Abstract

3D deconvolution is an established technique in microscopy that may be useful for low-cost high-resolution imaging of the retina. We report on a myopic 3D deconvolution method developed in a Bayesian framework. This method uses a 3D imaging model, a noise model that accounts for both photon and detector noises, a regularization term that is appropriate for objects that are a mix of sharp edges and smooth areas, a positivity constraint, and a smart parameterization of the point-spread function (PSF) by the pupil phase. It estimates the object and the PSF jointly. The PSF parameterization through the pupil phase constrains the inversion by dramatically reducing the number of unknowns. The joint deconvolution is further constrained by an additional longitudinal support constraint derived from a 3D interpretation of the phase-diversity technique. This method is validated by simulated retinal images.

© 2007 Optical Society of America

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    [CrossRef]
  2. L. M. Mugnier, C. Robert, J.-M. Conan, V. Michau, and S. Salem, "Myopic deconvolution from wave-front sensing," J. Opt. Soc. Am. A 18, 862-872 (2001).
    [CrossRef]
  3. D. Catlin and C. Dainty, "High-resolution imaging of the human retina with a Fourier deconvolution technique," J. Opt. Soc. Am. A 19, 1515-1523 (2002).
    [CrossRef]
  4. G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, "First diffraction-limited astronomical images with adaptive optics," Astron. Astrophys. 230, 29-32 (1990).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  12. J.Idier, ed., Approche Bayésienne pour les Problèmes Inverses (Hermès, 2001).
  13. L. M. Mugnier, T. Fusco, and J.-M. Conan, "MISTRAL: a myopic edge-preserving image restoration method, with application to astronomical adaptive optics-corrected long-exposure images," J. Opt. Soc. Am. A 21, 1841-1854 (2004).
    [CrossRef]
  14. J.-M. Conan, L. M. Mugnier, T. Fusco, V. Michau, and G. Rousset, "Myopic deconvolution of adaptive optics images by use of object and point-spread function power spectra," Appl. Opt. 37, 4614-4622 (1998).
    [CrossRef]
  15. D. Gratadour, D. Rouan, L. M. Mugnier, T. Fusco, Y. Clénet, E. Gendron, and F. Lacombe, "Near-infrared adaptive optics dissection of the core of NGC 1068 with NaCo," Astron. Astrophys. 446, 813-825 (2006).
    [CrossRef]
  16. A. Blanc, L. M. Mugnier, and J. Idier, "Marginal estimation of aberrations and image restoration by use of phase diversity," J. Opt. Soc. Am. A 20, 1035-1045 (2003).
    [CrossRef]
  17. P. J. Green, "Bayesian reconstructions from emission tomography data using a modified EM algorithm," IEEE Trans. Med. Imaging 9, 84-93 (1990).
    [CrossRef]
  18. C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Process. 2, 296-310 (1993).
    [CrossRef]
  19. J. Idier and L. Blanc-Féraud, "Deconvolution en imagerie," in Approche Bayésienne pour les Problèmes Inverses, J.Idier, ed. (Hermès, 2001), Chap. 6.
  20. W. J. J. Rey, Introduction to Robust and Quasi-Robust Statistical Methods (Springer-Verlag, 1983).
    [CrossRef]
  21. S. Brette and J. Idier, "Optimized single site update algorithms for image deblurring," in Proceedings of the International Conference on Image Processing (IEEE Computer Society, 1996), pp. 65-68.
  22. T. J. Schulz, "Multiframe blind deconvolution of astronomical images," J. Opt. Soc. Am. A 10, 1064-1073 (1993).
    [CrossRef]
  23. E. Thiébaut and J.-M. Conan, "Strict a priori constraints for maximum-likelihood blind deconvolution," J. Opt. Soc. Am. A 12, 485-492 (1995).
    [CrossRef]
  24. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. A 66, 207-211 (1976).
    [CrossRef]
  25. E. Thiébaut, "Optimization issues in blind deconvolution algorithms," in Astronomical Data Analysis. II, J.-L. Starck and F. D. Murtagh, eds., Proc. SPIE 4847, 174-183 (2002).
    [CrossRef]
  26. G. Chenegros, L. M. Mugnier, and F. Lacombe, "3D deconvolution of adaptive-optics corrected retinal images," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, J.-A. Conchello, C. J. Cogswell, and T. Wilson, eds., Proc. SPIE 6090, 60900P (2006).
    [CrossRef]
  27. R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).
  28. L. M. Mugnier, A. Blanc, and J. Idier, "Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging," in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elsevier, 2006), Vol. 141, Chap. 1.
    [CrossRef]
  29. M. F. Reiley, R. G. Paxman, J. R. Fienup, K. W. Gleichman, and J. C. Marron, "3D reconstruction of opaque objects from Fourier intensity data," in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE 3170, 76-87 (1997).
    [CrossRef]
  30. R. G. Paxman, J. H. Seldin, J. R. Fienup, and J. C. Marron, "Use of an opacity constraint in three-dimensional imaging," in Inverse Optics III, M. A. Fiddy, ed., Proc. SPIE 2241, 116-126 (1994).
    [CrossRef]

