Abstract

The interferometry technique is commonly used to obtain the phase information of an object in optical metrology. The obtained wrapped phase is subject to a 2π ambiguity. To remove the ambiguity and obtain the correct phase, phase unwrapping is essential. Conventional phase unwrapping approaches are time-consuming and noise sensitive. To address those issues, we propose a new approach, where we transfer the task of phase unwrapping into a multi-class classification problem and introduce an efficient segmentation network to identify classes. Moreover, a noise-to-noise denoised network is integrated to preprocess noisy wrapped phase. We have demonstrated the proposed method with simulated data and in a real interferometric system.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [Crossref]
  2. T. J. Flynn, “Consistent 2-d phase unwrapping guided by a quality map,” in Geoscience and Remote Sensing Symposium, vol. 4 (IEEE, 1996), pp. 2057–2059.
  3. M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50, 6214–6224 (2011).
    [Crossref] [PubMed]
  4. H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote. Sens. Lett. 8, 364–368 (2011).
    [Crossref]
  5. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692–2701 (1997).
    [Crossref]
  6. J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote. Sens. Lett. 13, 666–670 (2016).
    [Crossref]
  7. D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
    [Crossref]
  8. R. Juarez-Salazar, C. Robledo-Sanchez, and F. Guerrero-Sanchez, “Phase-unwrapping algorithm by a rounding-least-squares approach,” Opt. Eng. 53, 024102 (2014).
    [Crossref]
  9. C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Direct continuous phase demodulation in digital holography with use of the transport-of-intensity equation,” Opt. Commun. 309, 221–226 (2013).
    [Crossref]
  10. N. Pandey, A. Ghosh, and K. Khare, “Two-dimensional phase unwrapping using the transport of intensity equation,” Appl. Opt. 55, 2418–2425 (2016).
    [Crossref] [PubMed]
  11. J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Fast and accurate phase-unwrapping algorithm based on the transport of intensity equation,” Appl. Opt. 56, 7079–7088 (2017).
    [Crossref] [PubMed]
  12. W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, “Two-dimensional phase unwrapping using neural networks,” in Proceedings of IEEE Conference on Image Analysis and Interpretation, (IEEE, 2000), pp. 274–277.
  13. G. Dardikman and N. T. Shaked, “Phase unwrapping using residual neural networks,” in Imaging and Applied Optics 2018, (Optical Society of America, 2018), pp. CW3B–5.
  14. R. Liang, J. Zhang, X. Tian, and J. Shao, “Phase unwrapping using segmentation,” (2018). U.S. Provisional Patent Application No. 62/768,624.
  15. G. Spoorthi, S. Gorthi, and R. K. S. S. Gorthi, “Phasenet: A deep convolutional neural network for two-dimensional phase unwrapping,” IEEE Signal Process. Lett. 26, 54–58 (2019).
    [Crossref]
  16. V. Badrinarayanan, A. Kendall, and R. Cipolla, “Segnet: A deep convolutional encoder-decoder architecture for image segmentation,” IEEE Transactions on Pattern Analysis Mach. Intell. 39, 2481–2495 (2017).
    [Crossref]
  17. S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” in Proceedings of International Conference on Machine Learning, (2015), pp. 448–456.
  18. V. Nair and G. E. Hinton, “Rectified linear units improve restricted boltzmann machines,” in Proceedings of the 27th international conference on machine learning, (2010), pp. 807–814.
  19. K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” arXiv preprint arXiv:1409.1556 (2014).
  20. K. Janocha and W. M. Czarnecki, “On loss functions for deep neural networks in classification,” arXiv preprint arXiv:1702.05659 (2017).
  21. K. He, X. Zhang, S. Ren, and J. Sun, “Delving deep into rectifiers: Surpassing human-level performance on imagenet classification,” in Proceedings of IEEE International Conference on Computer Vision, (2015), pp. 1026–1034.
  22. D. P. Kingma and J. L. Ba, “Adam: A method for stochastic optimization,” in Proceedings of International Conference for Learning Representations, (2015).
  23. F. Sawaf and R. M. Groves, “Phase discontinuity predictions using a machine-learning trained kernel,” Appl. Opt. 53, 5439–5447 (2014).
    [Crossref] [PubMed]
  24. J. Lehtinen, J. Munkberg, J. Hasselgren, S. Laine, T. Karras, M. Aittala, and T. Aila, “Noise2noise: Learning image restoration without clean data,” arXiv preprint arXiv:1803.04189 (2018).
  25. R. Olaf, F. Philipp, and B. Thomas, “U-net: Convolutional networks for biomedical image segmentation,” in Proceedings of International Conference on Medical Image Computing and Computer-assisted Intervention, (Springer, 2015), pp. 234–241.
  26. D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping: theory, algorithms, and software (Wiley-Interscience, 1998).
  27. X. Tian, X. Tu, J. Zhang, O. Spires, N. Brock, S. Pau, and R. Liang, “Snapshot multi-wavelength interference microscope,” Opt. Express 26, 18279–18291 (2018).
    [Crossref] [PubMed]

