Digitization scheme that assures faithful reconstruction of plane figures

Abstract

In the present extensive work, we propose a digitization scheme for a broad class of plane figures. It includes arbitrary polygons (possibly, non-convex and with holes) and plane sets whose boundary consists of smooth curves or straight segments. Our approach is based on an appropriate scaling of the original continuous real object so that the obtained magnified object and its (appropriately constructed) digitization feature analogous geometric properties. As a bi-product of the presented theory we prove the strong NP-hardness of the problem of obtaining an optimal (i.e., with a minimal number of facets) polyhedral reconstruction in which the facets are trapezoids or triangles. This result implies the strong NP-hardness of the general polyhedral reconstruction problem, which was a long-standing open problem. As another major result, we show that the proposed digitization scheme allows faithful reconstruction of plane figures from the considered general class. The reconstructed set features the basic properties of the original object, such as location of curve and straight segments and inflection points of its boundary.