Abstract

Traditional digital holographic imaging algorithms need multiple iterations to obtain focused reconstructed image, which is time-consuming. In terms of phase retrieval, there is also the problem of phase compensation in addition to focusing task. Here, a new method is proposed for fast digital focus, where we use U-type convolutional neural network (U-net) to recover the original phase of microscopic samples. Generated data sets are used to simulate different degrees of defocused image, and verify that the U-net can restore the original phase to a great extent and realize phase compensation at the same time. We apply this method in the construction of real-time off-axis digital holographic microscope and obtain great breakthroughs in imaging speed.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (4)

2017 (8)

T. Nguyen, V. Bui, V. Lam, C. B. Raub, L. C. Chang, and G. Nehmetallah, “Automatic phase aberration compensation for digital holographic microscopy based on deep learning background detection,” Opt. Express 25(13), 15043–15057 (2017).
[Crossref] [PubMed]

A. Sinha, G. Barbastathis, J. Lee, and S. Li, “Lensless computational imaging through deep learning,” Optica 4(9), 1117–1125 (2017).
[Crossref]

A. Ozcan, H. Günaydin, H. Wang, Y. Rivenson, Y. Zhang, and Z. Göröcs, “Deep learning microscopy,” Optica 4(11), 1437–1443 (2017).
[Crossref]

A. Sevastopolsky, “Optic disc and cup segmentation methods for glaucoma detection with modification of U-Net convolutional neural network,” Pattern Recognit. Image Anal. 27(3), 618–624 (2017).
[Crossref]

E. Shelhamer, J. Long, and T. Darrell, “Fully convolutional networks for semantic segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 39(4), 640–651 (2017).
[Crossref] [PubMed]

Y. Yoon, H. G. Jeon, D. Yoo, J. Y. Lee, and I. S. Kweon, “Light-Field Image Super-Resolution Using Convolutional Neural Network,” IEEE Signal Process. Lett. 24(6), 848–852 (2017).
[Crossref]

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” Light Sci. Appl. 7(2), 17141 (2017).
[Crossref]

J. Zheng, P. Gao, and X. Shao, “Opposite-view digital holographic microscopy with autofocusing capability,” Sci. Rep. 7(1), 4255 (2017).
[Crossref] [PubMed]

2016 (2)

2015 (1)

N. Pavillon and P. Marquet, “Cell volume regulation monitored with combined epifluorescence and digital holographic microscopy,” Methods Mol. Biol. 1254(1254), 21–32 (2015).
[Crossref] [PubMed]

2014 (2)

I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative Adversarial Networks,” Adv. Neural Inf. Process. Syst. 3, 2672–2680 (2014).

L. Williams, P. P. Banerjee, G. Nehmetallah, and S. Praharaj, “Holographic volume displacement calculations via multiwavelength digital holography,” Appl. Opt. 53(8), 1597–1603 (2014).
[Crossref] [PubMed]

2013 (2)

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Phase aberration compensation in digital holographic microscopy based on principal component analysis,” Opt. Lett. 38(10), 1724–1726 (2013).
[Crossref] [PubMed]

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

2012 (2)

N. Pavillon, J. Kühn, C. Moratal, P. Jourdain, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Early Cell Death Detection with Digital Holographic Microscopy,” PLoS One 7(1), e30912 (2012).
[Crossref] [PubMed]

G. Nehmetallah and P. P. Banerjee, “Applications of digital and analog holography in three-dimensional imaging,” Adv. Opt. Photonics 4(4), 472 (2012).
[Crossref]

2011 (1)

2009 (2)

J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009).
[Crossref] [PubMed]

B. Rappaz, E. Cano, T. Colomb, J. Kühn, C. Depeursinge, V. Simanis, P. J. Magistretti, and P. Marquet, “Noninvasive characterization of the fission yeast cell cycle by monitoring dry mass with digital holographic microscopy,” J. Biomed. Opt. 14(3), 034049 (2009).
[Crossref] [PubMed]

2008 (1)

2006 (3)

2001 (1)

C. E. A. Shannon, “A mathematical theory of communication,” Mob. Comput. Commun. Rev. 5(1), 3–55 (2001).
[Crossref]

1999 (1)

1993 (1)

P. B. Gibbons and R. Mathon, “The use of hill‐climbing to construct orthogonal steiner triple systems,” J. Comb. Des. 1(1), 27–50 (1993).
[Crossref]

Abdulkadir, A.

Ö. Çiçek, A. Abdulkadir, S. S. Lienkamp, T. Brox, and O. Ronneberger, “3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation,” in International Conference on Medical Image Computing and Computer-Assisted Intervention (2016), pp. 424–432.

Abidi, B.

Y. Yao, B. Abidi, N. Doggaz, and M. Abidi, “Evaluation of sharpness measures and search algorithms for the auto focusing of high-magnification images,” Proc. SPIE 6246(6424), 62460G (2006).
[Crossref]

Abidi, M.