2019 (1)

G. Spoorthi, S. Gorthi, and R. K. S. S. Gorthi, “Phasenet: A deep convolutional neural network for two-dimensional phase unwrapping,” IEEE Signal Process. Lett. 26, 54–58 (2019).
[Crossref]

2018 (1)

2017 (2)

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Fast and accurate phase-unwrapping algorithm based on the transport of intensity equation,” Appl. Opt. 56, 7079–7088 (2017).
[Crossref] [PubMed]

V. Badrinarayanan, A. Kendall, and R. Cipolla, “Segnet: A deep convolutional encoder-decoder architecture for image segmentation,” IEEE Transactions on Pattern Analysis Mach. Intell. 39, 2481–2495 (2017).
[Crossref]

2016 (2)

J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote. Sens. Lett. 13, 666–670 (2016).
[Crossref]

N. Pandey, A. Ghosh, and K. Khare, “Two-dimensional phase unwrapping using the transport of intensity equation,” Appl. Opt. 55, 2418–2425 (2016).
[Crossref] [PubMed]

2014 (2)

R. Juarez-Salazar, C. Robledo-Sanchez, and F. Guerrero-Sanchez, “Phase-unwrapping algorithm by a rounding-least-squares approach,” Opt. Eng. 53, 024102 (2014).
[Crossref]

F. Sawaf and R. M. Groves, “Phase discontinuity predictions using a machine-learning trained kernel,” Appl. Opt. 53, 5439–5447 (2014).
[Crossref] [PubMed]

2013 (1)

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Direct continuous phase demodulation in digital holography with use of the transport-of-intensity equation,” Opt. Commun. 309, 221–226 (2013).
[Crossref]

2011 (2)

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50, 6214–6224 (2011).
[Crossref] [PubMed]

H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote. Sens. Lett. 8, 364–368 (2011).
[Crossref]

1997 (1)

1994 (1)

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Aila, T.

J. Lehtinen, J. Munkberg, J. Hasselgren, S. Laine, T. Karras, M. Aittala, and T. Aila, “Noise2noise: Learning image restoration without clean data,” arXiv preprint arXiv:1803.04189 (2018).

Aittala, M.

J. Lehtinen, J. Munkberg, J. Hasselgren, S. Laine, T. Karras, M. Aittala, and T. Aila, “Noise2noise: Learning image restoration without clean data,” arXiv preprint arXiv:1803.04189 (2018).

An, D.

J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote. Sens. Lett. 13, 666–670 (2016).
[Crossref]

Asundi, A.

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Direct continuous phase demodulation in digital holography with use of the transport-of-intensity equation,” Opt. Commun. 309, 221–226 (2013).
[Crossref]

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50, 6214–6224 (2011).
[Crossref] [PubMed]

Ba, J. L.

D. P. Kingma and J. L. Ba, “Adam: A method for stochastic optimization,” in Proceedings of International Conference for Learning Representations, (2015).

Badrinarayanan, V.

V. Badrinarayanan, A. Kendall, and R. Cipolla, “Segnet: A deep convolutional encoder-decoder architecture for image segmentation,” IEEE Transactions on Pattern Analysis Mach. Intell. 39, 2481–2495 (2017).
[Crossref]

Bovik, A. C.

W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, “Two-dimensional phase unwrapping using neural networks,” in Proceedings of IEEE Conference on Image Analysis and Interpretation, (IEEE, 2000), pp. 274–277.

