Geodesic Gaussian Processes for the Parametric Reconstruction of a Free-Form Surface

Reconstructing a free-form surface from 3-dimensional (3D) noisy measurements is a central problem in inspection, statistical quality control, and reverse engineering. We present a new method for the statistical reconstruction of a free-form surface patch based on 3D point cloud data. The surface is represented parametrically, with each of the three Cartesian coordinates (<i>x</i>, <i>y</i>, <i>z</i>) a function of surface coordinates (<i>u</i>, <i>v</i>), a model form compatible with computer-aided-design (CAD) models. This model form also avoids having to choose one Euclidean coordinate (say, <i>z</i>) as a “response” function of the other two coordinate “locations” (say, <i>x</i> and <i>y</i>), as commonly used in previous Euclidean kriging models of manufacturing data. The (<i>u</i>, <i>v</i>) surface coordinates are computed using parameterization algorithms from the manifold learning and computer graphics literature. These are then used as locations in a spatial Gaussian process model that considers correlations between two points on the surface a function of their <i>geodesic</i> distance on the surface, rather than a function of their Euclidean distances over the <i>xy</i> plane. We show how the proposed geodesic Gaussian process (GGP) approach better reconstructs the true surface, filtering the measurement noise, than when using a standard Euclidean kriging model of the “heights”, that is, <i>z</i>(<i>x</i>, <i>y</i>). The methodology is applied to simulated surface data and to a real dataset obtained with a noncontact laser scanner. Supplementary materials are available online.

从三维（3D）带噪测量数据中重构自由曲面是检测、统计质量控制以及逆向工程领域的核心问题。本文提出了一种基于三维点云数据的自由曲面片统计重构新方法。该曲面采用参数化形式表示，三个笛卡尔坐标（*x*, *y*, *z*）均为曲面坐标（*u*, *v*）的函数，这种模型形式与计算机辅助设计（CAD）模型兼容。相较于此前针对制造业数据常用的欧氏克里金（kriging）模型，该模型形式无需选取某一欧氏坐标（例如*z*）作为另外两个坐标“位置”（例如*x*与*y*）的“响应”函数。曲面坐标（*u*, *v*）可通过流形学习与计算机图形学领域文献中的参数化算法计算得到。随后将其作为空间高斯过程模型的位置参数，该模型会将曲面上两点的相关性建模为二者在曲面上的测地线（geodesic）距离的函数，而非二者在*xy*平面上的欧氏距离的函数。我们证明，相较于采用标准“高度”型欧氏克里金模型（即*z*(*x*, *y*)）的方法，所提出的测地线高斯过程（GGP）方法能够更优地重构真实曲面并滤除测量噪声。该方法被应用于模拟曲面数据与非接触式激光扫描仪采集的真实数据集。补充材料可在线获取。