## Imaging through turbid media based on wave transport model approach

Optics Express, Vol. 16, Issue 26, pp. 21389-21400 (2008)

http://dx.doi.org/10.1364/OE.16.021389

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### Abstract

Here a transport model is used to simulate amplitude-only imaging and intensity-based quantitative phase imaging in a turbid medium. We derive an optical transfer function for propagation through a scattering medium. We also show that, as expected, scattering leads to a degradation in the spatial resolution in both forms of imaging, while the magnitude of the phase retrieved using a solution of the transport-of-intensity equation decreases as the optical density of the scattering medium increases.

© 2008 Optical Society of America

## 1. Introduction

1. S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. **42**, 841–853 (1997). [CrossRef] [PubMed]

5. H. W. Lewis, “Multiple scattering in an infinite medium,” Phys. Rev. **78**, 526–529 (1950). [CrossRef]

6. C.-C. Cheng and M. G. Raymer, “Propagation of transverse optical coherence in random multiple scattering media,” Phys. Rev. A **62**, 023811 (2000). [CrossRef]

7. A. Wax and J. E. Thomas, “Measurement of smoothed Wigner phase-space distribution for small-angle scattering in a turbid medium,” J. Opt. Soc. Am. A **15**, 1896–1908 (1998). [CrossRef]

8. F. Dubois, L. Joannes, and J.-C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. **38**, 7085–7094 (1999). [CrossRef]

9. F. Dubois, M.-L. N. Requena, C. Minetti, O. Monnom, and E. Istasse, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. **43**, 1131–1139 (2004). [CrossRef] [PubMed]

10. C.-C. Cheng and M. G. Raymer, “Long range saturation of spatial decoherence in wave-field transport in random multiple scattering media,” Phys. Rev. Lett. **82**, 4807–4810 (1999). [CrossRef]

11. F. Dubois, M.-L. N. Requena, and C. Minetti, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. **43**, 1131–1139 (2004). [CrossRef] [PubMed]

12. N. A. Beaudry and T. D. Milster, “Effects of object roughness on partially coherent image formation,” Opt. Lett. **25**, 454–456 (2000). [CrossRef]

13. D. M. Marks, R. A. Stack, and D. J. Brady, “Astigmatic coherence sensor for digital imaging,” Opt. Lett. **25**, 1726–1728 (2000). [CrossRef]

14. A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express **11**, 2303–2314 (2003). [CrossRef] [PubMed]

15. M. Alrubaiee, M. Xu, S. K. Gayen, M. Brito, and R. R. Alfano, “Three-dimensional optical tomographic imaging of scattering objects in tissue-simulating turbid media using independent component analysis,” Appl. Phys. Lett. **87**, 19112 (2005). [CrossRef]

16. E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. **206**, 194–203 (2002). [CrossRef] [PubMed]

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

## 2. The correlation functions

*G*(

*r*⃑

_{1},

*r*⃑

_{2}), is defined [18] as:

*r*⃑

_{1}and

*r*⃑

_{2}are the coordinates of two points in the field in the plane z. Another correlation function called the spatial coherence function,

*J*(

*x*⃑,

*s*⃑,

*z*⃑), is defined in terms of the mean,

*x*=⃑(

*r*⃑

_{1}+

*r*⃑

_{2})/2 and the difference,

*s*⃑=

*r*⃑

_{1}-

*r*⃑

_{2}, coordinates [18] as:

19. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A **3**, 1227–1238 (1986). [CrossRef]

*u*⃗ is a variable conjugate to the difference coordinate

*s*⃑. Finally, the ambiguity function [20

20. K.-H. Brenner and J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta **31**, 213–223 (1984). [CrossRef]

21. M. J. Bastiaans and T. Alieva, “Wigner distribution moments in fractional fourier transform systems,” J. Opt. Soc. Am. A **19**, 1763–1773 (2002). [CrossRef]

20. K.-H. Brenner and J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta **31**, 213–223 (1984). [CrossRef]

*z*can be written:

*k*is the wavenumber in the medium. Thus the ambiguity function is sheared in a four-dimensional space as the field propagates.