2006 (2)

D. Gratadour, D. Rouan, L. M. Mugnier, T. Fusco, Y. Clénet, E. Gendron, and F. Lacombe, "Near-infrared adaptive optics dissection of the core of NGC 1068 with NaCo," Astron. Astrophys. 446, 813-825 (2006).
[CrossRef]

G. Chenegros, L. M. Mugnier, and F. Lacombe, "3D deconvolution of adaptive-optics corrected retinal images," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, J.-A. Conchello, C. J. Cogswell, and T. Wilson, eds., Proc. SPIE 6090, 60900P (2006).
[CrossRef]

2004 (2)

2003 (1)

2002 (2)

D. Catlin and C. Dainty, "High-resolution imaging of the human retina with a Fourier deconvolution technique," J. Opt. Soc. Am. A 19, 1515-1523 (2002).
[CrossRef]

E. Thiébaut, "Optimization issues in blind deconvolution algorithms," in Astronomical Data Analysis. II, J.-L. Starck and F. D. Murtagh, eds., Proc. SPIE 4847, 174-183 (2002).
[CrossRef]

2001 (1)

1999 (1)

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, "Three-dimensional imaging by deconvolution microscopy," Methods 19, 373-385 (1999).
[CrossRef] [PubMed]

1998 (1)

1997 (1)

M. F. Reiley, R. G. Paxman, J. R. Fienup, K. W. Gleichman, and J. C. Marron, "3D reconstruction of opaque objects from Fourier intensity data," in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE 3170, 76-87 (1997).
[CrossRef]

1995 (1)

1994 (1)

R. G. Paxman, J. H. Seldin, J. R. Fienup, and J. C. Marron, "Use of an opacity constraint in three-dimensional imaging," in Inverse Optics III, M. A. Fiddy, ed., Proc. SPIE 2241, 116-126 (1994).
[CrossRef]

1993 (2)

C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Process. 2, 296-310 (1993).
[CrossRef]

T. J. Schulz, "Multiframe blind deconvolution of astronomical images," J. Opt. Soc. Am. A 10, 1064-1073 (1993).
[CrossRef]

1991 (1)

1990 (3)

J. Primot, G. Rousset, and J.-C. Fontanella, "Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images," J. Opt. Soc. Am. A 7, 1598-1608 (1990).
[CrossRef]

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, "First diffraction-limited astronomical images with adaptive optics," Astron. Astrophys. 230, 29-32 (1990).

P. J. Green, "Bayesian reconstructions from emission tomography data using a modified EM algorithm," IEEE Trans. Med. Imaging 9, 84-93 (1990).
[CrossRef]

1989 (1)

G. Demoment, "Image reconstruction and restoration: overview of common estimation structures and problems," IEEE Trans. Acoust., Speech, Signal Process. 37, 2024-2036 (1989).
[CrossRef]

1982 (1)

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).

1976 (1)

R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. A 66, 207-211 (1976).
[CrossRef]

Arsenin, V.

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

Blanc, A.

A. Blanc, L. M. Mugnier, and J. Idier, "Marginal estimation of aberrations and image restoration by use of phase diversity," J. Opt. Soc. Am. A 20, 1035-1045 (2003).
[CrossRef]

L. M. Mugnier, A. Blanc, and J. Idier, "Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging," in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elsevier, 2006), Vol. 141, Chap. 1.
[CrossRef]

Blanc-Féraud, L.

J. Idier and L. Blanc-Féraud, "Deconvolution en imagerie," in Approche Bayésienne pour les Problèmes Inverses, J.Idier, ed. (Hermès, 2001), Chap. 6.

