two distortion measures and
d_{1}
dominates
d_{2}
, then a source coding system which meets the distortion criterion
(d_{l}, \Delta)
simply yields a system meeting the distortion criterion
(d_{2}, \Delta)
by a simple change of decoder. It is shown that a very simple characterization of dominance can be given in terms of the characteristic sets of the distortion measures. The characteristic set of a distortion matrix is defined as the sum of the convex hull of the columns of the matrix and the nonnegative orthant. If two distortion measures have the same characteristic set they are regarded as equivalent. It is shown that the definition of dominance is robust in the sense that some alternative definitions are equivalent to the given one. The definition of dominance is extended to the case where a pair of distortion measures dominates a third distortion measure. Here a simple characterization is given in terms of the characteristic sets of the distortion measures.