## 3. Model

*J*

_{0}(

*x*⃗,

*s*⃗) (and ambiguity function X

_{0}(

*q*⃗,

*s*⃗) is incident upon a thin object with a complex transmission coefficient

*O*(

*x*⃗)=

*A*(

*x*⃗)exp(

*iφ*(

*x*⃗)). The light then passes through a turbid medium of thickness

*ℓ*. In order to study the influence of scattering on image formation we wish to determine the intensity of the field leaving the scattering medium (i.e. at

*z*=0) and, in order to investigate its impact on phase imaging utilizing the transport-of-intensity equation, we also wish to find the derivative of the intensity at this plane with respect to

*z*.

6. C.-C. Cheng and M. G. Raymer, “Propagation of transverse optical coherence in random multiple scattering media,” Phys. Rev. A **62**, 023811 (2000). [CrossRef]

*J*(

_{S}*x*⃗,

*s*⃗,0) and it has been shown [6

6. C.-C. Cheng and M. G. Raymer, “Propagation of transverse optical coherence in random multiple scattering media,” Phys. Rev. A **62**, 023811 (2000). [CrossRef]

22. C. K. Aruldoss, N. M. Dragomir, and A. Roberts, “Non-interferometric characterization of partially coherent scalar wavefields and application to scattered light,” J. Opt. Soc. Am. A **24**, 3189–3197 (2007). [CrossRef]

*θ*is the width of a Gaussian fitted to the Mie differential scattering cross-section for the given scattering particles [23

_{0}23. C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (Wiley1998). [CrossRef]

*σ*is the scattering cross section integrated over the angle defined by the collection angle of the optical system used,

_{N}*N*is the number density of scattering particles,

*k*is the wavenumber in the medium and

_{med}*µ*is the total extinction coefficient.

_{T}*x*⃗, leads to the ambiguity function [20

20. K.-H. Brenner and J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta **31**, 213–223 (1984). [CrossRef]

*X*is the ambiguity function of the input field. Hence the term

_{0}*S*(

*q*⃗,

*s*⃗) acts as an optical transfer function describing the influence of only scattering on the propagation. This observation forms the key conclusion of the present paper. The shearing of the ambiguity function due to propagation in a homogeneous medium is described by the second factor. In the remainder of the paper, we investigate implications for imaging both amplitude and phase objects.

*O*(

*x*)=

*A*(

*x*)exp(-

*iφ*(

*x*)), is written

*J*

_{0}(

*x*⃗,

*s*⃗), is the spatial coherence function of the field incident on the object.

*J*

_{0}(

*x*⃗,

*s*⃗)=constant, and weak phase gradients, the ambiguity function describing the transmitted light takes the form:

*J*

_{obj}(

*x*⃗,

*s*⃗)≈

*A*

^{2}(

*x*⃗)exp(

*is*⃗•

*∇*φ(

*x*⃗)). Using Eq. (5) the ambiguity function a distance

*z*from the output face of the scattering medium is given by:

*z*from the output face of the scattering medium is given by:

*z*=0 plane (i.e. at the output face) by:

*S*(

*q*⃗,0) acts as the amplitude transfer function.

*denotes the gradient with respect to the second coordinate,*

_{s}*I*̃

_{0}(

*q*⃗) is defined in Eq. (16) and

*G*̃(

*q*⃗)=

*iq*⃗•∫

*dx*⃗

*x*′

*A*

^{2}(

*x*⃗′)∇φ(

*x*⃗′)exp(-

*iq*⃗•

*x*⃗′) is the Fourier transform of ∇•(

*A*

^{2}(

*x*⃗′)∇

*φ*(

*x*⃗′)). In the case of a phase-only object and uniform, coherent illumination,

*I*̃

_{0}(

*q*⃗)=

*I*δ(

_{0}*q*⃗) and the second term in Eq. (17) is zero so

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

25. 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**, 1942–1946 (1995). [CrossRef]

*S*(

*q*⃗,0) which is given by:

## 4. Transfer functions

*S*(

*q*⃗,

*s*⃗) as being an optical transfer-like function describing scattering, it is instructive to plot this function. In all simulations, the refractive index of the spherical particles relative to the medium has been chosen to be 1.2 and the free-space wavelength 632.8 nm. The near-forward scattering cross-section,

*σN*is also assumed to be equal to the total scattering cross-section. A typical transfer function is shown in Fig. 2. In this case the concentration of spheres is 1.6×10