Brock, N.

Chen, M.

H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote. Sens. Lett. 8, 364–368 (2011).
[Crossref]

Chen, Q.

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Direct continuous phase demodulation in digital holography with use of the transport-of-intensity equation,” Opt. Commun. 309, 221–226 (2013).
[Crossref]

Cipolla, R.

V. Badrinarayanan, A. Kendall, and R. Cipolla, “Segnet: A deep convolutional encoder-decoder architecture for image segmentation,” IEEE Transactions on Pattern Analysis Mach. Intell. 39, 2481–2495 (2017).
[Crossref]

Czarnecki, W. M.

K. Janocha and W. M. Czarnecki, “On loss functions for deep neural networks in classification,” arXiv preprint arXiv:1702.05659 (2017).

Dardikman, G.

G. Dardikman and N. T. Shaked, “Phase unwrapping using residual neural networks,” in Imaging and Applied Optics 2018, (Optical Society of America, 2018), pp. CW3B–5.

Evans, B. L.

W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, “Two-dimensional phase unwrapping using neural networks,” in Proceedings of IEEE Conference on Image Analysis and Interpretation, (IEEE, 2000), pp. 274–277.

Falaggis, K.

Flynn, T. J.

T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692–2701 (1997).
[Crossref]

T. J. Flynn, “Consistent 2-d phase unwrapping guided by a quality map,” in Geoscience and Remote Sensing Symposium, vol. 4 (IEEE, 1996), pp. 2057–2059.

Ghiglia, D. C.

Ghosh, A.

Ghosh, J.

W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, “Two-dimensional phase unwrapping using neural networks,” in Proceedings of IEEE Conference on Image Analysis and Interpretation, (IEEE, 2000), pp. 274–277.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Gorthi, R. K. S. S.

G. Spoorthi, S. Gorthi, and R. K. S. S. Gorthi, “Phasenet: A deep convolutional neural network for two-dimensional phase unwrapping,” IEEE Signal Process. Lett. 26, 54–58 (2019).
[Crossref]

Gorthi, S.

G. Spoorthi, S. Gorthi, and R. K. S. S. Gorthi, “Phasenet: A deep convolutional neural network for two-dimensional phase unwrapping,” IEEE Signal Process. Lett. 26, 54–58 (2019).
[Crossref]

Groves, R. M.

Guerrero-Sanchez, F.

R. Juarez-Salazar, C. Robledo-Sanchez, and F. Guerrero-Sanchez, “Phase-unwrapping algorithm by a rounding-least-squares approach,” Opt. Eng. 53, 024102 (2014).
[Crossref]

Hasselgren, J.

J. Lehtinen, J. Munkberg, J. Hasselgren, S. Laine, T. Karras, M. Aittala, and T. Aila, “Noise2noise: Learning image restoration without clean data,” arXiv preprint arXiv:1803.04189 (2018).

He, K.

K. He, X. Zhang, S. Ren, and J. Sun, “Delving deep into rectifiers: Surpassing human-level performance on imagenet classification,” in Proceedings of IEEE International Conference on Computer Vision, (2015), pp. 1026–1034.

Hinton, G. E.

V. Nair and G. E. Hinton, “Rectified linear units improve restricted boltzmann machines,” in Proceedings of the 27th international conference on machine learning, (2010), pp. 807–814.

Huang, L.

Huang, X.

J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote. Sens. Lett. 13, 666–670 (2016).
[Crossref]

Ioffe, S.

S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” in Proceedings of International Conference on Machine Learning, (2015), pp. 448–456.

Janocha, K.

K. Janocha and W. M. Czarnecki, “On loss functions for deep neural networks in classification,” arXiv preprint arXiv:1702.05659 (2017).

Juarez-Salazar, R.

R. Juarez-Salazar, C. Robledo-Sanchez, and F. Guerrero-Sanchez, “Phase-unwrapping algorithm by a rounding-least-squares approach,” Opt. Eng. 53, 024102 (2014).
[Crossref]

Karras, T.

J. Lehtinen, J. Munkberg, J. Hasselgren, S. Laine, T. Karras, M. Aittala, and T. Aila, “Noise2noise: Learning image restoration without clean data,” arXiv preprint arXiv:1803.04189 (2018).

