## Fast iterative reconstruction of differential phase contrast X-ray tomograms |

Optics Express, Vol. 21, Issue 5, pp. 5511-5528 (2013)

http://dx.doi.org/10.1364/OE.21.005511

Acrobat PDF (2120 KB)

### Abstract

Differential phase-contrast is a recent technique in the context of X-ray imaging. In order to reduce the specimen’s exposure time, we propose a new iterative algorithm that can achieve the same quality as FBP-type methods, while using substantially fewer angular views. Our approach is based on 1) a novel spline-based discretization of the forward model and 2) an iterative reconstruction algorithm using the alternating direction method of multipliers. Our experimental results on real data suggest that the method allows to reduce the number of required views by at least a factor of four.

© 2013 OSA

## 1. Introduction

1. V. Ingal and E. Beliaevskaya, “X-ray plane-wave tomography observation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys **28**, 2314–2317 (1995). [CrossRef]

3. D. Chapman, S. Patel, and D. Fuhrman, “Diffraction enhanced X-ray imaging,” Phys., Med. and Bio. **42**, 2015–2025 (1997). [CrossRef]

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

6. 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 **13**, 6296–6304 (2005). [CrossRef] [PubMed]

7. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelekov, “On the possibilities of X-ray phase-contrast microimaging by coherent high-energy synchroton radiation,” Rev. Sci. Instrum. **66**, 5486–5492 (1997). [CrossRef]

9. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nat. **384**, 335–338 (1996). [CrossRef]

*et al*[6

6. 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 **13**, 6296–6304 (2005). [CrossRef] [PubMed]

*et al*[10

10. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray talbot interferometry,” Jap. Jour. of Appl. Phys. **42**, L866–L868 (2003). [CrossRef]

*e*/nm

^{3}[11

11. F. Pfieffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Inst. and Meth. in Phys. Res. **580.2**, 925–928 (2007). [CrossRef]

6. 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 **13**, 6296–6304 (2005). [CrossRef] [PubMed]

11. F. Pfieffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Inst. and Meth. in Phys. Res. **580.2**, 925–928 (2007). [CrossRef]

12. M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. **46**, 801–806 (2011). [CrossRef]

### 1.1. Contributions

- The proposal of a new discretization scheme for derivatives of the Radon transform based on B-spline calculus. We show that the model lends itself to an analytical treatment, yielding a fast and accurate implementation.
- The design of an iterative reconstruction algorithm taking advantage of the proposed discretization and using a combination of TV and Tikhonov regularization. Our method follows an augmented-Lagrangian optimization principle and makes use of the conjugate gradient method to solve the linear step in the alternating direction method of multipliers (ADMM).
- The application of a problem-specific preconditioner that speeds up the convergence of the linear optimization step considerably.
- The experimental evaluation of the method using real data.

### 1.2. Related work

14. M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Imag. Proc. **20**, 681–695 (2011). [CrossRef]

15. S. Ramani and J. A. Fessler, “A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction,” IEEE Trans. Med. Imag. **31.3**, 677–688 (2012). [CrossRef]

18. Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, and M. A. Anastasio, “Investigation of discrete imaging models and iterative image reconstruction in differential X-ray phase-contrast tomography,” Opt. Express **20**, 10724–10749 (2012). [CrossRef] [PubMed]

*et al*. [16] express the derivative in the projection domain using derivatives along the two axes in the image domain. Then they implement these derivatives with finite differences. A basic steepest descent algorithm is used to solve the corresponding TV problem.

### 1.3. Outline

## 2. Physical model

**13**, 6296–6304 (2005). [CrossRef] [PubMed]

21. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray talbot interferometry for
biological imaging,” Jpn. J. Appl. Phys. **45**, 5254–5262 (2006). [CrossRef]

22. F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, “Neutron phase imaging and tomography,” Phys. Rev. Lett. **96**, 215505-1–4 (2006). [CrossRef]

*π*on the incident wave.

**= (cos**

*θ**θ*, sin

*θ*) is a unit vector that is orthogonal to the projection direction and

*y*is the projection coordinate depicted in Figure 1. The spatial coordinate of the 2-D object is denoted by

**= (**

*x**x*

_{1},

*x*

_{2}).