Y. Yao, B. Abidi, N. Doggaz, and M. Abidi, “Evaluation of sharpness measures and search algorithms for the auto focusing of high-magnification images,” Proc. SPIE 6246(6424), 62460G (2006).
[Crossref]

Aspert, N.

Asundi, A.

Aylo, R.

Banerjee, P. P.

L. Williams, P. P. Banerjee, G. Nehmetallah, and S. Praharaj, “Holographic volume displacement calculations via multiwavelength digital holography,” Appl. Opt. 53(8), 1597–1603 (2014).
[Crossref] [PubMed]

G. Nehmetallah and P. P. Banerjee, “Applications of digital and analog holography in three-dimensional imaging,” Adv. Opt. Photonics 4(4), 472 (2012).
[Crossref]

Barbastathis, G.

Bengio, Y.

I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative Adversarial Networks,” Adv. Neural Inf. Process. Syst. 3, 2672–2680 (2014).

Bourquin, S.

Breton, B.

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

Brox, T.

O. Ronneberger, P. Fischer, and T. Brox, “U-Net: Convolutional Networks for Biomedical Image Segmentation,” in International Conference on Medical Image Computing and Computer-Assisted Intervention (2015), pp. 234–241.
[Crossref]

Ö. Çiçek, A. Abdulkadir, S. S. Lienkamp, T. Brox, and O. Ronneberger, “3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation,” in International Conference on Medical Image Computing and Computer-Assisted Intervention (2016), pp. 424–432.

Bui, V.

Cano, E.

B. Rappaz, E. Cano, T. Colomb, J. Kühn, C. Depeursinge, V. Simanis, P. J. Magistretti, and P. Marquet, “Noninvasive characterization of the fission yeast cell cycle by monitoring dry mass with digital holographic microscopy,” J. Biomed. Opt. 14(3), 034049 (2009).
[Crossref] [PubMed]

Chambon, M.

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

Chang, L. C.

Charrie, F.

Charrière, F.

Chen, Q.

Çiçek, Ö.

Ö. Çiçek, A. Abdulkadir, S. S. Lienkamp, T. Brox, and O. Ronneberger, “3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation,” in International Conference on Medical Image Computing and Computer-Assisted Intervention (2016), pp. 424–432.

Colomb, T.

Corwin, A. D.

Courville, A.

I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative Adversarial Networks,” Adv. Neural Inf. Process. Syst. 3, 2672–2680 (2014).

Cuche, E.

Darrell, T.

E. Shelhamer, J. Long, and T. Darrell, “Fully convolutional networks for semantic segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 39(4), 640–651 (2017).
[Crossref] [PubMed]

Depeursinge, C.

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

N. Pavillon, J. Kühn, C. Moratal, P. Jourdain, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Early Cell Death Detection with Digital Holographic Microscopy,” PLoS One 7(1), e30912 (2012).
[Crossref] [PubMed]

J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009).
[Crossref] [PubMed]

B. Rappaz, E. Cano, T. Colomb, J. Kühn, C. Depeursinge, V. Simanis, P. J. Magistretti, and P. Marquet, “Noninvasive characterization of the fission yeast cell cycle by monitoring dry mass with digital holographic microscopy,” J. Biomed. Opt. 14(3), 034049 (2009).
[Crossref] [PubMed]

T. Colomb, F. Montfort, J. Kãn, N. Aspert, E. Cuche, A. Marian, F. Charrie, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23(12), 3177–3190 (2006).
[Crossref]

T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, P. Marquet, and C. Depeursinge, “Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation,” Appl. Opt. 45(5), 851–863 (2006).
[Crossref] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38(34), 6994–7001 (1999).
[Crossref] [PubMed]

Distante, C.

Dixon, E. L.

Doggaz, N.

Y. Yao, B. Abidi, N. Doggaz, and M. Abidi, “Evaluation of sharpness measures and search algorithms for the auto focusing of high-magnification images,” Proc. SPIE 6246(6424), 62460G (2006).
[Crossref]

Dong, H.

H. Dong, G. Yang, F. Liu, Y. Mo, and Y. Guo, “Automatic Brain Tumor Detection and Segmentation Using U-Net Based Fully Convolutional Networks,” in Conference on Medical Image Understanding and Analysis (2017), pp. 506–517.
[Crossref]

Emery, Y.

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

Fauser, S.

Ferraro, P.

Filkins, R. J.

Finizio, A.

Fischer, P.

O. Ronneberger, P. Fischer, and T. Brox, “U-Net: Convolutional Networks for Biomedical Image Segmentation,” in International Conference on Medical Image Computing and Computer-Assisted Intervention (2015), pp. 234–241.
[Crossref]

Gao, P.

J. Zheng, P. Gao, and X. Shao, “Opposite-view digital holographic microscopy with autofocusing capability,” Sci. Rep. 7(1), 4255 (2017).
[Crossref] [PubMed]

Gibbons, P. B.

P. B. Gibbons and R. Mathon, “The use of hill‐climbing to construct orthogonal steiner triple systems,” J. Comb. Des. 1(1), 27–50 (1993).
[Crossref]

Goodfellow, I. J.

