Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 528215, 12 pages http://dx.doi.org/10.1155/2013/528215 Research Article Firefly Algorithm for Explicit B-Spline Curve Fitting to Data Points Akemi Gálvez 1 and Andrés Iglesias 1,2 1 Department of Applied Mathematics and Computational Sciences, E.T.S.I. Caminos, Canales y Puertos, University of Cantabria, Avenida de los Castros s/n, 39005 Santander, Spain 2 Department of Information Science, Faculty of Sciences, Toho University, 2-2-1 Miyama, Funabashi 274-8510, Japan Correspondence should be addressed to Andr´ es Iglesias; iglesias@unican.es Received 11 May 2013; Revised 8 September 2013; Accepted 13 September 2013 Academic Editor: Yang Xu Copyright © 2013 A. G´ alvez and A. Iglesias. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tis paper introduces a new method to compute the approximating explicit B-spline curve to a given set of noisy data points. Te proposed method computes all parameters of the B-spline ftting curve of a given order. Tis requires to solve a difcult continuous, multimodal, and multivariate nonlinear least-squares optimization problem. In our approach, this optimization problem is solved by applying the frefy algorithm, a powerful metaheuristic nature-inspired algorithm well suited for optimization. Te method has been applied to three illustrative real-world engineering examples from diferent felds. Our experimental results show that the presented method performs very well, being able to ft the data points with a high degree of accuracy. Furthermore, our scheme outperforms some popular previous approaches in terms of diferent ftting error criteria. 1. Introduction Te problem of recovering the shape of a curve/surface, also known as curve/surface reconstruction, has received much attention in the last few years [1–14]. A classical approach in this feld is to construct the curve as a cross-section of the surface of an object. Tis is a typical problem in many research and application areas such as medical science and biomedical engineering, in which a dense cloud of data points of the surface of a volumetric object (an internal organ, for instance) is acquired by means of noninvasive techniques such as computer tomography, magnetic resonance imaging, and ultrasound imaging. Te primary goal in these cases is to obtain a sequence of cross-sections of the object in order to construct the surface passing through them, a process called surface skinning. Another diferent approach consists of reconstructing the curve directly from a given set of data points, as it is typically done in reverse engineering for computer-aided design and manufacturing (CAD/CAM), by using 3D laser scanning, tactile scanning, or other digitizing devices [15, 16]. Depending on the nature of these data points, two diferent approaches can be employed: interpolation and approxima- tion. In the former, a parametric curve is constrained to pass through all input data points. Tis approach is typically employed for sets of data points that are sufciently accurate and smooth. On the contrary, approximation does not require the ftting curve to pass through the data points, but just close to them, according to prescribed distance criteria. Such a distance is usually measure along the normal vector to the curve at that point. Te approximation approach is partic- ularly well suited when data are not exact but subjected to measurement errors. Because this is the typical case in many real-world industrial problems, in this paper we focus on the approximation scheme to a given set of noisy data points. Tere two key components for a good approximation: a proper choice of the approximating function and a suitable parameter tuning. Te usual models for curve approximation are the free-form curves, such as B´ ezier, B-spline, and NURBS [17–26]. In particular, B-splines are the preferred