^{11}/m

^{3}which corresponds to a mean free path of 10 mm, i.e. equal to the thickness of the medium. Hence, this medium has an optical density, OD=

*NσTℓ*, of 1. Figure 3 shows

*S*(

*q*⃗,0) for media containing spherical particles of diameter (a) 5 µm and (b) 20µm. As expected, the transfer functions have a maximum at

*q*=0 and can be regarded as consisting of two components associated with the ballistic and scattered components of the field. The scattered component contributes a component that is peaked around

*q*=0. The width of this peak depends on the size of the scattering spheres: larger spheres produce a broader peak consistent with the narrower differential scattering cross-section while smaller spheres produce a narrower scattered contribution to

*S*(

*q*⃗,0). It can be seen that the ballistic contribution decreases as the optical density increases which would correspond to a decrease in spatial resolution for images containing spatial frequencies greater than approximately 2 mm

^{-1}for 5 µm spheres and 8 mm

^{-1}for 20 µm spheres. These values correspond to the 1/

*e*-half-widths of Gaussians fitted to OD=5 data.

## 5. Simulated images

*a*spatial coherence function given by

*π*radians was placed in the centre of a scattering medium of total thickness 10 mm. The light incident on the medium had a field width of 2 mm and was spatially coherent. The intensity arising from free-space propagation of the resulting coherence function through distances of Δ

*z*=±1 mm on either side of an in-focus image of the sample was then calculated. These images were then used with an algorithm [26

26. D. Paganin and K. A. Nugent, “Non-interferometric phase imaging using partially coherent light,” Phys. Rev. Lett. **80**, 2586–2589.(1998). [CrossRef]

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

## 6. Conclusion

## Acknowledgments

## References and links

1. | S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. |

2. | R. Chandrasekhar, |

3. | A. Ishimaru, |

4. | J. J. Duderstadt and L. J. Hamilton, |

5. | H. W. Lewis, “Multiple scattering in an infinite medium,” Phys. Rev. |

6. | C.-C. Cheng and M. G. Raymer, “Propagation of transverse optical coherence in random multiple scattering media,” Phys. Rev. A |

7. | A. Wax and J. E. Thomas, “Measurement of smoothed Wigner phase-space distribution for small-angle scattering in a turbid medium,” J. Opt. Soc. Am. A |

8. | F. Dubois, L. Joannes, and J.-C. Legros, “Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence,” Appl. Opt. |

9. | F. Dubois, M.-L. N. Requena, C. Minetti, O. Monnom, and E. Istasse, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. |

10. | C.-C. Cheng and M. G. Raymer, “Long range saturation of spatial decoherence in wave-field transport in random multiple scattering media,” Phys. Rev. Lett. |

11. | F. Dubois, M.-L. N. Requena, and C. Minetti, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. |

12. | N. A. Beaudry and T. D. Milster, “Effects of object roughness on partially coherent image formation,” Opt. Lett. |

13. | D. M. Marks, R. A. Stack, and D. J. Brady, “Astigmatic coherence sensor for digital imaging,” Opt. Lett. |

14. | A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express |

15. | M. Alrubaiee, M. Xu, S. K. Gayen, M. Brito, and R. R. Alfano, “Three-dimensional optical tomographic imaging of scattering objects in tissue-simulating turbid media using independent component analysis,” Appl. Phys. Lett. |

16. | E. D. Barone-Nugent, A. Barty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. |

17. | A. Barty, K.A. Nugent, D. Paganin, and A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. , |

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

19. | M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A |

20. | K.-H. Brenner and J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta |

21. | M. J. Bastiaans and T. Alieva, “Wigner distribution moments in fractional fourier transform systems,” J. Opt. Soc. Am. A |

22. | C. K. Aruldoss, N. M. Dragomir, and A. Roberts, “Non-interferometric characterization of partially coherent scalar wavefields and application to scattered light,” J. Opt. Soc. Am. A |

23. | C. F. Bohren and D. R. Huffman, |

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

25. | 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 |

26. | D. Paganin and K. A. Nugent, “Non-interferometric phase imaging using partially coherent light,” Phys. Rev. Lett. |

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

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(110.4980) Imaging systems : Partial coherence in imaging

