## The transport of intensity equation for optical path length recovery using partially coherent illumination |

Optics Express, Vol. 21, Issue 12, pp. 14430-14441 (2013)

http://dx.doi.org/10.1364/OE.21.014430

Acrobat PDF (1467 KB)

### Abstract

We investigate the measurement of a thin sample’s optical thickness using the transport of intensity equation (TIE) and demonstrate a version of the TIE, valid for partially coherent illumination, that allows the measurement of a sample’s optical path length by the removal of illumination effects.

© 2013 OSA

## 1. Introduction

*U*may be represented in terms of its spatial distribution of intensity

*I*, which is directly measurable by a detector, and its phase

*ϕ*, which is not. Phase retrieval is of interest, particularly in imaging systems, because the intensity image carries information about the attenuation of light through the sample, while phase carries information about the optical path length through the sample. In microscopy of unstained biological samples or wavefront characterization, for example, much of the useful information may be contained in phase. Techniques to quantitatively recover the phase of a coherent field are often based upon interferometry of the field of interest with a second field, rendering the phase differences between the two fields visible as an intensity modulation. An alternative set of techniques relies on the interference of the field with itself upon propagation. One such propagation–based method relies upon guessing the initial unknown phase, and uses repeated iterations of computational propagation between two (or more) planes and the enforcement of prior knowledge about the field in order to reduce the error in the estimated phase [1

1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982) [CrossRef] [PubMed] .

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

*z*axis. This field may be represented over space in the complex representation,

*U*: with Here,

**x**represents position transverse to the optical axis and

*I*and

*ϕ*are real–valued intensity and spatially–varying phase. The physics of paraxial propagation under this representation are governed by the paraxial wave equation: where

*D*Laplacian operator over

**x**.

*I*and phase

*ϕ*may be derived by applying Eq (3) to Eq. (2), and separating the real and imaginary parts. The cost of eliminating the complex–valued field

*U*is that propagation is described by a pair of coupled, nonlinear differential equations: where ∇

**is the 2D gradient operator over**

_{x}**x**, ⊗ denotes the dyadic product between two vectors, and the notation

**v**(

_{x}**x**,

*z*) = ∇

_{x}*ϕ*(

**x**,

*z*)/

*k*has been adopted for notational simplicity. This pair of equations represents a coupled set of continuity equations expressing the conservation of intensity

*I*and transverse flux density

*I*

**v**upon propagation, and were initially introduced in quantum mechanics [3

3. E. Madelung, “Quantentheorie in hydrodynamische Form,”Z. für Phys. **40**, 322–326 (1926) [CrossRef] .

4. D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev. **85**, 166–179 (1952) [CrossRef] .

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

*k*

**v**= ∇

_{x}*ϕ*, Eq. (4b) can be simplified to a phase rather than flux density transport equation,

*∂ϕ/∂z*, ([see, for example, Eq. (3) of Ref. [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**, 1932–1941 (1995) [CrossRef] .

**v**(

**x**,

*z*) = ∇

_{x}*ϕ*(

**x**,

*z*)/

*k*so that Eq. (4a) becomes the TIE: Equation (5) is useful because it relates the phase

*ϕ*to changes in intensity

*∂I/∂z*through a single, linear differential equation. The derivative along

*z*can be approximated by finite differences between intensity measurements at two or more closely spaced planes [7

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

10. C. Zuo, Q. Chen, Y. Yu, and A. Asundi, “Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter–theory and applications,” Opt. Express **21**, 5346–5362 (2013) [CrossRef] [PubMed] .

11. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature **384**, 335–338 (1996) [CrossRef] .

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

13. T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy **108**, 805–815 (2008) [CrossRef] [PubMed] .

14. 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**, 158–159 (2000) [CrossRef] [PubMed] .

*z*= 0, characterized through a transmission function where

*T*and

*ψ*are real–valued. Under the thin–sample approximation, the total phase accrued upon propagation through the object is equal to the product of the optial path length (OPL) through the object with the wave number of the illumination,

*ψ*=

*k*OPL. If the sample consists of a uniform material of known index of refraction

*n*, its thickness is given by OPL/

*n*. When the sample is illuminated coherently, the complex field immediately after passing through the sample is given by

*∂*|

*U*|

^{2}/

*∂z*can be estimated by defocusing the detector, and the total phase

*ϕ*(

**x**, 0) +

*ψ*(

**x**) can be retrieved from the TIE. The sample’s OPL can be recovered if

*ϕ*(

**x**, 0) is known. In this manuscript, we consider the case in which partially coherent light illuminates a thin sample and show that although a partially coherent field does not have a well–defined 2D phase in the sense of Eq. (2), it is possible to characterize the incident illumination such that a TIE–type measurement yields

*ψ*for a thin sample.

