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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 21 — Oct. 10, 2011
  • pp: 20244–20250
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Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes

Bindang Xue, Shiling Zheng, Linyan Cui, Xiangzhi Bai, and Fugen Zhou  »View Author Affiliations


Optics Express, Vol. 19, Issue 21, pp. 20244-20250 (2011)
http://dx.doi.org/10.1364/OE.19.020244


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Abstract

A method based on the transport of intensity equation (TIE) for phase retrieval is presented, which can retrieve the optical phase from intensity measurements in multiple unequally-spaced planes in the near-field region. In this method, the intensity derivative in the TIE is represented by a linear combination of intensity measurements, and the coefficient of the combination can be expressed by explicitly analytical form related to the defocused distances. The proposed formula is a generalization of the TIE with high order intensity derivatives. The numerical experiments demonstrate that the proposed method can improve the accuracy of phase retrieval with higher-order intensity derivatives and is more convenient for practical application.

© 2011 OSA

1. Introduction

The TIE can relate the object-plane phase to the axial derivative of the intensity distribution in the Fresnel region [1

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

], and if the intensity distribution has no zeros, the TIE has a unique solution (within an additive constant) for coherent and partially coherent fields [2

2. 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]

]. Many effective numerical methods have been proposed for solving the TIE. Therefore, the TIE has been widely used for the recovery of optical phase from multiple intensity measurements, and phase recovery has been an essential imaging technique in many fields, e.g., optics [3

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

], electron [4

4. K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54(3), 191–197 (2005). [CrossRef] [PubMed]

], neutrons [5

5. B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000). [CrossRef] [PubMed]

] and x-ray microscopy [6

6. 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]

].

In this study, a generalization of the TIE is proposed, which uses multiple unevenly sampled intensity measurements for phase retrieval. Based on the higher-order Taylor expansions of the measured intensities, the axial intensity derivative in the TIE is expressed by a linear combination of the multiple unevenly sampled intensity measurements. Numerical simulations are conducted to test the method.

2. Unequally-spaced intensity measurements for phase retrieval

An optical plane wave uz(r) in the x-y plane propagating along the Z axis can be expressed as:
uz(r)=Iz1/2(r)exp(iϕz(r)),
(1)
where r=(x,y) is the spatial coordinates in the x-y plane, perpendicular to the Z axis. Iz(r) and ϕz(r) are respectively the intensity and the phase distributions. The focal plane is located in the z=0plane. With the paraxial approximation, the transport of intensity equation (TIE) states that [1

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

]:
·[Iz(r)ϕz(r)]=kIz(r)/z,
(2)
where k=2π/λ is the wave number, λ is the wavelength, and=(x,y) denotes the two-dimensional gradient operator in the x-y plane.

Assume that intensities measured in z={0,z1,z2,zM} planes are available for solving the TIE. ExpandingIz(r) with Taylor series at z=0as:
Iz(r)I0(r)=zI0(r)+n=2znnI0(r)/n!,
(3)
where n! denotes the factorial of n(n=2,3,...,M). nI0(r) is the nthorder axial intensity derivative evaluated at the point z=0. In the traditional TIE, the second term on the right side of Eq. (3) is ignored, which produce a linear Taylor approximation respect toz. To improve the accuracy, we should estimate and compensate the higher-order intensity derivative in Eq. (3). This can be done by constructing a linear system from Eq. (3) as:

[z1z12z1Mz2z22z2MzMzM2zMM][I0(r)2I0(r)/2!MI0(r)/M!]=[Iz1(r)I0(r)Iz2(r)I0(r)IzM(r)I0(r)].
(4)

When the intensities are equally-spaced and symmetrically distributed relative to the focal plane, Eq. (4) is equivalent to the equation in [15

15. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010). [CrossRef] [PubMed]

] (Eq. (7) in [15

15. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010). [CrossRef] [PubMed]

]). WhenM=4, Eq. (4) is reduced to the Eq. (7) in [16

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

].

