Abstract
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.
Highlights
Surface approximation is a recurrent problem in geometric modeling, data analysis, image processing, and many other engineering applications [1]
An experimental setup was configured for each test function as follows: the function is evaluated at 1024 points in a square grid of 32 × 32 over the interval [0, 1] × [0, 1]
For each experimental setup, a collection of 100 noisy data sets is generated at three different signal noise ratios (SNR) 2, 3, and 4, respectively, where SNR is defined as SD(f)/σ
Summary
Surface approximation is a recurrent problem in geometric modeling, data analysis, image processing, and many other engineering applications [1]. The choice of the number and positions of the knots of a spline is fundamental, as well as the method used to solve the system of equations Both tasks are critical and troublesome, leading to a hard continuous multimodal and multivariate nonlinear optimization problem with many local optimal solutions. Since we consider a spline based approach, we remark the fact that the main issue associated with the surface approximation through splines is to find the best set of knots, where the term “best” implies an adequate choice in the number and location of the knots To perform this task, in [5], the author provides a survey on the main algorithms used to carry out this task, which are based on regression spline methods and their respective optimizations.
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