A trust region algorithm for parametric curve and surface fitting

Let a family of curves or surfaces be given in parametric form via the model equation x =J‘(s, /I), where x E R”, p E KY’, and s E S c [wd. d < n. We present an algorithm for solving the problem: Find a shape cec’tor p* such that the manlfold M* = (f(s, /I*): s E S) is a best fir to scattered data :z, j ,‘= , c R” in the Sense that, jbr some is* )y=, , the sum of the squared least distances )-y=, ;I :, -j (ST. fi*) 1): ,f rom the datu points to the munifold M* is minimal umong all such mun$olds. For robustness. our algorithm uses a globally convergent trust region approach in which, at each iteration, an approximation to the objective function is minimized in a given neighborhood of the current iterate. The set S may be all of Rd or a closed, convex subset, In particular, it may be chosen so that our theory is applicable to splines.