Implicit polynomials, orthogonal distance regression, and the closest point on a curve

Implicit polynomials (i.e., multinomials) have a number of properties that make them attractive for modeling curves and surfaces in computer vision. The paper considers the problem of finding the best fitting implicit polynomial (or algebraic curve) to a collection of points in the plane using an orthogonal distance metric. Approximate methods for orthogonal distance regression have been shown by others to be prone to the problem of cusps in the solution and this is confirmed here. Consequently, this work focuses on exact methods for orthogonal distance regression. The most difficult and costly part of exact methods is computing the closest point on the algebraic curve to an arbitrary point in the plane. The paper considers three methods for achieving this in detail. The first is the standard Newton's method, the second is based on resultants which are making a resurgence in computer graphics, and the third is a novel technique based on successive circular approximations to the curve. It is shown that Newton's method is the quickest, but that it can fail sometimes even with a good initial guess. The successive circular approximation algorithm is not as fast, but is robust. The resultant method is the slowest of the three, but does not require an initial guess. The driving application of this work was the fitting of implicit quartics in two variables to thinned oblique ionogram traces.