The Vector Heat Method

TL;DR

The paper introduces a fast, robust method for parallel transporting vectors on curved surfaces using heat flow and the connection Laplacian, enabling various geometric computations on different data structures.

Contribution

It extends the vector heat method to efficiently compute parallel transport on curved manifolds via solving linear systems, applicable to diverse geometric data.

Findings

Converges under refinement empirically.

Applicable to point clouds and polygon meshes.

Enables efficient geometric computations like medians and Voronoi diagrams.

Abstract

This paper describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Fr\'{e}chet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations (iDT) so that they can be used in the context of tangent vector field processing.