(110.0113) Imaging systems : Imaging through turbid media

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: September 4, 2008

Revised Manuscript: December 9, 2008

Manuscript Accepted: December 10, 2008

Published: December 11, 2008

**Virtual Issues**

Vol. 4, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

C. K. Aruldoss and A. Roberts, "Imaging through turbid media based on wave transport model approach," Opt. Express **16**, 21389-21400 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-26-21389

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### References

- S. R. Arridge and J. C. Hebden, "Optical imaging in medicine: II. Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997). [CrossRef] [PubMed]
- R. Chandrasekhar, Radiative Transfer, (Oxford, 1950).
- A. Ishimaru, Wave propagation and scattering in random media Volume 1: Single scattering and transport theory (New York: Academic, 1978).
- J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).
- H. W. Lewis, "Multiple scattering in an infinite medium," Phys. Rev. 78, 526-529 (1950). [CrossRef]
- C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple scattering media," Phys. Rev. A 62, 023811 (2000). [CrossRef]
- A. Wax and J. E. Thomas, "Measurement of smoothed Wigner phase-space distribution for small-angle scattering in a turbid medium," J. Opt. Soc. Am. A 15, 1896-1908 (1998). [CrossRef]
- F. Dubois, L. Joannes and J.-C. Legros, "Improved three-dimensional imaging with a digital holography microscope with a source of partial spatial coherence," Appl. Opt. 38, 7085-7094 (1999). [CrossRef]
- F. Dubois, M.-L. N. Requena, C. Minetti, O. Monnom, and E. Istasse, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004). [CrossRef] [PubMed]
- C.-C. Cheng and M. G. Raymer, "Long range saturation of spatial decoherence in wave-field transport in random multiple scattering media," Phys. Rev. Lett. 82, 4807-4810 (1999). [CrossRef]
- F. Dubois, M.-L. N. Requena, and C. Minetti, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004). [CrossRef] [PubMed]
- N. A. Beaudry and T. D. Milster, "Effects of object roughness on partially coherent image formation," Opt. Lett. 25, 454-456 (2000). [CrossRef]
- D. M. Marks, R. A. Stack, and D. J. Brady, "Astigmatic coherence sensor for digital imaging," Opt. Lett. 25, 1726-1728 (2000). [CrossRef]
- A. Momose, "Phase-sensitive imaging and phase tomography using X-ray interferometers," Opt. Express 11, 2303-2314 (2003). [CrossRef] [PubMed]
- M. Alrubaiee, M. Xu, S. K. Gayen, M. Brito, and R. R. Alfano, "Three-dimensional optical tomographic imaging of scattering objects in tissue-simulating turbid media using independent component analysis," Appl. Phys. Lett. 87, 19112 (2005). [CrossRef]
- E. D. Barone-Nugent, A. Barty and, K. A. Nugent, "Quantitative phase-amplitude microscopy I: optical microscopy," J. Microsc. 206, 194-203 (2002). [CrossRef] [PubMed]
- A. Barty, K. A. Nugent, D. Paganin, and A. Roberts, "Quantitative optical phase microscopy," Opt. Lett. 23, 817-819 (1998). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).
- M. J. Bastiaans, "Application of the Wigner distribution function to partially coherent light," J. Opt. Soc. Am. A 3, 1227-1238 (1986). [CrossRef]
- K.-H. Brenner and J. Ojeda-Castaneda, "Ambiguity function and Wigner distribution function applied to partially coherent imagery," Opt. Acta 31, 213-223 (1984). [CrossRef]
- M. J. Bastiaans and T. Alieva, "Wigner distribution moments in fractional fourier transform systems," J. Opt. Soc. Am. A 19, 1763-1773 (2002). [CrossRef]
- C. K. Aruldoss, N. M. Dragomir, and A. Roberts, "Non-interferometric characterization of partially coherent scalar wavefields and application to scattered light," J. Opt. Soc. Am. A 24, 3189-3197 (2007). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998). [CrossRef]
- M. R. Teague, "Deterministic phase retrieval: a Green’s function solution," J. Opt. Soc. Am. 731434-1441 (1983). [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, 1942-1946 (1995). [CrossRef]
- D. Paganin and K. A. Nugent, "Non-interferometric phase imaging using partially coherent light," Phys. Rev. Lett. 80, 2586-2589 (1998). [CrossRef]
- D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, "Quantitative phase-amplitude microscopy III. The effects of noise," J. Microsc. 214, 51-61 (2004). [CrossRef] [PubMed]

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