Geometric Inference for Probability Measures
Geometric Inference for Probability Measures
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DOI：
10.1007/s10208-011-9098-0
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发表时间：
2011-12-01
影响因子：
3
通讯作者：
Merigot, Quentin
中科院分区：
文献类型：
--
作者：
Chazal, Frederic;Cohen-Steiner, David;Merigot, Quentin
数据通常以点云的形式出现，该点云是从欧几里德空间的未知紧凑子集中采样的。几何推理的总体目标是从近似点云数据中恢复该子集的几何和拓扑特征（例如，贝蒂数、法线）。看来，距离函数的研究可以成功地解决其中许多问题。然而，该框架的主要局限性之一是它不能很好地应对异常值或背景噪声。在本文中，我们展示了如何扩展距离函数的框架来克服这个问题。用度量代替紧凑子集，我们将距离函数的概念引入 Rd 中的概率分布。这些函数与经典距离函数共享许多属性，这使得它们适合推理目的。特别是，通过考虑这些距离函数的适当水平集，我们表明即使存在异常值，也可以在拓扑保证的情况下重建采样形状的偏移。此外，在考虑经验测量的情况下，可以轻松评估这些函数，使它们具有特别的实际意义。
Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal of geometric inference is then to recover geometric and topological features (e.g., Betti numbers, normals) of this subset from the approximating point cloud data. It appears that the study of distance functions allows one to address many of these questions successfully. However, one of the main limitations of this framework is that it does not cope well with outliers or with background noise. In this paper, we show how to extend the framework of distance functions to overcome this problem. Replacing compact subsets by measures, we introduce a notion of distance function to a probability distribution in Rd. These functions share many properties with classical distance functions, which make them suitable for inference purposes. In particular, by considering appropriate level sets of these distance functions, we show that it is possible to reconstruct offsets of sampled shapes with topological guarantees even in the presence of outliers. Moreover, in settings where empirical measures are considered, these functions can be easily evaluated, making them of particular practical interest.