## 2. The TIE under illumination of any state of coherence

**F**and

*S*is insufficient to characterize

*W*. However, it is possible to obtain the optical thickness of samples using Eq. (9) under certain illumination conditions. Consider a field with cross–spectral density

*W*

_{inc}(

**x**

_{1},

**x**

_{2},

*z*) and spectral density

*S*

_{inc}(

**x**,

*z*) passing through a thin sample described by Eq. (6). Immediately after the sample, the cross–spectral density is given by

*τ*

^{*}(

**x**

_{1})

*τ*(

**x**

_{2})

*W*

_{inc}(

**x**

_{1},

**x**

_{2},

*z*), with spectral density

*S*

_{tot}=

*T*(

**x**)

*S*

_{inc}(

**x**,

*z*) and the subscript “inc” denotes properties of the incident illumination. The PC–TIE for the total spectral density in the detector plane in this case is The result in Eq. (12) assumes that the displacement of the detector along

*z*is infinitesimal. In reality, one is limited to estimating

*∂S/∂z*from finite displacements in

*z*, where the physics embodied by Eq. (7) can more accurately be represented by a pair of Fresnel propagation integrals. A more detailed derivation of the PC–TIE, examining the validitity of the TIE approximation used to reduce the propagation integrals to Eq. (12) is given in Appendix A.

**F**

_{inc}=

**0**,

*i.e.*the incident illumination is symmetric with respect to the optical axis, Eq. (12) simplifies to which has the same form as the coherent TIE [24

24. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. **49**, 6–10 (1984) [CrossRef] .

**F**

_{inc}=

**0**include the case of coherent, plane wave illumination of a sample (for which

*S*is also uniform), and also the case of imaging about the waist of a partially coherent Gaussian Schell–model beam or using symmetric Köhler illumination.

**· [**

_{x}*T*(

**x**)

**F**

_{inc}(

**x**)]. This characterizes the intensity change along the axis due to interaction of the transmissivity

*T*, of the sample and flux of the incident illumination. The term on the right–hand side of Eq. (12) represents the flux induced by the sample,

*T*∇

_{x}*ψ*/

*k*, carrying the spectral density of the illumination. If

**F**

_{inc}≠

**0**, then

*ψ*may still be recovered provided that

**F**

_{inc}and

*S*

_{inc}, properties of the illumination, are known. This may be achieved by first measuring

*∂S*

_{inc}/

*∂z*with no sample in place, to characterize the scalar phase of

**F**

_{inc}, and then solving Eq. (12). Since the PC–TIE is valid for illumination in any state of coherence, it also includes the case of the coherent limit through the substitution

**F**

_{inc}(

**x**,

*z*) =

*k*

^{−1}

*I*

_{inc}(

**x**,

*z*)∇

_{x}*ϕ*

_{inc}(

**x**,

*z*), where

*I*

_{inc}and

*ϕ*

_{inc}are the intensity and phase of the coherent incident illumination in the sample plane, respectively.

## 3. Simulation results

*R*= 250

*μ*m emitting monochromatic light with wavelength of 620 nm. The collimating lens is taken to have a focal length of

*f*= 5 cm. We use a 512×512 pixel detector with 2.2

*μ*m pixels. The field modulation mask consists of a transparency with five vertical bars of OPL of 0.3

*μ*m. The field is numerically propagated (using Fresnel propagation integrals) a distance of 200

*μ*m to the in–focus plane and to an additional pair of defocused image planes at ±100

*μ*m from this plane. This is performed with and without a sample in place. Although, strictly speaking, such a system does not obey the TIE approximation for all spatial frequencies present in the image, we used a test object for which the majority of spatial frequency content does satisfy the TIE approximation. We added minor Poisson noise (SNR≈ 45 dB) to each image, such that we did not have to worry about significant denoising of the resulting images (considerably more noise is present in the experimental results presented in the following section). In order to solve both the TIE and PC–TIE, an FFT–based Poisson solver was used for direct inversion of the linear equations [26

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

*S*

_{tot}=

*S*

_{tot}(

**x**,

*z*) −

*S*

_{tot}(

**x**, −

*z*) is illustrated in Fig. 2 (d). The bottom row illustrates reconstructed optical thickness using a variety of techniques. In Fig. 2(e), Eq. (5) is used to construct OPL from the image, which clearly combines information about the illumination and sample. Figure 2(f) is the reconstructed scalar phase of the illumination. Figures 2(g) and 2(h) illustrate two ways of compensating for this background. In Fig. 2(g), Eq. (14) was used, which clearly does not compensate well for the illumination, leaving artifacts in regions of high transmittance variation of the sample such as the feet. In Fig. 2(h), the the PC–TIE is employed, using the previously characterized background flux in Eq. (12). Because the PC–TIE includes a term compensating for the interaction of the illumination’s flux with the sample’s transmittance, this method more effectively removes artifacts from regions in which both of these quantities are non-negligible, such as the feet. The remaining discrepancies between Figs. 2(h) and 2(a) are likely due to noise as well as spatial frequency components for which the TIE approximation does not hold. Notice the presence of low–frequency noise in all reconstructed images, since we only used extremely weak Tikhonov regularization in the inversion process to reduce the noise.

