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Journal of Applied Geodesy

Editor-in-Chief: Kahmen, Heribert / Rizos, Chris

CiteScore 2018: 1.61

SCImago Journal Rank (SJR) 2018: 0.532
Source Normalized Impact per Paper (SNIP) 2018: 1.064

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Volume 13, Issue 4


Fast converging elitist genetic algorithm for knot adjustment in B-spline curve approximation

Johannes Bureick / Hamza Alkhatib / Ingo Neumann
Published Online: 2019-08-23 | DOI: https://doi.org/10.1515/jag-2018-0015


B-spline curve approximation is a crucial task in many applications and disciplines. The most challenging part of B-spline curve approximation is the determination of a suitable knot vector. The finding of a solution for this multimodal and multivariate continuous nonlinear optimization problem, known as knot adjustment problem, gets even more complicated when data gaps occur. We present a new approach in this paper called an elitist genetic algorithm, which solves the knot adjustment problem in a faster and more precise manner than existing approaches. We demonstrate the performance of our elitist genetic algorithm by applying it to two challenging test functions and a real data set. We demonstrate that our algorithm is more efficient and robust against data gaps than existing approaches.

Keywords: B-spline curve; approximation; knot adjustment; genetic algorithm; data gaps; Monte Carlo


  • [1]

    K. R. Koch, Fitting free-form surfaces to laserscan data by NURBS, Allgemeine Vermessungs-Nachrichten (AVN) 116 (4) (2009) 134–140.Google Scholar

  • [2]

    C. Harmening, H. B. Neuner, A constraint-based parameterization technique for B-spline surfaces, Journal of Applied Geodesy 9 (3) (2015) 143–161. doi:.CrossrefWeb of ScienceGoogle Scholar

  • [3]

    J. Bureick, H. B. Neuner, C. Harmening, I. Neumann, Curve and surface approximation of 3D point clouds, Allgemeine Vermessungs-Nachrichten (AVN) 123 (11–12) (2016) 315–327.Google Scholar

  • [4]

    J. Bureick, H. Alkhatib, I. Neumann, Robust spatial approximation of laser scanner point clouds by means of free-form curve approaches in deformation analysis, Journal of Applied Geodesy 10 (1) (2016) 27–35. doi:.CrossrefWeb of ScienceGoogle Scholar

  • [5]

    X. Xu, X. Zhao, H. Yang, I. Neumann, TLS-based feature extraction and 3D modeling for arch structures, Journal of Sensors 2017 (2017) 1–8. doi:.CrossrefWeb of ScienceGoogle Scholar

  • [6]

    X. Xu, J. Bureick, H. Yang, I. Neumann, TLS-based composite structure deformation analysis validated with laser tracker, Composite Structures. doi:.CrossrefWeb of ScienceGoogle Scholar

  • [7]

    X. Xu, B. Kargoll, J. Bureick, H. Yang, H. Alkhatib, I. Neumann, TLS-based profile model analysis of major composite structures with robust B-spline method, Composite Structures 184 (2018) 814–820. doi:.CrossrefWeb of ScienceGoogle Scholar

  • [8]

    P. Dierckx, Curve and Surface Fitting with Splines, repr Edition, Oxford science publications, Clarendon Press, Oxford [i. a.], 1993.Google Scholar

  • [9]

    A. Gálvez, A. Iglesias, A. Avila, C. Otero, R. Arias, C. Manchado, Elitist clonal selection algorithm for optimal choice of free knots in B-spline data fitting, Applied Soft Computing 26 (2015) 90–106. doi:.CrossrefWeb of ScienceGoogle Scholar

  • [10]

    L. A. Piegl, W. Tiller, The NURBS Book, 2nd Edition, Monographs in visual communications, Springer, Berlin and New York, 1997.Google Scholar

  • [11]

    H. Park, J. H. Lee, B-spline curve fitting based on adaptive curve refinement using dominant points, Computer-Aided Design 39 (6) (2007) 439–451. doi:.CrossrefWeb of ScienceGoogle Scholar

  • [12]

