Skip to main content
SpringerLink
Log in

Hybridizing mesh adaptive search algorithm and artificial immune systems for discrete rational Bézier curve approximation

  • Original Article
  • Published:
The Visual Computer Aims and scope Submit manuscript

Abstract

This paper is an extension of a previous one presented at the conference Cyberworlds 2014. In that work we addressed the problem of obtaining the rational Bézier curve that fits a given set of data points better in the least-squares sense. Our approach was based on the clonal selection theory principles to compute all parameters of the problem, namely, the control points of the approximating curve, their corresponding weights, and a suitable parameterization of data points. Although we were able to obtain results with good accuracy, this scheme can still be significantly improved by hybridizing it with an efficient local search procedure. This is the approach proposed in this paper. In particular, we consider the mesh adaptive search algorithm, a direct search method aimed at improving the local search step to refine the quality of the solution. This hybrid strategy has been applied to six illustrative free-form shapes exhibiting challenging features, including the three examples in previous paper. A comparative analysis of our results with respect to the previous methodology is also reported. Our experimental results show that this hybrid scheme performs extremely well. It also outperforms the previous approach for all instances in our benchmark.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Audet, C., Dennis Jr, J.E.: Mesh adaptive direct search algorithms for constrained optimization. SIAM J. Optim. 17(1), 188–217 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barhak, J., Fischer, A.: Parameterization and reconstruction from 3D scattered points based on neural network and PDE techniques. IEEE Trans. Vis. Computer Graph. 7(1), 1–16 (2001)

    Article  MATH  Google Scholar 

  3. Barnhill, R.E.: Geometric Processing for Design and Manufacturing. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  4. Castillo, E., Iglesias, A.: Some characterizations of families of surfaces using functional equations. ACM Trans. Graph. 16(3), 296–318 (1997)

    Article  Google Scholar 

  5. Dasgupta, D. (ed.): Artificial Immune Systems and Their Applications. Springer, Berlin (1999)

  6. De Castro, L.N., Timmis, J.: Artificial Immune Systems: A New Computational Intelligence Approach. Springer, London (2002)

    MATH  Google Scholar 

  7. De Castro, L.N., Von Zuben, F.J.: Artificial Immune Systems: Part I—Basic Theory and Applications. Technical Report-RT DCA 01/99 (1999)

  8. De Castro, L.N., Von Zuben, F.J.: Learning and optimization using the clonal selection principle. IEEE Trans. Evol. Comput. 6(3), 239–251 (2002)

    Article  Google Scholar 

  9. Dierckx, P.: Curve and Surface Fitting with Splines. Oxford University Press, Oxford (1993)

    MATH  Google Scholar 

  10. Echevarría, G., Iglesias, A., Gálvez, A.: Extending neural networks for B-spline surface reconstruction. Lectures Notes in Computer Science, vol 2330, pp. 305–314 (2002)

  11. Farin, G.: Curves and surfaces for CAGD, 5th edn. Morgan Kaufmann, San Francisco (2002)

    Google Scholar 

  12. Fister Jr, I., Perc, M., Ljubic, K., Kamal, S.M., Iglesias, A., Fister, I.: Particle swarm optimization for automatic creation of complex graphic characters. Chaos Solitons Fractals 73, 29–35 (2015)

    Article  MathSciNet  Google Scholar 

  13. Gálvez, A., Cobo, A., Puig-Pey, J., Iglesias, A.: Particle swarm optimization for Bézier surface reconstruction. Lectures Notes in Computer Science, vol. 5102, pp. 116–125 (2008)

  14. Gálvez, A., Iglesias, A.: Efficient particle swarm optimization approach for data fitting with free knot B-splines. Computer-Aided Design 43(12), 1683–1692 (2011)

    Article  Google Scholar 

  15. Gálvez, A., Iglesias, A.: Particle swarm optimization for non-uniform rational B-spline surface reconstruction from clouds of 3D data points. Inf. Sci. 192(1), 174–192 (2012)

    Article  Google Scholar 

  16. Gálvez, A., Iglesias, A.: A new iterative mutually-coupled hybrid GA-PSO approach for curve fitting in manufacturing. Appl. Soft Comput. 13(3), 1491–1504 (2013)

    Article  Google Scholar 

  17. Gálvez A., Iglesias A.: Firefly algorithm for polynomial Bzier surface parameterization. J. Appl. Math. 9 (2013) (Article ID 237984)

