## Tomographic reconstruction of the refractive index with hard X-rays: an efficient method based on the gradient vector-field approach |

Optics Express, Vol. 22, Issue 5, pp. 5216-5227 (2014)

http://dx.doi.org/10.1364/OE.22.005216

Acrobat PDF (500 KB)

### Abstract

The refractive-index gradient vector field approach establishes a connection between a tomographic data set of differential phase contrast images and the distribution of the partial spatial derivatives of the refractive index in an object. The reconstruction of the refractive index in a plane requires the integration of its gradient field. This work shows how this integration can be efficiently performed by converting the problem to the Poisson equation, which can be accurately solved even in the case of noisy and large datasets. The performance of the suggested method is discussed and demonstrated experimentally by computing the refractive index distribution in both a simple plastic phantom and a complex biological sample. The quality of the reconstruction is evaluated through the direct comparison with other commonly used methods. To this end, the refractive index is retrieved from the same data set using also (1) the filtered backprojection algorithm for gradient projections, and (2) the regularized phase-retrieval procedure. Results show that the gradient vector field approach combined with the developed integration technique provides a very accurate depiction of the sample internal structure. Contrary to the two other techniques, the considered method does not require a preliminary phase-retrieval and can be implemented with any advanced computer tomography algorithm. In this work, analyzer-based phase contrast images are used for demonstration. Results, however, are generally valid and can be applied for processing differential phase-contrast tomographic data sets obtained with other phase-contrast imaging techniques.

© 2014 Optical Society of America

## 1. Introduction

1. A. Momose, “Demonstration of phase-contrast X-ray
computed tomography using an X-ray interferometer,”
Nucl. Instr. Meth. Phys. Res. A **352**, 622–628
(1995). [CrossRef]

2. V.A. Bushuev and A.A. Sergeev, “Inverse problem in the X-ray phase
contrast method,” Technical Phys.
Lett. **25**, 83–85
(1999). [CrossRef]

3. A. Maksimenko, M. Ando, S. Hiroshi, and T. Yausa, “Computed tomographic reconstruction
based on x-ray refraction contrast,” Appl.
Phys. Lett. **86**, 124105 (2005). [CrossRef]

4. F. Pfeiffer, C. Kottler, O. Bunk, and C. Davis, “Hard X-Ray Phase Tomography with
Low-Brilliance Sources,” Phys. Rev.
Lett. **98**, 108105 (2007). [CrossRef] [PubMed]

5. A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from
pre-clinical applications towards clinics,”
Phys. Med. Biol **58**, R1–R35
(2013). [CrossRef]

*E*> 50 keV). By now several algorithms have been developed to extract refraction-contrast information from the XPCI measurements [6

6. F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of
refraction using the diffraction enhanced imaging
method,” Phys. Med. Biol. **45**, 933–946
(2000). [CrossRef] [PubMed]

8. P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreissi, R.D. Speller, and A. Olivo, “A quantitative, non-interferometric
X-ray phase contrast imaging techniques,”
Opt. Express **21**, 647–661
(2012). [CrossRef]

- a CT data set of phase-contrast projections is recorded with a selected XPCI technique. The intensity in these images depends on both the attenuation and the refraction (i.e. angular deflection) of the X-ray beam induced by the interaction with the sample. In this work we focus on the so-called “in plane” XPCI CT arrangement [9], in which the phase-contrast signal depends on the amount of the X-ray deflections in the CT reconstruction plane.
9. T. Yuasa, A. Maksimenko, E. Hashimoto, H. Sugiyama, K. Hyodo, T. Akatsuka, and M. Ando, “Hard-x-ray region tomographic reconstruction of the refractive-index gradient vector field: imaging principles and comparisons with diffraction-enhanced-imaging-based computed tomography,” Opt. Lett.

**31**, 1818–1820 (2006). [CrossRef] [PubMed] - The deflection angles of the X-rays are calculated in the object exit plane according to the XPCI method employed in the first step. In the geometrical optics approximation the angular deflection at a point in the object exit plane is linearly proportional to the local spatial derivative of the wave phase. That is why projections of the deflection angle are also referred to as differential phase contrast images.
- The index of refraction distribution inside the object is reconstructed from the calculated projections of the deflection angle.