I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative Adversarial Networks,” Adv. Neural Inf. Process. Syst. 3, 2672–2680 (2014).

Göröcs, Z.

Günaydin, H.

A. Ozcan, H. Günaydin, H. Wang, Y. Rivenson, Y. Zhang, and Z. Göröcs, “Deep learning microscopy,” Optica 4(11), 1437–1443 (2017).
[Crossref]

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” Light Sci. Appl. 7(2), 17141 (2017).
[Crossref]

Guo, Y.

H. Dong, G. Yang, F. Liu, Y. Mo, and Y. Guo, “Automatic Brain Tumor Detection and Segmentation Using U-Net Based Fully Convolutional Networks,” in Conference on Medical Image Understanding and Analysis (2017), pp. 506–517.
[Crossref]

Han, B.

H. Noh, S. Hong, and B. Han, “Learning Deconvolution Network for Semantic Segmentation,” in 2015 IEEE International Conference on Computer Vision (2015),pp. 1520–1528.
[Crossref]

He, K.

K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” in IEEE Conference on Computer Vision and Pattern Recognition (2016), pp. 770–778.

He, Q.

He, Y.

Hinton, G. E.

A. Krizhevsky, I. Sutskever, and G. E. Hinton, “ImageNet classification with deep convolutional neural networks,” in International Conference on Neural Information Processing Systems (2012), pp. 1097–1105.

Hong, S.

H. Noh, S. Hong, and B. Han, “Learning Deconvolution Network for Semantic Segmentation,” in 2015 IEEE International Conference on Computer Vision (2015),pp. 1520–1528.
[Crossref]

Hoyng, C.

Huang, G.

G. Huang, Z. Liu, L. d. Maaten, and K. Q. Weinberger, “Densely Connected Convolutional Networks,” in IEEE Conference on Computer Vision and Pattern Recognition (2017), pp. 2261–2269.

Javidi, B.

Jeon, H. G.

Y. Yoon, H. G. Jeon, D. Yoo, J. Y. Lee, and I. S. Kweon, “Light-Field Image Super-Resolution Using Convolutional Neural Network,” IEEE Signal Process. Lett. 24(6), 848–852 (2017).
[Crossref]

Ji, Y.

Jia, J.

H. Zhao, J. Shi, X. Qi, X. Wang, and J. Jia, “Pyramid Scene Parsing Network,” in IEEE Conference on Computer Vision and Pattern Recognition (2017), pp. 6230–6239.

John, R.

Jourdain, P.

N. Pavillon, J. Kühn, C. Moratal, P. Jourdain, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Early Cell Death Detection with Digital Holographic Microscopy,” PLoS One 7(1), e30912 (2012).
[Crossref] [PubMed]

Kãn, J.

Kenny, K. B.

Kim, D. H.

H. N. D. Le, M. S. Kim, and D. H. Kim, “Comparison of Singular Value Decomposition and Principal Component Analysis applied to Hyperspectral Imaging of biofilm,” in Photonics Conference (2012), pp. 6–7.
[Crossref]

Kim, M. S.

H. N. D. Le, M. S. Kim, and D. H. Kim, “Comparison of Singular Value Decomposition and Principal Component Analysis applied to Hyperspectral Imaging of biofilm,” in Photonics Conference (2012), pp. 6–7.
[Crossref]

Krizhevsky, A.

A. Krizhevsky, I. Sutskever, and G. E. Hinton, “ImageNet classification with deep convolutional neural networks,” in International Conference on Neural Information Processing Systems (2012), pp. 1097–1105.

Kühn, J.

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

N. Pavillon, J. Kühn, C. Moratal, P. Jourdain, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Early Cell Death Detection with Digital Holographic Microscopy,” PLoS One 7(1), e30912 (2012).
[Crossref] [PubMed]

J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009).
[Crossref] [PubMed]

B. Rappaz, E. Cano, T. Colomb, J. Kühn, C. Depeursinge, V. Simanis, P. J. Magistretti, and P. Marquet, “Noninvasive characterization of the fission yeast cell cycle by monitoring dry mass with digital holographic microscopy,” J. Biomed. Opt. 14(3), 034049 (2009).
[Crossref] [PubMed]

T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, P. Marquet, and C. Depeursinge, “Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation,” Appl. Opt. 45(5), 851–863 (2006).
[Crossref] [PubMed]

Kweon, I. S.

Y. Yoon, H. G. Jeon, D. Yoo, J. Y. Lee, and I. S. Kweon, “Light-Field Image Super-Resolution Using Convolutional Neural Network,” IEEE Signal Process. Lett. 24(6), 848–852 (2017).
[Crossref]

Lam, E. Y.

Lam, V.

Le, H. N. D.

H. N. D. Le, M. S. Kim, and D. H. Kim, “Comparison of Singular Value Decomposition and Principal Component Analysis applied to Hyperspectral Imaging of biofilm,” in Photonics Conference (2012), pp. 6–7.
[Crossref]

Lee, J.

Lee, J. Y.

