## Contrast enhancement in X-ray phase contrast tomography |

Optics Express, Vol. 22, Issue 15, pp. 18020-18026 (2014)

http://dx.doi.org/10.1364/OE.22.018020

Acrobat PDF (1700 KB)

### Abstract

We demonstrate phase contrast enhancement of X-ray computed tomography derived from propagation based imaging. In this method, the absorption and phase components are assumed to be correlated, allowing for phase retrieval from a single image. Experimental results are shown for liquid samples. Signal-to-noise ratio is greatly enhanced relative to pure attenuation based imaging.

© 2014 Optical Society of America

## 1. Introduction

1. S. Singh and M. Singh, “Explosives detection systems (EDS) for aviation security,” Signal Process. **83**, 3155 (2003). [CrossRef]

2. K. M. Hasebroock and N. J. Serkova, “Toxicity of MRI and CT contrast agents,” Expert Opin. Drug Metab. Toxicol. **5**, 403–416 (2009). [CrossRef] [PubMed]

3. E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology **157**, 49–58 (1991). [CrossRef] [PubMed]

4. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature **373**, 595–598 (1995). [CrossRef]

6. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat Phys **2**, 258–261 (2006). [CrossRef]

7. J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the fresnel region,” Opt. Lett. **32**, 1617–1619 (2007). [CrossRef] [PubMed]

8. R. C. Chen, L. Rigon, and R. Longo, “Comparison of single distance phase retrieval algorithms by considering different object composition and the effect of statistical and structural noise,” Opt. Express **21**, 7384–7399 (2013). [CrossRef] [PubMed]

*per se*[9

9. J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express **21**, 14430 (2013). [CrossRef] [PubMed]

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

*I*(

*x*,

*y*,

*z*) is the intensity image at the propagation distance

*z*,

*ϕ*(

*x*,

*y*) is the phase map (equivalently, OPL) of the object, and

*k*is the wavenumber. By solving the TIE at multiple angles of exposure, the refractive index distribution of an object can be obtained through tomographic reconstruction. In this way, TIE tomographic reconstruction can be thought of as a two-step problem.

11. A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A **19**, 472–480 (2002). [CrossRef]

13. T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first born- and rytov-type approximations,” Appl. Opt. **43**, 2418–2430 (2004). [CrossRef] [PubMed]

*et al.*propose inversion of the TIE using prior knowledge that the sample consists of a single material of known refractive index [14

14. G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, “Phase-contrast tomography of single-material objects from few projections,” Opt. Express **16**, 908–919 (2008). [CrossRef] [PubMed]

15. E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory **52**, 489–509 (2006). [CrossRef]

16. T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM J. Imaging Sci. **2**, 323–343 (2009). [CrossRef]

17. L. Tian, J. C. Petruccelli, Q. Miao, H. Kudrolli, V. Nagarkar, and G. Barbastathis, “Compressive x-ray phase tomography based on the transport of intensity equation,” Opt. Lett. **38**, 3418–3421 (2013). [CrossRef] [PubMed]

18. X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett. **30**, 379–381 (2005). [CrossRef] [PubMed]

## 2. Reconstruction process

*λ*is located at the plane

*z*= −

*z*

_{0}, and a detector is located at the plane

*z*=

*d*. The sample, centered at

*z*= 0, is characterized by a complex refractive index

*n*(

*x*,

*y*,

*z*;

*λ*) = 1 −

*δ*(

*x*,

*y*,

*z*;

*λ*) +

*iβ*(

*x*,

*y*,

*z*;

*λ*), where 1 −

*δ*(

*x*,

*y*,

*z*;

*λ*) and

*β*(

*x*,

*y*,

*z*;

*λ*) are the real and imaginary parts of the refractive index, respectively. We assume that the geometry of our experiment is such that the beam passing through the object is approximated by a plane wave oriented along the optical axis, and that the interaction between the sample and the field follows the projection approximation.

17. L. Tian, J. C. Petruccelli, Q. Miao, H. Kudrolli, V. Nagarkar, and G. Barbastathis, “Compressive x-ray phase tomography based on the transport of intensity equation,” Opt. Lett. **38**, 3418–3421 (2013). [CrossRef] [PubMed]

*N*×

*N*grid of square pixels of side length

*M*Δ. Let Θ denote the number of angular projections. Then the measured projections

*g*at all angles may be arranged into a real-valued vector

**g**of length

*N*

^{2}Θ. The refractive index of the object will also be discretized into a

*N*×

*N*×

*N*cube width side length Δ, packed into a complex valued vector

**n**. Then let

**P**and

**R**denote operators corresponding to the discretized forms of Eqs. (2) and (3), respectively, such that where the operator

**A**is the cascade of the linear operators

**P**and

**R**. If

**n**is sparse in some basis, as it often is the case in tomography, then Eq. (6) can be solved using compressive reconstruction [15

15. E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory **52**, 489–509 (2006). [CrossRef]

16. T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM J. Imaging Sci. **2**, 323–343 (2009). [CrossRef]

22. J. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Transactions on Image Processing **16**, 2992–3004 (2007). [CrossRef] [PubMed]

*, ∇*

_{x}*, and ∇*

_{y}*are the finite difference operators in Cartesian coordinates [23*

_{z}23. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena **60**, 259–268 (1992). [CrossRef]

22. J. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Transactions on Image Processing **16**, 2992–3004 (2007). [CrossRef] [PubMed]

## 3. Experimental application to discrimination of liquids

*k*in phase reconstruction.

