Two-step optimization of knots in B-spline curve approximation

This paper presents a framework for optimizing B-spline knot placement in curve fitting. We show that the perturbation introduced during knot removal increases with the magnitude of the derivative jumps at the removed knots. Based on this observation, we employ polynomial trend filtering to detect abrupt changes in the higher-order discrete derivatives of the sample data, which in turn guides effective knot selection. The proposed framework consists of two main steps: (1) Starting from a densely placed initial knot vector, we optimize the coefficients of the 0-degree B-splines using a generalized lasso model. Knots corresponding to significant changes in discrete derivatives of a selected order are identified as active; (2) A higher-order B-spline approximation is then constructed using these active knots. Redundant knots are iteratively removed while maintaining the approximation quality. We validate the method on several functions and parameter curve fitting tasks. Results show that the proposed approach yields B-spline approximations with a similar number of knots as existing methods, while achieving comparable or improved accuracy.