Untangling parasitic reflection in phase measuring deflectometry by multi-frequency phase-shifting

Phase measuring deflectometry (PMD) is a technique that reconstructs the three-dimensional (3D) profiles of specular surfaces. When the object under test is transparent, its bottom surface creates a parasitic reflection that superimposes with the primary reflection created by the top surface. The superimposed reflections cause phase error in decoding of the fringe patterns and reduce the reconstruction accuracy. To accurately reconstruct the 3D profile of transparent objects, the superimposed reflections must first be untangled. In this paper, a multi-frequency phase-shifting approach is proposed to untangle the superimposed signals. Based on the principle of phase-shifting, a mathematical model is developed. The unknown phase angles in the mathematical model are solved by an optimization technique with input data obtained from fringe patterns at different spatial frequencies. A procedure is also developed to obtain the set of initial conditions for the optimization process. Both simulation and experiments were conducted to verify the proposed method. The results show that the proposed method can accurately untangle the phase angles corresponding to primary and parasitic reflections. The surface reconstruction result was compared to a reference measurement given by an interferometer, and a root-mean-square error of 32.95 nm was recorded. The accuracy achieved by the proposed method is compatible with another existing multi-frequency approach, while using roughly eight times fewer images. With the proposed method, better time efficiency can be achieved, and the computer's memory requirement can be lowered.