Abstract

Phase retrieval in differential X-ray phase contrast imaging involves a one dimensional integration step. In the presence of noise, standard integration methods result in image blurring and streak artifacts. This work proposes a regularized integration method which takes the availability of two dimensional data as well as the integration-specific frequency-dependent noise amplification into account. In more detail, a Fourier-domain algorithm is developed comprising a frequency-dependent minimization of the total variation orthogonal to the direction of integration. For both simulated and experimental data, the novel method yielded strong artefact reduction without increased blurring superior to the results obtained by standard integration methods or regularization techniques in the image domain.

© 2014 Optical Society of America

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2012

W. Cong, A. Momose, G. Wang, “Fourier transform-based iterative method for differential phase-contrast computed tomography,” Opt. Letters 37, 1784–1786 (2012).
[CrossRef]

K. Khare, C. J. Hardy, K. F. King, P. A. Turski, L. Marinelli, “Accelerated MR imaging using compressive sensing with no free parameters,” Magn. Reson. Med. 68, 1450(2012).
[CrossRef] [PubMed]

2011

T. Thüring, P. Modregger, B. Pinzer, Z. Wang, M. Stampanoni, “Non-linear regularized phase retrieval for unidirectional X-ray differential phase contrast radiography,” Opt. Express 19, 25545–25558 (2011).
[CrossRef]

L. Ritschl, F. Bergner, C. Fleischmann, M. Kachelrieß, “Improved total variation-based CT image reconstruction applied to clinical data,” Phys. Med. Biol. 56, 1545–1561 (2011).
[CrossRef] [PubMed]

2010

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

2009

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

D. L. Donoho, A. Maleki, A. Montanari, “Message-passing algorithms for compressed sensing,” PNAS 106, 18914–18919 (2009).
[CrossRef] [PubMed]

H. Yu, G. Wang, “Compressed sensing based interior tomography,” Phys. Med. Biol. 54, 2791–2805 (2009).
[CrossRef] [PubMed]

J. Herzen, T. Donath, F. Pfeiffer, O. Bunk, C. Padeste, F. Beckmann, A. Schreyer, C. David, “Quantitative phase-contrast tomography of a liquid phantom using a conventional X-ray tube source,” Opt. Express 17, 10010–10018 (2009).
[CrossRef] [PubMed]

2008

E. Y. Sidky, X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef] [PubMed]

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

M. de Jonge, B. Hornberger, C. Holzner, D. Legnini, D. Paterson, I. McNulty, C. Jacobsen, S. Vogt, “Quantitative phase imaging with a scanning transmission X-ray microscope,” Phys. Rev. Lett. 100, 163902 (2008).
[CrossRef] [PubMed]

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

2007

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, 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]

J. M. Bioucas-Dias, M. A. T. Figueiredo, “A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef] [PubMed]

C. Kottler, C. David, F. Pfeiffer, 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]

2006

M. Elad, “Why simple shrinkage is still relevant for redundant representations,” IEEE Trans. Inf. Theory 52, 5559–5569 (2006).
[CrossRef]

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. 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]

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

2005

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

M. O. Hasnah, C. Parham, E. D. Pisano, Z. Zhong, O. Oltulu, D. Chapman, “Mass density images from the diffraction enhanced imaging technique,” Med. Phys. 32, 54952 (2005).
[CrossRef]

P. L. Combettes, V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” SIAM J. Multiscale Model. Simul. 4, 1168–1200 (2005).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, E. Ziegler, “Quantitative X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[CrossRef] [PubMed]

2004

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

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

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imag. Vis. 20, 89–97 (2004).
[CrossRef]

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

2003

M. A. T. Figueiredo, R. D. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process. 12, 906–916 (2003).
[CrossRef]

A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11, 2303–2314 (2003).
[CrossRef] [PubMed]

2001

K. M. Pavlov, C. M. Kewish, J. R. Davis, M. J. Morgan, “A variant on the geometrical optics approximation in diffraction enhanced tomography,” J. Phys. D: Appl. Phys. 34, A168–A172 (2001).
[CrossRef]

2000

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53, 23–27 (2000).
[CrossRef]

1998

Z. Kam, “Microscopic differential interference contrast image processing by line integration (lid) and deconvolution,” Bioimaging. 6, 166176 (1998).
[CrossRef]

