## An iterative phase retrieval algorithm for in-line x-ray phase imaging

Optics Express, Vol. 15, Issue 13, pp. 8383-8390 (2007)

http://dx.doi.org/10.1364/OE.15.008383

Acrobat PDF (548 KB)

### Abstract

A general theoretical formulism for in-line phase x-ray imaging was presented with a corresponding linear formula in previous works. In this report, an iterative approach is introduced for phase retrieval with a nonlinear formula. The results of simulation showed that the iterative approach can retrieve the phase map more effectively with high efficiency and flexibility.

© 2007 Optical Society of America

## 1. Introduction

08. X. Wu and H. Liu, “Phase-space formulation for phase-contrast x-ray imaging,” Appl. Opt. **44**, 5847–5854 (2005). [CrossRef] [PubMed]

## 2. Theoretical background

*T*(

*x,y*) =

*A*(

*x,y*)exp(

*iϕ*(

*x,y*)). The objective of phase retrieval is to obtain the phase map

*ϕ*(

*x,y*) from an attenuation-based image

*I*

_{1}taken with

*M*= 1 and a phase-contrast image

*I*

_{2}taken with

*M*> 1. Here

*M*= (

*R*

_{1}+

*R*

_{2})/

*R*

_{1}is the geometrical magnification,

*R*

_{1}and

*R*

_{2}are the source-object-distance (SOD) and the object-image-distance (OID).

*I*

_{bp}(

*x*) =

*M*

^{2}

*I*(

*Mx*), the formulation of phase imaging by Wu and Liu [5] can be written as:

*I*

_{0}is the incident x-ray intensity, λ is the wavelength of the x-ray,

*u*is the spatial frequency,

*[∙] means Fourier transform. This formulation is obtained through the moderate variation conditions of the attenuation and phase maps:*ℱ ̂

*u*of the detectors. As has been estimated for a typical mammography system [5], λ

*R*

_{2}

*u*/

*M*is less than 0.62 micron, which is much smaller than the typical size of the breast structures to be imaged. In such cases, the moderate variation conditions can be met. This formula Eq. (1) considers the most general cases in clinical applications in a very concise form, providing high flexibility in the implementation of phase retrieval algorithms. The formulation was further extended to include the effects of polychromatic x-ray source with finite focal spot size and real detectors with finite spatial resolution [14

14. X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions.” Med. Phys. **31**, 2378–2384 (2004). [CrossRef] [PubMed]

_{G.U}, and the detector, OTF

_{det}.

*ϕ*(

*x,y*) from Eq. (1), it was linearized regarding the Fourier transform of

*A*

^{2}(

*x,y*)ϕ(

*x,y*) as [5]

*[*ℱ ̂

*A*

^{2}(η)

*ϕ*(η)] and hence

*ϕ*can be solved and calculated explicitly.

## 3. The iterative algorithm

*T*

_{1}through

*T*

_{4}. The assumption of

*A*(η) ≈

*A*(η ± λ

*R*

_{2}

*u*/

*M*) used for the linearization [5] could be valid, however, it cannot assure that

*T*

_{2}≫

*T*

_{4}since sin (λ

*R*

_{2}

*u*

^{2}/

*M*) can be much smaller than cos (λ

*R*

_{2}

*u*

^{2}/

*M*).

*T*

_{1}through

*T*

_{4}are dependent on the actual distribution of both

*A*(η) and ϕ(

*η*) and therefore no general conclusion can be made about their magnitudes for all cases. Here we just give some very rough estimations.

*R*

_{2}

*u*

^{2}/

*M*≫ 1 holds true for typical clinical applications,

*T*

_{1}through

*T*

_{4}are approximately proportional to λ

^{0}λ

^{1}, λ

^{2}and λ

^{1}, respectively, which leads to the estimation that

*T*

_{2}and

*T*

_{4}are of the same order, at least for some cases. Therefore, in phase retrieval studies, of the four terms of Eq. (1),

*T*

_{3}can be neglected while

*T*

_{4}should be kept. With the effects of the x-ray focal spot size and the detector resolution taken into account as presented by Wu and Liu [8

08. X. Wu and H. Liu, “Phase-space formulation for phase-contrast x-ray imaging,” Appl. Opt. **44**, 5847–5854 (2005). [CrossRef] [PubMed]

*T*

_{4}is a requirement of physical correctness. Without

*T*

_{4}, the image will be dependent on the choice of the zero point of

*ϕ*(

*η*), which is physically incorrect, while keeping

*T*

_{4}in the formula can avoid the ambiguity.