*f*(

**) =**

*x**n*(

**) − 1 with**

*x**n*(

**) is the real part of the object’s refractive index,**

*x**θ*is the angle of view of the monochromatic wave illuminating the object and

*ℛ*{

*f*(

**)}(**

*x**y*,

*θ*) is the Radon transform of

*f*(

**).**

*x***Definition 1.**[23] The (2-dimensional) Radon transform

*ℛ : L*

_{2}(ℝ

^{2}) →

*L*

_{2}(ℝ× [0,

*π*]) maps a function on ℝ

^{2}into the set of its integrals over the lines of ℝ

^{2}. Specifically, if

**= (cos**

*θ**θ*, sin

*θ*) and

*y*∈ ℝ, then where

*θ*^{⊥}is perpendicular to

**.**

*θ**y*: where

*α*(

*y*,

*θ*) is the refraction angle in radians, and

*λ*is the wavelength

**13**, 6296–6304 (2005). [CrossRef] [PubMed]

*x*.

_{g}*d*is the distance between the two gratings. Combining this relation with (1) and (3) yields For our purpose, this equation characterizes the forward model of DPCI:

*f*is the quantity of interest and

*g*is the physically measurable signal.

## 3. Review of the derivative variants of the Radon transform and generalized filtered back-projection

**Definition 2.**The Hilbert transform along the

*x*

_{2}axis,

*ℋ*

_{2}:

*L*

_{2}(ℝ

^{2}) →

*L*

_{2}(ℝ

^{2}), is defined in the Fourier domain as where (

*ω*

_{1},

*ω*

_{2}) are the spatial frequency coordinates corresponding to (

*x*

_{1},

*x*

_{2}).

**Proposition 1.**The sequential application of the Radon transform, the

*n*th derivative operator and the adjoint of the Radon transform on function

*f*(

**) ∈**

*x**L*

_{2}(ℝ

^{2}) is where

**‖ and**

*ω**n*times application of this operator.

*n*th derivative of

*g*(

*y*,

*θ*) with respect to

*y*is (

*iω*)

*(*

^{n}ĝ*ω*,

*θ*). Thus, where |

*ω*| = ‖

**‖ with**

*ω***= (**

*ω**ω*

_{1},

*ω*

_{2}) =

*ω*(cos

*θ*, sin

*θ*). For

*θ*∈ (0,

*π*), sgn(

*ω*) = sgn(

*ω*sin

*θ*) = sgn(

*ω*

_{2}). The space-domain equivalent is Therefore we have which, owing to the property that

*ℛ*

^{*}

*ℛ*= (−Δ)

^{1/2}, yields the desired results.

*ℛ*

^{(n)}is the adjoint of the

*n*-th derivative of the Radon transform and the transfer function of

*q*(

*y*) is: (17) is the basis for the generalized filtered back projection (GFBP). The full procedure is described in Algorithm 1.

## 4. Discretization scheme and implementation of the forward model

### 4.1. Discrete model of the Radon transform and its derivatives

19. M. Unser, “Sampling–50 years after Shannon,” Proc. IEEE **88**, 254104–1–3 (2000). [CrossRef]

*φ*(

**) ∈ L**

*x*_{2}(ℝ

^{2}) denote a generating Riesz basis function for the approximation space

**V**as:

*f*(

**) on the reconstruction space**

*x***V**is where

*φ*

_{k}(

**) =**

*x**φ*(

**−**

*x***) and**

*k***c**

_{k}= 〈

*f*,

*φ̃*(· −

**)〉 in which**

*k**φ̃*(

**) is the dual of**

*x**φ*(

**) with Fourier transform**

*x**f*(

**) is where and**

*x**ℛ*{

*φ*}(

*y*,

*θ*) =

*ℱ*

^{−1}{

*φ̂*(

*ω*

**)}(**

*θ**y*). Likewise, we obtain the derivatives of the Radon transform: where Based on Eqs. (22) and (24) we need to choose an appropriate generating function and to determine its Radon transform analytically. In this work we use a tensor-product B-spline, which provides a good compromise between support size and approximation order [24

24. H. Meijering, J. Niessen, and A. Viergever, “Quantitative evaluation of convolution-based methods for medical image interpolation,” Med. Imag. Anal. **5**, 111–126 (2001). [CrossRef]

*β*is a centered univariate B-spline of degree

^{m}*m*: where

*n*-fold iteration of the finite-difference operator Δ

*(*

_{h}f*y*) = (

*f*(

*y*+

*h*/2) −

*f*(

*y*−

*h*/2))/

*h*.

