Differential Evolution Optimization for Bezier Curve Fitting Priza Pandunata Soft Computing Research Group Universiti Teknologi Malaysia Skudai, Malaysia priza.pandunata@gmail.com Siti Mariyam Hj Shamsuddin Soft Computing Research Group Universiti Teknologi Malaysia Skudai, Malaysia mariyam@utm.my Abstract—In this study, we propose an alternative solution for Bezier curve fitting with Differential Evolution (DE) algorithm. DE algorithm is conducted by randomly generating the control points of the Bezier curve. These generated control points are used to calculate the Bezier curve point. The fitness function of DE algorithm is computed to search for the minimum error. From the experiments, we found that the results of our proposed method achieve the minimum error value significantly. Keywords- DE Algorithm, Curve Fitting, Bezier. I. INTRODUCTION Development of computer graphics technology grows rapidly in recent decades. Several studies have been done on the potential used of Artificial Intelligence (AI) in improving the effectiveness and efficiency of the curve fitting [1,2,5,6]. One of the important issues in computer graphics modeling is curve fitting with various applications such as approximating data, curve and surface fairing [2,9] and image processing on digitized data [1]. In the implementation of curve fitting techniques, there are some important issues need to be considered. Among them is how to get the correct parameter value of each data point while calculating the control points. Another issue is related to time duration and computational processes of large data dimensions, and the complexity of the techniques used. Some studies have been done to solve these issues. These include various methods such as simmulated annealing, genetic algorithm and swarm optimization [5,6]. Recent findings show that the use of simmulated annealing and genetic algorithms have problems in terms of computation time. Hence, this paper proposes an alternative method by implementing Differential Evolution (DE) algorithm to solve the curve fitting problem. This paper is organized as follows: Section 2 describes the theory of Bezier curve and Bezier curve fitting. Section 3 explains the DE algorithm. Section 4 discusses on the methodology and follows by the experimental results, conclusions and future work. II. BEZIER CURVE Bezier curve has several properties that some of them are useful in our proposed method. The advantages are the basis function are real; the degree of the polynomial is one less than the number of control polygon points; the first and last points on the curve are coincident with the first and last points of the control polygon. A parametric Bezier curve at parameter t is defined as: , 0 () () n i ni i Pt BJ t = = ∑ 0 1 t ≤ ≤ (1) where the n is degree of the curve, J n,i (t) is the Bernstein polynomial basis function, and the t is the parameter. Normally, t has numeric value between 0 and 1. A. Bernstein Basis Function The Bernstein basis function can be defined as: , () (1 ) i ni ni n J t t t i - ⎛ ⎞ = - ⎜ ⎟ ⎝ ⎠ 0 (0) 1 ≡ , (2) With: ( ) ! ! ! n n i i n i ⎛ ⎞ = ⎜ ⎟ - ⎝ ⎠ . 0! 1 ≡ (3) J n,i (t) is the nth-order Bernstein basis function similar to in equation (1). Here, n is the degree of the Bernstein basis function and thus of the polynomial curve segment, is one less than the number of points in the Bezier polygon. B. Parameterization Method Parameterization process needs to be done because the curve is generally defined in parametric form, rather than explicit form. This process explains how the numeric values assigned to each point of Bezier curve. So that each data point will get a numeric value that describes the position of the data points on Bezier curve. There are several parameterization methods such as chordal length parameterization method [5,7,9], centripetal parameterization method [9], uniform parameterization method [6], and hybrid parameterization method [5,10]. This study used the chordal, in the range of [0,1] as defined by: 2010 Seventh International Conference on Computer Graphics, Imaging and Visualization 978-0-7695-4166-2/10 $26.00 © 2010 IEEE DOI 10.1109/CGIV.2010.18 68 2010 Seventh International Conference on Computer Graphics, Imaging and Visualization 978-0-7695-4166-2/10 $26.00 © 2010 IEEE DOI 10.1109/CGIV.2010.18 68