During a surface acquisition process using 3D scanners, noise is inevitable and an important step in geometry processing is to remove these noise components from these surfaces (given as points-set or triangulated mesh). The noise-removal process (denoising) can be performed by filtering the surface normals first and by adjusting the vertex positions according to filtered normals afterwards. Therefore, in many available denoising algorithms, the computation of noise-free normals is a key factor. A variety of filters have been introduced for noise-removal from normals, with different focus points like robustness against outliers or large amplitude of noise. Although these filters are performing well in different aspects, a unified framework is missing to establish the relation between them and to provide a theoretical analysis beyond the performance of each method. In this paper, we introduce such a framework to establish relations between a number of widely-used nonlinear filters for face normals in mesh denoising and vertex normals in point set denoising. We cover robust statistical estimation with M-smoothers and their application to linear and non-linear normal filtering. Although these methods originate in different mathematical theories - which include diffusion-, bilateral-, and directional curvature-based algorithms - we demonstrate that all of them can be cast into a unified framework of robust statistics using robust error norms and their corresponding influence functions. This unification contributes to a better understanding of the individual methods and their relations with each other. Furthermore, the presented framework provides a platform for new techniques to combine the advantages of known filters and to compare them with available methods.

中文翻译：

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鲁棒统计框架中基于法线滤波的表面去噪

在使用 3D 扫描仪的表面采集过程中，噪声是不可避免的，几何处理中的一个重要步骤是从这些表面（作为点集或三角网格给出）去除这些噪声分量。噪声去除过程（去噪）可以通过首先过滤表面法线，然后根据过滤后的法线调整顶点位置来执行。因此，在许多可用的去噪算法中，无噪声法线的计算是一个关键因素。已经引入了各种滤波器来从法线去除噪声，具有不同的重点，例如对异常值的鲁棒性或大的噪声幅度。虽然这些过滤器在不同方面都表现良好，缺少统一的框架来建立它们之间的关系并提供超出每种方法性能的理论分析。在本文中，我们引入了这样一个框架来建立网格去噪中用于面部法线的许多广泛使用的非线性滤波器与点集去噪中的顶点法线之间的关系。我们介绍了 M 平滑器的稳健统计估计及其在线性和非线性法线滤波中的应用。尽管这些方法起源于不同的数学理论——包括基于扩散、双边和方向曲率的算法——我们证明了所有这些方法都可以使用稳健的误差范数及其相应的影响函数转化为稳健统计的统一框架。这种统一有助于更好地理解各个方法及其相互之间的关系。此外，所提出的框架为新技术提供了一个平台，以结合已知过滤器的优点并将它们与可用方法进行比较。