Abstract

The phase retrieval process is a nonlinear ill-posed problem. The Fresnel diffraction patterns obtained with hard x-ray synchrotron beam can be used to retrieve the phase contrast. In this work, we present a convergence comparison of several nonlinear approaches for the phase retrieval problem involving regularizations with sparsity constraints. The phase solution is assumed to have a sparse representation with respect to an orthonormal wavelets basis. One approach uses alternatively a solution of the nonlinear problem based on the Fréchet derivative and a solution of the linear problem in wavelet coordinates with an iterative thresholding. A second method is the one proposed by Ramlau and Teschke which generalizes to a nonlinear problem the classical thresholding algorithm. The algorithms were tested on a 3D Shepp–Logan phantom corrupted by white Gaussian noise. The best simulation results are obtained by the first method for the various noise levels and initializations investigated. The reconstruction errors are significantly decreased with respect to the ones given by the classical linear phase retrieval approaches.

© 2013 Optical Society of America

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    [CrossRef]

2012 (4)

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Nonlinear phase retrieval using projection operator and iterative wavelet thresholding,” IEEE Signal Process. Lett. 19, 579–582 (2012).
[CrossRef]

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “CPRL—an extension of compressive sensing to the phase retrieval problem,” Adv. Neural Inf. Process. Syst. 25, 1376–1384 (2012).

T. Gaass, G. Potdevin, P. B. Nol, A. Tapfer, M. Willner, J. Herzen, and A. Haase, “Compressed sensing for phase contrast CT,” AIP Conf. Proc. 1466, 150–154 (2012).
[CrossRef]

M. C. Newton, “Compressed sensing for phase retrieval,” Phys. Rev. E 85, 056706 (2012).
[CrossRef]

2011 (1)

2010 (4)

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2428–2436 (2010).
[CrossRef]

J. Moosmann, R. Hofmann, A. V. Bronnikov, and T. Baumbach, “Nonlinear phase retrieval from single-distance radiograph,” Opt. Express 18, 25771–25785 (2010).
[CrossRef]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105, 248102 (2010).
[CrossRef]

A. Alpers, G. T. Herman, H. Poulsen, and S. Schmidt, “Phase retrieval for superposed signals from multiple binary objects,” J. Opt. Soc. Am. A 27, 1927–1937 (2010).
[CrossRef]

2009 (2)

F. Dupe, J. M. Fadili, and J. L. Starck, “A proximal iteration for deconvolving poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18, 310–321(2009).
[CrossRef]

L. Chaâri, N. Pustelnik, C. Chaux, and J. C. Pesquet, “Solving inverse problems with overcomplete transforms and convex optimization techniques,” Proc. SPIE 7446, 74460U (2009).
[CrossRef]

2008 (2)

I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008).

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008).
[CrossRef]

2007 (2)

G. Teschke and R. Ramlau, “An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vector-valued regimes and an application to color image impainting,” Inverse Probl. 23, 1851–1870 (2007).
[CrossRef]

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32, 1617–1619 (2007).
[CrossRef]

2006 (2)

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509(2006).
[CrossRef]

R. Ramlau and G. Teschke, “A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints,” Numer. Math. 104, 177–203 (2006).
[CrossRef]

2005 (1)

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

2004 (1)

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef]

2003 (2)

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]

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

2002 (1)

R. Ramlau, “A steepest descent algorithm for the global minimization of the Tikhonov functional,” Inverse Probl. 18, 381–403 (2002).
[CrossRef]

1999 (2)

V. Dicken, “A new approach towards simultaneous activity and attenuation reconstruction in emission tomography,” Inverse Probl. 15, 931–960 (1999).
[CrossRef]

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-rays quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

1997 (1)

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

1996 (4)

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

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

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

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

1995 (1)

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]

1985 (1)

C. R. Vogel, “Numerical solution of a non-linear ill-posed problem arising in inverse scattering,” Inverse Probl. 1, 393–403 (1985).
[CrossRef]

1982 (1)

1965 (1)

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

Alpers, A.

Arfelli, F.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

Barrett, R.

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

Baruchel, J.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

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

Baumbach, T.

Beleggia, M.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef]

Boistel, R.

Bonse, U.

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1997).

Bronnikov, A. V.

Candes, E. J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509(2006).
[CrossRef]

Chaâri, L.

L. Chaâri, N. Pustelnik, C. Chaux, and J. C. Pesquet, “Solving inverse problems with overcomplete transforms and convex optimization techniques,” Proc. SPIE 7446, 74460U (2009).
[CrossRef]

Chapman, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Chaux, C.