Y. Yoon, H. G. Jeon, D. Yoo, J. Y. Lee, and I. S. Kweon, “Light-Field Image Super-Resolution Using Convolutional Neural Network,” IEEE Signal Process. Lett. 24(6), 848–852 (2017).
[Crossref]

Li, D.

Li, S.

Liefers, B.

Lienkamp, S. S.

Ö. Çiçek, A. Abdulkadir, S. S. Lienkamp, T. Brox, and O. Ronneberger, “3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation,” in International Conference on Medical Image Computing and Computer-Assisted Intervention (2016), pp. 424–432.

Liu, F.

H. Dong, G. Yang, F. Liu, Y. Mo, and Y. Guo, “Automatic Brain Tumor Detection and Segmentation Using U-Net Based Fully Convolutional Networks,” in Conference on Medical Image Understanding and Analysis (2017), pp. 506–517.
[Crossref]

Liu, Z.

G. Huang, Z. Liu, L. d. Maaten, and K. Q. Weinberger, “Densely Connected Convolutional Networks,” in IEEE Conference on Computer Vision and Pattern Recognition (2017), pp. 2261–2269.

Long, J.

E. Shelhamer, J. Long, and T. Darrell, “Fully convolutional networks for semantic segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 39(4), 640–651 (2017).
[Crossref] [PubMed]

Maaten, L. d.

G. Huang, Z. Liu, L. d. Maaten, and K. Q. Weinberger, “Densely Connected Convolutional Networks,” in IEEE Conference on Computer Vision and Pattern Recognition (2017), pp. 2261–2269.

Magistretti, P.

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

Magistretti, P. J.

N. Pavillon, J. Kühn, C. Moratal, P. Jourdain, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Early Cell Death Detection with Digital Holographic Microscopy,” PLoS One 7(1), e30912 (2012).
[Crossref] [PubMed]

B. Rappaz, E. Cano, T. Colomb, J. Kühn, C. Depeursinge, V. Simanis, P. J. Magistretti, and P. Marquet, “Noninvasive characterization of the fission yeast cell cycle by monitoring dry mass with digital holographic microscopy,” J. Biomed. Opt. 14(3), 034049 (2009).
[Crossref] [PubMed]

Marian, A.

Marquet, P.

N. Pavillon and P. Marquet, “Cell volume regulation monitored with combined epifluorescence and digital holographic microscopy,” Methods Mol. Biol. 1254(1254), 21–32 (2015).
[Crossref] [PubMed]

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

N. Pavillon, J. Kühn, C. Moratal, P. Jourdain, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Early Cell Death Detection with Digital Holographic Microscopy,” PLoS One 7(1), e30912 (2012).
[Crossref] [PubMed]

J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009).
[Crossref] [PubMed]

B. Rappaz, E. Cano, T. Colomb, J. Kühn, C. Depeursinge, V. Simanis, P. J. Magistretti, and P. Marquet, “Noninvasive characterization of the fission yeast cell cycle by monitoring dry mass with digital holographic microscopy,” J. Biomed. Opt. 14(3), 034049 (2009).
[Crossref] [PubMed]

T. Colomb, F. Montfort, J. Kãn, N. Aspert, E. Cuche, A. Marian, F. Charrie, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23(12), 3177–3190 (2006).
[Crossref]

T. Colomb, E. Cuche, F. Charrière, J. Kühn, N. Aspert, F. Montfort, P. Marquet, and C. Depeursinge, “Automatic procedure for aberration compensation in digital holographic microscopy and applications to specimen shape compensation,” Appl. Opt. 45(5), 851–863 (2006).
[Crossref] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38(34), 6994–7001 (1999).
[Crossref] [PubMed]

Mathews, S.

Mathon, R.

P. B. Gibbons and R. Mathon, “The use of hill‐climbing to construct orthogonal steiner triple systems,” J. Comb. Des. 1(1), 27–50 (1993).
[Crossref]

Memmolo, P.

Mena, J.

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

Mirza, M.

I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative Adversarial Networks,” Adv. Neural Inf. Process. Syst. 3, 2672–2680 (2014).

Mo, Y.

H. Dong, G. Yang, F. Liu, Y. Mo, and Y. Guo, “Automatic Brain Tumor Detection and Segmentation Using U-Net Based Fully Convolutional Networks,” in Conference on Medical Image Understanding and Analysis (2017), pp. 506–517.
[Crossref]

Montfort, F.

Moratal, C.

N. Pavillon, J. Kühn, C. Moratal, P. Jourdain, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Early Cell Death Detection with Digital Holographic Microscopy,” PLoS One 7(1), e30912 (2012).
[Crossref] [PubMed]

J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009).
[Crossref] [PubMed]

Nehmetallah, G.

Nguyen, T.

Noh, H.

H. Noh, S. Hong, and B. Han, “Learning Deconvolution Network for Semantic Segmentation,” in 2015 IEEE International Conference on Computer Vision (2015),pp. 1520–1528.
[Crossref]

Ozair, S.

I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative Adversarial Networks,” Adv. Neural Inf. Process. Syst. 3, 2672–2680 (2014).

Ozcan, A.