10. M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, Ch. 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]

*δ*(

**) in the object (**

*r*_{⊥}**indicates the Euclidian coordinates in a reconstruction plane). Refraction-based CT imaging without preliminary phase-retrieval can be also performed. In this case two other methods can be applied: (a) the filtered backprojection algorithm (FBP) for gradient projections [12**

*r*_{⊥}12. G. W. Faris and R. L. Byer, “Three-dimensional beam deflection
optical tomography of a supersonic jet,”
Appl. Opt. **27**, 5202–5212
(1988). [CrossRef] [PubMed]

3. A. Maksimenko, M. Ando, S. Hiroshi, and T. Yausa, “Computed tomographic reconstruction
based on x-ray refraction contrast,” Appl.
Phys. Lett. **86**, 124105 (2005). [CrossRef]

3. A. Maksimenko, M. Ando, S. Hiroshi, and T. Yausa, “Computed tomographic reconstruction
based on x-ray refraction contrast,” Appl.
Phys. Lett. **86**, 124105 (2005). [CrossRef]

9. T. Yuasa, A. Maksimenko, E. Hashimoto, H. Sugiyama, K. Hyodo, T. Akatsuka, and M. Ando, “Hard-x-ray region tomographic
reconstruction of the refractive-index gradient vector field: imaging
principles and comparisons with diffraction-enhanced-imaging-based computed
tomography,” Opt. Lett. **31**, 1818–1820
(2006). [CrossRef] [PubMed]

*δ*(

**) is obtained (similarly to the case in which the phase-retrieval is first applied), rather than only the fluctuating component of the**

*r*_{⊥}*δ*(

**) that is instead provided by FBP-algorithm for gradient projections.**

*r*_{⊥}**86**, 124105 (2005). [CrossRef]

**31**, 1818–1820
(2006). [CrossRef] [PubMed]

## 2. Description of the suggested refraction-based CT method

*s*is an elementary interval along the ray;

**is the spatial coordinate,**

*r***=**

*r***(**

*r**s*) is the ray trajectory,

**(**

*t***) is a unit vector tangential to the ray at the point**

*r***, and**

*r**n*(

**) is the distribution of refractive index. Further we assume that: (1) the sensitivity plane of the XPCI method we use is defined as**

*r**x*=

*const*, i.e. the phase-contrast signal does not depend on the amount of X-ray deflections out of the

*y*–

*z*plane (see Fig. 2); (2) the object rotation axis is perpendicular to the

*y*–

*z*plane (i.e. it corresponds to the

*x*axis). The differentiation of Eq. (1) gives two relations: where

**is a unit vector normal to**

*v***and**

*t**α*is the angular deflection of an X-ray from the original propagation direction. Equations (2) can be integrated separately (see [9

**31**, 1818–1820
(2006). [CrossRef] [PubMed]

*y′*,

*z′*) axes are rotated by an angle

*θ*with respect to the (

*y*,

*z*) axes, and ray trajectories are approximated as straight lines. Equations (3) are the main result of the vector field approach for refractive index CT [3

**86**, 124105 (2005). [CrossRef]

**31**, 1818–1820
(2006). [CrossRef] [PubMed]

**= (**

*f**∂n/∂y*,

*∂n/∂z*) is the solution of Eq. (3). In practice, functions

*∂n/∂y*and

*∂n/∂z*and their first derivatives are non-smooth, particularly at the interfaces between an object and air. Moreover, because of the dose and (or) acquisition time constraints, the experimentally recorded images may have a very poor signal to noise ratio (for instance in the case of large and highly absorbing samples). Thus a direct integration of the Eq. (4) in the CT reconstruction plane can be very inefficient. We found that a robust and accurate way to solve Eq. (4) is to apply the divergence operator to its both sides. The divergence of the gradient is the Laplacian, thus Eq. (4) becomes: This is the Poisson equation that can be solved since the boundary condition of the first-type (Dirichlet) is known:

*δ*(

*y,z*)|

_{(}

_{y,z}_{)∈Π}= 0, i.e. the decrement of the index of refraction

*δ*(

*n*= 1 −

*δ*) is equal to zero at points outside the object, for instance, on the edges of a square Π bounding the reconstruction area in the

*y*–

*z*plane. The boundary conditions can be also defined for

*n*as

*n*(

*y,z*)|

_{(}

_{y,z}_{)∈Π}= 1, i.e. the index of refraction of air outside the sample is set to one.

