Abstract

4Pi-microscopy doubles the aperture of the imaging system by coherent addition of the wavefronts for illumination and/or detection through opposing objective lenses. This improves the axial resolution 3–7 fold, but the raw data usually features ghost images which have to be removed by image reconstruction. This straightforward procedure is sometimes precluded by imperfect alignment of the instrument or a specimen with strong variations of its refractive index, because the image formation process now depends on the space-variant phase difference between the counter-propagating wavefronts. Here we present a computationally fast method of parametric blind deconvolution that allows for automatic and robust simultaneous estimation of both the object and the phase function in such cases. We verify the performance of our approach on both synthetic and real data. Because the method does not require a-priori knowledge of the phase function it is a major step towards reliable 4Pi-imaging and automatic image restoration by non-expert users.

© 2010 Optical Society of America

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References

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  1. S. W. Hell, "Double-scanning confocal microscope," European Patent (ISSN 0491289) (1990).
  2. C. Cremer and T. Cremer, "Considerations on a laser-scanning-microscope with high resolution and depth of field," Microsc. Acta 81(1), 31-44 (1978).
    [PubMed]
  3. S. W. Hell and A. Schonle, "Nanoscale resolution in far-field fluorescence microscopy," in Science of Microscopy, P. W. Hawkes and J. C. H. Spence, eds., (Springer, 2007), pp. 790-834.
    [CrossRef]
  4. M. Nagorni and S. W. Hell, "Coherent use of opposing lenses for axial resolution increase in fluorescence microscopy. I. Comparative study of concepts," J. Opt. Soc. Am. A 18(1), 36-48 (2001).
    [CrossRef]
  5. G. Vicidomini, S. W. Hell, and A. Schonle, "Automatic deconvolution of 4Pi-microscopy data with arbitrary phase," Opt. Lett. 34(22), 3583-3585 (2009).
    [CrossRef] [PubMed]
  6. C. M. Blanca, J. Bewersdorf, and S. W. Hell, "Determination of the unknown phase difference in 4Pi-confocal microscopy through the image intensity," Opt. Commun. 206(4-6), 281-285 (2002).
    [CrossRef]
  7. S. W. Hell, C. M. Blanca, and J. Bewersdorf, "Phase determination in interference-based superresolving microscopes through critical frequency analysis," Opt. Lett. 27(11), 888-890 (2002).
    [CrossRef]
  8. D. Baddeley, C. Carl, and C. Cremer, "4Pi microscopy deconvolution with a variable point-spread function," Appl. Opt. 45(27), 7056-7064 (2006).
    [CrossRef] [PubMed]
  9. T. Staudt, M. C. Lang, R. Medda, J. Engelhardt, and S. W. Hell, "2,2prime-Thiodiethanol: A new water soluble mounting medium for high resolution optical microscopy," Microsc. Res. Tech. 70, 1-9 (2007).
    [CrossRef]
  10. M. C. Lang, T. Staudt, J. Engelhardt, and S. W. Hell, "4Pi microscopy with negligible sidelobes," New J. Phys. 10, 043041 (2008).Q1
    [CrossRef]
  11. M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal Imaging in Fixed Biological Specimens," Biophys. J. 75(4), 1659-1668 (1998).
    [CrossRef] [PubMed]
  12. J. Enderlein, "Theoretical study of detection of a dipole emitter through an objective with high numerical aperture," Opt. Lett. 25(9), 634-636 (2000).
    [CrossRef]
  13. B. Richards and E. Wolf, "Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System," Proc. R. Soc. Lond. A 253(1274), 358-379 (1959).
    [CrossRef]
  14. M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, "Image deblurring with Poisson data: from cells to galaxies," Inverse Probl. 25, 123006 (2009).
    [CrossRef]
  15. A. Egner, M. Schrader, and S. W. Hell, "Refractive index mismatch induced intensity and phase variations in fluorescence confocal, multiphoton and 4Pi-microscopy," Opt. Commun. 153(4-6), 211-217 (1998).
    [CrossRef]
  16. A. Egner, S. Verrier, A. Goroshkov, H.-D. Soling, and S. W. Hell, "4Pi-microscopy of the Golgi apparatus in live mammalian cells," J. Struct. Biol. 147(1), 70-76 (2004).
    [CrossRef] [PubMed]
  17. G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, "Application of the split-gradient method to 3D image deconvolution in fluorescence microscopy," J. Microsc. 234(1), 47-61 (2008).
    [CrossRef]
  18. L. B. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974).
    [CrossRef]
  19. W. H. Richardson, "Bayesian-Based Iterative Method of Image Restoration," J. Opt. Soc. Am. 62(1), 55-59 (1972).
    [CrossRef]
  20. C. Kelley, Iterative Method for Optimization, (SIAM, Philadelphia, 1999) Vol. 18.
  21. D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization," Proc. Natl. Acad. Sci. U.S.A. 100(5), 2197-2202 (2003).
    [CrossRef]
  22. H. Lantéri, M. Roche, O. Cuevas, and C. Aime, "A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints," Signal Process. 81, 945-974 (2001).
    [CrossRef]
  23. I. Csiszár "Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problem," Ann. Stat. 19, 2032-2066 (1991).
    [CrossRef]

