Abstract

In optical phase shift profilometry (PSP), parallel fringe patterns are projected onto an object and the deformed fringes are captured using a digital camera. It is of particular interest in real time three- dimensional (3D) modeling applications because it enables 3D reconstruction using just a few image captures. When using this approach in a real life environment, however, the noise in the captured images can greatly affect the quality of the reconstructed 3D model. In this paper, a new image enhancement algorithm based on the oriented two-dimenional dual-tree complex wavelet transform (DT-CWT) is proposed for denoising the captured fringe images. The proposed algorithm makes use of the special analytic property of DT-CWT to obtain a sparse representation of the fringe image. Based on the sparse representation, a new iterative regularization procedure is applied for enhancing the noisy fringe image. The new approach introduces an additional preprocessing step to improve the initial guess of the iterative algorithm. Compared with the traditional image enhancement techniques, the proposed algorithm achieves a further improvement of 7.2dB on average in the signal-to-noise ratio (SNR). When applying the proposed algorithm to optical PSP, the new approach enables the reconstruction of 3D models with improved accuracy from 6 to 20dB in the SNR over the traditional approaches if the fringe images are noisy.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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  23. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004(2007).
    [CrossRef] [PubMed]
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  25. I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury, “The dual-tree complex wavelet transform,” IEEE Signal Process. 123–151 (2005).
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  26. H. Shi, B. Hu, and J. Q. Zhang, “A novel scheme for the design of approximate Hilbert transform pairs of orthonormal wavelet bases,” IEEE Trans. Signal Process. 56, 2289–2297(2008).
    [CrossRef]
  27. D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995).
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2010 (1)

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

2009 (2)

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

S. Li, X. Su, and W. J. Chen, “Spatial carrier fringe pattern phase demodulation by use of a two-dimensional real wavelet,” Appl. Opt. 48, 6893–6906 (2009).
[CrossRef] [PubMed]

2008 (4)

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolute phase estimation: Adaptive local denoising and global unwrapping,” Appl. Opt. 47, 5358–5369 (2008).
[CrossRef] [PubMed]

H. Shi, B. Hu, and J. Q. Zhang, “A novel scheme for the design of approximate Hilbert transform pairs of orthonormal wavelet bases,” IEEE Trans. Signal Process. 56, 2289–2297(2008).
[CrossRef]

T. W. Hui and G. K. H. Pang, “3-D measurement of solder paste using two-step phase shift profilometry,” IEEE Trans. Electron. Packag. Manufact. 31, 306–315 (2008).
[CrossRef]

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (PhaseLa) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833–846 (2008).
[CrossRef] [PubMed]

2007 (6)

K. Qian, “Windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[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, 113603 (2007).
[CrossRef]

J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef] [PubMed]

K. Qian, H. N. Le Tran, F. Lin, and H. S. Seah, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412–7418 (2007).
[CrossRef]

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111–119 (2007).
[CrossRef]

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004(2007).
[CrossRef] [PubMed]

2006 (3)

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722–8732 (2006).
[CrossRef] [PubMed]

M. Elad, “Why simple shrinkage is still relevant for redundant representations?” IEEE Trans. Inf. Theory 52, 5559–5569(2006).
[CrossRef]

M. Elad, B. Matalon, and M. Zibulevsky, “Image denoising with shrinkage and redundant representations,” 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol.  2, pp. 1924–1931.

2005 (2)

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

J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30, 2560–2562 (2005).
[CrossRef] [PubMed]

2004 (4)

2002 (1)

C. Lopez-Martinez and X. Fabregas, “Modeling and reduction of SAR interferometric phase noise in the wavelet domain,” IEEE Trans. Geosci. Remote Sens. 40, 2553–2566 (2002).
[CrossRef]

2001 (1)

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

1999 (1)

S. G. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).

1998 (2)

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

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

1995 (1)

D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995).
[CrossRef]

1984 (1)

P. Duhamel and H. Hollmann, “‘Split radix’ FFT algorithm,” Electron. Lett. 20, 14–16 (1984).
[CrossRef]

1983 (1)

Ainsworth, T. L.

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Astola, J.

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (PhaseLa) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833–846 (2008).
[CrossRef] [PubMed]

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolute phase estimation: Adaptive local denoising and global unwrapping,” Appl. Opt. 47, 5358–5369 (2008).
[CrossRef] [PubMed]

Asundi, A. K.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Baraniuk, R. G.

