Abstract

360 degrees (360°) digitalization of three dimensional (3D) solids using a projected light-strip is a well-established technique in academic and commercial profilometers. These profilometers project a light-strip over the digitizing solid while the solid is rotated a full revolution or 360-degrees. Then, a computer program typically extracts the centroid of this light-strip, and by triangulation one obtains the shape of the solid. Here instead of using intensity-based light-strip centroid estimation, we propose to use Fourier phase-demodulation for 360° solid digitalization. The advantage of Fourier demodulation over strip-centroid estimation is that the accuracy of phase-demodulation linearly-increases with the fringe density, while in strip-light the centroid-estimation errors are independent. Here we proposed first to construct a carrier-frequency fringe-pattern by closely adding the individual light-strip images recorded while the solid is being rotated. Next, this high-density fringe-pattern is phase-demodulated using the standard Fourier technique. To test the feasibility of this Fourier demodulation approach, we have digitized two solids with increasing topographic complexity: a Rubik's cube and a plastic model of a human-skull. According to our results, phase demodulation based on the Fourier technique is less noisy than triangulation based on centroid light-strip estimation. Moreover, Fourier demodulation also provides the amplitude of the analytic signal which is a valuable information for the visualization of surface details.

© 2016 Optical Society of America

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References

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  1. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
    [Crossref]
  2. M. Halioua, R. S. Krishnamurthy, H. C. Liu, and F. P. Chiang, “Automated 360 ° profilometry of 3-D diffuse objects,” Appl. Opt. 24(14), 2193–2196 (1985).
    [Crossref] [PubMed]
  3. X. X. Cheng, X. Y. Su, and L. R. Guo, “Automated measurement method for 360 ° profilometry of 3-D diffuse objects,” Appl. Opt. 30(10), 1274–1278 (1991).
    [Crossref] [PubMed]
  4. A. Asundi, C. S. Chan, and M. R. Sajan, “360 deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33(8), 2760–2769 (1994).
    [Crossref]
  5. M. Chang and W. C. Tai, “360-deg profile noncontact measurement using a neural network,” Opt. Eng. 34(12), 3572–3576 (1995).
    [Crossref]
  6. A. S. Gomes, L. A. Serra, A. S. Lage, and A. Gomes, “Automated 360 degree profilometry of human trunk for spinal deformity analysis,” in Proceedings of Three Dimensional Analysis of Spinal Deformities, M. D’Amico, A. Merolli, and G. C. Santambrogio, eds. (IOS, 1995), pp. 423–429.
  7. Y. Song, H. Zhao, W. Chen, and Y. Tan, “360 degree 3D profilometry,” Proc. SPIE 3204, 204–208 (1997).
    [Crossref]
  8. A. Asundi and W. Zhou, “Mapping algorithm for 360-deg profilometry with time delayed integration imaging,” Opt. Eng. 38(2), 339–344 (1999).
    [Crossref]
  9. X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
    [Crossref]
  10. X. Zhang, P. Sun, and H. Wang, “A new 360 rotation profilometry and its application in engine design,” Proc. SPIE 4537, 265–268 (2002).
    [Crossref]
  11. J. A. Muñoz-Rodríguez, A. Asundi, and R. Rodriguez-Vera, “Recognition of a light line pattern by Hu moments for 3-D reconstruction of a rotated object,” Opt. Laser Technol. 37(2), 131–138 (2005).
    [Crossref]
  12. G. Trujillo-Schiaffino, N. Portillo-Amavisca, D. P. Salas-Peimbert, L. Molina-de la Rosa, S. Almazan-Cuellar, and L. F. Corral-Martinez, “Three-dimensional profilometry of solid objects in rotation,” AIP Proc. 992, 924–928 (2008).
  13. B. Shi, B. Zhang, F. Liu, J. Luo, and J. Bai, “360° Fourier transform profilometry in surface reconstruction for fluorescence molecular tomography,” IEEE J. Biomed. Health Inform. 17(3), 681–689 (2013).
    [Crossref] [PubMed]
  14. Y. Zhang and G. Bu, “Automatic 360-deg profilometry of a 3D object using a shearing interferometer and virtual grating,” Proc. SPIE 2899, 162–169 (1996).
    [Crossref]
  15. M. Servin, G. Garnica, J. C. Estrada, and J. M. Padilla, “High-resolution low-noise 360-degree digital solid reconstruction using phase-stepping profilometry,” Opt. Express 22(9), 10914–10922 (2014).
    [Crossref] [PubMed]
  16. Y. Long, S. Wang, W. Wu, X. Yang, G. Jeon, and K. Liu, “Decoding line structured light patterns by using Fourier analysis,” Opt. Eng. 54(7), 073109 (2015).
    [Crossref]
  17. V. Bianco, M. Paturzo, and P. Ferraro, “Spatio-temporal scanning modality for synthesizing interferograms and digital holograms,” Opt. Express 22(19), 22328–22339 (2014).
    [Crossref] [PubMed]
  18. V. Bianco, M. Paturzo, V. Marchesano, I. Gallotta, E. Di Schiavi, and P. Ferraro, “Optofluidic holographic microscopy with custom field of view (FoV) using a linear array detector,” Lab Chip 15(9), 2117–2124 (2015).
    [Crossref] [PubMed]
  19. M. Servin, M. Padilla, and G. Garnica, “Fourier phase-demodulation applied to light-strip 360-degrees profilometry of 3D solids; theoretical principles,” http://arxiv.org/abs/1510.04587 .
  20. M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms and Applications (Wiley-VCH, 2014).

