A characterization of digital disks by discrete moments

Abstract

In this paper our studies are focused on the digital disks and problems of their characterization (coding) with an appropriate number of bits, and reconstruction of the original disk from the code that is used. Even though the digital disks appear very often in practice of the computer vision and image processing, only the problem of their recognition has been solved till now. In this paper a representation by constant number of integers, requireing optimal number of bits, is presented. One-to-one correspondence between the digital disks and their proposed codes, consisting of: - the number of points of the digital disk, - the sum of x-coordinates of the points of digital disk, - the sum of y-coordinates of the points of digital disk, is proved. The efficiency of the reconstruction of the original disk from the proposed code is analysed. It is shown that the errors in estimating the radius of the disk, and the coordinates of its center, tend to zero while the radius of the disk tends to infinity. More precisely, if a disk, having the radius equal to r, is digitized and proposed coding scheme is applied, then the radius and the center position of the original disk can be reconstructed (from the obtained code)with relative errors bounded by Ο (formula presented), and absolute errors bounded by O (formula presented). The numerical data strongly confirm the theoretical results. The illustration by several experimental results is given.