Sifting Convolution on the Sphere

TL;DR

This paper introduces a new spherical convolution method called sifting convolution, which supports directional kernels, remains on the sphere, and is computationally efficient, enabling advanced anisotropic filtering on spherical data.

Contribution

It proposes a novel spherical convolution based on the Dirac delta sifting property, offering desirable features lacking in previous definitions.

Findings

Supports directional kernels for anisotropic filtering

Maintains output on the sphere

Demonstrates efficiency and practical application on Earth topography

Abstract

A novel spherical convolution is defined through the sifting property of the Dirac delta on the sphere. The so-called sifting convolution is defined by the inner product of one function with a translated version of another, but with the adoption of an alternative translation operator on the sphere. This translation operator follows by analogy with the Euclidean translation when viewed in harmonic space. The sifting convolution satisfies a variety of desirable properties that are lacking in alternate definitions, namely: it supports directional kernels; it has an output which remains on the sphere; and is efficient to compute. An illustration of the sifting convolution on a topographic map of the Earth demonstrates that it supports directional kernels to perform anisotropic filtering, while its output remains on the sphere.