*μ*m. For a centimeters-thick sample, the dimensions of the computational grid in the CT reconstruction plane can be very large. For instance, in the current work the integration was performed on a 765 × 765 pixels grid. To address this problem and to avoid using Fourier transforms, we applied a numerically stable and rapid approach to solve Eq. (5) based on a multigrid technique [17

17. R. P. Fedorenko, “A relaxation method for solving
elliptic difference equations,” USSR Comput.
Math. Math. Phys. **1**, 1092 (1961). [CrossRef]

19. J. M. Hyman and M. Shashkov, ”Natural discretizations for the
divergence, gradient, and curl on logically rectangular
grids”, Computers Math. Applic. **33**, 81–104
(1997). [CrossRef]

*et. al*[3

**86**, 124105 (2005). [CrossRef]

## 3. Brief description of the alternative refractive-index CT reconstruction techniques

### 3.1. Filtered backprojection algorithm for gradient projections

*x*=

*x*

_{0}through the object: which for the rotated coordinates shown in Fig. 2 becomes According to the work by Faris and Bayer [12

12. G. W. Faris and R. L. Byer, “Three-dimensional beam deflection
optical tomography of a supersonic jet,”
Appl. Opt. **27**, 5202–5212
(1988). [CrossRef] [PubMed]

*g*(

*y′*), is the inverse Fourier transform of the function

*sgn*(

*Y′*) is the sign function in the reciprocal space. Equation (8) is the FBP algorithm for gradient projections. Note that instead of the index of refraction decrement, only its fluctuating component

*δ̃*is recovered by Eq. (8). Although the difference operator in Eq. (3) removes the constant component of

*δ*, this term can be recovered by taking into account an additional prior information about the boundary values [10

10. M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, Ch. 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]

14. F. J. Harris, “On the Use of Windows for Harmonic
Analysis with the Discrete Fourier Transform,”
Proc. IEEE , **66**,
51–83 (1978). [CrossRef]

*α*(

*y′*). In this way, the ring artifacts and aliasing errors in the reconstructed images were efficiently suppressed [15

15. A. J. Devaney, “A Computer Simulation study of
Diffraction Tomography,” IEEE Trans. Biomed.
Eng. **30**377–386 (1983). [CrossRef] [PubMed]

### 3.2. The regularized phase-retrieval procedure

*y′*-axis). Frequently the phase projections obtained by direct integration could not be used neither to recognize fine object features nor for the tomographic reconstruction. In order to suppress the streaking artifacts Wernick

*et al*. [10

10. M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, Ch. 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]

*et al*. [11] suggested to convert Eq. (10) to the constrained minimization problem. The constraint (i.e. a regularization term) is introduced to minimize variations of the phase in the direction perpendicular to the integration (

*x*-axis in our notation). In the current work we used a slightly different approach to this problem. The initial value problem of Eq. (10) is converted to the boundary value problem by differentiating the left and right sides over

*y′*once more: and afterward the regularization term is added to the left-hand side: Just as in the case of the minimization approach [10

**51**, 1769–1778
(2006). [CrossRef] [PubMed]

*ϕ*. The value of the regularization factor

*γ*is chosen according to each particular case. It must be sufficiently large to suppress streaks, but not to large to avoid deviation of the approximate solution from the original function (to process the data acquired in the current work

*γ*= 0.05 was used).

