Abstract

The transport-of-intensity equation (TIE) has been proven as a standard approach for phase retrieval. Some high efficiency solving methods for the TIE, extensively used in many works, is based on a Fourier transform (FT). However, several assumptions have to be made to solve the TIE by these methods. A common assumption is that there are no zero values for the intensity distribution allowed. The two most widespread Fourier-based approaches have further restrictions. One of these requires the uniformity of the intensity distribution and the other assumes the parallelism of the intensity and phase gradients. In this paper, we present an approach, which does not need any of these assumptions and consequently extends the application domain of the TIE.

© 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 (1)

2017 (4)

2016 (4)

2015 (4)

2014 (11)

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Optimum measurement criteria for the axial derivative intensity used in transport of intensity-equation-based solvers,” Opt. Lett. 39(2), 182–185 (2014).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, M. Jozwik, and T. Kozacki, “Comparison of phase retrieval techniques based on the transport of intensity equation using equally and unequally spaced plane separation criteria,” Proc. SPIE 9204, 92040G (2014).
[Crossref]

Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of Intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22(9), 10661–10674 (2014).
[Crossref] [PubMed]

Y. Zhu, A. Shanker, L. Tian, L. Waller, and G. Barbastathis, “Low-noise phase imaging by hybrid uniform and structured illumination transport of intensity equation,” Opt. Express 22(22), 26696–26711 (2014).
[Crossref] [PubMed]

C. Zuo, Q. Chen, L. Huang, and A. Asundi, “Phase discrepancy analysis and compensation for fast Fourier transform based solution of the transport of intensity equation,” Opt. Express 22(14), 17172–17186 (2014).
[Crossref] [PubMed]

A. Shanker, L. Tian, M. Sczyrba, B. Connolly, A. Neureuther, and L. Waller, “Transport of intensity phase imaging in the presence of curl effects induced by strongly absorbing photomasks,” Appl. Opt. 53(34), J1–J6 (2014).
[Crossref] [PubMed]

J. Martínez-Carranza, K. Falaggis, and T. Kozacki, “Optimum phase retrieval using the transport of intensity equation,” Proc. SPIE 9132, 91320T (2014).
[Crossref]

K. Falaggis, T. Kozacki, and M. Kujawinska, “Optimum plane selection criteria for single-beam phase retrieval techniques based on the contrast transfer function,” Opt. Lett. 39(1), 30–33 (2014).
[Crossref] [PubMed]

C. Zuo, Q. Chen, and A. Asundi, “Boundary-artifact-free phase retrieval with the transport of intensity equation: fast solution with use of discrete cosine transform,” Opt. Express 22(8), 9220–9244 (2014).
[Crossref] [PubMed]

R. Yazdani, M. Hajimahmoodzadeh, and H. R. Fallah, “Application of the transport of intensity equation in determination of nonlinear refractive index,” Appl. Opt. 53(35), 8295–8301 (2014).
[Crossref] [PubMed]

Z. Jingshan, R. A. Claus, J. Dauwels, L. Tian, and L. Waller, “Transport of Intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes,” Opt. Express 22(9), 10661–10674 (2014).
[Crossref] [PubMed]

2013 (4)

2012 (2)

R. Shomali, A. Darudi, and S. Nasiri, “Application of irradiance transport equation in aspheric surface testing,” Optik (Stuttg.) 123(14), 1282–1286 (2012).
[Crossref]

R. Bie, X.-H. Yuan, M. Zhao, and L. Zhang, “Method for estimating the axial intensity derivative in the TIE with higher order intensity derivatives and noise suppression,” Opt. Express 20(7), 8186–8191 (2012).
[Crossref] [PubMed]

2011 (2)

B. Xue, S. Zheng, L. Cui, X. Bai, and F. Zhou, “Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes,” Opt. Express 19(21), 20244–20250 (2011).
[Crossref] [PubMed]

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matterwave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[Crossref]

2010 (2)

2007 (2)

2004 (1)

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

2003 (2)

2002 (1)

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33(5), 411–416 (2002).
[Crossref] [PubMed]

1998 (1)

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

1993 (1)

1988 (1)

K. Creath, “V Phase-Measurement Interferometry Techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

1983 (1)

Abeywickrema, U.

Acosta, E.

Amézquita-Orozco, R.

Amiri, J.

Asundi, A.

Bai, X.

Banerjee, P. P.

Barbastathis, G.

Bartels, M.

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

Basunia, M.

Bie, R.

Bostan, E.

E. Bostan, E. Froustey, B. Rappaz, E. Shaffer, D. Sage, and M. Unser, “Phase retrieval by using transport-of-intensity equation and differential interference contrast microscopy,” in Proceedings of 2014 IEEE International Conference on Image Processing (ICIP) (IEEE, 2014), pp. 3939–3943.
[Crossref]

Chakraborty, T.

Chen, Q.

Claus, R. A.

Connolly, B.

Creath, K.

K. Creath, “V Phase-Measurement Interferometry Techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

Cui, L.

Darudi, A.

Dauwels, J.

De Graef, M.

