Multiresolution on Spherical Curves

Abstract

In this paper, we present a simple multiresolution framework for curves on the surface of a sphere. Multiresolution by subdivision and reverse subdivision allows one to decrease and restore the resolution of a curve, and is typically defined by affine combinations of points in Euclidean space. However, translating such combinations to spherical space is challenging. Several works perform such operations in an intermediate Euclidean space instead using some mapping (e.g. the exponential map), but such mappings cause distortions and are often complicated. We use a simple geometric construction for a multiresolution scheme on the sphere that does not require the use of an intermediate space, which is based on a modified Lane-Riesenfeld algorithm (point duplication followed by repeated averaging) that features an invertible averaging step. Such a multiresolution scheme allows one to simplify/compress and reconstruct curves on the surface of a sphere-like object — such as the Earth — simply, efficiently, and without distortion.