Achieving three-dimensional (3D) imaging with high flexibility and high robustness has long been a challenge for fringe projection profilometry (FPP). In order to address this challenge, a three-frequency number-theoretical temporal phase unwrapping (TFNT-TPU) method for phase-differencing profilometry (PDP) is proposed. By utilizing the isophase closure feature of the wrapped phase difference just caused by the measured object itself and the loose constraint relationship among the gratings with three different frequencies, 3D imaging with high flexibility can be achieved. Furthermore, to further enhance the error resistance and robustness of the proposed TFNT-TPU, an optimized design scheme for the parameters is established to suppress random errors. Additionally, a self-correction strategy is also proposed to suppress computing errors caused by arctangent operation. A series of theoretical analyses and experiments demonstrate that the 3D imaging with high flexibility and high robustness can be achieved by the proposed method.

Phase unwrapping is a critical step in fringe projection profilometry, essential for achieving accurate and efficient three-dimension (3D) imaging. Temporal phase unwrapping is the most widely utilized to improve robustness and the reconstruction quality. Unfortunately, due to abrupt phase discontinuities at boundaries, misalignment between the wrapped phases, and unreliable shadow regions, fringe order errors may occur. To address these challenges, this study presents a generalized multi-feature-guided progressive order correction algorithm (GMP-OCA) for high-quality 3D imaging. The algorithm integrates global coarse detection, incremental line-wise optimization, and regional precision scanning to progressively correct fringe orders. Static and dynamic experimental results demonstrate that GMP-OCA effectively eliminates the systematic errors inherent in various phase unwrapping methods, producing high-quality 3D imaging results.

Temporal phase unwrapping based on single auxiliary binary coded pattern has been proven to be effective for high-speed 3D measurement. However, in traditional spatial binary coding, it often leads to an imbalance between the number of periodic divisions and codewords. To meet this challenge, a large codewords orthogonal spatial binary coding method is proposed in this paper. By expanding spatial multiplexing from 1D to 2D orthogonal direction, it goes beyond the traditional 8 codewords to 27 codewords at three-level periodic division. In addition, a novel full-period connected domain segmentation technique based on local localization is proposed to avoid the time-consuming global iterative erosion and complex anomaly detection in traditional methods. For the decoding process, a purely spatial codewords recognition and a spatial-temporal hybrid codewords recognition methods are established to better suppress the percentage offset caused by static defocusing and dynamic motion, respectively. Obviating the need for intricate symbol recognition, the decoding process in our proposed method encompasses a straightforward analysis of statistical distribution. Building upon the development of special spatial binary coding, we have achieved a well-balance between low periodic division and large codewords for the first time. The experimental results verify the feasibility and validity of our proposed whole image processing method in both static and dynamic measurements.

In fringe projection profilometry (FPP) based on temporal phase unwrapping (TPU), reducing the number of projecting patterns has become one of the most important works in recent years. To remove the $2\pi $ ambiguity independently, this paper proposes a TPU method based on unequal phase-shifting code. Wrapped phase is still calculated from $N$ -step conventional phase-shifting patterns with equal phase-shifting amount to guarantee the measuring accuracy. Particularly, a series of different phase-shifting amounts relative to the first phase-shifting pattern are set as codewords, and encoded to different periods to generate one coded pattern. When decoding, Fringe order with a large number can be determined from the conventional and coded wrapped phases. In addition, we develop a self-correction method to eliminate the deviation between the edge of fringe order and the $2\pi $ discontinuity. Thus, the proposed method can achieve TPU but need to only project one additional coded pattern (e. g. 3+1), which can significantly benefit dynamic 3D shape reconstruction. The theoretical and experimental analysis verify that the proposed method performs high robustness on the reflectivity of the isolated object while ensuring the measuring speed.