Bouman, C.

C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Process. 2, 296-310 (1993).
[CrossRef]

Boyer, C.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, "First diffraction-limited astronomical images with adaptive optics," Astron. Astrophys. 230, 29-32 (1990).

Brette, S.

S. Brette and J. Idier, "Optimized single site update algorithms for image deblurring," in Proceedings of the International Conference on Image Processing (IEEE Computer Society, 1996), pp. 65-68.

Catlin, D.

Chenegros, G.

G. Chenegros, L. M. Mugnier, and F. Lacombe, "3D deconvolution of adaptive-optics corrected retinal images," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, J.-A. Conchello, C. J. Cogswell, and T. Wilson, eds., Proc. SPIE 6090, 60900P (2006).
[CrossRef]

Christou, J. C.

Clénet, Y.

D. Gratadour, D. Rouan, L. M. Mugnier, T. Fusco, Y. Clénet, E. Gendron, and F. Lacombe, "Near-infrared adaptive optics dissection of the core of NGC 1068 with NaCo," Astron. Astrophys. 446, 813-825 (2006).
[CrossRef]

Conan, J. M.

J. M. Conan, P. Y. Madec, and G. Rousset, "Image formation in adaptive optics partial correction," in Active and Adaptive Optics, F.Merkle, ed., Vol. 48 of ESO Conference and Workshop Proceeding (European Southern Observatory/International Commission of Optics, 1994).

Conan, J.-M.

Conchello, J. A.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, "Three-dimensional imaging by deconvolution microscopy," Methods 19, 373-385 (1999).
[CrossRef] [PubMed]

Cooper, J.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, "Three-dimensional imaging by deconvolution microscopy," Methods 19, 373-385 (1999).
[CrossRef] [PubMed]

Dainty, C.

Demoment, G.

G. Demoment, "Image reconstruction and restoration: overview of common estimation structures and problems," IEEE Trans. Acoust., Speech, Signal Process. 37, 2024-2036 (1989).
[CrossRef]

Fienup, J. R.

M. F. Reiley, R. G. Paxman, J. R. Fienup, K. W. Gleichman, and J. C. Marron, "3D reconstruction of opaque objects from Fourier intensity data," in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE 3170, 76-87 (1997).
[CrossRef]

R. G. Paxman, J. H. Seldin, J. R. Fienup, and J. C. Marron, "Use of an opacity constraint in three-dimensional imaging," in Inverse Optics III, M. A. Fiddy, ed., Proc. SPIE 2241, 116-126 (1994).
[CrossRef]

Fontanella, J.-C.

J. Primot, G. Rousset, and J.-C. Fontanella, "Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images," J. Opt. Soc. Am. A 7, 1598-1608 (1990).
[CrossRef]

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, "First diffraction-limited astronomical images with adaptive optics," Astron. Astrophys. 230, 29-32 (1990).

Fusco, T.

Gaffard, J.-P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, "First diffraction-limited astronomical images with adaptive optics," Astron. Astrophys. 230, 29-32 (1990).

Gendron, E.

D. Gratadour, D. Rouan, L. M. Mugnier, T. Fusco, Y. Clénet, E. Gendron, and F. Lacombe, "Near-infrared adaptive optics dissection of the core of NGC 1068 with NaCo," Astron. Astrophys. 446, 813-825 (2006).
[CrossRef]

Gigan, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, "First diffraction-limited astronomical images with adaptive optics," Astron. Astrophys. 230, 29-32 (1990).

Gleichman, K. W.

M. F. Reiley, R. G. Paxman, J. R. Fienup, K. W. Gleichman, and J. C. Marron, "3D reconstruction of opaque objects from Fourier intensity data," in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE 3170, 76-87 (1997).
[CrossRef]

Gonsalves, R. A.

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).

Gratadour, D.

D. Gratadour, D. Rouan, L. M. Mugnier, T. Fusco, Y. Clénet, E. Gendron, and F. Lacombe, "Near-infrared adaptive optics dissection of the core of NGC 1068 with NaCo," Astron. Astrophys. 446, 813-825 (2006).
[CrossRef]

Green, P. J.

P. J. Green, "Bayesian reconstructions from emission tomography data using a modified EM algorithm," IEEE Trans. Med. Imaging 9, 84-93 (1990).
[CrossRef]

Idier, J.

