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Show 55 the knots, but it is a linear problem with respect to the B-spline coefficients. This can be used to reduce the complexity of the nonlinear minimization problem, by solving the nonlinear problem only as a function of the nonlinear variables, and then computing the corresponding values of the linear variables, which simply requires solving a linear system [1 5]. The form of the reduced nonlinear model is of great importance, however. Experiments with a method which, for each iteration, optimized knot locations using the previous B-spline coefficients, were disappointing. Although a method with q-superlinear convergence was used, the observed convergence was much slower. The model did not reflect the variable nature of the B-spline coefficients, so the steps were always much too conservative. It seems reasonable that larger steps would be made if these additional degrees of freedom were taken into account. Golub and Pereyra [1 11 developed the mathematics for a reduced model which reflects all of these degrees of freedom, and thus reduces the number of variables in the nonlinear problem without degrading the performance of the algorithm. A Levenberg-Marquardt algorithm that is q-superlinearly convergent can be implemented using this model. A simple version was implemented, without all of the optimizations that Golub and Pereyra mention, and adapted to parametric curves. Note that approximation with variable knots optimizes the knot locations in such a way as to reduce the L2 error of the curve, computed from the pseudo-distance. So variable knot techniques alone cannot be expected to reduce the error based on the true distance. By combining knot optimization with reparametrization, however, we can expect the pseudo-distance to become a better estimator of the true distance, so that variable knots would also tend to reduce the true distance. It is difficult to devise a fair comparison of reparametrization and knot optimization, each alone, and in combination. Either an inappropriate parametrization or knot vector can induce a large error in an approximation. Permitting the knots to vary cannot compensate for a poor parametrization, and reparametriza- |
