The Vector Heat Method

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/Fréchet 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. Moreover, 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.

Parallel and Scalable Heat Methods for Geodesic Distance Computation

“…The heat method proposed by Crane et al [1] adopts a different strategy: rather than solving the distance function directly, it first computes a unit vector field that approximates its gradient, then integrates this vector field by solving a Poisson equation; this only requires solving two linear systems and is highly efficient. This approach was recently generalized to compute parallel transport vector-valued data on a curved manifold [26]. Our approach is also based on the heat method, but using iterative solvers for the linear systems instead of the direct solvers in [1].…”

A Smoothness Energy without Boundary Distortion for Curved Surfaces

“…A different discretization, reminiscent of finite differences, can be found in the work of Knöppel et al [2015], where it is used to compute stripe patterns on surfaces. The same discretization is also used by Sharp et al [2018] to compute the parallel transport of vectors. The work of Sharp et al [2018] also features the Weitzenböck identity that we use to derive the natural boundary conditions of our Hessian energy: they use it to construct a Dirichlet energy on the covector bundle.…”

“…The same discretization is also used by Sharp et al [2018] to compute the parallel transport of vectors. The work of Sharp et al [2018] also features the Weitzenböck identity that we use to derive the natural boundary conditions of our Hessian energy: they use it to construct a Dirichlet energy on the covector bundle. Liu et al [2016] discretize the covariant derivative using the notion of discrete connections.…”

“…Moreover, our simple Crouzeix-Raviart discretization of the oneform Dirichlet energy containing covariant derivatives from Section 6.2 offers an interesting approach to discretize the vector Dirichlet energy in a wide variety of applications. Potential applications include vector field design [Knöppel et al 2013], parallel transport of vectors [Sharp et al 2018], and many more [Azencot et al 2015;Corman and Ovsjanikov 2019;Liu et al 2016]. Stein et al [2018] (α = 2.5⋅10 -9 ) our Hessian energy (α = 1.8⋅10 -9 )…”

Efficient Computation of Smoothed Exponential Maps

“…In concurrent work, Sharp et al . [SSC18] introduce a heat‐based method to perform parallel transport of vector valued data on meshes. The basic idea of their method is similar to ours, yet they approach the development from the perspective of generalizing the heat method.…”