C. R. Vogel, M. E. Oman, “Fast, robust total variation-based reconstruction of noisy, blurred images,” IEEE Trans. Image Process. 7, 813–824 (1998).
[CrossRef]

1997

E. B. van Munster, L. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope.” J. Microsc. 188, 149157 (1997).
[CrossRef]

1996

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–337 (1996).
[CrossRef]

Y. Li, F. Santosa, “A computational algorithm for minimizing total variation in image restoration,” IEEE Trans. Image Process. 5, 987–995 (1996).
[CrossRef] [PubMed]

1995

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, 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]

D. L. Donoho, I. M. Johnstone, “Adapting to the unknown smoothness via wavelet shrinkage,” JASA 90, 1200–1224 (1995).
[CrossRef]

1992

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

Ando, M.

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

Arnison, M.

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

Aten, J. A.

E. B. van Munster, L. van Vliet, J. A. Aten, “Reconstruction of optical pathlength distributions from images obtained by a wide-field differential interference contrast microscope.” J. Microsc. 188, 149157 (1997).
[CrossRef]

Baumann, J.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, 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]

Bech, M.

M. Bech, “X-ray imaging with a grating interferometer,” Ph.D. thesis, University of Copenhagen (2009).

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]

Beckmann, F.

Bergner, F.

L. Ritschl, F. Bergner, C. Fleischmann, M. Kachelrieß, “Improved total variation-based CT image reconstruction applied to clinical data,” Phys. Med. Biol. 56, 1545–1561 (2011).
[CrossRef] [PubMed]

Bioucas-Dias, J. M.

J. M. Bioucas-Dias, M. A. T. Figueiredo, “A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[CrossRef] [PubMed]

Bunk, O.

J. Herzen, T. Donath, F. Pfeiffer, O. Bunk, C. Padeste, F. Beckmann, A. Schreyer, C. David, “Quantitative phase-contrast tomography of a liquid phantom using a conventional X-ray tube source,” Opt. Express 17, 10010–10018 (2009).
[CrossRef] [PubMed]

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

C. Kottler, C. David, F. Pfeiffer, 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]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, 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]

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

Caselles, V.

A. Chambolle, V. Caselles, D. Cremers, M. Novaga, T. Pock, “An introduction to total variation for image analysis,” in “Theoretical Foundations and Numerical Methods for Sparse Recovery,” (De Gruyter, 2010).

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imag. Vis. 20, 89–97 (2004).
[CrossRef]

A. Chambolle, V. Caselles, D. Cremers, M. Novaga, T. Pock, “An introduction to total variation for image analysis,” in “Theoretical Foundations and Numerical Methods for Sparse Recovery,” (De Gruyter, 2010).

Chapman, D.

M. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. 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]

M. O. Hasnah, C. Parham, E. D. Pisano, Z. Zhong, O. Oltulu, D. Chapman, “Mass density images from the diffraction enhanced imaging technique,” Med. Phys. 32, 54952 (2005).
[CrossRef]

Cloetens, P.

Cogswell, C.

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

Combettes, P. L.

P. L. Combettes, V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” SIAM J. Multiscale Model. Simul. 4, 1168–1200 (2005).
[CrossRef]

Cong, W.

W. Cong, A. Momose, G. Wang, “Fourier transform-based iterative method for differential phase-contrast computed tomography,” Opt. Letters 37, 1784–1786 (2012).
[CrossRef]

Cremers, D.

A. Chambolle, V. Caselles, D. Cremers, M. Novaga, T. Pock, “An introduction to total variation for image analysis,” in “Theoretical Foundations and Numerical Methods for Sparse Recovery,” (De Gruyter, 2010).

Daubechies, I.

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

David, C.

J. Herzen, T. Donath, F. Pfeiffer, O. Bunk, C. Padeste, F. Beckmann, A. Schreyer, C. David, “Quantitative phase-contrast tomography of a liquid phantom using a conventional X-ray tube source,” Opt. Express 17, 10010–10018 (2009).
[CrossRef] [PubMed]

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

C. Kottler, C. David, F. Pfeiffer, 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]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, 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]

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

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, E. Ziegler, “Quantitative X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[CrossRef] [PubMed]

Davis, J. R.

K. M. Pavlov, C. M. Kewish, J. R. Davis, M. J. Morgan, “A variant on the geometrical optics approximation in diffraction enhanced tomography,” J. Phys. D: Appl. Phys. 34, A168–A172 (2001).
[CrossRef]

de Jonge, M.