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

Bioucas-Dias, J.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004(2007).
[CrossRef] [PubMed]

Bruckstein, A. M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

Burton, D. R.

Chen, W.

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

Chen, W. J.

Daubechies, I.

I. Daubechies, M. Defrise, and C. D. Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, pp. 1413–1457(2004).
[CrossRef]

Defrise, M.

I. Daubechies, M. Defrise, and C. D. Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, pp. 1413–1457(2004).
[CrossRef]

Donoho, D. L.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995).
[CrossRef]

Duhamel, P.

P. Duhamel and H. Hollmann, “‘Split radix’ FFT algorithm,” Electron. Lett. 20, 14–16 (1984).
[CrossRef]

Egiazarian, K.

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolute phase estimation: Adaptive local denoising and global unwrapping,” Appl. Opt. 47, 5358–5369 (2008).
[CrossRef] [PubMed]

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (PhaseLa) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833–846 (2008).
[CrossRef] [PubMed]

Elad, M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

M. Elad, “Why simple shrinkage is still relevant for redundant representations?” IEEE Trans. Inf. Theory 52, 5559–5569(2006).
[CrossRef]

M. Elad, B. Matalon, and M. Zibulevsky, “Image denoising with shrinkage and redundant representations,” 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol.  2, pp. 1924–1931.

Fabregas, X.

C. Lopez-Martinez and X. Fabregas, “Modeling and reduction of SAR interferometric phase noise in the wavelet domain,” IEEE Trans. Geosci. Remote Sens. 40, 2553–2566 (2002).
[CrossRef]

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004(2007).
[CrossRef] [PubMed]

Frigo, M.

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111–119 (2007).
[CrossRef]

Gdeisat, M. A.

Ghiglia, D. C.

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

Grunes, M. R.

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Hollmann, H.

P. Duhamel and H. Hollmann, “‘Split radix’ FFT algorithm,” Electron. Lett. 20, 14–16 (1984).
[CrossRef]

Hu, B.

H. Shi, B. Hu, and J. Q. Zhang, “A novel scheme for the design of approximate Hilbert transform pairs of orthonormal wavelet bases,” IEEE Trans. Signal Process. 56, 2289–2297(2008).
[CrossRef]

Huang, L.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Hui, T. W.

T. W. Hui and G. K. H. Pang, “3-D measurement of solder paste using two-step phase shift profilometry,” IEEE Trans. Electron. Packag. Manufact. 31, 306–315 (2008).
[CrossRef]

Johnson, S. G.

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111–119 (2007).
[CrossRef]

Johnstone, I. M.

D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995).
[CrossRef]

Katkovnik, V.

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolute phase estimation: Adaptive local denoising and global unwrapping,” Appl. Opt. 47, 5358–5369 (2008).
[CrossRef] [PubMed]

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (PhaseLa) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833–846 (2008).
[CrossRef] [PubMed]

Kemao, Q.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Kingsbury, N. G.

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

Lalor, M. J.

Le Tran, H. N.

Lee, J. S.

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Li, S.

Lin, F.

Lopez-Martinez, C.

C. Lopez-Martinez and X. Fabregas, “Modeling and reduction of SAR interferometric phase noise in the wavelet domain,” IEEE Trans. Geosci. Remote Sens. 40, 2553–2566 (2002).
[CrossRef]

Mallat, S. G.

S. G. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).

Matalon, B.

M. Elad, B. Matalon, and M. Zibulevsky, “Image denoising with shrinkage and redundant representations,” 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol.  2, pp. 1924–1931.

Mol, C. D.

I. Daubechies, M. Defrise, and C. D. Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, pp. 1413–1457(2004).
[CrossRef]

Mutoh, K.

Pan, B.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Pang, G. K. H.

T. W. Hui and G. K. H. Pang, “3-D measurement of solder paste using two-step phase shift profilometry,” IEEE Trans. Electron. Packag. Manufact. 31, 306–315 (2008).
[CrossRef]

Papathanassiou, K. P.

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Pritt, M. D.

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

Qian, K.

Reigber, A.

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Seah, H. S.

Selesnick, I. W.

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

Shi, H.

