Geodesic Gaussian Process for the Reconstruction of Micro-Pins Surfaces

We consider a reconstruction of a surface from noisy measurements. This is a common industrial problem given the high availability of data coming from non-contact measurement sensors, which provide 3D points cloud data. In geostatistics, Kriging and Gaussian Processes (GP) have been used for modeling spatially distributed data of some scalar field under the assumption that a response factor z is a function of two parameters (x and y). Our goal is to describe the (x, y, z) data occurring on a non-euclidean surface to reconstruct the real 3D surface. We present a method, called Geodesic Gaussian Process (GGP), based on a two-step procedure. Firstly, a proper parameterization of the surface data is found with the aim of representing distances on the surface on a Euclidean space. In particular, a mesh parameterization algorithm (As Rigid As Possible) is applied to find the new parameter space (u and v). Then, three spatial Gaussian Processes are fit to model each of the three Cartesian coordinates as a function of the two newly defined surface parameters (u and v). Advantages of the proposed method are shown with reference to a real test case, dealing with the reconstruction of micro-pin surfaces.