Efficiency of characterizing ellipses and ellipsoids by discrete moments

Abstract

In this paper, our studies are focused on ellipses and problems related to their representation and reconstruction from the data resulting from their digitization. The main result of the paper is that a finite number of discrete moments, corresponded to digital ellipses, is in one-to-one correspondence with digital ellipses, which enables coding of digital ellipses with an asymptotically optimal amount of memory. In addition, the problem of reconstruction, based on the same parameters, is considered. Since the digitization of real shapes causes an inherent loss of information about the original objects, the precision of the original shape estimation from the corresponding digital data is limited. We derive a sharp upper bound for the errors in reconstruction of the center position and half-axes of the ellipse, in function of the applied picture resolution (i.e., the number of pixels per unit). An extension of these results to the 3D case is also given.