    F. Yoshimoto, T. Harada, Y. Yoshimoto, Data fitting with a spline using a real-coded genetic algorithm, Computer-Aided Design 35 (8) (2003) 751–760. doi:.CrossrefGoogle Scholar

  • [13]

    X. Zhao, C. Zhang, B. Yang, P. Li, Adaptive knot placement using a GMM-based continuous optimization algorithm in B-spline curve approximation, Computer-Aided Design 43 (6) (2011) 598–604. doi:.CrossrefWeb of ScienceGoogle Scholar

  • [14]

    M. G. Cox, The numerical evaluation of B-Splines, IMA Journal of Applied Mathematics 10 (2) (1972) 134–149. doi:.CrossrefGoogle Scholar

  • [15]

    C. de Boor, On calculating with B-splines, Journal of Approximation Theory 6 (1) (1972) 50–62. doi:.CrossrefGoogle Scholar

  • [16]

    F. Yoshimoto, M. Moriyama, T. Harada, Automatic knot placement by a genetic algorithm for data fitting with a spline, in: Proceedings of the International Conference on Shape Modeling and Applications, Aizu-Wakamatsu, IEEE Computer Society Press, 1999, pp. 162–169.Google Scholar

  • [17]

    C. Harmening, H. B. Neuner, Choosing the optimal number of B-spline control points (Part 1: Methodology and approximation of curves), Journal of Applied Geodesy 10 (3) (2016) 139–157. doi:.CrossrefWeb of ScienceGoogle Scholar

  • [18]

    W. Ma, J. P. Kruth, Parameterization of randomly measured points for least squares fitting of B-spline curves and surfaces, Computer-Aided Design 27 (9) (1995) 663–675. doi:.CrossrefGoogle Scholar

  • [19]

    J. Hoschek, Intrinsic parametrization for approximation, Computer Aided Geometric Design 5 (1) (1988) 27–31. doi:.CrossrefGoogle Scholar

  • [20]

    B. Sarkar, C. H. Menq, Parameter optimization in approximating curves and surfaces to measurement data, Computer Aided Geometric Design 8 (4) (1991) 267–290. doi:.CrossrefGoogle Scholar

  • [21]

    J. R. Rice, The Approximation of Function, 2nd Edition, Addison-Wesley Publishing Company, Reading, MA, 1969.Google Scholar

  • [22]

    D. L. Jupp, Approximation to data by spline with free knots, SIAM Journal on Numerical Analysis 15 (2) (1978) 328–343.CrossrefGoogle Scholar

  • [23]

    L. A. Piegl, W. Tiller, Surface approximation to scanned data, The Visual Computer 16 (7) (2000) 386–395. doi:.CrossrefGoogle Scholar

  • [24]

    M. Sarfraz, S. A. Raza, Capturing outline of fonts using genetic algorithm and splines, in: E. Banissi (Ed.), Proceedings Fifth International Conference on Information Visualisation, IEEE Computer Soc., Los Alamitos, Calif. [i. a.], 2001, pp. 738–743.Google Scholar

  • [25]

    E. Ülker, A. Arslan, Automatic knot adjustment using an artificial immune system for B-spline curve approximation, Information Sciences 179 (10) (2009) 1483–1494. doi:.CrossrefWeb of ScienceGoogle Scholar

  • [26]

    A. A. Adewuya, New methods in genetic search with real-valued chromosomes, Ph. D. thesis, Massachusetts Institute of Technology, (1996).Google Scholar

  • [27]

    D. Dennig, J. Bureick, J. Link, D. Diener, C. Hesse, I. Neumann, Comprehensive and highly accurate measurements of crane runways, profiles and fastenings, Sensors 17 (5) (2017) 1118. doi:.CrossrefWeb of ScienceGoogle Scholar

About the article

Received: 2018-04-16

Accepted: 2019-08-07

Published Online: 2019-08-23

Published in Print: 2019-10-25

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: NE 1453/5-1

This work was partly funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – NE 1453/5-1.

Citation Information: Journal of Applied Geodesy, Volume 13, Issue 4, Pages 317–328, ISSN (Online) 1862-9024, ISSN (Print) 1862-9016, DOI: https://doi.org/10.1515/jag-2018-0015.

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