  To measure the quality of a set of vector quantization points a means of
measuring the distance between a random point and its quantization is required.
Common metrics such as the {\em Hamming} and {\em Euclidean} metrics, while
mathematically simple, are inappropriate for comparing natural signals such as
speech or images. In this paper it is shown how an {\em environment} of
functions on an input space $X$ induces a {\em canonical distortion measure}
(CDM) on X. The depiction 'canonical" is justified because it is shown that
optimizing the reconstruction error of X with respect to the CDM gives rise to
optimal piecewise constant approximations of the functions in the environment.
The CDM is calculated in closed form for several different function classes. An
algorithm for training neural networks to implement the CDM is presented along
with some encouraging experimental results.
