Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees
Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees
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从有保证的噪声和非均匀近似平滑流形重建
DOI：
10.1016/j.comgeo.2007.07.001
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
A. Lieutier
中科院分区：
文献类型：
--
作者：
F. Chazal;A. Lieutier
给定 Rk 的一个子流形 S 的平滑紧致余维和 S 的紧致近似 K，我们证明可以使用以 K 为中心的球并集来重构 S 并在拓扑保证下逼近 S 的中轴。我们考虑噪声近似的两种概念，它们概括了 Amenta 等人引入的采样条件。和戴伊等人。第一个概括了基于局部特征尺寸最小值的均匀采样。第二个基于 S 的局部特征尺寸函数概括了非均匀采样。近似的密度和噪声受局部特征尺寸函数的常数倍的限制。该常数不依赖于表面 S。我们的结果基于距离函数的临界点理论。对于这两个近似条件，我们证明以 K 为中心的球并集边界的连通分量与 S 是同位素。我们考虑使用均匀半径的球以及半径随流形局部细节级别变化的球。对于第一个近似条件，我们证明 Rk∖K 的中轴的子集（称为 λ-中轴）与 S 的中轴同伦等价。我们的结果推广到任何余维的 Rk 的平滑紧致子流形 S。
Given a smooth compact codimension one submanifold S of Rkand a compact approximation K of S, we prove that it is possible to reconstruct S and to approximate the medial axis of S with topological guarantees using unions of balls centered on K. We consider two notions of noisy-approximation that generalize sampling conditions introduced by Amenta et al. and Dey et al. The first one generalizes uniform sampling based on the minimum value of the local feature size. The second one generalizes non-uniform sampling based on the local feature size function of S. The density and noise of the approximation are bounded by a constant times the local feature size function. This constant does not depend on the surface S. Our results are based upon critical point theory for distance functions. For the two approximation conditions, we prove that the connected components of the boundary of unions of balls centered on K are isotopic to S. We consider using both balls of uniform radius and also balls whose radii vary with the local level of detail of the manifold. For the first approximation condition, we prove that a subset (known as the λ-medial axis) of the medial axis of Rk∖K is homotopy equivalent to the medial axis of S. Our results generalize to smooth compact submanifolds S of Rkof any codimension.