| OCR Text |
Show 12 least squares fitting with piecewise polynomials tends to weight knot intervals with many data points more heavily than intervals with few data points. This is particularly important in approximating digitized points, since the user tends to digitize points more densely in regions of high curvature. This results in an approximation which undulates and does not fit the shape well in flatter regions of the curve. By using local least squares, these problems may be avoided. However, the local methods force interpolation at some of the data points, and require estimates of tangents to maintain slope continuity between sections. Enforced continuity of the second derivative can potentially cause instability in a local method [13], so continuous curvature may not be attainable. It appears, therefore, that global methods will be more useful for design applications. The poor fit in regions of low curvature can be alleviated by estimating weights for least squares approximation to compensate for it. Dierck.x [6] proposes a different global approximation method, simultaneously minimizing the L2 error of approximation and the discontinuity at knot values in the k'th derivative of a B-spline of degree k, supplying a factor controlling the balance between fit and "smoothness". However, determining this factor is a nonlinear problem, and can be time-consuming. 1.3.4 Evaluation of Quality of Fit Many methods compute the error of an approximating curve, compared to a user-specified error bound, to determine whether more degrees of freedom should be added to improve the fit. A good heuristic for detecting an acceptable approximation would free the user from specifying a parameter to control the closeness of fit. The criterion of Powell [20], as well the cross-validation proposed by Craven and Wahba [2], address this problem for explicit curves. But, in general, error varies depending upon how the data was obtained, and neither method takes this into account. Since the heuristics use statistical tests to decide whether the approximation is close enough, the algorithms are not necessarily suitable for all applications [6]. |
