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Four Adaptive Memetic Bat Algorithm Schemes for Bézier Curve Parameterization

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Transactions on Computational Science XXVIII

Part of the book series: Lecture Notes in Computer Science ((TCOMPUTATSCIE,volume 9590))

Abstract

This paper is an extension of a previous one presented at the conference Cyberworlds 2015. In that work we addressed the problem to fit a given set of data points in the least-square sense by using a polynomial Bézier curve. This problem arises in many scientific and industrial domains, such as numerical analysis, statistical regression, computer-aided design and manufacturing, computer graphics, virtual reality, etc. A critical issue to address this least-squares minimization problem is that of curve parameterization. In our previous work we solve it by applying a powerful nature-inspired optimization method called the bat algorithm. Although we obtained pretty good results on a number of examples, the method can still be further improved by considering a memetic approach, in which the global search bat algorithm is hybridized with a local search procedure to enhance the exploitation phase of the minimization process. In this paper we extend our previous method through two local search strategies: Luus-Jaakola and ASSRS. In both cases, the adaptive and self-adaptive versions are considered, leading to four memetic schemes. A comparative analysis of our results on the previous benchmark for these four memetic schemes and our previous method has been carried out. It shows that the memetic approaches improve the efficiency of the previous method at different extent for all instances in our benchmark.

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References

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

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    MATH  Google Scholar 

  5. Echevarría, G., Iglesias, A., Gálvez, A.: Extending neural networks for B-spline surface reconstruction. In: Sloot, P.M.A., Tan, C.J.K., Dongarra, J., Hoekstra, A.G. (eds.) ICCS-ComputSci 2002, Part II. LNCS, vol. 2330, pp. 305–314. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

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

    Google Scholar 

  7. Gálvez, A., Cobo, A., Puig-Pey, J., Iglesias, A.: Particle swarm optimization for Bézier surface reconstruction. In: Bubak, M., Albada, G.D., Dongarra, J., Sloot, P.M.A. (eds.) ICCS 2008, Part II. LNCS, vol. 5102, pp. 116–125. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

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

    Article  Google Scholar 

  9. 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 

  10. 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 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  15. Gálvez, A., Iglesias, A.: Particle-based meta-model for continuous breakpoint optimization in smooth local-support curve fitting. Appl. Math. Comput. 275, 195–212 (2016)

    MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  18. 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 

  19. 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. In: Gervasi, O., Gavrilova, M.L. (eds.) ICCSA 2007. LNCS, vol. 4706, pp. 680–693. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  20. 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 

  21. Gopalakrishnan, G.: On the convergence of the LJ search method. J. Optim. Theory Appl. 28(3), 429–434 (1979)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Hoos, H.H., Stutzle, T.: Stochastic Local Search: Foundations and Applications. Morgan Kaufmann Series in Artificial Intelligence, Burlington (2005)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  26. Iglesias, A., Gálvez, A.: A new artificial intelligence paradigm for computer-aided geometric design. In: Campbell, J., Roanes-Lozano, E. (eds.) AISC 2000. LNCS (LNAI), vol. 1930, p. 200. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  27. 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)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Iglesias, A., Gálvez, A.: Memetic firefly algorithm for data fitting with rational curves. In: Proceedings of Congress on Evolutionary Computation-CEC 2015, Sendai, Japan, pp. 507–514. IEEE CS Press, Los Alamitos (2015)

    Google Scholar 

  30. Iglesias, A., Gálvez, A., Avila, A.: Hybridizing mesh adaptive search algorithm and artificial immune systems for discrete rational Bézier curve approximation. Vis. Comput. (in press)

    Google Scholar 

  31. Iglesias, A., Gálvez, A., Collantes, M.: Bat algorithm for curve parameterization in data fitting with polynomial Bézier curves. In: Proceedings of Cyberworlds 2015, Visby, Sweden, pp. 107–114. IEEE Computer Society Press, Los Alamitos (2015)

    Google Scholar 

  32. Iglesias, A., Gálvez, A., Loucera, C.: Two simulated annealing optimization schemas for rational Bézier curve fitting in the presence of noise. Math. Prob. Eng. 2016, 17 (2016). Article ID 8241275

    Article  Google Scholar 

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

    Google Scholar 

  34. 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 

  35. 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 

  36. Luus, R., Jaakola, T.H.I.: Optimization by direct search and systematic reduction of the size of search region. Am. Inst. Chem. Eng. J. (AIChE) 19(4), 760–766 (1973)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  38. Marsaglia, G.: Choosing a point from the surface of a sphere. Ann. Math. Stat. 43, 645–646 (1972)

    Article  MATH  Google Scholar 

  39. Muller, M.E.: A note on a method for generating points uniformly on n-dimensional spheres. Commun. ACM 2(4), 19–20 (1959)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

  45. 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 

  46. Rastrigin, L.A.: The convergence of the random search method in the extremal control of a many parameter system. Autom. Remote Control 24(10), 1337–1342 (1963)

    Google Scholar 

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

    MATH  Google Scholar 

  48. 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)

    Google Scholar 

  49. Schumer, M.A., Steiglitz, K.: Adaptive step size random search. IEEE Trans. Autom. Control 13(3), 270–276 (1968)

    Article  Google Scholar 

  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 Science, 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. Graphics 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. Comput. Aided Des. 36, 639–652 (2004)

    Article  Google Scholar 

  54. Yang, X.-S.: A new metaheuristic bat-inspired algorithm. In: González, J.R., Pelta, D.A., Cruz, C., Terrazas, G., Krasnogor, N. (eds.) NICSO 2010. SCI, vol. 284, pp. 65–74. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  55. Yang, X.S., Gandomi, A.H.: Bat algorithm: a novel approach for global engineering optimization. Eng. Comput. 29(5), 464–483 (2012)

    Article  Google Scholar 

  56. Yang, X.S.: Bat algorithm: literature review and applications. Int. J. Bio-Inspired Comput. 5(3), 141–149 (2013)

    Article  Google Scholar 

  57. Yoshimoto, F., Moriyama, M., Harada, T.: Automatic knot adjustment by a genetic algorithm for data fitting with a spline. In: Proceeding of Shape Modeling International 1999, pp. 162–169, IEEE Computer Society Press (1999)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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Acknowledgements

This research 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.

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Authors and Affiliations

  1. Department of Applied Mathematics and Computational Sciences, University of Cantabria, Avda. de los Castros, s/n, 39005, Santander, Spain

    Andrés Iglesias, Akemi Gálvez & Marta Collantes

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

    Andrés Iglesias & Akemi Gálvez

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  1. Andrés Iglesias
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  2. Akemi Gálvez
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Correspondence to Andrés Iglesias .

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  1. University of Calgary , Calgary, Alberta, Canada

    Marina L. Gavrilova

  2. Sardina Systems , Tallinn, Estonia

    C.J. Kenneth Tan

  3. Nanyang Technological University , Singapore, Singapore

    Alexei Sourin

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Iglesias, A., Gálvez, A., Collantes, M. (2016). Four Adaptive Memetic Bat Algorithm Schemes for Bézier Curve Parameterization. In: Gavrilova, M., Tan, C., Sourin, A. (eds) Transactions on Computational Science XXVIII. Lecture Notes in Computer Science(), vol 9590. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53090-0_7

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