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    Cascading Upper Bounds for Triangle Soup Pompeiu-Hausdorff Distance
    (The Eurographics Association and John Wiley & Sons Ltd., 2024) Sacht, Leonardo; Jacobson, Alec; Hu, Ruizhen; Lefebvre, Sylvain
    We propose a new method to accurately approximate the Pompeiu-Hausdorff distance from a triangle soup A to another triangle soup B up to a given tolerance. Based on lower and upper bound computations, we discard triangles from A that do not contain the maximizer of the distance to B and subdivide the others for further processing. In contrast to previous methods, we use four upper bounds instead of only one, three of which newly proposed by us. Many triangles are discarded using the simpler bounds, while the most difficult cases are dealt with by the other bounds. Exhaustive testing determines the best ordering of the four upper bounds. A collection of experiments shows that our method is faster than all previous accurate methods in the literature.
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    Optimized Dual-Volumes for Tetrahedral Meshes
    (The Eurographics Association and John Wiley & Sons Ltd., 2024) Jacobson, Alec; Hu, Ruizhen; Lefebvre, Sylvain
    Constructing well-behaved Laplacian and mass matrices is essential for tetrahedral mesh processing. Unfortunately, the de facto standard linear finite elements exhibit bias on tetrahedralized regular grids, motivating the development of finite-volume methods. In this paper, we place existing methods into a common construction, showing how their differences amount to the choice of simplex centers. These choices lead to satisfaction or breakdown of important properties: continuity with respect to vertex positions, positive semi-definiteness of the implied Dirichlet energy, positivity of the mass matrix, and unbiased-ness on regular grids. Based on this analysis, we propose a new method for constructing dual-volumes which explicitly satisfy all of these properties via convex optimization.