Abstract

Phase sensitive x-ray imaging extends standard x-ray microscopy techniques by offering up to a thousand times higher sensitivity than absorption-based techniques. If an object is illuminated with a sufficiently coherent beam, phase contrast is achieved by moving the detector downstream from the object. There is a quantitative relationship between the phase shift induced by the object and the recorded intensity. This relationship can be used to retrieve the phase shift induced by the object through the solution of an inverse problem. Since the phase shift can be considered as a projection through the 3D refractive index, the latter can be reconstructed using standard tomographic inversion techniques. However, the determination of the phase shift from the recorded intensity is an ill-posed inverse problem. We investigate the application of Fourier-wavelet regularized deconvolution (ForWaRD) to this problem. The method is evaluated using simulated and experimental data and is shown to increase the quality of reconstructions, in terms of normalized RMS error and compared with standard Tikhonov regularization, at a three times increase in computational cost.

© 2009 Optical Society of America

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  1. A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. (N.Y.) 2, 473-475 (1996).
  2. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and 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]
  3. J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik (Stuttgart) 46, 121-125 (1977).
  4. K. Nugent, T. Gureyev, D. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77, 2961-2964 (1996).
    [CrossRef] [PubMed]
  5. P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
    [CrossRef]
  6. X. Wu and H. Liu, “A general theoretical formalism for X-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33-42 (2003).
  7. 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] [PubMed]
  8. 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, 4556-4566 (2008).
    [CrossRef] [PubMed]
  9. R. Neelamani, H. Choi, and R. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418-433 (2004).
    [CrossRef]
  10. A. N. Tikhonov and V. A. Arsenin, Solution of Ill-Posed Problems (Winston, 1977).
  11. 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, 1-7 (2005).
    [CrossRef]
  12. D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika 81, 425-455 (1994).
    [CrossRef]
  13. D. L. Donoho, “Denoising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613-627 (1995).
    [CrossRef]
  14. D. L. Donoho and I. M. Johnstone, “Asymptotic minimaxity of wavelet estimators with sampled data,” Stat. Sin. 9, 1-32 (1999).
  15. S. Ghael, A. M. Sayed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997).
    [CrossRef]
  16. S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, 1999).
  17. I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
    [CrossRef]
  18. J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007).
    [CrossRef] [PubMed]

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, 4556-4566 (2008).
[CrossRef] [PubMed]

2007

J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007).
[CrossRef] [PubMed]

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] [PubMed]

2005

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, 1-7 (2005).
[CrossRef]

2004

R. Neelamani, H. Choi, and R. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418-433 (2004).
[CrossRef]

2003

X. Wu and H. Liu, “A general theoretical formalism for X-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33-42 (2003).

1999

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

D. L. Donoho and I. M. Johnstone, “Asymptotic minimaxity of wavelet estimators with sampled data,” Stat. Sin. 9, 1-32 (1999).

1997

S. Ghael, A. M. Sayed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997).
[CrossRef]

1996

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

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. (N.Y.) 2, 473-475 (1996).

1995

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and 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, “Denoising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613-627 (1995).
[CrossRef]

1994

D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika 81, 425-455 (1994).
[CrossRef]

1977

J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik (Stuttgart) 46, 121-125 (1977).

Arsenin, V. A.

A. N. Tikhonov and V. A. Arsenin, Solution of Ill-Posed Problems (Winston, 1977).

Baraniuk, R.

R. Neelamani, H. Choi, and R. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418-433 (2004).
[CrossRef]

Baraniuk, R. G.

S. Ghael, A. M. Sayed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997).
[CrossRef]

Barnea, Z.

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

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, 1-7 (2005).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Boistel, R.

Carreiras, D. F.

J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007).
[CrossRef] [PubMed]

Choi, H.

R. Neelamani, H. Choi, and R. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418-433 (2004).
[CrossRef]

Cloetens, 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, 4556-4566 (2008).
[CrossRef] [PubMed]

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] [PubMed]

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, 1-7 (2005).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Cookson, D.

