## Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiography |

Optics Express, Vol. 19, Issue 25, pp. 25545-25558 (2011)

http://dx.doi.org/10.1364/OE.19.025545

Acrobat PDF (7109 KB)

### Abstract

Phase retrieval from unidirectional radiographic differential phase contrast images requires integration of noisy data. A method is presented, which aims to suppress stripe artifacts arising from direct image integration. It is purely algorithmic and therefore, compared to alternative approaches, neither additional alignment nor an increased scan time is required. We report on the theory of this method and present results using numerical as well as experimental data. The method shows significant improvements on the phase retrieval accuracy and enhances contrast in the phase image. Due to its general applicability, the proposed method provides a valuable tool for various 2D imaging applications using differential data.

© 2011 OSA

## 1. Introduction

*in-situ*studies, and materials science. The interaction of X-rays with matter can be described by the complex index of refraction

*n*= 1 –

*δ*+

*iβ*, where

*δ*characterizes the phase shift and

*β*the attenuation properties of the material. In the diagnostic energy range of X-rays (10 – 100keV),

*δ*is typically three orders of magnitude larger than

*β*. The consequently high interaction cross section makes the phase shift measurement a favorable imaging modality. It has been demonstrated that phase images can provide higher contrast than absorption-based images and contain complementary information about the sample [1

1. R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today **53**, 23–26 (2000). [CrossRef]

2. A. Momose and J. Fukuda, “Phase-contrast radiographs of nonstained rat cerebellar specimen,” Med. Phys. **22**, 375–379 (1995). [CrossRef] [PubMed]

3. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. **66**, 5486–5492 (1995). [CrossRef]

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

6. U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. **6**, 155–156 (1965). [CrossRef]

7. A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. **2**, 473–475 (1996). [CrossRef] [PubMed]

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

*ϕ*), its first spatial derivative (∇

*ϕ*) or its second derivative (∇

^{2}

*ϕ*). Many of the reported methods require highly coherent X-rays, only available at synchrotron sources. A recently developed technique based on grating interferometry has established itself as a suitable technique for phase contrast imaging on synchrotron sources [13

13. C. David, B. Nöhammer, H. Solak, and E. Ziegler, “Differential X-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. **81**, 3287–3289 (2002). [CrossRef]

14. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. **42**, L866–L868 (2003). [CrossRef]

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

16. M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus X-ray source,” Appl. Phys. Lett. **90**, 224101 (2007). [CrossRef]

17. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer” Nature Mater. **7**, 134–137 (2008). [CrossRef]

*ϕ*), the method is also referred to as differential phase contrast (DPC) imaging. Grating interferometry has mainly been demonstrated in unidirectional mode, where the gratings have a periodic line pattern and the phase gradient can only be measured in the perpendicular direction to these lines. In bidirectional grating interferometry, “checkerboard” and “mesh” like grating patterns are used, and the DPC acquisition can be extended to both directions [18

18. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-Dimensional X-Ray Grating Interferometer,” Phys. Rev. Lett. **105** (2010). [CrossRef]

19. D. Paganin, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. **234**, 87–105 (2004). [CrossRef]

*ϕ*, ∇

*φ*, ∇

^{2}

*φ*), further post processing (integration filter) is necessary to retrieve the quantitative phase signal

*ϕ*. Phase retrieval from the second derivative has mainly been discussed in association with propagation based methods [20

20. D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. **206**, 33–40 (2002). [CrossRef] [PubMed]

21. A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography” Opt. Express **14**, 8103–8110 (2006). [CrossRef] [PubMed]

22. M. de Jonge, B. Hornberger, C. Holzner, D. Legnini, D. Paterson, I. McNulty, C. Jacobsen, and S. Vogt, “Quantitative Phase Imaging with a Scanning Transmission X-Ray Microscope,” Phys. Rev. Lett. **100**, 163902 (2008). [CrossRef] [PubMed]

23. M. Arnison, K. Larkin, C. Sheppard, N. Smith, and C. Cogswell, “Linear phase imaging using differential interference contrast microscopy” J. microsc. **214**, 7–12 (2004). [CrossRef] [PubMed]

