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

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 ( 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 two 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.