Lévy Flight-Driven Simulated Annealing for B-spline Curve Fitting

  • Carlos Loucera
  • Andrés IglesiasEmail author
  • Akemi Gálvez
Part of the Studies in Computational Intelligence book series (SCI, volume 744)


Point cloud approximation by spline models, also called curve and surface reconstruction, is an active research field in computer-aided design and manufacturing (CAD/CAM). Due to the physical and mechanical processes used to obtain the data, the measurements are often affected by noise and other distortions. Obtaining a suitable spline model to reconstruct the underlying shape of the data while maintaining a low design complexity leads to a multivariate and highly non-linear optimization problem, also known to be non-convex and multi-modal. In this work, we propose a method to fit a given point cloud by means of a B-spline curve model. Our approach to solve the optimization problem is based on a powerful thermodynamics-driven metaheuristic known as the Simulated Annealing. We compute the model parameters by combining traditional SA techniques with Lévy flights (random walks based on the Lévy distribution). The ability to perform such a flight allows the algorithm to escape from local minima and energy plateaus, a strong requirement when dealing with highly multi-modal problems. The performance and robustness of our algorithm is tested against three illustrative examples. Our experimental results show that our method is able to reconstruct the underlying shape of the data, even in the presence of noise, with acceptable accuracy and in a completely automated way.


Data fitting Curve reconstruction Reverse engineering CAD/CAM B-spline functions Metaheuristic techniques Simulated annealing Lévy flights 



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. We also thank the anonymous reviewers who helped us to improve the chapter with their constructive comments and suggestions. A special recognition is also owe to our editor, Prof. Xin-She Yang, for his kind assistance and encouraging support during the process of writing this chapter.


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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Carlos Loucera
    • 1
  • Andrés Iglesias
    • 2
    • 3
    Email author
  • Akemi Gálvez
    • 2
    • 3
  1. 1.Department of Communications EngineeringUniversidad de CantabriaSantanderSpain
  2. 2.Faculty of Sciences, Department of Information ScienceToho UniversityFunabashiJapan
  3. 3.Department of Applied Mathematics and Computational SciencesUniversity of CantabriaSantanderSpain

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