24. B. Hornberger, M. Feser, and C. Jacobsen, “Quantitative amplitude and phase contrast imaging in a scanning transmission X-ray microscope” Ultramicroscopy **107**, 644–655 (2007). [CrossRef] [PubMed]

25. B. Hornberger, M. de Jonge, M. Feser, P. Holl, C. Holzner, C. Jacobsen, D. Legnini, D. Paterson, P. Rehak, L. Strüder, and S. Vogt, “Differential phase contrast with a segmented detector in a scanning X-ray microprobe” J. Synchrotron Radiat. **15**, 355–362 (2008). [CrossRef] [PubMed]

26. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy” Science **321**, 379–382 (2008). [CrossRef] [PubMed]

*et al.*also discussed this issue and proposed a bidirectional scanning approach [27

27. C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard X-rays,” Opt. Express **15**, 1175–1181 (2007). [CrossRef] [PubMed]

18. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-Dimensional X-Ray Grating Interferometer,” Phys. Rev. Lett. **105** (2010). [CrossRef]

28. M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images” Phys. Med. Biol. **51**, 1769–1778 (2006). [CrossRef] [PubMed]

## 2. Methods

### 2.1. Experimental setup and phase retrieval

*π*to the wavefront. This generates an interference pattern downstream, with maximum intensity differences at fractional Talbot distances, given by

*m*is an odd integer,

*g*

_{1}the period of the phase grating and

*λ*the wavelength [29

29. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express **12**, 6296–6304 (2005). [CrossRef]

*ϕ*(

*x*,

*y*), which is given by the line integral In this equation,

*λ*is the wavelength and

*δ*(

*x*,

*y*,

*z*) is the refractive index decrement, associated to the real part of the complex index of refraction

*n*= 1 –

*δ*+

*iβ*. The partial derivative of the phase profile with respect to

*x*is proportional to the beam refraction angle

*α*[29

29. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express **12**, 6296–6304 (2005). [CrossRef]

*x*-direction (see Fig. 1). This fringe shift, measured in radians with respect to the fringe period

*g*

_{2}, corresponds to the DPC signal and is given by Here,

*d*is the propagation distance downstream of the phase grating and Eq. (2) was used to obtain the right part of Eq. (3). The fringe period

*g*

_{2}is typically a few microns and therefore, the interference pattern cannot generally be resolved sufficiently by the X-ray detector. For the determination of

*φ*, an absorption grating, having the same period as the fringes (

*g*

_{2}) and acting as an analyzer mask, is positioned at a fractional Talbot order distance. The acquisition protocol consists in a so called “phase stepping scan”, moving the absorption grating in equidistant steps in

*x*direction and acquiring an image at each position [29

29. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express **12**, 6296–6304 (2005). [CrossRef]

*a*), the differential phase (

*φ*), and a scattering signal (

*v*) can be calculated by a Fourier analysis: where

*a*is the magnitude and

_{i}*φ*is the phase of the

_{i}*i*-th Fourier component of the PSC. Signals from reference and object scan are labeled by “obj” and “ref”, respectively. The scattering signal

*v*corresponds to the reduced fringe visibility

*V*

_{obj}/

*V*

_{ref}.

*φ*. According to Eq. (3), the retrieval of the phase image

*ϕ*(

*x*,

*y*) requires a one-dimensional integration in

*x*-direction, given by Due to noise in the DPC image, this integration accumulates the errors and therefore the noise variance. The typical result are strong stripe artifacts, reducing image contrast and impeding an exact phase retrieval. Figure 2 shows a noisy DPC image generated from the modified Shepp-Logan phantom [30] and the phase retrieval obtained from an integration in the horizontal direction. Prior to the integration step, the mean value was removed in every line of the DPC image to enforce a cumulative sum of zero. However, this simple correction approach could not remove strong horizontal stripe artifacts which still appear across the image.

