Phase retrieval from reflective fringe patterns of double-sided transparent objects

'Ghosted' fringe patterns simultaneously reflected from both the upper and lower sides of a transparent target in the fringe reflection technique are captured for transparent surface 3D shape measurement, but the phase retrieval from the captured 'ghosted' fringe patterns is still not solved. A novel method is proposed to solve this issue by using two sets of phase-shifted fringe patterns with slightly different frequencies. The nonlinear least-squares method is used to estimate the fringe phase and modulation from both front and rear interfaces. Several simulations are done to show the feasibility of the proposed method. The influence of fringe noise on the algorithm is studied as well, which indicates that the proposed method is able to retrieve the phase from double-sided reflective fringe patterns with fringe noise equivalent to that in practical measurements. The merits and limitations of the method are discussed and recommendations for future studies are made.