## 4. Performance of the suggested method and comparison of the three techniques

### 4.1. Acquisition of X-rays deflection angle projections

**86**, 124105 (2005). [CrossRef]

21. V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation
of the phase contrast from a non-crystalline
object,” J. Phys. D **28**, 2314–2318
(1995). [CrossRef]

*E/E*∼ 10

^{−4}), quasi-parallel X-ray beam. The photon energy was set to 52 keV. The XPCI signal was produced by means of a 3 cm thick and symmetrically cut perfect Bragg Si (333) crystal (the “analyzer”) placed between the sample and the detector. The crystal rocking curve had a full width half maximum of about 1.8

*μ*rad. The sample was rotated around the axis perpendicular to the plane containing the X-rays incident on and reflected from the analyzer crystal. A charge couple device detector with an effective pixel size of 96×96

*μ*m

^{2}was used to register the images. The complete tomography data set consisted of 250 views (projections) uniformly sampling the range of

*θ*= [0..

*π*) rad. In such conditions two images at angular offsets of

*φ*

_{1,2}≈ ±0.9

*μ*rad from the Bragg angle were obtained for every projection. The deflection angles were calculated using a non-linear extension for the diffraction enhanced imaging algorithm [22

22. A. Maksimenko, “Nonlinear extension of the X-ray
diffraction enhanced imaging,” Appl. Phys.
Lett. **90**, 154106 (2007). [CrossRef]

### 4.2. Testing of the reconstruction methods on a plastic phantom

*δ*were calculated using the well-known electrodynamic equation

*δ*= 2

*πρe*

^{2}/(

*mω*

^{2}) that relates material density

*ρ*to the index of refraction decrement (

*e*and

*m*are the mass and the charge of the electron and

*ω*is the X-ray wave frequency). The obtained values are

*ρ*=0.93 g/cm

^{3},

*δ*=8.60×10

^{−8}and

*ρ*=1.14 g/cm

^{3},

*δ*=1.06×10

^{−7}for the outer and the inner cylinder correspondingly.

*δ*along the vertical line crossing the center of each image. Comparing the analytically obtained image (Fig. 3(d)) with the experimental data (Figs. 3(a)–3(c)) one can notice blurring at the edges of the sample and a slight asymmetry in both the images and corresponding profiles. The latter effect appeared due to the uncertainty in the angular position of the analyzer crystal. This systematic error was suppressed and did not influence results obtained for the breast sample.

*δ*experiences the largest variation: the typical values of the refraction angles there are one order of magnitude greater than everywhere inside the object and the measured phase-contrast signal is therefore greatly affected by the detector blurring. Correspondingly the reconstruction is spoiled near the outer sample border. None of the common deconvolution methods that we have tried was able to suppress this error without a global amplification of the high frequency noise. Nonetheless, it can be seen that the air interface blurring is rather localized and its effect is much less severe at the interfaces between materials inside the sample.

### 4.3. Imaging of a biomedical sample: results and discussion

*δ*to the material density was done using standard tables that contain chemical composition of different biomedical materials [23, 24

24. G. R. Hammerstein, D. W. Miller, D. R. White, M. E. Masterson, H. Q. Woodard, and J. S. Laughlin, “Absorbed radiation dose in
mammography,” Radiology **130**, 485–491
(1979). [PubMed]

^{3}and they were normalized on the total number of pixels in the image in order to show the relative amount of each material. For each image the absolute density values were computed by means of a multiplication by a normalization factor. Normalization factors for each image were estimated as the ratio between the reference formalin value and the average value of formalin computed in a 40 × 40 pixels region in the respective image. The reference value is the density of a 4% formalin solution (

*ρ*= 1.08 g/cm

^{3}, the corresponding peak can be seen in all three distributions). One can immediately notice that apart from the peak representing the formalin (almost the 45% of the slice consists of formalin) histograms in Figs. 4(g) and 4(j) have two distinct peaks that correspond to: (1) the adipose tissue, with

*ρ*≈ 0.96 and

*ρ*≈ 0.95 g/cm

^{3}for Figs. 4(b) and 4(c) respectively and (2) the small maximum at the density of ≈ 1.16 g/cm

^{3}in both images that represents the skin. The glandular tissue, whose expected density value is

*ρ*≈ 1.04 g/cm, is not clearly distinguishable with respect to the other tissues/materials in any histograms (while it is very well recognizable in Fig. 4(b)).

