## Non-linear iterative phase retrieval based on Frechet derivative |

Optics Express, Vol. 19, Issue 23, pp. 22809-22819 (2011)

http://dx.doi.org/10.1364/OE.19.022809

Acrobat PDF (4308 KB)

### Abstract

Several methods of phase retrieval for in-line phase tomography have already been investigated based on the linearization of the relation between the phase shift induced by the object and the diffracted intensity. In this work, we present a non-linear iterative approach using the Frechet derivative of the intensity recorded at a few number of propagation distances. A Landweber type iterative method with an analytic calculation of the Frechet derivative adjoint is proposed. The inverse problem is regularized with the smoothing *L*_{2} norm of the phase gradient and evaluated for several different implementations. The evaluation of the method was performed using a simple phase map, both with and without noise. Our approach outperforms the linear methods on simulated noisy data up to high noise levels and thanks to the proposed analytical calculation is suited to the processing of large experimental image data sets.

© 2011 OSA

## 1. Introduction

1. G. R. Davis and S. L. Wong, “X-ray microtomography of bones and teeth,” Physiol. Meas. **17**, 121–146 (1996). [CrossRef] [PubMed]

7. G. J. Kazakia, A. J. Burghardt, S. Cheung, and S. Majumdar, “Assessment of bone tissue mineralization by conventional X-ray microcomputed tomography,” Med. Phys. **33**, 3170–3179 (2008). [CrossRef]

10. P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D.:Appl. Phys. **29**, 133–146 (1996). [CrossRef]

12. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. **77**, 2961–2964 (1996). [CrossRef]

13. 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**(9), 5878–5886 (1997). [CrossRef]

15. T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. **68**, S13–S17 (2008). [CrossRef] [PubMed]

16. D. M. Paganin, *Coherent X-Ray Optics* (Oxford University Press, 2006). [CrossRef]

17. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A **13**, 1670–1682 (1996). [CrossRef]

18. T.E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. **220**, 49–58 (2003). [CrossRef]

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

10. P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D.:Appl. Phys. **29**, 133–146 (1996). [CrossRef]

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

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

*I*(

*φ*) relating the measurements to the phase map. Since the inverse operator

*I*

^{−1}(

*φ*) is not bounded, the phase retrieval problem considered in this work is ill-posed. Thus Tikhonov regularization is included. The problem is expressed as the minimization of a non-linear regularized functional with the Frechet derivative of

*I*(

*φ*). An analytic expression of the adjoint of the Frechet derivative allows to speed up the algorithm and reduce memory requirements.

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

## 2. The direct problem and the linearized inverse problem phase contrast imaging

### 2.1. The direct problem

*λ*, its interaction with beam can be described by its complex refractive index, usually written as [23]: where

*δ*is the refractive index decrement and

_{r}*β*is the absorption index for the spatial coordinate (

*x,y,z*). In the following,

*z*denotes the propagation direction of the x-rays. Assuming a thin object, the diffraction within the object is neglected and so the interaction of x-rays with the object can be described by a transmittance function

*T*of the coordinates

**x**= (

*x,y*) in a plane perpendicular to the propagation direction

*z*.

*a*(

**x**), and phase shift

*φ*(

**x**) induced by the object can be considered as projections of the absorption and refraction index respectively:

*D*is given by the squared modulus of the exit wave: where * denotes the 2D convolution of the transmitance with the Fresnel propagator,

*D*being the propagation distance along

*z*. The direct problem can also be written in terms of Fourier transforms [24

24. 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**, 4556–4565 (2008). [CrossRef] [PubMed]

*ℱ*of a function

*g*(

**x**) is defined as: where

**f**= (

*f*,

_{x}*f*) is the spatial frequency and

_{y}**x**·

**f**denotes the scalar product.

### 2.2. The linear inverse problem

**32**, 1617–1619 (2007). [CrossRef] [PubMed]

*D*if the phase was zero and

*I*

_{0}is the intensity at exit surface of the object. This approach seems to be more accurate and robust to noise than other methods for mixed absorption and phase objects, like Contrast Transfer Function (CTF) or Transport of Intensity Equation(TIE) methods which rely on weak absorption or short propagation distance assumptions [24

