On Limit Properties in Digitization Schemes

Continuity of curve digitizations is considered. It is pointed out that no proposed digitization scheme satisfies continuity requirements. A general definition of digitization scheme is given, having those schemes in the literature as special cases, and it is proved that for practical purposes no digitization scheme is possible in which limit curves always have unambiguous digitizations. A weaker definition of continuity is then given, and this property is proved to hold for a large class of digitization schemes. These concepts apply when a curve with some extremal property must be reconstructed from the digitization. The cases of a least perimeter polygon and of a least energy rod are considered in detail. In the first case, necessary and sufficient conditions are developed that assure a one-to-one correspondence between the given digitization and the minimal polygon. The convergence of an algorithm for finding the minimal polygon is proved in the convex case.