For many profilometry techniques, phase unwrapping is one of the most challenging process. In order to sidestep the phase unwrapping process, Perciante et. al [Appl Opt 2015; 54(10):3018-23] proposed a wrapping-free method based on the direct integration of the spatial derivatives of the patterns to retrieve the phase. But it is only applicable for the case of the phase continuity for the tested object, which means it may fail to handle fringe patterns containing complicated singularities, such as noise, shadow, shears and surface discontinuity. In view of this problems, a robust wrapping-free phase retrieval method is proposed in this paper, which is based on combined Perciante's method and weighted least-squares method. Two partial derivatives of the desired phase is obtained from the fringe patterns, meanwhile the carrier is eliminated using direct phase difference method. The phase singularities are determined using derivative variance correlation map (DVCM), and the weighting coefficient is obtained from the binary mask of the reverse DVCM. Simulations and experiments are conducted to prove the validity of the proposed method. Results are analyzed and compared with those of Perciante's method, demonstrating that the proposed method can be available for measuring objects with some kinds of singularities sources.

Conventional dual-frequency fringe projection algorithm often suffers from phase unwrapping failure when the frequency ratio between the high frequency and the low one is too large. Zhang et.al. proposed an enhanced two-frequency phase-shifting method to use geometric constraints of digital fringe projection(DFP) to reduce the noise impact due to the large frequency ratio. However, this method needs to calibrate the DFP system and calculate the minimum phase map min  at the nearest position from the camera perspective, these procedures are are relatively complex and more time-cosuming. In this paper, we proposed an improved method, which eliminates the system calibration and min  determination in Zhang's method,meanwhile does not need to use the low frequency fringe pattern. In the proposed method,we only need a set of high frequency fringe patterns to measure the object after the high frequency min  is directly estimated by the experiment. Thus the proposed method can simplify the procedure and improve the speed. Finally, the experimental evaluation is conducted to prove the validity of the proposed method.The results demonstrate that the proposed method can overcome the main disadvantages encountered by Zhang's method.

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