Automatic knot adjustment using an artificial immune system for B-spline curve approximation Erkan Ülker * , Ahmet Arslan Selçuk University, Engineering and Architecture Faculty, Department of Computer Engineering, 42075 Konya, Turkey article info Article history: Received 27 July 2007 Accepted 25 November 2008 Keywords: Artificial immune system B-spline curve approximation Clonal selection Knot adjustment abstract Reverse engineering transforms real parts into engineering concepts or models. First, sam- pled points are mapped from the object’s surface by using tools such as laser scanners or cameras. Then, the sampled points are fitted to a free-form surface or a standard shape by using one of the geometric modeling techniques. The curves on the surface have to be modeled before surface modeling. In order to obtain a good B-spline curve model from large data, the knots are usually respected as variables. A curve is then modeled as a con- tinuous, nonlinear and multivariate optimization problem with many local optima. For this reason it is very difficult to reach a global optimum. In this paper, we convert the original problem into a discrete combinatorial optimization problem like in Yoshimoto et al. [F. Yoshimoto, M. Moriyama, T. Harada, Automatic knot placement by a genetic algorithm for data fitting with a spline, in: Proceedings of the International Conference on Shape Modeling and Applications, IEEE Computer Society Press, 1999, pp. 162–169] and Sarfraz et al. [M. Sarfraz, S.A. Raza, Capturing outline of fonts using genetic algorithm and splines, in: Fifth International Conference on Information Visualisation (IV’01), 2001, pp. 738–743]. Then, we suggest a new method that solves the converted problem by artificial immune systems. We think the candidates of the locations of knots as antibodies. We define the affinity measure benefit from Akaike’s Information Criterion (AIC). The proposed method determines the appropriate location of knots automatically and simultaneously. Further- more, we do not need any subjective parameter or good population of initial location of knots for a good iterative search. Some examples are also given to demonstrate the effi- ciency and effectiveness of our method. Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction The problem of recovering the 2D/3D shape of a curve/surface, also known as curve or surface reconstruction, has received much attention in the last few years. In literature, there are two different orientations [24]. In first orientation the authors address the problem of obtaining the curve or surface model from a set of given cross-section or points. This is a typical prob- lem in most research and application areas such as CAD/CAM, biomedical and medical science, in which usually an object (acquired from 3D laser scanning, ultrasound imaging, magnetic resonance imaging, computer tomography, etc.) is defined as a sequence of 2D cross-sections. The other different approaches, in the second orientation, include the reconstructing curves/surfaces from a given set of data points. Two different approaches are employed depend on the nature of these data points: interpolation and approximation. In the fitting form by interpolation, parametric curve is constrained to pass through all the given set of data points. The technique of fitting of data parametrically by interpolation is suitable when the data 0020-0255/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2008.11.037 * Corresponding author. E-mail addresses: eulker@selcuk.edu.tr (E. Ülker), ahmetarslan@selcuk.edu.tr (A. Arslan). Information Sciences 179 (2009) 1483–1494 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins