Abstract

In this paper we introduce a new reconstruction algorithm for X-ray differential phase-contrast Imaging (DPCI). Our approach is based on 1) a variational formulation with a weighted data term and 2) a variable-splitting scheme that allows for fast convergence while reducing reconstruction artifacts. In order to improve the quality of the reconstruction we take advantage of higher-order total-variation regularization. In addition, the prior information on the support and positivity of the refractive index is considered, which yields significant improvement. We test our method in two reconstruction experiments involving real data; our results demonstrate its potential for in-vivo and medical imaging.

© 2013 Optical Society of America

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  1. C. David, B. Nohammer, H. H Solak, E. Ziegler, “Differential X-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
    [CrossRef]
  2. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, Y. Suzuki, “Demonstration of X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
    [CrossRef]
  3. F. Pfeiffer, O. Bunk, C. Kottler, C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. 580(2), 925–928 (2007).
    [CrossRef]
  4. M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, M. Unser, “Fast iterative reconstruction of differential phase contrast X-ray tomograms,” Opt. Express 21, 5511–5528 (2013).
    [CrossRef] [PubMed]
  5. Z. Qi, J. Zambelli, N. Bevins, G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. SPIE 7258, 72584A (2009).
    [CrossRef]
  6. T. Köhler, B. Brendel, E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. 38, 4542–4545 (2011).
    [CrossRef] [PubMed]
  7. Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, 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]
  8. M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, M. Unser, “Iterative FBP for improved reconstruction of X-ray differential phase-contrast tomograms,” in Proc. of ISBI’13(2013), pp. 1248–1251.
  9. Y. Wang, J. Yang, W. Yin, Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
    [CrossRef]
  10. A. Beck, M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
    [CrossRef]
  11. S. Lefkimmiatis, J. P. Ward, M. Unser, “Hessian Schatten-norm regularization for linear inverse problems,” IEEE Trans. Imaging 22, 1873–1888 (2013).
  12. Z. Wang, A. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9, 81–84 (2002).
    [CrossRef]

2013

S. Lefkimmiatis, J. P. Ward, M. Unser, “Hessian Schatten-norm regularization for linear inverse problems,” IEEE Trans. Imaging 22, 1873–1888 (2013).

M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, M. Unser, “Fast iterative reconstruction of differential phase contrast X-ray tomograms,” Opt. Express 21, 5511–5528 (2013).
[CrossRef] [PubMed]

2012

2011

T. Köhler, B. Brendel, E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. 38, 4542–4545 (2011).
[CrossRef] [PubMed]

2009

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

Z. Qi, J. Zambelli, N. Bevins, G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. SPIE 7258, 72584A (2009).
[CrossRef]

2008

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

2007

F. Pfeiffer, O. Bunk, C. Kottler, C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. 580(2), 925–928 (2007).
[CrossRef]

2003

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

2002

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

Z. Wang, A. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9, 81–84 (2002).
[CrossRef]

Anastasio, M. A.

Beck, A.

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

Bevins, N.

Z. Qi, J. Zambelli, N. Bevins, G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. SPIE 7258, 72584A (2009).
[CrossRef]

Bovik, A.

Z. Wang, A. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9, 81–84 (2002).
[CrossRef]

Brendel, B.

T. Köhler, B. Brendel, E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. 38, 4542–4545 (2011).
[CrossRef] [PubMed]

Bunk, O.

F. Pfeiffer, O. Bunk, C. Kottler, C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. 580(2), 925–928 (2007).
[CrossRef]

Chen, G.

Z. Qi, J. Zambelli, N. Bevins, G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. SPIE 7258, 72584A (2009).
[CrossRef]

David, C.

F. Pfeiffer, O. Bunk, C. Kottler, C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. 580(2), 925–928 (2007).
[CrossRef]

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

Hamaishi, Y.

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

Kawamoto, S.

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

Köhler, T.

T. Köhler, B. Brendel, E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. 38, 4542–4545 (2011).
[CrossRef] [PubMed]

Kottler, C.

F. Pfeiffer, O. Bunk, C. Kottler, C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. 580(2), 925–928 (2007).
[CrossRef]

Koyama, I.

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

Lefkimmiatis, S.

S. Lefkimmiatis, J. P. Ward, M. Unser, “Hessian Schatten-norm regularization for linear inverse problems,” IEEE Trans. Imaging 22, 1873–1888 (2013).

Modregger, P.

Momose, A.

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

Nilchian, M.

M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, M. Unser, “Fast iterative reconstruction of differential phase contrast X-ray tomograms,” Opt. Express 21, 5511–5528 (2013).
[CrossRef] [PubMed]

M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, M. Unser, “Iterative FBP for improved reconstruction of X-ray differential phase-contrast tomograms,” in Proc. of ISBI’13(2013), pp. 1248–1251.

Nohammer, B.

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

Pan, X.

Pfeiffer, F.

F. Pfeiffer, O. Bunk, C. Kottler, C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. 580(2), 925–928 (2007).
[CrossRef]

Qi, Z.

Z. Qi, J. Zambelli, N. Bevins, G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. SPIE 7258, 72584A (2009).
[CrossRef]

Roessl, E.

T. Köhler, B. Brendel, E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. 38, 4542–4545 (2011).
[CrossRef] [PubMed]

Sidky, E. Y.

Solak, H. H

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

Stampanoni, M.

Suzuki, Y.

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

Takai, K.

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

Teboulle, M.

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

Unser, M.

S. Lefkimmiatis, J. P. Ward, M. Unser, “Hessian Schatten-norm regularization for linear inverse problems,” IEEE Trans. Imaging 22, 1873–1888 (2013).