H. Shi, B. Hu, and J. Q. Zhang, “A novel scheme for the design of approximate Hilbert transform pairs of orthonormal wavelet bases,” IEEE Trans. Signal Process. 56, 2289–2297(2008).
[CrossRef]

Su, X.

Takeda, M.

Valadao, G.

J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef] [PubMed]

Weng, J.

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, 113603 (2007).
[CrossRef]

Zhang, J. Q.

H. Shi, B. Hu, and J. Q. Zhang, “A novel scheme for the design of approximate Hilbert transform pairs of orthonormal wavelet bases,” IEEE Trans. Signal Process. 56, 2289–2297(2008).
[CrossRef]

Zhang, S.

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, 113603 (2007).
[CrossRef]

Zhong, J.

Zibulevsky, M.

M. Elad, B. Matalon, and M. Zibulevsky, “Image denoising with shrinkage and redundant representations,” 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol.  2, pp. 1924–1931.

Appl. Opt. (8)

Commun. Pure Appl. Math. (1)

I. Daubechies, M. Defrise, and C. D. Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, pp. 1413–1457(2004).
[CrossRef]

Electron. Lett. (1)

P. Duhamel and H. Hollmann, “‘Split radix’ FFT algorithm,” Electron. Lett. 20, 14–16 (1984).
[CrossRef]

IEEE Signal Process. (1)

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

IEEE Trans. Electron. Packag. Manufact. (1)

T. W. Hui and G. K. H. Pang, “3-D measurement of solder paste using two-step phase shift profilometry,” IEEE Trans. Electron. Packag. Manufact. 31, 306–315 (2008).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

C. Lopez-Martinez and X. Fabregas, “Modeling and reduction of SAR interferometric phase noise in the wavelet domain,” IEEE Trans. Geosci. Remote Sens. 40, 2553–2566 (2002).
[CrossRef]

IEEE Trans. Image Process. (3)

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (PhaseLa) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833–846 (2008).
[CrossRef] [PubMed]

J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[CrossRef] [PubMed]

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004(2007).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

M. Elad, “Why simple shrinkage is still relevant for redundant representations?” IEEE Trans. Inf. Theory 52, 5559–5569(2006).
[CrossRef]

IEEE Trans. Signal Process. (2)

H. Shi, B. Hu, and J. Q. Zhang, “A novel scheme for the design of approximate Hilbert transform pairs of orthonormal wavelet bases,” IEEE Trans. Signal Process. 56, 2289–2297(2008).
[CrossRef]

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111–119 (2007).
[CrossRef]

J. Am. Stat. Assoc. (1)

D. L. Donoho and I. M. Johnstone, “Adapting to unknown smoothness via wavelet shrinkage,” J. Am. Stat. Assoc. 90, 1200–1224 (1995).
[CrossRef]

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, 113603 (2007).
[CrossRef]

Opt. Lasers Eng. (3)

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

K. Qian, “Windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Opt. Lett. (1)

SIAM Rev. (1)

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

Other (3)

M. Elad, B. Matalon, and M. Zibulevsky, “Image denoising with shrinkage and redundant representations,” 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol.  2, pp. 1924–1931.

S. G. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).

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

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Figures (15)

Fig. 1
Fig. 1

Setup for optical PSP.

Fig. 2
Fig. 2

Test object.

Fig. 3
Fig. 3

3D plot of the original shapes used in the simulations: (a) Peaks and (b) Awl.

Fig. 4
Fig. 4

Undecimated 2D DWT of a noisy fringe image, level 2: H, V, and D subbands.

Fig. 5
Fig. 5

Clean fringe images.

Fig. 6
Fig. 6

Row 256 for Awl fringe pattern.

Fig. 7
Fig. 7

(a) Level 1 and (b) level 2 DT-CWT coefficients for row 256 of the Awl fringe pattern.

Fig. 8
Fig. 8

Oriented 2D DT-CWT (two levels are shown).

Fig. 9
Fig. 9

Oriented 2D DT-CWT magnitude response of clean fringe image, level 2 of the R H 1 , R H 2 , R V 1 , R V 2 , R D 1 , and R D 2 subbands (from left to right, respectively).

Fig. 10
Fig. 10

SNR of the enhanced fringe image using different enhancement approaches.

Fig. 11
Fig. 11

Enhanced fringe images using different algorithms, noise σ = 2.5 (a) M-NLF, (b) wavelet shrinkage, (c) DTISE-1, (d) DTISE-5.