24. 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**, 4556–4565 (2008). [CrossRef] [PubMed]

24. 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**, 4556–4565 (2008). [CrossRef] [PubMed]

26. M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. **19**, 2425–2436 (2010). [CrossRef]

*ψ*(

**x**) =

*I*

_{0}(

**x**)

*φ*(

**x**), the phase retrieval problem can be solved iteratively as: with and where

*α*is a regularizing parameter and

*ψ*

^{(n)}(

**x**) is the phase-absorption product at iteration

*n*. This iterative approximate solution lacks an optimality proof but it will be used as the starting point of our non-linear inverse problem approach.

## 3. The non-linear inverse phase retrieval problem

### 3.1. A Landweber type iterative method

*φ*,

*I*(

_{D}*φ*). The operator

*I*(

_{D}*φ*) can be considered as a non-linear operator which is Frechet differentiable in its domain. In the following, we will consider the phase having a Lipschitz bounded support Ω and that the domain

*𝒟*[

*I*(

_{D}*φ*)] of the operator

*I*(

_{D}*φ*) belongs to the functional Sobolev space

_{L2(Ω)}denotes the

*L*

_{2}(Ω) norm,

*I*approximates the exact data

_{δ}*I*with the accuracy

_{D}*δ*In order to regularize the problem, we introduce a Tikhonov’s functional of the following form: where

*α*is a regularizing parameter. The stabilizing norm is thus a Sobolev type regularizing term based on the gradient of the phase to be retrieved.

*I*′(

_{D}*φ*), where 〈,〉 denotes the scalar product or with the adjoint

*I*′ (

*φ*)

^{*}of the Frechet derivative The optimality condition is then:

*τ*is chosen in order to minimize the Tikhonov’s functional along the descent direction: where

_{k}*δ*=

_{k}*I′*(

_{D}*φ*)*[

_{k}*I*(

_{D}*φ*) –

_{k}*I*] –

_{δ}*α*Δ

_{k}*φ*is the descent direction. An approximate value is obtained with a dichotomy strategy.

_{k}*φ*

_{k}_{+1}is obtained from the iterate

*φ*with:

_{k}*φ*at the iteration

_{k}*k*, a linear search procedure is introduced with a variable step

*τ*yielding the following modification of the standard Landweber method

_{k}*J*(

*φ*). These equations correspond to a Landweber method with a regularizing smoothing term and an adaptative step length. This type of method has been used in several works on non-linear ill-posed problems [28

28. M. Hanke, A. Neubauer, and O. Scherzer, “A convergence analysis of the landweber iteration for nonlinear ill-posed problems,” Numer. Math. **72**, 21–37 (1995). [CrossRef]

29. I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. **14**, 764–792 (2008). [CrossRef]

### 3.2. Update of iterates

#### 3.2.1. *Analytical expression of the adjoint of the diffracted intensity derivative*

*I*(

_{D}*φ*) at the point

*φ*is the linear operator

_{k}*G*defined by the relation:

_{k}*I*′(

_{D}*φ*)(

_{k}*ɛ*) =

*G*(

_{k}*ɛ*) can be calculated explicitly as: or

#### 3.2.2. Numerical calculation by implicit filtering

*I*′ (

_{D}*φ*) are thus approximated by centered differences formulas. It is noteworthy that this finite difference method requires 2N evaluations of the Tikhonov functional, where N is the phase vector dimension and it leads to a squared matrix of size

*N*

^{2}. In this work, the phase increment is set to 0.05 rad which is the estimated noise level on the phase. This phase increment must be small enough so that the linearization is valid and higher than the noise level.