## 4. Experimental results for a pure–phase object

*μ*m. We use the previously described Köhler illumination system to create a quasi–uniform spectral density distribution after the first lens, and then illuminate either one or two objects. The first object, an MIT logo, is used as the field modulation mask, while the second, the MIT beaver, is the sample being imaged. Both samples were etched onto a glass microscope slide (

*n*≈ 1.5) to a depth of approximately 100 nm. Since the samples are of known refractive index, we reconstruct thickness rather than OPL. After the sample, the light passes through a 4f system to produce an in-focus image at unity magnification on a detector. To provide a baseline measurement for comparison, measurements were first taken with only the phase sample in place and quasi–uniform illumination [

*S*(

**x**) ≈ constant,

**F**(

**x**) ≈

**0**]. In order to test the PC–TIE, Eq. (12), measurements were taken with only the field modulating mask in place, and with both sample and mask in place. For each case, an underfocused, overfocused and in–focus images were taken with axial displacement in steps of 250

*μ*m.

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

28. L. Tian, J. C. Petruccelli, and G. Barbastathis, “Nonlinear diffusion regularization for transport of intensity phase imaging,” Opt. Lett. **37**, 4131–4133 (2012) [CrossRef] [PubMed] .

*i.e.*is it nearly pure–phase, so that the term ∇

**· [**

_{x}*T*(

**x**)

**F**

_{inc}(

**x**,

*z*)] that differentiates Eqs. (12) and (14) is negligible.

## 5. Concluding remarks

## A. Derivation of the TIE from finite propagation distances

19. T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. **259**, 569–580 (2006) [CrossRef] .

*S*produced at a propagation distance

*z*due to a field with cross–spectral density

*W*described over an initial plane at

*z*= 0 is described by a pair of Fresnel propagation interals acting on the initial

*W*, We are interested in is the case of partially coherent illumination of a thin sample, taken without loss of generality to be located at

*z*= 0, such that the cross–spectral density immediately after the sample is given by where

*W*

_{inc}represents the incident field’s cross–spectral density at the sample plane, and

**x′**→

*λz*

**u′**to simplify the results, the spectral density after propagation can be written as

*λz*

**u′**can be well–approximated by first–order Taylor expansions in

*λz*

**u′**, in which case

**F**

_{inc}, is defined in terms of

*W*

_{inc}by Eq. (10). Factors of

**u′**in the integrand may be removed by expressing them as gradient operators on the exponential, since Both integrations can then be performed, resulting in a forward-difference approximation of the PC–TIE

**x**to spatial frequency

**u**, and performing the integration over

**u′**in closed form, we can examine the validity of the TIE approximation in terms of the spatial frequency content of the image formed at the detector plane,

*Ŝ*is the 2D Fourier transform of

*S*measured by the defocused detector from transverse position

**x**to spatial frequency

**u**. Equation (24) results again from a first–order Taylor expansion with respect to

*λz*

**u**. In theory, one could always pick displacement

*z*small enough so that this expansion holds for all spatial frequencies present in the system. However, notice that the OPL of the sample is present through the Taylor expansion of

*ψ*in only one term on the right–hand side of Eq. (22), while the expansion of

*W*

_{inc}is present through

**F**

_{inc}in a separate term. If the limit on

*z*for the expansion of

*W*

_{inc}is significantly more restrictive than that of

*ψ*, the term containing

**F**

_{inc}will likely be significantly larger than that containing

*ψ*, making the OPL of the sample difficult to detect, especially in the presence of noise. If a larger propagation distance is used, the TIE reconstruction will no longer be accurate for higher spatial frequencies since

*W*is not accurately represented by this expansion. To accurately reconstruct those spatial frequencies in the sample’s OPL, a more accurate expression for propagated intensity based on Eq. (25) must be employed.

