Biharmonic Subdivision on Riemannian Manifolds

TL;DR

This paper develops a biharmonic subdivision framework on Riemannian manifolds, extending classical Euclidean schemes to curved surfaces like the sphere and hyperbolic plane, with improved smoothness and fairness properties.

Contribution

It introduces a novel biharmonic interpolatory subdivision method on Riemannian manifolds, including explicit rules and hierarchy for higher smoothness, with theoretical analysis and numerical validation.

Findings

The six-point scheme achieves lower fairness energy than classical schemes.

The scheme maintains fourth-order smoothness and better curvature profiles.

Numerical experiments show improved smoothness and reduced ringing on non-uniform data.

Abstract

This paper introduces a biharmonic interpolatory subdivision framework on Riemannian manifolds. In the Euclidean setting, the six-point Deslauriers-Dubuc stencil is characterised as the unique minimiser of a discrete curvature-variation energy under symmetric six-point support and degree-five polynomial reproduction conditions, linking a classical interpolatory rule to a first-principles fairness criterion. Exact symbol analysis establishes fourth-order smoothness. The construction extends to the two-sphere and the hyperbolic plane via a second-order reduced governing ODE derived from the biharmonic Euler-Lagrange equation on constant-curvature surfaces. This reduced model yields closed-form insertion rules, and proximity analysis confirms that the manifold scheme satisfies the Wallner-Dyn second-order condition, preserving fourth-order smoothness. A hierarchy of biharmonic stencils achieving higher smoothness orders is also described. Numerical experiments demonstrate that the six-point scheme delivers lower fairness energy and smoother curvature profiles than the classical four-point Dyn-Gregory-Levin scheme, while remaining more local and exhibiting less ringing on non-uniform data than the eight-point variant.