Fig. 12
Fig. 12

Reconstruction using Goldstein’s phase unwrapping method from noisy fringe images (top left), enhanced fringe images using M-NLF (median filter plus nonlinear filtering approach) [2] (top right), and proposed DTISE: initial estimate (bottom left) and after five iterations (bottom right) for noise σ = 1.5 .

Fig. 13
Fig. 13

Average reconstruction SNR using different enhancement approaches.

Fig. 14
Fig. 14

Fringe images captured at (a) ISO100, exposure 1 / 10 s ; (b) ISO1600, exposure 1 / 320 s ; (c) result of using M-NLF [2] on (b); and (d) result of using the proposed DTISE (five iterations) on (b).

Fig. 15
Fig. 15

Reconstructed 3D model using Goldstein’s phase unwrapping method from fringe images captured at (a) ISO100, exposure 1 / 10 s ; (b) ISO1600, exposure 1 / 320 s ; (c) result of using M-NLF [2] on (b); and (d) result of using the proposed DTISE (five iterations) on (b).

Tables (2)

Tables Icon

Table 1 Approximate Number of Floating Point Operations (Flops) of the Proposed DTISE (Five Iterations) and WFF for Various Fringe Image Sizes and Parameters

Tables Icon

Table 2 Execution Time for Enhancing a 512 × 512 “Peaks” Fringe Image

Equations (38)