L. Chaâri, N. Pustelnik, C. Chaux, and J. C. Pesquet, “Solving inverse problems with overcomplete transforms and convex optimization techniques,” Proc. SPIE 7446, 74460U (2009).
[CrossRef]

Cloetens, P.

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2428–2436 (2010).
[CrossRef]

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008).
[CrossRef]

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32, 1617–1619 (2007).
[CrossRef]

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

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

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

Daubechies, I.

I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008).

David, C.

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105, 248102 (2010).
[CrossRef]

Davidoiu, V.

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Nonlinear phase retrieval using projection operator and iterative wavelet thresholding,” IEEE Signal Process. Lett. 19, 579–582 (2012).
[CrossRef]

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Non-linear phase retrieval based on Fréchet derivative,” Opt. Express 19, 22809–22819 (2011).
[CrossRef]

Davis, T.

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]

Dicken, V.

V. Dicken, “A new approach towards simultaneous activity and attenuation reconstruction in emission tomography,” Inverse Probl. 15, 931–960 (1999).
[CrossRef]

Donath, T.

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105, 248102 (2010).
[CrossRef]

Dong, R.

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “CPRL—an extension of compressive sensing to the phase retrieval problem,” Adv. Neural Inf. Process. Syst. 25, 1376–1384 (2012).

Dupe, F.

F. Dupe, J. M. Fadili, and J. L. Starck, “A proximal iteration for deconvolving poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18, 310–321(2009).
[CrossRef]

Fadili, J. M.

F. Dupe, J. M. Fadili, and J. L. Starck, “A proximal iteration for deconvolving poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18, 310–321(2009).
[CrossRef]

Fienup, J. R.

Fornasier, M.

I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008).

Gaass, T.

T. Gaass, G. Potdevin, P. B. Nol, A. Tapfer, M. Willner, J. Herzen, and A. Haase, “Compressed sensing for phase contrast CT,” AIP Conf. Proc. 1466, 150–154 (2012).
[CrossRef]

Gao, D.

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

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]

Gmur, N.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Grasmair, M.

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variational Methods in Imaging (Springer-Verlag, 2008).

Grossauer, H.

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variational Methods in Imaging (Springer-Verlag, 2008).

Guigay, J. P.

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008).
[CrossRef]

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32, 1617–1619 (2007).
[CrossRef]

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

Guigay, J.-P.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

Gureyev, T.

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]

Gureyev, T. E.

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-rays quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

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

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

Haase, A.

T. Gaass, G. Potdevin, P. B. Nol, A. Tapfer, M. Willner, J. Herzen, and A. Haase, “Compressed sensing for phase contrast CT,” AIP Conf. Proc. 1466, 150–154 (2012).
[CrossRef]

Haltmeier, M.

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variational Methods in Imaging (Springer-Verlag, 2008).

Hamaishi, Y.

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]

Hansen, P. C.

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1987).

Hart, M.

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

Herman, G. T.

Herzen, J.

T. Gaass, G. Potdevin, P. B. Nol, A. Tapfer, M. Willner, J. Herzen, and A. Haase, “Compressed sensing for phase contrast CT,” AIP Conf. Proc. 1466, 150–154 (2012).
[CrossRef]

Hirano, K.

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

Hofmann, R.

Itai, Y.

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

Johnston, R. E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Kawamoto, S.

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]

Koyama, I.

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]

Langer, M.

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Nonlinear phase retrieval using projection operator and iterative wavelet thresholding,” IEEE Signal Process. Lett. 19, 579–582 (2012).
[CrossRef]

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Non-linear phase retrieval based on Fréchet derivative,” Opt. Express 19, 22809–22819 (2011).
[CrossRef]

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2428–2436 (2010).
[CrossRef]

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008).
[CrossRef]

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “A mixed contrast transfer and transport of intensity approach for phase retrieval in the Fresnel region,” Opt. Lett. 32, 1617–1619 (2007).
[CrossRef]

Lenzen, F.

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variational Methods in Imaging (Springer-Verlag, 2008).

Loris, I.

I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008).

Menk, R.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Momose, A.

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]

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

Moosmann, J.

Mukherjee, S.

S. Mukherjee and C. S. Seelamantula, “An iterative algorithm for phase retrieval with sparsity constraints: application to frequency domain optical coherence tomography,” In 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2012), pp. 553–556.

Newton, M. C.

M. C. Newton, “Compressed sensing for phase retrieval,” Phys. Rev. E 85, 056706 (2012).
[CrossRef]

Nol, P. B.

T. Gaass, G. Potdevin, P. B. Nol, A. Tapfer, M. Willner, J. Herzen, and A. Haase, “Compressed sensing for phase contrast CT,” AIP Conf. Proc. 1466, 150–154 (2012).
[CrossRef]

Nugent, K. A.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

Ohlsson, H.

