Discretization in Hausdorff space
In this paper, a new approach to the discretization of n-dimensional Euclidean figures is studied: the discretization of a compact Euclidean set K is a discrete set S whose Hausdorff distance to K is minimal; in particular such a discretization depends on the choice of a metric in the Euclidean space, for example the Euclidean or a chamfer distance. We call such a set S a Hausdorff discretizing set of K. The set of Hausdorff discretizing sets of K is nonvoid, finite, and closed under union; we consider thus in particular the greatest one among such sets, which we call the maximal Hausdorff discretizarion of K. We give a mathematical description of Hausdorff discretizing sets: it is related to the discretization by dilation considered by Heijmans and Toet and the cover discretization studied by Andres. We have a bound on the Hausdorff distance between a compact set and its maximal Hausdorff discretization, and the latter converges (for the Hausdorff metric) to the compact set when the spacing of the discrete grid tends to zero. Such a convergence result holds also for the discretization by dilation when the structuring element satisfies the covering assumption. Our approach is here the most general possible. In a next paper we will consider the case where the underlying metric on points satisfies some general constraints in relation to the cells associated to the discrete points, and we will then see that these constraints guarantee that the usual supercover and cover discretizations give indeed Hausdorff discretizing sets.
discretization distances dilations compact sets hausdorff metric metric spaces