Abstract

The noise problem is generally inevitable for phase retrieval by solving the transport of intensity equation (TIE). The noise effect can be alleviated by using multiple intensities to estimate the axial intensity derivative in the TIE. In this study, a method is proposed for estimating the intensity derivative by using multiple unevenly-spaced noisy measurements. The noise-minimized intensity derivative is approximated by a linear combination of the intensity data, in which the coefficients are obtained by solving a constrained optimization problem. The performance of the method is investigated by both the error analysis and the numerical simulations, and the results show that the method can reduce the noise effect on the retrieved phase. In addition, guidelines for the choice of the number of the intensity planes are given.

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References

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    [CrossRef]
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    [CrossRef]
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2011 (3)

2010 (4)

2007 (1)

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]

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]

2001 (1)

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[CrossRef]

1998 (2)

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998).
[CrossRef] [PubMed]

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

1997 (1)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[CrossRef]

1996 (1)

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

1995 (2)

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. 12(9), 1932–1941 (1995).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995).
[CrossRef]

1993 (1)

1988 (1)

1983 (1)

Acosta, E.

Alimi, R.

Allen, L. J.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[CrossRef]

Altmeyer, S.

Bai, X.

Barbastathis, G.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

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]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998).
[CrossRef] [PubMed]

Cong, W. X.

W. X. Cong and G. Wang, “Higher-order phase shift reconstruction approach,” Med. Phys. 37(10), 5238–5242 (2010).
[CrossRef] [PubMed]

Cookson, D. J.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Cui, L.

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]

Eliezer, S.

Frank, J.

Gureyev, T. E.

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. 12(9), 1932–1941 (1995).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995).
[CrossRef]

Horstmann, J.

Matrisch, J.

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]

Nugent, K. 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]

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

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998).
[CrossRef] [PubMed]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. 12(9), 1932–1941 (1995).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12(9), 1942–1946 (1995).
[CrossRef]

Oxley, M. P.

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[CrossRef]

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. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23(11), 817–819 (1998).
[CrossRef] [PubMed]

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

Perelmutter, L.

Pinhasi, S. V.

Ponda, S.

Roberts, A.

Roddier, C.

Roddier, F.

Soto, M.

Teague, M. R.

Tian, L.

Tsang, M.

Volkov, V. V.

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]

Waller, L.

Wang, G.

W. X. Cong and G. Wang, “Higher-order phase shift reconstruction approach,” Med. Phys. 37(10), 5238–5242 (2010).
[CrossRef] [PubMed]

Wernicke, G.

Xue, B.

Yang, S. Y.

Zheng, S.

Zhou, F.

Zhu, Y.

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]

Appl. Opt. (3)

J. Microsc. (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]

J. Opt. Soc. Am. (2)

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983).
[CrossRef]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. 12(9), 1932–1941 (1995).
[CrossRef]

J. Opt. Soc. Am. A (4)

Med. Phys. (1)

W. X. Cong and G. Wang, “Higher-order phase shift reconstruction approach,” Med. Phys. 37(10), 5238–5242 (2010).
[CrossRef] [PubMed]

Micron (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]

Opt. Commun. (2)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997).
[CrossRef]

L. J. Allen and M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. Lett. (2)

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[CrossRef] [PubMed]

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

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), pp. 55–61.

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Figures (10)

Fig. 1
Fig. 1

Simulated phase distribution in the z=0 plane.

Fig. 2
Fig. 2

Simulated noisy diffraction intensities in the z=19μm plane with different SNRs of (a) 70db ; (b) 60db ; (c) 55db ; (d) 50db .

Fig. 3
Fig. 3

The average NMSEs of 60 times for each noise level are plotted as a function of M and SNR. The noise in the intensities is assumed to be Gaussian white noise.

