Digitizations Preserving Topological and Differential Geometric Properties

In this paper we present conditions which guarantee that every digitization process preserves important topological and di erential geometric properties. These conditions also allow us to determine the correct digitization resolution for a given class of real objects. Knowing that these properties are invariant under digitization, we can then use them in feature-based recognition. Moreover, these conditions imply that only a few digital patterns can occur as neighborhoods of boundary points in the digitization. This is very useful for noise detection, since if the neighborhood of a boundary point does not match one of these patterns, it must be due to noise. Our de nition of a digitization approximates many real digitization processes. The digitization process is modeled as a mapping from continuous sets representing real objects to discrete sets represented as digital images. We show that an object A and the digitization of A are homotopy equivalent. This, for example, implies that the digitization of A preserves connectivity of the object and its complement. Moreover, we show that the digitization of A will not change the qualitative di erential geometric properties of the boundary of A, i.e. a boundary point which is locally convex cannot be digitized to a locally concave pixel and a boundary point which is locally concave cannot be digitized to a locally convex pixel.