Abstract

When phase shifting profilometry (PSP) is employed for 3-D shape measurement, the object must be kept static during the projection and acquisition of the multiple fringe patterns. Errors will occur when the object moves and if the projection and capture of fringe patterns are not fast enough. In this paper, a new approach is proposed to tackle the problem, consisting of two steps. Firstly, the rotation matrix and translation vector describing the movement of the object are estimated using a set of marks placing on the surface of the object. Then the expressions of the fringe patterns under the influence of 2-D object movement are derived, which are employed to determine the correct phase map, leading to accurate measurement of the profile. Simulations and experiments are presented to verify the effectiveness of the proposed algorithm.

© 2013 Optical Society of America

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  1. S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng.48(2), 149–158 (2010).
    [CrossRef]
  2. Y. Ding, J. Xi, Y. Yu, and J. Chicharo, “Recovering the absolute phase maps of two fringe patterns with selected frequencies,” Opt. Lett.36(13), 2518–2520 (2011).
    [CrossRef] [PubMed]
  3. S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express18(9), 9684–9689 (2010).
    [CrossRef] [PubMed]
  4. Y. Hu, J. Xi, J. Chicharo, and Z. Yang, “Blind color isolation for color-channel-based fringe pattern profilometry using digital projection,” J. Opt. Soc. Am. A24(8), 2372–2382 (2007).
    [CrossRef] [PubMed]
  5. K. Liu, Y. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Gamma model and its analysis for phase measuring profilometry,” J. Opt. Soc. Am. A27(3), 553–562 (2010).
    [CrossRef] [PubMed]
  6. X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng.36(1), 49–64 (2001).
    [CrossRef]
  7. Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express13(8), 3110–3116 (2005).
    [CrossRef] [PubMed]
  8. S. Zhang and S.-T. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng.46(11), 113603 (2007).
    [CrossRef]
  9. E. Hu and Y. He, “Surface profile measurement of moving objects by using an improved π phase-shifting Fourier transform profilometry,” Opt. Lasers Eng.47(1), 57–61 (2009).
    [CrossRef]
  10. Y. Hu, J. Xi, J. F. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas.58(9), 3305–3314 (2009).
    [CrossRef]
  11. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell.22(11), 1330–1334 (2000).
    [CrossRef]
  12. P. Meer, D. Mintz, A. Rosenfeld, and D. Y. Kim, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vis.6(1), 59–70 (1991).
    [CrossRef]
  13. K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-Squares fitting of two 3-D point sets,” IEEE Trans. Pattern Anal. Mach. Intell.9(5), 698–700 (1987).
    [CrossRef] [PubMed]
  14. B. K. P. Horn, “Closed-form solution of absolute orientation using unit quaternions,” J. Opt. Soc. Am. A4(4), 629–642 (1987).
    [CrossRef]
  15. A. Trujillo-Pino, K. Krissian, M. Alemán-Flores, and D. Santana-Cedrés, “Accurate subpixel edge location based on partial area effect,” Image Vis. Comput.31(1), 72–90 (2013).
    [CrossRef]

2013 (1)

A. Trujillo-Pino, K. Krissian, M. Alemán-Flores, and D. Santana-Cedrés, “Accurate subpixel edge location based on partial area effect,” Image Vis. Comput.31(1), 72–90 (2013).
[CrossRef]

2011 (1)

2010 (3)

2009 (2)

E. Hu and Y. He, “Surface profile measurement of moving objects by using an improved π phase-shifting Fourier transform profilometry,” Opt. Lasers Eng.47(1), 57–61 (2009).
[CrossRef]

Y. Hu, J. Xi, J. F. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas.58(9), 3305–3314 (2009).
[CrossRef]

2007 (2)

S. Zhang and S.-T. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng.46(11), 113603 (2007).
[CrossRef]

Y. Hu, J. Xi, J. Chicharo, and Z. Yang, “Blind color isolation for color-channel-based fringe pattern profilometry using digital projection,” J. Opt. Soc. Am. A24(8), 2372–2382 (2007).
[CrossRef] [PubMed]

2005 (1)

2001 (1)

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng.36(1), 49–64 (2001).
[CrossRef]

2000 (1)

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell.22(11), 1330–1334 (2000).
[CrossRef]

1991 (1)

P. Meer, D. Mintz, A. Rosenfeld, and D. Y. Kim, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vis.6(1), 59–70 (1991).
[CrossRef]

1987 (2)

K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-Squares fitting of two 3-D point sets,” IEEE Trans. Pattern Anal. Mach. Intell.9(5), 698–700 (1987).
[CrossRef] [PubMed]

B. K. P. Horn, “Closed-form solution of absolute orientation using unit quaternions,” J. Opt. Soc. Am. A4(4), 629–642 (1987).
[CrossRef]

Alemán-Flores, M.

