A multiscale approach to connectivity

The concept of connectivity is fundamental in image analysis and computer vision problems, and particularly in problems of image segmentation and object detection. In this paper, we introduce a novel theory of connectivity, which considers traditional concepts in a multiscale framework. The proposed theory includes, as a single-scale special case, the notion of connectivity classes in complete lattices. Following mathematical preliminaries, we introduce multiscale connectivity by means of two alternative, but equivalent, approaches. The first approach is based on the notion of a connectivity measure, which quantifies the degree of connectivity of a given object, whereas the second approach is based on the notion of a connectivity pyramid. We also introduce the notion of σ-connectivity openings and show that these operators define multiscale connectivities. Moreover, we introduce the notion of σ-reconstruction operators and show that, under certain conditions, these operators define multiscale connectivities as well. Based on the proposed theory, we show that fuzzy topological and fuzzy graph-theoretic connectivities are multiscale analogs of the classical notions of topological and graph-theoretic connectivity, respectively. We also discuss a generalization of the proposed multiscale connectivity concept which leads to the notion of multiscale level connectivity for grayscale images. Examples illustrate several key points of our approach.