V. V. Volkov, Y. Zhu, and M. De Graef, “A new symmetrized solution for phase retrieval using the transport of intensity equation,” Micron 33(5), 411–416 (2002).
[Crossref] [PubMed]

Dorrer, C.

Falaggis, K.

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Fast and accurate phase-unwrapping algorithm based on the transport of intensity equation,” Appl. Opt. 56(25), 7079–7088 (2017).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Multi-filter transport of intensity equation solver with equalized noise sensitivity,” Opt. Express 23(18), 23092–23107 (2015).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Optimum measurement criteria for the axial derivative intensity used in transport of intensity-equation-based solvers,” Opt. Lett. 39(2), 182–185 (2014).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, M. Jozwik, and T. Kozacki, “Comparison of phase retrieval techniques based on the transport of intensity equation using equally and unequally spaced plane separation criteria,” Proc. SPIE 9204, 92040G (2014).
[Crossref]

K. Falaggis, T. Kozacki, and M. Kujawinska, “Optimum plane selection criteria for single-beam phase retrieval techniques based on the contrast transfer function,” Opt. Lett. 39(1), 30–33 (2014).
[Crossref] [PubMed]

J. Martínez-Carranza, K. Falaggis, and T. Kozacki, “Optimum phase retrieval using the transport of intensity equation,” Proc. SPIE 9132, 91320T (2014).
[Crossref]

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).
[Crossref]

Fallah, H. R.

Froustey, E.

E. Bostan, E. Froustey, B. Rappaz, E. Shaffer, D. Sage, and M. Unser, “Phase retrieval by using transport-of-intensity equation and differential interference contrast microscopy,” in Proceedings of 2014 IEEE International Conference on Image Processing (ICIP) (IEEE, 2014), pp. 3939–3943.
[Crossref]

Ghosh, A.

Greenaway, A. H.

Guan, X.

Gureyev, T. E.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matterwave fields: Validity of Teague’s method for solution of the transport-of-intensity equation,” Phys. Rev. A 84(2), 023808 (2011).
[Crossref]

Hajimahmoodzadeh, M.

He, W.

Huang, L.

Idir, M.

Ishikawa, K.

Jingshan, Z.

Jozwik, M.

J. Martinez-Carranza, K. Falaggis, M. Jozwik, and T. Kozacki, “Comparison of phase retrieval techniques based on the transport of intensity equation using equally and unequally spaced plane separation criteria,” Proc. SPIE 9204, 92040G (2014).
[Crossref]

Khare, K.

Kozacki, T.

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Fast and accurate phase-unwrapping algorithm based on the transport of intensity equation,” Appl. Opt. 56(25), 7079–7088 (2017).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Multi-filter transport of intensity equation solver with equalized noise sensitivity,” Opt. Express 23(18), 23092–23107 (2015).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Optimum measurement criteria for the axial derivative intensity used in transport of intensity-equation-based solvers,” Opt. Lett. 39(2), 182–185 (2014).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, M. Jozwik, and T. Kozacki, “Comparison of phase retrieval techniques based on the transport of intensity equation using equally and unequally spaced plane separation criteria,” Proc. SPIE 9204, 92040G (2014).
[Crossref]

K. Falaggis, T. Kozacki, and M. Kujawinska, “Optimum plane selection criteria for single-beam phase retrieval techniques based on the contrast transfer function,” Opt. Lett. 39(1), 30–33 (2014).
[Crossref] [PubMed]

J. Martínez-Carranza, K. Falaggis, and T. Kozacki, “Optimum phase retrieval using the transport of intensity equation,” Proc. SPIE 9132, 91320T (2014).
[Crossref]

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).
[Crossref]

Krenkel, M.

Kujawinska, M.

K. Falaggis, T. Kozacki, and M. Kujawinska, “Optimum plane selection criteria for single-beam phase retrieval techniques based on the contrast transfer function,” Opt. Lett. 39(1), 30–33 (2014).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).
[Crossref]

Li, J.

C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7(1), 7654 (2017).
[Crossref] [PubMed]

Liu, F.

Martinez-Carranza, J.

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Fast and accurate phase-unwrapping algorithm based on the transport of intensity equation,” Appl. Opt. 56(25), 7079–7088 (2017).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Multi-filter transport of intensity equation solver with equalized noise sensitivity,” Opt. Express 23(18), 23092–23107 (2015).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, M. Jozwik, and T. Kozacki, “Comparison of phase retrieval techniques based on the transport of intensity equation using equally and unequally spaced plane separation criteria,” Proc. SPIE 9204, 92040G (2014).
[Crossref]

J. Martinez-Carranza, K. Falaggis, and T. Kozacki, “Optimum measurement criteria for the axial derivative intensity used in transport of intensity-equation-based solvers,” Opt. Lett. 39(2), 182–185 (2014).
[Crossref] [PubMed]

J. Martinez-Carranza, K. Falaggis, T. Kozacki, and M. Kujawinska, “Effect of imposed boundary conditions on the accuracy of transport of intensity equation based solvers,” Proc. SPIE 8789, 87890N (2013).
[Crossref]

Martínez-Carranza, J.

