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Full Description
Concerned with the mathematical theory of surfaces, objects and their boundaries in discrete spaces, this text provides a self-contained and mathematically precise introduction to the field. It is application-oriented and presents problems of visualization and analysis of multidimensional data sets. The primary areas of mathematics used are graph theory and topology.
Contents
Cloning files on sugar cubes - what is our game? a methodology for extracting object boundaries, files in flatland, components determined by binary relations, so, what does a flat fly do? back to the cuberille, algorithms for flat files, diagraphs, so, what can a fat fly do? algorithms for cloning flies, an efficient implementation, exercises; enhancing the cube - why study noncubic grids? other spaces, exercises; digital spaces - the basic definitions, interiors and exteriors, connectedness in digital spaces, isomorphisms between digital spaces, exercises; topological digital spaces - what is a topology? some topological digital spaces, many digital spaces are not topological, connectedness of topological interiors, exercises; binary pictures - digital pictures, fuzzy segmentation, boundaries in binary pictures, Jordan pairs of spel-adjacencies, new Jordan pairs from old ones; simply connected digital spaces - N-simply connected digital spaces, locally-Jordan surfaces, applications to finding Jordan pairs, 1-simply connected digital spaces, exercises; Jordan graphs - the theory of (strong) Jordan graphs, Jordan surfaces, spel-manifolds, exercises; boundary tracking - tracking in finitary 1-simply connected spaces, efficient tracking of boundary elements, boundary tracking on hypercubes, proofs of the boundary-tracking claims, boundary tracking in the FCC grid, pointers to further reading, exercises.