## 4. Simulations

### 4.1. Simulation of the diffracted intensity

**35**, 4556–4565 (2008). [CrossRef] [PubMed]

*δ*and

*β*, for different materials at 24 keV were used (

*λ*= 0.5166

*Å*) for the phantom. Propagation in free-space was simulated using Eq. (4). The original phase map to be retrieved is displayed in Fig. 1(a), together with the absorption map in Fig. 1(b) and the corresponding Fresnel diffraction pattern for D=1.4 m without noise in Fig. 1(c). The convolution product was calculated by Fourier transforms and the intensity has been obtained as the squared modulus in the spatial domain of this convolution. Using the free-space propagation equation, images were calculated for the eight propagation distances 0.2 m, 0.4 m, 0.6 m, 0.8 m, 1.0 m, 1.2 m, 1.4 m and 1.6 m. These distances are suitable for testing and comparing the phase retrieval methods.

^{75×75}. The Frechet derivative

*G*calculated with the finite difference method at the point

_{k}*φ*is a matrix of ℝ

_{k}^{5475×5475}. The phase contrast images were all corrupted with additive Gaussian white noise with various peak-to-peak signal to noise ratios (PPSNR), between 24 dB and 0 dB. The peak-to-peak signal to noise ratio is defined by: where

*f*is the maximum signal amplitude and

_{max}*n*is the maximum noise amplitude.

_{max}### 4.2. Initialization and stopping conditions

*L*

_{2}(Ω) norm. If

*φ*

^{*}is the phase to be recovered and

*φ*the current estimate, the NMSE is calculated as:

_{k}*φ*

_{k}_{+1}is accepted if the following two conditions are satisfied: and the iterations are terminated when with the residual value

*δ*equal to the noise level. A divergence of the iterates away from the solution is obtained if these stopping conditions are not imposed.

## 5. Results and discussion

*A*_{1}: Kaczmarz type finite difference Landweber method with a regularization term or without any regularization (*α*= 0).*A*_{2}: Sequential type finite difference Landweber method with a regularization term or without regularization term performed with N=100 iterations for each distance.*A*_{3}: Kaczmarz type analytic Landweber method with the former stopping conditions with a regularization term or without any regularization (*α*= 0).

*A*

_{1}and

*A*

_{2}is the way of the cyclic iteration in the intensity maps is performed. The last algorithm

*A*

_{3}represents the Kaczmarz type analytic Landweber method with the stopping conditions (28), (29) and (30) both without regularization term or with the regularization method.

*A*

_{1}(PPSNR=24 dB),

*A*

_{2}(noisy-free) and

*A*

_{3}(PPSNR=24 dB). The regularization parameter was set to

*α*= 0.01. In order to have more quantitative information about the convergence rates and to compare the algorithms, we have studied NMSE for the phase shift as a function of the number of iterations. Figure 3 shows the evolution of the NMSE as a function of the number of iterations for the different algorithms on the noise-free and noisy data (PPSNR=24 dB) for

*α*= 0.01.

*A*

_{2}. Kaczmarz type methods are thus to be preferred.

*α*= 0.01) smoothing regularization yields good phase retrieval convergence results.

*A*

_{3}has good convergence properties. The algorithm

*A*

_{3}is also much faster since the large scale matrix used in the finite difference methods is replaced by the analytic expression of the adjoint of the Frechet derivative. It should be noted that at the end of the iterations, the condition of Eq. (30) is fulfilled and that we have to stop the iterations considering the noise level.

*A*

_{3}for the noise-free data and noisy data (PP-SNR=24 dB) with a smooth regularization

*α*= 0.01 are displayed in Figs. 4(a) and 4(b) respectively. The final NMSE are 0.095 for noisy-data (PPSNR=24 dB) and 0.09 for noise-free data.

*φ*||

_{L2}are well-known. An isotropic smoothing effect is obtained and the boundaries are not well preserved. This is obvious in Fig. 2(c). The noise is suppressed but the high values of the gradient are too greatly penalized on the edge. In future works, the gradient ∇

*φ*may be replaced by a non-linear functional of

*φ*as in semi-quadratic regularization or by the bounded variation semi-norm [27] or by anisotropic terms.

