Estimating phase shifts from three fringe patterns by use of cross spectrum

In phase-shifting technique, using self-calibrating algorithms allows us to determine phases and phase shifts simultaneously, thus eliminating the errors caused by miscalibrations of phase shifters. However, it is difficult to estimate phase shifts when only three fringe patterns are available, because in this case the problem becomes underdetermined. In this paper, we analyze the effects of phase-shift errors on the calculated phases, and find that the phase-shift errors introduce correlations between different frequency components of the calculated phases. We measure these correlations by calculating the cross spectrum of cis functions between the calculated phases and their trebles, and further define a single-valued objective function. A gradient-guided search strategy is used for minimizing this objective function, so that the phase shifts are estimated from three fringe patterns. The simulation and experimental results demonstrate that the newly proposed algorithm, in comparison with the existing correlation-based algorithms, has several advantages, such as being insensitive to the nonuniformities of the background intensities and the modulations, having a high stability, and offering improved computational efficiency.