Fitting the most probable curve to noisy observations

Abstract

Orthogonal distance regression (ODR) is widely used to fit a curve through scattered points in the plane. It generally produces good fits, but intuition and practice suggest it can give biased results when fitting closed convex curves or corners. ODR gives the maximum likelihood estimate of the best curve under a straightforward stochastic model for data generation. The paper proposes an extension of this model to include a reasonable partial prior for the distribution of samples along the curve. Although the associated optimisation problem for the maximum likelihood estimator under the extended model no longer has a simple form, it lends itself naturally to asymptotic approximation. The first order approximation is essentially ODR: the second order approximation has an extra contribution involving the local curvature. Finally the results are applied to the illustrative problem of fitting a circle through scattered data: ODR is shown to indeed give a biased estimate of the true radius, and this bias is reduced under the second order estimator.