Statistical efficiency of curve fitting algorithms

We study the problem of fitting parametrized curves to noisy data. Under certain assumptions (known as Cartesian and radial functional models), we derive asymptotic expressions for the bias and the covariance matrix of the parameter estimates. We also extend Kanatani's version of the Cramer-Rao lower bound, which he proved for unbiased estimates only, to more general estimates that include many popular algorithms (most notably, the orthogonal least squares and algebraic fits). We then show that the gradient-weighted algebraic fit is statistically efficient and describe all other statistically efficient algebraic fits.Comment: 17 pages, 3 figure

Least squares fitting of circles and lines

We study theoretical and computational aspects of the least squares fit (LSF) of circles and circular arcs. First we discuss the existence and uniqueness of LSF and various parametrization schemes. Then we evaluate several popular circle fitting algorithms and propose a new one that surpasses the existing methods in reliability. We also discuss and compare direct (algebraic) circle fits.Comment: 26 pages, 14 figures, submitte

On Covering a Solid Sphere with Concentric Spheres in Z3{\mathbb Z}^3

We show that a digital sphere, constructed by the circular sweep of a digital semicircle (generatrix) around its diameter, consists of some holes (absentee-voxels), which appear on its spherical surface of revolution. This incompleteness calls for a proper characterization of the absentee-voxels whose restoration will yield a complete spherical surface without any holes. In this paper, we present a characterization of such absentee-voxels using certain techniques of digital geometry and show that their count varies quadratically with the radius of the semicircular generatrix. Next, we design an algorithm to fill these absentee-voxels so as to generate a spherical surface of revolution, which is more realistic from the viewpoint of visual perception. We further show that covering a solid sphere by a set of complete spheres also results in an asymptotically larger count of absentees, which is cubic in the radius of the sphere. The characterization and generation of complete solid spheres without any holes can also be accomplished in a similar fashion. We furnish test results to substantiate our theoretical findings