## 6. Conclusion

*L*

_{2}norm of the phase gradient. The best convergence rates are found when the various distances are treated with a Kaczmarz type method where one descent iteration is performed for each distance. The evaluation of the method was performed using a simple phase map, both with and without noise. For the simulated data, the normalized mean square error was measured. Tikhonov regularization based on linear filtering used in this work has some well-known drawbacks since it does not only smooth noise but also blurs important features such as edges. To avoid these shortcomings, non-linear partial differential diffusion equations may be useful. Therefore, other regularization methods will be tested in future work. Our approach outperforms the linear methods on simulated noisy data for PPSNR above 20dB and the nonlinearities of the Fresnel diffraction are well taken into account. The analytic calculation of the adjoint of the Frechet derivative speeds up the calculations and overcomes memory limitations due to the Frechet derivative matrix. Thus this method opens promising perspectives to process experimental data in various applications.

## References and links

1. | G. R. Davis and S. L. Wong, “X-ray microtomography of bones and teeth,” Physiol. Meas. |

2. | M. Salomé, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel, and P. Spanne, “A synchrotron radiation microtomography system for the analysis of trabecular bone samples,” Med. Phys. |

3. | S. Nuzzo, F. Peyrin, P. Cloetens, J. Baruchel, and G. Boivin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Med. Phys. |

4. | S. Bayat, L. Apostol, E. Boller, T. Brochard, and F. Peyrin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Nucl. Instr. Meth. Phys. Res. A |

5. | C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault, and F. Peyrin, “Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microachitecture of human femoral heads,” Med. Phys. |

6. | M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. |

7. | G. J. Kazakia, A. J. Burghardt, S. Cheung, and S. Majumdar, “Assessment of bone tissue mineralization by conventional X-ray microcomputed tomography,” Med. Phys. |

8. | J. Baruchel, E. Marire, P. Merle, and G. Peix, |

9. | U. Bonse, “Developments in X-ray tomography II,” Proc. SPIE |

10. | P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D.:Appl. Phys. |

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

12. | S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. |

13. | 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. |

14. | A. Momose, T. Takeda, Y. Itai, A. Yoneyama, and K. Hirano, “Phase-contrast tomographic imaging using an X-ray interferometer,” J. Synchrotron. Rad. |

15. | T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. |

16. | D. M. Paganin, |

17. | T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A |

18. | T.E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. |

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

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

21. | S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X-rays,” Rev. Sci. Instrum |

22. | J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. |

23. | M. Born and E. Wolf, |

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

25. | M. Langer, P. Cloetens, and F. Peyrin, “Fourier-wavelet regularization of phase retrieval in X-ray in-line phase tomography,” J. Opt. Soc. Am. A |

26. | M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. |

27. | O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, |

28. | M. Hanke, A. Neubauer, and O. Scherzer, “A convergence analysis of the landweber iteration for nonlinear ill-posed problems,” Numer. Math. |

29. | I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. |

30. | C. T. Kelley and P. Gilmore, “An implicit filtering algorithm for optimization of functions with many local minima,” SIAM J. Optm. |

31. | D. Stoneking, G. L. Bilbro, R. Trew, P. Gilmore, and C. T. Kelley, “Yield optimization using a gaas process simulator coupled to a physical device model,” IEEE Trans. Microwave Theory Tech. |

32. | C. T. Kelley, “Iterative methods for optimization,” |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(100.5070) Image processing : Phase retrieval

(110.7440) Imaging systems : X-ray imaging

(180.7460) Microscopy : X-ray microscopy

**ToC Category:**

Image Processing

**History**

Original Manuscript: May 4, 2011

Revised Manuscript: June 22, 2011

Manuscript Accepted: June 24, 2011

Published: October 26, 2011

**Virtual Issues**

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

**Citation**

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, "Non-linear iterative phase retrieval based on Frechet derivative," Opt. Express **19**, 22809-22819 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22809

