Abstract

In this paper, we discuss a deterministic regularization algorithm to handle the missing cone problem of three-dimensional optical diffraction tomography (ODT). The missing cone problem arises in most practical applications of ODT and is responsible for elongation of the reconstructed shape and underestimation of the value of the refractive index. By applying positivity and piecewise-smoothness constraints in an iterative reconstruction framework, we effectively suppress the missing cone artifact and recover sharp edges rounded out by the mis sing cone, and we significantly improve the accuracy of the predictions of the refractive index. We also show the noise-handling capability of our algorithm in the reconstruction process.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. Dasari, and M. Feld, “Optical diffraction tomography for high-resolution live-cell imaging,” Opt. Express 17, 266–277 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
  5. G. Tsihrintzis and A. Devaney, “Higher-order (nonlinear) diffraction tomography: reconstruction algorithms and computer simulation,” IEEE Trans. Image Process. 9, 1560–1572 (2000).
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  6. M. Defrise and C. De Mol, “A regularized iterative algorithm for limited-angle inverse Radon transform,” J. Mod. Opt. 30, 403–408 (1983).
    [CrossRef]
  7. A. Delaney and Y. Bresler, “Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography,” IEEE Trans. Image Process. 7, 204–221 (1998).
    [CrossRef]
  8. M. Persson, D. Bone, and H. Elmqvist, “Total variation norm for three-dimensional iterative reconstruction in limited view angle tomography,” Phys. Med. Biol. 46, 853 (2001).
    [CrossRef] [PubMed]
  9. A. Zunino, F. Benvenuto, E. Armadillo, M. Bertero, and E. Bozzo, “Iterative deconvolution and semiblind deconvolution methods in magnetic archaeological prospecting,” Geophysics 74, L43 (2009).
    [CrossRef]
  10. M. Bronstein, A. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging 21, 1395–1401 (2002).
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  11. K. Belkebir and A. Sentenac, “High-resolution optical diffraction microscopy,” J. Opt. Soc. Am. A 20, 1223–1229 (2003).
    [CrossRef]
  12. G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
    [CrossRef] [PubMed]
  13. K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
    [CrossRef]
  14. F. Charrière, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. 31, 178–180 (2006).
    [CrossRef] [PubMed]
  15. K. Lada and A. Devaney, “Iterative methods in geophysical diffraction tomography,” Inverse Probl. 8, 119–132 (1992).
    [CrossRef]
  16. O. Haeberlé, K. Belkebir, H. Giovaninni, and A. Sentenac, “Tomographic diffractive microscopy: basics, techniques, and perspectives,” J. Mod. Opt. 57, 686–699 (2010).
    [CrossRef]
  17. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
    [CrossRef] [PubMed]
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  19. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).
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    [CrossRef]
  21. C. Vogel, Computational Methods for Inverse Problems (Society for Industrial Mathematics, 2002).
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  22. P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311(1997).
    [CrossRef] [PubMed]
  23. R. Barer and S. Tkaczyk, “Refractive index of concentrated protein solutions,” Nature 173, 821–822 (1954).
    [CrossRef] [PubMed]
  24. R. Gerchberg, “Super-resolution through error energy reduction,” J. Mod. Opt. 21, 709–720 (1974).
    [CrossRef]
  25. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742(1975).
    [CrossRef]
  26. A. Baussard, K. Belkebir, and D. Premel, “A markovian regularization approach of the modified gradient method for solving a two-dimensional inverse scattering problem,” J. Electromag. Waves Appl. 17, 989–1008 (2003).
    [CrossRef]
  27. H. Ayasso, B. Duchêne, and A. Mohammad-Djafari, “Bayesian inversion for optical diffraction tomography,” J. Mod. Opt. 57, 765–776 (2010).
    [CrossRef]
  28. K. Ladas and A. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
    [CrossRef]
  29. M. Schabel, “3D Shepp-Logan phantom,” MATLAB Central File Exchange (2006).
  30. E. Mudry, P. Chaumet, K. Belkebir, G. Maire, and A. Sentenac, “Mirror-assisted tomographic diffractive microscopy with isotropic resolution,” Opt. Lett. 35, 1857–1859 (2010).
    [CrossRef] [PubMed]
  31. A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

2010

O. Haeberlé, K. Belkebir, H. Giovaninni, and A. Sentenac, “Tomographic diffractive microscopy: basics, techniques, and perspectives,” J. Mod. Opt. 57, 686–699 (2010).
[CrossRef]

