Parallel and Scalable Heat Method
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The heat method is a popular approach to computing geodesic distances on discrete domains. It first performs short-time heat diffusion to approximate the gradient direction of the geodesic distance function, followed by least-squares recovery of the actual distance. It only requires solving two sparse linear systems, which can be done efficiently using matrix prefactorization. However, it does no scale well to large models, due to the large memory consumption for matrix factorization and the difficulty in its parallelization. In this paper, we propose an improved version of the heat method which overcomes its limitation in scalability. Our key observation is that the recovery of geodesic distance from approximate gradient directions can be reformulated as gradient optimization problem that enforces integrability, which can be solved using a parallel fixed-order method that requires no linear system solving and converges quickly. Afterwards, the geodesic distance are efficiently recovered by parallel integration of the optimized gradients in breadth-first order. Moreover, a similar breadth-first strategy is developed for a parallel Gauss-Seidel solver for heat diffusion. 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 our method can efficiently compute geodesic distance on meshes with up to 99 million vertices on a workstation with a workstation with an octa-core CPU and 128GB RAM, outperforming the original heat method and other state-of-the-art geodesic distance solvers.
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