*T*

_{4}is kept, the formula remains nonlinear and is not explicitly solvable, and therefore, direct retrieval is no longer applicable. To treat this problem, we present an iterative retrieval algorithm as follows:

- Calculate the attenuation map
*A*^{2}from the attenuation-based image*I*_{1}by direct de-convolution of the OTFs of the x-ray source and the detector. - Set initial distribution
*A*^{2}*ϕ*= 0 or any other distribution reasonable. It is proved that the algorithm does not require a precise estimation. - Calculate
*T*_{4}using the*A*^{2}*ϕ*distribution. In Eq. (6),**44**, 5847–5854 (2005). [CrossRef] [PubMed]*T*_{4}has been rewritten to be explicitly dependent on*A*^{2}*ϕ* - Retrieve
*A*^{2}*ϕ*from Eq. (6) using the phase-contrast image*I*_{2}and the*T*_{4}calculated in Step 3.

*A*

^{2}

*ϕ*is obtained. Dividing

*A*

^{2}

*ϕ*by

*A*

^{2}gives the phase map

*ϕ*

*A*

^{2}in Step 4, which cannot be obtained by dividing 2 sin (πλ

*R*

_{2}

*u*

_{2}/

*M*) since sin(0) = 0. This problem was solved by choosing the zero point of phase map

*ϕ*and hence that of

*A*

^{2}

*ϕ*. Practically, one can choose a “region of zero phase” (RZP) in the object plane, in which both

*ϕ*and

*A*

^{2}

*ϕ*are assumed to be zero. The choice of RZP is arbitrary as long as

*ϕ*is approximately a constant (not necessarily zero) in it. In the algorithm, after each Step 4, a constant equal to the average value in RZP is subtracted from

*A*

^{2}

*ϕ*Thus, the zero-frequency amplitude is determined. It should be pointed that, though the choice of RZP does not affect the distribution of

*ϕ*, it does on

*A*

^{2}

*ϕ*because

*A*

^{2}(

*ϕ*+

*c*) will generally differ from

*A*

^{2}

*ϕ*for a constant

*c*.

## 4. Tests and results

*R*

_{1}=

*R*

_{2}= 1m. The virtual sample is of an attenuation map “cameraman” as shown in Fig. 1(a) and a phase map “Lena” as shown in Fig. 1(b). With the distinct attenuation and phase map, one can see more evidently the effectiveness of the retrieval algorithm. The RZP is chosen as the 100 by 100 square on the upper-left corner of the sample. In the RZP,

*ϕ*= 5.

*I*

_{1}and the phase-contrast map

*I*

_{2}obtained by the simulation program are shown in Fig. 2. One can see that, due to the short wavelength of the x-ray and hence weak diffraction, the phase contrast effects are almost unnoticeable in

*I*

_{2}, but can be seen in the difference image

*M*

^{2}

*I*

^{2}-

*I*

_{1}. The standard deviation (STD) of the difference image is 2.8×10

^{-4}, three orders of magnitude smaller than that of the attenuation map.

*A*

^{2}

*ϕ*contains information of both attenuation and phase, however, it is clearly not the product of the two. Moreover, as discussed above, there exists ambiguity in the retrieval and the phase map cannot be uniquely determined. Direct division of the obtained

*A*

^{2}

*ϕ*by

*A*

^{2}generally tends to give a phase map with bad or even wrong distribution as shown in Fig. 3(b).

*ϕ*between the retrieved map and the original map used for simulation. The STD of Δ

*ϕ*is plotted against the iteration count in Fig. 5.

*A*

^{2}

*ϕ*= 0 leads to the increase of the STD for the first 4 iterations. Even though, the algorithm is robust enough to find a convergent solution quickly after 13 iterations. The algorithm is also of high efficiency. Actually, the 20 iterations was done on an ordinary PC in several minutes with large scope for further optimization.

## 5. Discussions and conclusions

*A*

^{2}

*ϕ*= 0, the linearized formula is actually the same as the first iteration of the iterative approach. Furthermore, for “pure-phase” objects with constant attenuation, the iterative formula degrades to the linear one. Therefore, the iterative algorithm can be seen as an extension to the linearized algorithm, with higher precision and flexibility. Nevertheless, from another point of view, it is more than just an extension. Traditional iterative phase retrieval approaches used in crystallography such as the Gerchberg-Saxton (GS) scheme [15] are proved to be less deterministic, robust and computationally efficient, whereas popular linear retrieval algorithms based on the paraxial Fresnel-Kirchoff diffraction integral are not so successful in handling complex objects. Based on an extensive and general theoretical formulation for in-line phase imaging, our iterative algorithm has the merits of both quantitativeness and efficiency, therefore has great potential for complicated samples in clinical applications.