### 2.2. Measurement and noise model

*D*is a forward operator which models the relation between

_{x}*ϕ*and

*φ*in absence of noise and

*w*is a random image modelling the noise in measured data. The noise standard deviation derived by Engel et al. [31

31. K. Engel, D. Geller, T. Köhler, G. Martens, S. Schusser, G. Vogtmeier, and E. Rössl, “Contrast-to-noise in X-ray differential phase contrast imaging,” Nucl. Instrum. Methods A **648**, 202–207 (2010). [CrossRef]

*V*is the visibility and

*N*is the number of photons in the measurement.

*D*models the first derivative and can be implemented as a finite differences transform operator, given by where

_{x}*i*and

*j*are now discrete coordinates (pixel coordinates) and

*N*is the image size in

_{i}*x*-direction.

### 2.3. Regularized image integration method

*φ*causes the typical horizontal stripe artifacts. The phase retrieval can be improved by solving a constrained optimization problem. While maintaining consistency with the measured DPC data, the solution is retrieved by minimizing a cost function. The constrained optimization problem for the above model can be written as Essentially, this optimization problem seeks for the minimal

*ℓ*norm of

_{p}*D*

_{y}*f*for any

*f*being consistent with the measurement data

*φ*in a least squares sense.

*W*is a weighting operator, multiplying each pixel with its inverse noise standard deviation (1/

*σ*) to account for the noise model. This approach is also known under the name penalized weighted least squares (PWLS) [32

32. K. Sauer and C. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process. **41**, 534–548 (1993). [CrossRef]

*D*and

_{x}*D*are finite differences transforms in the

_{y}*x*- and

*y*-direction, respectively,

*f*is the solution of the optimization problem and

*ɛ*is a boundary for the noise. The

*ℓ*norm of an image is defined as For instance, the most often used

_{p}*ℓ*

_{2}norm corresponds to the Euclidean norm. In this case, problem (9) is linear and an explicit inversion for

*f*exists. Here, we focus on the use of the

*ℓ*

_{1}norm, which is the sum of the magnitude of all pixel values. The

*ℓ*

_{1}norm minimization of the finite differences transform is well-known in denoising applications and often referred to as total variation denoising (two-dimensional finite differences) [33

33. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D Nonlinear Phenom. **60**, 259–268 (1992). [CrossRef]

*ℓ*

_{1}norm minimization typically shows strong edge preserving characteristics and causes less blurring than the

*ℓ*

_{2}norm. The better performance comes at the expense of a non-linear optimization problem, having no explicit inversion formula.

*p*= 1, the problem is first reformulated to an unconstrained form: The data consistency constraint and the optimization function are now expressed in a single objective function

*F*(

*f*). A Lagrange multiplier

*λ*(regularization parameter) is used for weighting the two terms. The parameter

*ɛ*from Eq. (9) is now implicitly chosen by adjusting

*λ*. This parameter is strongly data-dependent and in general difficult to determine. Here, it will be determined by evaluating well-known image quality metrics.

*a-priori*knowledge about the sample is available. Minimum norm constraints with any norm number

*p*and for any linear transform

*Tf*can be added. An example for regularized integration using two regularization terms will be given in the next section.

34. J. Nocedal and S. Wright, *Numerical optimization* (Springer verlag, 1999). [CrossRef]

*p*= 2) and the non-linear (

*p*= 1) case and thus provides a convenient way for a comparison. The implementation was done in Matlab (MathWorks, Version R2010a) and run on a 64-bit Linux machine (Intel Xeon 2.83GHz processor).

## 3. Results

### 3.1. Numerical data

*ϕ*was the known (ground-truth) image, i.e. the Shepp-Logan phantom, representing the line integral of

*δ*(real part of the refractive index) through an object. Noise has been generated by using the previously discussed model for DPC measurements (Eq. (7)). The calculation of a noise standard deviation map required the simulation of a transmission and a visibility image. For the transmission image, we assumed a constant ratio

*δ*/

*β*= 10 (

*β*is the imaginary part of the refractive index), yielding the transmission image directly from the known phase image. For the visibility image, a high scattering signal at strong edges of the object was assumed and small angle scattering was neglected. Therefore, an edge enhanced image was generated by applying a discrete two-dimensional Laplace filter to the phantom image, from which the visibility map was determined. Figure 3 shows the simulated transmission and visibility image and the combined noise standard deviation map. Using the noise standard deviation map, noisy DPC data, as shown in Fig. 2(b), was generated as input for the regularized image integration.