Kemao, Q.

Kendall, A.

V. Badrinarayanan, A. Kendall, and R. Cipolla, “Segnet: A deep convolutional encoder-decoder architecture for image segmentation,” IEEE Transactions on Pattern Analysis Mach. Intell. 39, 2481–2495 (2017).
[Crossref]

Khare, K.

Kingma, D. P.

D. P. Kingma and J. L. Ba, “Adam: A method for stochastic optimization,” in Proceedings of International Conference for Learning Representations, (2015).

Kozacki, T.

Laine, S.

J. Lehtinen, J. Munkberg, J. Hasselgren, S. Laine, T. Karras, M. Aittala, and T. Aila, “Noise2noise: Learning image restoration without clean data,” arXiv preprint arXiv:1803.04189 (2018).

Lehtinen, J.

J. Lehtinen, J. Munkberg, J. Hasselgren, S. Laine, T. Karras, M. Aittala, and T. Aila, “Noise2noise: Learning image restoration without clean data,” arXiv preprint arXiv:1803.04189 (2018).

Liang, R.

X. Tian, X. Tu, J. Zhang, O. Spires, N. Brock, S. Pau, and R. Liang, “Snapshot multi-wavelength interference microscope,” Opt. Express 26, 18279–18291 (2018).
[Crossref] [PubMed]

R. Liang, J. Zhang, X. Tian, and J. Shao, “Phase unwrapping using segmentation,” (2018). U.S. Provisional Patent Application No. 62/768,624.

Martinez-Carranza, J.

Milner, T. E.

W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, “Two-dimensional phase unwrapping using neural networks,” in Proceedings of IEEE Conference on Image Analysis and Interpretation, (IEEE, 2000), pp. 274–277.

Munkberg, J.

J. Lehtinen, J. Munkberg, J. Hasselgren, S. Laine, T. Karras, M. Aittala, and T. Aila, “Noise2noise: Learning image restoration without clean data,” arXiv preprint arXiv:1803.04189 (2018).

Nair, V.

V. Nair and G. E. Hinton, “Rectified linear units improve restricted boltzmann machines,” in Proceedings of the 27th international conference on machine learning, (2010), pp. 807–814.

Olaf, R.

R. Olaf, F. Philipp, and B. Thomas, “U-net: Convolutional networks for biomedical image segmentation,” in Proceedings of International Conference on Medical Image Computing and Computer-assisted Intervention, (Springer, 2015), pp. 234–241.

Pandey, N.

Pau, S.

Philipp, F.

R. Olaf, F. Philipp, and B. Thomas, “U-net: Convolutional networks for biomedical image segmentation,” in Proceedings of International Conference on Medical Image Computing and Computer-assisted Intervention, (Springer, 2015), pp. 234–241.

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping: theory, algorithms, and software (Wiley-Interscience, 1998).

Qu, W.

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Direct continuous phase demodulation in digital holography with use of the transport-of-intensity equation,” Opt. Commun. 309, 221–226 (2013).
[Crossref]

Ren, S.

K. He, X. Zhang, S. Ren, and J. Sun, “Delving deep into rectifiers: Surpassing human-level performance on imagenet classification,” in Proceedings of IEEE International Conference on Computer Vision, (2015), pp. 1026–1034.

Robledo-Sanchez, C.

R. Juarez-Salazar, C. Robledo-Sanchez, and F. Guerrero-Sanchez, “Phase-unwrapping algorithm by a rounding-least-squares approach,” Opt. Eng. 53, 024102 (2014).
[Crossref]

Romero, L. A.

Sawaf, F.

Schwartzkopf, W.

W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, “Two-dimensional phase unwrapping using neural networks,” in Proceedings of IEEE Conference on Image Analysis and Interpretation, (IEEE, 2000), pp. 274–277.

Shaked, N. T.

G. Dardikman and N. T. Shaked, “Phase unwrapping using residual neural networks,” in Imaging and Applied Optics 2018, (Optical Society of America, 2018), pp. CW3B–5.

Shao, J.

R. Liang, J. Zhang, X. Tian, and J. Shao, “Phase unwrapping using segmentation,” (2018). U.S. Provisional Patent Application No. 62/768,624.

