Quad meshes as a surface representation have many conceptual advantages over triangle meshes. Their edges can naturally be aligned to principal curvatures of the underlying surface and they have the flexibility to create strongly anisotropic cells without causing excessively small inner angles. While in recent years a lot of progress has been made towards generating high quality uniform quad meshes for arbitrary shapes, their adaptive and anisotropic refinement remains difficult since a single edge split might propagate across the entire surface in order to maintain consistency. In this paper we present a novel refinement technique which finds the optimal trade‐off between number of resulting elements and inserted singularities according to a user prescribed weighting. Our algorithm takes as input a quad mesh with those edges tagged that are prescribed to be refined. It then formulates a binary optimization problem that minimizes the number of additional edges which need to be split in order to maintain consistency. Valence 3 and 5 singularities have to be introduced in the transition region between refined and unrefined regions of the mesh. The optimization hence computes the optimal trade‐off and places singularities strategically in order to minimize the number of consistency splits — or avoids singularities where this causes only a small number of additional splits. When applying the refinement scheme iteratively, we extend our binary optimization formulation such that previous splits can be undone if this prevents degenerate cells with small inner angles that otherwise might occur in anisotropic regions or in the vicinity of singularities. We demonstrate on a number of challenging examples that the algorithm performs well in practice.

中文翻译：

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成本最小化局部各向异性四边形网格细化

与三角形网格相比，四边形网格作为表面表示具有许多概念上的优势。它们的边缘可以自然地与下表面的主曲率对齐，并且它们具有创建强各向异性单元的灵活性，而不会导致过小的内角。虽然近年来在为任意形状生成高质量均匀四边形网格方面取得了很大进展，但它们的自适应和各向异性细化仍然很困难，因为单个边缘分裂可能会在整个表面上传播以保持一致性。在本文中，我们提出了一种新的细化技术，该技术根据用户规定的权重在结果元素数量和插入的奇点之间找到最佳权衡。我们的算法将四边形网格作为输入，这些网格的边被标记为需要细化。然后，它制定了一个二元优化问题，以最小化需要拆分以保持一致性的附加边的数量。必须在网格的细化和未细化区域之间的过渡区域中引入价 3 和 5 奇点。因此，优化计算最佳权衡并战略性地放置奇点，以最大限度地减少一致性拆分的数量——或者避免仅导致少量额外拆分的奇点。当迭代地应用细化方案时，我们扩展了我们的二元优化公式，如果这可以防止具有小内角的退化单元格，否则可能会发生在各向异性区域或奇点附近，则可以撤消先前的拆分。我们在许多具有挑战性的例子中证明了该算法在实践中表现良好。