Abstract

In three-dimensional fluorescence microscopy the point spread function (PSF) changes with depth, inducing errors in the restored images when these variations are neglected during the deconvolution of thick specimens. Some deconvolution algorithms have been developed to take the depth variations of the PSF into consideration. For these algorithms, the accuracy of the estimated structures depends on the knowledge of the PSF at various depths. We propose an alternative to measuring all required PSFs at different depths. The needed PSFs are interpolated from a limited measured PSF set using a method based on Zernike moments. The proposed method offers the possibility to obtain an accurate PSF interpolation at different depths using only a few measured ones.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. G. McNally, C. Preza, J.-A. Conchello, and L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A 11, 1056–1067 (1994).
    [CrossRef]
  2. J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999).
    [CrossRef] [PubMed]
  3. P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006).
    [CrossRef]
  4. R. Neelamani, “ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems,” IEEE Trans. Signal Process. 52, 418–433 (2004).
    [CrossRef]
  5. C. Vonesch and M. Unser, “A fast thresholded landweber algorithm for wavelet-regularized multidimensional deconvolution,” IEEE Trans. Image Process. 17, 539–549 (2008).
    [CrossRef] [PubMed]
  6. S. F. Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A 9, 154–166 (1992).
    [CrossRef] [PubMed]
  7. C. Preza and J.-A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21, 1593–1601 (2004).
    [CrossRef]
  8. J. G. Nagy and D. P. O’Leary, “Fast iterative image restoration with a spatially-varying PSF,” Proc. SPIE 3162, 388–399 (1997).
    [CrossRef]
  9. C. Preza and J.-A. Conchello, “Image estimation accounting for point-spread function depth-variation in three-dimensional fluorescence microscopy,” Proc. SPIE 4964, 135–142 (2003).
    [CrossRef]
  10. M. Arigovindan, J. Shaevitz, J. McGowan, J. W. Sedat, and D. A. Agard, “A parallel product-convolution approach for representing the depth varying point spread functions in 3D widefield microscopy based on principalcomponent analysis,” Opt. Express 18, 6461–6476 (2010).
    [CrossRef] [PubMed]
  11. J. W. Shaevitz and D. A. Fletcher, “Enhanced three-dimensional deconvolution microscopy using a measured depth-varying point-spread function,” J. Opt. Soc. Am. A 24, 2622–2627(2007).
    [CrossRef]
  12. S. F. Gibson and F. Lanni, “Diffraction by a circular aperture as a model for three-dimensional optical microscopy,” J. Opt. Soc. Am. A 6, 1357–1367 (1989).
    [CrossRef] [PubMed]
  13. P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  14. A. Chomik, A. Dieterlen, C. Xu, O. Haeberle, J. J. Meyer, and S. Jacquey, “Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation,” J. Opt. 28, 225–233 (1997).
    [CrossRef]
  15. A. Diaspro, F. Federici, and M. Robello, “Influence of refractive-index mismatch in high-resolution three-dimensional confocal microscopy,” Appl. Opt. 41, 685–690 (2002).
    [CrossRef] [PubMed]
  16. O. Haeberlé, F. Bicha, C. Simler, A. Dieterlen, C. Xu, B. Colicchio, S. Jacquey, and M.-P. Gramain, “Identification of acquisition parameters from the point spread function of a fluorescence microscope,” Opt. Commun. 196, 109–117 (2001).
    [CrossRef]
  17. F. Aguet, D. Van de ville, and M. Unser, “An accurate PSF model with few parameters for axially shift-variant deconvolution,” in IEEE International Symposium on Biomedical Imaging: From Nano to Macro (IEEE, 2008), pp. 157–160.
    [CrossRef]
  18. Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98, 3790–3795 (2001).
    [CrossRef] [PubMed]
  19. B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
    [CrossRef] [PubMed]
  20. J. B. de Monvel, E. Scarfone, S. Le Calvez, and M. Ulfendahl, “Image-adaptive deconvolution for three-dimensional deep biological imaging,” Biophys. J. 85, 3991–4001 (2003).
    [CrossRef] [PubMed]
  21. M. Von Tiedemann, A. Fridberger, M. Ulfendahl, I. Tomo, J. B. de Monvel, and J. B. De Monvel, “Image adaptive point-spread function estimation and deconvolution for in vivo confocal microscopy,” Microsc. Res. Tech. 69, 10–20 (2006).
    [CrossRef] [PubMed]
  22. S. X. Liao and M. Pawlak, “Image analysis with Zernike moment descriptors,” in 1997 IEEE Canadian Conference on Electrical and Computer Engineering (IEEE, 1997), Vol. 2, pp. 700–703.
  23. C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation invariants of Zernike moments,” Pattern Recogn. 36, 1765–1773(2003).
    [CrossRef]
  24. A. Dieterlen, A. De Meyer, P. Kessler, B. Colicchio, and J. De Mey, “On the use of Zernike polynomials in PSF processing: 4-D wide field fast microscopy images deconvolution,” presented at the International Conference Focus on Microscopy 2005, Jena, Germany (March 20–23, 2005).
  25. A. De Meyer, “Contribution à l’amélioration des outils de restauration d’image et de caractérisation de l’instrument en microscopie 3D par fluorescence,” Ph.D. dissertation (Université de Mulhouse, 2008).
  26. N. Becherer, H. Jodicke, G. Schlosser, J. Hesser, F. Zeilfelder, and R. Manner, “On soft clipping of Zernike moments for deblurring and enhancement of optical point spread functions,” Proc. SPIE 6065, 60650C-11 (2006).
    [CrossRef]
  27. O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: conventional microscopy,” Opt. Commun. 216, 55–63 (2003).
    [CrossRef]
  28. M. R. Teague, “Image analysis via the general theory of moments*,” J. Opt. Soc. Am. 70, 920–930 (1980).
    [CrossRef]
  29. M. Pawlak and S. X. Liao, “On the recovery of a function on a circular domain,” IEEE Trans. Inf. Theory 48, 2736–2753 (2002).
    [CrossRef]
  30. A.-R. Mohammed and J. Yang, “Practical fast computation of Zernike moments,” J. Comp. Sci. Technol. 17, 181–188 (2002).
    [CrossRef]
  31. J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Pattern Recogn. 35, 2905–2911 (2002).
    [CrossRef]

