First Order Geometric Distance (The Myth of Sampsonus)

Abstract

There is a prevalent myth in Computer Vision that the gradient weighted algebraic distance, the so-called “Sampson Error,” is a first order approximation of the distance from a point to a curve or surface. In truth, however, it is the exact geometric distance to the first order approximation of the curve. The linguistic difference is subtle, but mathematically, the two statements are at odds. In this paper, we derive the actual first order approximation of the Mahalanobis distance to a curve, a special case of which is the geometric distance. Furthermore, we show that it too, like the Sampson error, is a weighted algebraic distance. The first order distance introduces an increase in computational effort (total “flops”), which is the inevitable cost of a better approximation; however, since it too is an explicit expression, it has the same computational complexity as the Sampson error. Numerical testing shows that the first order distance performs an order of magnitude better than the Sampson error in terms of relative error with respect to the geometric and Mahalanobis distances. Our results suggest that the first order distance is an exceptional candidate cost function for approximate maximum likelihood fitting.