## Acknowledgments

## References and links

1. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

2. | M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A |

3. | E. Madelung, “Quantentheorie in hydrodynamische Form,”Z. für Phys. |

4. | D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev. |

5. | D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. |

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 |

7. | K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. , |

8. | T. C. Petersen, V. Keast, K. Johnson, and S. Duvall, “TEM based phase retrieval of p-n junction wafers using the transport of intensity equation,” Phil. Mag. |

9. | L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express |

10. | C. Zuo, Q. Chen, Y. Yu, and A. Asundi, “Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter–theory and applications,” Opt. Express |

11. | S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature |

12. | K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. |

13. | T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy |

14. | 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 |

15. | L. Mandel and E. Wolf, |

16. | T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. |

17. | A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. |

18. | A. M. Zysk, R. W. Schoonover, P. S. Carney, and M. A. Anastasio, “Transport of intensity and spectrum for partially coherent fields,” Opt. Lett. |

19. | T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. |

20. | K. A. Nugent, “Wave field determination using three–dimensional intensity information,” Phys. Rev. Lett. |

21. | M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. |

22. | L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express |

23. | L. Mandel and E. Wolf, |

24. | N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. |

25. | L. Mandel and E. Wolf, |

26. | T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. |

27. | D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. of Microscopy |

28. | L. Tian, J. C. Petruccelli, and G. Barbastathis, “Nonlinear diffusion regularization for transport of intensity phase imaging,” Opt. Lett. |

29. | J. Petruccelli, L. Tian, and G. Barbastathis, “Source diversity for transport of intensity phase imaging,” to appear in to |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(100.5070) Image processing : Phase retrieval

(110.4980) Imaging systems : Partial coherence in imaging

(350.5030) Other areas of optics : Phase

**ToC Category:**

Image Processing

**History**

Original Manuscript: April 10, 2013

Revised Manuscript: May 24, 2013

Manuscript Accepted: May 27, 2013

Published: June 10, 2013

**Virtual Issues**

July 12, 2013 *Spotlight on Optics*

**Citation**

Jonathan C. Petruccelli, Lei Tian, and George Barbastathis, "The transport of intensity equation for optical path length recovery using partially coherent illumination," Opt. Express **21**, 14430-14441 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-14430

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### References

- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21, 2758–2769 (1982). [CrossRef] [PubMed]
- M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A73, 1434–1441 (1983). [CrossRef]
- E. Madelung, “Quantentheorie in hydrodynamische Form,”Z. für Phys.40, 322–326 (1926). [CrossRef]
- D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev.85, 166–179 (1952). [CrossRef]
- D. Paganin and K. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80, 2586–2589 (1998). [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. A12, 1932–1941 (1995). [CrossRef]
- K. Ishizuka and B. Allman, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc., 54191–197 (2005). [CrossRef]
- T. C. Petersen, V. Keast, K. Johnson, and S. Duvall, “TEM based phase retrieval of p-n junction wafers using the transport of intensity equation,” Phil. Mag.87, 3565–3578 (2007). [CrossRef]
- L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express18, 12552–12561 (2010). [CrossRef] [PubMed]
- C. Zuo, Q. Chen, Y. Yu, and A. Asundi, “Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter–theory and applications,” Opt. Express21, 5346–5362 (2013). [CrossRef] [PubMed]
- S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996). [CrossRef]
- K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett.77, 2961–2964 (1996). [CrossRef] [PubMed]
- T. C. Petersen, V. J. Keast, and D. M. Paganin, “Quantitative TEM-based phase retrieval of MgO nano-cubes using the transport of intensity equation,” Ultramicroscopy108, 805–815 (2008). [CrossRef] [PubMed]
- 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, 158–159 (2000). [CrossRef] [PubMed]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). Ch. 4. [CrossRef]
- T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93, 068103 (2004). [CrossRef] [PubMed]
- A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett.95, 043904 (2005). [CrossRef] [PubMed]
- A. M. Zysk, R. W. Schoonover, P. S. Carney, and M. A. Anastasio, “Transport of intensity and spectrum for partially coherent fields,” Opt. Lett.35, 2239–2241 (2010). [CrossRef] [PubMed]
- T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun.259, 569–580 (2006). [CrossRef]
- K. A. Nugent, “Wave field determination using three–dimensional intensity information,” Phys. Rev. Lett.68, 2261–2264 (1992). [CrossRef] [PubMed]
- M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994). [CrossRef] [PubMed]
- L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express20, 8296–8308 (2012). [CrossRef] [PubMed]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), Sec. 5.7.1. [CrossRef]
- N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun.49, 6–10 (1984). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995), pp. 188–193.
- T. Gureyev and K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun.133, 339–346 (1997). [CrossRef]
- D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. of Microscopy214, 51–61 (2004). [CrossRef]
- L. Tian, J. C. Petruccelli, and G. Barbastathis, “Nonlinear diffusion regularization for transport of intensity phase imaging,” Opt. Lett.37, 4131–4133 (2012). [CrossRef] [PubMed]
- J. Petruccelli, L. Tian, and G. Barbastathis, “Source diversity for transport of intensity phase imaging,” to appear in to Computational Optical Sensing and Imaging (Optical Society of America, 2013), June2013, paper CTu2C.3.

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