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This book introduces computational proximity (CP) as an algorithmic approach to finding nonempty sets of points that are either close to each other or far apart. Typically in computational proximity, the book starts with some form of proximity space (topological space equipped with a proximity relation) that has an inherent geometry. In CP, two types of near sets are 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 the topology of digital images taught over the past several years. Many of the students have provided important insights and valuable suggestions. The topics in this monograph introduce many forms of proximities with a computational flavour (especially, what has become known as the strong contact relation), many nuances of topological spaces, and point-free geometry.
List of contents
Computational Proximity.- Proximities Revisited.- Distance and Proximally Continuous.-Image Geometry and Nearness Expressions for Image and Scene Analysis.- Homotopic Maps, Shapes and Borsuk-Ulam Theorem.- Visibility, Hausdorffness, Algebra and Separation Spaces.- Strongly Near Sets and Overlapping Dirichlet TessellationRegions.- Proximal Manifolds.-Watershed, Smirnov Measure, Fuzzy Proximity and Sorted Near Sets.- Strong Connectedness Revisited.- Helly's Theorem and Strongly Proximal Helly Theorem.- Nerves and Strongly Near Nerves.- Connnectedness Patterns.- Nerve Patterns- Appendix A: Mathematica and Matlab Scripts.- Appendix B: Kuratowski Closure Axioms.- Appendix C: Sets. A Topological Perspective.- Appendix D: Basics of Proximities.- Appendix E: Set Theory Axioms, Operations and Symbols.- Appendix F: Topology of Digital Images.
Summary
This book introduces computational proximity (CP) as an algorithmic approach to finding nonempty sets of points that are either close to each other or far apart. Typically in computational proximity, the book starts with some form of proximity space (topological space equipped with a proximity relation) that has an inherent geometry. In CP, two types of near sets are 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 the topology of digital images taught over the past several years. Many of the students have provided important insights and valuable suggestions. The topics in this monograph introduce many forms of proximities with a computational flavour (especially, what has become known as the strong contact relation), many nuances of topological spaces, and point-free geometry.
