Digital Approximation of Moments of Convex Regions

Representation of real regions by corresponding digital pictures causes an inherent loss of information. There are infinitely many different real regions with an identical corresponding digital picture. So, there are limitations in the reconstruction of the originals and their properties from digital pictures. The problem which will be studied here is the impact of a digitization process on the efficiency in the reconstruction of the basic geometric properties of a planar convex region from the corresponding digital picture: position (usually described by the gravity center or centroid), orientation (usually described by the axis of the least second moment), and elongation (usually calculated as the ratio of the minimal and maximal second moments values w.r.t. the axis of the least second moment). Note that the size (area) estimation of the region (mostly estimated as the number of digital points belonging to the considered region) is a problem with an extensive history in number theory. We start with smooth convex regions, i.e., regions, whose boundaries have a continuous third-order derivative and positive curvature (at every point), and show that if such a planar convex region is represented by a binary picture with resolution r, then the mentioned features can be reconstructed with an absolute upper error bound of $\text{O}\text{(}\text{1}\text{r}{}^{\mathrm{15/11-ϵ}}\text{)≈}\text{O}\text{(}\text{1}\text{r}{}^{\mathrm{1.3636...}}\text{),}$ in the worst case. Since r is the number of pixels per unit, $\text{1}\text{r}$ is the pixel size. This result can be extended to regions which may be obtained from the previously described convex regions by finite applications of unions, intersections, or set differences. The upper error bound remains the same and converges to zero with increases in grid resolution. The given description of the speed of convergence is very sharp. Only smooth, curved regions are studied because if the considered region contains a straight section, the worst-case errors in the above estimations have $\text{1}\text{r}$ as their order of magnitude. This is a trivial result—The derivation is based on the estimation of the difference between the real moments (of the first and second order) and the corresponding discrete moments. The derived estimation can be a necessary mathematical tool in the evaluation of other procedures in the area of digital image analysis based on moment calculations.