Hermite-Birkhoff Interpolation on Arbitrarily Distributed Data on the Sphere and Other Manifolds
This provides a novel interpolation technique for scattered data on manifolds, potentially useful in geophysics, computer graphics, and machine learning, but the abstract does not provide concrete performance numbers, so the significance is unclear.
The paper proposes a new method for Hermite-Birkhoff interpolation on scattered data on the sphere and other manifolds, using basis functions that are orthonormal with respect to point-evaluation functionals and have vanishing derivatives at interpolation points, without requiring linear system solutions.
We consider the problem of interpolating a function given on scattered points using Hermite-Birkhoff formulas on the sphere and other manifolds. We express each proposed interpolant as a linear combination of basis functions, the combination coefficients being incomplete Taylor expansions of the interpolated function at the interpolation points. The basis functions have the following features: (i) depend on the geodesic distance; (ii) are orthonormal with respect to the point-evaluation functionals; and (iii) have all derivatives equal zero up to a certain order at the interpolation points. Moreover, the construction of such interpolants, which belong to the class of partition of unity methods, takes advantage of not requiring any solution of linear systems.