PGGC: phase-gradient Gray code with temporal pattern sharing for 3D measurement

Temporal phase unwrapping commonly projects phase-shifted fringes and auxiliary codewords separately, which increases the acquisition burden in dynamic fringe projection profilometry. We propose phase-gradient Gray code (PGGC), which represents four Gray-code bits through controlled reversals of phase-gradient polarity. Successive exchanges of orthogonal sine and cosine fringe pairs form a 12-pattern cyclic sequence, from which five overlapping demodulation groups recover one reference phase and four Gray-code-bearing wrapped phases. Distance-adaptive gradient voting, half-period connectivity, and stable-region voting are then used to decode and refine the fringe orders before phase-polarity restoration and reference-consistency correction. The phase-shifting patterns thereby support both wrapped-phase retrieval and fringe-order decoding without a separate intensity-codeword sequence. In the low-reflectivity experiment, PGGC achieved a fringe-order disagreement rate of 0.118% relative to complementary Gray code (CGC), the lowest among the non-reference methods tested. Static and high-spatial-frequency experiments further demonstrated the reconstruction procedure, and continuous acquisition was evaluated with a model rotating at approximately 4 rpm. At 120 projected patterns/s, each 12-pattern cycle yields five temporally shared phase outputs, corresponding to an average of 50 phase maps/s. Reference-consistency correction supports this reconstruction under limited intra-cycle physical phase variation.