Phase demodulation using adaptive windowed Fourier transform based on Hilbert-Huang transform |
Optics Express, Vol. 20, Issue 16, pp. 18459-18477 (2012)
http://dx.doi.org/10.1364/OE.20.018459
Acrobat PDF (2078 KB)
Abstract
The phase demodulation method of adaptive windowed Fourier transform (AWFT) is proposed based on Hilbert-Huang transform (HHT). HHT is analyzed and performed on fringe pattern to obtain instantaneous frequencies firstly. These instantaneous frequencies are further analyzed based on the condition of AWFT to locate local stationary areas where the fundamental spectrum will not be interfered by high-order spectrum. Within each local stationary area, the fundamental spectrum can be extracted accurately and adaptively by using AWFT with the background, which has been determined previously with the presented criterion during HHT, being eliminated to remove the zero-spectrum. This method is adaptive and unconstrained by any precondition for the measured phase. Experiments demonstrate its robustness and effectiveness for measuring the object with discontinuities or complex surface.
© 2012 OSA
1. Introduction
1. S. Gai and F. Da, “A novel phase-shifting method based on strip marker,” Opt. Lasers Eng. 48(2), 205–211 (2010). [CrossRef]
2. 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]
3. J. Zhong and H. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe patterns,” Appl. Opt. 46(14), 2670–2675 (2007). [CrossRef] [PubMed]
4. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43(13), 2695–2702 (2004). [CrossRef] [PubMed]
5. J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30(19), 2560–2562 (2005). [CrossRef] [PubMed]
6. S. Li, X. Su, and W. Chen, “Wavelet ridge techniques in optical fringe pattern analysis,” J. Opt. Soc. Am. A 27(6), 1245–1254 (2010). [CrossRef] [PubMed]
7. S. Özder, Ö. Kocahan, E. Coşkun, and H. Göktaş, “Optical phase distribution evaluation by using an S-transform,” Opt. Lett. 32(6), 591–593 (2007). [CrossRef] [PubMed]
8. M. Zhong, W. Chen, and M. Jiang, “Application of S-transform profilometry in eliminating nonlinearity in fringe pattern,” Appl. Opt. 51(5), 577–587 (2012). [CrossRef] [PubMed]
9. S. Zheng, W. Chen, and X. Su, “Adaptive Windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45(6), 063601 (2006). [CrossRef]
3. J. Zhong and H. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe patterns,” Appl. Opt. 46(14), 2670–2675 (2007). [CrossRef] [PubMed]
10. J. Zhong and J. Weng, “Generalized Fourier analysis for phase retrieval of fringe pattern,” Opt. Express 18(26), 26806–26820 (2010). [CrossRef] [PubMed]
11. Z. Wang, J. Ma, and M. Vo, “Recent progress in two-dimensional continuous wavelet transform technique for fringe pattern analysis,” Opt. Lasers Eng. 50(8), 1052–1058 (2012). [CrossRef]
12. S. Fernandez, M. A. Gdeisat, J. Salvi, and D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284(12), 2797–2807 (2011). [CrossRef]
13. X. Zhou, H. Zhao, and T. Jiang, “Adaptive analysis of optical fringe patterns using ensemble empirical mode decomposition algorithm,” Opt. Lett. 34(13), 2033–2035 (2009). [CrossRef] [PubMed]
14. W. Gao and Q. Kemao, “Statistical analysis for windowed Fourier ridge algorithm in fringe pattern analysis,” Appl. Opt. 51(3), 328–337 (2011). [CrossRef] [PubMed]
2. The key issue of using the AWFT method in phase demodulation of fringe pattern
15. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22(24), 3977–3982 (1983). [CrossRef] [PubMed]
9. S. Zheng, W. Chen, and X. Su, “Adaptive Windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45(6), 063601 (2006). [CrossRef]
10. J. Zhong and J. Weng, “Generalized Fourier analysis for phase retrieval of fringe pattern,” Opt. Express 18(26), 26806–26820 (2010). [CrossRef] [PubMed]
3. HHT applied in the AWFT method
3.1 HHT
16. N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998). [CrossRef]
17. S. Equis and P. Jacquot, “The empirical mode decomposition: a must-have tool in speckle interferometry?” Opt. Express 17(2), 611–623 (2009). [CrossRef] [PubMed]
19. G. Rilling and P. Flandrin, “One or two frequencies? The empirical mode decomposition answers,” IEEE Trans. Signal Process. 56(1), 85–95 (2008). [CrossRef]
16. N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998). [CrossRef]
3.2 Locating local stationary areas with HHT
3.2.1 Determining the needed instantaneous frequencies through analysis of HHT
