Computational Proximity

This book introduces computational proximity (CP) as an algorithmic approachto finding nonempty sets of points thatare either close to each other or far apart.Typically in computational proximity, the bookstarts with some form of proximity space (topological space equipped with aproximity relation) that has an inherent geometry. In CP,two types of near setsare considered, namely, spatially near sets and descriptivelynear sets. It is shown that connectedness, boundedness, mesh nerves, convexity, shapes and shape theory are principal topics in the study of nearness and separation of physical aswell as abstract sets. CP has a hefty visual content. Applications of CP in computer vision, multimedia, brain activity, biology, social networks, and cosmology are included.The book has been derived from the lectures of the author in a graduate course on thetopology of digital images taught over the past several years. Many of thestudents have provided important insights and valuable suggestions.The topics in this monograph introduce many forms of proximities with acomputational flavour (especially, what has become known as the strong contactrelation), many nuances of topological spaces, and point-free geometry.