Optimal spline fitting to planar shape

Parametric spline models are used extensively in representing and coding planar curves. For many applications, it is desirable to be able to derive the spline representation from a set of sample points of the planar shape. The problem we address in this paper is to nd a cubic spline model to optimally approximate a given planar shape. We solve this problem by treating the control points which de ne the spline as variables and apply an optimization technique to minimize an error norm so as to nd the best locations of the control points. The error norm, which is de ned as the total squared distance of the curve sample points from the spline model, re ects the discrepancy between the spline and the original curve. The objective function for the optimization process is the error norm plus a term which ensures convergence to the correct solution. The initial locations of the control points are selected heuristically. We also describe an extension of this method, which allows the addition of control points to the spline model based on local error measures if the initial set of control points fails to represent the given shape within a prespeci ed tolerance. Funding for this work was provided by an operating grant from the Natural Sciences and Engineering Research Council of Canada.