A. Blanc, L. M. Mugnier, and J. Idier, "Marginal estimation of aberrations and image restoration by use of phase diversity," J. Opt. Soc. Am. A 20, 1035-1045 (2003).
[CrossRef]

J. Idier and L. Blanc-Féraud, "Deconvolution en imagerie," in Approche Bayésienne pour les Problèmes Inverses, J.Idier, ed. (Hermès, 2001), Chap. 6.

S. Brette and J. Idier, "Optimized single site update algorithms for image deblurring," in Proceedings of the International Conference on Image Processing (IEEE Computer Society, 1996), pp. 65-68.

L. M. Mugnier, A. Blanc, and J. Idier, "Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging," in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elsevier, 2006), Vol. 141, Chap. 1.
[CrossRef]

Jagourel, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, "First diffraction-limited astronomical images with adaptive optics," Astron. Astrophys. 230, 29-32 (1990).

Karpova, T.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, "Three-dimensional imaging by deconvolution microscopy," Methods 19, 373-385 (1999).
[CrossRef] [PubMed]

Kern, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, "First diffraction-limited astronomical images with adaptive optics," Astron. Astrophys. 230, 29-32 (1990).

Lacombe, F.

G. Chenegros, L. M. Mugnier, and F. Lacombe, "3D deconvolution of adaptive-optics corrected retinal images," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, J.-A. Conchello, C. J. Cogswell, and T. Wilson, eds., Proc. SPIE 6090, 60900P (2006).
[CrossRef]

D. Gratadour, D. Rouan, L. M. Mugnier, T. Fusco, Y. Clénet, E. Gendron, and F. Lacombe, "Near-infrared adaptive optics dissection of the core of NGC 1068 with NaCo," Astron. Astrophys. 446, 813-825 (2006).
[CrossRef]

Léna, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, "First diffraction-limited astronomical images with adaptive optics," Astron. Astrophys. 230, 29-32 (1990).

Madec, P. Y.

J. M. Conan, P. Y. Madec, and G. Rousset, "Image formation in adaptive optics partial correction," in Active and Adaptive Optics, F.Merkle, ed., Vol. 48 of ESO Conference and Workshop Proceeding (European Southern Observatory/International Commission of Optics, 1994).

Marron, J. C.

M. F. Reiley, R. G. Paxman, J. R. Fienup, K. W. Gleichman, and J. C. Marron, "3D reconstruction of opaque objects from Fourier intensity data," in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE 3170, 76-87 (1997).
[CrossRef]

R. G. Paxman, J. H. Seldin, J. R. Fienup, and J. C. Marron, "Use of an opacity constraint in three-dimensional imaging," in Inverse Optics III, M. A. Fiddy, ed., Proc. SPIE 2241, 116-126 (1994).
[CrossRef]

McNally, J. G.

J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, "Three-dimensional imaging by deconvolution microscopy," Methods 19, 373-385 (1999).
[CrossRef] [PubMed]

Merkle, F.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, "First diffraction-limited astronomical images with adaptive optics," Astron. Astrophys. 230, 29-32 (1990).

Michau, V.

Mugnier, L. M.

D. Gratadour, D. Rouan, L. M. Mugnier, T. Fusco, Y. Clénet, E. Gendron, and F. Lacombe, "Near-infrared adaptive optics dissection of the core of NGC 1068 with NaCo," Astron. Astrophys. 446, 813-825 (2006).
[CrossRef]

G. Chenegros, L. M. Mugnier, and F. Lacombe, "3D deconvolution of adaptive-optics corrected retinal images," in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing XIII, J.-A. Conchello, C. J. Cogswell, and T. Wilson, eds., Proc. SPIE 6090, 60900P (2006).
[CrossRef]

L. M. Mugnier, T. Fusco, and J.-M. Conan, "MISTRAL: a myopic edge-preserving image restoration method, with application to astronomical adaptive optics-corrected long-exposure images," J. Opt. Soc. Am. A 21, 1841-1854 (2004).
[CrossRef]

A. Blanc, L. M. Mugnier, and J. Idier, "Marginal estimation of aberrations and image restoration by use of phase diversity," J. Opt. Soc. Am. A 20, 1035-1045 (2003).
[CrossRef]

L. M. Mugnier, C. Robert, J.-M. Conan, V. Michau, and S. Salem, "Myopic deconvolution from wave-front sensing," J. Opt. Soc. Am. A 18, 862-872 (2001).
[CrossRef]

J.-M. Conan, L. M. Mugnier, T. Fusco, V. Michau, and G. Rousset, "Myopic deconvolution of adaptive optics images by use of object and point-spread function power spectra," Appl. Opt. 37, 4614-4622 (1998).
[CrossRef]

L. M. Mugnier, A. Blanc, and J. Idier, "Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging," in Advances in Imaging and Electron Physics, P.Hawkes, ed. (Elsevier, 2006), Vol. 141, Chap. 1.
[CrossRef]

Noll, R. J.