20. Z. Wu and N. E. Huang, “Ensemble empirical mode decomposition: a noise assisted data analysis method,” Adv. Adapt. Data Anal. 1(1), 1–41 (2009). [CrossRef]
16. N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998). [CrossRef]
3.2.2 Locating local stationary signals with the selected instantaneous frequencies
1. S. Gai and F. Da, “A novel phase-shifting method based on strip marker,” Opt. Lasers Eng. 48(2), 205–211 (2010). [CrossRef]
2. 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]
3. J. Zhong and H. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe patterns,” Appl. Opt. 46(14), 2670–2675 (2007). [CrossRef] [PubMed]
3.3 Determining background through analysis of HHT
21. 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(5), 1195–1201 (2009). [CrossRef]
3.4 The flow chart of the proposed method
4. Simulation and experiments
4.1 Simulation
10. J. Zhong and J. Weng, “Generalized Fourier analysis for phase retrieval of fringe pattern,” Opt. Express 18(26), 26806–26820 (2010). [CrossRef] [PubMed]
10. J. Zhong and J. Weng, “Generalized Fourier analysis for phase retrieval of fringe pattern,” Opt. Express 18(26), 26806–26820 (2010). [CrossRef] [PubMed]
10. J. Zhong and J. Weng, “Generalized Fourier analysis for phase retrieval of fringe pattern,” Opt. Express 18(26), 26806–26820 (2010). [CrossRef] [PubMed]
22. S. Zhang, X. Li, and S. T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46(1), 50–57 (2007). [CrossRef] [PubMed]
4.2 Experiments
23. S. Gai and F. Da, “Fringe image analysis based on the amplitude modulation method,” Opt. Express 18(10), 10704–10719 (2010). [CrossRef] [PubMed]
5. Conclusion
Acknowledgments
References and links
1. | S. Gai and F. Da, “A novel phase-shifting method based on strip marker,” Opt. Lasers Eng. 48(2), 205–211 (2010). [CrossRef] |
2. | 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] |
3. | J. Zhong and H. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe patterns,” Appl. Opt. 46(14), 2670–2675 (2007). [CrossRef] [PubMed] |
4. | Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43(13), 2695–2702 (2004). [CrossRef] [PubMed] |
5. | J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30(19), 2560–2562 (2005). [CrossRef] [PubMed] |
6. | S. Li, X. Su, and W. Chen, “Wavelet ridge techniques in optical fringe pattern analysis,” J. Opt. Soc. Am. A 27(6), 1245–1254 (2010). [CrossRef] [PubMed] |
7. | S. Özder, Ö. Kocahan, E. Coşkun, and H. Göktaş, “Optical phase distribution evaluation by using an S-transform,” Opt. Lett. 32(6), 591–593 (2007). [CrossRef] [PubMed] |
8. | M. Zhong, W. Chen, and M. Jiang, “Application of S-transform profilometry in eliminating nonlinearity in fringe pattern,” Appl. Opt. 51(5), 577–587 (2012). [CrossRef] [PubMed] |
9. | S. Zheng, W. Chen, and X. Su, “Adaptive Windowed Fourier transform in 3-D shape measurement,” Opt. Eng. 45(6), 063601 (2006). [CrossRef] |
10. | J. Zhong and J. Weng, “Generalized Fourier analysis for phase retrieval of fringe pattern,” Opt. Express 18(26), 26806–26820 (2010). [CrossRef] [PubMed] |
11. | Z. Wang, J. Ma, and M. Vo, “Recent progress in two-dimensional continuous wavelet transform technique for fringe pattern analysis,” Opt. Lasers Eng. 50(8), 1052–1058 (2012). [CrossRef] |
12. | S. Fernandez, M. A. Gdeisat, J. Salvi, and D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284(12), 2797–2807 (2011). [CrossRef] |
13. | X. Zhou, H. Zhao, and T. Jiang, “Adaptive analysis of optical fringe patterns using ensemble empirical mode decomposition algorithm,” Opt. Lett. 34(13), 2033–2035 (2009). [CrossRef] [PubMed] |
14. | W. Gao and Q. Kemao, “Statistical analysis for windowed Fourier ridge algorithm in fringe pattern analysis,” Appl. Opt. 51(3), 328–337 (2011). [CrossRef] [PubMed] |
15. | M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22(24), 3977–3982 (1983). [CrossRef] [PubMed] |
16. | N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A 454(1971), 903–995 (1998). [CrossRef] |
17. | S. Equis and P. Jacquot, “The empirical mode decomposition: a must-have tool in speckle interferometry?” Opt. Express 17(2), 611–623 (2009). [CrossRef] [PubMed] |
18. | G. Rilling, P. Flandrin, and P. Goncalves, “On empirical mode decomposition and its algorithms,” in IEEE-EURASIP Workshop on Nonlinear signal and Image Processing, NSTP-03, GRADO (I) (2003). |
19. | G. Rilling and P. Flandrin, “One or two frequencies? The empirical mode decomposition answers,” IEEE Trans. Signal Process. 56(1), 85–95 (2008). [CrossRef] |
