Intrinsic Geometrical Methods for Statistical Process Control of Complex Data Objects
Open Access
- Author:
- Zhao, Xueqi
- Graduate Program:
- Industrial Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- December 03, 2021
- Committee Members:
- Eunhye Song, Major Field Member
Enrique Del Castillo, Chair & Dissertation Advisor
Necdet Aybat, Major Field Member
Ethan Fang, Outside Unit, Field & Minor Member
Steven Landry, Program Head/Chair - Keywords:
- Differential geometry
Laplace-Beltrami operator
Noncontact sensor
Permutation test
Spectral methods
Part localization
Helmholtz equation
Persistent homology - Abstract:
- This dissertation presents new intrinsic geometrical methods for Statistical Process Control (SPC) of industrial processes from sequences of 3-dimensional (3D) metrology part data. We assume the part data can be collected from non-contact sensors or CT scans, describing two or three-dimensional manifolds embedded in the three-dimensional Euclidean space, and constituting datasets in the form of surface meshes (triangulations) or voxel arrays. The ultimate goal is to efficiently determine whether or not the manufacturing process is in a state of statistical control by comparing the shape of the newly produced parts against a reference set of parts previously determined to be produced under "in-control" process conditions. Here, the shape of an object is understood as the geometrical information that is independent of translations and rotations. Previous SPC approaches for 3D part data discounted such transformations by implementing a registration step, which superimposes the data from different parts so that they have the same location and orientation. This step, however, results in a non-convex optimization problem with a computational complexity of O(n^2), where n is the average sample size. Instead, our methods avoid the difficult registration problem by only utilizing intrinsic geometrical properties. Given that intrinsic features are calculated using information from the manifold (part) but not from the ambient space, they are naturally independent of any rigid transformation, making the registration step unnecessary. Furthermore, the methods proposed are shown to outperform registration-based methods in terms of their ability to detect out-of-control states. We first consider the estimation of the Laplace-Beltrami (LB) spectrum and its use as a monitoring statistic in a distribution-free SPC chart for control of 3D parts. The LB spectrum, consisting of the non-decreasing eigenvalues of the LB operator, is an intrinsic operator closely related to the heat equation, which encodes the geometry of the object in which the heat diffusion takes place, a fact used in machine learning and computer graphics. We study two different types of LB estimators, one based on the heat kernel, applicable to mesh (surface) scans, and another based on finite element methods (FEM) applied to the heat differential equation in 3 dimensions, applicable to both mesh and volumetric (voxel) metrology datasets. We show that the proposed SPC methods based on the LB spectrum are more sensitive to small shape deviations and are more robust to increments in noise than registration-based SPC methods. Furthermore, the FEM LB spectrum extends the applicability of our method to open meshes, which are very common in practice given the nature of the scanning process. Moreover, by taking advantage of symmetry and sparseness in the resulting matrices, the computational complexity of the FEM LB spectrum method is reduced to O(n). When in regular production a part has triggered an SPC alarm, it is necessary to locate the region on the part causing the shape deviations to allow for further investigation of the causes of the problem. A first simple approach we consider for solving this fault localization problem is to use registration between the defective part and the CAD model. However, to avoid the problems created by poor registration, and in order to be consistent with the intrinsic techniques developed in the rest of this thesis, we also develop an intrinsic defect localization method that uses the LB eigenvectors of the scanned parts based on the idea of finding a functional map. Since the method is based on multiple pointwise comparisons, a thresholding technique that filters out false positives is presented. We also show how to focus the comparison between parts on a particular region of interest, improving the performance of the method. The last part of this dissertation is devoted to the study of topological properties of objects, evidently intrinsic, too, and their use as statistical process control tools. In particular, we study persistent homology to monitor the topological properties of 3D lattice structures, important in a wide variety of additive manufacturing applications, from aerospace to medical bone implants. We propose to use homological features of dimensions zero and one, representing connected components and one-dimensional loops, respectively, to monitor the geometry of the holes in 3D printed lattice structures. A non-parametric hypothesis test and an SPC scheme are developed based on these features. It is shown how TDA methods are ideally suited for this type of hollowed structure, and they share the advantages of the other intrinsic geometrical SPC methods presented earlier.