We propose a novel method for fitting planar B-spline curves to unorganized data points. In traditional methods, optimization of control points and foot points are performed in two very time-consuming steps in each iteration: 1) control points are updated by setting up and solving a linear system of equations; and 2) foot points are computed by projecting each data point onto a B-spline curve. Our method uses the L-BFGS optimization method to optimize control points and foot points simultaneously and therefore it does not need to perform either matrix computation or foot point projection in every iteration. As a result, our method is much faster than existing methods.
All the above methods update control points @math and location parameters @math in two interleaving steps. The main difference of our new method with these existing methods is that in every iteration we update @math and @math simultaneously. In this sense the most closely related work is @cite_15 which also optimizes control points and location parameters simultaneously in every iteration. However, that method uses the Gauss-Newton optimization and therefore still needs to valuate and store the Jacobian matrices of the objective function, whose size depends on the number of data points and control points @cite_15 , as well as to solve a linear system of equations. In contrast, our approach based on L-BFGS does not need to formulate and solve any linear equations and is therefore faster than the method in @cite_15 , as we are going to demonstrate in later experiments.
@cite_15
1994853132
Abstract For approximation of a set of points P j ϵ R d by a parametric curve X ( t ) the choice of parameters t j is essential. None of the parametrization strategies is optimal. To obtain good approximation results, reparametrization (parameter correction) of the points is necessary. In general, these reparametrization methods work only locally. We present a global reparametrization method which leads to dramatically better results.