Background: Phase measuring deflectometry is a highly precise and full field metrology technique for specular surfaces based on the distortion of known reference patterns observed as a reflection at the surface under test. Typically, liquid crystal displays are employed to provide the required patterns. Due to a lack of research, these displays are used without sufficient calibration. Methods: In this work, we present an enhanced calibration for phase measuring deflectometry, taking flatness deviations of the display surface into account. The display shape is modelled as a polynomial surface whose coefficients are determined by minimizing the retrace error in a global optimization procedure during calibration. This approach does not require any additional measurements or hardware. Improvements due to the enhanced calibration model are qualified experimentally using a flat and a spherical concave mirror. Results and conclusion: The model-based parameterization of the display surface yields significant improvement on both samples. The peak to valley (PV) of measured deviations on the plane mirror are reduced by 67% to 0.55 μm. Measuring the spherical sample without the display parameterization leads to a rather large shape deviation of 33.40 μm PV which is reduced by 94% to 1.98 μm. The viability of our approach confirms the dominant role of flatness deviations of the display surface as an error source in absolute shape measurement using phase measuring deflectometry.
Vision ray calibration provides imaging properties of cameras for application in optical metrology by identifying an independent vision ray for each sensor pixel. Due to this generic description of imaging properties, setups of multiple cameras can be considered as one imaging device. This enables holistic calibration of such setups with the same algorithm that is used for the calibration of a single camera. Obtaining reference points for the calculation of independent vision rays requires knowledge of the parameters of the calibration setup. This is achieved by numerical optimization which comes with high computational effort due to the large amount of calibration data. Using the collinearity of reference points corresponding to individual sensor pixels as the measure of accuracy of system parameters, we derived a cost function that does not require explicit calculation of vision rays. We analytically derived formulae for gradient and Hessian matrix of this cost function to improve computational efficiency of vision ray calibration. Fringe projection measurements using a holistically vision ray calibrated system of two cameras demonstrate the effectiveness of our approach. To the best of our knowledge, neither any explicit description of vision ray calibration calculations nor the application of vision ray calibration in holistic camera system calibration can be found in literature.
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