A fast algorithm for cubic B-spline curve fitting

Based on the matrix perturbation technique. a fast-fitting algorithm using uniform cubic B-spline curves is presented. Our algorithm entails much less floating-point operations when compared with Gaussian elimination method. In addition, our result can be applied to solve the closed cubic B-spline curve-fitting problem. Experimental results are included for a practical version. These experimental values confirm our theoretic results. I. INTRODU(JTION Curve fitting is important in computer graphics, computer-aided design, pattern recognition, and picture processing[ 11, 13, 151. The task of curve fitting is to construct a smooth curve that fits a set of given points in the space. In practice, a curve-fitting algorithm should meet two criterions: First, adjusting a control point of the curve affects only its vicinity, and second, it should be fast enough to be incorporated into an interactive program. The cubic B-spline curve interpolation[ I I] is a good fitting tool to meet the first criterion. Gaussian elimination has been used to solve the cubic B-spline curve-fitting problem. How to speed up the computation of the cubic B-spline curve-fitting in order to meet the second criterion is a very interesting research problem. This paper only considers the uniform cubic B-spline case[2]. Based on the matrix perturbation technique, a fasthtting algorithm using uniform cubic B-spline curves is presented. Given n points, the number of floatingpoint (FP for short) operations required for our algorithm is about Sn. While using Gaussian elimination, it takes about 7n FP operations to solve the same fitting problem. Our algorithm entails much less FP operations when compared with Gaussian elimination method. In addition, our result can be applied to solve the closed cubic B-spline curve-fitting problem, and the number of FP operations used in our algorithm over the number of FP operations used in Gaussian elimination is about one-third. Experimental results are included for a practical version. These experimental values confirm our theoretic results derived in this paper. 2. THE CUBIC B-SPLINE CURVE FIITING Suppose we are given a set of points, B, = (hj”, hj”. hl”) for I i i 5 n. According to [2], for an uniform cubic B-spline curve, each given point can be expressed by a weighted average of three control points: B, = t(C,_, + 4C, + C,,,), 1 I i 5 n, where C, = (cl , i’) d2’, cl”). They form a system of n equations in n + 2 unknowns for all given points. In order to completely solve the system, we need the following two additional equations to specify how the boundary control points are interpolated: Co = C,; C,, , = C,,. For simplicity, we only consider b = (h,, b2, . , h,)’ = (h\‘), by’. . . , b!:‘)’ and c = (cl, c2, . . , c,)’ = (cd,“, c$“, . , c’,“‘)’ throughout this paper. In what follows, matrices are represented by bold uppercase letters, vectors by bold lowercase letters, and scalars by plain lowercase letters. Thus, the above system of equations can be equivalently transformed into