Hausdorff Discretization for Cellular Distances and Its Relation to Coverand Supercover Discretizations

Morphological Sampling of Closed Sets

“…In 2001); Tajine and Ronse (2002), the authors introduced a new approach to discretization, based on the idea that the discretized set must be "close" to its Euclidean counterpart, the "closeness" between the two sets being measured by the Hausdorff distance. Now the latter is a metric (it satisfies the axioms of a distance) on the family of all compact sets of a metric space.…”

“…the coordinate axes; for instance d can be the L p metric, which includes as particular cases the cityblock (p = 1), chessboard (p = ∞), and Euclidean (p = 2) distances. Then for every nonvoid compact subset K of E, its supercover discretization ∆ SC (K) is a Hausdorff discretiziation of K (Ronse and Tajine, 2001). …”

“…Let us stress that the standard morphological approach to sampling (Heijmans, 1994;Heijmans and Toet, 1991), as well as our Hausdorff metric approach 2001;Tajine and Ronse, 2002), are not models of the way image acquisition devices make discrete images from physical phenomena in the (Euclidean) real world. Indeed, a sensor measures the amount of light energy, so this means that for a binary object X, the discretization of X is made of all pixels p such that the area of X ∩ C(p) exceeds some threshold; this discretization calls for a probabilistic formulation (Hall and Molchanov, 1999).…”

“…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").…”

“…Then we summarize the main results of our metric theory of sampling 2001;Tajine and Ronse, 2002), which can be used to analyse several sampling schemes, in particular the morphological one. We extend it to the sampling of closed sets (instead of compact ones), and to the case where the sampling subspace is boundedly compact (instead of boundedly finite), and obtain new results on morphological sampling.…”