General phase-shifting algorithm for hybrid errors suppression using variable-frequency fringes

In measurements based on phase-shifting fringe pattern analysis, residual ripple-like artifacts often appear due to the co-influence of several error sources, e.g., phase-shifting errors, temporal intensity fluctuations and high-order fringe harmonics, when existing algorithms are adopted to retrieve phase using limited number of fringe patterns. To overcome this issue, a general phase-shifting algorithm for hybrid errors suppression by variable-frequency fringes is proposed in this paper for what we believe to be the first time. A corresponding fringe model is deduced to represent real patterns more accurately under the co-influence of these error factors. Variable-frequency fringes are introduced to provide a least and sufficient system of equations, while a least-squares iterative technique with a grouped step-by-step strategy is adopted for stable calculating a larger number of desired parameters in the constructed model. For the phase jump problem caused by non-full rank matrices at certain sampling points, a regularization combined with constraints between coefficients of high-order fringe harmonics is further proposed for identification and processing. Simulations and experimental results have shown that compared with the prior techniques, the accuracies of the proposed algorithm have been significantly enhanced at least 2.1 (simulations) and 1.5 (experiments) times respectively using bi-frequency equal three-step as an example in the study.