Geodesics in heat: A new approach to computing distance based on heat flow

Authors:
Keenan Crane;Clarisse Weischedel;Max Wardetzky
Affiliations:
Caltech, Pasadena, CA;University of Göttingen, Gottingen, Germany;University of Göttingen, Gottingen, Germany
Venue:
ACM Transactions on Graphics (TOG)
Year:
2013

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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.