## Journal of Applied Mathematics

- J. Appl. Math.
- Volume 2013, Special Issue (2013), Article ID 237984, 9 pages.

### Firefly Algorithm for Polynomial Bézier Surface Parameterization

Akemi Gálvez and Andrés Iglesias

**Full-text: Open access**

#### Abstract

A classical issue in many applied fields is to obtain an approximating surface to a given set of data points. This problem arises in Computer-Aided Design and Manufacturing (CAD/CAM), virtual reality, medical imaging, computer graphics, computer animation, and many others. Very often, the preferred approximating surface is polynomial, usually described in parametric form. This leads to the problem of determining suitable parametric values for the data points, the so-called surface parameterization. In real-world settings, data points are generally irregularly sampled and subjected to measurement noise, leading to a very difficult nonlinear continuous optimization problem, unsolvable with standard optimization techniques. This paper solves the parameterization problem for polynomial Bézier surfaces by applying the firefly algorithm, a powerful nature-inspired metaheuristic algorithm introduced recently to address difficult optimization problems. The method has been successfully applied to some illustrative examples of open and closed surfaces, including shapes with singularities. Our results show that the method performs very well, being able to yield the best approximating surface with a high degree of accuracy.

#### Article information

**Source**

J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 237984, 9 pages.

**Dates**

First available in Project Euclid: 14 March 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.jam/1394807368

**Digital Object Identifier**

doi:10.1155/2013/237984

**Zentralblatt MATH identifier**

06950572

#### Citation

Gálvez, Akemi; Iglesias, Andrés. Firefly Algorithm for Polynomial Bézier Surface Parameterization. J. Appl. Math. 2013, Special Issue (2013), Article ID 237984, 9 pages. doi:10.1155/2013/237984. https://projecteuclid.org/euclid.jam/1394807368

#### References

- R. E. Barnhill,
*Geometry Processing for Design and Manufacturing*, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1992. - G. Echevarra, A. Iglesias, and A. Galvez, “Extending neural networks for \emphB-spline surface reconstruction,” in
*Computational Science–-ICCS 2002*, vol. 2330 of*Lecture Notes in Computer Science*, pp. 305–314, 2002. - A. Gálvez, A. Iglesias, and J. Puig-Pey, “Iterative two-step genetic-algorithm-based method for efficient polynomial \emphB-spline surface reconstruction,”
*Information Sciences*, vol. 182, pp. 56–76, 2012.Mathematical Reviews (MathSciNet): MR2847002

Digital Object Identifier: doi:10.1016/j.ins.2010.09.031 - A. Gálvez and A. Iglesias, “Particle swarm optimization for non-uniform rational \emphB-spline surface reconstruction from clouds of 3D data points,”
*Information Sciences*, vol. 192, pp. 174–192, 2012. - A. Iglesias and A. Galvez, “A new artificial intelligence paradigm for computer aided geometric design,” in
*Artificial Intelligence and Symbolic Computation*, vol. 1930, pp. 200–213, Lecture Notes in Computer Science, 2001. - A. Iglesias, G. Echevarría, and A. Gálvez, “Functional networks for \emphB-spline surface reconstruction,”
*Future Generation Computer Systems*, vol. 20, no. 8, pp. 1337–1353, 2004. - H. Pottmann, S. Leopoldseder, M. Hofer, T. Steiner, and W. Wang, “Industrial geometry: recent advances and applications in CAD,”
*Computer Aided Design*, vol. 37, no. 7, pp. 751–766, 2005. - T. Varady and R. Martin, “Reverse engineering,” in
*Handbook of Computer Aided Geometric Design*, pp. 651–681, North-Holland, Amsterdam, The Netherlands, 2002.Mathematical Reviews (MathSciNet): MR1928559

Zentralblatt MATH: 0995.68152

Digital Object Identifier: doi:10.1016/B978-044451104-1/50027-7 - N. M. Patrikalakis and T. Maekawa,
*Shape Interrogation for Computer Aided Design and Manufacturing*, Springer, Berlin, Germany, 2002.Mathematical Reviews (MathSciNet): MR1891533 - A. Gálvez and A. Iglesias, “Efficient particle swarm optimization approach for data fitting with free knot \emphB-splines,”
*Computer Aided Design*, vol. 43, no. 12, pp. 1683–1692, 2011. - A. Galvez and A. Iglesias, “A new iterative mutually-coupled hybrid GA-PSO approach for curve fitting in manufacturing,”
*Applied Soft Computing*, vol. 13, no. 3, pp. 1491–1504, 2013. - J. Ling and S. Li, “Fitting \emphB-spline curves by least squares support vector machines,” in
*Proceedings of the International Conference on Neural Networks and Brain Proceedings (ICNNB '05)*, pp. 905–909, Beijing, China, October 2005. - D. L. B. Jupp, “Approximation to data by splines with free knots,”
*SIAM Journal on Numerical Analysis*, vol. 15, no. 2, pp. 328–343, 1978.Mathematical Reviews (MathSciNet): MR488884