R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. A 66, 207-211 (1976).
[CrossRef]

Paxman, R. G.

M. F. Reiley, R. G. Paxman, J. R. Fienup, K. W. Gleichman, and J. C. Marron, "3D reconstruction of opaque objects from Fourier intensity data," in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE 3170, 76-87 (1997).
[CrossRef]

R. G. Paxman, J. H. Seldin, J. R. Fienup, and J. C. Marron, "Use of an opacity constraint in three-dimensional imaging," in Inverse Optics III, M. A. Fiddy, ed., Proc. SPIE 2241, 116-126 (1994).
[CrossRef]

Primot, J.

Reiley, M. F.

M. F. Reiley, R. G. Paxman, J. R. Fienup, K. W. Gleichman, and J. C. Marron, "3D reconstruction of opaque objects from Fourier intensity data," in Image Reconstruction and Restoration II, T. J. Schulz, ed., Proc. SPIE 3170, 76-87 (1997).
[CrossRef]

Rey, W. J. J.

W. J. J. Rey, Introduction to Robust and Quasi-Robust Statistical Methods (Springer-Verlag, 1983).
[CrossRef]

Rigaut, F.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, "First diffraction-limited astronomical images with adaptive optics," Astron. Astrophys. 230, 29-32 (1990).

Robert, C.

Roggemann, M. C.

Roorda, A.

Rouan, D.

D. Gratadour, D. Rouan, L. M. Mugnier, T. Fusco, Y. Clénet, E. Gendron, and F. Lacombe, "Near-infrared adaptive optics dissection of the core of NGC 1068 with NaCo," Astron. Astrophys. 446, 813-825 (2006).
[CrossRef]

Rousset, G.

J.-M. Conan, L. M. Mugnier, T. Fusco, V. Michau, and G. Rousset, "Myopic deconvolution of adaptive optics images by use of object and point-spread function power spectra," Appl. Opt. 37, 4614-4622 (1998).
[CrossRef]

J. Primot, G. Rousset, and J.-C. Fontanella, "Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images," J. Opt. Soc. Am. A 7, 1598-1608 (1990).
[CrossRef]

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Léna, C. Boyer, P. Jagourel, J.-P. Gaffard, and F. Merkle, "First diffraction-limited astronomical images with adaptive optics," Astron. Astrophys. 230, 29-32 (1990).

J. M. Conan, P. Y. Madec, and G. Rousset, "Image formation in adaptive optics partial correction," in Active and Adaptive Optics, F.Merkle, ed., Vol. 48 of ESO Conference and Workshop Proceeding (European Southern Observatory/International Commission of Optics, 1994).

Salem, S.

Sauer, K.

C. Bouman and K. Sauer, "A generalized Gaussian image model for edge-preserving MAP estimation," IEEE Trans. Image Process. 2, 296-310 (1993).
[CrossRef]

Schulz, T. J.

Seldin, J. H.

R. G. Paxman, J. H. Seldin, J. R. Fienup, and J. C. Marron, "Use of an opacity constraint in three-dimensional imaging," in Inverse Optics III, M. A. Fiddy, ed., Proc. SPIE 2241, 116-126 (1994).
[CrossRef]

Thiébaut, E.

E. Thiébaut, "Optimization issues in blind deconvolution algorithms," in Astronomical Data Analysis. II, J.-L. Starck and F. D. Murtagh, eds., Proc. SPIE 4847, 174-183 (2002).
[CrossRef]

E. Thiébaut and J.-M. Conan, "Strict a priori constraints for maximum-likelihood blind deconvolution," J. Opt. Soc. Am. A 12, 485-492 (1995).
[CrossRef]

Tikhonov, A.

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

Williams, D. R.