M. de Jonge, B. Hornberger, C. Holzner, D. Legnini, D. Paterson, I. McNulty, C. Jacobsen, S. Vogt, “Quantitative phase imaging with a scanning transmission X-ray microscope,” Phys. Rev. Lett. 100, 163902 (2008).
[CrossRef] [PubMed]

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

Defrise, M.

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

Diaz, A.

Dierolf, M.

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

Donath, T.

Donoho, D. L.

D. L. Donoho, A. Maleki, A. Montanari, “Message-passing algorithms for compressed sensing,” PNAS 106, 18914–18919 (2009).
[CrossRef] [PubMed]

D. L. Donoho, I. M. Johnstone, “Adapting to the unknown smoothness via wavelet shrinkage,” JASA 90, 1200–1224 (1995).
[CrossRef]

Elad, M.

M. Elad, “Why simple shrinkage is still relevant for redundant representations,” IEEE Trans. Inf. Theory 52, 5559–5569 (2006).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

The modified FORBILD phantom (A), the noisy dXPC signal (B) (10% noise level), and reconstructed images for Algorithms 1 to 4 (C–F).

Fig. 2
Fig. 2

Noise patterns of the reconstructed images for the dXPC simulation of the modified FORBILD phantom at 10% noise level for Algorithms 1 to 4 (A–D) calculated by subtracting the ground truth Fig. 1(A). Corresponding close-ups of the inner ear region indicated by the rectangle in C are shown in E–H, respectively.

Fig. 3
Fig. 3

Quantitative error analysis of the reconstructed images for the dXPC simulation of the modified FORBILD phantom at various noise levels.

Fig. 4
Fig. 4

Cost-functions terms vs. iteration t for Algorithm 3 (left) and Algorithm 4 (right) applied to the modified FORBILD phantom: normalized residual norm (||zt||, blue) and TV ( TV y ε ( y t ) and TV y ε ( W y t ) , respectively, green). The vertical line (red) indicates the iteration for which the convergence criterion Eq. (18) was met.

Fig. 5
Fig. 5

The absorption image of the salami (A), the differential phase image (B), and the reconstructed phase images for Algorithms 1 to 4 (C–F).

Tables (1)

Tables Icon

Table 1 System specifications of the dXPC setup.

Equations (19)

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Ω M , N : = { ( x m , y n ) | x m = m h x , y n = n h y , m = 1 M , j = 1 N }
ψ = D x ϕ ,
D x : M × N M × N , ( D x ϕ ) m , n = ϕ m , n ϕ m 1 , n h x
I x , c : M × N M × N , ( I x , c ψ ) m , n = i = 1 m ψ i , n + c n ,
x ψ = k x ϕ
k : M × N M × N , ( k a ) m , n = j ω m a m , n ,
ω m = 2 π m M / 2 M , m = 1 M ,
k ˜ : M × N M × N , ( k ˜ a ) m , n = { 0 if ω m = 0 j ω m a m , n else .
min ϕ M × N D x ϕ ψ 2 .
min a M × N x 1 ( k a ) ψ 2 .
D y s : M × N M × N , ( D y s ϕ ) m , n = ϕ m , n + 1 ϕ m , n 1 2 h y ,
TV y ε : M × N , TV y ε ( ϕ ) = m = 1 M n = 1 N ( | ( D y s ϕ ) m , n | 2 + ε 2 ) 1 / 2 = : s m , n ( ϕ ) .
min ϕ M × N D x ϕ ψ 2 + λ TV y ε ( ϕ )
min a M × N x 1 ( k a ) ψ 2 + λ TV y ε ( W a )
W : M × N M × N , ( W a ) m n = w ( ω m ) a m n
ϕ m n TV y ε ( ϕ ) = ϕ m n ( s m , n + 1 1 / 2 ( ϕ ) + s m , n 1 1 / 2 ( ϕ ) ) = ϕ m , n ϕ m , n + 2 2 h y s m , n + 1 1 / 2 ( ϕ ) + ϕ m , n ϕ m , n 2 2 h y s m , n 1 1 / 2 ( ϕ )
TV y ε : M × N M × N , ( TV y ε ( ϕ ) ) m , n = ϕ m n TV y ε ( ϕ ) .
z t 1 z t < TOL ψ .
w ( ω ) = { 1 if | ω | 2 / M 0 else ,

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