Hausdorff Sampling of Closed Sets into a Boundedly Compact Space

Morphological Sampling of Closed Sets

“…That is why in 2001) we restricted our discretization to compact subsets of E. However the Hausdorff distance can also be defined on closed sets, and it satisfies then all axioms of a metric, except that it can take an infinite value; we say thus that it is a generalized metric. It is not hard to generalize our theory of Hausdorff discretization to closed sets: this was done in Tajine and Ronse (2002) in the case where E = R n and D = (ρZ n ), and we have given in Ronse and Tajine (2002) similar results in a more general framework (we will summarize them in Section "Hausdorff sampling").…”

“…One of the purposes of this paper is to extend the methodology of Ronse and Tajine (2002) to morphological discretization. As we will see, the main results of extend to any closed set F, in particular the fact that the Hausdorff distance between F and its discretization by dilation with a covering windowing function W is bounded by the radius r W of that windowing function.…”

“…This means that D is not necessarily a discrete space, so we will speak here of "sampling" instead of "discretization". We already used this general framework in Ronse and Tajine (2002) in order to study Hausdorff sampling (as a generalization of the Hausdorff discretization studied in ), and we will apply it here to the study of morphological sampling as an extension of morphological discretization.…”

“…Section "Topology and closed sets in a metric space" summarizes the essentials about closed and compact sets in a metric space, boundedly compact or boundedly finite spaces, and proximinal sets (see Section 2 of Ronse and Tajine (2002) for more details). It recalls also the Hausdorff metric on compact sets and its extension to closed sets, as well as the Hausdorff-Busemann metric.…”

“…It consists mainly in recalling more or less known facts from metric topology; the reader is referred to Barnsley (1993); Busemann (1955);Choquet (1966); Hocking and Young (1988); Ronse and Tajine (2002) for more details. In Subsection "Boundedly compact, boundedly finite, and proximinal sets" we give a nomenclature of concepts related to closed sets: boundedly compact and boundedly finite spaces, and proximinal sets.…”