Appl. Opt. (2)

Astron. Astrophys. (2)

D. Gratadour, D. Rouan, L. M. Mugnier, T. Fusco, Y. Clénet, E. Gendron, and F. Lacombe, "Near-infrared adaptive optics dissection of the core of NGC 1068 with NaCo," Astron. Astrophys. 446, 813-825 (2006).
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

Illustration of the 3D image formation for three object planes. The object is on the left, and the image is on the right. The system is composed of the eye and the optical system (including the AO). In image i 1 , object o 1 is focused; o 2 and o 3 are defocused. Images i 2 and i 3 are not represented here.

Fig. 2
Fig. 2

Perspective view of the 3D object used for the simulations.

Fig. 3
Fig. 3

Five object layers [black corresponds to 0 photoelectrons per pixel (ph/pix)].

Fig. 4
Fig. 4

Five image layers.

Fig. 5
Fig. 5

Five estimated object layers with L 2 regularization without the positivity constraint and using the true PSF.

Fig. 6
Fig. 6

Five estimated object layers with L 2 regularization under the positivity constraint and using the true PSF (black corresponds to 0 ph pix ).

Fig. 7
Fig. 7

Five estimated object layers with L 2 L 1 regularization under the positivity constraint and using the true PSF (black corresponds to 0 ph pix ).

Fig. 8
Fig. 8

Deconvolution with a wrong (unaberrated) PSF. The different object planes are not correctly disentangled because of the mismatch between true PSF and assumed PSF (black corresponds to 0 ph pix ).

Fig. 9
Fig. 9

Estimated aberrations with and without the positivity constraint (PC).

Fig. 10
Fig. 10

Criterion surface with the positivity constraint: the criterion is strictly convex. A is the position of the global minimum; it is close to the position of the true aberrations B.

Fig. 11
Fig. 11

Criterion surface without the positivity constraint: the criterion is nonconvex. A is the position of the global minimum, and B is the position of the true aberrations.

Fig. 12
Fig. 12

Principle of phase diversity: two images differing by a known aberration (here defocus) are used to estimate the pupil phase.

Fig. 13
Fig. 13

Ten object layers where five layers are empty (black corresponds to 0 ph pix ).

Fig. 14
Fig. 14

Ten simulated image layers.

Fig. 15
Fig. 15

Estimated aberrations with and without the Z support constraint (ZSC).

Fig. 16
Fig. 16

Five estimated object layers with L 2 L 1 regularization under the positivity constraint and ZSC (black corresponds to 0 ph pix ).

Equations (16)

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i = h * o + n ,
i k = ( l = 0 N 1 h k 1 * o l ) + n k ,
o ̂ = arg max o p ( o i ) = arg max o p ( i o ) × p ( o ) .
J ( o ) = J i ( o ) + J o ( o ) ,
J i ( o ) = 1 2 k = 0 N 1 p , q = 0 N p i x 1 ( 1 σ k 2 ( p , q ) i k ( p , q ) t = 0 N 1 ( h k l ( p , q ) * o l ( p , q ) ) 2 ) ,
PSD ( f ) = E [ o ( f ) 2 ] o m ( f ) 2 = k [ 1 + ( f f 0 ) p ] o m ( f ) 2 ,
h k ( φ ) = FT 1 { P ( x , y ) exp ( j ( φ ( x , y ) + φ d k ( x , y ) ) ) } 2 , φ ( x , y ) = m = 5 M a m Z m ( x , y ) ,
φ d k ( x , y , δ z ) = a 4 d ( δ z ) . Z 4 ( x , y ) ,
a 4 d ( δ z ) = π δ z 8 . 3 . λ n . ( f . n D ) 2 ,
[ o ̂ , φ ̂ ] = arg max o , φ p ( o , φ i ) = arg max o , φ p ( i o , φ ) × p ( o ) × p ( φ ) .
J ( o , φ ) = J i ( o , φ ) + J o ( o ) + J φ ( φ ) ,
J φ ( φ ) = 1 2 ( φ φ ¯ ) t C φ 1 ( φ φ ¯ ) ,
J i ( o , φ ) = 1 2 σ n 2 k = 0 N 1 i k l = 0 N 1 ( h k l ( φ ) * o l ) 2 .
i f = h ( φ ) * o + n ,
i d = h ( φ + φ d ) * o + n ,
J i ( o , φ ) = 1 2 σ n 2 k = 0 N 1 i k t = 0 l S o N 1 ( h k l ( φ ) * o l ) 2 ,

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