An algorithm for curve and surface fitting using B-splines

The problem of curve and surface fitting using B-splines is addressed. B-splines are particularly attractive interpolants due to such properties as optimal smoothness, variation diminishing, local control, and convex hull and the existence of good evaluation algorithms. The technique uses involves a geometric approach from a signal processing perspective. It starts with a straight line approximation to the given data (which corresponds to multiple knots at each point in the B-spline representation). The knots are the positions at which the piecewise polynomials meet and are initially the given data points for the interpolation. The next step in the algorithm is to reduce the number of discontinuous derivatives without perturbing the spline beyond a given tolerance. This is accomplished by removing knots so that the successive curves lie in the subspace of the original polynomial space defined by the original curve. This procedure is attractive in its ability to produce an interpolating curve that retains extremely high accuracy with a minimal number of knots or data to represent the curve. A sample curve and the spectrum of the resulting fitting error are presented as are some extensions to tensor product surface fitting