H. Ayasso, B. Duchêne, and A. Mohammad-Djafari, “Bayesian inversion for optical diffraction tomography,” J. Mod. Opt. 57, 765–776 (2010).
[CrossRef]

E. Mudry, P. Chaumet, K. Belkebir, G. Maire, and A. Sentenac, “Mirror-assisted tomographic diffractive microscopy with isotropic resolution,” Opt. Lett. 35, 1857–1859 (2010).
[CrossRef] [PubMed]

2009

A. Zunino, F. Benvenuto, E. Armadillo, M. Bertero, and E. Bozzo, “Iterative deconvolution and semiblind deconvolution methods in magnetic archaeological prospecting,” Geophysics 74, L43 (2009).
[CrossRef]

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. Dasari, and M. Feld, “Optical diffraction tomography for high-resolution live-cell imaging,” Opt. Express 17, 266–277 (2009).
[CrossRef] [PubMed]

2007

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

2006

2005

2003

K. Belkebir and A. Sentenac, “High-resolution optical diffraction microscopy,” J. Opt. Soc. Am. A 20, 1223–1229 (2003).
[CrossRef]

A. Baussard, K. Belkebir, and D. Premel, “A markovian regularization approach of the modified gradient method for solving a two-dimensional inverse scattering problem,” J. Electromag. Waves Appl. 17, 989–1008 (2003).
[CrossRef]

2002

M. Bronstein, A. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging 21, 1395–1401 (2002).
[CrossRef]

2001

M. Persson, D. Bone, and H. Elmqvist, “Total variation norm for three-dimensional iterative reconstruction in limited view angle tomography,” Phys. Med. Biol. 46, 853 (2001).
[CrossRef] [PubMed]

2000

G. Tsihrintzis and A. Devaney, “Higher-order (nonlinear) diffraction tomography: reconstruction algorithms and computer simulation,” IEEE Trans. Image Process. 9, 1560–1572 (2000).
[CrossRef]

1998

A. Delaney and Y. Bresler, “Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography,” IEEE Trans. Image Process. 7, 204–221 (1998).
[CrossRef]

1997

M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Probl. 13, 441–463 (1997).
[CrossRef]

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311(1997).
[CrossRef] [PubMed]

1992

K. Lada and A. Devaney, “Iterative methods in geophysical diffraction tomography,” Inverse Probl. 8, 119–132 (1992).
[CrossRef]

1991

K. Ladas and A. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
[CrossRef]

1983

M. Defrise and C. De Mol, “A regularized iterative algorithm for limited-angle inverse Radon transform,” J. Mod. Opt. 30, 403–408 (1983).
[CrossRef]

1982

A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).
[CrossRef]

1981

1975

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742(1975).
[CrossRef]

1974

R. Gerchberg, “Super-resolution through error energy reduction,” J. Mod. Opt. 21, 709–720 (1974).
[CrossRef]

1969

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1954

R. Barer and S. Tkaczyk, “Refractive index of concentrated protein solutions,” Nature 173, 821–822 (1954).
[CrossRef] [PubMed]

Armadillo, E.

A. Zunino, F. Benvenuto, E. Armadillo, M. Bertero, and E. Bozzo, “Iterative deconvolution and semiblind deconvolution methods in magnetic archaeological prospecting,” Geophysics 74, L43 (2009).
[CrossRef]

Aubert, G.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311(1997).
[CrossRef] [PubMed]

Ayasso, H.

H. Ayasso, B. Duchêne, and A. Mohammad-Djafari, “Bayesian inversion for optical diffraction tomography,” J. Mod. Opt. 57, 765–776 (2010).
[CrossRef]

Azhari, H.

M. Bronstein, A. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging 21, 1395–1401 (2002).
[CrossRef]

Badizadegan, K.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. Dasari, and M. Feld, “Optical diffraction tomography for high-resolution live-cell imaging,” Opt. Express 17, 266–277 (2009).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

Barer, R.

R. Barer and S. Tkaczyk, “Refractive index of concentrated protein solutions,” Nature 173, 821–822 (1954).
[CrossRef] [PubMed]

Barlaud, M.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311(1997).
[CrossRef] [PubMed]

Baussard, A.

A. Baussard, K. Belkebir, and D. Premel, “A markovian regularization approach of the modified gradient method for solving a two-dimensional inverse scattering problem,” J. Electromag. Waves Appl. 17, 989–1008 (2003).
[CrossRef]

Belkebir, K.

