Jacobi-PIA Algorithm for Non-uniform Cubic B-Spline Curve Interpolation

Based on the Jacobi iterative method for solving the system of linear equations, we propose a progressive iterative approximation method for interpolating a set of points by non-uniform cubic B-spline curves,（abbr. Jacobi-PIA）. In Jacobi-PIA, the control points of the initial cubic B-spline curve are set as the points to be interpolated, then control points of the interpolation cubic B-spline curve are derived in iteration manner. In each iteration, we define the displacement vector as the difference of the point to be interpolated and its corresponding point on the cubic B-spline curve in the previous level, and then the control points of the current level are derived by those of previous level adding the corresponding displacement vector. Jacobi-PIA algorithm has less computation than existed PIA algorithm because of cutting down a subtraction in updating the control points. Theoretical analysis shows that Jacobi-PIA algorithm is convergent. Numerical examples show that the rate of convergence of Jacobi-PIA is faster than that of PIA algorithm, and is almost equal to that of the weighted PIA algorithm with the best weight.