A Registration-free approach for Statistical Process Control of 3D scanned objects via FEM

Recent work in on-line Statistical Process Control (SPC) of manufactured 3-dimensional (3-D) objects has been proposed based on the estimation of the spectrum of the Laplace-Beltrami (LB) operator, a differential operator that encodes the geometrical features of a manifold and is widely used in Machine Learning (i.e., Manifold Learning). The resulting spectra are an intrinsic geometrical feature of each part, and thus can be compared between parts avoiding the part to part registration (or "part localization") pre-processing or the need for equal size meshes, characteristics which are required in previous approaches for SPC of 3D parts. The recent spectral SPC methods, however, are limited to monitoring surface data from objects such that the scanned meshes have no boundaries, holes, or missing portions. In this paper, we extend spectral methods by first considering a more accurate and general estimator of the LB spectrum that is obtained by application of Finite Element Methods (FEM) to the solution of Helmholtz's equation with boundaries. It is shown how the new spectral FEM approach, while it retains the advantages of not requiring part localization/registration or equal size datasets scanned from each part, it provides more accurate spectrum estimates, which results in faster detection of out of control conditions than earlier methods, can be applied to both mesh or volumetric (solid) scans, and furthermore, it is shown how it can be applied to partial scans that result in open meshes (surface or volumetric) with boundaries, increasing the practical applicability of the methods. The present work brings SPC methods closer to contemporary research in Computer Graphics and Manifold Learning. MATLAB code that reproduces the examples of this paper is provided in the supplementary materials.