Fuzzy Connected Object Delineation: Axiomatic Path Strength Definition and the Case of Multiple Seeds

Abstract

This paper presents an extension of previously published theory and algorithms for object definition based on fuzzy connectedness. In this approach, a strength of connectedness is assigned between every pair of image elements. This is done by finding the strongest among all possible connecting paths between the two elements in each pair. The strength assigned to a particular path is defined as the weakest affinity between successive pairs of elements along the path. Affinity specifies the degree to which elements hang together locally in the image and is determined considering how close the elements are in space and in intensity features. A fuzzy connected object containing a particular seed element specified within the object is computed via dynamic programming. In all reported works so far, the minimum of affinities has been considered to define path strength and the maximum of path strengths has been used to define 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 for defining fuzzy connectedness. In the past, a single fuzzy connected object was specified through a single seed element indicated interactively within the object region. When many objects are to be identified in a single image, interactive seed specification becomes cumbersome. Further, selecting exactly one element in each region automatically is more difficult than identifying a set of elements. Hence the theory and algorithms for the single-element case need to be generalized to multiple elements for more effective practical use. This is the second main contribution of this paper. The importance of multiple seeded fuzzy connected object definition from the considerations of both practicality and automation in image segmentation are described and illustrated with examples taken from several real medical applications.