A. Ozcan, H. Günaydin, H. Wang, Y. Rivenson, Y. Zhang, and Z. Göröcs, “Deep learning microscopy,” Optica 4(11), 1437–1443 (2017).
[Crossref]

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” Light Sci. Appl. 7(2), 17141 (2017).
[Crossref]

Pandiyan, V. P.

Parent, J.

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

Paturzo, M.

Pavillon, N.

N. Pavillon and P. Marquet, “Cell volume regulation monitored with combined epifluorescence and digital holographic microscopy,” Methods Mol. Biol. 1254(1254), 21–32 (2015).
[Crossref] [PubMed]

N. Pavillon, J. Kühn, C. Moratal, P. Jourdain, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Early Cell Death Detection with Digital Holographic Microscopy,” PLoS One 7(1), e30912 (2012).
[Crossref] [PubMed]

J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009).
[Crossref] [PubMed]

Pouget-Abadie, J.

I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative Adversarial Networks,” Adv. Neural Inf. Process. Syst. 3, 2672–2680 (2014).

Praharaj, S.

Qi, X.

H. Zhao, J. Shi, X. Qi, X. Wang, and J. Jia, “Pyramid Scene Parsing Network,” in IEEE Conference on Computer Vision and Pattern Recognition (2017), pp. 6230–6239.

Qu, W.

Rappaz, B.

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

J. Kühn, F. Montfort, T. Colomb, B. Rappaz, C. Moratal, N. Pavillon, P. Marquet, and C. Depeursinge, “Submicrometer tomography of cells by multiple-wavelength digital holographic microscopy in reflection,” Opt. Lett. 34(5), 653–655 (2009).
[Crossref] [PubMed]

B. Rappaz, E. Cano, T. Colomb, J. Kühn, C. Depeursinge, V. Simanis, P. J. Magistretti, and P. Marquet, “Noninvasive characterization of the fission yeast cell cycle by monitoring dry mass with digital holographic microscopy,” J. Biomed. Opt. 14(3), 034049 (2009).
[Crossref] [PubMed]

Raub, C.

Raub, C. B.

Ren, S.

K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” in IEEE Conference on Computer Vision and Pattern Recognition (2016), pp. 770–778.

Ren, Z.

Rivenson, Y.

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” Light Sci. Appl. 7(2), 17141 (2017).
[Crossref]

A. Ozcan, H. Günaydin, H. Wang, Y. Rivenson, Y. Zhang, and Z. Göröcs, “Deep learning microscopy,” Optica 4(11), 1437–1443 (2017).
[Crossref]

Ronneberger, O.

O. Ronneberger, P. Fischer, and T. Brox, “U-Net: Convolutional Networks for Biomedical Image Segmentation,” in International Conference on Medical Image Computing and Computer-Assisted Intervention (2015), pp. 234–241.
[Crossref]

Ö. Çiçek, A. Abdulkadir, S. S. Lienkamp, T. Brox, and O. Ronneberger, “3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation,” in International Conference on Medical Image Computing and Computer-Assisted Intervention (2016), pp. 424–432.

Sánchez, C. I.

Schreur, V.

Sevastopolsky, A.

A. Sevastopolsky, “Optic disc and cup segmentation methods for glaucoma detection with modification of U-Net convolutional neural network,” Pattern Recognit. Image Anal. 27(3), 618–624 (2017).
[Crossref]

Shaffer, E.

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

Shannon, C. E. A.

C. E. A. Shannon, “A mathematical theory of communication,” Mob. Comput. Commun. Rev. 5(1), 3–55 (2001).
[Crossref]

Shao, X.

J. Zheng, P. Gao, and X. Shao, “Opposite-view digital holographic microscopy with autofocusing capability,” Sci. Rep. 7(1), 4255 (2017).
[Crossref] [PubMed]

Shelhamer, E.

E. Shelhamer, J. Long, and T. Darrell, “Fully convolutional networks for semantic segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 39(4), 640–651 (2017).
[Crossref] [PubMed]

Shen, Z.

Shi, J.

H. Zhao, J. Shi, X. Qi, X. Wang, and J. Jia, “Pyramid Scene Parsing Network,” in IEEE Conference on Computer Vision and Pattern Recognition (2017), pp. 6230–6239.

Simanis, V.

B. Rappaz, E. Cano, T. Colomb, J. Kühn, C. Depeursinge, V. Simanis, P. J. Magistretti, and P. Marquet, “Noninvasive characterization of the fission yeast cell cycle by monitoring dry mass with digital holographic microscopy,” J. Biomed. Opt. 14(3), 034049 (2009).
[Crossref] [PubMed]

Sinha, A.

Sun, J.

K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” in IEEE Conference on Computer Vision and Pattern Recognition (2016), pp. 770–778.

Sutskever, I.

A. Krizhevsky, I. Sutskever, and G. E. Hinton, “ImageNet classification with deep convolutional neural networks,” in International Conference on Neural Information Processing Systems (2012), pp. 1097–1105.

Tasimi, K.

Teng, D.