**Proposition 2.**The

*n*-th derivative of the Radon transform of the tensor-product B-spline is given by

*Proof.*We define

*g*(

*y*,

*θ*) =

*ℛ*

^{(n)}{

*β*}(

*y*,

*θ*). By an application of the Fourier-slice theorem let us now focus on the Fourier-domain expression Taking the inverse Fourier transform of this expression yields the desired result.

### 4.2. Matrix formulation of the forward model

**c**is a vector whose entries are the B-spline coefficients

*c*

_{k},

**g**is the output vector and

**H**is the system matrix defined by: The corresponding adjoint operator is represented by the conjugate transpose of the system matrix.

### 4.3. Fast implementation

**H**cannot be stored explicitly due to its size. To circumvent this problem we exploit the translation-invariance of the Radon transform. This property implies that all the matrix entries in Eq. (32) can be derived from a single derivative of the Radon transform, namely that of the generating function

*φ*:

*ℛ*

^{(n)}

*φ*(

*y*,

*θ*) on a fine grid

*Y*× Θ with 100 samples along each angular direction and store the values in a lookup table

**L**. To compute the matrix entries we define a mapping with

*K*is the number of samples along each direction,

*P*is the number of projections and (

*Y*(

*j*), Θ(

*i*)) is the sample in

*Y*× Θ that is nearest to (

*y*,

*θ*). Therefore, we have

### 4.4. B-spline-based discretization scheme vs Kaiser Bessel window functions

**Proposition 3.**The derivative of the Radon transform of the isotropic function where

*Proof.*Since the function is isotropic, its Radon transform is independent of the direction. For simplicity we consider

*θ*= 0 whose projection coordinate is parallel to the

*x*

_{1}axis. Thus we have which yields the desired result.

*α*∈ ℝ and

**x**

*∈ ℝ*

_{k}^{2}. Figure 2(a) shows the analytic phantom that we consider as the object of interest. It was generated using 10 random radiuses

*a*, 10 random locations

_{k}**x**

*and 10 random intensities*

_{k}*α*. Its differentiated sinogram is displayed in Figure 2(b) with 1800 viewing angles that are chosen uniformly between 0 and

_{k}*π*.

18. Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, and M. A. Anastasio, “Investigation of discrete imaging models and iterative image reconstruction in differential X-ray phase-contrast tomography,” Opt. Express **20**, 10724–10749 (2012). [CrossRef] [PubMed]

18. Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, and M. A. Anastasio, “Investigation of discrete imaging models and iterative image reconstruction in differential X-ray phase-contrast tomography,” Opt. Express **20**, 10724–10749 (2012). [CrossRef] [PubMed]

**H**is the discretized forward operator,

**c**contains the expansion coefficient in the chosen basis and

**g**is the measurement vector. We use the sampled version of the object on a 1024 × 1024 grid as the ground truth. Since the data is correctly sampled and noise-free, there is not need for regularization and the reconstruction was performed using the standard CG algorithm. We used the same basis functions as in our first experiment. The results are shown in the last row of Table 1 and again demonstrate the advantage of our spline-based formulation. They confirm that the partition-of-unity condition provides a significant advantage in terms of reconstruction quality. This condition is necessary and sufficient for the approximation error to vanish as the grid size tends to zero [19

19. M. Unser, “Sampling–50 years after Shannon,” Proc. IEEE **88**, 254104–1–3 (2000). [CrossRef]

26. P. Thvenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imag. **19.7**, 739–758 (2000). [CrossRef]

## 5. Image reconstruction

**g**is the measurement vector and the regularization term Ψ(

**c**) is chosen based on the following considerations. First we observe that the null space of the derivative of the Radon transform contain the zero frequency (constant). To compensate for this we incorporate the energy of the coefficients into the regularization term. Second, to enhance the edges in the reconstructed image we impose a sparsity constraint in the gradient domain. This leads to where

*λ*

_{1}and

*λ*

_{2}are regularization parameters and {

**Lc**}

_{k}∈ ℝ

^{2}denotes the gradient of the image at position

**. More precisely, the gradient is computed based on our continuous-domain image model using the following property.**

*k***Proposition 4.**Let

*f*(

**) =∑**

*x*_{k ∈2}

*c*

_{k}

*β*(

^{n}**−**

*x***). The gradient of**

*k**f*on the Cartesian grid is

*k*

_{1},

*k*

_{2}∈ ,

*b*[

_{i}*k*

_{1},

*k*

_{2}] =

*β*(

^{n}*k*) for

_{i}*i*= 1, 2.

