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Mathematical Problems in Engineering
Volume 2014, Article ID 706247, 11 pages
Research Article

Hierarchical Genetic Algorithm for B-Spline Surface Approximation of Smooth Explicit Data

1Centro de Investigaciones en Óptica, A.C., Loma del Bosque 115, Col. Lomas del Campestre, 37150 León, GT, Mexico
2Instituto Tecnológico Superior de Irapuato, Carretera Irapuato Silao Km 12.5, 36821 Irapuato, GT, Mexico
3División de Ingenierías Campus Irapuato-Salamanca, Universidad de Guanajuato, Carretera Salamanca-Valle de Santiago Km 3.5+1.8 Km, Com. Palo Blanco s/n, 36885 Salamanca, GT, Mexico

Received 8 January 2014; Revised 12 May 2014; Accepted 14 May 2014; Published 5 June 2014

Academic Editor: K. M. Liew

Copyright © 2014 C. H. Garcia-Capulin et al. This 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.


B-spline surface approximation has been widely used in many applications such as CAD, medical imaging, reverse engineering, and geometric modeling. Given a data set of measures, the surface approximation aims to find a surface that optimally fits the data set. One of the main problems associated with surface approximation by B-splines is the adequate selection of the number and location of the knots, as well as the solution of the system of equations generated by tensor product spline surfaces. In this work, we use a hierarchical genetic algorithm (HGA) to tackle the B-spline surface approximation of smooth explicit data. The proposed approach is based on a novel hierarchical gene structure for the chromosomal representation, which allows us to determine the number and location of the knots for each surface dimension and the B-spline coefficients simultaneously. The method is fully based on genetic algorithms and does not require subjective parameters like smooth factor or knot locations to perform the solution. In order to validate the efficacy of the proposed approach, simulation results from several tests on smooth surfaces and comparison with a successful method have been included.