2015 (2)

Y. Long, S. Wang, W. Wu, X. Yang, G. Jeon, and K. Liu, “Decoding line structured light patterns by using Fourier analysis,” Opt. Eng. 54(7), 073109 (2015).
[Crossref]

V. Bianco, M. Paturzo, V. Marchesano, I. Gallotta, E. Di Schiavi, and P. Ferraro, “Optofluidic holographic microscopy with custom field of view (FoV) using a linear array detector,” Lab Chip 15(9), 2117–2124 (2015).
[Crossref] [PubMed]

2014 (2)

2013 (1)

B. Shi, B. Zhang, F. Liu, J. Luo, and J. Bai, “360° Fourier transform profilometry in surface reconstruction for fluorescence molecular tomography,” IEEE J. Biomed. Health Inform. 17(3), 681–689 (2013).
[Crossref] [PubMed]

2005 (1)

J. A. Muñoz-Rodríguez, A. Asundi, and R. Rodriguez-Vera, “Recognition of a light line pattern by Hu moments for 3-D reconstruction of a rotated object,” Opt. Laser Technol. 37(2), 131–138 (2005).
[Crossref]

2002 (1)

X. Zhang, P. Sun, and H. Wang, “A new 360 rotation profilometry and its application in engine design,” Proc. SPIE 4537, 265–268 (2002).
[Crossref]

2001 (1)

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[Crossref]

1999 (1)

A. Asundi and W. Zhou, “Mapping algorithm for 360-deg profilometry with time delayed integration imaging,” Opt. Eng. 38(2), 339–344 (1999).
[Crossref]

1997 (1)

Y. Song, H. Zhao, W. Chen, and Y. Tan, “360 degree 3D profilometry,” Proc. SPIE 3204, 204–208 (1997).
[Crossref]

1996 (1)

Y. Zhang and G. Bu, “Automatic 360-deg profilometry of a 3D object using a shearing interferometer and virtual grating,” Proc. SPIE 2899, 162–169 (1996).
[Crossref]

1995 (1)

M. Chang and W. C. Tai, “360-deg profile noncontact measurement using a neural network,” Opt. Eng. 34(12), 3572–3576 (1995).
[Crossref]

1994 (1)

A. Asundi, C. S. Chan, and M. R. Sajan, “360 deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33(8), 2760–2769 (1994).
[Crossref]

1991 (1)

1985 (1)

1982 (1)

Asundi, A.

J. A. Muñoz-Rodríguez, A. Asundi, and R. Rodriguez-Vera, “Recognition of a light line pattern by Hu moments for 3-D reconstruction of a rotated object,” Opt. Laser Technol. 37(2), 131–138 (2005).
[Crossref]

A. Asundi and W. Zhou, “Mapping algorithm for 360-deg profilometry with time delayed integration imaging,” Opt. Eng. 38(2), 339–344 (1999).
[Crossref]

A. Asundi, C. S. Chan, and M. R. Sajan, “360 deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33(8), 2760–2769 (1994).
[Crossref]

Bai, J.

B. Shi, B. Zhang, F. Liu, J. Luo, and J. Bai, “360° Fourier transform profilometry in surface reconstruction for fluorescence molecular tomography,” IEEE J. Biomed. Health Inform. 17(3), 681–689 (2013).
[Crossref] [PubMed]

Bianco, V.

