Data fusion of surface normals and point coordinates for deflectometric measurements

Abstract

Measuring specular surfaces can be realized by means of deflectometric measurement systems with at least two reference planes as proposed proposed by Petz and Tutsch (2004). The results are the point coordinates and the normal direction of each valid measurement point. The typical evaluation strategy for continuous surfaces involves an integration or regularization of the measured normals. This method yields smooth results of the surface with deviations in the nanometer range but it is sensitive to systematic deviations. The measured point coordinates are robust against systematic deviations but the noise level is in the order of micrometers. As an alternative evaluation strategy a data-fusion process that combines both the normal direction and the point coordinates has been developed. A linear fitting technique is proposed to increase the accuracy of the point coordinate measurements by forming an objective functional as the mean squared misfit of the gradients with respect to the point coordinates on the one hand and to the normals on the other hand. Moreover, a constraint on the maximal change of the coordinate measurements is added to the optimization problem. To minimize to objective under the constraint a projected gradient method is used. The results show that the proposed method is able to adjust the point coordinate measurement to the measured normals and hence decrease the spatial noise level by more than an order of magnitude.