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g ( x , y ) = r ( x , y ) + c ( x , y ) cos [ 2 π f 0 x + ϕ ( x , y ) ] ,
W ψ g y ( a , b ) = c ( x , y ) cos [ w 0 x + ϕ ( x , y ) ] ψ ( b x a ) ( 1 a ) d x
W ψ g y ( a , b ) = c y ( a , b ) cos [ w 0 x + ϕ ( x , y ) ] ψ ( b x a ) ( 1 a ) d x .
W ψ g ( a , b ) = c ( a , b ) 2 e j w ϕ ( a , b ) w 0 e j w b [ δ ( w w 0 ) + δ ( w + w 0 ) ] ψ ^ ( a w ) d w ,
W ψ g ( a , b ) = c ( a , b ) 2 | ψ ^ ( a w 0 ) | e j ( w 0 b ϕ ( a , b ) + ϕ ψ ^ ( a w 0 ) = : B e j θ ,
g ( x , y ) = c ( x , y ) cos [ u 0 x + v 0 y + ϕ r ( x , y ) ] ,
g ^ ( u , v ) = c 2 e j ϕ r u / u 0 [ δ ( u u 0 ) δ ( v v 0 ) + δ ( u + u 0 ) δ ( v + v 0 ) ] .
W ψ g ( a , b x , b y ) = c ( x , y ) cos [ u 0 x + v 0 y + ϕ r ( x , y ) ] ψ ( b x x a , b y y a ) ( 1 a 2 ) d x d y = c ( a , b x , b y ) 2 e j ϕ r ( a , b x , b y ) u / u 0 [ δ ( u u 0 ) δ ( v v 0 ) + δ ( u + u 0 ) δ ( v + v 0 ) ] · ψ ^ ( a u , a v ) e j u b x e j v b y d u d v = c ( a , b x , b y ) 2 e j ( u 0 b x + v 0 b y + ϕ r ( a , b x , b y ) ψ ^ ( a u 0 , a v 0 ) = c ( a , b x , b y ) 2 | ψ ^ ( a u 0 , b v 0 ) | e j ( u 0 b x + v 0 b y ϕ r ( a , b x , b y ) + ϕ ψ ^ ( a u 0 , b v 0 ) = : B e j θ ,
R H 1 = ( H p p H q q ) + j ( H q p + H p q ) ;
R V 1 = ( V p p V q q ) + j ( V q p + V p q ) ;
R D 1 = ( D p p D q q ) + j ( D q p + D p q ) ;
R H 2 = ( H p p + H q q ) + j ( H q p H p q ) ;
R V 2 = ( V p p + V q q ) + j ( V q p V p q ) ;
R D 2 = ( D p p + D q q ) + j ( D q p D p q ) .
y = g + n = W + g w + W + n w ,
arg min g w 1 2 W + g w y 2 + λ · ρ ( g w ) ,
g w k + 1 = η ( W ( y W + g w k ) + g w k , λ ) ) ,
η ( x , λ ) = max ( 0 , | x | λ ) exp ( j . x ) ,
g w 0 = med ( | η ( W g , λ ) ) | , N x , N y ) . exp ( j . W g ) ,
y ^ = W + ( g w 0 ) .
g w k + 1 = η ( W ( y ^ W + g w k ) + g w k , λ ) ) .
f ( x ) = B cos ( w 0 x + ϕ ) u ( x ) = g ( x ) u ( x ) ,
F { f } ( w ) = F { u } ( y ) F { g } ( w y ) d y = B 2 { e j ϕ [ δ ( w w 0 ) + 1 j π ( w w 0 ) ] + e j ϕ [ δ ( w + w 0 ) + 1 j π ( w + w 0 ) ] } .
W ψ A { f } ( a , b ) = F { f } ( w ) e j b w ψ ^ ( a w ) d w = B | ψ ^ A ( a w 0 ) | e j [ w 0 b ϕ + ϕ ψ ( a w 0 ) ] + B 2 1 j π ( e j ϕ w w 0 + e j ϕ w + w 0 ) e j w b ψ ^ ( a w ) d w .
B 2 j π 1 w e j w b { e j ϕ + j w 0 b ψ ^ A [ a ( w + w 0 ) ] + e j ϕ j w 0 b ψ ^ A [ a ( w w 0 ) ] } d w .
B 2 π e j w b { e j ϕ + j w 0 b ψ ^ A 2 [ a ( w + w 0 ) ] + e j ϕ j w 0 b ψ ^ A 2 [ a ( w w 0 ) ] } d w ,
C ODT 2 D CWT ( N , M , L ) 4 C DWT ( N , M , L ) + N M ( 1 1 2 2 L ) C subband 4 C DWT ( N , M , L ) + N M C subband ,
C DWT ( N , M , L ) 1 4 L 1 4 1 N M ( N f C mult + ( N f 1 ) C add ) 4 3 N M ( N f C mult + ( N f 1 ) C add ) ,
C IODT 2 D CWT ( N , M , L ) 4 C IDWT ( N , M , L ) + N M C I subband ,
C d n 0 ( N , M , L ) 2 N M ( 1 1 2 2 L ) ( C shrink + C median Filter ) .
C d n 1 ( N , M , L ) 2 N M ( 1 1 2 2 L ) C shrink + 4 N M ( 1 1 2 2 L ) C add + NMC add .
C DTISE 0 = C ODT 2 D CWT ( N , M , L ) + C IODT 2 D CWT ( N , M , L ) + C d n 0 ( N , M , L ) NMC add ( 32 3 ( N f 1 ) + 11 ) + N M C mult ( 32 3 N f + 8 ) + 2 N M ( C sin + C cos + C atan 2 + C sqrt + C median Filer ) .
C DTISE 1 = C ODT 2 D CWT ( N , M , L ) + C IODT 2 D CWT ( N , M , L ) + C d n 1 ( N , M , L ) N M C add ( 32 3 ( N f 1 ) + 16 ) + NMC mult ( 32 3 N f + 8 ) + 2 N M ( C sin + C cos + C atan 2 + C sqrt ) .
C DTISE 0 NMC op ( 32 3 ( N f 1 ) + 11 + 32 3 N f + 8 + 2 ( 5 + 5 + 10 + 20 + 50 ) ) = NMC op ( 64 3 N f 32 3 + 199 ) ,
C DTISE 1 NMC op ( 32 3 ( N f 1 ) + 16 + 32 3 N f + 8 + 2 ( 5 + 5 + 10 + 20 ) ) = NMC op ( 64 3 N f 32 3 + 104 ) .
C 2 D WFT ( N , M , N w x , N w y ) C FFT + N w x N w y ( C FFT + N M C zmult + C IFFT ) ,
C 2 D - IWFT ( N , M , N w x , N w y ) N w x N w y ( C FFT + NMC zmult + C IFFT ) + ( N w x N w y 1 ) NMC add .
C WFF = C 2 D WFT ( N , M , N w x , N w y ) + C 2 D IWFT ( N , M , N w x , N w y ) + C WFF thresh = C op ( 1 + 4 N w x N w y ) ( 4 N M lg ( N M ) 12 N M + 8 ( M + N ) ) + C op N M ( 14 N w x N w y 1 ) .

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