V. Bianco, M. Paturzo, V. Marchesano, I. Gallotta, E. Di Schiavi, and P. Ferraro, “Optofluidic holographic microscopy with custom field of view (FoV) using a linear array detector,” Lab Chip 15(9), 2117–2124 (2015).
[Crossref] [PubMed]

V. Bianco, M. Paturzo, and P. Ferraro, “Spatio-temporal scanning modality for synthesizing interferograms and digital holograms,” Opt. Express 22(19), 22328–22339 (2014).
[Crossref] [PubMed]

Bu, G.

Y. Zhang and G. Bu, “Automatic 360-deg profilometry of a 3D object using a shearing interferometer and virtual grating,” Proc. SPIE 2899, 162–169 (1996).
[Crossref]

Chan, C. S.

A. Asundi, C. S. Chan, and M. R. Sajan, “360 deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33(8), 2760–2769 (1994).
[Crossref]

Chang, M.

M. Chang and W. C. Tai, “360-deg profile noncontact measurement using a neural network,” Opt. Eng. 34(12), 3572–3576 (1995).
[Crossref]

Chen, W.

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[Crossref]

Y. Song, H. Zhao, W. Chen, and Y. Tan, “360 degree 3D profilometry,” Proc. SPIE 3204, 204–208 (1997).
[Crossref]

Cheng, X. X.

Chiang, F. P.

Di Schiavi, E.

V. Bianco, M. Paturzo, V. Marchesano, I. Gallotta, E. Di Schiavi, and P. Ferraro, “Optofluidic holographic microscopy with custom field of view (FoV) using a linear array detector,” Lab Chip 15(9), 2117–2124 (2015).
[Crossref] [PubMed]

Estrada, J. C.

Ferraro, P.

V. Bianco, M. Paturzo, V. Marchesano, I. Gallotta, E. Di Schiavi, and P. Ferraro, “Optofluidic holographic microscopy with custom field of view (FoV) using a linear array detector,” Lab Chip 15(9), 2117–2124 (2015).
[Crossref] [PubMed]

V. Bianco, M. Paturzo, and P. Ferraro, “Spatio-temporal scanning modality for synthesizing interferograms and digital holograms,” Opt. Express 22(19), 22328–22339 (2014).
[Crossref] [PubMed]

Gallotta, I.

V. Bianco, M. Paturzo, V. Marchesano, I. Gallotta, E. Di Schiavi, and P. Ferraro, “Optofluidic holographic microscopy with custom field of view (FoV) using a linear array detector,” Lab Chip 15(9), 2117–2124 (2015).
[Crossref] [PubMed]

Garnica, G.

Guo, L. R.

Halioua, M.

Ina, H.

Jeon, G.

Y. Long, S. Wang, W. Wu, X. Yang, G. Jeon, and K. Liu, “Decoding line structured light patterns by using Fourier analysis,” Opt. Eng. 54(7), 073109 (2015).
[Crossref]

Kobayashi, S.

Krishnamurthy, R. S.

Liu, F.

B. Shi, B. Zhang, F. Liu, J. Luo, and J. Bai, “360° Fourier transform profilometry in surface reconstruction for fluorescence molecular tomography,” IEEE J. Biomed. Health Inform. 17(3), 681–689 (2013).
[Crossref] [PubMed]

Liu, H. C.

Liu, K.

Y. Long, S. Wang, W. Wu, X. Yang, G. Jeon, and K. Liu, “Decoding line structured light patterns by using Fourier analysis,” Opt. Eng. 54(7), 073109 (2015).
[Crossref]

Long, Y.

Y. Long, S. Wang, W. Wu, X. Yang, G. Jeon, and K. Liu, “Decoding line structured light patterns by using Fourier analysis,” Opt. Eng. 54(7), 073109 (2015).
[Crossref]

Luo, J.

B. Shi, B. Zhang, F. Liu, J. Luo, and J. Bai, “360° Fourier transform profilometry in surface reconstruction for fluorescence molecular tomography,” IEEE J. Biomed. Health Inform. 17(3), 681–689 (2013).
[Crossref] [PubMed]

Marchesano, V.

V. Bianco, M. Paturzo, V. Marchesano, I. Gallotta, E. Di Schiavi, and P. Ferraro, “Optofluidic holographic microscopy with custom field of view (FoV) using a linear array detector,” Lab Chip 15(9), 2117–2124 (2015).
[Crossref] [PubMed]

Muñoz-Rodríguez, J. A.

