This paper addresses the problem of improving the query performance of the triangular expansion algorithm (TEA) for computing visibility regions by finding the most advantageous instance of the triangular mesh, the preprocessing structure. The TEA recursively traverses the mesh while keeping track of the visible region, the set of all points visible from a query point in a polygonal world. We show that the measured query time is approximately proportional to the number of triangle edge expansions during the mesh traversal. We propose a new type of triangular mesh that minimizes the expected number of expansions assuming the query points are drawn from a known probability distribution. We design a heuristic method to approximate the mesh and evaluate the approach on many challenging instances that resemble real-world environments. The proposed mesh improves the mean query times by 12-16% compared to the reference constrained Delaunay triangulation. The approach is suitable to boost offline applications that require computing millions of queries without addressing the preprocessing time. The implementation is publicly available to replicate our experiments and serve the community.

翻译：本文旨在通过寻找最有利的三角网格实例（即预处理结构）来提升三角扩展算法（TEA）计算可见区域的查询性能。TEA算法在递归遍历网格的同时，持续追踪可见区域——即多边形世界中从查询点可见的所有点的集合。我们证明，实测的查询时间大致与网格遍历过程中的三角形边扩展次数成正比。我们提出了一种新型三角网格，该网格在假设查询点服从已知概率分布的前提下，能够最小化预期的扩展次数。我们设计了一种启发式方法来近似构造该网格，并在多个模拟真实环境的复杂实例上评估了该方法的性能。与参考的约束 Delaunay 三角剖分相比，所提出的网格将平均查询时间提升了12-16%。该方法适用于需要计算数百万次查询且不关注预处理时间的离线应用场景。相关实现已公开，以便复现我们的实验并为社区提供服务。