Filtered kriging for improved interpolation of periodic manufacturing surfaces

High-resolution characterization of surfaces is essential in a variety of quality control tasks in modern manufacturing, such as surface quality inspection and tooling maintenance. However, direct high-resolution surface measurements often come with high cost and/or long measurement time. Interpolation based on spatial process models, especially kriging-type methods, has been used to obtain denser estimations from low-resolution and cheaper measurements. Periodic spatial correlations, which commonly exist in manufacturing applications, cannot be adequately captured by conventional spatial models, thereby causing potential performance degradation or numerical issues. To address these challenges, we propose a new procedure termed as filtered kriging (FK), which separates the periodic component using a bandpass pre-filter, such that the residual can be well fitted with common models. Through frequency-domain analysis, conditions under which FK is effective are identified, and a practical bandpass filter design strategy is devised. A new theorem is proven to show that, when measurements are free from aliasing, perfect reconstruction guaranteed by the Nyquist-Shannon sampling theorem is achieved by FK estimations under certain assumptions. Finally, the effectiveness of FK is demonstrated by case studies using real-world periodic surfaces from two-photon lithography and ultrasonic metal welding. FK is shown to capture spatial correlation more adequately than conventional methods, and achieves better interpolation accuracy.