Surface Spline Approximation on SO(3)

TL;DR

This paper introduces a new class of kernels on SO(3) for approximation and interpolation, providing a framework with explicit error estimates that match the smoothness of target functions.

Contribution

It develops a novel kernel-based approximation method on SO(3) with explicit error bounds and practical implementation advantages.

Findings

Kernels are conditionally positive definite with closed-form expressions.

Approximation schemes achieve error rates matching target function smoothness.

Method enables direct implementation via interpolation or least-squares approximation.

Abstract

The purpose of this article is to introduce a new class of kernels on SO(3) for approximation and interpolation, and to estimate the approximation power of the associated spaces. The kernels we consider arise as linear combinations of Green's functions of certain differential operators on the rotation group. They are conditionally positive definite and have a simple closed-form expression, lending themselves to direct implementation via, e.g., interpolation, or least-squares approximation. To gauge the approximation power of the underlying spaces, we introduce an approximation scheme providing precise L_p error estimates for linear schemes, namely with L_p approximation order conforming to the L_p smoothness of the target function.