Second-harmonic injection for nonlinear phase-ripple suppression in four-step fringe projection profilometry

Projector-camera nonlinearity produces carrier-locked phase ripple in four-step fringe projection profilometry. We introduce second-harmonic injection, whose coefficient controls the third-fifth harmonic imbalance through nonlinear intermodulation, while the injected harmonic is rejected by ideal four-step demodulation. A two-stage spatial-reuse calibration selects the coefficient directly from measured ripple, evaluating 22 candidates using eight images. Across eight independent reference-plane projection-capture repetitions, the method achieved a phase RMSE of 0.00617 ± 0.00009 rad (mean ± sample standard deviation), compared with 0.02243 ± 0.00038 rad without correction. Sculpture-model measurements further demonstrated ripple suppression on a complex surface. The method provides efficient active predistortion for repeated four-step profilometry under a stable, calibrated response.