A. Trujillo-Pino, K. Krissian, M. Alemán-Flores, and D. Santana-Cedrés, “Accurate subpixel edge location based on partial area effect,” Image Vis. Comput.31(1), 72–90 (2013).
[CrossRef]

Arun, K. S.

K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-Squares fitting of two 3-D point sets,” IEEE Trans. Pattern Anal. Mach. Intell.9(5), 698–700 (1987).
[CrossRef] [PubMed]

Blostein, S. D.

K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-Squares fitting of two 3-D point sets,” IEEE Trans. Pattern Anal. Mach. Intell.9(5), 698–700 (1987).
[CrossRef] [PubMed]

Chao, Y.

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng.36(1), 49–64 (2001).
[CrossRef]

Chen, W.

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng.36(1), 49–64 (2001).
[CrossRef]

Cheng, W.

Y. Hu, J. Xi, J. F. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas.58(9), 3305–3314 (2009).
[CrossRef]

Chicharo, J.

Chicharo, J. F.

Y. Hu, J. Xi, J. F. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas.58(9), 3305–3314 (2009).
[CrossRef]

Ding, Y.

Hao, Q.

Hassebrook, L. G.

He, Y.

E. Hu and Y. He, “Surface profile measurement of moving objects by using an improved π phase-shifting Fourier transform profilometry,” Opt. Lasers Eng.47(1), 57–61 (2009).
[CrossRef]

Horn, B. K. P.

Hu, E.

E. Hu and Y. He, “Surface profile measurement of moving objects by using an improved π phase-shifting Fourier transform profilometry,” Opt. Lasers Eng.47(1), 57–61 (2009).
[CrossRef]

Hu, Y.

Y. Hu, J. Xi, J. F. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas.58(9), 3305–3314 (2009).
[CrossRef]

Y. Hu, J. Xi, J. Chicharo, and Z. Yang, “Blind color isolation for color-channel-based fringe pattern profilometry using digital projection,” J. Opt. Soc. Am. A24(8), 2372–2382 (2007).
[CrossRef] [PubMed]

Huang, T. S.

K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-Squares fitting of two 3-D point sets,” IEEE Trans. Pattern Anal. Mach. Intell.9(5), 698–700 (1987).
[CrossRef] [PubMed]

Kim, D. Y.

P. Meer, D. Mintz, A. Rosenfeld, and D. Y. Kim, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vis.6(1), 59–70 (1991).
[CrossRef]

Krissian, K.

A. Trujillo-Pino, K. Krissian, M. Alemán-Flores, and D. Santana-Cedrés, “Accurate subpixel edge location based on partial area effect,” Image Vis. Comput.31(1), 72–90 (2013).
[CrossRef]

Lau, D. L.

Liu, K.

Meer, P.

P. Meer, D. Mintz, A. Rosenfeld, and D. Y. Kim, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vis.6(1), 59–70 (1991).
[CrossRef]

Mintz, D.

P. Meer, D. Mintz, A. Rosenfeld, and D. Y. Kim, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vis.6(1), 59–70 (1991).
[CrossRef]

Oliver, J.

Rosenfeld, A.

P. Meer, D. Mintz, A. Rosenfeld, and D. Y. Kim, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vis.6(1), 59–70 (1991).
[CrossRef]

Santana-Cedrés, D.

A. Trujillo-Pino, K. Krissian, M. Alemán-Flores, and D. Santana-Cedrés, “Accurate subpixel edge location based on partial area effect,” Image Vis. Comput.31(1), 72–90 (2013).
[CrossRef]

Su, X.

Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express13(8), 3110–3116 (2005).
[CrossRef] [PubMed]

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng.36(1), 49–64 (2001).
[CrossRef]

Trujillo-Pino, A.

A. Trujillo-Pino, K. Krissian, M. Alemán-Flores, and D. Santana-Cedrés, “Accurate subpixel edge location based on partial area effect,” Image Vis. Comput.31(1), 72–90 (2013).
[CrossRef]

Van Der Weide, D.

Wang, Y.

Xi, J.

Yang, Z.

Y. Hu, J. Xi, J. F. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas.58(9), 3305–3314 (2009).
[CrossRef]

Y. Hu, J. Xi, J. Chicharo, and Z. Yang, “Blind color isolation for color-channel-based fringe pattern profilometry using digital projection,” J. Opt. Soc. Am. A24(8), 2372–2382 (2007).
[CrossRef] [PubMed]

Yau, S.-T.

S. Zhang and S.-T. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng.46(11), 113603 (2007).
[CrossRef]

Yu, Y.

Zhang, Q.

Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express13(8), 3110–3116 (2005).
[CrossRef] [PubMed]

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng.36(1), 49–64 (2001).
[CrossRef]

Zhang, S.