2009 (2)

G. Vicidomini, S. W. Hell, and A. Schonle, "Automatic deconvolution of 4Pi-microscopy data with arbitrary phase," Opt. Lett. 34(22), 3583-3585 (2009).
[CrossRef] [PubMed]

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, "Image deblurring with Poisson data: from cells to galaxies," Inverse Probl. 25, 123006 (2009).
[CrossRef]

2008 (2)

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, "Application of the split-gradient method to 3D image deconvolution in fluorescence microscopy," J. Microsc. 234(1), 47-61 (2008).
[CrossRef]

M. C. Lang, T. Staudt, J. Engelhardt, and S. W. Hell, "4Pi microscopy with negligible sidelobes," New J. Phys. 10, 043041 (2008).Q1
[CrossRef]

2007 (1)

T. Staudt, M. C. Lang, R. Medda, J. Engelhardt, and S. W. Hell, "2,2prime-Thiodiethanol: A new water soluble mounting medium for high resolution optical microscopy," Microsc. Res. Tech. 70, 1-9 (2007).
[CrossRef]

2006 (1)

2004 (1)

A. Egner, S. Verrier, A. Goroshkov, H.-D. Soling, and S. W. Hell, "4Pi-microscopy of the Golgi apparatus in live mammalian cells," J. Struct. Biol. 147(1), 70-76 (2004).
[CrossRef] [PubMed]

2003 (1)

D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization," Proc. Natl. Acad. Sci. U.S.A. 100(5), 2197-2202 (2003).
[CrossRef]

2002 (2)

C. M. Blanca, J. Bewersdorf, and S. W. Hell, "Determination of the unknown phase difference in 4Pi-confocal microscopy through the image intensity," Opt. Commun. 206(4-6), 281-285 (2002).
[CrossRef]

S. W. Hell, C. M. Blanca, and J. Bewersdorf, "Phase determination in interference-based superresolving microscopes through critical frequency analysis," Opt. Lett. 27(11), 888-890 (2002).
[CrossRef]

2001 (2)

M. Nagorni and S. W. Hell, "Coherent use of opposing lenses for axial resolution increase in fluorescence microscopy. I. Comparative study of concepts," J. Opt. Soc. Am. A 18(1), 36-48 (2001).
[CrossRef]

H. Lantéri, M. Roche, O. Cuevas, and C. Aime, "A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints," Signal Process. 81, 945-974 (2001).
[CrossRef]

2000 (1)

1998 (2)

A. Egner, M. Schrader, and S. W. Hell, "Refractive index mismatch induced intensity and phase variations in fluorescence confocal, multiphoton and 4Pi-microscopy," Opt. Commun. 153(4-6), 211-217 (1998).
[CrossRef]

M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal Imaging in Fixed Biological Specimens," Biophys. J. 75(4), 1659-1668 (1998).
[CrossRef] [PubMed]

1991 (1)

I. Csiszár "Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problem," Ann. Stat. 19, 2032-2066 (1991).
[CrossRef]

1978 (1)

C. Cremer and T. Cremer, "Considerations on a laser-scanning-microscope with high resolution and depth of field," Microsc. Acta 81(1), 31-44 (1978).
[PubMed]

1974 (1)

L. B. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974).
[CrossRef]

1972 (1)

1959 (1)

B. Richards and E. Wolf, "Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System," Proc. R. Soc. Lond. A 253(1274), 358-379 (1959).
[CrossRef]

Aime, C.

