Morphological Sampling of Closed Sets

Abstract

We briefly survey the standard morphological approach (Heijmans, 1994) to the sampling (or discretization) of sets. 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. In the original morphological theory for sampling (Heijmans, 1994), some reconstruction of the sampling was shown to converge, relative to the Fell topology (or Hausdorff-Busemann metric), to the original closed set when the resolution of the sampling space tends to zero. As we explain here, this reconstruction step is artificial and unnecessary, the sampling itself converges to the original closed set under the Hausdorff metric, which is a stronger convergence than for the Hausdorff-Busemann metric.

Correspondence between Topological and Discrete Connectivities in Hausdorff Discretization

“…We will recall some known concepts and results, first about connections and partial connections [11], then about some families of closed subsets in a metric space [14,15], and finally in the theory of Hausdorff discretization [12,13]. We also summarize related works.…”

“…The reader is assumed to be familiar with basic topological and metric notions, such as a metric (or distance function), a metric space, a bounded set, an open set, a closed set, the relative topology on a subset, a compact space and a compact set, see [6]. We recall some more advanced definitions and results [14,15].…”

“…Our notation is summarized in Table 1. Section 2 recalls the theoretical background: first partial connections and connections [11], then some special families of closed sets [14,15], and finally Hausdorff discretization [12,13]. It also discusses related works by other authors.…”

“…For instance, we showed that when digital cells constitute the Voronoi diagram of D, every cover discretization is a Hausdorff discretization. On the other hand, in morphological discretization by dilation, provided that the dilation of D is E (every point of E belongs to the structuring element associated to a point of D), the Hausdorff distance between a compact and its discretization is bounded by the radius of the structuring element.In [14,15] we extended our theoretical framework to the discretization of non-empty closed subsets of E instead of compact ones.In Section 5 of [22], we analysed the preservation of connectivity by Hausdorff discretization in the case where E = R 2 , D = Z 2 and the metric d is induced by a norm N such that for (x 1 , x 2 ) ∈ R 2 and ε 1 , ε 2 = ±1, N(ε 1 x 1 , ε 2 x 2 ) = N(x 1 , x 2 ) = N(x 2 , x 1 ), for example the Lp norm (1 ≤ p ≤ ∞), see (1) next page. We showed then that for a non-empty connected closed subset F of E, 1. for N ≠ L 1 , every Hausdorff discretization of F is 8-connected; 2. the greatest Hausdorff discretization of F is 4-connected.There was an error in [22]: we overlooked the condition N ≠ L 1 in item 1; it was later pointed out by D. Wagner (private communication), and indeed for N = L 1 we show a counterexample in Section 4 ( Figure 8).…”

“…In [14,15] we extended our theoretical framework to the discretization of non-empty closed subsets of E instead of compact ones.…”