E. Mudry, P. Chaumet, K. Belkebir, G. Maire, and A. Sentenac, “Mirror-assisted tomographic diffractive microscopy with isotropic resolution,” Opt. Lett. 35, 1857–1859 (2010).
[CrossRef] [PubMed]

O. Haeberlé, K. Belkebir, H. Giovaninni, and A. Sentenac, “Tomographic diffractive microscopy: basics, techniques, and perspectives,” J. Mod. Opt. 57, 686–699 (2010).
[CrossRef]

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[CrossRef]

K. Belkebir and A. Sentenac, “High-resolution optical diffraction microscopy,” J. Opt. Soc. Am. A 20, 1223–1229 (2003).
[CrossRef]

A. Baussard, K. Belkebir, and D. Premel, “A markovian regularization approach of the modified gradient method for solving a two-dimensional inverse scattering problem,” J. Electromag. Waves Appl. 17, 989–1008 (2003).
[CrossRef]

Benvenuto, F.

A. Zunino, F. Benvenuto, E. Armadillo, M. Bertero, and E. Bozzo, “Iterative deconvolution and semiblind deconvolution methods in magnetic archaeological prospecting,” Geophysics 74, L43 (2009).
[CrossRef]

Bertero, M.

A. Zunino, F. Benvenuto, E. Armadillo, M. Bertero, and E. Bozzo, “Iterative deconvolution and semiblind deconvolution methods in magnetic archaeological prospecting,” Geophysics 74, L43 (2009).
[CrossRef]

M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Probl. 13, 441–463 (1997).
[CrossRef]

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).
[CrossRef]

Blanc-Feraud, L.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311(1997).
[CrossRef] [PubMed]

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Taylor & Francis, 1998).
[CrossRef]

Bone, D.

M. Persson, D. Bone, and H. Elmqvist, “Total variation norm for three-dimensional iterative reconstruction in limited view angle tomography,” Phys. Med. Biol. 46, 853 (2001).
[CrossRef] [PubMed]

Bozzo, E.

A. Zunino, F. Benvenuto, E. Armadillo, M. Bertero, and E. Bozzo, “Iterative deconvolution and semiblind deconvolution methods in magnetic archaeological prospecting,” Geophysics 74, L43 (2009).
[CrossRef]

Bresler, Y.

A. Delaney and Y. Bresler, “Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography,” IEEE Trans. Image Process. 7, 204–221 (1998).
[CrossRef]

Bronstein, A.

M. Bronstein, A. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging 21, 1395–1401 (2002).
[CrossRef]

Bronstein, M.

M. Bronstein, A. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging 21, 1395–1401 (2002).
[CrossRef]

Charbonnier, P.

P. Charbonnier, L. Blanc-Feraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311(1997).
[CrossRef] [PubMed]

Charrière, F.

Chaumet, P.

Chaumet, P. C.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[CrossRef]

Choi, W.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. Dasari, and M. Feld, “Optical diffraction tomography for high-resolution live-cell imaging,” Opt. Express 17, 266–277 (2009).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

Colomb, T.

Cuche, E.

Dasari, R.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. Dasari, and M. Feld, “Optical diffraction tomography for high-resolution live-cell imaging,” Opt. Express 17, 266–277 (2009).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

De Mol, C.

M. Defrise and C. De Mol, “A regularized iterative algorithm for limited-angle inverse Radon transform,” J. Mod. Opt. 30, 403–408 (1983).
[CrossRef]

Defrise, M.

M. Defrise and C. De Mol, “A regularized iterative algorithm for limited-angle inverse Radon transform,” J. Mod. Opt. 30, 403–408 (1983).
[CrossRef]

Delaney, A.

A. Delaney and Y. Bresler, “Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography,” IEEE Trans. Image Process. 7, 204–221 (1998).
[CrossRef]

Depeursinge, C.

Devaney, A.

G. Tsihrintzis and A. Devaney, “Higher-order (nonlinear) diffraction tomography: reconstruction algorithms and computer simulation,” IEEE Trans. Image Process. 9, 1560–1572 (2000).
[CrossRef]

K. Lada and A. Devaney, “Iterative methods in geophysical diffraction tomography,” Inverse Probl. 8, 119–132 (1992).
[CrossRef]

K. Ladas and A. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
[CrossRef]

A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).
[CrossRef]

A. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981).
[CrossRef] [PubMed]

Drsek, F.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

Duchêne, B.