*λ*. First, this allows a direct comparison of regularized and conventional direct integration and second, it provides support for the choice of an optimal

*λ*, although this is of course dependent on the chosen metric. In order to compare the the non-linear

*ℓ*

_{1}and the linear

*ℓ*

_{2}norm minimization, the simulations are performed for

*p*∈ {1, 2}. For all simulations, the same convergence criterion in the NLCG algorithm is applied.

*f*

_{1}and

*f*

_{2}is defined as where

*N*and

_{i}*N*are the horizontal and vertical image sizes, respectively. Figure 4(e) displays the RMSE as a function of the regularization parameter

_{j}*λ*and with

*p*∈ {1, 2}, for regularized integration (RI), and the RMSE for direct integration (DI). For all values of

*λ*within the investigated range, a smaller RMSE could be measured for regularized integration. The minimum was achieved with the

*ℓ*

_{1}norm minimization at

*λ*

_{RMSE}= 5 · 10

^{−3}, yielding the optimal solution (displayed in Fig. 4(b)).

*σ*in Fig. 4(c) and (d), is smaller for the regularized integration, confirming the improved image quality.

*S*and

*σ*are the mean value and the standard deviation, respectively, within a region-of-interest (ROI) of the object (obj) and the background (bg). Figure 4(f) shows the CNR as a function of

*λ*, calculated by using the object and background regions marked with the red boxes in Fig. 4(a). The

*ℓ*

_{1}norm minimization again performed better than the

*ℓ*

_{2}norm and the optimal regularization parameter is

*λ*

_{CNR}= 4 · 10

^{−2}, which is almost an order of magnitude larger than the optimal value in terms of RMSE. This is mainly because high regularization parameters significantly suppress noise, resulting in a higher CNR. On the other hand, a high

*λ*can smooth the image and reduce edged sharpness. For this reason, optimizing the CNR is not always the best approach and may lead to an overestimation of

*λ*.

*ℓ*

_{1}norm minimization performed clearly better than the

*ℓ*

_{2}norm for both metrics. For this reason, in the examples of the following sections, only the

*ℓ*

_{1}norm minimization will be used for the regularized phase retrieval.

### 3.2. Experimental data

35. M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: The SLS experience,” Proc. SPIE **6318**, 63180M (2006). [CrossRef]

^{rd}fractional Talbot order. The source-to-phase grating distance was 25m, the inter-grating distance was

*d*= 120mm and the grating periods were

*g*

_{1}= 3.98

*μ*m and

*g*

_{2}= 2.0

*μ*m. The materials of the absorption and phase grating were gold and silicon, respectively [36

36. C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. **84**, 1172–1177 (2007). [CrossRef]

*n*= 1 –

*δ*+

*iβ*at 25keV of each material was known (available at http://sergey.gmca.aps.anl.gov), allowing for a calculation of the ground-truth image (calculated analytically by assuming a known sample profile).

*a-priori*knowledge about the image boundaries, assuming a background signal of zero. This assumption holds because the DPC image is flat-field corrected, essentially meaning that in a background pixel, the phase signals of the object and the reference scan should be the same. The optimization problem, consisting now of two regularization terms, is defined as: In the second regularization term, the matrix

*M*operates as a binary mask for the pixels in the image

*f*at the left and right edges. For the norm parameter,

*p*= 2 was chosen, because this constrains the image to have low energy at the edge pixels. While the first regularization parameter

*λ*

_{1}was varied, the second regularization parameter was kept constant at

*λ*

_{2}= 10

^{−2}. Since the second regularization term only constrains parts of the image, the phase retrieval is not very sensitive to

*λ*

_{2}, making an exact determination of an optimum for this parameter unnecessary. There is a high certainty that a background signal of zero is a true assumption, therefore the value should be kept high. On the other hand, in order to avoid a too strong influence of this term to the result,

*λ*

_{2}should at the same time be kept small. The choice of

*λ*

_{2}= 10

^{−2}is purely empirical and does not represent a unique optimum.

