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]

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]

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]

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]

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]

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]

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]

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]

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

[Crossref]

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]

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]

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]

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]

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

[Crossref]

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

[Crossref]

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]

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]

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. 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]

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]

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

[Crossref]

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

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]

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]

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]

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]

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]

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. 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).

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

[Crossref]

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]

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]

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

[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]

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).

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]

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]

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).

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]

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]

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]

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

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

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]

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]

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]

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]

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]

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

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]

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

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

[Crossref]

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]

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]

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]

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]

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, 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]

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]

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

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).

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]

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]

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.

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

[Crossref]

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]

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]

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).

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]

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

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]

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]

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]

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. 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]

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]

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]

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]

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]

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

[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).

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]

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

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]

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]

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.

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]

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]

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]

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]

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]

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, 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]

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]

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]

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]

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]

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

[Crossref]

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]

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]

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

[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]

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]

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

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).

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]

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

[Crossref]

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]

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]

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]

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

[Crossref]

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]

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]

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]

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]

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).

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]

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]

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]

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]

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]

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]

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

[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]

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

[Crossref]

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]

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

[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]

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

[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]

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]

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]

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).