Curve approximation by adaptive neighborhood simulated annealing and piecewise Bézier curves

• Published in: Soft computing (Berlin, Germany) Vol. 24; no. 24; pp. 18821 - 18839
• Main Authors: Ueda, E. K., Sato, A. K., Martins, T. C., Takimoto, R. Y., Rosso, R. S. U., Tsuzuki, M. S. G.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2020; Springer Nature B.V
• Subjects: Algorithms; Approximation; Artificial Intelligence; CAD/CAM; Computational Intelligence; Control; Crystallization; Curves; Data structures; Engineering; Mathematical Logic and Foundations; Mechatronics; Methodologies and Application; Optimization; Regularization; Robotics; Simulated annealing; Curve approximation; Piecewise Bézier curve; Adaptive neighborhood simulated annealing; Multi-objective simulated annealing
• ISSN: 1432-7643; 1433-7479
• DOI: 10.1007/s00500-020-05114-0

The curve approximation problem is widely researched in CAD/CAM and geometric modelling. The problem consists in determining an approximating curve from a given sequence of points. The usual approach is the minimization of the discrepancy between the approximating curve and the given sequence of points. However, the minimization of just the discrepancy leads to the overfitting problem, in which the solution is not unique. A new approach is proposed to overcome this problem, in which the length of the approximating curve is used as a regularization increasing the algorithm stability. Another new proposal is the discrepancy determination, in which a method that has the best ratio between accuracy and processing time is proposed. A new simulated annealing (SA) approach is used to minimize the problem, in which the next candidate is determined by a probability distribution controlled by the crystallization factor. The crystallization factor is low for higher temperatures ensuring the exploration of the domain. The crystallization factor is high for lower temperatures, corresponding the refinement phase of the SA. The approximating curve is represented as a piecewise cubic Bézier curve, which is a sequence of several connected cubic Bézier curves. The piecewise Bézier curve supports a new proposed data structure that improves the proposed algorithm. A comparison is also made between the used single-objective SA and the AMOSA multi-objective SA. The results showed that the proposed single-objective SA finds a solution which is not dominated by the Pareto front determined by AMOSA. The results also showed that the regularization stabilized the algorithm, in which the increase in parameters does not lead to the overfitting problem. The proposed algorithm can process even complex curves with self-intersections and higher curvature.