M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, M. Unser, “Fast iterative reconstruction of differential phase contrast X-ray tomograms,” Opt. Express 21, 5511–5528 (2013).
[CrossRef] [PubMed]

M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, M. Unser, “Iterative FBP for improved reconstruction of X-ray differential phase-contrast tomograms,” in Proc. of ISBI’13(2013), pp. 1248–1251.

Vonesch, C.

M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, M. Unser, “Fast iterative reconstruction of differential phase contrast X-ray tomograms,” Opt. Express 21, 5511–5528 (2013).
[CrossRef] [PubMed]

M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, M. Unser, “Iterative FBP for improved reconstruction of X-ray differential phase-contrast tomograms,” in Proc. of ISBI’13(2013), pp. 1248–1251.

Wang, Y.

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

Wang, Z.

Z. Wang, A. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9, 81–84 (2002).
[CrossRef]

Ward, J. P.

S. Lefkimmiatis, J. P. Ward, M. Unser, “Hessian Schatten-norm regularization for linear inverse problems,” IEEE Trans. Imaging 22, 1873–1888 (2013).

Xu, Q.

Yang, J.

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

Yin, W.

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

Zambelli, J.

Z. Qi, J. Zambelli, N. Bevins, G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. SPIE 7258, 72584A (2009).
[CrossRef]

Zhang, Y.

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

Ziegler, E.

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

Appl. Phys. Lett.

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

IEEE Signal Process. Lett.

Z. Wang, A. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9, 81–84 (2002).
[CrossRef]

IEEE Trans. Imaging

S. Lefkimmiatis, J. P. Ward, M. Unser, “Hessian Schatten-norm regularization for linear inverse problems,” IEEE Trans. Imaging 22, 1873–1888 (2013).

Jpn. J. Appl. Phys.

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

Med. Phys.

T. Köhler, B. Brendel, E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. 38, 4542–4545 (2011).
[CrossRef] [PubMed]

Nucl. Instrum. Methods Phys. Res.

F. Pfeiffer, O. Bunk, C. Kottler, C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. 580(2), 925–928 (2007).
[CrossRef]

Opt. Express

Proc. SPIE

Z. Qi, J. Zambelli, N. Bevins, G. Chen, “A novel method to reduce data acquisition time in differential phase contrast computed tomography using compressed sensing,” Proc. SPIE 7258, 72584A (2009).
[CrossRef]

SIAM J. Imaging Sci.

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

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

Other

M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, M. Unser, “Iterative FBP for improved reconstruction of X-ray differential phase-contrast tomograms,” in Proc. of ISBI’13(2013), pp. 1248–1251.

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Figures (3)

Fig. 1
Fig. 1

Two reference samples (a) and (b).

Fig. 2
Fig. 2

Scaffold reconstruction with 250 projections using (a) FBP, (b) ADMM-PCG, (c) CRWN with HS regularization and (d) CRWN with TV regularization.

Fig. 3
Fig. 3

The reconstruction performances concerning speed and quality is shown in (a) and (b), respectively.

Tables (3)

Tables Icon

Algorithm 1: Denoising algorithm

Tables Icon

Algorithm 2: DPCI-constrained regularized reconstruction with weighted norm (CRWN).

Tables Icon

Table 1 Performance of different reconstruction techniques which have been applied on Phantom and Scaffold samples.

Equations (19)

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g ( y , θ ) = ( 1 ) { f } ( y , θ ) = { f } ( y , θ ) y ,
f = ( 1 ) * { h * y g } .
f ( x ) = k c k β n ( x k ) ,
c 0 = argmin c 𝒞 { J ( c ) 1 2 Hc g W 2 + λ 1 Ψ 1 ( c ) + λ 2 Ψ 2 ( c ) } ,
μ ( c , u , α ) = 1 2 Hu g W 2 + λ 1 Ψ 1 ( u ) + λ 2 Ψ 2 ( c ) + α T ( u c ) + μ 2 u c 2 2 ,
{ ( c k + 1 , u k + 1 ) argmin c 𝒞 , u μ ( c , u , α k ) α k + 1 α k + μ ( u k + 1 c k + 1 ) .
{ u k + 1 argmin u μ ( c k , u , α k ) ( Step 1 ) c k + 1 argmin c 𝒞 μ ( c , u k + 1 , α k ) ( Step 2 ) α k + 1 α k + μ ( u k + 1 c k + 1 ) . ( Step 3 )
μ ( c k , u , α k ) = ( H T WH + ( μ + λ 1 ) I ) u ( H T Wg ( α k μ c k ) ) .
argmin c 𝒞 { μ ( c , u k + 1 , α k ) α k T ( u k + 1 c ) + μ 2 u k + 1 c 2 2 + λ 2 ψ 2 ( c ) } = argmin c 𝒞 { 1 2 u k + 1 + α k μ c 2 2 + λ 2 μ Ψ 2 ( c ) } .
Ψ 2 ( c ) = Rc ,
prox , λ , 𝒫 𝒞 ( z ) = argmin c 𝒞 { 1 2 z c 2 2 + λ Rc } ,
Rc = max p R T p , u ,
p * = argmin p f ( p ) + 1 ,
Ψ 2 ( c ) = Lc 1 , 1
, = { p = [ p 1 T , p 2 T , , p N T ] T N × 2 : p i 1 , i = 1 , 2 , , N } .
[ y ˜ i ] j = sgn ( [ y i ] j ) min ( | [ y i ] j | , 1 ) , i = 1 , 2 , , N , j = 1 , 2 ,
Ψ 2 ( c ) = Hc 1 , 𝒮 1 ,
, 𝒮 = { p = [ p 1 T , p 2 T , , p N T ] T N × 2 × 2 : p i 𝒮 1 , i = 1 , 2 , , N } ,
L = λ 2 × λ max ( RR T ) ,

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