Volume 40, Issue 4
Knot Placement for B-Spline Curve Approximation via $L_{∞, 1}$-Norm and Differential Evolution Algorithm

J. Comp. Math., 40 (2022), pp. 589-606.

Published online: 2022-04

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• 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.

65D17, 68U07

khm@suda.edu.cn (Hongmei Kang)

yangzw@ustc.edu.cn (Zhouwang Yang)

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@Article{JCM-40-589, author = {Luo , JiaqiKang , Hongmei and Yang , Zhouwang}, title = {Knot Placement for B-Spline Curve Approximation via $L_{∞, 1}$-Norm and Differential Evolution Algorithm}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {4}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2012-m2020-0203}, url = {http://global-sci.org/intro/article_detail/jcm/20502.html} }
TY - JOUR T1 - Knot Placement for B-Spline Curve Approximation via $L_{∞, 1}$-Norm and Differential Evolution Algorithm AU - Luo , Jiaqi AU - Kang , Hongmei AU - Yang , Zhouwang JO - Journal of Computational Mathematics VL - 4 SP - 589 EP - 606 PY - 2022 DA - 2022/04 SN - 40 DO - http://doi.org/10.4208/jcm.2012-m2020-0203 UR - https://global-sci.org/intro/article_detail/jcm/20502.html KW - B-spline curve approximation, Knot placement, $l_{\infty,1}$-norm, Differential Evolution algorithm. AB -

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.

Jiaqi Luo, Hongmei Kang & Zhouwang Yang. (2022). Knot Placement for B-Spline Curve Approximation via $L_{∞, 1}$-Norm and Differential Evolution Algorithm. Journal of Computational Mathematics. 40 (4). 589-606. doi:10.4208/jcm.2012-m2020-0203
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