Gravity Centres of Smooth Planar Conves Regions from Digital Pictures

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. 1 The University of Auckland, Computer Science Department, CITR, Tamaki Campus (Building 731), Glen Innes, Auckland, New Zealand 2 Faculty of Engineering, University of Novi Sad, Trg D. Obradovica 6, 21000 Novi Sad, Yugoslavia Gravity Centers of Smooth Planar Convex Regions from Digital Pictures Reinhard Klette 1 and Jovi sa Zuni c 2 1 CITR, University of Auckland Tamaki Campus, Building 731, Auckland, New Zealand 2 Faculty of Engineering, University of Novi Sad, Trg D. Obradovi ca 6, 21000 Novi Sad, Yugoslavia Abstract: Representations of real objects by corresponding digital pictures cause an inherent loss of information. There are in nitely many di erent real shapes with an identical corresponding digital picture. The problem we are interested in is how e ciently 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 show that if such a smooth planar convex region is represented in a binary picture with resolution r, then the coordinates of its center of gravity can be reconstructed with an absolute upper error bound of O 1 r 15 11 " Representations of real objects by corresponding digital pictures cause an inherent loss of information. There are in nitely many di erent real shapes with an identical corresponding digital picture. The problem we are interested in is how e ciently 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 show that if such a smooth planar convex region is represented in a binary picture with resolution r, then the coordinates of its center of gravity can be reconstructed with an absolute upper error bound of O 1 r 15 11 " 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 nite applications of unions, intersections or set di erences. The upper error bound remains the same which converges to zero with increase in grid resolution.