Results and Discussions For N-step PSP, M-coded (in this paper, M=3) patterns can mark N^M fringe orders, which can effectively improve the number of fringe orders. In this paper, four conventional sinusoidal phase-shifting patterns provide a high-quality wrapped phase, and three quasi-sinusoidal coded patterns can be used to extract the corresponding fringe order map independently. The results (Table 1) of several planes with known heights show that the reconstructed planes with different heights are very close to the corresponding standard planes, which verifies that the proposed temporal phase unwrapping method can accurately obtain the corresponding fringe orders of wrapped phases. To obtain fringe orders, both Gray-code and phase-code methods require rounding operation, which generates round-off errors. Overall, the existing stair-shaped codes have high sensitivity and great difficulty in determining numerous fringe orders due to complex shapes or texture variations. Facing complex surfaces with a large range of reflectivity, the proposed N-ary sinusoidal codewords are considerably self-adaptable to non-uniform reflectivity due to the loose quantification processing. The analysis of theory and experimental results indicates that the proposed coding method has higher coding efficiency and robustness than intensity-code methods, and is more flexible and concise than phase-code methods while breaking the limitation of the total number of fringe orders to recover the absolute phase more accurately. The validity and superiority of the proposed method are confirmed by experiments.

Phase-shifting profilometry (PSP) has been widely used in various three-dimensional (3D) scenes due to its high accuracy and robustness. In fringe projection profilometry (FPP), unwrapping the phase map in (-π, π] is an inevitable consideration. Phase unwrapping algorithms are commonly divided into two types: spatial phase unwrapping algorithms (SPUAs) and temporal phase unwrapping algorithms (TPUAs). Conventionally, TPUAs are more suitable for measuring discontinuous objects as they can identify fringe orders pixel by pixel. TPUAs are also employed in real-time 3D measurement because of the development of hardware and defocusing systems. Numerous TPUAs have developed at a fast pace in the past few decades, mainly including multi-frequency, intensity-code, and phase-code ones. Multi-frequency methods suffer the low accuracy of the low-frequency patterns and the complicated selection among different frequencies. Intensity-code methods, mainly N-ary Gray codes with concision and high efficiency, directly use intensity information to generate the fringe order map, but they can barely measure colorful objects. In phase-code methods, massive codewords are coded into the phase domain with a depth of only 2π, and the difference between adjacent quantized phases may be too small to ensure the correct decoded codewords for a large number of fringe orders. Essentially, the proposed sinusoidal codewords are directly extracted from N-step phase-shifting patterns to replace additional stair-shaped codewords in intensity-code and phase-code methods. By contrast, the proposed fringe-order encoding method based on N-ary sinusoidal codewords performs outstanding coding flexibility and efficiency while breaking through the limitations of the number of fringe orders in phase-code methods and overcoming the sensitivity to reflectivity in Gray-code methods.

A temporal phase unwrapping method based on N-ary coding is proposed to realize the 3D measurement of colorfully complex objects. When measuring an object with a large range of overall surface reflectivity, traditional stair-shaped patterns face the difficulty of quantization of an excessive number of fringe orders. During encoding, N-ary sinusoidal codewords are successively extracted from sinusoidal phase-shifting patterns to replace traditional quantized gray codewords. It is worth noting that the edges of the extracted N-ary codewords coincide with the 2π discontinuities of wrapped phases to reduce the mismatch. By numeral system conversion, N-ary sinusoidal codewords are embedded into different periods of projected patterns to achieve the encoding of fringe orders. During decoding, the differences between coded patterns and N-step sinusoidal phase-shifting patterns can be first used to calculate N-ary quantized patterns by a loose operation [Eq. (5)], and then a unique fringe order can be obtained by reverse numeral system conversion. To remove the mismatches caused by the defocusing and noise of the system, a fringe-order self-correction method [Eq. (10)] is used to correct the jump errors around the 2π discontinuities of wrapped phases. Finally, the absolute phase can be obtained by the collation of the corrected fringe orders and wrapped phases. In this paper, an object [Fig. 5 (a)] with several planes of known height is measured to verify the feasibility of the proposed method, and some colorfully complex scenes [Figs. 8 (a)-(c)] in our daily life are further measured to demonstrate its high performance.

A temporal phase unwrapping method based on N-ary coding is proposed. By extracting the codewords from phase-shifting patterns to replace additional stair-shaped codewords in the existing TPUAs, the proposed method makes encoding and decoding more flexible and efficient. The experimental results demonstrate that the proposed self-coding method features high robustness for measuring sharply discontinuous and colorful objects in practice.