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “CPRL—an extension of compressive sensing to the phase retrieval problem,” Adv. Neural Inf. Process. Syst. 25, 1376–1384 (2012).

Paganin, D.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

Paganin, D. M.

D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).

Pesquet, J. C.

L. Chaâri, N. Pustelnik, C. Chaux, and J. C. Pesquet, “Solving inverse problems with overcomplete transforms and convex optimization techniques,” Proc. SPIE 7446, 74460U (2009).
[CrossRef]

Peyrin, F.

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Nonlinear phase retrieval using projection operator and iterative wavelet thresholding,” IEEE Signal Process. Lett. 19, 579–582 (2012).
[CrossRef]

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Non-linear phase retrieval based on Fréchet derivative,” Opt. Express 19, 22809–22819 (2011).
[CrossRef]

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2428–2436 (2010).
[CrossRef]

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008).
[CrossRef]

Pisano, E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Pogany, A.

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

Potdevin, G.

T. Gaass, G. Potdevin, P. B. Nol, A. Tapfer, M. Willner, J. Herzen, and A. Haase, “Compressed sensing for phase contrast CT,” AIP Conf. Proc. 1466, 150–154 (2012).
[CrossRef]

Poulsen, H.

Pustelnik, N.

L. Chaâri, N. Pustelnik, C. Chaux, and J. C. Pesquet, “Solving inverse problems with overcomplete transforms and convex optimization techniques,” Proc. SPIE 7446, 74460U (2009).
[CrossRef]

Ramlau, R.

G. Teschke and R. Ramlau, “An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vector-valued regimes and an application to color image impainting,” Inverse Probl. 23, 1851–1870 (2007).
[CrossRef]

R. Ramlau and G. Teschke, “A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints,” Numer. Math. 104, 177–203 (2006).
[CrossRef]

R. Ramlau, “A steepest descent algorithm for the global minimization of the Tikhonov functional,” Inverse Probl. 18, 381–403 (2002).
[CrossRef]

Raven, C.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-rays quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

Romberg, J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509(2006).
[CrossRef]

Rutishauser, S.

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105, 248102 (2010).
[CrossRef]

Sastry, S. S.

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “CPRL—an extension of compressive sensing to the phase retrieval problem,” Adv. Neural Inf. Process. Syst. 25, 1376–1384 (2012).

Sayers, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Scherzer, O.

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variational Methods in Imaging (Springer-Verlag, 2008).

Schlenker, M.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

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

Schmidt, S.

Schofield, M. A.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef]

Seelamantula, C. S.

S. Mukherjee and C. S. Seelamantula, “An iterative algorithm for phase retrieval with sparsity constraints: application to frequency domain optical coherence tomography,” In 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2012), pp. 553–556.

Sixou, B.

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Nonlinear phase retrieval using projection operator and iterative wavelet thresholding,” IEEE Signal Process. Lett. 19, 579–582 (2012).
[CrossRef]

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Non-linear phase retrieval based on Fréchet derivative,” Opt. Express 19, 22809–22819 (2011).
[CrossRef]

Snigirev, A.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-rays quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

Snigireva, I.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-rays quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

Starck, J. L.

F. Dupe, J. M. Fadili, and J. L. Starck, “A proximal iteration for deconvolving poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18, 310–321(2009).
[CrossRef]

Stevenson, A.

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]

Stevenson, A. W.

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

Suzuki, Y.

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]

Takai, K.

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]

Takeda, T.

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

Tao, T.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509(2006).
[CrossRef]

Tapfer, A.

T. Gaass, G. Potdevin, P. B. Nol, A. Tapfer, M. Willner, J. Herzen, and A. Haase, “Compressed sensing for phase contrast CT,” AIP Conf. Proc. 1466, 150–154 (2012).
[CrossRef]

Teschke, G.

G. Teschke and R. Ramlau, “An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vector-valued regimes and an application to color image impainting,” Inverse Probl. 23, 1851–1870 (2007).
[CrossRef]

R. Ramlau and G. Teschke, “A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints,” Numer. Math. 104, 177–203 (2006).
[CrossRef]

Thomlinson, W.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Vogel, C. R.

C. R. Vogel, “Numerical solution of a non-linear ill-posed problem arising in inverse scattering,” Inverse Probl. 1, 393–403 (1985).
[CrossRef]

Volkov, V. V.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef]

Washburn, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Weitkamp, T.

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105, 248102 (2010).
[CrossRef]

Wilkins, S.

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]

Wilkins, S. W.

