Memetic firefly algorithm for data fitting with rational curves

This paper concerns the problem of obtaining a smooth fitting curve to a given set of (noisy) data points. This problem arises frequently in several industrial fields, such as computer-aided design and manufacturing (construction of car bodies, ship hulls, airplane fuselage), computer graphics and animation, medicine, and many others. The classical approach relies on polynomial functions to solve this problem. It has been noticed, however, that some shapes cannot be properly approximated through this polynomial scheme. In this paper, we address this issue by using rational functions, particularly the rational Bernstein basis functions. This poses an additional challenge: we have not only to compute the poles of the resulting rational Bézier fitting curve but also to obtain their corresponding weights and a suitable parameterization of data points. Overall, this leads to a continuous multivariate nonlinear optimization problem that cannot be solved through traditional mathematical optimization techniques. Our approach to tackle this issue is based on a memetic firefly algorithm combining a powerful metaheuristic technique (the firefly algorithm) for global optimization with a local search method. The performance of our scheme is illustrated through its application to four illustrative examples of free-form synthetic shapes. Our experimental results show that our memetic approach performs very well, and allows us to reconstruct the underlying shape of data points automatically with high accuracy. A comparative analysis on our benchmark shows that our approach outperforms some alternative methods reported in the literature for this problem.