J. A. Muñoz-Rodríguez, A. Asundi, and R. Rodriguez-Vera, “Recognition of a light line pattern by Hu moments for 3-D reconstruction of a rotated object,” Opt. Laser Technol. 37(2), 131–138 (2005).
[Crossref]

Padilla, J. M.

Paturzo, M.

V. Bianco, M. Paturzo, V. Marchesano, I. Gallotta, E. Di Schiavi, and P. Ferraro, “Optofluidic holographic microscopy with custom field of view (FoV) using a linear array detector,” Lab Chip 15(9), 2117–2124 (2015).
[Crossref] [PubMed]

V. Bianco, M. Paturzo, and P. Ferraro, “Spatio-temporal scanning modality for synthesizing interferograms and digital holograms,” Opt. Express 22(19), 22328–22339 (2014).
[Crossref] [PubMed]

Rodriguez-Vera, R.

J. A. Muñoz-Rodríguez, A. Asundi, and R. Rodriguez-Vera, “Recognition of a light line pattern by Hu moments for 3-D reconstruction of a rotated object,” Opt. Laser Technol. 37(2), 131–138 (2005).
[Crossref]

Sajan, M. R.

A. Asundi, C. S. Chan, and M. R. Sajan, “360 deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33(8), 2760–2769 (1994).
[Crossref]

Servin, M.

Shi, B.

B. Shi, B. Zhang, F. Liu, J. Luo, and J. Bai, “360° Fourier transform profilometry in surface reconstruction for fluorescence molecular tomography,” IEEE J. Biomed. Health Inform. 17(3), 681–689 (2013).
[Crossref] [PubMed]

Song, Y.

Y. Song, H. Zhao, W. Chen, and Y. Tan, “360 degree 3D profilometry,” Proc. SPIE 3204, 204–208 (1997).
[Crossref]

Su, X.

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[Crossref]

Su, X. Y.

Sun, P.

X. Zhang, P. Sun, and H. Wang, “A new 360 rotation profilometry and its application in engine design,” Proc. SPIE 4537, 265–268 (2002).
[Crossref]

Tai, W. C.

M. Chang and W. C. Tai, “360-deg profile noncontact measurement using a neural network,” Opt. Eng. 34(12), 3572–3576 (1995).
[Crossref]

Takeda, M.

Tan, Y.

Y. Song, H. Zhao, W. Chen, and Y. Tan, “360 degree 3D profilometry,” Proc. SPIE 3204, 204–208 (1997).
[Crossref]

Wang, H.

X. Zhang, P. Sun, and H. Wang, “A new 360 rotation profilometry and its application in engine design,” Proc. SPIE 4537, 265–268 (2002).
[Crossref]

Wang, S.

Y. Long, S. Wang, W. Wu, X. Yang, G. Jeon, and K. Liu, “Decoding line structured light patterns by using Fourier analysis,” Opt. Eng. 54(7), 073109 (2015).
[Crossref]

Wu, W.

Y. Long, S. Wang, W. Wu, X. Yang, G. Jeon, and K. Liu, “Decoding line structured light patterns by using Fourier analysis,” Opt. Eng. 54(7), 073109 (2015).
[Crossref]

Yang, X.

Y. Long, S. Wang, W. Wu, X. Yang, G. Jeon, and K. Liu, “Decoding line structured light patterns by using Fourier analysis,” Opt. Eng. 54(7), 073109 (2015).
[Crossref]

Zhang, B.

B. Shi, B. Zhang, F. Liu, J. Luo, and J. Bai, “360° Fourier transform profilometry in surface reconstruction for fluorescence molecular tomography,” IEEE J. Biomed. Health Inform. 17(3), 681–689 (2013).
[Crossref] [PubMed]

Zhang, X.

X. Zhang, P. Sun, and H. Wang, “A new 360 rotation profilometry and its application in engine design,” Proc. SPIE 4537, 265–268 (2002).
[Crossref]

Zhang, Y.

Y. Zhang and G. Bu, “Automatic 360-deg profilometry of a 3D object using a shearing interferometer and virtual grating,” Proc. SPIE 2899, 162–169 (1996).
[Crossref]

Zhao, H.

Y. Song, H. Zhao, W. Chen, and Y. Tan, “360 degree 3D profilometry,” Proc. SPIE 3204, 204–208 (1997).
[Crossref]

Zhou, W.