To solve Eq. (4), we denote the coefficient matrix in the left hand of it as A, and the inverse matrix of Aas B. Since A is an Mthorder Vandermonde matrix, the specific expression of Bis [17

17. A. E. Knuth, The Art of Computer Programming: Volume 1: Fundamental Algorithms (Addison-Wesley Professional, 1997), pp. 38.

]:

B=[b]M,bij=1k1<...<kMiMk1,...,kMij(1)i1zk1...zkMi/[zj1kMkj(zkzj)].
(5)

The solution to Eq. (4) can be obtained by inverting the matrixA:
[I0(r)2I0(r)/2!MI0(r)/M!]=B[Iz1(r)I0(r)Iz2(r)I0(r)IzM(r)I0(r)].
(6)
Substituting Eq. (6) into Eq. (3), the Mthorder approximation to the intensity derivatives can be expressed with the form:
I0(r)=m=1Mcm[Izm(r)I0(r)].
(7)
Where the weight coefficients have the form
cm=b1j=1k1<...<kM1Mk1,...,kM1mzk1...zkM1/[zm1kMkm(zkzm)],m=j.
(8)
Substituting Eq. (7) into Eq. (2), we can obtain a new generalization of the TIE with high order intensity derivatives:

·[I0(r)ϕ0(r)]=km=1Mcm[Izm(r)I0(r)].
(9)

In this study, the forward formula is used to estimate the intensity derivative. Table 1

Table 1. Comparison of the weight coefficients for the given order of accuracy.

table-icon
View This Table
shows the difference between the weight coefficients cm,(m=1,...,4) with unequally-spaced planes and that with equally-spaced planes [15

15. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010). [CrossRef] [PubMed]

]. If the intensity measurement planes are equally-spaced, i.e.zm=mΔz, the proposed expressions of the coefficients are reduced to that in [15

15. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010). [CrossRef] [PubMed]

].

3. Simulations

Numerical simulation experiments are conducted to demonstrate the feasibility of the proposed phase retrieval approach using intensity measurements in multiple unequally-spaced planes. A simulated optical fiber is chosen as the true phase profile (see Fig. 1
Fig. 1 Simulated phase in the focal plane.
).The dimensions of the image are 512 × 512 pixels with an assigned pixel size of 4μm. The phase value is between 0 and 0.5π radians. The radiation wavelength is chosen to be 550nm. The intensity distribution in the input plane is uniform. The root-mean-square-error (RMSE) is employed to quantify the accuracy of the retrieved phase:
RMSE=x,y(θ0(x,y)θ0(x,y)a)2/(X×Y)a=x,y(θ0(x,y)θ0(x,y))/(X×Y),
(10)
where θ0(x,y)and θ0(x,y) denote the calculated phase distribution and the true phase distribution at pixel(x,y) respectively. X×Y is the sampled pixels of the phase image.

The diffraction images are calculated by the angular spectrum method [9

9. 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]

,18

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

] as follows:
Izm=uzm(r)uzm*(r),uzm(r)=F1{F{u0(r)}Hzm(g)},Hzm(g)=exp(ikzm(1λ2g2)1/2).
(11)
Where F{} and F1{} respectively denote the Fourier and inverse Fourier transforms. g=(gx,gy)is the spatial frequency, conjugate to r. The superscript * denotes conjugation. zm(m=1,...,M) denotes the defocused distance. Figure 2
Fig. 2 An example of the simulated intensity focal stack. The defocused distances for the intensities are respectively (from left to right): 0,4.02μm,8.03μm,12.00μm,16.00μm,20.03μm,24.07μm.
shows an example of the simulated diffraction images, which can be used to solve the TIE for phase retrieval. It should be mentioned that the diffraction images are sampled in unequally-spaced planes.

