Knot Placement for B-Spline Curve Approximation via $L_{∞, 1}$-Norm and Differential Evolution Algorithm
Year: 2022
Author: Jiaqi Luo, Hongmei Kang, Zhouwang Yang
Journal of Computational Mathematics, Vol. 40 (2022), Iss. 4 : pp. 589–606
Abstract
In this paper, we consider the knot placement problem in B-spline curve approximation. A novel two-stage framework is proposed for addressing this problem. In the first step, the $l_{\infty, 1}$-norm model is introduced for the sparse selection of candidate knots from an initial knot vector. By this step, the knot number is determined. In the second step, knot positions are formulated into a nonlinear optimization problem and optimized by a global optimization algorithm — the differential evolution algorithm (DE). The candidate knots selected in the first step are served for initial values of the DE algorithm. Since the candidate knots provide a good guess of knot positions, the DE algorithm can quickly converge. One advantage of the proposed algorithm is that the knot number and knot positions are determined automatically. Compared with the current existing algorithms, the proposed algorithm finds approximations with smaller fitting error when the knot number is fixed in advance. Furthermore, the proposed algorithm is robust to noisy data and can handle with few data points. We illustrate with some examples and applications.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2012-m2020-0203
Journal of Computational Mathematics, Vol. 40 (2022), Iss. 4 : pp. 589–606
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 18
Keywords: B-spline curve approximation Knot placement $l_{\infty 1}$-norm Differential Evolution algorithm.