A. Asundi and W. Zhou, “Mapping algorithm for 360-deg profilometry with time delayed integration imaging,” Opt. Eng. 38(2), 339–344 (1999).
[Crossref]

Appl. Opt. (2)

IEEE J. Biomed. Health Inform. (1)

B. Shi, B. Zhang, F. Liu, J. Luo, and J. Bai, “360° Fourier transform profilometry in surface reconstruction for fluorescence molecular tomography,” IEEE J. Biomed. Health Inform. 17(3), 681–689 (2013).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

Lab Chip (1)

V. Bianco, M. Paturzo, V. Marchesano, I. Gallotta, E. Di Schiavi, and P. Ferraro, “Optofluidic holographic microscopy with custom field of view (FoV) using a linear array detector,” Lab Chip 15(9), 2117–2124 (2015).
[Crossref] [PubMed]

Opt. Eng. (4)

Y. Long, S. Wang, W. Wu, X. Yang, G. Jeon, and K. Liu, “Decoding line structured light patterns by using Fourier analysis,” Opt. Eng. 54(7), 073109 (2015).
[Crossref]

A. Asundi and W. Zhou, “Mapping algorithm for 360-deg profilometry with time delayed integration imaging,” Opt. Eng. 38(2), 339–344 (1999).
[Crossref]

A. Asundi, C. S. Chan, and M. R. Sajan, “360 deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33(8), 2760–2769 (1994).
[Crossref]

M. Chang and W. C. Tai, “360-deg profile noncontact measurement using a neural network,” Opt. Eng. 34(12), 3572–3576 (1995).
[Crossref]

Opt. Express (2)

Opt. Laser Technol. (1)

J. A. Muñoz-Rodríguez, A. Asundi, and R. Rodriguez-Vera, “Recognition of a light line pattern by Hu moments for 3-D reconstruction of a rotated object,” Opt. Laser Technol. 37(2), 131–138 (2005).
[Crossref]

Opt. Lasers Eng. (1)

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[Crossref]

Proc. SPIE (3)

X. Zhang, P. Sun, and H. Wang, “A new 360 rotation profilometry and its application in engine design,” Proc. SPIE 4537, 265–268 (2002).
[Crossref]

Y. Song, H. Zhao, W. Chen, and Y. Tan, “360 degree 3D profilometry,” Proc. SPIE 3204, 204–208 (1997).
[Crossref]

Y. Zhang and G. Bu, “Automatic 360-deg profilometry of a 3D object using a shearing interferometer and virtual grating,” Proc. SPIE 2899, 162–169 (1996).
[Crossref]

Other (4)

G. Trujillo-Schiaffino, N. Portillo-Amavisca, D. P. Salas-Peimbert, L. Molina-de la Rosa, S. Almazan-Cuellar, and L. F. Corral-Martinez, “Three-dimensional profilometry of solid objects in rotation,” AIP Proc. 992, 924–928 (2008).

M. Servin, M. Padilla, and G. Garnica, “Fourier phase-demodulation applied to light-strip 360-degrees profilometry of 3D solids; theoretical principles,” http://arxiv.org/abs/1510.04587 .

M. Servin, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms and Applications (Wiley-VCH, 2014).

A. S. Gomes, L. A. Serra, A. S. Lage, and A. Gomes, “Automated 360 degree profilometry of human trunk for spinal deformity analysis,” in Proceedings of Three Dimensional Analysis of Spinal Deformities, M. D’Amico, A. Merolli, and G. C. Santambrogio, eds. (IOS, 1995), pp. 423–429.

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Figures (13)

Fig. 2
Fig. 2

This figure schematically shows how a carrier-frequency fringe-pattern is constructed or synthesized from N individual Gaussian light-strip images bounded by (y) .

Fig. 3
Fig. 3

Cross-section at z0 of a dented solid ρ = ρ(φ,z0) having 6 discontinuities. At the specific rotation angle φ shown, the light-strip projector cast a self-occluding shadow (in blue) over the discontinuity as seen from the CCD camera. The blue-zones in the estimated signal ρ = ρ(φ,z0) denote zero intensity (shadows) where the recovered object is undefined.

Fig. 4
Fig. 4

Photographs of (a) the Rubik’s cube, and (b) a human-skull model used in our experiment. In both cases the ambient light is turned-on to see the object; however this ambient light is normally turned-off. If the ambient light cannot be turned-off, the energy of the background signal would increase and the fringes’ contrast would decrease, resulting in noisier phase measurements [20].

Fig. 5
Fig. 5

Digitized phase-modulated strips of the 2 solids under study. These come from 5 consecutive azimuthal rotation angles. Panel (a) shows light-strips over the Rubik’s cube. Panel (b) shows the light-strips over the plastic skull.