H. Ayasso, B. Duchêne, and A. Mohammad-Djafari, “Bayesian inversion for optical diffraction tomography,” J. Mod. Opt. 57, 765–776 (2010).
[CrossRef]

Elmqvist, H.

M. Persson, D. Bone, and H. Elmqvist, “Total variation norm for three-dimensional iterative reconstruction in limited view angle tomography,” Phys. Med. Biol. 46, 853 (2001).
[CrossRef] [PubMed]

Fang-Yen, C.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. Dasari, and M. Feld, “Optical diffraction tomography for high-resolution live-cell imaging,” Opt. Express 17, 266–277 (2009).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

Feld, M.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. Dasari, and M. Feld, “Optical diffraction tomography for high-resolution live-cell imaging,” Opt. Express 17, 266–277 (2009).
[CrossRef] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

Gerchberg, R.

R. Gerchberg, “Super-resolution through error energy reduction,” J. Mod. Opt. 21, 709–720 (1974).
[CrossRef]

Giovaninni, H.

O. Haeberlé, K. Belkebir, H. Giovaninni, and A. Sentenac, “Tomographic diffractive microscopy: basics, techniques, and perspectives,” J. Mod. Opt. 57, 686–699 (2010).
[CrossRef]

Giovannini, H.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

Girard, J.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Haeberlé, O.

O. Haeberlé, K. Belkebir, H. Giovaninni, and A. Sentenac, “Tomographic diffractive microscopy: basics, techniques, and perspectives,” J. Mod. Opt. 57, 686–699 (2010).
[CrossRef]

Kak, A.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Konan, D.

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

Kuehn, J.

Lada, K.

K. Lada and A. Devaney, “Iterative methods in geophysical diffraction tomography,” Inverse Probl. 8, 119–132 (1992).
[CrossRef]

Ladas, K.

K. Ladas and A. Devaney, “Generalized ART algorithm for diffraction tomography,” Inverse Probl. 7, 109–125 (1991).
[CrossRef]

Lue, N.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

Maire, G.

E. Mudry, P. Chaumet, K. Belkebir, G. Maire, and A. Sentenac, “Mirror-assisted tomographic diffractive microscopy with isotropic resolution,” Opt. Lett. 35, 1857–1859 (2010).
[CrossRef] [PubMed]

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

Marian, A.

Marquet, P.

Mohammad-Djafari, A.

H. Ayasso, B. Duchêne, and A. Mohammad-Djafari, “Bayesian inversion for optical diffraction tomography,” J. Mod. Opt. 57, 765–776 (2010).
[CrossRef]

Montfort, F.

Mudry, E.

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742(1975).
[CrossRef]

Persson, M.

M. Persson, D. Bone, and H. Elmqvist, “Total variation norm for three-dimensional iterative reconstruction in limited view angle tomography,” Phys. Med. Biol. 46, 853 (2001).
[CrossRef] [PubMed]

Piana, M.

M. Piana and M. Bertero, “Projected Landweber method and preconditioning,” Inverse Probl. 13, 441–463 (1997).
[CrossRef]

Premel, D.

A. Baussard, K. Belkebir, and D. Premel, “A markovian regularization approach of the modified gradient method for solving a two-dimensional inverse scattering problem,” J. Electromag. Waves Appl. 17, 989–1008 (2003).
[CrossRef]

Schabel, M.

M. Schabel, “3D Shepp-Logan phantom,” MATLAB Central File Exchange (2006).

Sentenac, A.

E. Mudry, P. Chaumet, K. Belkebir, G. Maire, and A. Sentenac, “Mirror-assisted tomographic diffractive microscopy with isotropic resolution,” Opt. Lett. 35, 1857–1859 (2010).
[CrossRef] [PubMed]

O. Haeberlé, K. Belkebir, H. Giovaninni, and A. Sentenac, “Tomographic diffractive microscopy: basics, techniques, and perspectives,” J. Mod. Opt. 57, 686–699 (2010).
[CrossRef]

G. Maire, F. Drsek, J. Girard, H. Giovannini, A. Talneau, D. Konan, K. Belkebir, P. C. Chaumet, and A. Sentenac, “Experimental demonstration of quantitative imaging beyond Abbe’s limit with optical diffraction tomography,” Phys. Rev. Lett. 102, 213905 (2009).
[CrossRef] [PubMed]

K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[CrossRef]

K. Belkebir and A. Sentenac, “High-resolution optical diffraction microscopy,” J. Opt. Soc. Am. A 20, 1223–1229 (2003).
[CrossRef]

Slaney, M.

