Gravity Centers of Smooth Planar Convex Regions from Digital Pictures

Abstract

Representations of real objects by corresponding digital pictures cause an inherent loss of information. There are infinitely many different real shapes with an identical corresponding digital picture. The problem we are interested in is how effciently the gravity center (or centroid) of a planar convex region whose boundary has a continuos third derivative and positive curvature (at every point) can be reconstructed from its digital picture. We derive an absolute upper error bound if such a smooth planar convex region is represented in a binary picture with resolution r, where r is the number of pixels per unit. This result can be extended to regions which may be obtained from smooth planar convex regions by finite applications of unions, intersections or set differences. The upper error bound remains the same which converges to zero with increase in grid resolution.