S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express18(9), 9684–9689 (2010).
[CrossRef] [PubMed]

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng.48(2), 149–158 (2010).
[CrossRef]

S. Zhang and S.-T. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng.46(11), 113603 (2007).
[CrossRef]

Zhang, Z.

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell.22(11), 1330–1334 (2000).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

Y. Hu, J. Xi, J. F. Chicharo, W. Cheng, and Z. Yang, “Inverse function analysis method for fringe pattern profilometry,” IEEE Trans. Instrum. Meas.58(9), 3305–3314 (2009).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell.22(11), 1330–1334 (2000).
[CrossRef]

K. S. Arun, T. S. Huang, and S. D. Blostein, “Least-Squares fitting of two 3-D point sets,” IEEE Trans. Pattern Anal. Mach. Intell.9(5), 698–700 (1987).
[CrossRef] [PubMed]

Image Vis. Comput. (1)

A. Trujillo-Pino, K. Krissian, M. Alemán-Flores, and D. Santana-Cedrés, “Accurate subpixel edge location based on partial area effect,” Image Vis. Comput.31(1), 72–90 (2013).
[CrossRef]

Int. J. Comput. Vis. (1)

P. Meer, D. Mintz, A. Rosenfeld, and D. Y. Kim, “Robust regression methods for computer vision: a review,” Int. J. Comput. Vis.6(1), 59–70 (1991).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Eng. (1)

S. Zhang and S.-T. Yau, “High-speed three-dimensional shape measurement system using a modified two-plus-one phase-shifting algorithm,” Opt. Eng.46(11), 113603 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (3)

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng.48(2), 149–158 (2010).
[CrossRef]

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3-D shape measurement method based on FTP,” Opt. Lasers Eng.36(1), 49–64 (2001).
[CrossRef]

E. Hu and Y. He, “Surface profile measurement of moving objects by using an improved π phase-shifting Fourier transform profilometry,” Opt. Lasers Eng.47(1), 57–61 (2009).
[CrossRef]

Opt. Lett. (1)

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Figures (12)

Fig. 1
Fig. 1

The structure of PSP system.

Fig. 2
Fig. 2

Hemisphere simulation. (a) Fringe patterns of the hemisphere for the first step of PSP; (b) Reconstructed result by Mesh display; (c) Front view of Fig. 2(b); (d) The cross section of the dash line in Fig. 2(c) where x = 300.

Fig. 3
Fig. 3

Reconstructed results of traditional PSP when the object has oblique movement. (a) Reconstructed result by Mesh display; (b) Front view of Fig. 3(a); (c) The cross section of the dash line in Fig. 3(b) where x = 300.

Fig. 4
Fig. 4

Reconstructed results of the proposed algorithm when the object has oblique movement. (a) Reconstructed result by Mesh display; (b) Front view of Fig. 4(a); (c) The cross section of the dash line in Fig. 4(b) where x = 300.

Fig. 5
Fig. 5

Reconstructed results of the traditional PSP when the object has rotation movement. (a) Reconstructed result by Mesh display; (b) Front view of Fig. 5(a); (c) The cross section of the dash line in Fig. 5(b) where x = 300.

Fig. 6
Fig. 6

Reconstructed results of the proposed algorithm when the object has rotation movement. (a) Reconstructed result by Mesh display; (b) Front view of Fig. 6(a); (c) The cross section of the dash line in Fig. 6(b) where x = 300.

Fig. 7
Fig. 7

Object with marks

Fig. 8
Fig. 8

Reconstructed results of the traditional PSP when the object is static. (a) Fringe patterns of the first step of PSP; (b) Reconstructed result by Mesh display; (c) Front view of Fig. 8(b); (d) The cross section of the dash line in Fig. 8(c) where x = 115.

Fig. 9
Fig. 9

Reconstructed results of the traditional PSP when the object has oblique movement. (a) Reconstructed result by mesh display; (b) Front view of Fig. 9(a); (c) The cross section of the dash line in Fig. 9(b) where x = 115.

Fig. 10
Fig. 10

Reconstructed results of the proposed algorithm when the object has oblique movement. (a) Reconstructed result by mesh display; (b) Front view of Fig. 10(a); (c) The cross section of the dash line in Fig. 10(b) where x = 115.

Fig. 11
Fig. 11

Reconstructed results of the traditional PSP when the object has rotation movement. (a) Reconstructed result by mesh display; (b) Front view of Fig. 11(a); (c) The cross section of the dash line in Fig. 11(b) where x = 120.

Fig. 12
Fig. 12

Reconstructed results of the proposed algorithm when the object has rotation movement. (a) Reconstructed result by mesh display; (b) Front view of Fig. 12(a); (c) The cross section of the dash line in Fig. 12(b) where x = 120.