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” Light Sci. Appl. 7(2), 17141 (2017).
[Crossref]

Theelen, T.

Turcatti, G.

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

van Asten, F.

van Ginneken, B.

Venhuizen, F. G.

Wang, H.

Wang, X.

H. Zhao, J. Shi, X. Qi, X. Wang, and J. Jia, “Pyramid Scene Parsing Network,” in IEEE Conference on Computer Vision and Pattern Recognition (2017), pp. 6230–6239.

Warde-Farley, D.

I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative Adversarial Networks,” Adv. Neural Inf. Process. Syst. 3, 2672–2680 (2014).

Weinberger, K. Q.

G. Huang, Z. Liu, L. d. Maaten, and K. Q. Weinberger, “Densely Connected Convolutional Networks,” in IEEE Conference on Computer Vision and Pattern Recognition (2017), pp. 2261–2269.

Williams, L.

Xu, B.

I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative Adversarial Networks,” Adv. Neural Inf. Process. Syst. 3, 2672–2680 (2014).

Xu, Z.

Yang, G.

H. Dong, G. Yang, F. Liu, Y. Mo, and Y. Guo, “Automatic Brain Tumor Detection and Segmentation Using U-Net Based Fully Convolutional Networks,” in Conference on Medical Image Understanding and Analysis (2017), pp. 506–517.
[Crossref]

Yao, Y.

Y. Yao, B. Abidi, N. Doggaz, and M. Abidi, “Evaluation of sharpness measures and search algorithms for the auto focusing of high-magnification images,” Proc. SPIE 6246(6424), 62460G (2006).
[Crossref]

Yazdanfar, S.

Yoo, D.

Y. Yoon, H. G. Jeon, D. Yoo, J. Y. Lee, and I. S. Kweon, “Light-Field Image Super-Resolution Using Convolutional Neural Network,” IEEE Signal Process. Lett. 24(6), 848–852 (2017).
[Crossref]

Yoon, Y.

Y. Yoon, H. G. Jeon, D. Yoo, J. Y. Lee, and I. S. Kweon, “Light-Field Image Super-Resolution Using Convolutional Neural Network,” IEEE Signal Process. Lett. 24(6), 848–852 (2017).
[Crossref]

Zhang, G.

Zhang, X.

K. He, X. Zhang, S. Ren, and J. Sun, “Deep Residual Learning for Image Recognition,” in IEEE Conference on Computer Vision and Pattern Recognition (2016), pp. 770–778.

Zhang, Y.

A. Ozcan, H. Günaydin, H. Wang, Y. Rivenson, Y. Zhang, and Z. Göröcs, “Deep learning microscopy,” Optica 4(11), 1437–1443 (2017).
[Crossref]

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” Light Sci. Appl. 7(2), 17141 (2017).
[Crossref]

Zhao, H.

H. Zhao, J. Shi, X. Qi, X. Wang, and J. Jia, “Pyramid Scene Parsing Network,” in IEEE Conference on Computer Vision and Pattern Recognition (2017), pp. 6230–6239.

Zheng, J.

J. Zheng, P. Gao, and X. Shao, “Opposite-view digital holographic microscopy with autofocusing capability,” Sci. Rep. 7(1), 4255 (2017).
[Crossref] [PubMed]

Zuo, C.

Adv. Neural Inf. Process. Syst. (1)

I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative Adversarial Networks,” Adv. Neural Inf. Process. Syst. 3, 2672–2680 (2014).

Adv. Opt. Photonics (1)

G. Nehmetallah and P. P. Banerjee, “Applications of digital and analog holography in three-dimensional imaging,” Adv. Opt. Photonics 4(4), 472 (2012).
[Crossref]

Appl. Opt. (5)

Assay Drug Dev. Technol. (1)

J. Kühn, E. Shaffer, J. Mena, B. Breton, J. Parent, B. Rappaz, M. Chambon, Y. Emery, P. Magistretti, C. Depeursinge, P. Marquet, and G. Turcatti, “Label-Free Cytotoxicity Screening Assay by Digital Holographic Microscopy,” Assay Drug Dev. Technol. 11(2), 101–107 (2013).
[Crossref] [PubMed]

Biomed. Opt. Express (1)

IEEE Signal Process. Lett. (1)

Y. Yoon, H. G. Jeon, D. Yoo, J. Y. Lee, and I. S. Kweon, “Light-Field Image Super-Resolution Using Convolutional Neural Network,” IEEE Signal Process. Lett. 24(6), 848–852 (2017).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

E. Shelhamer, J. Long, and T. Darrell, “Fully convolutional networks for semantic segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 39(4), 640–651 (2017).
[Crossref] [PubMed]

J. Biomed. Opt. (1)

B. Rappaz, E. Cano, T. Colomb, J. Kühn, C. Depeursinge, V. Simanis, P. J. Magistretti, and P. Marquet, “Noninvasive characterization of the fission yeast cell cycle by monitoring dry mass with digital holographic microscopy,” J. Biomed. Opt. 14(3), 034049 (2009).
[Crossref] [PubMed]

