Interpolation and scattered data fitting on manifolds using projected Powell-Sabin splines

Abstract

We present methods either for interpolating data or for fitting scattered data on a 2D smooth manifold Ω. The methods are based on a local bivariate Powell–Sabin interpolation scheme, and make use of a family of charts {(Uξ, ϕξ)}ξ∈ Ω satisfying certain conditions of smooth dependence on ξ. If Ω is a C2-manifold embedded into ℝ3, then projections into tangent planes can be employed. The data-fitting method is a two-stage method. We prove that the resulting function on the manifold is continuously differentiable, and establish error bounds for both methods for the case when the data are generated by a smooth function.