On Limit Properties in Digitization Schemes

Con t inu i ty of curve digi t iza t ions is considered. I t is po in ted out t h a t no proposed digi t iza t ion scheme satisfies con t inu i ty requi rements . A general definition of digi t izat ion scheme is given, hav ing those schemes in the l i t e ra tu re as special cases, and it is proved tha t for pract ical purposes no digi t izat ion scheme is possible in which l imit curves always have unambiguous digi t izat ions . A weaker definit ion of con t inu i ty is then given, and this proper ty is proved to hold for a large class of digi t izat ion schemes. These concepts apply when a curve w i th some extremal proper ty mus t be recons t ruc ted from the digi t izat ion. The cases of a least per imeter polygon and of a least energy rod are considered in detail . In the first case, necessary and sufficient condi t ions are developed t ha t assure a one-to-one correspondence between the given digi t iza t ion and the minimal polygon. The convergence of an a lgor i thm for finding the minimal polygon is proved in the convex case. KEY "WORDS AND PHRASES: digi t izat ion, cont inui ty , chain encoding, plane curve digi t izat ion, s t a n d a r d curve, recons t ruc ted curve, minimal per imeter polygon, nonl inear programming, Gauss-Seidel , convex curve, convex minimal polygon CR CATEGORIES: 3.63, 3.69, 5.13, 5.41