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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 7 — Mar. 26, 2012
  • pp: 8186–8191
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Method for estimating the axial intensity derivative in the TIE with higher order intensity derivatives and noise suppression

Rui Bie, Xiu-Hua Yuan, Ming Zhao, and Li Zhang  »View Author Affiliations


Optics Express, Vol. 20, Issue 7, pp. 8186-8191 (2012)
http://dx.doi.org/10.1364/OE.20.008186


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Abstract

It is an effective scheme to the phase retrieval for axial intensity derivative computing. In this paper, we demonstrate a method for estimating the axial intensity derivative and improving the calculation accuracy in the transport of intensity equation (TIE) from multiple intensity measurements. The method takes both the higher-order intensity derivatives and the noise into account, and minimizes the impact of detecting noise. The simulation results demonstrate that the proposed method can effectively reduce the error of intensity derivative computing.

© 2012 OSA

1. Introduction

Teague introduced the transport of intensity equation (TIE) which relate the object-plane to the axial derivative of the intensity distribution,I(x,y,z)/z, in the Fresnel region to us, and if the intensity distribution has no zeros, the TIE has a unique solution (with an additive arbitrary constant). Many effective methods such as Fast Fourier Transform (FFT) [2

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

, 3

3. 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]

], the multigrid (MG) [4

4. 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]

], the Zernike polynomial expansion methods [5

5. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996). [CrossRef]

, 6

6. 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. A 12(9), 1932–1942 (1995). [CrossRef]

], and the Green’s function based method [1

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

] have been proposed for solving it. And all these methods require two images that are defocused from each other in order for the axial derivation of intensity derivative. However, there are tradeoffs among the amount of defocus, the accuracy of the result and noise considerations.

In recent years, some algorithms for derivation of intensity derivative from intensity measurements in multiple planes are proposed: In 2007, Marcos Soto [7

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

] evaluated the variance of the detecting noise and the axial second-order intensity derivative with multi-plane intensity distribution, based on which a method minimizing the error is proposed; and in 2010 Laura Waller [8

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

] of MIT took the impact of higher order derivative into account and described it with a matrix transformation; a year later, Bindang Xue [9

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

, 10

10. S. L. Zheng, B. D. Xue, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express 20(2), 972–985 (2012). [CrossRef] [PubMed]

] of Beihang University in China extended Laura Waller’s work to the situation with intensity distribution measured in unequally-spaced planes.

This paper proposed a novel method to evaluate intensity derivative, which takes both the detecting noise and higher order derivative into consideration. Based on Taylor expansions of the measured intensities, the axial intensity derivative in the TIE is expressed by a linear combination of the multiple intensity measurements, which is used to estimate the difference from the actual value. Numerical simulations are conducted to test it.

2. Multiple intensity measurements for phase retrieval

An optical plane wave u(r) on the x-y plane can be described by optical intensity and phase as
u(r)=[I(r)]1/2exp[iϕ(r)],
(1)
where r=(x,y) denotes the coordinate in the x-y plane, perpendicular to the z axis, I(r) and ϕ(r) are the intensity and phase distribution in the plane respectively. In this paper, the focal plane is assumed to be located atz=0.

Under the paraxial approximation, the transport of intensity equation (TIE) can be derived as
kIz=(Iϕ),
(2)
where k=2π/λ is the wave number, and λ denotes the wavelength, ={x,y} is the two dimensional gradient operator in the x-y plane.

Assume the intensity distribution of m x-y planes located at z1,z2,zm is available, based on the Maclaurin formula, I(zi) (ignore the x-y dimension) can be expanded as
I(zi)=I(0)+I(0)zzi+k=2nzikk!kI(0)zk+zin+1(n+1)!n+1I(ξi)zn+1,
(3)
where kzkdenotes the kth order derivative, and ξi is some number located between 0 and zi. As a result, we get m equations. Multiply these equations with a number ai separately and sum their left and right hand side respectively, we can get the following expression

i=1maiI(zi)=I(0)i=1mai+I(0)zi=1maizi+k=2n1k!kI(0)zki=1maizik+1(n+1)!i=1maizin+1n+1I(ξi)zn+1.
(4)

Let
i=1mai=0,i=1maizi=1,i=1maizik=0k=2,3...n,
(5)
we can derive the derivative of intensity

I(0)z=i=1maiI(zi)1(n+1)!i=1maizin+1n+1I(ξi)zn+1.
(6)

Ignoring the second term on the right hand of Eq. (6),the intensity derivative can be calculated in practical system like

[zI]c=i=1maiI(zi).
(7)

If the detecting noise is taken into consideration, the difference from Eq. (6) is
[zI]cI(0)z=i=1maiN(zi)+1(n+1)!i=1maizin+1n+1I(ξi)zn+1,
(8)
where N(zi) denotes the intensity noise at zi.Under the assumption of noise statistic independence, the mean square can be written as
ε2=D(N)+D(ξ),
(9)
where
D(N)=σ2i=1mai2,
(10)
D(ξ)=[1(n+1)!i=1maizin+1n+1I(ξi)zn+1]2,
(11)
and σ2denotes the variance of the noise. Because of the uncertainty of high-order item, we only evaluate the impact of detecting noise i.e. Equation (10). Our mission is to minimize it under the condition of Eq. (5).

Generally, Eq. (5) has another description form

[1111z1z2z3zmz12z22z32zm2z1nz2nz3nzmn]·[a1a2a3am]=[0100].
(12)

Obviously, the first matrix of left hand side called transformation matrix in this paper is a Vandermonde matrix.

