On the complexity of curve fitting algorithms

We study a popular algorithm for fitting polynomial curves to scattered data based on the least squares with gradient weights. We show that sometimes this algorithm admits a substantial reduction of complexity, and, furthermore, find precise conditions under which this is possible. It turns out that this is, indeed, possible when one fits circles but not ellipses or hyperbolas. In many applications one needs to fit a curve described by a polynomial equation P (x, y; Θ) = 0 (here Θ denotes the vector of unknown parameters) to experimental data (x i , y i), i = 1,. .. , n. In this equation P is a polynomial in x and y, and its coefficients are either unknown parameters or functions of unknown parameters. For example, a number of recent publications [5, 6, 9] are devoted to the problem of fitting quadrics Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, in which case Θ = (A, B, C, D, E, F) is the parameter vector. The problem of fitting circles, given by equation (x − a) 2 + (y − b) 2 − R 2 = 0 with three parameters a, b, R, also arises in practice [2, 8]. It is standard to assume that the data (x i , y i) are noisy measurements of some true (but unknown) points (¯ x i , ¯ y i) on the curve, see [1, 4, 7, 8] for details. The noise vectors e i = (x i − ¯ x i , y i − ¯ y i) are then assumed to be independent gaussian vectors with zero mean and a scalar covariance matrix, σ 2 I. In this case the maximum likelihood estimate of Θ is given by the orthogonal least squares fit (OLSF), which is based on the minimization of the function F (Θ) = n i=1 d 2 i (1) where d i denotes the distance from the point (x i , y i) to the curve P (x, y; Θ) = 0. Under these assumptions the OLSF is statistically optimal – it provides estimates of Θ whose covariance matrix attains its Rao-Cramer lower bound [4, 7, 8]. The OLSF is 1