Relative fuzzy connectedness and object definition: Theory, algorithms, and applications in image segmentation

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 1,000s of images in a variety of applications. In the previous framework, a fuzzy connected object is defined with a threshold on the strength of connectedness. In this paper, we introduce the notion of relative connectedness that overcomes the need for a threshold and that leads to more effective segmentations. The central idea is that an object gets defined in an image because of the presence of other co-objects. Each object is initialized by a seed element. An image element c is considered to belong to that object with respect to whose reference image element c has the highest strength of connectedness. In this fashion, objects compete among each other utilizing fuzzy connectedness to grab membership of image elements. We present a theoretical and algorithmic framework for defining objects via relative connectedness and demonstrate utilizing the theory that the objects defined are independent of reference elements chosen as long as they are not in the fuzzy boundary between objects. An iterative strategy is also introduced wherein the strongest relative connected core parts are first defined and iteratively relaxed to conservatively capture the more fuzzy parts subsequently. Examples from medical imaging are presented to illustrate visually the effectiveness of relative fuzzy connectedness. A quantitative mathematical phantom study involving 160 images is conducted to demonstrate objectively the effectiveness of relative fuzzy connectedness.