Approximate Connectivity and Mathematical Morphology

Abstract

As known from the works of Serra, Ronse, and Haralick and Shapiro, the connectivity relations are found to be useful in filtering binary images. But it can be used also to find roadmaps in robot motion planning, i.e. to build discrete networks of simple paths connecting points in the robot’s configuration space capturing the connectivity of this space. This paper generalises and puts together the notion of a connectivity class and the notion of a separation relation. This gives an opportunity to introduce approximate epsilon-connectivity, and thus we show the relation between our approach and the Epsilon Geometry introduced by Guibas, Salesin and Stolfi. Ronse and Serra have defined connectivity analogues on complete lattices with certain properties. As a particular case of their work we consider the connectivity of fuzzy compact sets, which is a natural way to study the connectivity of greyscale images. This idea can be transferred also in planning robot trajectories in the presence of uncertainties. Since based on fuzzy sets theory, our approach is intuitively closer to the classical set oriented approach, used for binary images and robot path planning in known environment with obstacles.