*λ*

_{1,RMSE}=

*λ*

_{1,CNR}= 5 · 10

^{−4}). Again, the suppression of stripe artifacts in the regularized integration is clearly visible, if Fig. 5(b) and (c) are compared. According to Fig. 5(e), the quantitative phase recovery could be improved over the whole investigated range of the regularization parameter, although a complete agreement to the calculated ground-truth was not achieved (Fig. 5(f)). However, it is important to notice that the result of Fig. 5(f) must be taken with care due to the following important facts: In this example, RI was compared with DI after a vertical summation of the images, not by looking at line profiles (as it was done in the simulations). This summation also smoothes out the stripe artifacts in DI and therefore, the DI line in Fig. 5(f) would be expected to highly match the ground-truth line, which is not the case. This inconsistency might result from several random effects, as for example inhomogeneities in the material, inexact numbers for the material densities and the refractive indeces or inconsistencies of the DPC measurements resulting from grating drift or grating imperfections from the fabrication process. Due to these facts, it is impossible to state that RI was indeed quantitatively more accurate than DI, simply because the ground-truth is actually unknown. On the other hand, due to the removal of the stripe artifacts, it is very likely that a line profile from RI is quantitatively more accurate than a line profile (no vertical summation) of DI.

## 4. Discussion

*ℓ*

_{1}regularized image integration can significantly improve the quality of phase images obtained from noisy, unidirectional DPC measurements without increasing the radiation dose. Longer exposure times, additional phase steps, or bidirectional measurements can be avoided. The method outperformed direct integration in the given examples in terms of RMSE compared to ground-truth data, noise pattern and CNR. The improved RMSE promises more accurate results for a quantitative data analysis, while a higher CNR will facilitate data post processing as for instance image segmentation.

*λ*, being the key parameter of the method. It determines the weighting between the data consistency and the optimization function and can be interpreted as a tuning parameter for the resulting image quality. If

*λ*is chosen too high, overregularization occurs, generating new image artifacts. A typical overregularization effect when using the

*ℓ*

_{1}norm minimization of the finite differences transform is the generation of piecewise constant areas in the image, looking similar to median filtering. If

*λ*is below its optimum, the regularization term is too weak and the stripe artifacts will not vanish. Naturally,

*λ*is data dependent and the optimum is not always trivial to determine. The evaluation of image metrics (CNR, RMSE) for the optimization of

*λ*is one possible approach amongst others. Another well known method is the L-curve analysis, where the data constraint term and the minimization terms are evaluated after each iteration [37

37. P. Hansen, “Analysis of Discrete Ill-Posed Problems by Means of the L-Curve,” SIAM Rev. **34**, 561–580 (1992). [CrossRef]

*λ*. Fortunately, this relaxes the requirement to exactly find an optimum value for

*λ*. Eventually, finding the optimum solution will remain a task dependent issue.

*p*. It has been demonstrated, that for the minimization of the finite differences transform, the

*ℓ*

_{1}norm performed better than the

*ℓ*

_{2}norm in terms of the evaluated metrics. The

*ℓ*

_{1}norm has also been subject to many investigations in the field of tomographic image reconstruction, where an image is assumed to have a sparse representation in a linear transform domain. The minimization of the

*ℓ*

_{1}norm in such a domain enforces sparsity and may improve image reconstruction. This concept is also consistent to our problem, since the finite difference transform in the y-direction is expected to yield a sparse image representation. The benefit of minimizing the

*ℓ*

_{1}norm instead of the

*ℓ*

_{2}norm comes at the expense of a non-linear problem formulation. For the finite difference transform, the

*ℓ*

_{1}norm has been shown to perform clearly better, however for other transform domains, the

*ℓ*

_{2}norm may be a better choice.

