THE INERIA MATRIX IN LEAST-SQUARES LINEAR FITTING

摘要

Least squares fitting of point sets to lines, planes, curves and surfaces is carried out using eigenvalues and eigenvectors to find the major principal moment of inertia axis of a point set taken as representing the mass distribution of a rigid body. This engineering geometric approach produces identical results when compared to methods of conventional minimization using partial derivatives with respect to linear equation coefficients. Extending the approach to the fitting of conies and quadrics achieves great computational advantage over conventional least-squares optimization of Euclidean, as opposed to algebraic distance. The results, though imperfect, provide a starting point for iterations that will converge rapidly. Often, if enough points are given and these do not deviate wildly from the fit shape type selected, the result is satisfactory without resorting to further improvement.