## Acknowledgement

## References and links

01. | A. Snigirev and I. Snigireva, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. |

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

03. | A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. |

04. | F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, “Mammography with synchrotron radiation: Phase-detection techniques,” Radiology |

05. | X. Wu and H. Liu, “A general theoretical formalism for X-ray phase contrast imaging,” J. X-ray Sci. Tech. |

06. | X. Wu and H. Liu, “Clinical implementation of x-ray phase-contrast imaging: Theoretical foundations and design considerations,” Med. Phys. |

07. | X. Wu and H. Liu, “A dual detector approach for X-ray attenuation and phase imaging,” J. X-ray Sci. Tech. |

08. | X. Wu and H. Liu, “Phase-space formulation for phase-contrast x-ray imaging,” Appl. Opt. |

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

10. | Y. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. |

11. | B. D. Arhatari, K. A. Nugent, A. G. Peele, and J. Thornton, “Phase contrast radiography. II. Imaging of complex objects,” Rev. Sci. Instrum. |

12. | T. E. Gureyev, Y. L. 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. |

13. | T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, “Quantitative in-line phase-contrast imaging with multienergy X rays.” Phys. Rev. Lett. |

14. | X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions.” Med. Phys. |

15. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik(Stuttgart) |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

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

**ToC Category:**

Image Processing

**History**

Original Manuscript: December 22, 2006

Revised Manuscript: April 2, 2007

Manuscript Accepted: May 21, 2007

Published: June 20, 2007

**Citation**

Fanbo Meng, Hong Liu, and Xizeng Wu, "An iterative phase retrieval algorithm for in-line x-ray phase imaging," Opt. Express **15**, 8383-8390 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8383

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### References

- A. Snigirev and I. Snigireva, "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation," Rev. Sci. Instrum. 66, 5486-5492 (1995). [CrossRef]
- 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]
- A. Pogany, D. Gao, and S.W. Wilkins, "Contrast and resolution in imaging with a microfocus x-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997). [CrossRef]
- F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, M. Fabrizioli, R. Longo, R. H. Menk, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, M. Ratti, L. Rigon, G. Tromba, A. Vacchi, E. Vallazza, and F. Zanconati, "Mammography with synchrotron radiation: Phasedetection techniques," Radiology 215, 286-293 (2000).
- X. Wu and H. Liu, "A general theoretical formalism for X-ray phase contrast imaging," J. X-ray Sci.Tech. 11, 33-42 (2003).
- X. Wu and H. Liu, "Clinical implementation of x-ray phase-contrast imaging: Theoretical foundations and design considerations," Med. Phys. 30, 2169-2179 (2003). [CrossRef] [PubMed]
- X. Wu and H. Liu, "A dual detector approach for X-ray attenuation and phase imaging," J. X-ray Sci.Tech. 12, 35-42 (2004).
- X. Wu and H. Liu, "Phase-space formulation for phase-contrast x-ray imaging," Appl. Opt. 44, 5847-5854 (2005). [CrossRef] [PubMed]
- X. Wu, H. Liu, and A. M. Yan, "X-ray phase-attenuation duality and phase retrieval," Opt. Lett. 30, 379-381 (2005). [CrossRef] [PubMed]
- Y. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093,706 (2005). [CrossRef]
- B. D. Arhatari, K. A. Nugent, A. G. Peele, and J. Thornton, "Phase contrast radiography. II. Imaging of complex objects," Rev. Sci. Instrum. 76, 113,704 (2005). [CrossRef]
- T. E. Gureyev, Y. L. 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]
- T. E. Gureyev, S. Mayo, S. W. Wilkins, D. Paganin, and A. W. Stevenson, "Quantitative in-line phase-contrast imaging with multienergy X rays." Phys. Rev. Lett. 86, 5827-5830 (2001). [CrossRef] [PubMed]
- X. Wu and H. Liu, "A new theory of phase-contrast x-ray imaging based on Wigner distributions." Med. Phys. 31, 2378-2384 (2004). [CrossRef] [PubMed]
- R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik(Stuttgart) 35, 237-246 (1972).

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