论文标题
同质形态歧管重建的平坦Delaunay综合体
Flat Delaunay Complexes for Homeomorphic Manifold Reconstruction
论文作者
论文摘要
鉴于欧几里得空间的平滑子序，有限的点云和比例参数，我们引入了一种构造，我们称之为平坦的Delaunay综合体（FDC）。这是Boissonnat等人引入的切向Delaunay综合体（TDC）的变体。在他们的工作中，我们提供了一个简短而直接的证据，即当点云样本足够很好地表达了子手机时，并且足够安全并且非常安全（我们在论文中定义了一个概念），我们的构造对Submanifold是同理的。由于证明即使在数据点很嘈杂的情况下也有效，因此这使我们能够提出一个扰动方案，该方案将点云作为输入足够好，并返回一个点云，此外，该点云还足够安全。同样重要的是，我们的构建提供了我们在同伴论文中提出的重建问题的各种公式的框架。
Given a smooth submanifold of the Euclidean space, a finite point cloud and a scale parameter, we introduce a construction which we call the flat Delaunay complex (FDC). This is a variant of the tangential Delaunay complex (TDC) introduced by Boissonnat et al.. Building on their work, we provide a short and direct proof that when the point cloud samples sufficiently nicely the submanifold and is sufficiently safe (a notion which we define in the paper), our construction is homeomorphic to the submanifold. Because the proof works even when data points are noisy, this allows us to propose a perturbation scheme that takes as input a point cloud sufficiently nice and returns a point cloud which in addition is sufficiently safe. Equally importantly, our construction provides the framework underlying a variational formulation of the reconstruction problem which we present in a companion paper.
