| OCR Text |
Show 56 tion cannot eliminate error that results from a poor choice of knots. This seems consistent with the accompanying examples, in which three methods are compared: one that reparametrizes the data 40 times, one that optimizes knot locations 40 times, and one that alternately optimizes knots and reparametrizes, each 20 times. Note that, eventually, the combined method reduces the error of approximation far more than either method alone. Such results seem to be typical. The data used in each of the first two examples, in Figures 12 and 13, was sampled from parametric 8-spline curves without added noise. The knots and parameters used in these examples have been only slightly perturbed, so it is unlikely that any of the methods will converge to a local minimum which is not global in these examples. From the steps in the graph of the error per iteration for the combined method, it seems that much greater improvement is achieved by reparametriza-tion than by variable knots; in fact, variable knots can increase the true distance of the approximation from the data, particularly in the first few iterations, before the data has been reparametrized many times."' Yet it is also apparent that use of the variable knots with reparametrization reduces the final error. It is inter-esting to note, however, that reparametrization alone always seems to reduce the error of approximation more than optimal knots alone. There are always more degrees of freedom available by varying the parameters than by varying knots, so it may be possible to achieve more error reduction by varying parameters rather than knots. The second example represents an effort to use a similar number of knots and data points (four and nine, respectively), but reparametrization alone still reduces the error much more than knot optimiza-tion alone. It is clear that reparametrization is a very important type of op-timization in parametric approximation. * This can be seen in the second example, both in knot optimization alone, and in the combined method. |
