Discrete representations of spatial objects in computer vision
Discrete representation of spatial ojects in computer vision is a book based on the author's post doctoral dissertation. It deals with the difficult task of representing continuous objects in the discrete world of the computer. The target application is computer vision where objects are represented as ``discrete sets''. Discrete sets are obtained by the process of sampling which is realized by digitizing using cameras or scanners. The main question the book addresses is this: which properties inferred from discrete representations correspond to properties of the original object? That is, given a sampling, however way it was obtained, how can we relate properties of objects to those of the samples? A more interesting question would be: given an object, how to sample this object so that its important properties are representable in its discrete decomposition? Four approaches are presented: Graph-based approach: the properties of continuous concepts are compared to discrete theories, i.e. graph theory. Axiomatic approach: an axiomatic theory that describes both continuous and discrete structures is developed. Embedding approach: the discrete structure is embedded into a known continuous structure, e.g. the Euclidean space. Digitization approach: a continuous object is mapped into a discrete graph structure. The book gives a decent coverage of the above techniques following a Definition-Theory-Proof style presentation. This makes the book extremely boring to read. There are very few practical examples or algorithms, and the illustrations are presented to understand the proofs only. Obviously, the author is a member of the group of researchers practicing the philosophy: ``a proof is worth a thousand pictures''.
scientific article; zbMATH DE number 1120178