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-rays quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

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

Willner, M.

T. Gaass, G. Potdevin, P. B. Nol, A. Tapfer, M. Willner, J. Herzen, and A. Haase, “Compressed sensing for phase contrast CT,” AIP Conf. Proc. 1466, 150–154 (2012).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1997).

Yang, A. Y.

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “CPRL—an extension of compressive sensing to the phase retrieval problem,” Adv. Neural Inf. Process. Syst. 25, 1376–1384 (2012).

Zabler, S.

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

Zanette, I.

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105, 248102 (2010).
[CrossRef]

Zhong, Z.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Zhu, Y.

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef]

Adv. Neural Inf. Process. Syst. (1)

H. Ohlsson, A. Y. Yang, R. Dong, and S. S. Sastry, “CPRL—an extension of compressive sensing to the phase retrieval problem,” Adv. Neural Inf. Process. Syst. 25, 1376–1384 (2012).

AIP Conf. Proc. (1)

T. Gaass, G. Potdevin, P. B. Nol, A. Tapfer, M. Willner, J. Herzen, and A. Haase, “Compressed sensing for phase contrast CT,” AIP Conf. Proc. 1466, 150–154 (2012).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

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

IEEE Signal Process. Lett. (1)

V. Davidoiu, B. Sixou, M. Langer, and F. Peyrin, “Nonlinear phase retrieval using projection operator and iterative wavelet thresholding,” IEEE Signal Process. Lett. 19, 579–582 (2012).
[CrossRef]

IEEE Trans. Image Process. (2)

F. Dupe, J. M. Fadili, and J. L. Starck, “A proximal iteration for deconvolving poisson noisy images using sparse representations,” IEEE Trans. Image Process. 18, 310–321(2009).
[CrossRef]

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19, 2428–2436 (2010).
[CrossRef]

IEEE Trans. Inf. Theory (1)

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509(2006).
[CrossRef]

Inverse Probl. (4)

C. R. Vogel, “Numerical solution of a non-linear ill-posed problem arising in inverse scattering,” Inverse Probl. 1, 393–403 (1985).
[CrossRef]

V. Dicken, “A new approach towards simultaneous activity and attenuation reconstruction in emission tomography,” Inverse Probl. 15, 931–960 (1999).
[CrossRef]

G. Teschke and R. Ramlau, “An iterative algorithm for nonlinear inverse problems with joint sparsity constraints in vector-valued regimes and an application to color image impainting,” Inverse Probl. 23, 1851–1870 (2007).
[CrossRef]

R. Ramlau, “A steepest descent algorithm for the global minimization of the Tikhonov functional,” Inverse Probl. 18, 381–403 (2002).
[CrossRef]

J. Fourier Anal. Appl. (1)

I. Daubechies, M. Fornasier, and I. Loris, “Accelerated projected gradient method for linear inverse problems with sparsity constraints,” J. Fourier Anal. Appl. 14, 764–792 (2008).

J. Opt. Soc. Am. A (1)

J. Phys. D (2)

T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-rays quantitative non-interferometric phase-contrast microscopy,” J. Phys. D 32, 563–567 (1999).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

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]

Med. Phys. (1)

M. Langer, P. Cloetens, J. P. Guigay, and F. Peyrin, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556–4565 (2008).
[CrossRef]

Nat. Med. (1)

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

Nature (2)

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]

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

Numer. Math. (1)

R. Ramlau and G. Teschke, “A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints,” Numer. Math. 104, 177–203 (2006).
[CrossRef]

Opt. Commun. (1)

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun. 220, 49–58 (2003).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (1)

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42, 2015–2025 (1997).
[CrossRef]

Phys. Rev. E (1)

M. C. Newton, “Compressed sensing for phase retrieval,” Phys. Rev. E 85, 056706 (2012).
[CrossRef]

Phys. Rev. Lett. (2)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x rays,” Phys. Rev. Lett. 77, 2961–2964 (1996).
[CrossRef]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105, 248102 (2010).
[CrossRef]

Proc. SPIE (1)

L. Chaâri, N. Pustelnik, C. Chaux, and J. C. Pesquet, “Solving inverse problems with overcomplete transforms and convex optimization techniques,” Proc. SPIE 7446, 74460U (2009).
[CrossRef]

Rev. Sci. Instrum. (1)

S. Zabler, P. Cloetens, J.-P. Guigay, J. Baruchel, and M. Schlenker, “Optimization of phase contrast imaging using hard x-rays,” Rev. Sci. Instrum. 76, 073705 (2005).
[CrossRef]

Ultramicroscopy (1)

M. Beleggia, M. A. Schofield, V. V. Volkov, and Y. Zhu, “On the transport of intensity technique for phase retrieval,” Ultramicroscopy 102, 37–49 (2004).
[CrossRef]

Other (8)

D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1997).