H. Lantéri, M. Roche, O. Cuevas, and C. Aime, "A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints," Signal Process. 81, 945-974 (2001).
[CrossRef]

Baddeley, D.

Bahlmann, K.

M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal Imaging in Fixed Biological Specimens," Biophys. J. 75(4), 1659-1668 (1998).
[CrossRef] [PubMed]

Bertero, M.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, "Image deblurring with Poisson data: from cells to galaxies," Inverse Probl. 25, 123006 (2009).
[CrossRef]

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, "Application of the split-gradient method to 3D image deconvolution in fluorescence microscopy," J. Microsc. 234(1), 47-61 (2008).
[CrossRef]

Bewersdorf, J.

S. W. Hell, C. M. Blanca, and J. Bewersdorf, "Phase determination in interference-based superresolving microscopes through critical frequency analysis," Opt. Lett. 27(11), 888-890 (2002).
[CrossRef]

C. M. Blanca, J. Bewersdorf, and S. W. Hell, "Determination of the unknown phase difference in 4Pi-confocal microscopy through the image intensity," Opt. Commun. 206(4-6), 281-285 (2002).
[CrossRef]

Blanca, C. M.

C. M. Blanca, J. Bewersdorf, and S. W. Hell, "Determination of the unknown phase difference in 4Pi-confocal microscopy through the image intensity," Opt. Commun. 206(4-6), 281-285 (2002).
[CrossRef]

S. W. Hell, C. M. Blanca, and J. Bewersdorf, "Phase determination in interference-based superresolving microscopes through critical frequency analysis," Opt. Lett. 27(11), 888-890 (2002).
[CrossRef]

Boccacci, P.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, "Image deblurring with Poisson data: from cells to galaxies," Inverse Probl. 25, 123006 (2009).
[CrossRef]

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, "Application of the split-gradient method to 3D image deconvolution in fluorescence microscopy," J. Microsc. 234(1), 47-61 (2008).
[CrossRef]

Carl, C.

Cremer, C.

D. Baddeley, C. Carl, and C. Cremer, "4Pi microscopy deconvolution with a variable point-spread function," Appl. Opt. 45(27), 7056-7064 (2006).
[CrossRef] [PubMed]

C. Cremer and T. Cremer, "Considerations on a laser-scanning-microscope with high resolution and depth of field," Microsc. Acta 81(1), 31-44 (1978).
[PubMed]

Cremer, T.

C. Cremer and T. Cremer, "Considerations on a laser-scanning-microscope with high resolution and depth of field," Microsc. Acta 81(1), 31-44 (1978).
[PubMed]

Csiszár, I.

I. Csiszár "Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problem," Ann. Stat. 19, 2032-2066 (1991).
[CrossRef]

Cuevas, O.

H. Lantéri, M. Roche, O. Cuevas, and C. Aime, "A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints," Signal Process. 81, 945-974 (2001).
[CrossRef]

Desiderà, G.

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, "Image deblurring with Poisson data: from cells to galaxies," Inverse Probl. 25, 123006 (2009).
[CrossRef]

Diaspro, A.

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, "Application of the split-gradient method to 3D image deconvolution in fluorescence microscopy," J. Microsc. 234(1), 47-61 (2008).
[CrossRef]

Donoho, D. L.

D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization," Proc. Natl. Acad. Sci. U.S.A. 100(5), 2197-2202 (2003).
[CrossRef]

Egner, A.