We carry out the phase retrieval simulations by solving the TIE with the FFT method [8

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

]. The first simulation assumes that the intensities are measured in the planeszm=mΔz+εm, (m=1,2,...,M). εm is a random disturbance between 0 and 0.1μm, which denotes the measurement errors of the defocused distance. This numerical experiment includes the following two cases:

Case1: Ignoringεm, and the equally-spaced formula [15

15. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010). [CrossRef] [PubMed]

] is used to solve the TIE.

Case2: Consideringεm, and the unequally-spaced formula is used to solve the TIE.

Figure 3
Fig. 3 Recovered phases (radians) and phase errors (radians) for case 2: The first row are listed the recovered phases forM=1,2,6. The second row are listed the corresponding phase errors relative to the true phase.
shows the recovered phases and the corresponding phase errors for case 2. In this numerical experiment, Δz=4μmandM=6, the defocused distances for the six intensities are respectively assigned to be 4.02,8.03,12.00,16.00,20.03μm and24.07μm. The visual impression of the reconstructed phases is in good agreement with the true phase, and from the phase error images we can see that the visual differences between the reconstructed phases and the true phase tend to be smaller with the increase of the number of intensity measurements.

In the other simulation experiment, the intensities are measured in Mrandom planes between[z1,zM]. Here we choose z1=4μm andzM=16μm. The intervals between the adjacent defocused planes are randomly chosen. In this simulation, the defocused distances of the six intensities are respectively set to be 4,7,9,13,15μmand16μm. We name the second simulation as case 3.

Figure 4
Fig. 4 RMSEs as a function of the number of intensity measurements with a logarithmic scale. (a) case 1 and 2; (b) case 2 and case 3.
shows the RMSE as a function of M on a logarithmic scale. Figure 4(a) shows that the defocused distance errors can effect the oscillation of the RMSE curve, which corresponds to case 1. In this case, εmis ignored and the equally-spaced formula [15

15. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010). [CrossRef] [PubMed]

] is used to solve the TIE. Because the defocused distance errors are ignored, the accuracy of phase reconstruction cannot be improved by including higher order intensity derivatives. In case 2, we have considered the measurement errors, and the unequally-spaced formulas are used for phase retrieval. Consequently, the accuracy of the retrieved phases can be improved with the increase of the number of intensity measurements. The difference of the RMSE between case 2 and case 3 shown in Fig. 4(b) is due to the defocused distances. Since the noise is not considered, the smaller the defocused distances, the lower the RMSE are. Therefore, the RMSE in case 2 is larger than that in case 3.

4. Conclusions

We demonstrate a phase retrieval method based on the TIE from multiple unequally-spaced intensities, which helps relax the limitation of equally-spaced intensity measurements. Therefore, the proposed phase imaging method is more convenient for practical application. The numerical simulation results illustrate that the measurement error of the defocused distances effect the accuracy of phase retrieval and the proposed method can eliminate this influence. It can also improve the accuracy of the phase reconstruction with higher-order intensity derivative components just as the method with equally-spaced intensity measurements. The method has a potential use in the optical measurement system [13

13. J. Frank, G. Wernicke, J. Matrisch, S. Wette, J. Beneke, and S. Altmeyer, “Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique,” Proc. SPIE 8082, 80820N, 80820N-9 (2011). [CrossRef]

] for retrieving the phase with high accuracy in real time. However, the method presented in this paper does not consider the noise effect. More work will be done to simultaneously minimize the effects of the noise and the higher-order intensity derivatives on the retrieved phase from multiple unequally-spaced intensity measurements in the future.

Acknowledgments

This work is partly supported by a grant from the National Natural Science Foundation of China (No.60502018, No.61171005), the Innovation Fund and the Scholarship Award of BUAA for PhD Students, the Fundamental Research Fund for the Central Universities and the Aeronautical Special Fund of AVIC (No.CXY2010BH02).

References and links

1.

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

2.

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]

3.

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

4.