Fig. 4
Fig. 4

Retrieved phases from M=1 (the first row) and M=15 (the second row) Gaussian-noise damaged intensity measurements with different SNRs of 70db, 60db, 55db and 50db (from left to right). The RMSEs for the phase images in the first row are respectively 0.042, 0.180, 0.330 and 0.458, and for those in the second row are respectively 0.017, 0.043, 0.139 and 0.169.

Fig. 5
Fig. 5

Simulated noisy diffraction intensities in the z=19μm plane with Poisson noise. The average photons for each pixel of the images are respectively: (a) 100,000; (b) 80,000; (c) 32,000; and (d) 10,000.

Fig. 6
Fig. 6

The NMSEs between the estimated intensity derivative and the ideal intensity derivative are plotted as a function of M and the average photons per pixel in the diffraction images. The noise in the intensities are assumed to be Poisson noise.

Fig. 7
Fig. 7

Retrieved phases from M=1 (the first row) and M=15 (the second row) Poisson-noise damaged intensity measurements with average photons per pixel of 100,000, 80,000, 32,000, and 10,000 (from left to right). The phase images in the first row have the RMSEs of 0.542, 0.422, 0.730, and 1.682 respectively, and those in the second row have the RMSEs of 0.223, 0.223, 0.300, and 0.750 respectively.

Fig. 8
Fig. 8

Simulated diffraction intensities corrupted with mixed noise in the z=19μm plane. (a) case 1; (b) case 2; (c) case 3; and (d) case 4.

Fig. 9
Fig. 9

The NMSEs between the estimated intensity derivative and the ideal intensity derivative are plotted as a function of M for the four cases (as indicated in the main text). The noise in the intensities are assumed to be mixed noise.

Fig. 10
Fig. 10

Retrieved phases from M=1 (the first row) and M=15 (the second row) mixed-noise damaged intensity measurements for the four cases (please refer to the main text for details) (from left to right).

Tables (2)

Tables Icon

Table 1 Comparison of the errors between the estimated axial intensity derivatives and the reconstructed phases by solving TIE from different numbers of intensities with different levels of Gaussian noise.

Tables Icon

Table 2 Comparison of the errors for the estimated axial intensity derivatives and the reconstructed phases by solving TIE from different numbers of intensities with different levels of Poisson noise.

Equations (35)

