research-article

The Vector Heat Method

Authors:

Nicholas Sharp,

Yousuf Soliman,

Keenan CraneAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 38, Issue 3

Article No.: 24, Pages 1 - 19

https://doi.org/10.1145/3243651

Published: 07 June 2019 Publication History

Abstract

This article 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. 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 so that they can be used in the context of tangent vector field processing.

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Index Terms

The Vector Heat Method
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Shape analysis
2. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Partial differential equations
    2. Numerical analysis
      1. Discretization

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cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 38, Issue 3

June 2019

125 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3322934

Editor:
Marc Alexa
TU Berlin, Germany

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Copyright © 2019 ACM.

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Publication History

Published: 07 June 2019

Accepted: 01 March 2019

Revised: 01 February 2019

Received: 01 May 2018

Published in TOG Volume 38, Issue 3

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Liu SWang HYan DLi QLuo FTeng ZLiu X(2025)Spectral Descriptors for 3D Deformable Shape Matching: A Comparative SurveyIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2024.336808331:3(1677-1697)Online publication date: 1-Mar-2025
https://dl.acm.org/doi/10.1109/TVCG.2024.3368083
Li FZhu HLi WLiu CMa JLi L(2025)Generation of an optimal triangulated irregular network for topographic surface via optimal transport theoryGeo-spatial Information Science10.1080/10095020.2024.2446306(1-16)Online publication date: 17-Jan-2025
https://doi.org/10.1080/10095020.2024.2446306
Webb TSussman D(2025)curvedSpaceSim: A framework for simulating particles interacting along geodesicsComputer Physics Communications10.1016/j.cpc.2025.109545311(109545)Online publication date: Jun-2025
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King NSu HAanjaneya MRuuth SBatty C(2024)A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry ProcessingACM Transactions on Graphics10.1145/367365243:5(1-26)Online publication date: 9-Aug-2024
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