A. Egner, S. Verrier, A. Goroshkov, H.-D. Soling, and S. W. Hell, "4Pi-microscopy of the Golgi apparatus in live mammalian cells," J. Struct. Biol. 147(1), 70-76 (2004).
[CrossRef] [PubMed]

A. Egner, M. Schrader, and S. W. Hell, "Refractive index mismatch induced intensity and phase variations in fluorescence confocal, multiphoton and 4Pi-microscopy," Opt. Commun. 153(4-6), 211-217 (1998).
[CrossRef]

Elad, M.

D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization," Proc. Natl. Acad. Sci. U.S.A. 100(5), 2197-2202 (2003).
[CrossRef]

Enderlein, J.

Engelhardt, J.

M. C. Lang, T. Staudt, J. Engelhardt, and S. W. Hell, "4Pi microscopy with negligible sidelobes," New J. Phys. 10, 043041 (2008).Q1
[CrossRef]

T. Staudt, M. C. Lang, R. Medda, J. Engelhardt, and S. W. Hell, "2,2prime-Thiodiethanol: A new water soluble mounting medium for high resolution optical microscopy," Microsc. Res. Tech. 70, 1-9 (2007).
[CrossRef]

Giese, G.

M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal Imaging in Fixed Biological Specimens," Biophys. J. 75(4), 1659-1668 (1998).
[CrossRef] [PubMed]

Goroshkov, A.

A. Egner, S. Verrier, A. Goroshkov, H.-D. Soling, and S. W. Hell, "4Pi-microscopy of the Golgi apparatus in live mammalian cells," J. Struct. Biol. 147(1), 70-76 (2004).
[CrossRef] [PubMed]

Hell, S. W.

G. Vicidomini, S. W. Hell, and A. Schonle, "Automatic deconvolution of 4Pi-microscopy data with arbitrary phase," Opt. Lett. 34(22), 3583-3585 (2009).
[CrossRef] [PubMed]

M. C. Lang, T. Staudt, J. Engelhardt, and S. W. Hell, "4Pi microscopy with negligible sidelobes," New J. Phys. 10, 043041 (2008).Q1
[CrossRef]

T. Staudt, M. C. Lang, R. Medda, J. Engelhardt, and S. W. Hell, "2,2prime-Thiodiethanol: A new water soluble mounting medium for high resolution optical microscopy," Microsc. Res. Tech. 70, 1-9 (2007).
[CrossRef]

A. Egner, S. Verrier, A. Goroshkov, H.-D. Soling, and S. W. Hell, "4Pi-microscopy of the Golgi apparatus in live mammalian cells," J. Struct. Biol. 147(1), 70-76 (2004).
[CrossRef] [PubMed]

C. M. Blanca, J. Bewersdorf, and S. W. Hell, "Determination of the unknown phase difference in 4Pi-confocal microscopy through the image intensity," Opt. Commun. 206(4-6), 281-285 (2002).
[CrossRef]

S. W. Hell, C. M. Blanca, and J. Bewersdorf, "Phase determination in interference-based superresolving microscopes through critical frequency analysis," Opt. Lett. 27(11), 888-890 (2002).
[CrossRef]

M. Nagorni and S. W. Hell, "Coherent use of opposing lenses for axial resolution increase in fluorescence microscopy. I. Comparative study of concepts," J. Opt. Soc. Am. A 18(1), 36-48 (2001).
[CrossRef]

M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal Imaging in Fixed Biological Specimens," Biophys. J. 75(4), 1659-1668 (1998).
[CrossRef] [PubMed]

A. Egner, M. Schrader, and S. W. Hell, "Refractive index mismatch induced intensity and phase variations in fluorescence confocal, multiphoton and 4Pi-microscopy," Opt. Commun. 153(4-6), 211-217 (1998).
[CrossRef]

Lang, M. C.

M. C. Lang, T. Staudt, J. Engelhardt, and S. W. Hell, "4Pi microscopy with negligible sidelobes," New J. Phys. 10, 043041 (2008).Q1
[CrossRef]

T. Staudt, M. C. Lang, R. Medda, J. Engelhardt, and S. W. Hell, "2,2prime-Thiodiethanol: A new water soluble mounting medium for high resolution optical microscopy," Microsc. Res. Tech. 70, 1-9 (2007).
[CrossRef]

Lantéri, H.

