Axiomatics oriented connectivity

Christian Ronse

doi:10.1016/j.patrec.2014.03.020

Axiomatics oriented connectivity

2014-10-01

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Christian Ronse *

DOI：10.1016/j.patrec.2014.03.020

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Tankyevych et al. [17] considered, in a directed graph, any set S of vertices where there is some marker vertex p is an element of S such that for every vertex x is an element of S, there is a directed path from p to x included in S; the family of all such sets S was called a semi-connection. Their properties were briefly analysed and compared with connectivity and connected components in undirected graphs. We give an abstract algebraic formalization of this concept, following the same approach as that of Serra [14] and Ronse [9] for the notions of connection and partial connection, which generalize both topological and graph-theoretic connectivity. Here the sets S are unsufficient, one must associate to them their markers p; thus in a space E we consider a family R of ordered pairs (d, S) is an element of E x P(E), where the set S can be reached'' from marker p; this family, which we call a reach, must satisfy the three properties of union, transitivity and membership; a fourth point property leads to a full reach. As in [14,9], we give an equivalent definition in terms of a system of point openings (gamma(p),p is an element of E) satisfying some properties. The special case of symmetry, where S does not depend on the choice of the marker p subset of S, leads to a partial connection or a connection. Some examples are given. Possible applications of this new theory lie in the analysis of connected structures having an orientation, for instance vascular networks in medical imaging. One can also apply it to geodesic reconstruction and connected filtering. (C) 2014 Elsevier B. V. All rights reserved.

Keywords:

Oriented path

Oriented connectivity

Reach

Full reach

Partial connection

Connection

Axiomatics oriented connectivity

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