Zentralblatt MATH: 0403.65004

Digital Object Identifier: doi:10.1137/0715022 - T. C. M. Lee, “On algorithms for ordinary least squares regression spline fitting: a comparative study,”
*Journal of Statistical Computation and Simulation*, vol. 72, no. 8, pp. 647–663, 2002. - W. Li, S. Xu, G. Zhao, and L. P. Goh, “Adaptive knot placement in \emphB-spline curve approximation,”
*Computer Aided Design*, vol. 37, no. 8, pp. 791–797, 2005. - H. Park, “An error-bounded approximate method for representing planar curves in \emphB-splines,”
*Computer Aided Geometric Design*, vol. 21, no. 5, pp. 479–497, 2004.Mathematical Reviews (MathSciNet): MR2058393

Zentralblatt MATH: 1069.65528

Digital Object Identifier: doi:10.1016/j.cagd.2004.03.003 - H. Park and J. Lee, “\emphB-spline curve fitting based on adaptive curve refinement using dominant points,”
*Computer Aided Design*, vol. 39, no. 6, pp. 439–451, 2007. - G. Farin,
*Curves and Surfaces for CAGD*, Morgan Kaufmann, San Francisco, Calif, USA, 5th edition, 2002. - J. Hoschek and D. Lasser,
*Fundamentals of Computer Aided Geometric Design*, A K Peters, Wellesley, Mass, USA, 1993.Mathematical Reviews (MathSciNet): MR1258308 - L. Piegl and W. Tiller,
*The NURBS Book*, Springer, Berlin, Germany, 1997. - D. F. Rogers,
*An Introduction to NURBS: With His Historical Perspective*, Morgan Kaufmann, 2000. - G. E. Hölzle, “Knot placement for piecewise polynomial approximation of curves,”
*Computer-Aided Design*, vol. 15, no. 5, pp. 295–296, 1983. - W. Ma and J. Kruth, “Parameterization of randomly measured points for least squares fitting of \emphB-spline curves and surfaces,”
*Computer-Aided Design*, vol. 27, no. 9, pp. 663–675, 1995. - L. A. Piegl and W. Tiller, “Least-squares \emphB-spline curve approximation with arbitrary end derivatives,”
*Engineering with Computers*, vol. 16, no. 2, pp. 109–116, 2000. - T. Várady, R. R. Martin, and J. Cox, “Reverse engineering of geometric models–-an introduction,”
*Computer Aided Design*, vol. 29, no. 4, pp. 255–268, 1997. - W. P. Wang, H. Pottmann, and Y. Liu, “Fitting \emphB-spline curves to point clouds by curvaturebased squared distance minimization,”
*ACM Transactions on Graphics*, vol. 25, no. 2, pp. 214–238, 2006. - F. Yoshimoto, M. Moriyama, and T. Harada, “Automatic knot adjustment by a genetic algorithm for data fitting with a spline,” in
*Proceedings of the International Conference on Shape Modeling International and Applications*, pp. 162–169, IEEE Computer Society Press, 1999. - F. Yoshimoto, T. Harada, and Y. Yoshimoto, “Data fitting with a spline using a real-coded genetic algorithm,”
*Computer Aided Design*, vol. 35, no. 8, pp. 751–760, 2003. - J. Barhak and A. Fischer, “Parameterization and reconstruction from 3D scattered points based on neural network and PDE techniques,”
*IEEE Transactions on Visualization and Computer Graphics*, vol. 7, no. 1, pp. 1–16, 2001. - M. Alhanaty and M. Bercovier, “Curve and surface fitting and design by optimal control methods,”
*Computer Aided Design*, vol. 33, no. 2, pp. 167–182, 2001. - P. Dierckx,
*Curve and Surface Fitting with Splines*, Oxford University Press, Oxford, Miss, USA, 1993. - T. Lyche and K. Mørken, “Knot removal for parametric \emphB-spline curves and surfaces,”
*Computer Aided Geometric Design*, vol. 4, no. 3, pp. 217–230, 1987.Mathematical Reviews (MathSciNet): MR917781