  18. Gálvez A., Iglesias A.: Firefly algorithm for explicit B-Spline curve fitting to data points. Math. Problems Eng. 12 (2013) (Article ID 528215)

  19. Gálvez A., Iglesias A.: From nonlinear optimization to convex optimization through firefly algorithm and indirect approach with applications to CAD/CAM. Sci. World J. 10 (2013) (Article ID 283919)

  20. Gálvez A., Iglesias A.: Firefly algorithm for Bézier curve approximation. In: Proceedings of International Conference on Computational Science and Applications, ICCSA’2013 (2013)

  21. Gálvez A., Iglesias A.: Cuckoo search with Lévy flights for weighted Bayesian energy functional optimization in global-support curve data fitting. Sci. World J. 11 (2014) (Article ID 138760)

  22. Gálvez A., Iglesias A.: New memetic self-adaptive firefly algorithm for continuous optimization. Int. J. Bio-Inspired Comput. (in press)

  23. Gálvez, A., Iglesias, A., Avila, A.: Discrete Bézier curve fitting with artificial immune systems. Stud. Comput. Intell. 441, 59–75 (2013)

    Article  Google Scholar 

  24. Gálvez, A., Iglesias, A., Avila, A.: Immunological-based approach for accurate fitting of 3D noisy data points with Bézier surfaces. In: Proceedings of International Conference on Computational Science, ICCS (2013)

  25. Gálvez A., Iglesias A., Avila, A.: Applying clonal selectiontheory to data fitting with rational Bézier curves. In: Proceedings of Cyberworlds 2014, CW’2014, Santander (Spain), pp. 221–228. IEEE Computer Society Press, Los Alamitos (2014)

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

    Article  Google Scholar 

  27. Gálvez, A., Iglesias, A., Cobo, A., Puig-Pey, J., Espinola, J.: Bézier curve and surface fitting of 3D point clouds through genetic algorithms, functional networks and least-squares approximation. Lectures Notes in Computer Science, vol. 4706, pp. 680–693 (2007)

  28. Gálvez, A., Iglesias, A., Puig-Pey, J.: Iterative two-step genetic-algorithm method for efficient polynomial B-spline surface reconstruction. Inf. Sci. 182(1), 56–76 (2012)

    Article  MathSciNet  Google Scholar 

  29. Gu, P., Yan, X.: Neural network approach to the reconstruction of free-form surfaces for reverse engineering. Computer-Aided Design 27(1), 59–64 (1995)

    Article  Google Scholar 

  30. Hoffmann, M.: Numerical control of Kohonen neural network for scattered data approximation. Numer. Algorithms 39, 175–186 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Iglesias, A., Echevarría, G., Gálvez, A.: Functional networks for B-spline surface reconstruction. Future Gener. Computer Syst. 20(8), 1337–1353 (2004)

    Article  MATH  Google Scholar 

  32. Iglesias, A., Gálvez, A.: A new artificial intelligence paradigm for computer aided geometric design. Lectures Notes in Artificial Intelligence, vol. 2001, pp. 200–213 (1930)

  33. Iglesias, A., Gálvez, A.: Applying functional networks to fit data points from B-spline surfaces. In: Proceedings of the Computer Graphics International, CGI’2001, Hong-Kong (China), pp. 329–332. IEEE Computer Society Press, Los Alamitos (2001)

  34. Iglesias, A., Gálvez, A.: Curve fitting with RBS functional networks. In: Proceedings of International Conference on Convergence Information Technology-ICCIT’2008 (2008)

  35. Iglesias, A., Gálvez, A.: Hybrid functional-neural approach for surface reconstruction. Math. Problems Eng. 13 (2014) (Article ID 351648)

  36. Jing, L., Sun, L.: Fitting B-spline curves by least squares support vector machines. In: Proceedings of the 2nd International Conference on Neural Networks & Brain, pp. 905–909. Beijing (China). IEEE Press (2005)

  37. Jupp, D.L.B.: Approximation to data by splines with free knots. SIAM J. Numer. Anal. 15, 328–343 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  38. Knopf, G.K., Kofman, J.: Adaptive reconstruction of free-form surfaces using Bernstein basis function networks. Eng. Appl. Artif. Intell. 14(5), 577–588 (2001)