H. Lantéri, M. Roche, O. Cuevas, and C. Aime, "A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints," Signal Process. 81, 945-974 (2001).
[CrossRef]

Lucy, L. B.

L. B. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974).
[CrossRef]

Medda, R.

T. Staudt, M. C. Lang, R. Medda, J. Engelhardt, and S. W. Hell, "2,2prime-Thiodiethanol: A new water soluble mounting medium for high resolution optical microscopy," Microsc. Res. Tech. 70, 1-9 (2007).
[CrossRef]

Nagorni, M.

Richards, B.

B. Richards and E. Wolf, "Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System," Proc. R. Soc. Lond. A 253(1274), 358-379 (1959).
[CrossRef]

Richardson, W. H.

Roche, M.

H. Lantéri, M. Roche, O. Cuevas, and C. Aime, "A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints," Signal Process. 81, 945-974 (2001).
[CrossRef]

Schonle, A.

Schrader, M.

A. Egner, M. Schrader, and S. W. Hell, "Refractive index mismatch induced intensity and phase variations in fluorescence confocal, multiphoton and 4Pi-microscopy," Opt. Commun. 153(4-6), 211-217 (1998).
[CrossRef]

M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal Imaging in Fixed Biological Specimens," Biophys. J. 75(4), 1659-1668 (1998).
[CrossRef] [PubMed]

Soling, H.-D.

A. Egner, S. Verrier, A. Goroshkov, H.-D. Soling, and S. W. Hell, "4Pi-microscopy of the Golgi apparatus in live mammalian cells," J. Struct. Biol. 147(1), 70-76 (2004).
[CrossRef] [PubMed]

Staudt, T.

M. C. Lang, T. Staudt, J. Engelhardt, and S. W. Hell, "4Pi microscopy with negligible sidelobes," New J. Phys. 10, 043041 (2008).Q1
[CrossRef]

T. Staudt, M. C. Lang, R. Medda, J. Engelhardt, and S. W. Hell, "2,2prime-Thiodiethanol: A new water soluble mounting medium for high resolution optical microscopy," Microsc. Res. Tech. 70, 1-9 (2007).
[CrossRef]

Verrier, S.

A. Egner, S. Verrier, A. Goroshkov, H.-D. Soling, and S. W. Hell, "4Pi-microscopy of the Golgi apparatus in live mammalian cells," J. Struct. Biol. 147(1), 70-76 (2004).
[CrossRef] [PubMed]

Vicidomini, G.

G. Vicidomini, S. W. Hell, and A. Schonle, "Automatic deconvolution of 4Pi-microscopy data with arbitrary phase," Opt. Lett. 34(22), 3583-3585 (2009).
[CrossRef] [PubMed]

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, "Image deblurring with Poisson data: from cells to galaxies," Inverse Probl. 25, 123006 (2009).
[CrossRef]

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, "Application of the split-gradient method to 3D image deconvolution in fluorescence microscopy," J. Microsc. 234(1), 47-61 (2008).
[CrossRef]

Wolf, E.

B. Richards and E. Wolf, "Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System," Proc. R. Soc. Lond. A 253(1274), 358-379 (1959).
[CrossRef]

Ann. Stat. (1)

I. Csiszár "Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problem," Ann. Stat. 19, 2032-2066 (1991).
[CrossRef]

Appl. Opt. (1)

Astron. J. (1)

L. B. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974).
[CrossRef]

Biophys. J. (1)

M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal Imaging in Fixed Biological Specimens," Biophys. J. 75(4), 1659-1668 (1998).
[CrossRef] [PubMed]

Inverse Probl. (1)

M. Bertero, P. Boccacci, G. Desiderà, and G. Vicidomini, "Image deblurring with Poisson data: from cells to galaxies," Inverse Probl. 25, 123006 (2009).
[CrossRef]

