## Image reconstruction in phase-contrast tomography exploiting the second-order statistical properties of the projection data |

Optics Express, Vol. 19, Issue 24, pp. 24396-24410 (2011)

http://dx.doi.org/10.1364/OE.19.024396

Acrobat PDF (1061 KB)

### Abstract

X-ray phase-contrast tomography (PCT) methods seek to quantitatively reconstruct separate images that depict an object’s absorption and refractive contrasts. Most PCT reconstruction algorithms generally operate by explicitly or implicitly performing the decoupling of the projected absorption and phase properties at each tomographic view angle by use of a phase-retrieval formula. However, the presence of zero-frequency singularity in the Fourier-based phase retrieval formulas will lead to a strong noise amplification in the projection estimate and the subsequent refractive image obtained using conventional algorithms like filtered backprojection (FBP). Tomographic reconstruction by use of statistical methods can account for the noise model and *a priori* information, and thereby can produce images with better quality over conventional filtered backprojection algorithms. In this work, we demonstrate an iterative image reconstruction method that exploits the second-order statistical properties of the projection data can mitigate noise amplification in PCT. The autocovariance function of the reconstructed refractive images was empirically computed and shows smaller and shorter noise correlation compared to those obtained using the FBP and unweighted penalized least-squares methods. Concepts from statistical decision theory are applied to demonstrate that the statistical properties of images produced by our method can improve signal detectability.

© 2011 OSA

## 1. Introduction

4. R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. **49**, 3573–3583 (2004). URL http://stacks.iop.org/0031-9155/49/3573. [CrossRef] [PubMed]

6. F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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,” Radiol. **215**, 286–293 (2000).

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

9. P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. **83**, 1480–1482 (2003). [CrossRef]

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

12. M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. **35**, 4556–66 (2008). [CrossRef] [PubMed]

12. M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. **35**, 4556–66 (2008). [CrossRef] [PubMed]

15. C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. **37**, 270 (2010). [CrossRef] [PubMed]

14. C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line X-ray phase-contrast imaging,” Opt. Express **17**, 14,466–14,480 (2009). [CrossRef]

16. C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in x-ray in-line phase-contrast imaging,” in Medical Imaging 2008: Physics of Medical Imaging, J. Hsieh and E. Samei, eds., Proc. SPIE6913, 69131Z (2008). URL http://link.aip.org/link/?PSI/6913/69131Z/1.

17. C.-Y. Chou and M. A. Anastasio, “Statistical properties of X-ray phase-contrast tomography,” in Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2009. EMBC 2009 , pp. 6648 –6650 (2009). [CrossRef] [PubMed]

## 2. Imaging physics of propagation-based X-ray phase-contrast tomography

## 3. Image reconstruction

### 3.1. Phase retrieval

*P̃*(

*u,v*;

_{r}*θ*) denote

*Ã*(

_{m,n}*u, v*;

_{r}*θ*) or

*ϕ̃*(

_{m,n}*u, v*;

_{r}*θ*). If

*Ã*(

_{m,n}*u, v*;

_{r}*θ*) and

*ϕ̃*(

_{m,n}*u, v*;

_{r}*θ*) are assumed to be quasi-bandlimited, containing a maximum frequency

*Ã*(

*u,v*;

_{r}*θ*) or

*ϕ̃*(

*u,v*;

_{r}*θ*) described by Eq. (11) or (12) as

### 3.2. Tomographic reconstruction

*α*is the

_{n}*n*-th component of the coefficient vector

*α*,

*N*

^{2}corresponds to the number of expansion functions. In this work, we adopt the conventional voxel expansion as the choice for

*ψ*, and the corresponding coefficient component is given by where

_{n}*f*(

*r⃗*) describes the 3D refractive index distribution. Although the conventional voxel expansion function is adopted here, other sets of expansion functions [24

24. R. Lewitt, “Alternatives to voxels for image representation in iterative reconstruction algorithms,” Phys. Med. Biol. **37**, 705–716 (1992). [CrossRef] [PubMed]

25. M. Defrise and G. T. Gullberg, “Image reconstruction,” Phys. Med. Biol. **51**, R139 (2006). URL http://stacks.iop.org/0031-9155/51/i=13/a=R09. [CrossRef] [PubMed]

