Boundaries and Topological Algorithms

Abstract : This thesis develops a model for the topological structure of situations. In this model, the topological structure of space is altered by the presence or absence of boundaries, such as those at the edge of objects. This allows the intuitive meaning of topological concepts such as region connectivity, function continuity, and preservation of topological structure to be modelled using the standard mathematical definitions. The thesis shows that these concepts are important in a wide range of artificial intelligence problems, including low-level vision, high-level vision, natural language semantics, and high-level reasoning. A formal framework for manipulating space and boundaries is developed, called cellular topology. Combinatorial methods of representing the topological structure of digitized space are developed and used to develop formal models of the changes in space induced by boundaries. The cellular structure imposed on space restricts the form of representations in ways that are useful for artificial intelligence applications. The cell structure, together with descriptions of the support and error neighborhoods of functions, provides a convenient model for the scale or resolution of representations used in applications. Two algorithms were implemented for this thesis: an edge finder and a stereo matcher. The edge finder takes advantage of the topological structure of images to distinguish real features from camera noise. The stereo matcher constrains possible matches by requiring that they preserve the topological structure of the image. In informal tests, both algorithms show improvements over previous proposals. Thesis.