J. Microsc. (1)

G. Vicidomini, P. Boccacci, A. Diaspro, and M. Bertero, "Application of the split-gradient method to 3D image deconvolution in fluorescence microscopy," J. Microsc. 234(1), 47-61 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Struct. Biol. (1)

A. Egner, S. Verrier, A. Goroshkov, H.-D. Soling, and S. W. Hell, "4Pi-microscopy of the Golgi apparatus in live mammalian cells," J. Struct. Biol. 147(1), 70-76 (2004).
[CrossRef] [PubMed]

Microsc. Acta (1)

C. Cremer and T. Cremer, "Considerations on a laser-scanning-microscope with high resolution and depth of field," Microsc. Acta 81(1), 31-44 (1978).
[PubMed]

Microsc. Res. Tech. (1)

T. Staudt, M. C. Lang, R. Medda, J. Engelhardt, and S. W. Hell, "2,2prime-Thiodiethanol: A new water soluble mounting medium for high resolution optical microscopy," Microsc. Res. Tech. 70, 1-9 (2007).
[CrossRef]

New J. Phys. (1)

M. C. Lang, T. Staudt, J. Engelhardt, and S. W. Hell, "4Pi microscopy with negligible sidelobes," New J. Phys. 10, 043041 (2008).Q1
[CrossRef]

Opt. Commun. (2)

C. M. Blanca, J. Bewersdorf, and S. W. Hell, "Determination of the unknown phase difference in 4Pi-confocal microscopy through the image intensity," Opt. Commun. 206(4-6), 281-285 (2002).
[CrossRef]

A. Egner, M. Schrader, and S. W. Hell, "Refractive index mismatch induced intensity and phase variations in fluorescence confocal, multiphoton and 4Pi-microscopy," Opt. Commun. 153(4-6), 211-217 (1998).
[CrossRef]

Opt. Lett. (3)

Proc. Natl. Acad. Sci. U.S.A. (1)

D. L. Donoho and M. Elad, "Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization," Proc. Natl. Acad. Sci. U.S.A. 100(5), 2197-2202 (2003).
[CrossRef]

Proc. R. Soc. Lond. A (1)

B. Richards and E. Wolf, "Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System," Proc. R. Soc. Lond. A 253(1274), 358-379 (1959).
[CrossRef]

Signal Process. (1)

H. Lantéri, M. Roche, O. Cuevas, and C. Aime, "A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints," Signal Process. 81, 945-974 (2001).
[CrossRef]

Other (3)

S. W. Hell, "Double-scanning confocal microscope," European Patent (ISSN 0491289) (1990).

C. Kelley, Iterative Method for Optimization, (SIAM, Philadelphia, 1999) Vol. 18.

S. W. Hell and A. Schonle, "Nanoscale resolution in far-field fluorescence microscopy," in Science of Microscopy, P. W. Hawkes and J. C. H. Spence, eds., (Springer, 2007), pp. 790-834.
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Image restoration on simulated data (qualitative results): (a) phantom (136×136 pixels, pixel size 20×20nm in both lateral and axial direction); (b) cubic RPF; (c) simulated image obtained using the following parameters: objective NA 1.46 oil (RI 1.515), excitation wavelength λex = 632 nm, emission wavelength λem = 680 nm, pinhole 0.5 AU, constant background of 2 photon counts and expected photon counts of 100 in the bright pixel; (d) restored object (the RPF is known a-priori); (e) restored object and (f) reconstructed RPF for FPBR; (g) restored object (h) and reconstructed RPF for SPBR.

Fig. 2.
Fig. 2.

Image restoration on simulated data (quantitative results): (a) Kullback-Leibler divergences between test phantom and reconstructed object as a function of the number of cycles for FPBR vs. SPBR; (b) Euclidean distances between reconstructed and assumed RPF as a function of the number of cycles for FPBR and SPBR.

Fig. 3.
Fig. 3.

Image restoration on experimental data: (a) raw xz image of microtubules wrapping around the nucleus in a Vero cell (186 × 352 pixels); (b) deconvolved data assuming constant RPF; (c) restored object and (d) reconstructed RPF for FPBR; (e) restored object and (f) reconstructed RPF for SPBR. Scale bar 1μm.

