Algorithms and Data Structures for Computer Topology

The paper presents an introduction to computer topology with applications to image processing and computer graphics. Basic topological notions such as connectivity, frontier, manifolds, surfaces, combinatorial homeomorphism etc. are recalled and adapted for locally finite topological spaces. The paper describes data structures for explicitly representing classical topological spaces in computers and presents some algorithms for computing topological features of sets. Among them are: boundary tracing (n=2,3), filling of interiors (n=2,3,4), labeling of components, computing of skeletons and others. Introduction: Topology and Computers Topology plays an important role in computer graphics and image analysis. Connectedness, boundaries and inclusion of regions are topological features which are important for both rendering images and analyzing their contents. Computing these features is one of the tasks of the computer topology. We use this term rather than "computational topology" since our approach is analogous to that of digital geometry rather than to that of computational geometry: we are using models of topological spaces explicitly representing each element of a finite topological space as an element of the computer memory, defined by its integer coordinates. The other possible approach would be to think about the Euclidean space, to define objects by equations and inequalities in real coordinates and to approximate real coordinates on a computer by floating point variables. Computer topology may be of interest both for computer scientists who attempt to apply topological knowledge for analyzing digitized images, and for mathematicians who may use computers to solve complicated topological problems. Thus, for example, essential progress in investigating three-dimensional manifolds has been reached by means of computers (see e.g. [14]). Topological ideas are becoming increasingly important in modern theoretical physics where attempts to develop a unique theory of gravitation and quantum mechanics have led to the Topological Quantum Field Theory (see e.g. [2,13]), in which topology of multi-dimensional spaces plays a crucial role. This is one more possible application field for computer topology. Thus, computer topology is important both for applications in computer imagery and in basic research in mathematics and physics.