Axiomatic path strength definition for fuzzy connectedness and the case of multiple seeds

This paper presents an extension of the theory and algorithms for fuzzy connectedness. In this framework, a strength of connectedness is assigned to every pair of image elements by finding the strongest connecting path between them. The strength of a path is the weakest affinity between successive pairs of elements along the path. Affinity specifies the degree to which elements hang together locally in the image. A fuzzy connected object containing a particular seed element is computed via dynamic programming. In all reported works so far, the minimum of affinities has been considered for path strength and the maximum of path strengths for fuzzy connectedness. The question thus remained all along as to whether there are other valid formulations for fuzzy connectedness. One of the main contributions of this paper is a theoretical investigation under reasonable axioms to establish that maximum of path strengths of minimum of affinities along each path is indeed the one and only valid choice. The second contribution here is to generalize the theory and algorithms of fuzzy connectedness to the multi-seeded case. The importance of multi-seeded fuzzy connectedness is illustrated with examples taken from several real medical imaging applications.