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

Curfs, C.

J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007).
[CrossRef] [PubMed]

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
[CrossRef]

Donoho, D. L.

D. L. Donoho and I. M. Johnstone, “Asymptotic minimaxity of wavelet estimators with sampled data,” Stat. Sin. 9, 1-32 (1999).

D. L. Donoho, “Denoising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613-627 (1995).
[CrossRef]

D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika 81, 425-455 (1994).
[CrossRef]

Ferre, G. C.

J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007).
[CrossRef] [PubMed]

Ghael, S.

S. Ghael, A. M. Sayed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997).
[CrossRef]

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, 4556-4566 (2008).
[CrossRef] [PubMed]

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] [PubMed]

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, 1-7 (2005).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik (Stuttgart) 46, 121-125 (1977).

Gureyev, T.

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

Hirano, K.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. (N.Y.) 2, 473-475 (1996).

Homs, A.

J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007).
[CrossRef] [PubMed]

Itai, Y.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. (N.Y.) 2, 473-475 (1996).

Johnstone, I. M.

D. L. Donoho and I. M. Johnstone, “Asymptotic minimaxity of wavelet estimators with sampled data,” Stat. Sin. 9, 1-32 (1999).

D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika 81, 425-455 (1994).
[CrossRef]

Kohn, V.

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

Kuznetsov, S.

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

Labiche, J.-C.

J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007).
[CrossRef] [PubMed]

Langer, M.

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, 4556-4566 (2008).
[CrossRef] [PubMed]

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] [PubMed]

Liu, H.

X. Wu and H. Liu, “A general theoretical formalism for X-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33-42 (2003).

Ludwig, W.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, 1999).

Maton, O.

J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007).
[CrossRef] [PubMed]

Momose, A.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. (N.Y.) 2, 473-475 (1996).

Neelamani, R.

R. Neelamani, H. Choi, and R. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418-433 (2004).
[CrossRef]

Newton, M. A.

J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007).
[CrossRef] [PubMed]

Nugent, K.

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

Paganin, D.

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

Pascarelli, S.

J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007).
[CrossRef] [PubMed]

Peyrin, F.

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, 4556-4566 (2008).
[CrossRef] [PubMed]

Sayed, A. M.

S. Ghael, A. M. Sayed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997).
[CrossRef]

Schelokov, I.

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

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, 1-7 (2005).
[CrossRef]

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Snigirev, A.

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

Snigireva, I.

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

Takeda, T.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. (N.Y.) 2, 473-475 (1996).

Tikhonov, A. N.

A. N. Tikhonov and V. A. Arsenin, Solution of Ill-Posed Problems (Winston, 1977).

Van Dyck, D.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Van Landuyt, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Vaughan, G.

J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007).
[CrossRef] [PubMed]

Wu, X.

X. Wu and H. Liu, “A general theoretical formalism for X-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33-42 (2003).

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, 1-7 (2005).
[CrossRef]

Appl. Phys. Lett.

P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation X rays,” Appl. Phys. Lett. 75, 2912-2914 (1999).
[CrossRef]

Biometrika

D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika 81, 425-455 (1994).
[CrossRef]

IEEE Trans. Inf. Theory

D. L. Donoho, “Denoising by soft-thresholding,” IEEE Trans. Inf. Theory 41, 613-627 (1995).
[CrossRef]

IEEE Trans. Signal Process.

R. Neelamani, H. Choi, and R. Baraniuk, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418-433 (2004).
[CrossRef]

J. X-Ray Sci. Technol.

X. Wu and H. Liu, “A general theoretical formalism for X-ray phase contrast imaging,” J. X-Ray Sci. Technol. 11, 33-42 (2003).

Med. Phys.

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, 4556-4566 (2008).
[CrossRef] [PubMed]

Nat. Med. (N.Y.)

