(J. Franke, Kaiserslautern)
 

A common task in testing the quality of materials is the inspection of surfaces during production. There are several ad-hoc-algorithms which do not rely on a specific mathematical model at all or which are based on a quite simplified model for the structure of the surface an for the possible disturbances. Typically, the surface is partitioned into small segments, and from the data observed in these segments, a measure for the local regularity of the surface is calculated either directly with local smoothing procedures or indirectly following a suitable transformation (Fourier, Gabor, Wavelet, ...). If this measure of regularity assumes values outside some given tolerance limits, a defect in the surface is detected. Segmentation and choice of tolerance limits are based on simple heuristics. The goal of this project is the development of appropriate stochastic models for various regular surface structures and for local defects, which may be used for a systematic and generalizable construction of a data-adaptive segmentation and the choice of tolerance limits for local regularity measures. As candidates for such models, appropriate locally stationary stochastic processes on the integer lattice Z² are considered. On the basis of such models, an adaptive segmentation into areas of similar regularity can be developped following the example of a similar approach from regression analysis.
The rigourous models also allow for the construction of formal tests if there are significant defects in a given segment.