Fig. 1. Measured phase profiles of planes at different positions. (a) Circular fringes; (b) straight fringes

Fig. 2. Diagram of measurement principle

Fig. 3. Diagram of coordinate conversion

Fig. 4. Fringes. (a) Circular fringes in Cartesian coordinate system; (b) straight fringes in polar coordinate system

Fig. 5. Improved Fourier transform profilometry process of circular fringe

Fig. 6. Reconstruction results without noise. (a) Simulated object; (b) reference circular fringes; (c) deformed circular fringes; (d) reference fringes in polar coordinate system; (e) deformed fringes in polar coordinate system; (f) reference fringes after Gerchberg iteration; (g) deformed fringes after Gerchberg iteration; (h) calculated reference phase; (i) calculated deformation phase; (j) reconstructed surface; (k) reconstructed error

Fig. 7. Reconstruction results with 40 dB noise. (a) Deformed circular fringe; (b) deformed fringe in polar coordinate system;

Fig. 8. Comparison of reconstruction results of out-of-plane spherical dome obtained by circular fringe projection and straight fringe projection. (a) Measured dome; (b) reference circular fringe; (c) deformed circular fringe; (d) deformed fringe in polar coordinate system; (e) deformed fringe after Gerchberg iteration; (f) reconstructed out-of-plane dome by circular fringe projection; (g) reconstructed error of out-of-plane dome by circular fringe projection; (h) reference straight fringe; (i) deformed straight fringe; (j) reconstructed out-of-plane dome by straight fringe projection; (k) reconstructed error of out-of-plane dome by straight fringe projection; (l) profile of reconstructed out-of-plane dome by circular fringe projection and straight fringe projection

Fig. 9. Picture of experimental device

Fig. 10. Multi-plane calibration process. (a) Diagram of multi-plane calibration; (b) xm in central row of four planes relative to reference plane

Fig. 11. Reconstructed results of circular fringes. (a) Circular fringes of plane 1 to be tested; (b) circular fringes of plane 2 to be tested; (c) circular fringes of plane 3 to be tested; (d) fringes of plane 1 to be tested after Gerchberg iteration; (e) fringes of plane 2 to be tested after Gerchberg iteration; (f) fringe of plane 3 to be tested after Gerchberg iteration; (g) reconstructed height of measured planes; (h) height profiles of reconstructed planes

Fig. 12. Results of reconstructed plane with straight fringe. (a) Straight fringe of plane 1 to be tested; (b) straight fringe of plane 2 to be tested; (c) straight fringe of plane 3 to be tested; (d) reconstructed height of planes to be tested; (e) height profiles of reconstructed planes

Fig. 13. Experimental results of heart-shaped object. (a) Reference circular fringes; (b) deformed circular fringes; (c) deformed fringes before background normalization in polar coordinate system; (d) deformed fringes after background normalization in polar coordinate system; (e) deformed fringes after Gerchberg iteration; (f) reconstructed result of heart-shaped object by circular fringe; (g) reference straight fringes; (h) deformed straight fringes; (i) reconstructed result of heart-shaped object by straight fringe