Optimal smoothing spline surfaces with constraints on derivatives

In this paper, we consider the problem of constructing optimal smoothing spline surfaces with constraints on their derivatives. The spline surfaces are constituted by using normalized uniform B-splines as the basis functions. We then show that the derivatives of spline surface can be expressed by using B-splines of lower degree, and that the corresponding control points are computed as two-dimensional differences of original control point array. This enables us to treat systematically equality and/or inequality constraints over arbitrary knot point regions on partial derivatives of arbitrary degree. Then, the problem of optimal smoothing spline surfaces with constraints is reduced to convex quadratic programming problem. The performance is examined numerically by approximating monotone and concave surfaces.