Spline Smoothing on Surfaces

We present a method for estimating functions on topologically and/or geometrically complex surfaces from possibly noisy observations. Our approach is an extension of spline smoothing, using a nite element method. The paper has a substantial tutorial component: we start by reviewing smoothness measures for functions de ned on surfaces, simplicial surfaces and dierentiable structures on such surfaces, subdivison functions, and subdivision surfaces. After describing our method, weshow results of an experiment comparing nite element approximations to exact smoothing splines on the sphere, and we give examples suggesting that generalized cross-validation is an eective way of determining the optimal degree of smoothing for function estimation on surfaces