Contour tree connectivity of binary images from algebraic graph theory

We propose a novel feature for binary images that provides connectivity information by taking into account the proximity of connected components and cavities. We start by applying the Euclidean distance transform and then we compute the contour tree. Finally, we assign the normalized algebraic connectivity of a contour tree derivative as a feature for connectivity. Our algorithm can be applied to any dimensions of data as well as topology. And the resultant connectivity index is a single real number between 0 and 1. We test and demonstrate interesting properties of our approach on various 2D and 3D images. With its intriguing properties, the proposed index is widely applicable for studying binary morphology. Especially, it is complementary to Euler number for studying connectivity of microstructures of materials such as soil, paper, filter, food products as well as biomaterials and biological tissues.