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. (N.Y.) 2, 473-475 (1996).

Opt. Lett.

Optik (Stuttgart)

J. P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik (Stuttgart) 46, 121-125 (1977).

Phys. Rev. Lett.

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

Proc. SPIE

S. Ghael, A. M. Sayed, and R. G. Baraniuk, “Improved wavelet denoising via empirical Wiener filtering,” Proc. SPIE 3169, 389-399 (1997).
[CrossRef]

Rev. Sci. Instrum.

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, 1-7 (2005).
[CrossRef]

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

J.-C. Labiche, O. Maton, S. Pascarelli, M. A. Newton, G. C. Ferre, C. Curfs, G. Vaughan, A. Homs, and D. F. Carreiras, “The FReLoN camera as a versatile X-ray detector for time resolved dispersive EXAFS and diffraction studies of dynamic problems in materials science, chemistry, and catalysis,” Rev. Sci. Instrum. 091301 (2007).
[CrossRef] [PubMed]

Stat. Sin.

D. L. Donoho and I. M. Johnstone, “Asymptotic minimaxity of wavelet estimators with sampled data,” Stat. Sin. 9, 1-32 (1999).

Other

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, 1999).

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
[CrossRef]

A. N. Tikhonov and V. A. Arsenin, Solution of Ill-Posed Problems (Winston, 1977).

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Fig. 1
Fig. 1

Schematic of the experimental setup. The sample is mounted on a rotation stage and the detector on a translation stage. The sample is then illuminated with partially coherent x-rays and tomographic scans are recorded at several (typically four) sample-to-detector distances D.

Fig. 2
Fig. 2

Squared modulus of the transfer functions to distances D = ( 2 , 10 , 20 , 45 ) mm . The dashed curves are the transfer functions to three different distances ( A D 2 ( f ) ) and the solid curve is the combined transfer function ( 1 N ) D A D 2 ( f ) ) for all three distances. Propagation to a single distance has zeros in the transfer function, which is countered by using several propagation distances. The combined transfer function still goes to zero for low frequencies, which must be handled by regularization.

Fig. 3
Fig. 3

Central slices ( 512 × 512   pixels ) of the 3D phantom used. The images are generated from an analytical definition. Each point is represented by a complex number with the imaginary part shown in (a), which is the absorption index β, and the real part shown in (b), which is the refractive index δ r .

Fig. 4
Fig. 4

Reconstructed central slices of the simulated data. (a)–(c) were reconstructed using Tikhonov regularization. The PPSNR was (a) 12 dB , (b) 0 dB , (c) 12 dB . (d)–(f) were reconstructed using ForWaRD with the same noise levels as above.

Fig. 5
Fig. 5

Reconstructed slices of the experimental data. (a) Reconstruction using Tikhonov regularization. Note the characteristic swell-like low-frequency noise which renders analysis difficult. (b) Reconstruction using ForWaRD. The low-frequency noise is improved, but not alleviated.

Tables (1)

Tables Icon

Table 1 Normalized RMS Error [Eq. (31)] for Different PPSNR

Equations (31)