*P*< 0.010 for absorption,

*P*< 6.79 × 10

^{−7}for phase). Taking only 100 randomly sampled voxels from each CT (after segmentation), intensity based CT could only distinguish water and peroxide in 26% of tests, meaning rejection of the null hypothesis at the 5% significance level, while phase based CT could distinguish the two liquids in 100% of tests.

## 4. Discussion

## Acknowledgments

## References and links

1. | S. Singh and M. Singh, “Explosives detection systems (EDS) for aviation security,” Signal Process. |

2. | K. M. Hasebroock and N. J. Serkova, “Toxicity of MRI and CT contrast agents,” Expert Opin. Drug Metab. Toxicol. |

3. | E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology |

4. | T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature |

5. | A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phasecontrast xray computed tomography for observing biological soft tissues,” Nat Med |

6. | F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat Phys |

7. | J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the fresnel region,” Opt. Lett. |

8. | R. C. Chen, L. Rigon, and R. Longo, “Comparison of single distance phase retrieval algorithms by considering different object composition and the effect of statistical and structural noise,” Opt. Express |

9. | J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express |

10. | M. Reed Teague, “Deterministic phase retrieval: a green?s function solution,” J. Opt. Soc. Am. |

11. | A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A |

12. | A. Burvall, U. Lundstrm, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in x-ray phase-contrast imaging suitable for tomography,” Opt. Express |

13. | T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first born- and rytov-type approximations,” Appl. Opt. |

14. | G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, “Phase-contrast tomography of single-material objects from few projections,” Opt. Express |

15. | E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory |

16. | T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM J. Imaging Sci. |

17. | L. Tian, J. C. Petruccelli, Q. Miao, H. Kudrolli, V. Nagarkar, and G. Barbastathis, “Compressive x-ray phase tomography based on the transport of intensity equation,” Opt. Lett. |

18. | X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett. |

19. | T. E. Gureyev and S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A |

20. | D. Paganin, |

21. | S. Matej, J. Fessler, and I. Kazantsev, “Iterative tomographic image reconstruction using fourier-based forward and back-projectors,” IEEE Transactions on Medical Imaging |

22. | J. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Transactions on Image Processing |

23. | L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(100.5070) Image processing : Phase retrieval

(340.7440) X-ray optics : X-ray imaging

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: May 20, 2014

Revised Manuscript: July 2, 2014

Manuscript Accepted: July 9, 2014

Published: July 17, 2014

**Virtual Issues**

Vol. 9, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Adam Pan, Ling Xu, Jon C. Petruccelli, Rajiv Gupta, Bipin Singh, and George Barbastathis, "Contrast enhancement in X-ray phase contrast tomography," Opt. Express **22**, 18020-18026 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18020

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### References

- S. Singh and M. Singh, “Explosives detection systems (EDS) for aviation security,” Signal Process.83, 3155 (2003). [CrossRef]
- K. M. Hasebroock and N. J. Serkova, “Toxicity of MRI and CT contrast agents,” Expert Opin. Drug Metab. Toxicol.5, 403–416 (2009). [CrossRef] [PubMed]
- E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology157, 49–58 (1991). [CrossRef] [PubMed]
- T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature373, 595–598 (1995). [CrossRef]
- A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phasecontrast xray computed tomography for observing biological soft tissues,” Nat Med2, 473–475 (1996). [CrossRef] [PubMed]
- F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat Phys2, 258–261 (2006). [CrossRef]
- J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the fresnel region,” Opt. Lett.32, 1617–1619 (2007). [CrossRef] [PubMed]
- R. C. Chen, L. Rigon, and R. Longo, “Comparison of single distance phase retrieval algorithms by considering different object composition and the effect of statistical and structural noise,” Opt. Express21, 7384–7399 (2013). [CrossRef] [PubMed]
- J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express21, 14430 (2013). [CrossRef] [PubMed]
- M. Reed Teague, “Deterministic phase retrieval: a green?s function solution,” J. Opt. Soc. Am.73, 1434–1441 (1983). [CrossRef]
- A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A19, 472–480 (2002). [CrossRef]
- A. Burvall, U. Lundstrm, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in x-ray phase-contrast imaging suitable for tomography,” Opt. Express19, 10359–10376 (2011). [CrossRef] [PubMed]
- T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first born- and rytov-type approximations,” Appl. Opt.43, 2418–2430 (2004). [CrossRef] [PubMed]
- G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, “Phase-contrast tomography of single-material objects from few projections,” Opt. Express16, 908–919 (2008). [CrossRef] [PubMed]
- E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory52, 489–509 (2006). [CrossRef]
- T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM J. Imaging Sci.2, 323–343 (2009). [CrossRef]
- L. Tian, J. C. Petruccelli, Q. Miao, H. Kudrolli, V. Nagarkar, and G. Barbastathis, “Compressive x-ray phase tomography based on the transport of intensity equation,” Opt. Lett.38, 3418–3421 (2013). [CrossRef] [PubMed]
- X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett.30, 379–381 (2005). [CrossRef] [PubMed]
- T. E. Gureyev and S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A15, 579–585 (1998). [CrossRef]
- D. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006). [CrossRef]
- S. Matej, J. Fessler, and I. Kazantsev, “Iterative tomographic image reconstruction using fourier-based forward and back-projectors,” IEEE Transactions on Medical Imaging23, 401–412 (2004). [CrossRef] [PubMed]
- J. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Transactions on Image Processing16, 2992–3004 (2007). [CrossRef] [PubMed]
- L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena60, 259–268 (1992). [CrossRef]

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