Equicontinuous random functions

This paper proposes a class of random functions that models multidimensional scenes (microscopy, macroscopy, video sequences, etc.) in a particularly adequate way. In the triplet (Omega, sigma, P) that defines a random function, the sigma-algebra sigma, here, is that introduced by G. Matheron in his theory of the upper (or lower;) semi-continuous functions from a topological space E into (R) over bar. On the other hand, the set Omega of the mathematical objects is the class, of the equicontinuous functions of a given modulus, phi say, that map E, supposed to be metric, into (R) over bar or ((R) over bar)(n). For a comprehensive member of metrics on (R) over bar, class L-phi is a compact subset of the u.s.c. functions E-->(R) over bar, on which the topology reduces to that of the pointwise convergence. In addition, class L-phi is closed under the usual dilations, erosions and morphological filters, as well as for convolutions g such that f\g(dx) \ = 1. Examples of the soundness of the model are given. (C) 1997 SPIE and IS&T.