20. | Z. Wu and N. E. Huang, “Ensemble empirical mode decomposition: a noise assisted data analysis method,” Adv. Adapt. Data Anal. 1(1), 1–41 (2009). [CrossRef] |
21. | 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(5), 1195–1201 (2009). [CrossRef] |
22. | S. Zhang, X. Li, and S. T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46(1), 50–57 (2007). [CrossRef] [PubMed] |
23. | S. Gai and F. Da, “Fringe image analysis based on the amplitude modulation method,” Opt. Express 18(10), 10704–10719 (2010). [CrossRef] [PubMed] |
OCIS Codes
(100.5070) Image processing : Phase retrieval
(110.2960) Imaging systems : Image analysis
(120.2650) Instrumentation, measurement, and metrology : Fringe analysis
ToC Category:
Image Processing
History
Original Manuscript: May 30, 2012
Revised Manuscript: July 18, 2012
Manuscript Accepted: July 19, 2012
Published: July 27, 2012
Citation
Chenxing Wang and Feipeng Da, "Phase demodulation using adaptive windowed Fourier transform based on Hilbert-Huang transform," Opt. Express 20, 18459-18477 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18459
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References
- S. Gai and F. Da, “A novel phase-shifting method based on strip marker,” Opt. Lasers Eng.48(2), 205–211 (2010). [CrossRef]
- 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]
- J. Zhong and H. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe patterns,” Appl. Opt.46(14), 2670–2675 (2007). [CrossRef] [PubMed]
- Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt.43(13), 2695–2702 (2004). [CrossRef] [PubMed]
- J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett.30(19), 2560–2562 (2005). [CrossRef] [PubMed]
- S. Li, X. Su, and W. Chen, “Wavelet ridge techniques in optical fringe pattern analysis,” J. Opt. Soc. Am. A27(6), 1245–1254 (2010). [CrossRef] [PubMed]
- S. Özder, Ö. Kocahan, E. Coşkun, and H. Göktaş, “Optical phase distribution evaluation by using an S-transform,” Opt. Lett.32(6), 591–593 (2007). [CrossRef] [PubMed]
- M. Zhong, W. Chen, and M. Jiang, “Application of S-transform profilometry in eliminating nonlinearity in fringe pattern,” Appl. Opt.51(5), 577–587 (2012). [CrossRef] [PubMed]
- S. Zheng, W. Chen, and X. Su, “Adaptive Windowed Fourier transform in 3-D shape measurement,” Opt. Eng.45(6), 063601 (2006). [CrossRef]
- J. Zhong and J. Weng, “Generalized Fourier analysis for phase retrieval of fringe pattern,” Opt. Express18(26), 26806–26820 (2010). [CrossRef] [PubMed]
- Z. Wang, J. Ma, and M. Vo, “Recent progress in two-dimensional continuous wavelet transform technique for fringe pattern analysis,” Opt. Lasers Eng.50(8), 1052–1058 (2012). [CrossRef]
- S. Fernandez, M. A. Gdeisat, J. Salvi, and D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun.284(12), 2797–2807 (2011). [CrossRef]
- X. Zhou, H. Zhao, and T. Jiang, “Adaptive analysis of optical fringe patterns using ensemble empirical mode decomposition algorithm,” Opt. Lett.34(13), 2033–2035 (2009). [CrossRef] [PubMed]
- W. Gao and Q. Kemao, “Statistical analysis for windowed Fourier ridge algorithm in fringe pattern analysis,” Appl. Opt.51(3), 328–337 (2011). [CrossRef] [PubMed]
- M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt.22(24), 3977–3982 (1983). [CrossRef] [PubMed]
- N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. N. Zheng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998). [CrossRef]
- S. Equis and P. Jacquot, “The empirical mode decomposition: a must-have tool in speckle interferometry?” Opt. Express17(2), 611–623 (2009). [CrossRef] [PubMed]
- G. Rilling, P. Flandrin, and P. Goncalves, “On empirical mode decomposition and its algorithms,” in IEEE-EURASIP Workshop on Nonlinear signal and Image Processing, NSTP-03, GRADO (I) (2003).
- G. Rilling and P. Flandrin, “One or two frequencies? The empirical mode decomposition answers,” IEEE Trans. Signal Process.56(1), 85–95 (2008). [CrossRef]
- Z. Wu and N. E. Huang, “Ensemble empirical mode decomposition: a noise assisted data analysis method,” Adv. Adapt. Data Anal.1(1), 1–41 (2009). [CrossRef]
- 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. A26(5), 1195–1201 (2009). [CrossRef]
- S. Zhang, X. Li, and S. T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt.46(1), 50–57 (2007). [CrossRef] [PubMed]
- S. Gai and F. Da, “Fringe image analysis based on the amplitude modulation method,” Opt. Express18(10), 10704–10719 (2010). [CrossRef] [PubMed]
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