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.

Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Audet, C., Dennis Jr, J.E.: Mesh adaptive direct search algorithms for constrained optimization. SIAM J. Optim. 17(1), 188–217 (2006)
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)
Barnhill, R.E.: Geometric Processing for Design and Manufacturing. SIAM, Philadelphia (1992)
Castillo, E., Iglesias, A.: Some characterizations of families of surfaces using functional equations. ACM Trans. Graph. 16(3), 296–318 (1997)
Dasgupta, D. (ed.): Artificial Immune Systems and Their Applications. Springer, Berlin (1999)
De Castro, L.N., Timmis, J.: Artificial Immune Systems: A New Computational Intelligence Approach. Springer, London (2002)
De Castro, L.N., Von Zuben, F.J.: Artificial Immune Systems: Part I—Basic Theory and Applications. Technical Report-RT DCA 01/99 (1999)
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)
Dierckx, P.: Curve and Surface Fitting with Splines. Oxford University Press, Oxford (1993)
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)
Farin, G.: Curves and surfaces for CAGD, 5th edn. Morgan Kaufmann, San Francisco (2002)
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)
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)
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)
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)
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)
Gálvez A., Iglesias A.: Firefly algorithm for polynomial Bzier surface parameterization. J. Appl. Math. 9 (2013) (Article ID 237984)
Gálvez A., Iglesias A.: Firefly algorithm for explicit B-Spline curve fitting to data points. Math. Problems Eng. 12 (2013) (Article ID 528215)
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)
Gálvez A., Iglesias A.: Firefly algorithm for Bézier curve approximation. In: Proceedings of International Conference on Computational Science and Applications, ICCSA’2013 (2013)
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)
Gálvez A., Iglesias A.: New memetic self-adaptive firefly algorithm for continuous optimization. Int. J. Bio-Inspired Comput. (in press)
Gálvez, A., Iglesias, A., Avila, A.: Discrete Bézier curve fitting with artificial immune systems. Stud. Comput. Intell. 441, 59–75 (2013)
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)
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)
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)
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)
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)
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)
Hoffmann, M.: Numerical control of Kohonen neural network for scattered data approximation. Numer. Algorithms 39, 175–186 (2005)
Iglesias, A., Echevarría, G., Gálvez, A.: Functional networks for B-spline surface reconstruction. Future Gener. Computer Syst. 20(8), 1337–1353 (2004)
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)
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)
Iglesias, A., Gálvez, A.: Curve fitting with RBS functional networks. In: Proceedings of International Conference on Convergence Information Technology-ICCIT’2008 (2008)
Iglesias, A., Gálvez, A.: Hybrid functional-neural approach for surface reconstruction. Math. Problems Eng. 13 (2014) (Article ID 351648)
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)
Jupp, D.L.B.: Approximation to data by splines with free knots. SIAM J. Numer. Anal. 15, 328–343 (1978)
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)
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)
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)
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)
Park, H.: An error-bounded approximate method for representing planar curves in B-splines. Computer Aided Geom. Design 21, 479–497 (2004)
Park, H., Lee, J.H.: B-spline curve fitting based on adaptive curve refinement using dominant points. Computer-Aided Design 39, 439–451 (2007)
Patrikalakis, N.M., Maekawa, T.: Shape Interrogation for Computer Aided Design and Manufacturing. Springer, Heidelberg (2002)
Piegl, L., Tiller, W.: The NURBS Book. Springer, Berlin Heidelberg (1997)
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)
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)
Rice, J.R.: The Approximation of Functions, vol. 2. Addison-Wesley, Reading (1969)
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)
Ulker, E., Arslan, A.: Automatic knot adjustment using an artificial immune system for B-spline curve approximation. Inf. Sci. 179, 1483–1494 (2009)
Varady, T., Martin, R.: Reverse Engineering. In: Farin, G., Hoschek, J., Kim, M. (eds.) Handbook of Computer Aided Geometric Design. Elsevier, Amsterdam (2002)
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)
Yang, H.P., Wang, W.P., Sun, J.G.: Control point adjustment for B-spline curve approximation. Computer-Aided Design 36, 639–652 (2004)
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)
Yoshimoto, F., Harada, T., Yoshimoto, Y.: Data fitting with a spline using a real-coded algorithm. Computer-Aided Design 35, 751–760 (2003)
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)
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
Corresponding author
Rights and permissions
About this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00371-015-1181-0