MATHEMATICAL-MODELING OF REAL-WORLD IMAGES

The problem of finding appropriate mathematical objects to model image s is considered. Using the notion of a completed graph of a bounded fu nction, which is a closed and bounded point set in the three-dimension al Euclidean space R(3), and exploring the Hausdorff distance between these point sets, a metric space IM(D) of functions is defined. The ma in purpose is to show that the functions f is an element of IM(D), def ined on the square D = [0, 1](2), appropriate mathematical models of r eal world images. The properties of the metric space IM(D) are studied and methods of approximation for the purpose of image compression are presented. The metric space IM(D) contains the so-called pixel functi ons which are produced through digitizing images. It is proved that ev ery function f is an element of IM(D) may be digitized and represented by a pixel function p(n), with n pixels, in such a way that the dista nce between f and p(n) is no greater than 2n(-1/2). It is advocated th at the Hausdorff distance is the most natural one to measure the diffe rence between two pixel representations of a given image. This gives a natural mathematical measure of the quality of the compression produc ed through different methods.