Direct least-squares fitting of algebraic surfaces

Abstract

In the course of developing a system for fitting smooth curves to camera input we have developed several direct (i.e. noniterative) methods for fitting a shape (line, circle, conic, cubic, plane, sphere, quadric, etc.) to a set of points, namely exact fit, simple fit, spherical fit, and blend fit. These methods are all dimension-independent, being just as suitable for 3D surfaces as for the 2D curves they were originally developed for.Exact fit generalizes to arbitrary shapes (in the sense of the term defined in this paper) the well-known determinant method for planar exact fit. Simple fit is a naive reduction of the general overconstrained case to the exact case. Spherical fit takes advantage of a special property of circles and spheres that permits robust fitting; no prior direct circle fitters have been as robust, and there have been no previous sphere fitters. Blend fit finds the best fit to a set of points of a useful generalization of Middleditch-Sears blending curves and surfaces, via a nonpolynomial generalization of planar fit.These methods all require (am+bn)n2 operations for fitting a surface of order n to m points, with a = 2 and b = 1/3 typically, except for spherical fit where b is larger due to the need to extract eigenvectors. All these methods save simple fit achieve a robustness previously attained by direct algorithms only for fitting planes. All admit incremental batched addition and deletion of points at cost an2 per point and bn3 per batch.