Abstract

When reconstructing the three-dimensional (3D) object height profile using the fringe projection profilometry (FPP) technique, the light intensity reflected from the object surface can yield abruptly changing bias in the captured fringe image, which leads to severe reconstruction error. The traditional approach tries to remove the bias by suppressing the zero spectrum of the fringe image. It is based on the assumption that the aliasing between the frequency spectrum of the bias, which is around the zero frequency, and the frequency spectrum of the fringe is negligible. This, however, is not the case in practice. In this paper, we propose a novel (to our knowledge) technique to eliminate the bias in the fringe image using the dual-tree complex wavelet transform (DT-CWT). The new approach successfully identifies the features of bias, fringe, and noise in the DT-CWT domain, which allows the bias to be effectively extracted from a noisy fringe image. Experimental results show that the proposed algorithm is superior to the traditional methods and facilitates accurate reconstruction of objects’ 3D models.

© 2012 Optical Society of America

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References

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  1. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef]
  2. T. C. Hsung, D. P. K. Lun, and W. W. L. Ng, “efficient fringe image enhancement based on dual-tree complex wavelet transform,” Appl. Opt. 50, 3973–3986 (2011).
    [CrossRef]
  3. X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001).
    [CrossRef]
  4. J. Zhong and J. Weng, “Spatial carrier-fringe pattern analysis by means of wavelet transform,” Appl. Opt. 43, 4993–4998 (2004).
    [CrossRef]
  5. S. Li, X. Su, and W. J. Chen, “Spatial carrier fringe pattern demodulation by use of a two-dimensional real wavelet,” Appl. Opt. 48, 6893–6906 (2009).
    [CrossRef]
  6. K. Qian, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
    [CrossRef]
  7. I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Process. Mag. 22(6), 123–151 (2005).
    [CrossRef]
  8. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
    [CrossRef]
  9. S. Li, X. Su, W. Chen, and L. Xiang, “Eliminating the zero spectrum in Fourier transform profilometry using empirical mode decomposition,” J. Opt. Soc. Am. A 26, 1195–1201 (2009).
    [CrossRef]
  10. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

2011 (1)

2009 (2)

2006 (1)

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
[CrossRef]

2005 (1)

I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Process. Mag. 22(6), 123–151 (2005).
[CrossRef]

2004 (2)

2001 (1)

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

1983 (1)

Baraniuk, R. G.

I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Process. Mag. 22(6), 123–151 (2005).
[CrossRef]

Burton, D. R.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
[CrossRef]

Chen, W.

Chen, W. J.

Gdeisat, M. A.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Hsung, T. C.

Kingsbury, N. G.

I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Process. Mag. 22(6), 123–151 (2005).
[CrossRef]

Lalor, M. J.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
[CrossRef]

Li, S.

Lun, D. P. K.

Mutoh, K.

Ng, W. W. L.

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Qian, K.

Selesnick, I. W.

I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Process. Mag. 22(6), 123–151 (2005).
[CrossRef]

Su, X.

Takeda, M.

Weng, J.

Xiang, L.

Zhong, J.

Appl. Opt. (5)

IEEE Signal Process. Mag. (1)

I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Process. Mag. 22(6), 123–151 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
[CrossRef]

Opt. Lasers Eng. (1)

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

Other (1)

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

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Figures (19)

Fig. 1.
Fig. 1.

Testing object.

Fig. 2.
Fig. 2.

Computed fringe of the testing object.

Fig. 3.
Fig. 3.

Biased and noisy fringe ( β = 10 , σ 2 = 1 ).

Fig. 4.
Fig. 4.

Row #256 of Fig. 3.

Fig. 5.
Fig. 5.

Bias extracted using the proposed algorithm.

Fig. 6.
Fig. 6.

Reconstructed 3D model using the proposed algorithm.

Fig. 7.
Fig. 7.

Reconstructed 3D model using the FT approach.

Fig. 8.
Fig. 8.

Reconstructed 3D model using the EMD approach.

Fig. 9.
Fig. 9.

Testing the real object with the fringe projected.

Fig. 10.
Fig. 10.

Row #200 of the fringe image.

Fig. 11.
Fig. 11.

Reconstructed 3D model using the FT approach.

Fig. 12.
Fig. 12.

Reconstructed 3D model using the EMD approach.

Fig. 13.
Fig. 13.

Reconstructed 3D model using the proposed algorithm.

Fig. 14.
Fig. 14.

Reconstructed 3D model using the PSP approach. The object is static when capturing the two fringe images.

Fig. 15.
Fig. 15.

Reconstructed 3D model using the PSP approach. The object is slightly moved when capturing the two fringe images.

Fig. 16.
Fig. 16.

Ball object with fringe projected.

Fig. 17.
Fig. 17.

Reconstructed 3D model using the FT approach.

Fig. 18.
Fig. 18.

Reconstructed 3D model using the EMD approach.

Fig. 19.
Fig. 19.

Reconstructed 3D model using the proposed algorithm.

Tables (1)

Tables Icon

Table 1. Comparison of the Reconstructed 3D Models in Terms of Mean Square Error (mse) after Using Different Bias Removal Algorithms at Different Noise Levels and Bias Magnitudes

Equations (17)

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I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ 2 π f o x + φ ( x , y ) ] + n ( x , y ) ,
I = g + n = W + I w = W + ( g w + n w ) ,
̑ I = W + ( Ψ ( W I f ) ) = W + ( Ψ ( W ( I W + ϑ ( W I ) ) ) ) = W + ( Ψ ( g w + n w ϑ ( g w + n w ) ) ) ,
I w l = a w l + f w l + n w l ,
| I w l | 2 = | a w l + f w l + n w l | 2 = | a w l | 2 + | f w l | 2 + | n w l | 2 + 2 * real ( a w l f w l * + a w l n w l * + f w l n w l * ) ,
l min = log 2 ( 1 ( f o / f s ) ( 1 d / s ) ( Δ h ) max ) ,
l max = log 2 ( 1 2 ( f o / f s ) ( 1 d / s ) ( Δ h ) min ) ,
E { | I w l | 2 } = E { | a w l | 2 } + E { | n w l | 2 } ,
| I ^ w l | max ( | I w l | λ , 0 ) for l with negligible f w l .
λ = μ + 3 * std .
( med { | I w l | } ) 2 | f w l | 2 + E ( | n w l | 2 ) = | f w l | 2 + 4 σ 2 .
| I w l | 2 ( med { | I w l | } ) 2 + 4 σ 2 | a ˜ w l | 2 + | n w l | 2 + 2 × real ( f w l n w l * ) ,
I ^ w l = ϑ ( W I ) = | I ^ w l | exp ( j I w l ) .
BW ( g y ) { f 0 + 1 2 π ( φ ( x , y ) x ) min , f 0 + 1 2 π ( φ ( x , y ) x ) max } ,
BW ( w l V ) = { f s 2 l + 1 , f s 2 l } ,
f s 2 l min f 0 + 1 2 π ( ϕ x ) max , 1 2 l min 1 f s ( f 0 + 1 2 π ( x ( 2 π f 0 d · h ( x , y ) s ) ) max ) , l min = log 2 ( 1 ( f o / f s ) ( 1 d / s ) ( ( h ( x , y ) x ) ) max ) .
f s 2 l max + 1 f 0 + 1 2 π ( ϕ x ) min , l max = log 2 ( 1 2 ( f o / f s ) ( 1 d / s ) ( h ( x , y ) x ) min ) .

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