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.