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### References

- G. R. Davis and S. L. Wong, “X-ray microtomography of bones and teeth,” Physiol. Meas. 17, 121–146 (1996). [CrossRef] [PubMed]
- M. Salomé, F. Peyrin, P. Cloetens, C. Odet, A. M. Laval-Jeantet, J. Baruchel, and P. Spanne, “A synchrotron radiation microtomography system for the analysis of trabecular bone samples,” Med. Phys. 26, 2194–2204 (1999). [CrossRef] [PubMed]
- S. Nuzzo, F. Peyrin, P. Cloetens, J. Baruchel, and G. Boivin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Med. Phys. 29, 2672–2681 (2002). [CrossRef] [PubMed]
- S. Bayat, L. Apostol, E. Boller, T. Brochard, and F. Peyrin, “Quantification of the degree of mineralization of bone in three dimensions using synchrotron radiation microtomography,” Nucl. Instr. Meth. Phys. Res. A 548, 247–252 (2005). [CrossRef]
- C. Chappard, A. Basillais, L. Benhamou, A. Bonassie, N. Bonnet, B. Brunet-Imbault, and F. Peyrin, “Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microachitecture of human femoral heads,” Med. Phys. 33, 287–293 (2003).
- M. Ito, S. Ejiri, H. Jinnai, J. Kono, S. Ikeda, A. Nishida, K. Uesugi, N. Yagi, M. Tanaka, and K. Hayashi, “Bone structure and mineralization demonstrated using synchrotron radiation computed tomography (SR-CT) in animal models: preliminary findings,” J. Bone Miner. Metab. 21, 3568–3577 (2006).
- G. J. Kazakia, A. J. Burghardt, S. Cheung, and S. Majumdar, “Assessment of bone tissue mineralization by conventional X-ray microcomputed tomography,” Med. Phys. 33, 3170–3179 (2008). [CrossRef]
- J. Baruchel, E. Marire, P. Merle, and G. Peix, X-ray Tomography in Material Science (Hermes Science Publications, 2000).
- U. Bonse, “Developments in X-ray tomography II,” Proc. SPIE 3775 (1999).
- P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard X-ray imaging,” J. Phys. D.:Appl. Phys. 29, 133–146 (1996). [CrossRef]
- S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase contrast imaging using polychromatic hard X-rays,” Nature (London) 384, 335–338 (1996). [CrossRef]
- S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. 77, 2961–2964 (1996). [CrossRef]
- 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(9), 5878–5886 (1997). [CrossRef]
- A. Momose, T. Takeda, Y. Itai, A. Yoneyama, and K. Hirano, “Phase-contrast tomographic imaging using an X-ray interferometer,” J. Synchrotron. Rad. 5, 309–314 (1998). [CrossRef]
- T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. 68, S13–S17 (2008). [CrossRef] [PubMed]
- D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006). [CrossRef]
- T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996). [CrossRef]
- T.E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003). [CrossRef]
- J.R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
- J. P. Guigay, “Fourier transform analyis of Fresnel diffraction patterns in in-line holograms,” Optik 46, 12–125 (1977).
- S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard X-rays,” Rev. Sci. Instrum 76, 1–7 (2005). [CrossRef]
- J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32, 1617–1619 (2007). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1997).
- 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, 4556–4565 (2008). [CrossRef] [PubMed]
- M. Langer, P. Cloetens, and F. Peyrin, “Fourier-wavelet regularization of phase retrieval in X-ray in-line phase tomography,” J. Opt. Soc. Am. A 28, 1877–1882 (2009). [CrossRef]
- M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2425–2436 (2010). [CrossRef]
- O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variationnal Methods in Imaging (Springer Verlag, 2008).
- M. Hanke, A. Neubauer, and O. Scherzer, “A convergence analysis of the landweber iteration for nonlinear ill-posed problems,” Numer. Math. 72, 21–37 (1995). [CrossRef]
- I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008). [CrossRef]
- C. T. Kelley and P. Gilmore, “An implicit filtering algorithm for optimization of functions with many local minima,” SIAM J. Optm. 5, 269–285 (1985).
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