Correspondence between Topological and Discrete Connectivities in Hausdorff Discretization

Abstract : We consider Hausdorff discretization from a metric space E to a discrete subspace D, which associates to a closed subset F of E any subset S of D minimizing the Hausdorff distance between F and S; this minimum distance, called the Hausdorff radius of F and written rH (F), is bounded by the resolution of D. We call a closed set F separated if it can be partitioned into two non-empty closed subsets F1 and F2 whose mutual distances have a strictly positive lower bound. Assuming some minimal topological properties of E and D (satisfied in R n and Z n), we show that given a non-separated closed subset F of E, for any r > rH (F), every Hausdorff discretization of F is connected for the graph with edges linking pairs of points of D at distance at most 2r. When F is connected, this holds for r = rH (F), and its greatest Hausdorff discretization belongs to the partial connection generated by the traces on D of the balls of radius rH (F). However when the closed set F is separated, the Hausdorff discretizations are disconnected whenever the resolution of D is small enough. In the particular case where E = R n and D = Z n with norm-based distances , we generalize our previous results for n = 2. For a norm invariant under changes of signs of coordinates, the greatest Hausdorff discretization of a connected closed set is axially connected. For the so-called coordinate-homogeneous norms, which include the Lp norms, we give an adjacency graph for which all Hausdorff discretizations of a connected closed set are connected.
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Christian Ronse, Mohamed Tajine, Loïc Mazo. Correspondence between Topological and Discrete Connectivities in Hausdorff Discretization. Mathematical Morphology-Theory and Applications, De Gruyter, In press. ⟨hal-02309344⟩

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