Registration-Free Localization of Defects in Three-Dimensional Parts from Mesh Metrology Data Using Functional Maps

We consider a common problem occurring after using a statistical process control (SPC) method based on three-dimensional measurements: locate where on the surface of the part that triggered an out-of-control alarm there is a significant shape difference with respect to either an in-control part or its nominal (computer-aided design (CAD)) design. In the past, only registration-based solutions existed for this problem, which first orient and locate the part and its nominal design under the same frame of reference. Recently, spectral Laplacian methods have been proposed for the SPC of discrete parts and their measured surface meshes. These techniques provide an intrinsic solution to the SPC problem: that is, a solution exclusively based on data whose coordinates lie on the surfaces without making reference to their ambient space, thus avoiding registration. Registration-free methods avoid the computationally expensive, nonconvex registration step needed to align the parts as required by previous methods, eliminating registration errors, and they are important in industry because of the increasing use of portable noncontact scanners. In this paper, we first present a new registration-free solution to the post-SPC part defect localization problem. The approach uses a spectral decomposition of the Laplace–Beltrami operator in order to construct a functional map between the CAD and measured manifolds to locate defects on the suspected part. A computational complexity analysis demonstrates the approach scales better with the mesh size and is more stable than a registration-based approach. To reduce computational expense, a new mesh partitioning algorithm is presented to find a region of interest on the surface of the part where defects are more likely to exist. The functional map method involves a large number of point-to-point comparisons based on noisy measurements, and a new statistical thresholding method used to filter the false positives in the underlying massive multiple comparisons problem is also provided.