Digital Object Identifier: doi:10.1016/0167-8396(87)90013-6 - M. J. D. Powell, “Curve fitting by splines in one variable,” in
*Numerical Approximation to Functions and Data*, J. G. Hayes, Ed., Athlone Press, London, UK, 1970. - J. R. Rice,
*Numerical Methods, Software and Analysis*, Academic Press, New York, NY, USA, 2nd edition, 1993. - H. Yang, W. Wang, and J. Sun, “Control point adjustment for \emphB-spline curve approximation,”
*Computer Aided Design*, vol. 36, no. 7, pp. 639–652, 2004. - E. Castillo and A. Iglesias, “Some characterizations of families of surfaces using functional equations,”
*ACM Transactions on Graphics*, vol. 16, no. 3, pp. 296–318, 1997. - A. Gálvez, J. Puig-Pey, and A. Iglesias, “A differential method for parametric surface intersection,” in
*Computational science and its applications–-ICCSA 2004*, vol. 3044 of*Lecture Notes in Computer Science*, pp. 651–660, Springer, Berlin, Germany, 2004.Mathematical Reviews (MathSciNet): MR2189660

Zentralblatt MATH: 1127.65307

Digital Object Identifier: doi:10.1007/978-3-540-24709-8_69 - P. Gu and X. Yan, “Neural network approach to the reconstruction of freeform surfaces for reverse engineering,”
*Computer-Aided Design*, vol. 27, no. 1, pp. 59–64, 1995. - M. Hoffmann, “Numerical control of Kohonen neural network for scattered data approximation,”
*Numerical Algorithms*, vol. 39, no. 1–3, pp. 175–186, 2005.Mathematical Reviews (MathSciNet): MR2137750

Zentralblatt MATH: 1101.65309

Digital Object Identifier: doi:10.1007/s11075-004-3628-7 - G. K. Knopf and J. Kofman, “Free-form surface reconstruction using Bernstein basis function networks,” in
*Proceedings of the Artificial Neural Networks in Engineering Conference (ANNIE '99)*, vol. 9, pp. 797–802, ASME Press, November 1999. - A. Iglesias and A. Galvez, “Applying functional networks to fit data points from \emphB-spline surfaces,” in
*Proceedings of the Computer Graphics International (CGI '01)*, pp. 329–332, IEEE Computer Society Press, Hong Kong, China, 2001. - A. Galvez, A. Iglesias, A. Cobo, J. Puig-Pey, and J. Espinola, “Bézier curve and surface fitting of 3D point clouds through genetic algorithms, functional networks and least-squares approximation,” in
*Computational Science and Its Applications–-ICCSA 2007*, vol. 4706 of*Lecture Notes in Computer Science*, pp. 680–693, 2007. - M. Sarfraz and S. A. Raza, “Capturing outline of fonts using genetic algorithms and splines,” in
*Proceedings of the 5th International Conference on Information Visualization (IV '01)*, pp. 738–743, IEEE Computer Society Press, 2001. - A. Galvez, A. Iglesias, and A. Avila, “Immunological-based approach for accurate fitting of 3D noisy data points with Bezier surfaces,” in
*Proceedings of the International Conference on Computational Science (ICCS '13)*, vol. 18, pp. 50–59, Procedia Computer Science, 2013. - E. Ülker and A. Arslan, “Automatic knot adjustment using an artificial immune system for \emphB-spline curve approximation,”
*Information Sciences*, vol. 179, no. 10, pp. 1483–1494, 2009. - X. Zhao, C. Zhang, B. Yang, and P. Li, “Adaptive knot placement using a GMM-based continuous optimization algorithm in \emphB-spline curve approximation,”
*Computer Aided Design*, vol. 43, no. 6, pp. 598–604, 2011. - X.-S. Yang, “Firefly algorithms for multimodal optimization,” in
*Stochastic Algorithms: Foundations and Applications*, vol. 5792 of*Lectures Notes in Computer Science*, pp. 169–178, Springer, Berlin, Germany, 2009.Mathematical Reviews (MathSciNet): MR2580252

Zentralblatt MATH: 1260.90164

Digital Object Identifier: doi:10.1007/978-3-642-04944-6_14 - X. S. Yang, “Firey algorithm, stochastic test functions and design optimisation,”
*International Journal of Bio-Inspired Computation*, vol. 2, no. 2, pp. 78–84, 2010. - S. L. Tilahun and H. C. Ong, “Modified firefly algorithm,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 467631, 12 pages, 2012. - X.-S. Yang,
*Nature-Inspired Metaheuristic Algorithms*, Luniver Press, Frome, UK, 2nd edition, 2010. - X.-S. Yang,
*Engineering Optimization: An Introduction with Metaheuristic Applications*, Wiley & Sons, New Jersey, NJ, USA, 2010. - I. Fister, I. Fister Jr., X. S. Yang, and J. Brest, “A comprehensive review of firefly algorithms,”
*Swarm and Evolutionary Computation*. In press. - I. Fister, X. S. Yang, J. Brest, and I. Fister Jr., “Memetic self-adaptive firefly algorithm,” in
*Swarm Intelligence and Bio-Inspired Computation: Theory and Applications*, X. S. Yang, Z. Cui, R. Xiao, A. H. Gandomi, and M. Karamanoglu, Eds., pp. 73–102, Elsevier, 2013.

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