On optimizing knot positions for multi-dimensional B-spline models

In this paper, we present a new method for optimizing knot positions for a multi-dimensional B-spline model. Using the results from from univariate polynomial approximation theory, spline approximation theory and multivariate tensor product theory, we develop the algorithm in three steps. First, we derive a local upper bound for the L-infinity error in a multivariate B-spline tensor product approximation over a span. Second, we use this result to bound the approximation error for a multi-dimensional B-spline tensor product approximation. Third, we developed two knot position optimization methods based on the minimization of two global approximation errors: L-infinity global error and L-2 global error. We test our method with 2D surface fitting experiments using B-spline models defined in both 2D Cartesian and polar coordinates. Simulation results demonstrate that the optimized knots can fit a surface more accurately than fixed uniformly spaced knots.