Just the problem of interpolation in an arbitrary set Ξ of scattered data sites in ℝ𝑑 itself is a nontrivial matter. As the scattering makes some standard approaches like (higher order) polynomials or (to some degree) even splines hardly applicable, one would often rely on the so-called meshless methods therein (cf. [17, 40, 105]). Two of the perhaps most prominent examples of these are moving least squares and
radial basis functions. Both of these are capable of providing pleasant convergence
behaviour once the data gets denser and denser in terms of a decrease of the fill
distance
ℎΞ,Ω = sup 𝑥∈Ω
min
ξ∈Ξ ∣∣𝑥 − ξ∣∣2.
And even if the data is sparse, there are special cases of these methods that are capable of producing highly satisfactory results in ℝ𝑑: For example, in terms of moving least squares, one can rely on the very basic concept of inverse distance
weighting, i.e. moving least squares of order one. There, one simply determines
the extrapolation function as
ո Ξ,Y(𝑥) = ∑ ξ∈Ξ үξ n (∣∣𝑥 − ξ∣∣2) ∑ζ∈Ξ n (∣∣𝑥 − ζ∣∣2)
for an appropriate weight function n (⋅) that is related to the inverse distance of the two arguments (cf. [46, 48, 53, 70, 92, 105]). For radial basis functions, there are in particular the celebrated polyharmonic splines or thin-plate splines. These are defined as Ϸ𝑑,𝑚(𝑥, ξ) = ϸ𝑑,𝑚(∣∣𝑥 − ξ∣∣2)〈1〉 with ϸ𝑑,𝑚(𝑡) = ⎧ { ⎨ { ⎩ 𝑡2𝑚−𝑑 𝑑 odd 𝑡2𝑚−𝑑log(𝑡) 𝑑 even
and depend on the respective dimension 𝑑 and naturally require 𝑚 > 𝑑/2. They yield, under the assumption of sufficient polynomial unisolvency (cf. [105]), an interpolant of the form
∑ ξ∈Ξ
𝛼ξϷ𝑑,𝑚(𝑥, ξ) + 𝑃𝑚(𝑥),
for a suitable polynomial 𝑃𝑚 ∈ P𝑚(ℝ𝑑) and suitable coefficients 𝛼
ξ∈ ℝ (cf. [105]). The resulting interpolant will then even minimise the Euclidean version of the Hes- sian energy under interpolation constraints over all of ℝ𝑑in case 𝑚 = 2 (cf. [105]).
If we turn to the submanifold setting, we can obtain a solution to an interpolation
〈1〉We use the nonstandard letters ”Þ,þ” in Ϸ
𝑑,𝑚, ϸ𝑑,𝑚instead of the usual Փ𝑑,𝑚, 𝜙𝑑,𝑚to avoid confu-
sion with the compactly supported Wendland functions that are usually also abbreviated Փ𝑑,𝑚, 𝜙𝑑,𝑚.
These letters are the greek letters ”Sho” to express a ”sh”, and simultaneously the norse letters ”Thorn” that represent a ”th” like in the ”thin” of thin-plate spline.
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 0 1 2 3 4 5 6 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 2 2.5 3 3.5 4 4.5 5 5.5 1 1.5 2 2.5 3 3.5
Figure 6.1: Results of naive approach to extrapolation: Top left depicted is a closed curve with
intricate geometry and some data sites on it. Top right is the plot of restrictions of different direct interpolations to function values 1, 1, 0, 2, 2, 3, 3, 2, assigned counterclockwise with starting from point (1, 0) along the curve. Depicted is an arc-length proportional evaluation for the ”standard interval” [0, 2π] we use here and always in future for closed curves. Black dotted: ”Benchmark” of a suitable extrapolation, based on univariate periodic cubic splines. Blue: Linear triangulated spline interpolation based on a Delaunay triangulation of the points (cf. [6], as provided by Matlab®).
Orange: Interpolation for Wendland function 𝜙3,1and support radius 10.0 (cf. [105, Ch. 9]). Green: Interpolation for polyharmonic spline ϸ2,2. Bottom left, bottom right: Close-ups of the results that emphasise the artifacts.
problem on Ϻ by solving it in ℝ𝑑with one of the presented methods and restricting the solution. In case of RBF, this simple restriction of the solution to Ϻ will yield a reasonable approximation as soon as the intrinsic version of the fill distance
ℎΞ,Ϻ= sup 𝑥∈Ϻ
min
ξ∈Ξ dϺ(𝑥, ξ)
is sufficiently small (cf. [49]). However, the problem of any approach where the approximation is determined from the data sites without considering the underly- ing ESM is that it will become increasingly unreliable when the data gets sparser. In that case, the problem can no longer be considered as if the points were part of a
linear subspace locally, which is the case when the data is sufficiently dense. Now the geometry of the ESM between the data sites becomes increasingly relevant and can introduce undesirable artifacts (see Fig. 6.1).
6.1 Remark: (1) What can be said about approximation on ESMs in general does
of course hold in a sparse data setting in particular: Direct intrinsic function spaces will hardly be available in a reasonable way — so aside from sphere, torus and the like, we cannot hope for any intrinsic space to solve our extrapolation problem, and chart-based methods might even be exposed to a case where a couple of charts contain no data at all.
(2) Inverse distance weighting could of course also be employed with the intrinsic geodesic distance dϺ instead of the Euclidean distance of the ambient space, and thereby generalise to Ϻ without facing the danger of geometry-induced artifacts. But then one would have to consider only ESMs where calculation of the distance is possible for any two points, so effectively it would need to be complete. And even if we have that, the calculation of the distance to any point in Ξ would be required for any evaluation of the extrapolation, whereby the whole approach can become very costly.