Fig. 1. Principle of the proposed dual-frequency method: (a) four phase-shifting patterns of f_h; (b) four phase-shifting patterns of f_l; (c) wrapped phase ϕ_h; (d) wrapped phase ϕ_l; (e) phase difference ϕ_d; (f) phase sum ϕ_s; (g) one cross section of the unwrapped phase difference Φ_d, fringe order k_l, and wrapped phase ϕ_l; (h) one cross section of the phase sum ϕ_s, fringe order k_s, and unwrapped phase Φ_l.

Fig. 2. Chart of the computer simulation process of the proposed dual-frequency by FPP.

Fig. 3. Computer simulation results of the proposed dual-frequency in 3D shape measurement. (a), (b) One cross section of wrapped phases (T_l = 170 pixels, T_h = 150 pixels) and synthetic phases (T_d = 1275 pixels, T_s = 79.7 pixels); (c) one cross section of wrapped phase ϕ_l, phase difference ϕ_d, and fringe order k_l; (d) one cross section of wrapped phase ϕ_s, unwrapped phase Φ_l, and fringe order k_s; (e) one cross section of the restored object’s height; (f) one cross section of the restored height error.

Fig. 4. Trend chart of measurement accuracy with the high frequency of 1/30 and changing low frequency.

Fig. 5. Reconstruction phase error of reference plane (a) by the conventional dual-frequency method and (b)–(f) by the proposed dual-frequency method with different low frequencies (f_h = 1/45, and f_l = 1/50, 1/70, 1/90, 1/120, 1/140).

Fig. 6. Experimental results of measuring a simple object: (a), (b) one of the captured fringe patterns of f_h and f_l (T_h = 80 pixels, T_l = 85 pixels); (c), (d) wrapped phases ϕ_h and ϕ_l; (e) phase difference (T_d = 1360 pixels); (f) wrapped phase sum (T_s = 41.2 pixels); (g) fringe order k_l; (h) unwrapped phase Φ_l; (i) fringe order k_s; (j) unwrapped phase sum Φ_s.

Fig. 7. Reconstruction of the object’s absolute phase (a) by the conventional dual-frequency method and (b)–(f) by the proposed dual-frequency method with different low frequencies (f_h = 1/45, and f_l = 1/50, 1/70, 1/90, 1/120, 1/140).

Fig. 8. Reconstruction phase errors with different low frequencies corresponding to Fig. 7.

Fig. 9. Trend chart of measurement accuracy of actual experiments with a fixed high frequency and various low frequencies in measuring (a) the foam heart and (b) the petal.