## 5. Conclusions

## Acknowledgments

## References and links

1. | A. Momose, “Demonstration of phase-contrast X-ray
computed tomography using an X-ray interferometer,”
Nucl. Instr. Meth. Phys. Res. A |

2. | V.A. Bushuev and A.A. Sergeev, “Inverse problem in the X-ray phase
contrast method,” Technical Phys.
Lett. |

3. | A. Maksimenko, M. Ando, S. Hiroshi, and T. Yausa, “Computed tomographic reconstruction
based on x-ray refraction contrast,” Appl.
Phys. Lett. |

4. | F. Pfeiffer, C. Kottler, O. Bunk, and C. Davis, “Hard X-Ray Phase Tomography with
Low-Brilliance Sources,” Phys. Rev.
Lett. |

5. | A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from
pre-clinical applications towards clinics,”
Phys. Med. Biol |

6. | F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of
refraction using the diffraction enhanced imaging
method,” Phys. Med. Biol. |

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

8. | P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreissi, R.D. Speller, and A. Olivo, “A quantitative, non-interferometric
X-ray phase contrast imaging techniques,”
Opt. Express |

9. | T. Yuasa, A. Maksimenko, E. Hashimoto, H. Sugiyama, K. Hyodo, T. Akatsuka, and M. Ando, “Hard-x-ray region tomographic
reconstruction of the refractive-index gradient vector field: imaging
principles and comparisons with diffraction-enhanced-imaging-based computed
tomography,” Opt. Lett. |

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

11. | T. Thuering, P. Modregger, B. R. Pinzer, Z. Wang, and M. Stampanoni, “Non-linear regularized phase retrieval
for unidirectional X-ray differential phase contrast
radiography,” Opt. Express |

12. | G. W. Faris and R. L. Byer, “Three-dimensional beam deflection
optical tomography of a supersonic jet,”
Appl. Opt. |

13. | A. C. Kak and M. Slaney, |

14. | F. J. Harris, “On the Use of Windows for Harmonic
Analysis with the Discrete Fourier Transform,”
Proc. IEEE , |

15. | A. J. Devaney, “A Computer Simulation study of
Diffraction Tomography,” IEEE Trans. Biomed.
Eng. |

16. | A. Beck and M. Teboulle, “A fast Iterative Shrinkage-Thresholding
Algorithm for Linear Inverse Problems,” SIAM
J. Imaging Sciences |

17. | R. P. Fedorenko, “A relaxation method for solving
elliptic difference equations,” USSR Comput.
Math. Math. Phys. |

18. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

19. | J. M. Hyman and M. Shashkov, ”Natural discretizations for the
divergence, gradient, and curl on logically rectangular
grids”, Computers Math. Applic. |

20. | O. C. Zienkiewicz and K. Morgan, |

21. | V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation
of the phase contrast from a non-crystalline
object,” J. Phys. D |

22. | A. Maksimenko, “Nonlinear extension of the X-ray
diffraction enhanced imaging,” Appl. Phys.
Lett. |

23. | Tissue Substitutes in Radiation Dosimetry and Measurement, ICRU Report 44 (1989). |

24. | G. R. Hammerstein, D. W. Miller, D. R. White, M. E. Masterson, H. Q. Woodard, and J. S. Laughlin, “Absorbed radiation dose in
mammography,” Radiology |

**OCIS Codes**

(110.7440) Imaging systems : X-ray imaging

(120.5710) Instrumentation, measurement, and metrology : Refraction

(110.6955) Imaging systems : Tomographic imaging

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: November 28, 2013

Revised Manuscript: January 9, 2014

Manuscript Accepted: January 11, 2014

Published: February 27, 2014

**Virtual Issues**

Vol. 9, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Sergei Gasilov, Alberto Mittone, Emmanuel Brun, Alberto Bravin, Susanne Grandl, Alessandro Mirone, and Paola Coan, "Tomographic reconstruction of the refractive index with hard X-rays: an efficient method based on the gradient vector-field approach," Opt. Express **22**, 5216-5227 (2014)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-22-5-5216