Tables (1)

Tables Icon

Table 1 The RMS measurement error of the mask

Equations (36)

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s n (x,y)=a+bcos(ϕ(x,y)+ 2π(n1) N )
d n (x,y)=a+bcos(ϕ(x,y)+Φ(x,y)+ 2π(n1) N )
ϕ r (x,y)=ϕ(x,y)=arctan n=1 N s n (x,y) sin2π(n1)/N n=1 N s n (x,y) cos2π(n1)/N
ϕ o (x,y)=ϕ(x,y)+Φ(x,y)=arctan n=1 N d n (x,y) sin2π(n1)/N n=1 N d n (x,y) cos2π(n1)/N
Φ(x,y)= Φ o (x,y) Φ r (x,y)
h(x,y)= l 0 Φ(x,y) Φ(x,y)2π f 0 d 0
[ x y ]=R[ u v ]+T, [ u v ]= R ¯ [ x y ]+ T ¯ .
R=[ r 11 r 12 r 21 r 22 ], T=[ t 1 t 2 ] ,
R ¯ =[ r ¯ 11 r ¯ 12 r ¯ 21 r ¯ 22 ], T ¯ =[ t ¯ 1 t ¯ 2 ].
R ¯ = R 1 , T ¯ = R 1 T.
h ˜ xy (u,v)= h xy (x,y)
h ˜ xy (u,v)= h xy (x,y)= h xy (f(u,v),g(u,v))
f(u,v)= r 11 u+ r 12 v+ t 1 , g(u,v)= r 21 u+ r 22 v+ t 2 .
h ˜ xy (x,y)= h xy (f(x,y),g(x,y))
d ˜ xy n (x,y)=a+bcos(ϕ(x,y)+ Φ ˜ (x,y)+ 2π(n1) N )
Φ ˜ (x,y)=Φ(f(x,y),g(x,y))
d ˜ xy n (x,y)=a+bcos(ϕ(x,y)+Φ(f(x,y),g(x,y))+ 2π(n1) N )
[ x y ]= R ¯ [ ξ η ]+ T ¯
d ˜ ξη n (ξ,η)= d ˜ xy n ( f ¯ (ξ,η), g ¯ (ξ,η)) =a+bcos(ϕ( f ¯ (ξ,η), g ¯ (ξ,η))+Φ(ξ,η)+ 2π(n1) N )
f ¯ (ξ,η)= r ¯ 11 ξ+ r ¯ 12 η+ t ¯ 1 , g ¯ (ξ,η)= r ¯ 21 ξ+ r ¯ 22 η+ t ¯ 2 .
d ˜ n (x,y)= d ˜ xy n ( f ¯ (x,y), g ¯ (x,y)) =a+bcos(ϕ( f ¯ (x,y), g ¯ (x,y))+Φ(x,y)+ 2π(n1) N )
{ d ˜ 1 (x,y)=a+bcos(ϕ(x,y)+Φ(x,y)) d ˜ 2 (x,y)=a+bcos(ϕ( f ¯ 2 (x,y), g ¯ 2 (x,y))+Φ(x,y)+2π/N) d ˜ N (x,y)=a+bcos(ϕ( f ¯ N (x,y), g ¯ N (x,y))+Φ(x,y)+2π(N1)/N)
Φ(x,y)=arctan D A D B D C D D
D A = n=1 N d ˜ n (x,y)cos 2π(n1) N n=1 N cos(ϕ( f ¯ n (x,y), g ¯ n (x,y))+ 2π(n1) N )sin 2π(n1) N ,
D B = n=1 N d ˜ n (x,y)sin 2π(n1) N n=1 N cos(ϕ( f ¯ n (x,y), g ¯ n (x,y))+ 2π(n1) N )cos 2π(n1) N ,
D C = n=1 N d ˜ n (x,y)cos 2π(n1) N n=1 N sin(ϕ( f ¯ n (x,y), g ¯ n (x,y))+ 2π(n1) N )sin 2π(n1) N ,
D D = n=1 N d ˜ n (x,y)sin 2π(n1) N n=1 N sin(ϕ( f ¯ n (x,y), g ¯ n (x,y))+ 2π(n1) N )cos 2π(n1) N .
f ¯ 1 (x,y)=x, g ¯ 1 (x,y)=y.
P j =[ x j y j ], Q j =[ u j v j ], j=1,2,...,J.
Q j = R ¯ P j + T ¯ + V j
P = 1 J j=1 J P j , P cj = P j P ,
Q = 1 J j=1 J Q j , Q cj = Q j Q .
2 = j=1 J Q j R ¯ ^ P j T ¯ ^ 2 = j=1 J Q cj R ¯ ^ P cj 2 = j=1 J ( Q cj T Q cj + P cj T P cj 2 Q cj T R ¯ ^ P cj )
H= j=1 J P cj Q cj T
R ¯ ^ =V U T
T ¯ ^ = Q R ¯ ^ P

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