A global approach to the refinement of manifold data

TL;DR

This paper introduces a global method for refining manifold data by computing all elements simultaneously with geodesic averages, improving convergence analysis over traditional element-wise approaches.

Contribution

It presents a novel global refinement scheme for manifold data using geodesic averages and provides convergence conditions for repeated refinements.

Findings

The global approach computes all refined elements simultaneously.

Conditions for strong convergence of the refinement scheme are derived.

The method enhances multiresolution representations of manifold data.

Abstract

A refinement of manifold data is a computational process, which produces a denser set of discrete data from a given one. Such refinements are closely related to multiresolution representations of manifold data by pyramid transforms, and approximation of manifold-valued functions by repeated refinements schemes. Most refinement methods compute each refined element separately, independently of the computations of the other elements. Here we propose a global method which computes all the refined elements simultaneously, using geodesic averages. We analyse repeated refinements schemes based on this global approach, and derive conditions guaranteeing strong convergence.