Classification and construction of closed-form kernels for signal representation on the 2-sphere

TL;DR

This paper develops closed-form kernels for signal representation on the 2-sphere using RKHS, offering an alternative to wavelets with potential advantages in smoothness and computational efficiency.

Contribution

It constructs new classes of closed-form kernels, including isotropic, non-isotropic, and anisotropic, enabling efficient inner product computation on the sphere.

Findings

Introduces kernels based on von Mises-Fisher distribution.

Defines three classes of RKHS with spherical harmonic eigenfunctions.

Provides explicit formulas for efficient kernel evaluation.

Abstract

This paper considers the construction of Reproducing Kernel Hilbert Spaces (RKHS) on the sphere as an alternative to the conventional Hilbert space using the inner product that yields the L^2(S^2) function space of finite energy signals. In comparison with wavelet representations, which have multi-resolution properties on L^2(S^2), the representations that arise from the RKHS approach, which uses different inner products, have an overall smoothness constraint, which may offer advantages and simplifications in certain contexts. The key contribution of this paper is to construct classes of closed-form kernels, such as one based on the von Mises-Fisher distribution, which permits efficient inner product computation using kernel evaluations. Three classes of RKHS are defined: isotropic kernels and non-isotropic kernels both with spherical harmonic eigenfunctions, and general anisotropic kernels.