Objective  Phase Measuring Deflectometry (PMD) in optical 3D measurement is widely used in optical surface measurement, rapid detection and other fields because of its advantages of rapid speed, high precision, stability and anti-interference. Due to the refraction and reflection of the upper and lower surfaces of transparent objects, the camera collects mixed fringes with different surface reflections. It is difficult for traditional PMD to measure them effectively in three dimensions. The existing phase extraction methods require high accuracy of the initial phase and need to collect a large number of fringe patterns. In order to solve this problem, a PMD method based on multi-frequency phase-shifting is proposed to measure the 3D morphology of transparent objects surface.

  Methods  This study proposes a multi-frequency phase-shifting-based PMD for measuring the 3D surface morphology of transparent objects. Firstly, the display screen shows a variety of sinusoidal fringes with different frequencies combined with multi-step phase shift, and the camera collects the mixed fringes reflected and superimposed on the surface of the object from another angle. Subsequently, the mixed fringes are separated iteratively by the least square method, and the wrapped phase of the upper and lower surfaces are obtained, and the unwrapping phase is obtained by the optimum three-fringe selection method in temporal phase unwrapping. Then, the relationship between phase and gradient is determined by gradient calibration, and the gradient of the measured object relative to the reference plane is determined according to the unwrapped phase. Finally, the gradient integral is used to restore the 3D morphology of the transparent object surface.

  Results and Discussions  In order to prove the effectiveness of the proposed method, a glass plate with a thickness of 3 mm and a plano-convex lens with a radius of curvature of 515.09 mm were measured, and a comparative experiment was conducted between the multi-frequency phase-shifting method and the multi-frequency method to verify the effectiveness of the proposed phase separation method (Fig.14). The experimental results show that the proposed method can effectively measure the 3D morphology of transparent objects surface (Fig.16, Fig.19). Compared with the existing methods, the average error of measuring the upper surface of transparent glass plate is reduced from 32.4 μm to 5.1 μm (Tab.2). This method can effectively avoid the influence caused by the large deviation of initial phase value, shorten the calculation time, and is suitable for 3D shape measurement of transparent objects with different shapes.

  Conclusions  A method for measuring the 3D morphology of transparent objects surface based on multiple frequencies is proposed. This method makes up for the deficiency of phase separation in traditional phase deflection measurement and can effectively measure the 3D morphology of transparent objects surface. Through the combination of different frequency fringes and multi-step phase-shifting, the upper and lower surfaces of transparent objects are separated by using the least square method for multiple iterations, which reduces the accuracy requirements of initial phase values, improves the numerical stability, is easy to realize phase convergence operation and shortens the calculation time. Compared with the traditional methods, it can be concluded that the proposed method of measuring the 3D topography of transparent objects based on multiple frequencies improves not only the realizability and numerical stability of iterative algorithm, but also the accuracy of topography measurement.