Joint Research Conference
June 24-26, 2014
Reconstructing a free-form surface from 3-dimensional 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 3-dimensional point cloud data. The surface is represented parametrically, with each of the three Cartesian coordinates (x, y, z) a function of surface coordinates (u, v), a model form compatible with computer-aided-design (CAD) models. This model form also avoids having to choose one Euclidean coordinate (say, z) as a "response" function of the other 2 coordinate \locations" (say, x and y), as commonly used in previous Euclidean kriging models of manufacturing data. The (u, v) 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 geodesic distance on the surface, rather than a function of their Euclidean distances over the xy plane. It is shown 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', i.e., z(x, y). The methodology is applied to simulated surface data and to a real dataset obtained with a non-contact laser scanner.