**g**∈ ℝ

*corresponds to the lexicographically ordered collection of projection estimates,*

^{M}*α*∈ ℝ

^{N2}denotes the object function, and

**R**∈ ℝ

^{M×N2}represents the discrete approximation of 2D Radon transform. The application of Eqs. (11) and (12) followed by the inverse DFT in Eq. (13) will yield the estimation of

*A*(

*x*=

*r*Δ

*,*

_{d}*y*=

_{r}*s*Δ

_{r}*) and*

_{d}*ϕ*(

*x*=

*r*Δ

*,*

_{d}*y*=

_{r}*s*Δ

_{r}*) at each view angle, respectively. Subsequently,*

_{d}*β*[

*r,s,t*] and

*a*[

*r,s,t*] can be obtained by inverting 2D Radon transforms. This can be accomplished by applying a statistical reconstruction method aimed at finding the solution that minimizes the mismatch between the observed data and that projected by the estimated image. Unregularized reconstruction methods produce images of increasingly noisy appearance with iteration [26

26. D. Snyder and M. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. **32**, 3864–3872 (1985). [CrossRef]

*α*) that we aim to minimize takes on the following form [27

27. J. Fessler and S. Booth, “Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction,” IEEE Trans. Image Process. **8**, 688–699 (1999). [CrossRef]

**W**corresponds to the inverse of the autocovariance matrix of

**g**, † denotes adjoint operation,

*U*(

*α*) is the penalty function that smoothes the object function

*α*, and

*η*is the regularization parameter that controls the tradeoff between the noise and resolution. In this study we employed a simple quadratic smoothness penalty given by where 𝒩

*is the set of 4 neighbors of the*

_{n}*n*-th pixel. The objective of the reconstruction method is to find an image that minimizes the cost function. This can be accomplished by employing optimization algorithms such as conjugate-gradient (CG) methods [27

27. J. Fessler and S. Booth, “Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction,” IEEE Trans. Image Process. **8**, 688–699 (1999). [CrossRef]

*α̂*that satisfies In phase-contrast tomography, the elements of

**g**refer to the finite-dimensional representation of estimated

*λ*/2

*π*

*· A*[

_{m,n}*r, s*] or −

_{r}*λ*/2

*π*

*·*

*ϕ*[

_{m,n}*r, s*], and

_{r}*α*of those refers to the corresponding

*β*[

_{m,n}*r, s,t*] or

*a*[

_{m,n}*r, s,t*].

## 4. Noise model

### 4.1. Noise statistics of measured intensity and reconstructed images

*I*[

_{m}*r, s*] represents a stochastic quantity that contains the noiseless intensity plus noise. In order not to further complicate the noise analysis of the proposed iterative method, without loss of generality, we considered a simplified measurement model [13

_{r}13. 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 (2003). [CrossRef]

*n*[

_{m}*r, s*] denote the noiseless intensity data and an additive noise term, respectively. We assume that the noise satisfies where

_{r}*δ*denotes the Kronecker delta function, and Cov{

*n*[

_{m}*r, s*],

_{r}*n*

_{m′}[

*r*′,

*s*′

*]} and Var{*

_{r}*n*[

_{m}*r, s*]} denote the autocovariance and variance functions of the noise, respectively [28, 29].

_{r}14. C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line X-ray phase-contrast imaging,” Opt. Express **17**, 14,466–14,480 (2009). [CrossRef]

### 4.2. Weighting matrix

**K**is enormous, its inversion is not intractable. The inverse of a block diagonal matrix is another block diagonal matrix, composed of the inverse of each block, as follows:

_{g}*×*4k detector, the reconstruction involves the inversion of each block matrix with dimensions of 4000

*×*4000, which can be computed readily. Consequently, even though the PWLS method involves computation of matrix inversion, it is applicable to real data of such dimensions.

