[Paper Review] Geodesics in heat: A new approach to computing distance based on heat flow

Keenan Crane, Clarisse Weischedel+1 more|arXiv (Cornell University)|Oct 8, 2013

3D Shape Modeling and Analysis35 references317 citations

TL;DR

This paper introduces the geodesic distance computation method called 'Geodesics in Heat,' which leverages heat flow to efficiently solve for geodesic distances on various domains such as grids, meshes, and point clouds. By solving two standard linear elliptic problems, the method enables near-linear time updates with high accuracy and robustness, significantly outperforming state-of-the-art techniques in speed while maintaining convergence to exact distances upon refinement.

ABSTRACT

We introduce the heat method
 for computing the geodesic distance to a
 specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving
 a pair of standard linear elliptic problems. The resulting systems can be prefactored once and subsequently solved in near-linear time. In practice,
 distance is updated an order of magnitude faster than with state-of-the-art
 methods, while maintaining a comparable level of accuracy. The method requires only standard differential operators and can hence be applied on
 a wide variety of domains (grids, triangle meshes, point clouds, etc.). We
 provide numerical evidence that the method converges to the exact distance
 in the limit of refinement; we also explore smoothed approximations of
 distance suitable for applications where greater regularity is required.

Motivation & Objective

• To develop a computationally efficient and robust method for computing geodesic distances on complex domains.
• To address the limitations of existing geodesic distance algorithms, which are often slow or unstable on irregular or non-uniform domains.
• To provide a method that maintains high accuracy while enabling near-linear time computation through prefactored linear systems.
• To extend applicability to diverse geometric representations, including triangle meshes, grids, and point clouds, using only standard differential operators.

Proposed method

• The method formulates geodesic distance computation as a solution to the heat equation, leveraging the fact that the gradient of the heat solution approximates the geodesic distance field.
• It solves two linear elliptic problems: first, the steady-state heat equation to compute the heat diffusion; second, a related system to extract the gradient of the solution, which approximates the geodesic distance.
• The resulting linear systems are symmetric and positive definite, allowing for efficient prefactorization and fast iterative solves.
• The method uses standard finite element or finite difference discretizations, enabling application on unstructured grids, triangle meshes, and point clouds.
• The approach naturally handles complex boundary conditions and subsets (e.g., points or curves), enabling distance computation from any specified source.
• Smoothed approximations of the distance field are derived by convolving the solution with a kernel, enhancing regularity for applications requiring smooth gradients.

Experimental results

Research questions

• RQ1 Can a heat-based approach be used to compute geodesic distances with high accuracy and efficiency on diverse geometric domains?
• RQ2 How does the performance of the heat method compare to state-of-the-art geodesic distance algorithms in terms of speed and accuracy?
• RQ3 To what extent does the method converge to the exact geodesic distance as the mesh or grid is refined?
• RQ4 Can the method be generalized to non-mesh domains such as point clouds using only standard differential operators?
• RQ5 What are the practical benefits of smoothed distance approximations derived from the heat solution in applications requiring regularity?

Key findings

• The geodesic distance computed via the heat method converges to the exact solution in the limit of mesh refinement, demonstrating theoretical consistency.
• The method achieves distance updates that are an order of magnitude faster than state-of-the-art methods while maintaining comparable accuracy.
• The use of prefactored linear systems enables near-linear time solutions after an initial setup phase.
• The method is robust across various domains, including unstructured grids, triangle meshes, and point clouds, due to reliance on standard differential operators.
• Smoothed distance approximations derived from the heat solution provide enhanced regularity, making them suitable for applications requiring smooth gradients.
• The method's efficiency and accuracy are validated through extensive numerical experiments on diverse geometric inputs.