Registration-free localization of defects in 3-D parts from mesh metrology data using functional maps

TL;DR

This paper introduces a registration-free method using functional maps based on spectral Laplacian techniques for localizing defects on 3D parts from mesh data, improving efficiency and accuracy over registration-based approaches.

Contribution

It proposes a novel registration-free defect localization method using functional maps and a recursive partitioning algorithm for efficient surface analysis.

Findings

Effective localization of defects without registration.

Reduced computational cost through recursive partitioning.

Robustness to noisy measurements with statistical filtering.

Abstract

Spectral Laplacian methods, widely used in computer graphics and manifold learning, have been recently proposed for the Statistical Process Control (SPC) of a sequence of manufactured parts, whose 3-dimensional metrology is acquired with non-contact sensors. These techniques provide an {\em intrinsic} solution to the SPC problem, that is, a solution exclusively based on measurements on the scanned surfaces or 2-manifolds without making reference to their ambient space. These methods, therefore, avoid the computationally expensive, non-convex registration step needed to align the parts, as required by previous methods for SPC based on 3-dimensional measurements. Once a SPC mechanism triggers and out-of-control alarm, however, an additional problem remains: that of locating where on the surface of the part that triggered the SPC alarm there is a significant shape difference with respect to either an in-control part or its nominal (CAD) design. In the past, only registration-based solutions existed for this problem. In this paper, we present a new registration-free solution to the part localization problem. Our approach uses a functional map between the manifolds to be compared, that is, a map between functions defined on each manifold based on intrinsic differential operators, in particular, the Laplace-Beltrami operator, in order to construct a point to point mapping between the two manifolds and be able to locate defects on the suspected part. A recursive partitioning algorithm is presented to define a region of interest on the surface of the part where defects are likely to occur, which results in considerable computational advantages. The functional map method involves a very large number of point-to-point comparisons based on noisy measurements, and a statistical thresholding method is presented to filter the false positives in the underlying massive multiple comparisons problem.