Digitized circular arcs: characterization and parameter estimation

The digitization of a circular arc causes an inherent loss of geometrical information. Arcs with slightly different local curvature or position may lead to exactly the same digital pattern. In this paper we give a characterization of all centers and radii of circular arcs yielding the same digitization pattern. The radius of the arcs varies over the set. However, only one curvature or radius estimate can be assigned to the digital pattern. We derive an optimal estimator and give expressions for the bound on the precision of estimation. This bound due to digitization is the deterministic equivalent of the Cramér/Rao bound known from parameter estimation theory. Consider the estimation of the local curvature and local radius of a smooth object. Typically such parameters are estimated by moving a window along the digital boundary. Methods in literature show a poor precision in estimating curvature values, relative errors of over 40% are often found [34]. From the definition of curvature it follows that locally the curve can be considered a circular arc and hence the method presented in this paper can be applied to the pattern in the window giving estimates with optimal precision and a measure for the remaining error. On the practical side we present examples of the residual error due to the discrete grid. The estimation of the radius or curvature of a circular arc at random position with an estimation window containing 10 points (coded with nine Freemancodes) has a relative deviation exceeding 2%. For a full disk the deviation is below 1% when the radius r exceeds four grid units. The presented method is particularly useful for problems where some prior knowledge on the distribution of radii is known and where there is a noise-free sampling.