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·[ I z (r) ϕ z (r) ]=k I z (r) / z ,
I z j (r)= I ¯ z j (r)+ n z j (r).
I z j (r)= I ¯ z j (r)+ n z j (r).
I ^ 0 (r)= j=1 M a j z j z j ( I z j (r) I z j (r) ) .
ε 2 (r)=E{ [ I ^ 0 (r) I 0 (r) ] 2 },
I ¯ z j (r)= I ¯ 0 (r)+ z j I 0 (r)+ z j 2 2 I c j (r) /2 , c j [0, z j ].
I ¯ z j (r)= I ¯ 0 (r)+ z j I 0 (r)+ z j 2 2 I c j (r) /2 , c j [ z j ,0].
I ^ 0 (r)= j=1 M a j z j z j ( n z j (r) n z j (r) ) + I 0 (r) j=1 M a j + j=1 M a j z j z j ( z j 2 2 2 I c j (r) z j 2 2 2 I c j (r) ) .
j=1 M a j 1=0.
ε 2 (r)= ε n 2 (r)+ ε h 2 (r) is minimum,
ε n 2 (r)=2 σ 2 j=1 M a j 2 ( z j z j ) 2 ,
ε h 2 (r)= 1 4 { j=1 M a j z j z j [ z j 2 2 I c j (r) z j 2 2 I c j (r) ] } 2 .
j=1 M a j 2 ( z j z j ) 2 =minimum.
a j = ( z j z j ) 2 j=1 M ( z j z j ) 2 .
·[ I z (r) ϕ z (r) ]=k j=1 M s j ( I z j (r) I z j (r) ) ,
s j = ( z j z j ) j=1 M ( z j z j ) 2 .
z j =jΔz, z j =jΔz, j=1,2,...
s j = 3j M(M+1)(2M+1) , j=±1,±2,...,
ε n 2 (r)=2 σ 2 / j=1 M ( z j z j ) 2 ,
ε h 2 (r)= 1 4 { j=1 M z j z j j=1 M ( z j z j ) 2 [ z j 2 2 I c j (r) z j 2 2 I c j (r) ] } 2 .
ε h(lo) 2 ε h 2 (r) ε h(up) 2 ,
ε h(up) 2 = U 2 Z up 2 (M) /4 , Z up 2 (M)= [ j=1 M ( z j z j )( z j 2 + z j 2 ) ] 2 [ j=1 M ( z j z j ) 2 ] 2 , ε h(lo) 2 =0,
U= max r,c (| 2 I c (r)|), c[ z min , z max ], z min z j , z j z max ,j=1,2,...,M.
ε M+1 2 (r)< ε M 2 (r) ε h_(M+1) 2 (r) ε h_M 2 (r)< ε n_M 2 (r) ε n_(M+1) 2 (r),
ε h(up)_(M+1) 2 (r) ε h(up)_M 2 (r)< ε n_M 2 (r) ε n_(M+1) 2 (r).
{ z M+1 < z j+1 < z j , z M+1 | z ( j+1 ) |<| z j |, | z ( M+1 ) |<| z ( j+1 ) |, | z ( M+1 ) | z j+1 . (j=1,2,...,M1).
( z M+1 2 + z ( M+1 ) 2 )( z j+1 z ( j+1 ) )( z j+1 2 + z ( j+1 ) 2 )( z M+1 z ( M+1 ) ) = z M+1 z j+1 ( z M+1 z j+1 ) z M+1 z ( j+1 ) ( z M+1 + z ( j+1 ) ) + z ( M+1 ) z j+1 ( z ( M+1 ) + z j+1 ) z ( M+1 ) z ( j+1 ) ( z ( M+1 ) z ( j+1 ) )<0
Z up (M+1) Z up (M) = j=1 M+1 ( z j z j )( z j 2 + z j 2 ) j=1 M+1 ( z j z j ) 2 j=1 M ( z j z j )( z j 2 + z j 2 ) j=1 M ( z j z j ) 2 = j=1 M ( z M+1 z ( M+1 ) )( z j z j )[ ( z M+1 2 + z ( M+1 ) 2 )( z j z j )( z j 2 + z j 2 )( z M+1 z ( M+1 ) ) ] j=1 M ( z j z j ) 2 j=1 M+1 ( z j z j ) 2 <0.
{ z M+1 > z j+1 > z j , z M+1 | z ( j+1 ) |>| z j |, | z ( M+1 ) |>| z ( j+1 ) |, | z ( M+1 ) | z j+1 . (j=1,2,...,M1),
E 2 = x=1 X y=1 Y [ I ^ 0 (x,y) I 0 (x,y) ] 2 / x=1 X y=1 Y [ I 0 (x,y) ] 2 .
I 0 =2Im[ u 0 * (r) F 1 { F{ u 0 (r) }k (1 λ 2 g 2 ) 1/2 } ].
RMSE= x=1 X y=1 Y [ θ 0 (x,y) θ 0 (x,y)μ ] 2 /(X×Y) , μ= x=1 X y=1 Y [ θ 0 (x,y) θ 0 (x,y) ] /(X×Y)
u z (r)= F 1 { F{ u 0 (r) } H z (g) }, H z (g)=exp( jkz (1 λ 2 g 2 ) 1/2 ).
I z = z ( u z (r) u z * (r) )=2Re[ u z * (r) z u z (r) ]=2Re[ u z * (r) F 1 { F{ u 0 (r) }jk (1 λ 2 g 2 ) 1/2 H z (g) } ].
I 0 =2Im[ u 0 * (r) F 1 { F{ u 0 (r) }k (1 λ 2 g 2 ) 1/2 } ].

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