**H**

^{T}**H**. In our case, the derivative of the Radon transform is a very ill-conditioned operator leading to very slow convergence. To overcome this difficulty, we use a variable-splitting scheme so as to map our general optimization problem into simpler ones. Specifically, we introduce the auxiliary variable

**u**[14

14. M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Imag. Proc. **20**, 681–695 (2011). [CrossRef]

27. Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM Jour. on Imag. Sci. **1**, 248–272 (2008). [CrossRef]

**is a vector of Lagrange multipliers. To minimize (45), we adopt a cyclic update scheme also known as Alternating-Direction Method of Multipliers (ADMM), which consists of the iterations**

*α**ℒ*(

_{μ}**c**,

**u**

*,*

^{k}

*α**) is a quadratic function with respect to*

^{k}**c**whose gradient is We minimize

*ℒ*(

_{μ}**c**,

**u**

*,*

^{k}

*α**) iteratively using the conjugate gradient (CG) algorithm to solve the equation*

^{k}**Ac**=

**b**. Since

**A**has a large condition number, it is helpful to introduce a preconditioning matrix

**M**. This matrix is chosen such that

**M**

^{−1}(

**H**

^{T}**H**+

*μ*

**L**

^{T}**L**+

*λ*

_{1}I) has a condition number close to 1. To design this problem-specific preconditioner, we use the following property of the forward model.

**Proposition 5.**The successive application of the derivatives of the Radon transform and their adjoint is a highpass filter with frequency response ‖

**‖**

*ω*^{2}

^{n}^{−1}:

*Proof.*This follows from

*ℛ*

^{(1)*}

*ℛ*

^{(1)}has a Fourier transform that is proportional to ‖

**‖ while**

*ω***L**

^{T}**L**is a discretized Laplace operator whose continuous-domain frequency response is ‖

**‖**

*ω*^{2}. Thus, the preconditioner

**M**

^{−1}that we use in the discrete domain is the discrete filter that approximates the frequency response

*ℒ*(

_{μ}**c**

*,*

^{k}**u**,

*α**) with respect to*

^{k}**u**is given by the shrinkage function [31

31. I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math. **57**, 1413–1457 (2004). [CrossRef]

32. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Imag. Sci. **2**, 183–202 (2009). [CrossRef]

## 6. Experimental analysis

### 6.1. Performance metrics

33. Z. Wang and A. Bovik, “A universal image quality index,” IEEE Sig. Proc. Lett. **9**, 81 –84 (2002). [CrossRef]

34. Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Imag. Proc. **13**, 600–612 (2004). [CrossRef]

*et al*which compares the luminance, contrast and structure of images. SSIM is computed for a window of size

*R*×

*R*around each image pixel. The SSIM measure for two images

**x**and

**x̂**for the specified window is where

*C*

_{1}and

*C*

_{2}are small constants values to avoid instability.

*μ*and

_{x}*μ*

_{x}_{̂}denote the empirical mean of image

**x**and

**x̂**in the specified window, respectively.

*σ*and

_{x}*σ*

_{x}_{̂}are the empirical variance of the corresponding images and the covariance of two images is denoted by

*σ*

_{xx}_{̂}for the corresponding window. In our experiments, we choose

*C*

_{1}=

*C*

_{2}= (.001 *

*L*)

^{2}where

*L*is the dynamic range of the image pixel values. SSIM for the total image is obtained as the average of SSIM over all pixels. It takes values between 0 and 1 with 1 corresponding to the highest similarity.

### 6.2. Parameter selection

*λ*

_{1}= 10

^{−5}. Here we use

*λ*

_{2}= ‖

**g**‖

_{2}× 10

^{−3}. Based on our experience, the parameter

*μ*can be chosen ten times larger than

*λ*

_{2}. The number of inner iterations for solving the linear step plays no role in the convergence of the proposed technique, but affects speed; we suggest to choose it as small as possible, e.g., 2 or 3. We use cubic B-splines with

*m*= 3 as the basis functions.

### 6.3. Experimental result

35. S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” Sync. Rad. **16**, 562–572 (2009). [CrossRef]

*μ*m.

