  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.