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### References

- A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instr. Meth. Phys. Res. A 352, 622–628 (1995). [CrossRef]
- V.A. Bushuev, A.A. Sergeev, “Inverse problem in the X-ray phase contrast method,” Technical Phys. Lett. 25, 83–85 (1999). [CrossRef]
- A. Maksimenko, M. Ando, S. Hiroshi, T. Yausa, “Computed tomographic reconstruction based on x-ray refraction contrast,” Appl. Phys. Lett. 86, 124105 (2005). [CrossRef]
- F. Pfeiffer, C. Kottler, O. Bunk, C. Davis, “Hard X-Ray Phase Tomography with Low-Brilliance Sources,” Phys. Rev. Lett. 98, 108105 (2007). [CrossRef] [PubMed]
- A. Bravin, P. Coan, P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol 58, R1–R35 (2013). [CrossRef]
- F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000). [CrossRef] [PubMed]
- T. Weitkamp, A. Diaz, Ch. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express. 13, 6296–6304 (2005). [CrossRef] [PubMed]
- P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreissi, R.D. Speller, A. Olivo, “A quantitative, non-interferometric X-ray phase contrast imaging techniques,” Opt. Express 21, 647–661 (2012). [CrossRef]
- T. Yuasa, A. Maksimenko, E. Hashimoto, H. Sugiyama, K. Hyodo, T. Akatsuka, M. Ando, “Hard-x-ray region tomographic reconstruction of the refractive-index gradient vector field: imaging principles and comparisons with diffraction-enhanced-imaging-based computed tomography,” Opt. Lett. 31, 1818–1820 (2006). [CrossRef] [PubMed]
- M. N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, Ch. Parham, E. Pisano, Z. Zhong, “Computation of mass-density images from x-ray refraction-angle images,” Phys. Med. Biol. 51, 1769–1778 (2006). [CrossRef] [PubMed]
- T. Thuering, P. Modregger, B. R. Pinzer, Z. Wang, M. Stampanoni, “Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiography,” Opt. Express 19, 25542–25558 (2011).
- G. W. Faris, R. L. Byer, “Three-dimensional beam deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202–5212 (1988). [CrossRef] [PubMed]
- A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging(IEEE Press, 1988, Chap. 5).
- F. J. Harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform,” Proc. IEEE, 66, 51–83 (1978). [CrossRef]
- A. J. Devaney, “A Computer Simulation study of Diffraction Tomography,” IEEE Trans. Biomed. Eng. 30377–386 (1983). [CrossRef] [PubMed]
- A. Beck, M. Teboulle, “A fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems,” SIAM J. Imaging Sciences 2, 183–202 (2009). [CrossRef]
- R. P. Fedorenko, “A relaxation method for solving elliptic difference equations,” USSR Comput. Math. Math. Phys. 1, 1092 (1961). [CrossRef]
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical recipes in C, 2 (Cambridge University, 1992, pp. 871–872).
- J. M. Hyman, M. Shashkov, ”Natural discretizations for the divergence, gradient, and curl on logically rectangular grids”, Computers Math. Applic. 33, 81–104 (1997). [CrossRef]
- O. C. Zienkiewicz, K. Morgan, Finite Elements and Approximation (Dover Pubn. Inc., 2006, Chap. 3).
- V. N. Ingal, E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2318 (1995). [CrossRef]
- A. Maksimenko, “Nonlinear extension of the X-ray diffraction enhanced imaging,” Appl. Phys. Lett. 90, 154106 (2007). [CrossRef]
- Tissue Substitutes in Radiation Dosimetry and Measurement, ICRU Report 44 (1989).
- G. R. Hammerstein, D. W. Miller, D. R. White, M. E. Masterson, H. Q. Woodard, J. S. Laughlin, “Absorbed radiation dose in mammography,” Radiology 130, 485–491 (1979). [PubMed]

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