Fig. 6
Fig. 6

Carrier-frequency fringe-patterns (Eq. (3)) synthesized by adding individual light-strips for each rotation-step (see Fig. 5). Note that we have rotated the object more than a full-revolution; this excess rotation is useful to be far away from the left and right boundaries where the light-strip is bent by the 3D surface, and the Fourier demodulation does not work properly.

Fig. 7
Fig. 7

Frequency spectrum of the digitally constructed carrier-frequency fringes of the Rubik’s cube. The circle in red is the radius of the pass-band filter used to obtain the desired analytic signal. The spectral harmonics are due to the use of Gaussian-strip profiles instead of a sinusoidal profile.

Fig. 8
Fig. 8

Wrapped-phases of (a) the Rubik’s cube, and (b) the plastic model of the human-skull. These phases were obtained after filtering the analytic-signal (Fig. 7) and masking-out self-occluding shadow regions. Notice that the modulating phase in the Rubik’s cube is wrapped 6 times, while the skull phase is wrapped 7 times.

Fig. 9
Fig. 9

Unwrapped phases gρ(φ,z) in cylindrical coordinates masking-out low-amplitude fringe regions. From top to bottom we show: the cube with its analytic magnitude as texture, the skull phase without texture, and the skull phase with its analytic magnitude as texture.

Fig. 10
Fig. 10

Three-dimensional Cartesian renderings using the analytic magnitude as texture. At top, three perspectives of the Rubik’s cube. At bottom, the plastic skull showing self-occluding shadows inside the eye-basins where the light-strip is occluded from the camera view.

Fig. 11
Fig. 11

Comparison of the Fourier phase-demodulation against strip-light centroid-estimation for the Rubik’s cube digitalization. Panel (a) shows the Rubik’s cube with three z-cuts at z1, z2 and z3. These cuts are color-coded as level-curves for (b) line-centroid estimation, and (c) Fourier phase-demodulation. It is clear that we have a less noise using Fourier phase-demodulation.

Fig. 12
Fig. 12

A solid-sphere as imaged over the CCD camera using the 360° setups in Fig. 1. In panel 12(a), the fringe projector is located above the sphere as in Fig. 1(a) and we keep just the central column-pixels (marked in black). Panel 12(b) shows one out of 360 images from which we keep just the central column-pixel (see [15] and Fig. 1(b)). In panel 12(c), the strip projection as Fig. 1(c) shows; 360 of these small frames (inside the black rectangle) are needed. The ambient-light is turned-on to show the digitizing sphere.

Fig. 13
Fig. 13

Panel (a) shows the carrier-fringes constructed for the first time in [2]; the lower part of the sphere is occluded by self-shadow [2]. (b) 360° fringe pattern (without carrier) constructed following Servin et al. [15]; please note that we have no carrier, so one must collect 4 phase-shifted fringes requiring 4 solid’s revolutions [15]. Panel (c) shows 360° fringe-pattern construction using our herein proposed technique which requires a single fringe-pattern image.

Equations (10)

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G(y,z;φ)={ a1(y,z;φ)+b1(x,y;φ) e [ yρ(φ,z)sin θ 0 ] 2 2 σ 2 } (y) ;ρ[0,R],φ[0,2π),
(y)={ 1ify[ 0,max[y(φ,z)] ] 0otherwise ;φ[0,2π],z[L.L].
I(φ,z)= n=0 N1 G[ y(φnΔφ,z),z ] ;Δφ= 2π N .
I(φ,z)a(φ,z)+b(φ,z)cos[ ω 0 φ+gρ(φ,z) ];g= ω 0 sin( φ 0 ),
ω 0 = 2π(Numberofstripsintheφdirection) Numberofpixelsintheφdirection ( radians pixels ).
I(φ,z)=a(φ,z)+ b(φ,z) 2 e i[ ω 0 φ+gρ(φ,z) ] + b(φ,z) 2 e i[ ω 0 φ+gρ(φ,z) ] .
F[I(φ,z)]=A( ω φ , ω z )+C( ω φ ω 0 , ω z )+ C ( ω φ + ω 0 , ω z ).
F 1 [ C( ω φ , ω z ) ]= b(φ,z) 2 e igρ(φ,z) .
[ gρ(φ,z) ]mod2π=angle{ b(φ,z) 2 e igρ(φ,z) }.
Mask(φ,z)={ 1if| b(φ,z) |ε 0Otherwise .

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