## 5. Computer-simulation studies

### 5.1. Eigen analysis of Hessian matrices

**H**

_{W}^{†}

**H**) of Hessian matrix that exploits the second-order statistical properties of the projection data corresponding to where the weighting matrix

_{W}**W**was computed when the standard deviation of Gaussian noise was set as

*σ*= 1%. On the other hand, the system matrix of the unweighted least-squares method was also computed for comparison purpose. The unweighted system matrix is given by In Fig. 2, the normalized eigen spectra of the Hermitian operators

**H**

_{W}^{†}

**H**and

_{W}**H**

^{†}

**H**(

*η*= 0 for both cases) are respectively depicted by solid and dashed curves. The decay behavior of eigenvalues corresponding to the unweighted Hessian matrix is greatly improved when the inverse autocovariance function is included in

**H**[29,30

_{W}30. M. Bertero, *Introduction to inverse problems in imaging* (Taylor & Francis, 1998). [CrossRef]

### 5.2. Numerical phantom and simulated measurement data

*σ*= 1%. The reconstructed images of Phantom A by use of these algorithms are presented in Fig. 3. At higher noise level of

*σ*= 10%, we employed Phantom set B that consists of one small ellipsoid, the signal, embedded in one large ellipsoid, the background, as our numerical phantoms to investigate the second-order statistical properties of the reconstructed images and the associated signal detection studies. Additional description about Phantom set B is provided in Sec. 5.5.

*z*

_{1}= 9 and

*z*

_{2}= 38 mm behind the object, respectively. The detector was assumed to contain 128

*×*128 elements of dimension of 5

*μ*m. Noisy intensity data were produced according to Eq. (20), where the standard deviation of Gaussian noise was

*σ*= 1%. Estimates of the Fourier components of

*Ã*

_{1,2}(

*u, v*) and

_{r}*ϕ̃*

_{1,2}(

*u*,

*v*) were reconstructed by use of Eqs. (11) and (12), respectively. From these Fourier data, estimates of

_{r}*A*

_{1,2}(

*x*,

*y*) and

_{r}*ϕ*

_{1,2}(

*x*,

*y*) were obtained after application of the 2D inverse Fourier transform. A set of tomographic data was obtained by repeating the above procedures at 180 evenly distributed tomographic view angles

_{r}*θ*over [0

*π*).

14. C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line X-ray phase-contrast imaging,” Opt. Express **17**, 14,466–14,480 (2009). [CrossRef]

**W**in Eq. (17) were specified by inverting the computed autocovariance matrix, according to Eq. (25). In order to search for the solution that minimizes the objective function expressed in Eq. (17), the conjugate-gradient method was employed. In this work, we used the Polak-Ribiere CG method to calculate the search direction and solve the inverse problem iteratively [27

27. J. Fessler and S. Booth, “Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction,” IEEE Trans. Image Process. **8**, 688–699 (1999). [CrossRef]

31. E. Mumcuoglu, R. Leahy, S. R. Cherry, and Z. Zhou, “Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imag. **13**, 687–701 (1994). [CrossRef]

### 5.3. Reconstructed results

*ϕ*(

*x,y*) is contaminated by low frequency noise while

_{r}*A*(

*x, y*) is not [14

_{r}**17**, 14,466–14,480 (2009). [CrossRef]

15. C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. **37**, 270 (2010). [CrossRef] [PubMed]

*σ*= 1%, the reconstructed estimates of

*a*(

*r⃗*) corresponding to the transverse slice

*x*= 0 are presented in Fig. 3. The regularization parameter was chosen as

*η*= 0.0434 for the normalized matrices

**R**

^{†}

**WR**and

**R**

^{†}

**R**. Subfigures (a) and (b)–(d) of Fig. 3 correspond to the images reconstructed by use of the FBP and PWLS algorithms at the 10-th, 50-th, and 90-th iterations, respectively.

**a**

^{(ℓ)})|| and normalized mean square error of reconstructed images versus iteration number defined as ||

**â**

^{(ℓ)}–

**a**||/||

**a**|| were examined and are presented in Fig. 4, where

**â**

^{(ℓ)}is the estimate in the

*ℓ*-th iteration and ||·|| denotes L2 norm. The normalized mean square errors in Figs. 4(b) and 4(d) reveal that the PWLS and PLS algorithms converge differently. The proposed PWLS method, though converges slower than PLS one, reaches the solution of minimal error around the 80-th iteration. As for the PLS algorithm, whose minimal error solution occurs around the 3-rd iteration diverges soon after that. Based on the above observation, in all the simulation studies hereafter, we terminated the CG algorithm after 90 and 15 iterations for the PWLS and PLS algorithms, respectively.

