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

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Davydov, O. and Schumaker, L.L. (2008) Interpolation and scattered data fitting on manifolds using projected Powell–Sabin splines. IMA Journal of Numerical Analysis, 28 (4). pp. 785-805. ISSN 0272-4979 (https://doi.org/10.1093/imanum/drm033)

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Abstract

We present methods for either interpolating data or for fitting scattered data on a two-dimensional smooth manifold. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of a family of charts {(Uξ , ξ)}ξ∈ satisfying certain conditions of smooth dependence on ξ. If 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.

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