Relative Fuzzy Connectedness among Multiple Objects: Theory, Algorithms, and Applications in Image Segmentation

Abstract

The notion of fuzzy connectedness captures the idea of “hanging-togetherness” of image elements in an object by assigning a strength of connectedness to every possible path between every possible pair of image elements. This concept leads to powerful image segmentation algorithms based on dynamic programming whose effectiveness has been demonstrated on thousands of images in a variety of applications. In a previous framework, we introduced the notion of relative fuzzy connectedness for separating a foreground object from a background object. In this framework, an image element c is considered to belong to that among these two objects with respect to whose reference image element c has the higher strength of connectedness. In fuzzy connectedness, a local fuzzy relation called affinity is used on the image domain. This relation was required for theoretical reasons to be of fixed form in the previous framework. In the present paper, we generalize relative connectedness to multiple objects, allowing all objects (of importance) to compete among themselves to grab membership of image elements based on their relative strength of connectedness to reference elements. We also allow affinity to be tailored to the individual objects. We present a theoretical and algorithmic framework and demonstrate that the objects defined are independent of the reference elements chosen as long as they are not in the fuzzy boundary between objects. Examples from medical imaging are presented to illustrate visually the effectiveness of multiple object relative fuzzy connectedness. A quantitative evaluation based on 160 mathematical phantom images demonstrates objectively the effectiveness of relative fuzzy connectedness with object-tailored affinity relation.