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion” (2011), http://arxiv.org/abs/1109.0573 .

E. J. Candès and X. Li, “Solving quadratic equations via PhaseLift when there are about as many equations as unknowns” (2012), http://arxiv.org/abs/1208.6247 .

I. Waldspurger, A. D’Aspremont, and S. Mallat, “Phase recovery, maxcut, and complex semidefinite programming” (2012), http://arxiv.org/pdf/1206.0102.pdf .

S. Mukherjee and C. S. Seelamantula, “An iterative algorithm for phase retrieval with sparsity constraints: application to frequency domain optical coherence tomography,” In 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (IEEE, 2012), pp. 553–556.

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1987).

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, and F. Lenzen, Variational Methods in Imaging (Springer-Verlag, 2008).

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Figures (8)

Fig. 1.
Fig. 1.

(a) Ideal phase to be retrieved. (b) Fresnel diffraction intensity pattern for D=0.035m with PPSNR=12dB, (c) Absorption map with PPSNR=12dB. (d) Intensity profiles without noise and with PPSNR=12dB.

Fig. 2.
Fig. 2.

Evolution of the NMSE (%) as a function of iterations for different values of the regularization parameters R used in the RTS algorithm.

Fig. 3.
Fig. 3.

(a) Wavelet decomposition of the phase map and (b) low approximation wavelet coefficients used in WNL.

Fig. 4.
Fig. 4.

Evolution of the NMSE (%) as a function of initial PPSNR (decibels) for (a) the linear algorithms TIE (green line), CTF (blue line) and Mixed (red line). The phase maps obtained are used as initializations in the nonlinear schemes. Evolution of the NMSE (%) for different noise levels for the final nonlinear solutions obtained with NL (green line), RTS (red line) and WNL (blue line) methods using the (b) CTF, (c) TIE, and (d) Mixed solution as starting point.

Fig. 5.
Fig. 5.

NMSE for the phase versus iteration number for (a) WNL and (b) RTS approach initialized with CTF solution without noise.

Fig. 6.
Fig. 6.

Diagonal profiles for the Shepp–Logan phantom obtained with the nonlinear methods for different initializations: (a) TIE (PPSNR=48dB), (b) CTF (PPSNR=12dB), (c) Mixed (PPSNR=48dB), and (d) Mixed (PPSNR=12dB).

Fig. 7.
Fig. 7.

Error maps retrieved with (a) WNL and (b) RTS initialized with the CTF solution with PPSNR=12dB [Fig. 8(b)].

Fig. 8.
Fig. 8.

(a) True phase to be recovered, (b) phase map obtained for PPSNR=12dB with CTF method, and corresponding phase maps obtained using (b) as starting point with (c) WNL and (d) RTS methods.

Tables (1)

Tables Icon

Table 1. NMSE (%) for Different Algorithms and Noise Levelsa

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

n(x,y,z)=1δr(x,y,z)+iβ(x,y,z),
T(x)=exp[B(x)+iφ(x)]=a(x)exp[iφ(x)],
PD(x)=1iλDexp(iπλD|x|2).
ID(x)=|T(x)*PD(x)|2.
E(u(x))=exp(iu(x)).
ID(φ)=M(FDψ,FDψ¯)(φ)
εID(φ)(ϵ)=M(A,FDψ(φ)¯)+c.c.
Jα(φ)=12ID(φ)IδL2(Ω)2+α2φL2(Ω)2,
φk=φk1τk1{ID(φk1)*[ID(φk1)Iδ]+αφk1}.
Iδ=O(φ)+ε,
A1xL2(Ω)2λΛ|x,ϕλ|2A2xL2(Ω)2.
min{IδOW*(c)l222+ζcl1,cl2},
cn+1=Sζτ{cnτWO*[OW*(cn)]Iδ},
Sa(u)=sign(u)max(|u|a,0).
I=ID(φ)=ID(W*(c))=I˜D(c).
Jα(c)=I˜D(c)Iδ+αcl1,
Jαs(c,a)=Jα(c)+Rcal22ID(W*(c))ID(W*(a))l22,
ck+1=argmincJαs(c,ck).
0ID*(IδID(a))+R(ca)+αcl1.
cλ=SαR([WID(W*(c))*(IδID(W*(a)))/R+a]λ),
ID(φk)ID(φk1)L2(Ω)ωID(φk1)L2(Ω),
IδID(φk)L2(Ω)δ,
φkφk1L2(Ω)ωφkL2(Ω).

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