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Figures (9)

Fig. 1
Fig. 1

Flow chart for iterative reconstruction of optical diffraction tomography.

Fig. 2
Fig. 2

Modified 3D Shepp–Logan phantom and its scattering fields calculated for various illumination angles: (a) 3D rendered phantom with the wave vector of the incident wave shown as a thick purple line, (b) center cross section of the phantom refractive index, (c)–(e) scattered fields calculated for the incident angle of (c)  α x = 90 ° , α y = 90 ° , (d)  α x = 90 ° , α y = 49.5 ° , (e)  α x = 51.5 ° , α y = 78.3 ° . Color bar for (c)–(e) indicates the phase delay in radian after the phase unwrapping. Scale bar for (b), 2 μm ; scale bar for (c), 5 μm .

Fig. 3
Fig. 3

(a) Example of a scattering field corrupted with additive noise, (b) horizontal cross section of the reconstructed 3D image using direct mapping by the Fourier diffraction theorem, (c) horizontal cross section of the reconstructed 3D image by iterative reconstruction with the edge-preserving penalty. Scale bar, 2 μm .

Fig. 4
Fig. 4

Transfer function of (a) 2D diffraction tomography and (b) 3D diffraction tomography, (c) vertical cross section of the reconstructed refractive index map using Fourier mapping, (d) index profile of (c) along the dotted line. The sample used for the simulation is a homogeneous, spherical bead with the refractive index 1.37. ( U , V , W ) are the spatial frequency components corresponding to the spatial coordinates ( X , Y , Z ) . Scale bar for (c), 5 μm .

Fig. 5
Fig. 5

Simulation with a spherical bead: (a) comparison of index profile along the optical axis, (b) histogram of 3D refractive index map, which was reconstructed by various methods.

Fig. 6
Fig. 6

(a) Spherical bead squashed in the optical axis direction with the aspect ratio of 2 to 1, (b) spatial frequency spectrum of (a), in which the frequency support of the transfer function for the angular coverage from 60 ° to 60 ° is drawn as a dotted line. The color bar indicates the amplitude of the spectrum in the logarithmic scale with base 10.

Fig. 7
Fig. 7

Simulation with a squashed bead in Fig. 6a: (a) comparison of index profiles along the optical axis, (b) histograms of the 3D refractive index map reconstructed by various methods.

Fig. 8
Fig. 8

Nonhomogeneous sample consisting of two regions of different refractive index. The numbers in the figure are the values of the refractive index.

Fig. 9
Fig. 9

Simulation with a nonhomogeneous sample representing a biological cell: (a) comparison of index profiles along the optical axis, (b) histograms of the 3D refractive index map reconstructed by various methods.

Equations (12)

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f ( r ) = k 2 ( 1 ( n ( r ) / n 0 ) 2 ) ,
u ( r ) u in ( r ) 1 = 1 4 π u in ( r ) f ( r ) u in ( r ) G ( r r ) d 3 r ,
ln ( u ( r ) u in ( r ) ) = 1 4 π u in ( r ) f ( r ) u in ( r ) G ( r r ) d 3 r .
Φ μ ( f ; g ) = 1 2 n A n f g n 2 + α J ( f ) ,
f μ ( k + 1 ) = f μ ( k ) + τ n ( A n g n A n A n f μ ( k ) ) τ α J ( f μ ( k ) ) ,
J ( f ) = 1 / 2 ψ ( | f | 2 ) d V ,
f μ = D Ω f μ + D ¯ Ω P + f μ ,
A n f = 1 i 4 π w f ˜ ( U , V , W ) e i 2 π ( U x + V y ) d U d V ,
( g 1 , g 2 ) X = g 1 * ( x , y ) g 2 ( x , y ) d x d y , ( f 1 , f 2 ) Y = f 1 * ( x , y , z ) f 2 ( x , y , z ) d x d y d z ,
( A n f , g ) X = ( f , A n g ) Y ,
A n g = 1 i 4 π w g ˜ ( U , V ) e i 2 π ( U x + V y + W z ) d U d V .
A n A n f = 1 ( 4 π w ) 2 f ˜ ( U , V , W ) e i 2 π ( U x + V y + W z ) d U d V .

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