Conclusions

Abstract

The Least Squares Orthogonal Distance Fitting method can be represented as two nested sub-problems: a minimization of the distance from a point on the curve to be fit to the spline curve we are using to perform the fit, and a subsequent minimization with respect to changes in the parameters of the spline. The solutions can be characterized as being either local minima or saddle points of different degree. The classification is based on the eigenvalues of the second-order response matrix. This matrix will contain Hessian terms only when using Beta-splines for the curve fit. The single most difficult aspect of the curve fit is the smooth conversion from an asymmetric shape to a symmetric shape, which invariably causes discontinuities in the parameters of the fit, although not in the rms error. During this transformation process it is typical for solutions to merge and disappear, or to cross over each other, or to narrowly avoid such crossings. These events can be classified according to the behavior of the eigenvalues of the response matrix, as well as the augmented matrix obtained by calculating the response to the parameter that controls the shape of the curve to be fit.