J. Comb. Des. (1)

P. B. Gibbons and R. Mathon, “The use of hill‐climbing to construct orthogonal steiner triple systems,” J. Comb. Des. 1(1), 27–50 (1993).
[Crossref]

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

Light Sci. Appl. (1)

Y. Rivenson, Y. Zhang, H. Günaydın, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” Light Sci. Appl. 7(2), 17141 (2017).
[Crossref]

Methods Mol. Biol. (1)

N. Pavillon and P. Marquet, “Cell volume regulation monitored with combined epifluorescence and digital holographic microscopy,” Methods Mol. Biol. 1254(1254), 21–32 (2015).
[Crossref] [PubMed]

Mob. Comput. Commun. Rev. (1)

C. E. A. Shannon, “A mathematical theory of communication,” Mob. Comput. Commun. Rev. 5(1), 3–55 (2001).
[Crossref]

Opt. Eng. (1)

T. Nguyen, V. Bui, and G. Nehmetallah, “Computational Optical Tomography Using 3D Deep Convolutional Neural Networks (3D-DCNNs),” Opt. Eng. 57(4), 043111 (2018).

Opt. Express (2)

Opt. Lett. (4)

Optica (3)

Pattern Recognit. Image Anal. (1)

A. Sevastopolsky, “Optic disc and cup segmentation methods for glaucoma detection with modification of U-Net convolutional neural network,” Pattern Recognit. Image Anal. 27(3), 618–624 (2017).
[Crossref]

PLoS One (1)

N. Pavillon, J. Kühn, C. Moratal, P. Jourdain, C. Depeursinge, P. J. Magistretti, and P. Marquet, “Early Cell Death Detection with Digital Holographic Microscopy,” PLoS One 7(1), e30912 (2012).
[Crossref] [PubMed]

Proc. SPIE (1)

Y. Yao, B. Abidi, N. Doggaz, and M. Abidi, “Evaluation of sharpness measures and search algorithms for the auto focusing of high-magnification images,” Proc. SPIE 6246(6424), 62460G (2006).
[Crossref]

Sci. Rep. (1)

J. Zheng, P. Gao, and X. Shao, “Opposite-view digital holographic microscopy with autofocusing capability,” Sci. Rep. 7(1), 4255 (2017).
[Crossref] [PubMed]

Other (15)

A. Krizhevsky, I. Sutskever, and G. E. Hinton, “ImageNet classification with deep convolutional neural networks,” in International Conference on Neural Information Processing Systems (2012), pp. 1097–1105.

H. Noh, S. Hong, and B. Han, “Learning Deconvolution Network for Semantic Segmentation,” in 2015 IEEE International Conference on Computer Vision (2015),pp. 1520–1528.
[Crossref]

O. Ronneberger, P. Fischer, and T. Brox, “U-Net: Convolutional Networks for Biomedical Image Segmentation,” in International Conference on Medical Image Computing and Computer-Assisted Intervention (2015), pp. 234–241.
[Crossref]

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Figures (16)

Fig. 1
Fig. 1 off-axis DHM system. SLD, superluminescent diode; SMF, single-mode fiber; CL, collimator; LLF, laser line filter; BS, beam splitter; ODL optical delay line; OL, objective lens; TL, tube lens.
Fig. 2
Fig. 2 (a) The architecture of U-net, indicating the number of layers, nodes in each layer, etc. The input is the intensity and phase information of the image, the size of which is 512x512. The output is the reconstructed phase of the objective image. (b) Two scenarios for network input. The first row represents the input of intensity and phase diagrams; The second row represents the input of intensity, sin(phase) and cos(phase) diagrams.
Fig. 3
Fig. 3 Process of generating training sets. (a) object contour and texture pattern (b) The generated focused intensity of objective image. (c) The generated focused phase of objective image. (d) Additional background phase (e) The intensity and phase of the defocused objective image. (f) The intensity, cos(phase) and sin(phase) of the defocused objective image.
Fig. 4
Fig. 4 Reconstruction results of trained U-net. The baseline distance on which the network was trained is 15mm to 35 mm, and the stride is 0.05 mm.
Fig. 5
Fig. 5 Reconstruction results of our trained U-net under three different generated data sets. The baseline distance on which the network was trained is (a) z1 = 25 ± 10 mm, (b) z2 = 95 ± 10 mm, and (c) z3 = 155 ± 10 mm, respectively.
Fig. 6
Fig. 6 Quantitative analysis of the sensitivity of the trained U-net to three different generated data sets. The baseline distance on which the network was trained is (a) z1 = 25 ± 10 mm, (b) z2 = 95 ± 10 mm, and (c) z3 = 155 ± 10 mm, respectively
Fig. 7
Fig. 7 The phase trig function mapping of micro-quartz pieces
Fig. 8
Fig. 8 Reconstruction results of our trained U-net under three different generated data sets. The baseline distance on which the network was trained is −0.01 mm to 0.01 mm.
Fig. 9
Fig. 9 Poor reconstruction of the neural network (without distinguishing between background and target).
Fig. 10
Fig. 10 The training and testing error curve for the network trained on training set with background additional phase at distance dz = ± 0.01mm over 10 epochs.
Fig. 11
Fig. 11 The overall flow chart of the fast-focus algorithm. The green line represents the network training section. The red line represents the process from hologram to coarse focus. The blue line represents the trigonometric mapping process and the final result (1) Hologram received from CCD (2) The spectrum of the hologram (3) The spectrum of the separated light field (4) coarse - focused light field through Fresnel transform formula (5) Label of training set generation (6) trigonometric mapping process of training set (7) trigonometric mapping process of practical samples (8) U-net’s phase reconstruction of practical samples.
Fig. 12
Fig. 12 Rapid phase focusing of micro-quartz pieces. (a)is the hologram captured by CCD (b) and (c) are phase and intensity image of MQPs. (d) and (e) are the phase distribution after triangulation. (f) is the reconstructed phase.
Fig. 13
Fig. 13 The testing error curve for the network trained on training set without background additional phase for L1 and L2 loss.
Fig. 14
Fig. 14 Comparison of reconstruction results of two loss functions. (a) represents ground truth; (b) represents the reconstruction result by using L1 loss function; (c) represents the reconstruction result by using L2 loss function;
Fig. 15
Fig. 15 Comparison of reconstruction results by using additional loss. (a) represents ground truth; (b) represents the reconstruction result by using additional loss ; (c) represents the reconstruction result without additional loss;
Fig. 16
Fig. 16 phase focusing experiment of resolution plate. (a) defocused intensity image; (b) defocused phase image; (c) and (d) represent the phase operated by trig function; (e) reconstructed result of trained U-net; (f) Focused intensity images obtained through traditional methods; (h) Focused intensity images obtained through traditional methods.

