Economic Upper Bound Estimation in Hausdorff Distance Computation for Triangle Meshes

Abstract

The Hausdorff distance is one of the most fundamental metrics for comparing 3D shapes. To compute the Hausdorff distance efficiently from a triangular mesh A$A$ to another triangular mesh B$B$, one needs to cull the unnecessary triangles on A$A$ quickly. These triangles have no chance to improve the Hausdorff distance estimation, that is the parts with local upper bound smaller than the global lower bound. The local upper bound estimation should be tight, use fast distance computation, and involve a small number of triangles in B$B$ during the reduction phase for efficiency. In this paper, we propose to use point‐triangle distance, and only involve at most four triangles in B$B$ in the reduction phase. Comparing with the state‐of‐the‐art proposed by Tang et al. in 2009, which uses more costly triangle‐triangle distance and may involve a large number of triangles in reduction phase, our local upper bound estimation is faster, and with only a small impact on the tightness of the bound on error estimation. Such a more economic strategy boosts the overall performance significantly. Experiments on the Thingi10K dataset show that our method can achieve several (even over 20) times speedup on average. On a few models with different placements and resolutions, we show that close placement and large difference in resolution bring big challenges to Hausdorff distance computation, and explain why our method can achieve more significant speedup on challenging cases.