### 5.4. Empirical determination of reconstructed image statistics

15. C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. **37**, 270 (2010). [CrossRef] [PubMed]

*J*(=2000) reconstructed refractive images estimated up to 90 and 15 iterations from noisy realizations of intensity measurement pairs

*I*

_{1}[

*r, s*,

_{r}*k*] and

*I*

_{2}[

*r, s*,

_{r}*k*], as described in Eq. (20) with

*σ*= 1%.

*J*noisy sets of

*α*̂ were reconstructed by the PWLS and PLS methods and their autocovariance functions were empirically determined by where superscript

*i*denotes the

*i*-th noisy realization.

*σ*= 10%, the noise amplification in the images produced by use of the FBP and PLS methods are much more pronounced than in those produced by use of the PWLS method. The profiles of autocovariance functions of these results are presented in Figs. 7(a) and 7(b), in which the peak values of the profiles for the FBP and PLS results are about 100 times of their counterparts in Fig. 6, while the corresponding noise amplification by use of the PWLS method only increases by about 6-fold.

### 5.5. Signal detection studies

*t*(

*α*) =

**w**

^{†}

*α*for 2000 pairs of reconstructed refractive images, where

*ᾱ*

_{1}and

*ᾱ*

_{0}correspond to the average images of signal present and signal absent, respectively. In the signal detection simulation, Phantom set B was employed and the signal to be detected corresponds to one small ellipsoid that is embedded in another larger ellipsoid (i.e., the background). Here,

*α*refers to the refractive estimates. The numerical phantoms corresponding to signal present and signal absent cases are illustrated in Figs. 8(a) and (b). The figure of merit is the receiver operating characteristic (ROC) curve, which summarizes the performance of the mathematical observer.

*a*

_{1,2}[0,

*s*,

*t*] reconstructed from noisy measurement data with

*σ*= 10% are contained in Fig. 9, where solid, solid with circle marker and dashed curves denote the results obtained by use of the PWLS, PLS and FBP algorithms, respectively. Note that the solid curve and solid curve with circle marker in Fig. 9 correspond to the PWLS and PLS results obtained at the 50-th and 3-rd iterations. Even though only a single ROC curve corresponding to the PWLS algorithm is shown here, its differences with that evaluated after 70 or 90 iterations are negligible. The ROC curves produced by use of the PWLS algorithm that employed 70 and 90 iterations were found to overlap with the results obtained by use of 50 iterations shown in Fig. 9. The ROC curves indicate that the observer performed considerably better with images reconstructed by use of the PWLS method than those obtained by the conventional PLS and FBP algorithms.

## 6. Summary and conclusions

*a priori*information in PCT reconstruction. The addition of second-order statistical properties of the projection data not only enhanced the stability of tomographic reconstruction, but also mitigated the low frequency noise in PCT images, as well as the length of noise correlation. This is confirmed by comparing the empirically determined autocovariance profiles of the PWLS results with those estimated by the PLS and FBP algorithms. It is worth noted that the correlation length and magnitude of autocovariance functions increase as iterations proceed, but they both gradually converge when approaching the minimum error solution. The peak value and correlation length of the autocovariance profile of the PWLS results estimated at 90-th iteration are still lower and smaller than those of the FBP and PLS algorithms. The direct impact of such noise reduction in the practical usage of these images is improvement of signal detection, which is corroborated by our numerical observer study.

