An improved algorithm for algebraic curve and surface fitting

Recent years have seen an increasing interest in algebraic curves and surfaces of high degree as geometric models or shape descriptors for model-based computer vision tasks such as object recognition and position estimation. Although their invariant-theoretic properties them a natural choice for these tasks, fitting algebraic curves and surfaces to data sets is difficult, and fitting algorithms often suffer from instability, and numerical problems. One source of problems seems to be the performance function being minimized. Since minimizing the sum of the squares of the Euclidean distances from the data points to the curve or surface with respect to the coefficients of the defining polynomials is computationally impractical, because measuring the Euclidean distances require iterative processes, approximations are used. In the past we have used a simple first order approximation of the Euclidean distance from a point to an implicit curve or surface which yielded good results in the case of unconstrained algebraic curves or surfaces, and reasonable results in the case of bounded algebraic curves and surfaces. However, experiments with the exact Euclidean distance have shown the limitations of this simple approximation. In this paper we introduce a more complex, and better, approximation to the Euclidean distance from a point to an algebraic curve or surface. Evaluating this new approximate distance does not require iterative procedures either, and the fitting algorithm based on it produces results of the same quality as those based on the exact Euclidean distance.