Manifold-valued subdivision schemes based on geodesic inductive averaging

TL;DR

This paper introduces a new class of manifold-valued subdivision schemes utilizing geodesic inductive averaging, ensuring convergence on geodesically complete manifolds for approximating manifold-valued functions.

Contribution

It proposes a novel adaptation of linear subdivision schemes using geodesic inductive means, with intrinsic convergence analysis for manifold-valued data.

Findings

Convergence conditions for manifold-valued subdivision schemes are established.

The schemes are demonstrated on important examples, confirming their effectiveness.

The approach is intrinsic to the manifold, avoiding extrinsic assumptions.

Abstract

Subdivision schemes have become an important tool for approximation of manifold-valued functions. In this paper, we describe a construction of manifold-valued subdivision schemes for geodesically complete manifolds. Our construction is based upon the adaptation of linear subdivision schemes using the notion of repeated binary averaging, where as a repeated binary average we propose to use the geodesic inductive mean. We derive conditions on the adapted schemes which guarantee convergence from any initial manifold-valued sequence. The definition and analysis of convergence are intrinsic to the manifold. The adaptation technique and the convergence analysis are demonstrated by several important examples.