Simonyan, K.

K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” arXiv preprint arXiv:1409.1556 (2014).

Spires, O.

Spoorthi, G.

G. Spoorthi, S. Gorthi, and R. K. S. S. Gorthi, “Phasenet: A deep convolutional neural network for two-dimensional phase unwrapping,” IEEE Signal Process. Lett. 26, 54–58 (2019).
[Crossref]

Su, X.

Sun, J.

K. He, X. Zhang, S. Ren, and J. Sun, “Delving deep into rectifiers: Surpassing human-level performance on imagenet classification,” in Proceedings of IEEE International Conference on Computer Vision, (2015), pp. 1026–1034.

Szegedy, C.

S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” in Proceedings of International Conference on Machine Learning, (2015), pp. 448–456.

Tang, J.

H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote. Sens. Lett. 8, 364–368 (2011).
[Crossref]

Thomas, B.

R. Olaf, F. Philipp, and B. Thomas, “U-net: Convolutional networks for biomedical image segmentation,” in Proceedings of International Conference on Medical Image Computing and Computer-assisted Intervention, (Springer, 2015), pp. 234–241.

Tian, X.

X. Tian, X. Tu, J. Zhang, O. Spires, N. Brock, S. Pau, and R. Liang, “Snapshot multi-wavelength interference microscope,” Opt. Express 26, 18279–18291 (2018).
[Crossref] [PubMed]

R. Liang, J. Zhang, X. Tian, and J. Shao, “Phase unwrapping using segmentation,” (2018). U.S. Provisional Patent Application No. 62/768,624.

Tu, X.

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Xu, J.

J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote. Sens. Lett. 13, 666–670 (2016).
[Crossref]

Yi, P.

J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote. Sens. Lett. 13, 666–670 (2016).
[Crossref]

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Zhang, J.

X. Tian, X. Tu, J. Zhang, O. Spires, N. Brock, S. Pau, and R. Liang, “Snapshot multi-wavelength interference microscope,” Opt. Express 26, 18279–18291 (2018).
[Crossref] [PubMed]

R. Liang, J. Zhang, X. Tian, and J. Shao, “Phase unwrapping using segmentation,” (2018). U.S. Provisional Patent Application No. 62/768,624.

Zhang, Q.

Zhang, S.

H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote. Sens. Lett. 8, 364–368 (2011).
[Crossref]

Zhang, X.

K. He, X. Zhang, S. Ren, and J. Sun, “Delving deep into rectifiers: Surpassing human-level performance on imagenet classification,” in Proceedings of IEEE International Conference on Computer Vision, (2015), pp. 1026–1034.

Zhao, M.

Zhong, H.

H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote. Sens. Lett. 8, 364–368 (2011).
[Crossref]

Zisserman, A.

K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” arXiv preprint arXiv:1409.1556 (2014).

Zuo, C.

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Direct continuous phase demodulation in digital holography with use of the transport-of-intensity equation,” Opt. Commun. 309, 221–226 (2013).
[Crossref]

Appl. Opt. (4)

IEEE Geosci. Remote. Sens. Lett. (2)

J. Xu, D. An, X. Huang, and P. Yi, “An efficient minimum-discontinuity phase-unwrapping method,” IEEE Geosci. Remote. Sens. Lett. 13, 666–670 (2016).
[Crossref]

H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote. Sens. Lett. 8, 364–368 (2011).
[Crossref]

IEEE Signal Process. Lett. (1)

G. Spoorthi, S. Gorthi, and R. K. S. S. Gorthi, “Phasenet: A deep convolutional neural network for two-dimensional phase unwrapping,” IEEE Signal Process. Lett. 26, 54–58 (2019).
[Crossref]

IEEE Transactions on Pattern Analysis Mach. Intell. (1)

V. Badrinarayanan, A. Kendall, and R. Cipolla, “Segnet: A deep convolutional encoder-decoder architecture for image segmentation,” IEEE Transactions on Pattern Analysis Mach. Intell. 39, 2481–2495 (2017).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Direct continuous phase demodulation in digital holography with use of the transport-of-intensity equation,” Opt. Commun. 309, 221–226 (2013).
[Crossref]

Opt. Eng. (1)

R. Juarez-Salazar, C. Robledo-Sanchez, and F. Guerrero-Sanchez, “Phase-unwrapping algorithm by a rounding-least-squares approach,” Opt. Eng. 53, 024102 (2014).
[Crossref]

Opt. Express (1)

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Other (13)

T. J. Flynn, “Consistent 2-d phase unwrapping guided by a quality map,” in Geoscience and Remote Sensing Symposium, vol. 4 (IEEE, 1996), pp. 2057–2059.