Fig. 4.
Fig. 4.

Intensity profiles of the restored objects taken at the positions indicated by arrows in Fig. 3(a). (a) intensity profile at site 1; (b) intensity profile at site 2.

Fig. 5.
Fig. 5.

3D Parametric blind restoration: (a) raw data (228 × 334 × 40 voxels); (b) restored object and (c) reconstructed RPF for SPBR. The dotted lines indicate the position of the xz- and yz-slices shown for each of the stacks. Scale bar 1μm.

Equations (39)

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g_(r)=heff(r,s)f(s)ds
heff(r,s)=E1(rs)+exp((r))E2(rs)2hdet(rs)=heff(rs,ϕ(r))
g_(r)=[H(ϕ)f](r)=heff(rs,ϕ(r))f(s)ds.
H(ϕ)f=A0f+cos(ϕ)A1fsin(ϕ)A2f
H(ϕ)f=sin(ϕ)A1fcos(ϕ)A2f
A0f=[(E12+E22)hdet]f
A1f=[2(E1*+E2)hdet]f
A2f=[2(E1*+E2)hdet]f.
g_=H(ϕ)f+b=A0f+cos(ϕ)A1fsin(ϕ)A2f+b.
P(f,ϕg)=P(gf,ϕ)P(f)P(ϕ)/P(g).
P(gf,ϕ)=npoi[g(n)ḡ(n)]
P(X)=exp[μxJx(X)]/Z,
J(f,ϕg)=Jo(f,ϕg)+μfJf(f)+μϕJϕ(ϕ),
Jo(f,ϕg)=ng_(n)g(n)lnḡ(n).
Jϕ(ϕ)=12nm∈ηn[ϕ(m)ϕ(n)]2,
fl+1=argminf0J(f,ϕlg)
ϕl+1 =argminϕJ(fl+1,ϕg)
fl,i+1=fl,iH(ϕl)T1+μffl,i(H(ϕl)TgH(ϕl)fl,i+b).
ϕl,i+1=ϕl,iτl,iρ(ϕl,i).
ρ(ϕl,i)=(Cl,i)1ϕ[J(fl+1,ϕ|g)](ϕl,i)
ϕDψ=pψpφp
φp(r)=xp1yp2zp3
∣∣fl,i+1fl,i∣∣2εSGM ∣∣fl,i∣∣2 .
∣∣x[J(fl+1,ϕg)](Xl,i)∣∣2εQNM∣∣x[J(fl+1,ϕg)](Xl,0)∣∣2,
J(fl+1,ϕl+1g)J(fl,ϕlg)εAMMJ(fl,ϕlg)
DKL(f,fE)=n{f(n)ln[f(n)fE(n)]+[fE(n)f(n)]},
DE(ϕ,ϕE)=∣∣exp(iTϕE)exp(i)∣∣2,
f[Jo(f,ϕg)](f)=Uo(fϕ,g)Vo(fϕ,g)
f[Jf(f)](f)=Uf(f)Vf(f),
fi+1=fi(Uo(fiϕ,g)+μfUf(fi)Vo(fiϕ,g)+μfVf(fi)).
f[Jo(f,ϕg)](f)=H(ϕ)T1H(ϕ)Tg[H(ϕ)f+b]1,
Uo(fϕ,g)=H(ϕ)TgH(ϕ)f+b
Vo(fϕ,g)=H(ϕ)T1.
f[Jf(f)](f)=f,
ϕ[J(f,ϕ;g)](ϕ)=H′(ϕ)fgH(ϕ)f+bH′(ϕ)f+μϕ,
()(n)=mηn{ϕ(m)ϕ(n)},
ψp[J(f,Dψg)]=n[Hp(Dψ)fgH(Dψ)f+bHp(Dψ)f](n)+μψψp/[ψp2+δ]1/2
Hp(Dψ)f=ψp[H(Dψ)f].
Hp(Dψ)f=sin(Dψ)φpA1fcos(Dψ)φpA2f.

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