Spherical Splines for Data Interpolation and Fitting

Abstract

Abstract. We study minimal energy interpolation, discrete and penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines. Several numerical experiments are conducted to compare approximating properties of homogeneous and nonhomogeneous splines. Our numerical experiments show that nonhomogeneous splines have certain advantages over homogeneous splines.Key words. spherical splines, data fitting AMS subject classifications. 65D05, 65D07, 65D171. Introduction. Contemporary research in atmospheric sciences, geodesy and geophysics requires the use of global data heterogeneously distributed in space around the Earth. For spherical data interpolation/approximation tensor products of univariate splines are not a good choice, since data locations are not usually spaced over a regular grid. Radial basis functions are not good candidates either, since the data values may have no rotational symmetry.Spherical Bernstein-Bézier splines (introduced in [1] and studied in [2] and [7]) are well suited for scattered data interpolation/approximation problems. The spherical spline functions have many properties in common with classical polynomial splines over planar triangulations. Moreover, many spline interpolation and approximation methods for planar scattered data problems have analogs in spherical setting [2].One of the disadvantages of using homogeneous spherical splines is that spline spaces of even and odd degrees have only zero function in common due to homogeneity of the basis. More explicitly, the spline space S

Efficient solvers for the shallow water equations on a sphere

“…However, in the case of homogeneous spherical splines, the spline spaces of even and odd degrees have only zero function in common. Therefore, in such spaces we cannot reproduce polynomials of degree m d unless m ≡ d mod 2 [4]. This is the reason why we have better results for quadratic splines compared to cubic splines in Table 3.…”

The convergence of three L1 spline methods for scattered data interpolation and fitting

“…It is interesting and useful to continue the investigation to see if the L 1 spline methods are good for shape preservation using other spline functions. Recently, triangulated spherical splines for scattered data interpolation and fitting are studied in [3]. The convergence of the minimal energy method for spherical spline interpolation using the usual quadratic energy functional is studied in [1].…”

Convergence of discrete and penalized least squares spherical splines

“…For comparison of spherical splines with spherical radial basis functions, see [7]. In [6], an efficient iterative computational algorithm is combined with the global fitting methods outlined in [3] to reduce the size of linear systems involved and thus to decrease computational costs. Convergence of the minimal energy interpolating spherical splines was studied in [5].…”