Computational Homology

Description

In recent years, there has been a growing interest in applying homology to problems involving geometric data sets, whether obtained from physical measurements or generated through numerical simulations. This book presents a novel approach to homology that emphasizes the development of efficient algorithms for computation. As well as providing a highly accessible introduction to the mathematical theory, the authors describe a variety of potential applications of homology in fields such as digital image processing and nonlinear dynamics. The material is aimed at a broad audience of engineers, computer scientists, nonlinear scientists, and applied mathematicians. Mathematical prerequisites have been kept to a minimum and there are numerous examples and exercises throughout the text. The book is complemented by a website containing software programs and projects that help to further illustrate the material described within. Front Matter....Pages I-XVII Front Matter....Pages 1-1 Preview....Pages 3-37 Cubical Homology....Pages 39-92 Computing Homology Groups....Pages 93-141 Chain Maps and Reduction Algorithms....Pages 143-171 Preview of Maps....Pages 173-197 Homology of Maps....Pages 199-234 Computing Homology of Maps....Pages 235-254 Front Matter....Pages 255-255 Prospects in Digital Image Processing....Pages 257-278 Homological Algebra....Pages 279-306 Nonlinear Dynamics....Pages 307-376 Homology of Topological Polyhedra....Pages 377-393 Front Matter....Pages 395-395 Topology....Pages 397-418 Algebra....Pages 419-449 Syntax of Algorithms....Pages 451-464 Back Matter....Pages 465-482