Geodesics in Heat (1204.6216v2)

Abstract: We introduce the heat method for computing the shortest 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 method represents a significant breakthrough in the practical computation of distance on a wide variety of geometric domains, since the resulting linear systems can be prefactored once and subsequently solved in near-linear time. In practice, distance can be updated via the heat method an order of magnitude faster than with state-of-the-art methods while maintaining a comparable level of accuracy. We provide numerical evidence that the method converges to the exact geodesic distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where more regularity is required.

• The paper introduces the heat method, which computes geodesic distances by solving the heat diffusion and Poisson equations.
• The method achieves an order of magnitude faster performance than traditional fast marching techniques while maintaining accuracy.
• Its versatility across triangle meshes, polygonal domains, and point clouds enables broad applications in graphics, robotics, and simulations.

Geodesics in Heat

The paper "Geodesics in Heat" by Keenan Crane et al. introduces a novel method for efficiently calculating geodesic distances on various geometric domains. The authors present a new approach, termed the "heat method," which leverages the relationship between heat diffusion and geodesic distances, grounded in Varadhan's formula. This method is both practical and efficient, significantly improving the computational cost and applicability compared to existing techniques.

Methodology

The core of the heat method is its decomposition into three main stages:

1. Heat Diffusion: Solve the heat equation backward in time on the domain.
2. Gradient Normalization: Compute a unit vector field from the temperature gradient.
3. Poisson Equation: Solve for the potential function via a Poisson equation, obtaining the geodesic distance.

The method cleverly bypasses the challenges of directly applying Varadhan's formula by focusing on the direction of the gradient, thus ensuring robustness against numerical errors.

Advantages

The heat method contrasts favorably with traditional approaches, such as fast marching, by offering two primary advantages:

• General Applicability: It adapts to different data representations, including triangle meshes, polygonal meshes, and even point clouds, without requiring re-meshing or significant preprocessing.
• Computational Efficiency: Solutions to the involved linear systems can be prefactored, ensuring rapid distance updates, which is critical for applications requiring dynamic computations like shape matching or path planning.

Performance and Accuracy

Empirical evidence demonstrates that the heat method is capable of maintaining accuracy while being considerably more efficient. On average, it operates at least an order of magnitude faster than state-of-the-art methods, such as fast marching, without compromising accuracy. For a detailed comparison, the method shows competitive mean and maximum errors relative to exact geodesic computation, emphasizing its potential application in real-world scenarios.

Implications and Future Research

The introduction of the heat method has several implications:

• Practical Applications: The ease of integrating this method into existing frameworks makes it particularly suitable for graphics applications, robotics, and simulations requiring real-time distance computations.
• Theoretical Exploration: Future work could explore extensions to more complex Hamilton-Jacobi equations or weighted distance computations utilizing this framework.

Overall, the heat method represents a substantial advancement in computational geometry, providing both a practical tool for practitioners and a foundation for further theoretical exploration in the field.