Surface profiling using sequential sampling and inverse methods. Part I: Mathematical background

A novel method is presented for measuring the height profile of the surface of an object, even in the presence of relative motion between the surface and the height sensors. The method involves making height measurements of the surface using several sensors arranged along the scanned surface. The surface profile and the relative motions are mathematically separated by observing that the surface features appear sequentially among the sensor data, while the relative motions appear simultaneously. The proposed linear inverse technique overcomes several limitations of existing profiling methods based on curvature measurements. In contrast with other inverse calculations, the results are quite stable, with noise in the results only about twice the measurement noise. Regularization is introduced as a means of smoothing noise and for achieving solutions for ill-posed cases. This paper focuses on the profile calculation of one-sided objects, and initially uses the assumption that the rigid relative motion between object and sensors is purely translational.