J. Martínez-Carranza, K. Falaggis, and T. Kozacki, “Optimum phase retrieval using the transport of intensity equation,” Proc. SPIE 9132, 91320T (2014).
[Crossref]

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

Mejía-Barbosa, Y.

Moradi, A. R.

Nasiri, S.

R. Shomali, A. Darudi, and S. Nasiri, “Application of irradiance transport equation in aspheric surface testing,” Optik (Stuttg.) 123(14), 1282–1286 (2012).
[Crossref]

Nehmetallah, G.

Neureuther, A.

Nguyen, T.

Nugent, K.

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

Nugent, K. A.

K. A. Nugent, “Coherent methods in the X-ray sciences,” Adv. Phys. 59(1), 1–99 (2010).
[Crossref]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

Oikawa, Y.

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[Crossref]

Paganin, D. M.

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Figures (9)

Fig. 1
Fig. 1 Theoretical setup for the TIE with the object plane and the two intensity measurements in close proximity dz.
Fig. 2
Fig. 2 (a) and (b) are the intensity distributions in the first and the second plane respectively. (c) is the original phase in the first plane and (d) the phase distribution in the first plane reconstructed by the presented method. Both images are quadratic with a width and height of 1.135 mm and the pixel number is 1135 x 1135. The wavelength of the light in the simulation is 633 nm. According to [42], the paraxial condition requires a very small transverse spatial frequency f compared to the spatial frequency of the light f f( f=1/λ ). Thus, the maximum transverse spatial frequency of the complex amplitude u 1 = I 1 exp( i ϕ 1 ) was 0.1f. The used C according to Eq. (19) was 0.057.
Fig. 3
Fig. 3 (a) and (b) are the intensity distributions in the first and the second plane respectively without any zero-value pixels. (c) is the original phase in the first plane and (d) the reconstructed phase distribution in the first plane without any singularity points.
Fig. 4
Fig. 4 Correlation coefficient (a) and RMSE values (b) for the image in Fig. 2 in dependence of the SNR.
Fig. 5
Fig. 5 Stability of the algorithm, (a) and (b) are the correlation and RMSE values respectively for the used image in Fig. 2 dependent on the iteration number.
Fig. 6
Fig. 6 , circle distribution of the zero intensity values. (a) and (b) Intensity distributions in the first and the second plane, respectively. (c) and (d) Original and reconstructed phase. The amount of zero-value pixels is 15% of the total pixel number.
Fig. 7
Fig. 7 , grating distribution of the zero intensity values. (a) and (b) Intensity distributions in the first and the second plane, respectively. (c) and (d) Original and reconstructed phase. The amount of zero-value pixels is 3% of the total pixel number. Please note that although the blue area seems to be bigger than in Fig. 6(a) and 6(b), the number of pixels with zero intensity is smaller.
Fig. 8
Fig. 8 , band distribution of the zero intensity values. (a) and (b) Intensity distributions in the first and the second plane respectively. (c) and (d) The original and the reconstructed phase. The zero-value pixels are 23% of the total pixel number.
Fig. 9
Fig. 9 Dependence of the phase retrieval quality from the number of zero-value pixels, (a) and (b) are the correlation and RMSE values, respectively for the intensity distribution in Fig. 8(a) with an increased zero intensity area.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

( I ϕ )=k I z ,
I z I 2 I 1 dz .
( I+C ) 2 ϕ=k I z I ϕ+C 2 ϕ,
2 ϕ n+1 = k I z I ϕ n +C 2 ϕ n I+C .
4 π 2 ( f x 2 + f y 2 ) ϕ ˜ n+1 =FT{ k I z I ϕ n +C 2 ϕ n I+C }.
ϕ ˜ n+1 = 1 4 π 2 ( f x 2 + f y 2 ) FT{ k I z I ϕ n +C 2 ϕ n I+C }.
ϕ n+1 = 1 4 π 2 IFT{ 1 f x 2 + f y 2 FT{ k I z I ϕ n +C 2 ϕ n I+C } }.
f = r λz =f r z ,
k N A obj k.
k 1,x ϕ( x ) x , k 1,y ϕ( y ) y , k 2,x ϕ( x+h ) x , k 2,y ϕ( y+h ) y ,
| ϕ |N A obj k.
| 2 ϕ |=| 2 ϕ x 2 + 2 ϕ y 2 | 1 h | ϕ( x+h ) x ϕ( x ) x + ϕ( y+h ) y ϕ( y ) y |.
| 2 ϕ | 1 h | k 2,x k 1,x + k 2,y k 1,y | 1 h ( | k 1,x + k 1,y |+| k 2,x + k 2,y | ).
| ϕ | 2 k x 2 + k y 2 N A obj 2 k 2 | k x + k y | N A obj 2 k 2 +2 k x k y ,
| 2 ϕ | 1 h ( N A obj 2 k 2 +2 k 1,x k 1,y + N A obj 2 k 2 +2 k 2,x k 2,y ).
| 2 ϕ | 2 2 N A obj k h .
( I 1 +C ) 2 ϕ 1 =k I 1 z .
2 2 N A obj | I 1 +C |h| I 1 z |.
C h | I 1 z | max 2 2 N A obj .

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