Abstract

In this paper we describe a high-resolution, low-noise phase-shifting algorithm applied to 360 degree digitizing of solids with diffuse light scattering surface. A 360 degree profilometer needs to rotate the object a full revolution to digitize a three-dimensional (3D) solid. Although 360 degree profilometry is not new, we are proposing however a new experimental set-up which permits full phase-bandwidth phase-measuring algorithms. The first advantage of our solid profilometer is: it uses base-band, phase-stepping algorithms providing full data phase-bandwidth. This contrasts with band-pass, spatial-carrier Fourier profilometry which typically uses 1/3 of the fringe data-bandwidth. In addition phase-measuring is generally more accurate than single line-projection, non-coherent, intensity-based line detection algorithms. Second advantage: new fringe-projection set-up which avoids self-occluding fringe-shadows for convex solids. Previous 360 degree fringe-projection profilometers generate self-occluding shadows because of the elevation illumination angles. Third advantage: trivial line-by-line fringe-data assembling based on a single cylindrical coordinate system shared by all 360-degree perspectives. This contrasts with multi-view overlapping fringe-projection systems which use iterative closest point (ICP) algorithms to fusion the 3D-data cloud within a single coordinate system (e.g. Geomagic). Finally we used a 400 steps/rotation turntable, and a 640x480 pixels CCD camera. Higher 3D digitized surface resolutions and less-noisy phase measurements are trivial by increasing the angular-spatial resolution and phase-steps number without any substantial change on our 360 degree profilometer.

© 2014 Optical Society of America

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References

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  1. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, l56–l60 (1982).
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. M. Chang, 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. Damico et al. eds., (IOS, Burke, 1995), pp. 423–429.
  7. Y. Song, H. Zhao, W. Chen, Y. Tan, “360 degree 3D profilometry,” Proc. SPIE 3204, 204–208 (1997).
    [CrossRef]
  8. A. Asundi, W. Zhou, “Mapping algorithm for 360-deg profilometry with time delayed integration imaging,” Opt. Eng. 38(2), 339–344 (1999).
    [CrossRef]
  9. X. Su, W. Chen, “Fourier transform profilometry,” Opt. Lasers Eng. 35(5), 263–284 (2001).
    [CrossRef]
  10. X. Zhang, P. Sun, H. Wang, “A new 360 rotation profilometry and its application in engine design,” Proc. SPIE 4537, 265–268 (2002).
    [CrossRef]
  11. J. A. Munoz-Rodriguez, A. Asundi, 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. Tmjillo-Schiaffino, N. Portillo-Amavisca, D. P. Salas-Peimbert, L. Molina-de la Rosa, S. Almazan-Cuellarand, and L. F. Corral-Martinez, “Three-dimensional profilometry of solid objects in rotation,” in AIP Proceedings 992, 924–928 (2008).
  13. B. Shi, B. Zhang, F. Liu, J. Luo, 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, 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. Z. Zhang, H. Ma, S. Zhang, T. Guo, C. E. Towers, D. P. Towers, “Simple calibration of a phase-based 3D imaging system based on uneven fringe projection,” Opt. Lett. 36(5), 627–629 (2011).
    [CrossRef] [PubMed]
  16. M. Servin, J. C. Estrada, “Analysis and synthesis of phase shifting algorithms based on linear systems theory,” Opt. Lasers Eng. 50(8), 1009–1014 (2012).
    [CrossRef]

2013 (1)

B. Shi, B. Zhang, F. Liu, J. Luo, 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]

2012 (1)

M. Servin, J. C. Estrada, “Analysis and synthesis of phase shifting algorithms based on linear systems theory,” Opt. Lasers Eng. 50(8), 1009–1014 (2012).
[CrossRef]

2011 (1)

2005 (1)

J. A. Munoz-Rodriguez, A. Asundi, 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, H. Wang, “A new 360 rotation profilometry and its application in engine design,” Proc. SPIE 4537, 265–268 (2002).
[CrossRef]

2001 (1)

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

1999 (1)

A. Asundi, 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, Y. Tan, “360 degree 3D profilometry,” Proc. SPIE 3204, 204–208 (1997).
[CrossRef]

1996 (1)

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

1994 (1)

A. K. Asundi, “360-deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33(8), 2760–2769 (1994).
[CrossRef]

1991 (1)

1985 (1)

1982 (1)

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, l56–l60 (1982).

Asundi, A.

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

Asundi, A. K.

A. K. Asundi, “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, 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]

Bu, G.

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

Chang, M.

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

Chen, W.

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

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

Cheng, X. X.

Chiang, F. P.

Estrada, J. C.

M. Servin, J. C. Estrada, “Analysis and synthesis of phase shifting algorithms based on linear systems theory,” Opt. Lasers Eng. 50(8), 1009–1014 (2012).
[CrossRef]

Guo, L. R.

Guo, T.

Halioua, M.

Ina, H.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, l56–l60 (1982).

Kobayashi, S.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, l56–l60 (1982).

Krishnamurthy, R. S.

Liu, F.

B. Shi, B. Zhang, F. Liu, J. Luo, 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.

Luo, J.

B. Shi, B. Zhang, F. Liu, J. Luo, 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]

Ma, H.

Munoz-Rodriguez, J. A.

J. A. Munoz-Rodriguez, A. Asundi, 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]

Rodriguez-Vera, R.

J. A. Munoz-Rodriguez, A. Asundi, 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]

Servin, M.

M. Servin, J. C. Estrada, “Analysis and synthesis of phase shifting algorithms based on linear systems theory,” Opt. Lasers Eng. 50(8), 1009–1014 (2012).
[CrossRef]

Shi, B.

