Topology-preserving Hodge Decomposition in the Eulerian Representation

TL;DR

This paper introduces a new topology-preserving Hodge decomposition method in the Eulerian framework, unifying components and validating it through numerical experiments on complex objects.

Contribution

It presents a comprehensive 5-component Hodge decomposition that maintains topology in the Eulerian representation, addressing a longstanding challenge.

Findings

Validates the method on various objects including single-cell RNA velocity

Confirms $L^2$-orthogonality and accurate cohomology

Demonstrates effectiveness through numerical experiments

Abstract

The Hodge decomposition is a fundamental result in differential geometry and algebraic topology, particularly in the study of differential forms on a Riemannian manifold. Despite extensive research in the past few decades, topology-preserving Hodge decomposition of scalar and vector fields on manifolds with boundaries in the Eulerian representation remains a challenge due to the implicit incorporation of appropriate topology-preserving boundary conditions. In this work, we introduce a comprehensive 5-component topology-preserving Hodge decomposition that unifies normal and tangential components in the Cartesian representation. Implicit representations of planar and volumetric regions defined by level-set functions have been developed. Numerical experiments on various objects, including single-cell RNA velocity, validate the effectiveness of our approach, confirming the expected rigorous $L^2$-orthogonality and the accurate cohomology.