W. Schwartzkopf, T. E. Milner, J. Ghosh, B. L. Evans, and A. C. Bovik, “Two-dimensional phase unwrapping using neural networks,” in Proceedings of IEEE Conference on Image Analysis and Interpretation, (IEEE, 2000), pp. 274–277.

G. Dardikman and N. T. Shaked, “Phase unwrapping using residual neural networks,” in Imaging and Applied Optics 2018, (Optical Society of America, 2018), pp. CW3B–5.

R. Liang, J. Zhang, X. Tian, and J. Shao, “Phase unwrapping using segmentation,” (2018). U.S. Provisional Patent Application No. 62/768,624.

S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” in Proceedings of International Conference on Machine Learning, (2015), pp. 448–456.

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Figures (10)

Fig. 1
Fig. 1 The network architecture for phase unwrapping.
Fig. 2
Fig. 2 Phase unwrapping results on simulated clean data. (a) Wrapped phase (input), (b) output (integral multiple ), (c) ground-truth (integral multiple n), (d) reconstructed unwrapped phase, (e) ground-truth (unwrapped phase), (f) difference.
Fig. 3
Fig. 3 Phase unwrapping results based on post-processing. From top to bottom are: wrapped phases ((a1), (a2)), ground-truth (unwrapped phase, (b1), (b2)), reconstructed unwrapped phases ((c1), (c2)), post-processed unwrapped phases ((d1), (d2)) and differences ((e)=(c)-(b), (f)=(d)-(b)).
Fig. 4
Fig. 4 Phase discontinuity extraction ((a), (c)) and connected region labeling ((b), (d)).
Fig. 5
Fig. 5 The network architecture of denoising noisy wrapped phase.
Fig. 6
Fig. 6 Unwrapping result on simulated noisy data (SNR = 4.0 dB). (a) noisy wrapped phase, (b) denoised wrapped phase, (c) ground-truth (wrapped phase), (d) unwrapped phase, (e) ground-truth (unwrapped phase), (f) difference.
Fig. 7
Fig. 7 Unwrapping result on more badly corrupted data (SNR = 0.6 dB). (a) noisy wrapped phase, (b) denoised wrapped phase, (c) ground-truth (wrapped phase), (d) unwrapped phase, (e) ground-truth (unwrapped phase), (f) difference.
Fig. 8
Fig. 8 Unwrapping results of other methods. Unwrapped phases (a) and (e) are produced by Goldstein’s branch cut algorithm, (c) and (g) are obtained by Quality-guided path-following method, (b), (d), (f), and (h) are differences.
Fig. 9
Fig. 9 Experimental setup to demonstrate the phase unwrapping method with denoised and convolutional segmentation networks. L1: collimating lens; P: polarizer; PBS: polarized beam splitter; QWP1, QWP2, QWP3: quarter waveplate; DM: deformable mirror; L2: imaging lens.
Fig. 10
Fig. 10 Unwrapping results on real data. From left to right are: wrapped phases (input, (a), (e)), reconstructed unwrapped phases by our network ((b), (f)) and MG ((c), (g)), and differences ((d), (h)).

Tables (1)

Tables Icon

Table 1 RMSE results and running time of the different methods.

Equations (5)

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φ unw ( x , y ) = φ w ( x , y ) + 2 π * n ( x , y ) ,
φ ^ unw ( x , y ) = φ w ( x , y ) + 2 π * ( n ^ ( x , y ) C ) .
loss = 1 M k = 1 M x , y log ( p k , t ( x , y ) ) ,
I a = A + B ( φ unw + a * π / 2 ) + noise ,
φ w = arctan ( I 3 I 1 I 0 I 2 ) .

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