Approximation by NURB curves with free knots

This paper is concerned with approximating noisy samples by NURBS curves with special emphasis on free knots. We consider the knots as unknown parameters so as to find their optimal positions. The original problem which is linear with respect to the weights and the control points but is nonlinear with respect to the knots is reformulated such that the knots are the only variable set. We show how to set up the problem such that nonlinear optimization methods can be applied efficiently. This involves the introduction of penalizing terms in order to avoid undesired knot positions. We report on our implementation of the nonlinear optimization. The performance of our method are confirmed by several practical examples. The generalization to the surface case will be briefly described at the end.