Phase fusion in multi-frequency phase-shifting fringe projection profilometry for minimizing effects of random noise

In fringe projection profilometry, random noise is recognized as one of the most crucial error-inducing factors. This paper presents a phase fusion strategy for minimizing the effects of noise on measurement results in multi-frequency phase-shifting profilometry. The multi-frequency profilometry captures several sequences of phase-shifted fringe patterns with different fringe frequencies for phase-unwrapping purposes. In it, the phase map of a lower-frequency fringe pattern has a larger unambiguous phase range, thus being used to form a reference for unwrapping the phases of a higher-frequency fringe pattern, and only the phases of the highest-frequency fringe pattern are retained as the final phase-measuring result because of its highest phase-sensitivity to object height variations. However, this traditional approach is not the optimal choice for suppressing the impact of noise. To solve this problem, we analyze the statistics such as biasedness and variance of noise-induced phase errors in a general case about phase shifts. Using the phase error model, we suggest a phase fusion strategy that calculates a linear combination of all the available phase maps with different frequencies, instead of discarding the low-frequency phase maps, to minimize the effects of noise. The optimum weight for each phase map, which is related to the fringe frequency and phase shifts, is deduced, and the minimized phase error variance is derived. Simulation and experimental results demonstrate the correctness of this principle and the effectiveness of this method in improving measurement accuracies.