*a-priori*knowledge about the sample is available, are expected to further improve the phase retrieval. For instance, if the sample is known to consist mainly of piecewise constant parts, a total variation minimization would be appropriate. Another example, which is less sample specific but also often applicable, would be an “isolated specimen” constraint, meaning that there is a zero background signal all around the sample. Essentially, this would be an generalization of the proposed zero-background constraint, which assumed the signal to be zero only at the left and right edges. In any case, the potential improvement always comes at the expense of an additional free regularization parameter and an increased computational complexity. Additional regularization terms are certainly an important subject for future investigations.

## Acknowledgments

## References and links

1. | R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today |

2. | A. Momose and J. Fukuda, “Phase-contrast radiographs of nonstained rat cerebellar specimen,” Med. Phys. |

3. | A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. |

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

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

6. | U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. |

7. | A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. |

8. | T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature |

9. | D. Chapman, W. Thomlinson, R. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. |

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13. | C. David, B. Nöhammer, H. Solak, and E. Ziegler, “Differential X-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. |

14. | A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. |

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

16. | M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus X-ray source,” Appl. Phys. Lett. |

17. | F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer” Nature Mater. |

18. | I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-Dimensional X-Ray Grating Interferometer,” Phys. Rev. Lett. |

19. | D. Paganin, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. |

20. | D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. |

21. | A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography” Opt. Express |

22. | M. de Jonge, B. Hornberger, C. Holzner, D. Legnini, D. Paterson, I. McNulty, C. Jacobsen, and S. Vogt, “Quantitative Phase Imaging with a Scanning Transmission X-Ray Microscope,” Phys. Rev. Lett. |

23. | M. Arnison, K. Larkin, C. Sheppard, N. Smith, and C. Cogswell, “Linear phase imaging using differential interference contrast microscopy” J. microsc. |

24. | B. Hornberger, M. Feser, and C. Jacobsen, “Quantitative amplitude and phase contrast imaging in a scanning transmission X-ray microscope” Ultramicroscopy |

25. | B. Hornberger, M. de Jonge, M. Feser, P. Holl, C. Holzner, C. Jacobsen, D. Legnini, D. Paterson, P. Rehak, L. Strüder, and S. Vogt, “Differential phase contrast with a segmented detector in a scanning X-ray microprobe” J. Synchrotron Radiat. |

26. | P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy” Science |

27. | C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating based differential phase contrast imaging using hard X-rays,” Opt. Express |

28. | M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, and Z. Zhong, “Computation of mass-density images from X-ray refraction-angle images” Phys. Med. Biol. |

29. | T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express |

30. | L. Shepp and B. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. |

31. | K. Engel, D. Geller, T. Köhler, G. Martens, S. Schusser, G. Vogtmeier, and E. Rössl, “Contrast-to-noise in X-ray differential phase contrast imaging,” Nucl. Instrum. Methods A |

32. | K. Sauer and C. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. Signal Process. |

33. | L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D Nonlinear Phenom. |

34. | J. Nocedal and S. Wright, |

35. | M. Stampanoni, A. Groso, A. Isenegger, G. Mikuljan, Q. Chen, A. Bertrand, S. Henein, R. Betemps, U. Frommherz, P. Böhler, D. Meister, M. Lange, and R. Abela, “Trends in synchrotron-based tomographic imaging: The SLS experience,” Proc. SPIE |

36. | C. David, J. Bruder, T. Rohbeck, C. Grünzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard X-ray phase contrast imaging,” Microelectron. Eng. |

37. | P. Hansen, “Analysis of Discrete Ill-Posed Problems by Means of the L-Curve,” SIAM Rev. |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(110.7440) Imaging systems : X-ray imaging

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Image Processing

**Virtual Issues**

Vol. 7, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Thomas Thüring, Peter Modregger, Bernd R. Pinzer, Zhentian Wang, and Marco Stampanoni, "Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiography," Opt. Express **19**, 25545-25558 (2011)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-19-25-25545

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