  • (1) Whenn+1>m, Eq. (12) are an overdetermined equations. Under this condition, it pointless to evaluate the nth order derivative using intensity data of m planes.
  • (2) Whenn+1=m, it is the case that Bindang Xue described, the coefficient a1~am can be calculated definitely from Eq. (12), when the detecting noise cannot be suppressed.
  • (3) Whenn+1<m, Eq. (12) are an underdetermined equations, it is necessary to lay our attention on this situation which has not been considered.

We can divide the matrix of Eq. (12) as
[AB]·[CD]=E,
(13)
where, B consists of the last n + 1 columns of the transformation matrix, which is nonsingular. AndC, D is divided to match A and B.

Then D can be written as

D=B-1EB-1AC.
(14)

As a result,

D(N)=σ2[i=1n+1ai2+j=1mn1(fjGjC)2].
(15)

To minimize it,
akD(N)=σ2[2akj=1n+12(fjGjC)gjk]=0k=1,2(mn1),
(16)
where fj denotes the jth element of the vector F=B-1E, and Gj is jth row vector of matrix G=B-1A. Obviously, Eq. (16) is equivalent to

(GTG+I)C=GTF.
(17)

Combining with Eq. (13), we get

[GTG+I0AB]·[CD]=[GTFE].
(18)

In Eq. (18), GTG+I and B are both nonsingular matrix, so the coefficient a1~amcan be derived easily.

3. Simulations

We take the “Gauss Beam”, a widely used beam mode, to check the validity of the method described above.

That is the 0th order Gauss mode
u00(x,y,z)=cw(z)exp(r2w2(z))exp{i[k(z+r22R)arctan(zf)]},
(19)
where

k=2πλr2=x2+y2w(z)=w01+(zf)2R=R(z)=z[1+(fz)2]f=πw02λw0=λfπ.
(20)

As a result, the intensity distribution of the axis can be written as
I(z)=c2w2(z),
(21)
where

w0=0.005m,λ=1550nm,f=πw02λ=50.6708m,c=0.01.
(22)

The simulation error of Bindang Xue’s method (we call it high order derivative suppression method here) and the algorithm with noise suppression proposed in this paper are both shown in Fig. 1
Fig. 1 Axial intensity derivative error computed by noise suppression (described in this paper) and high order derivative suppression method (proposed by Bindang Xue) for different sampling number m: Each point is the average of 100 results. The blue lines show the results derived by noise suppression method for n = 4. (a)~(d) denotes the results for different sampling intervalsΔz.
.

When sampling interval Δzis large, the error raised by high order derivative impacts the final result significantly. Bindang Xue’s method plays a good role in this situation as shown in Fig. 1(a). But as it decreases, the influence of high order derivative vanishes gradually, while the detecting noise takes the dominated place step by step. As described in Fig. 1(a)1(d) the performance of noise suppression method improves significantly.

4. Conclusions

This paper proposed a novel method to derive the intensity derivative, which extends the work of Marcos Soto etc [7

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

10

10. S. L. Zheng, B. D. Xue, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express 20(2), 972–985 (2012). [CrossRef] [PubMed]

]. Including higher order intensity derivatives and detecting noise improves the accuracy of intensity derivative computing, which leads to enhancement of the algorithm performance based on TIE. By adopting correction for nonlinearities (higher order derivatives) and noise suppression, large stacks of data spanning longer distances can be use to estimate the derivative more precisely.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No.61077058).

References and links

1.

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

2.

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

3.

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]

4.

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]

5.

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996). [CrossRef]

6.

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. A 12(9), 1932–1942 (1995). [CrossRef]

7.

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

8.

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

9.

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

10.

S. L. Zheng, B. D. Xue, W. F. Xue, X. Z. Bai, and F. G. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express 20(2), 972–985 (2012). [CrossRef] [PubMed]

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

ToC Category:
Image Processing

History
Original Manuscript: February 17, 2012
Revised Manuscript: March 17, 2012
Manuscript Accepted: March 19, 2012
Published: March 23, 2012

Citation
Rui Bie, Xiu-Hua Yuan, Ming Zhao, and Li Zhang, "Method for estimating the axial intensity derivative in the TIE with higher order intensity derivatives and noise suppression," Opt. Express 20, 8186-8191 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-8186


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References

  1. M. Reed Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]
  2. D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998). [CrossRef]
  3. T. E. Gureyev, K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133(1-6), 339–346 (1997). [CrossRef]
  4. L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199(1-4), 65–75 (2001). [CrossRef]
  5. T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996). [CrossRef]
  6. T. E. Gureyev, A. Roberts, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12(9), 1932–1942 (1995). [CrossRef]
  7. M. Soto, E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. 46(33), 7978–7981 (2007). [CrossRef] [PubMed]
  8. L. Waller, L. Tian, G. Barbastathis, “Transport of intensity imaging with higher order derivatives,” Opt. Express 18(12), 12552–12561 (2010). [CrossRef] [PubMed]
  9. B. D. Xue, S. L. Zheng, L. Y. Cui, X. Z. Bai, F. G. Zhou, “Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes,” Opt. Express 19(21), 20244–20250 (2011). [CrossRef] [PubMed]
  10. S. L. Zheng, B. D. Xue, W. F. Xue, X. Z. Bai, F. G. Zhou, “Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes,” Opt. Express 20(2), 972–985 (2012). [CrossRef] [PubMed]

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