K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo) 54(3), 191–197 (2005). [CrossRef] [PubMed]

5.

B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature 408(6809), 158–159 (2000). [CrossRef] [PubMed]

6.

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]

7.

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]

8.

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

9.

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]

10.

W. Xiao, M. Heng, and Z. Dazun, “Phase retrieval based on intensity transport equation,” Acta Opt. Sin. 27, 2117–2122 (2007).

11.

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]

12.

S. V. Pinhasi, R. Alimi, L. Perelmutter, and S. Eliezer, “Topography retrieval using different solutions of the transport intensity equation,” J. Opt. Soc. Am. A 27(10), 2285–2292 (2010). [CrossRef] [PubMed]

13.

J. Frank, G. Wernicke, J. Matrisch, S. Wette, J. Beneke, and S. Altmeyer, “Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique,” Proc. SPIE 8082, 80820N, 80820N-9 (2011). [CrossRef]

14.

M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. 46(33), 7978–7981 (2007). [CrossRef] [PubMed]

15.

L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010). [CrossRef] [PubMed]

16.

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

17.

A. E. Knuth, The Art of Computer Programming: Volume 1: Fundamental Algorithms (Addison-Wesley Professional, 1997), pp. 38.

18.

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

OCIS Codes
(100.3010) Image processing : Image reconstruction techniques
(100.5070) Image processing : Phase retrieval

ToC Category:
Image Processing

History
Original Manuscript: July 25, 2011
Revised Manuscript: September 16, 2011
Manuscript Accepted: September 16, 2011
Published: September 30, 2011

Citation
Bindang Xue, Shiling Zheng, Linyan Cui, Xiangzhi Bai, and Fugen Zhou, "Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes," Opt. Express 19, 20244-20250 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20244


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References

  1. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am.73(11), 1434–1441 (1983). [CrossRef]
  2. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A12(9), 1942–1946 (1995). [CrossRef]
  3. A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett.23(11), 817–819 (1998). [CrossRef] [PubMed]
  4. K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. (Tokyo)54(3), 191–197 (2005). [CrossRef] [PubMed]
  5. B. E. Allman, P. J. McMahon, K. A. Nugent, D. Paganin, D. L. Jacobson, M. Arif, and S. A. Werner, “Phase radiography with neutrons,” Nature408(6809), 158–159 (2000). [CrossRef] [PubMed]
  6. 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]
  7. 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]
  8. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80(12), 2586–2589 (1998). [CrossRef]
  9. 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]
  10. W. Xiao, M. Heng, and Z. Dazun, “Phase retrieval based on intensity transport equation,” Acta Opt. Sin.27, 2117–2122 (2007).
  11. 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]
  12. S. V. Pinhasi, R. Alimi, L. Perelmutter, and S. Eliezer, “Topography retrieval using different solutions of the transport intensity equation,” J. Opt. Soc. Am. A27(10), 2285–2292 (2010). [CrossRef] [PubMed]
  13. J. Frank, G. Wernicke, J. Matrisch, S. Wette, J. Beneke, and S. Altmeyer, “Quantitative determination of the optical properties of phase objects by using a real-time phase retrieval technique,” Proc. SPIE8082, 80820N, 80820N-9 (2011). [CrossRef]
  14. M. Soto and E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt.46(33), 7978–7981 (2007). [CrossRef] [PubMed]
  15. L. Waller, L. Tian, and G. Barbastathis, “Transport of Intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express18(12), 12552–12561 (2010). [CrossRef] [PubMed]
  16. W. X. Cong and G. Wang, “Higher-order phase shift reconstruction approach,” Med. Phys.37(10), 5238–5242 (2010). [CrossRef] [PubMed]
  17. A. E. Knuth, The Art of Computer Programming: Volume 1: Fundamental Algorithms (Addison-Wesley Professional, 1997), pp. 38.
  18. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), pp. 55–61.

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