Parallel and Scalable Heat Methods for Geodesic Distance Computation

Abstract

In this paper, we propose a parallel and scalable approach for geodesic distance computation on triangle meshes. Our key observation is that the recovery of geodesic distance with the heat method [1] can be reformulated as optimization of its gradients subject to integrability, which can be solved using an efficient first-order method that requires no linear system solving and converges quickly. Afterward, the geodesic distance is efficiently recovered by parallel integration of the optimized gradients in breadth-first order. Moreover, we employ a similar breadth-first strategy to derive a parallel Gauss-Seidel solver for the diffusion step in the heat method. To further lower the memory consumption from gradient optimization on faces, we also propose a formulation that optimizes the projected gradients on edges, which reduces the memory footprint by about 50%. Our approach is trivially parallelizable, with a low memory footprint that grows linearly with respect to the model size. This makes it particularly suitable for handling large models. Experimental results show that it can efficiently compute geodesic distance on meshes with more than 200 million vertices on a desktop PC with 128GB RAM, outperforming the original heat method and other state-of-the-art geodesic distance solvers.

GeodesicEmbedding (GE): A High-Dimensional Embedding Approach for Fast Geodesic Distance Queries

“…Most algorithms focus on computing SSAD on triangular meshes. Representative algorithms are wavefront propagation methods [2], [14], [15], [16], [17], [18], [19], [20], [21], partial differential equation (PDE) methods [3], [22], [23], [24], [25], [26], and graph-based methods [1], [4], [27].…”

A fast numerical solver for local barycentric coordinates

“…Rabinovich et al (2017) present a scalable approach for the optimization of flip-preventing energies for mesh parameterization, by iteratively minimizing a simple proxy energy instead of the original distortion energy. Tao et al (2018) propose a scalable heat method for computer geodesic distance, by reformulating the original heat method from Crane et al (2013) as an optimization problem for the distance gradient. The reformulation of LBC in this paper is similar to the gradient-based approach taken by Tao et al (2018), but with a non-smooth target function and a different set of constraints to enforce their defining properties.…”

“…Tao et al (2018) propose a scalable heat method for computer geodesic distance, by reformulating the original heat method from Crane et al (2013) as an optimization problem for the distance gradient. The reformulation of LBC in this paper is similar to the gradient-based approach taken by Tao et al (2018), but with a non-smooth target function and a different set of constraints to enforce their defining properties.…”

A minimalistic approach for fast computation of geodesic distances on triangular meshes

“…However, the accuracy of the distance map computation is sensitive to the choice of a parameter. A parallel and scalable version of the Heat method was proposed by Tao et al [24]. Finally, Litman and Bronstein [25] proposed a method that also works on the spectral domain, called the Spectrometer.…”

“…In the case of CPU, the Cholmod library is more stable and we got a good result for the armadillo mesh. Recently, Tao et al [24] proposed a scalable version of the Heat Method, which has been tested only on CPU for large meshes. This method uses Gauss-Seidel iterations instead of solving the linear system using Cholesky decomposition.…”