Equations (18)

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F(dz,O):C(x,y;z+dz)=argmaxG(f(dz,O(x,y;z))).
C( x,y;z+dz )= e jkdz jλdz e jk( x 2 + y 2 ) 2dz ×fft{O( x,y;z ) e jk( x 2 + y 2 ) 2dz }.
C( x,y;z+dz )=ff t 1 {fft{O( x,y;z )}×fft{ e jkdz jλdz e jk( x 2 + y 2 ) 2dz }}.
C( x,y;z+dz )=ff t 1 {fft{O( x,y;z )} e jk 1 ( λ f x ) 2 ( λ f y ) 2 }.
C( x,y;z+dz )=O( x,y;z ) dz iλ e ik d z 2 + x 2 + y 2 d z 2 + x 2 + y 2 .
L= 1 wh ( a 1 w 1 + a 2 w 2 + a 3 )||YG| | 2 .
w 1 =mag(X)= [ ( X x ) 2 + ( X y ) 2 ] 1 2 G σ ( x,y ).
w 2 =mag(G)= [ ( G x ) 2 + ( G y ) 2 ] 1 2 G σ ( x,y ).
A 0 ( x,y,z )=I( x,y,z ) e iψ ( x,y,z ) .
B 0 ( x,y,z+dz )= A 0 ( x,y,z )h( x,y,dz )= e ikdz iλdz e f ik( x 2 + y 2 ) 2dz ft{ A 0 ( x 0 , y 0 ,z )e } ik( x 0 2 + y 0 2 ) 2dz .
h( x,y,dz )= e ikdz iλdz e . ik( x 2 + y 2 ) 2dz
MAE= 1 wh x y |arg( B 0 h( x,y,dz ) )( arg( B 0 ),abs( B 0 ) ) H FCN ( x,y )| = 1 wh x y |ψ( x,y,z )( arg( A 0 ( x,y,z )h( x,y,dz ) ),abs( A 0 ( x,y,z )h( x,y,dz ) ) ) H FCN ( x,y )|.
abs( A 0 (x,y,z)h(x,y,dz))abs( A 0 (x,y,z)h(x,y,dz+dt))
abs( e ikdz iλdz e f ik( x 2 + y 2 ) 2dz f t 1 ( A 0 ( x 0 , y 0 ,z )e ik( x 0 2 + y 0 2 ) 2dz ) )abs( e ik( dz+dt ) iλ( dz+dt ) e f ik( x 2 + y 2 ) 2( dz+dt ) f t 1 ( A 0 ( x 0 , y 0 ,z )e ik( x 0 2 + y 0 2 ) 2( dz+dt ) ) ) e ik( x 0 2 + y 0 2 ) 2dz e ik( x 0 2 + y 0 2 ) 2( dz+dt ) k( x 0 2 + y 0 2 ) 2dz 2π k( x 0 2 + y 0 2 ) 2( dz+dt )
g( dz,dt )= k( x 0 2 + y 0 2 ) 2dz 2π k( x 0 2 + y 0 2 ) 2( dz+dt ) =0.
g dz = k( x 0 2 + y 0 2 ) 2d z 2 + k( x 0 2 + y 0 2 ) 2 ( dz+dt ) 2 .
g dt = k( x 0 2 + y 0 2 ) 2 ( dz+dt ) 2 .
dz dt >0.

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