Surface Variation Modeling by Fusing Multiresolution Spatially Nonstationary Data Under a Transfer Learning Framework

High-definition metrology (HDM) has gained significant attention for surface quality inspection since it can reveal spatial surface variations in detail. Due to its cost and durability, such HDM measurements are occasionally implemented. The limitation creates a new research opportunity to improve surface variation characterization by fusing the insights gained from limited HDM data with widely available low-resolution surface data during quality inspections. A useful insight from state-of-the-art research using HDM is the revealed relationship and positive correlation between surface height and certain measurable covariates, such as material removal rate (MRR). Such a relationship was assumed spatially constant and integrated with surface measurements to improve surface quality modeling. However, this method encounters challenges when the covariates have nonstationary relationships with the surface height over different surface areas, i.e., the covariate-surface height relationship is spatially varying. Additionally, the nonstationary relationship can only be captured by HDM, adding to the challenge of surface modeling when most training data are measured at low resolution. This paper proposes a transfer learning (TL) framework to deal with these challenges by which the common information from a spatial model of an HDM-measured surface is transferred to a new surface where only low-resolution data are available. Under this framework, the paper develops and compares three surface models to characterize the nonstationary relationship including two varying coefficient-based spatial models and an inference rule-based spatial model. Real-world case studies were conducted to demonstrate the proposed methods for improving surface modeling.