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n ( x , y , z ) = 1 δ r ( x , y , z ) + i β ( x , y , z ) ,
T ( x ) = a ( x ) exp [ i ϕ ( x ) ] = exp [ B ( x ) + i ϕ ( x ) ] ,
B ( x ) = 2 π λ β ( x , y , z ) d z , ϕ ( x ) = ϕ 0 2 π λ δ r ( x , y , z ) d z ,
I D ( x ) = | Fr D [ T ( x ) ] | 2 .
Fr D [ T ( x ) ] = T ( x ) P D ( x )
P D ( x ) = 1 i λ D exp ( i π λ D | x | 2 ) .
g ̃ ( f ) = F { g } ( f ) = g ( x ) exp ( i 2 π x f ) d x ,
P ̃ D ( f ) = exp ( i π λ D | f | 2 ) .
I ̃ D ( f ) = T ( x λ D f 2 ) T * ( x + λ D f 2 ) exp ( i 2 π x f ) d x ,
I ̃ D ( f ) = I ̃ D ϕ = 0 ( f ) + 2 sin ( π λ D | f | 2 ) F { I 0 ϕ } ( f ) + cos ( π λ D | f | 2 ) λ D 2 π F { ( ϕ I 0 ) } ( f ) ,
ψ ̃ ( f ) = arg min ψ ̃ D | A D ( f ) ψ ̃ ( f ) [ I ̃ D ( f ) I ̃ D ϕ = 0 ( f ) Δ D ( f ) ] | 2 ,
ψ ̃ ( n + 1 ) ( f ) = D A D * ( f ) [ I ̃ D ( f ) I ̃ D ϕ = 0 ( f ) Δ D ( n ) ( f ) ] D | A D ( f ) | 2 .
p ( x ) = h ( x ) q ( x ) + n ( x ) ,
q ̂ ̃ ( f ) = p ̃ ( f ) h ̃ ( f ) n ̃ ( f ) h ̃ ( f ) ,
q ̂ ̃ ( f ) = p ̃ ( f ) h ̃ * ( f ) | q ̃ ( f ) | 2 | h ̃ ( f ) | 2 | q ̃ ( f ) | 2 + | n ̃ ( f ) | 2 .
q ̂ ̃ ( f ) = λ ( f ) p ̃ ( f ) h ̃ ( f ) ,
λ ( f ) = | h ̃ ( f ) | 2 | h ̃ ( f ) | 2 + | n ̃ ( f ) | 2 | q ̃ ( f ) | 2 .
λ ( f ) = | h ̃ ( f ) | 2 | h ̃ ( f ) | 2 + α .
q ̂ ̃ ( f ) = arg min q ̂ ̃ | h ̃ ( f ) q ̂ ̃ ( f ) p ̃ ( f ) | 2 + α | q ̂ ̃ ( f ) | 2 ,
p ( x ) = q ( x ) + n ( x )
g W ( j , l ) = W { g } ( j , l ) = g ( x ) 1 j x j y Ψ * ( x l x j x , y l y j y ) d x ,
λ W ( j , l ) = | q W ( j , l ) | 2 | q W ( j , l ) | 2 + σ j 2 ,
σ j 2 = σ 2 l | W { F 1 [ h ̃ | h ̃ | 2 + α ] } ( j , l ) | 2 .
λ W 1 T ( j , l ) = { 1 , if q ̂ W 1 ( j , l ) > T j σ j 0 , if q ̂ W 1 ( j , l ) T j σ j } ,
g pilot W 2 ( j , l ) = W 2 { W 1 1 [ λ W 1 T q ̂ W 1 ] } ( j , l ) .
g ̂ ̂ W 2 ( j , l ) = g ̂ W 2 ( j , l ) | q pilot W 2 ( j , l ) | 2 | q pilot W 2 ( j , l ) | 2 + σ j 2 .
ψ ̃ ( f ) = arg min ψ ̃ D | A D ( f ) ψ ̃ ( f ) [ I ̃ D ( f ) I ̃ D ϕ = 0 ( f ) Δ D ( f ) ] | 2 + α | ψ ̃ ( f ) | 2 ,
ψ ̃ ( n + 1 ) ( f ) = D A D * ( f ) [ I ̃ D ( f ) I ̃ D ϕ = 0 ( f ) Δ D ( n ) ( f ) ] D | A D ( f ) | 2 + α .
σ j = σ l | W 2 { F 1 [ D A ̃ D D A ̃ D 2 + α ] } ( j , l ) | .
λ W 2 ( j , l ) = | ψ ̂ pilot W 2 ( j , l ) | 2 | ψ ̂ pilot W 2 ( j , l ) | 2 + σ j 2 .
ε = ( | f i ( x ) f ( x ) | 2 | f i ( x ) | 2 ) 1 2 ,

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