    Article  Google Scholar 

  39. Loucera, C., Gálvez, A., Iglesias, A.: Simulated annealingalgorithm for Bezier curve approximation. In: Proceedings of Cyberworlds 2014, CW’2014, Santander (Spain), pp. 182–189. IEEE Computer Society Press, Los Alamitos (2014)

  40. Lyche, T., Morken, K.: A data-reduction strategy for splines with applications to the approximation of functions and data. IMA J. Numer. Anal. 8, 185–208 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  42. Park, H.: An error-bounded approximate method for representing planar curves in B-splines. Computer Aided Geom. Design 21, 479–497 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  44. Patrikalakis, N.M., Maekawa, T.: Shape Interrogation for Computer Aided Design and Manufacturing. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  45. Piegl, L., Tiller, W.: The NURBS Book. Springer, Berlin Heidelberg (1997)

    Book  MATH  Google Scholar 

  46. Pottmann, H., Leopoldseder, S., Hofer, M., Steiner, T., Wang, W.: Industrial geometry: recent advances and applications in CAD. Computer-Aided Design 37, 751–766 (2005)

    Article  Google Scholar 

  47. Powell, M.J.D.: Curve fitting by splines in one variable. In: Hayes, J.G. (ed.) Numerical Approximation to Functions and Data. Athlone Press, London (1970)

    Google Scholar 

  48. Rice, J.R.: The Approximation of Functions, vol. 2. Addison-Wesley, Reading (1969)

    MATH  Google Scholar 

  49. Sarfraz, M., Raza, S.A.: Capturing outline of fonts using genetic algorithms and splines. In: Proceedings of Fifth International Conference on Information Visualization IV’2001, pp. 738–743. IEEE Computer Society Press (2001)

  50. Ulker, E., Arslan, A.: Automatic knot adjustment using an artificial immune system for B-spline curve approximation. Inf. Sci. 179, 1483–1494 (2009)

    Article  Google Scholar 

  51. Varady, T., Martin, R.: Reverse Engineering. In: Farin, G., Hoschek, J., Kim, M. (eds.) Handbook of Computer Aided Geometric Design. Elsevier, Amsterdam (2002)

    Google Scholar 

  52. Wang, W.P., Pottmann, H., Liu, Y.: Fitting B-spline curves to point clouds by curvature-based squared distance minimization. ACM Trans. Graph. 25(2), 214–238 (2006)

    Article  Google Scholar 

  53. Yang, H.P., Wang, W.P., Sun, J.G.: Control point adjustment for B-spline curve approximation. Computer-Aided Design 36, 639–652 (2004)

    Article  Google Scholar 

  54. Yoshimoto F., Moriyama, M., Harada T.: Automatic knot adjustment by a genetic algorithm for data fitting with a spline. In: Proceedings of Shape Modeling International’99, pp. 162–169. IEEE Computer Society Press (1999)

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

    Article  Google Scholar 

  56. Zhao, X., Zhang, C., Yang, B., Li, P.: Adaptive knot adjustment using a GMM-based continuous optimization algorithm in B-spline curve approximation. Computer-Aided Design 43, 598–604 (2011)

    Article  Google Scholar 

Download references

Acknowledgments

This research work has been kindly supported by the Computer Science National Program of the Spanish Ministry of Economy and Competitiveness, Project Ref. #TIN2012-30768, Toho University (Funabashi, Japan), and the University of Cantabria (Santander, Spain). The authors are particularly grateful to the Department of Information Science of Toho University for all the facilities given to carry out this work.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Computational Sciences, E.T.S.I. Caminos, Canales y Puertos, University of Cantabria, Avda. de los Castros, s/n, 39005, Santander, Spain

    Andrés Iglesias, Akemi Gálvez & Andreina Avila

  2. Department of Information Science, Faculty of Science, Narashino Campus, Toho University, 2-2-1 Miyama, 274-8510, Funabashi, Japan

    Andrés Iglesias

Authors
  1. Andrés Iglesias
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akemi Gálvez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andreina Avila
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrés Iglesias.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Iglesias, A., Gálvez, A. & Avila, A. Hybridizing mesh adaptive search algorithm and artificial immune systems for discrete rational Bézier curve approximation. Vis Comput 32, 393–402 (2016). https://doi.org/10.1007/s00371-015-1181-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00371-015-1181-0

Keywords

Search

Navigation