*α*= 10.4) were essentially indistinguishable for those in Figure 4(b, e). On the other hand, the effect of choosing an optimized set of basis functions is more significant in the well-posed scenario without regularization as demonstrated in Section 4.4.

*h*(

*ω*) is a lowpass filter and

_{y}*k*is an exponent that acts as a smoothing parameter. We chose

*h*(

*ω*) to be the standard Hamming window. The reconstructions are shown in the bottom row of Figure 4. They suggest that with this smoothed version of GFBP there is a trade-off between artifacts and image contrast. Note that for these experiments the parameter

*k*was optimized so as to achieve the best SNR.

## 7. Conclusion

## Acknowledgment

## References and links

1. | V. Ingal and E. Beliaevskaya, “X-ray plane-wave tomography observation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys |

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

3. | D. Chapman, S. Patel, and D. Fuhrman, “Diffraction enhanced X-ray imaging,” Phys., Med. and Bio. |

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

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

6. | 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 |

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

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

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

10. | A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray talbot interferometry,” Jap. Jour. of Appl. Phys. |

11. | F. Pfieffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Inst. and Meth. in Phys. Res. |

12. | M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. Kubik-Huch, G. Singer, M. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Inves. radio. |

13. | M. Nilchian and M. Unser, “Differential phase-contrast X-ray computed tomography: From model discretization to image reconstruction,” Proc. of the Ninth IEEE Inter. Symp. on Biomed. Imag.: From Nano to Macro (ISBI’12), 90–93 (2012). |

14. | M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Imag. Proc. |

15. | S. Ramani and J. A. Fessler, “A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction,” IEEE Trans. Med. Imag. |

16. | Z. Qi, J. Zambelli, N. Bevins, and G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. of SPIE |

17. | T. Köhler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. phys. |

18. | Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, and M. A. Anastasio, “Investigation of discrete imaging models and iterative image reconstruction in differential X-ray phase-contrast tomography,” Opt. Express |

19. | M. Unser, “Sampling–50 years after Shannon,” Proc. IEEE |

20. | E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed
tomography by constrained, total-variation minimization,” Phys. Med.
Biol. |

21. | A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray talbot interferometry for
biological imaging,” Jpn. J. Appl. Phys. |

22. | F. Pfeiffer, C. Grünzweig, O. Bunk, G. Frei, E. Lehmann, and C. David, “Neutron phase imaging and tomography,” Phys. Rev. Lett. |

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24. | H. Meijering, J. Niessen, and A. Viergever, “Quantitative evaluation of convolution-based methods for medical image interpolation,” Med. Imag. Anal. |

25. | A. Entezari, M. Nilchian, and M. Unser, “A box spline calculus for the discretization of computed tomography reconstruction problems,” IEEE Trans. Med. Imag. |

26. | P. Thvenaz, T. Blu, and M. Unser, “Interpolation revisited [medical images application],” IEEE Trans. Med. Imag. |

27. | Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM Jour. on Imag. Sci. |

28. | T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM Jour. on Imag. Sci. |

29. | M. Ng, P. Weiss, and X. Yuan, “Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods,” SIAM Jour. on Sci. Comp. |

30. | B. Vandeghinste, B. Goossens, J. De Beenhouwer, A. Pizurica, W. Philips, S. Vandenberghe, and S. Staelens, “Split-bregman-based sparse-view CT reconstruction,” in “Fully 3D 2011 proc. ,” 431–434 (2011). |

31. | I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math. |

32. | A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Imag. Sci. |

33. | Z. Wang and A. Bovik, “A universal image quality index,” IEEE Sig. Proc. Lett. |

34. | Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Imag. Proc. |

35. | S. McDonald, F. Marone, C. Hintermuller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” Sync. Rad. |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(100.3190) Image processing : Inverse problems

(110.6960) Imaging systems : Tomography

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

**ToC Category:**

Image Processing

**History**

Original Manuscript: December 20, 2012

Revised Manuscript: January 24, 2013

Manuscript Accepted: January 25, 2013

Published: February 27, 2013

**Virtual Issues**

Vol. 8, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Masih Nilchian, Cédric Vonesch, Peter Modregger, Marco Stampanoni, and Michael Unser, "Fast iterative reconstruction of differential phase contrast X-ray tomograms," Opt. Express **21**, 5511-5528 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5511

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