B. Shi, B. Zhang, F. Liu, J. Luo, 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, Y. Tan, “360 degree 3D profilometry,” Proc. SPIE 3204, 204–208 (1997).
[CrossRef]

Su, X.

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

Su, X. Y.

Sun, P.

X. Zhang, P. Sun, 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, W. C. Tai, “360-deg profile noncontact measurement using a neural network,” Opt. Eng. 34(12), 3572–3576 (1995).
[CrossRef]

Takeda, M.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, l56–l60 (1982).

Tan, Y.

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

Towers, C. E.

Towers, D. P.

Wang, H.

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

Zhang, B.

B. Shi, B. Zhang, F. Liu, J. Luo, 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, S.

Zhang, X.

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

Zhang, Y.

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

Zhang, Z.

Zhao, H.

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

Zhou, W.

A. Asundi, 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, 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. A (1)

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, l56–l60 (1982).

Opt. Eng. (3)

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

A. K. Asundi, “360-deg profilometry: new techniques for display and acquisition,” Opt. Eng. 33(8), 2760–2769 (1994).
[CrossRef]

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

Opt. Laser Technol. (1)

J. A. Munoz-Rodriguez, A. Asundi, 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. (2)

M. Servin, J. C. Estrada, “Analysis and synthesis of phase shifting algorithms based on linear systems theory,” Opt. Lasers Eng. 50(8), 1009–1014 (2012).
[CrossRef]

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

Opt. Lett. (1)

Proc. SPIE (3)

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

X. Zhang, P. Sun, 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, Y. Tan, “360 degree 3D profilometry,” Proc. SPIE 3204, 204–208 (1997).
[CrossRef]

Other (2)

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. Damico et al. eds., (IOS, Burke, 1995), pp. 423–429.

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

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Figures (6)

Fig. 1
Fig. 1

Panel (a) shows the well known set-up when a linear grating projector is used. The linear grating is orthogonal to the z direction and its phase-sensitivity is proportional to tan(θ0). Panel (b) shows the experimental set-up for a stripe-light projected towards the solid under analysis. In light-stripe test (panel (b)) the phase-sensitivity is proportional to tan(φ0).

Fig. 2
Fig. 2

Proposed 4π steradians profilometer for 3D-surfaces within the space of ρ(z,φ)∈C1. The projected linear grating is in this case, along the z axis and orthogonal to previous 360-degree fringe-projection profilometers (Fig. 1(a)). As for the light-stripe (Fig. 1(b)) the phase-sensitivity of this 360-degree test is proportional to tan(φ0). This angle (φ0) must be kept large enough to increase the sensitivity of the test while avoiding lateral self-occluding shadowing.

Fig. 3
Fig. 3

Panel (a) shows the spatial linear fringes along the z direction (Fig. 2) projected over the simulated sphere (Eq. (8)) as imaged over the CCD-(x,z) plane. Panels (b), (c) and (d) show the fringe patterns obtained when N-lines at (x = 0,z) over the CCD are collected while the sphere is rotated 2π radians along the coordinate φ∈[0.2π) (Eq. (9)). The phase-shift among these 3 fringe patterns (see Eq. (9)) is 2π/3 radians, and are phase-demodulated using a 3-step least-squares phase-shifting algorithm to obtain (z,φ).

Fig. 4
Fig. 4

In panel (a) we show the photograph of the (white-painted) Rubik’s cube used as 3D test-object. Panel (b) shows the cube with the projected fringes along the z direction.

Fig. 5
Fig. 5

Panel (a) and (b) show two (out-of 4) phase-shifted, closed-fringe patterns of the Rubik’s cube in cylindrical coordinates (see Fig. 2). These two close-fringe patterns in (a) and (b) have 0, and π radian phase-shift respectively (the ones with π/2 and 3π/2 phase-shift are not shown). Panel (c) shows the wrapped phase of (z,φ) obtained by Eq. (10). Finally panel (d) shows the unwrapped phase of the cylindrical coordinate representation of the Rubik’s (z,φ).

Fig. 6
Fig. 6

This figure shows 3 perspectives of the 4π steradian (360 degree) digitized Rubik’s cube

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

ρ=ρ(z,φ),ρ= x 2 + y 2 ,z[L,L],φ[0,2π).
I(z,φ)=a(z,φ)+b(z,φ)cos[ ω 0 z+gρ(z,φ) ]; ω 0 = v 0 cos( θ 0 ),g= v 0 tan( θ 0 ).
I(x,z,φ)=δ[ xgρ(z,φ) ];g=tan( φ 0 ).
ρ=ρ(z,φ),ρ(z,φ) C 1 ,ρ= x 2 + y 2 ,z[L,L],φ[0,2π).
I(x,z)=a(x,z)+b(x,z)cos[ ω 0 x+gρ(x,z) ];φ=0, ω 0 = v 0 cos( φ 0 ),g= v 0 tan( φ 0 ).
I(0,z,φ)=a(0,z,φ)+b(0,z,φ)cos[ ω 0 0+gρ(0,z,φ) ];g= v 0 tan( φ 0 ),φ[0,2π).
I(z,φ)=a(z,φ)+b(z,φ)cos[ gρ(z,φ) ];x=0,z[L,L],φ[0,2π).
ρ(z,φ)= L 2 z 2 ;z[L,L],φ[0,2π),ρ(z,φ) C 1 .
I(z,φ,n)=a(z,φ)+b(z,φ)cos[ g L 2 z 2 +n 2π 3 ];z[L,L],φ[0,2π),n={0,1,2}.
A e igρ(z,φ) =I(z,φ,0)+ e i( 2π 4 ) I( z,φ,1 )+ e iπ I(z,φ,2)+ e i( 3 2π 4 ) I( z,φ,3 ).

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