## Acknowledgments

## References and links

1. | P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE |

2. | S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express |

3. | D. M. Paganin, |

4. | R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. |

5. | S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol. |

6. | F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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,” Radiol. |

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

8. | T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, and S. W. Wilkins, “Effects of incident illumination on in-line phase-contrast imaging,” J. Opt. Soc. Am. A |

9. | P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett. |

10. | K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett. |

11. | P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. |

12. | M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. |

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

14. | C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in quantitative in-line X-ray phase-contrast imaging,” Opt. Express |

15. | C.-Y. Chou and M. A. Anastasio, “Noise texture and signal detectability in propagation-based x-ray phase-contrast tomography,” Med. Phys. |

16. | C.-Y. Chou and M. A. Anastasio, “Influence of imaging geometry on noise texture in x-ray in-line phase-contrast imaging,” in Medical Imaging 2008: Physics of Medical Imaging, J. Hsieh and E. Samei, eds., Proc. SPIE6913, 69131Z (2008). URL http://link.aip.org/link/?PSI/6913/69131Z/1. |

17. | C.-Y. Chou and M. A. Anastasio, “Statistical properties of X-ray phase-contrast tomography,” in Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2009. EMBC 2009 , pp. 6648 –6650 (2009). [CrossRef] [PubMed] |

18. | P. Cloetens, “Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation: Principles, Implementation and Applications,” Ph.D. thesis, Vrije Universiteit Brussel (1999). |

19. | W. D. Stanley, G. R. Dougherty, and R. Dougherty, |

20. | J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik |

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

22. | C. J. Kotre and I. P. Birch, “Phase contrast enhancement of x-ray mammography: a design study,” Phys. Med. Biol. |

23. | P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE |

24. | R. Lewitt, “Alternatives to voxels for image representation in iterative reconstruction algorithms,” Phys. Med. Biol. |

25. | M. Defrise and G. T. Gullberg, “Image reconstruction,” Phys. Med. Biol. |

26. | D. Snyder and M. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. |

27. | J. Fessler and S. Booth, “Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction,” IEEE Trans. Image Process. |

28. | A. Papoulis and S. U. Pillai, |

29. | H. H. Barrettt and K. J. Myers, |

30. | M. Bertero, |

31. | E. Mumcuoglu, R. Leahy, S. R. Cherry, and Z. Zhou, “Fast gradient-based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imag. |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(110.4280) Imaging systems : Noise in imaging systems

(110.7440) Imaging systems : X-ray imaging

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: September 7, 2011

Revised Manuscript: October 15, 2011

Manuscript Accepted: October 23, 2011

Published: November 14, 2011

**Virtual Issues**

Vol. 7, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Cheng-Ying Chou and Pin-Yu Huang, "Image reconstruction in phase-contrast tomography exploiting the second-order statistical properties of the projection data," Opt. Express **19**, 24396-24410 (2011)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-24-24396

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### References

- P. Cloetens, W. Ludwig, E. Boller, L. Helfen, L. Salvo, R. Mache, and M. Schlenker, “Quantitative phase-contrast tomography using coherent synchrotron radiation,” in Developments in X-Ray Tomography III, U. Bonse, ed., Proc. SPIE4503, 82–91 (2002).
- S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express11, 2289–2302 (2003). [CrossRef] [PubMed]
- D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, New York, 2006).
- R. A. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol.49, 3573–3583 (2004). URL http://stacks.iop.org/0031-9155/49/3573 . [CrossRef] [PubMed]
- S. Fiedler, A. Bravin, J. Keyrilainen, M. Fernandaz, P. Suortti, W. Thomlinson, M. Tenhenun, P. Virkkunen, and M. Karjalainen-Lindsberg, “Imaging lobular breast carcinoma: comparison of synchrotron radiation CT-DEI technique with clinical CT, mammography and histology,” Phys. Med. Biol.49, 1–15 (2004). [CrossRef]
- F. Arfelli, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. D. Palma, M. D. 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,” Radiol.215, 286–293 (2000).
- A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A19, 472–480 (2002). [CrossRef]
- T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, and S. W. Wilkins, “Effects of incident illumination on in-line phase-contrast imaging,” J. Opt. Soc. Am. A23, 34–42 (2006). URL http://josaa.osa.org/abstract.cfm?URI=josaa-23-1-34 . [CrossRef]
- P. McMahon, A. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau, “X-ray tomographic imaging of the complex refractive index,” Appl. Phys. Lett.83, 1480–1482 (2003). [CrossRef]
- K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. M. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett.77, 2961–2964 (1996). [CrossRef] [PubMed]
- P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys.81, 5878–5886 (1997). [CrossRef]
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