2010 (1)

2008 (1)

C. Vonesch and M. Unser, “A fast thresholded landweber algorithm for wavelet-regularized multidimensional deconvolution,” IEEE Trans. Image Process. 17, 539–549 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (3)

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006).
[CrossRef]

M. Von Tiedemann, A. Fridberger, M. Ulfendahl, I. Tomo, J. B. de Monvel, and J. B. De Monvel, “Image adaptive point-spread function estimation and deconvolution for in vivo confocal microscopy,” Microsc. Res. Tech. 69, 10–20 (2006).
[CrossRef] [PubMed]

N. Becherer, H. Jodicke, G. Schlosser, J. Hesser, F. Zeilfelder, and R. Manner, “On soft clipping of Zernike moments for deblurring and enhancement of optical point spread functions,” Proc. SPIE 6065, 60650C-11 (2006).
[CrossRef]

2004 (3)

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

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[CrossRef] [PubMed]

C. Preza and J.-A. Conchello, “Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 21, 1593–1601 (2004).
[CrossRef]

2003 (4)

J. B. de Monvel, E. Scarfone, S. Le Calvez, and M. Ulfendahl, “Image-adaptive deconvolution for three-dimensional deep biological imaging,” Biophys. J. 85, 3991–4001 (2003).
[CrossRef] [PubMed]

C. Preza and J.-A. Conchello, “Image estimation accounting for point-spread function depth-variation in three-dimensional fluorescence microscopy,” Proc. SPIE 4964, 135–142 (2003).
[CrossRef]

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: conventional microscopy,” Opt. Commun. 216, 55–63 (2003).
[CrossRef]

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation invariants of Zernike moments,” Pattern Recogn. 36, 1765–1773(2003).
[CrossRef]

2002 (4)

M. Pawlak and S. X. Liao, “On the recovery of a function on a circular domain,” IEEE Trans. Inf. Theory 48, 2736–2753 (2002).
[CrossRef]

A.-R. Mohammed and J. Yang, “Practical fast computation of Zernike moments,” J. Comp. Sci. Technol. 17, 181–188 (2002).
[CrossRef]

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Pattern Recogn. 35, 2905–2911 (2002).
[CrossRef]

A. Diaspro, F. Federici, and M. Robello, “Influence of refractive-index mismatch in high-resolution three-dimensional confocal microscopy,” Appl. Opt. 41, 685–690 (2002).
[CrossRef] [PubMed]

2001 (2)

O. Haeberlé, F. Bicha, C. Simler, A. Dieterlen, C. Xu, B. Colicchio, S. Jacquey, and M.-P. Gramain, “Identification of acquisition parameters from the point spread function of a fluorescence microscope,” Opt. Commun. 196, 109–117 (2001).
[CrossRef]

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98, 3790–3795 (2001).
[CrossRef] [PubMed]

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]

1997 (2)

J. G. Nagy and D. P. O’Leary, “Fast iterative image restoration with a spatially-varying PSF,” Proc. SPIE 3162, 388–399 (1997).
[CrossRef]

A. Chomik, A. Dieterlen, C. Xu, O. Haeberle, J. J. Meyer, and S. Jacquey, “Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation,” J. Opt. 28, 225–233 (1997).
[CrossRef]

1995 (1)

1994 (1)

1992 (1)

1989 (1)

1980 (1)

Agard, D. A.

M. Arigovindan, J. Shaevitz, J. McGowan, J. W. Sedat, and D. A. Agard, “A parallel product-convolution approach for representing the depth varying point spread functions in 3D widefield microscopy based on principalcomponent analysis,” Opt. Express 18, 6461–6476 (2010).
[CrossRef] [PubMed]

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[CrossRef] [PubMed]

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Aguet, F.

F. Aguet, D. Van de ville, and M. Unser, “An accurate PSF model with few parameters for axially shift-variant deconvolution,” in IEEE International Symposium on Biomedical Imaging: From Nano to Macro (IEEE, 2008), pp. 157–160.
[CrossRef]

Arigovindan, M.

Becherer, N.

N. Becherer, H. Jodicke, G. Schlosser, J. Hesser, F. Zeilfelder, and R. Manner, “On soft clipping of Zernike moments for deblurring and enhancement of optical point spread functions,” Proc. SPIE 6065, 60650C-11 (2006).
[CrossRef]

Bicha, F.

O. Haeberlé, F. Bicha, C. Simler, A. Dieterlen, C. Xu, B. Colicchio, S. Jacquey, and M.-P. Gramain, “Identification of acquisition parameters from the point spread function of a fluorescence microscope,” Opt. Commun. 196, 109–117 (2001).
[CrossRef]

Booker, G. R.

Chomik, A.

A. Chomik, A. Dieterlen, C. Xu, O. Haeberle, J. J. Meyer, and S. Jacquey, “Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation,” J. Opt. 28, 225–233 (1997).
[CrossRef]

Chong, C.-W.

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation invariants of Zernike moments,” Pattern Recogn. 36, 1765–1773(2003).
[CrossRef]

Colicchio, B.

O. Haeberlé, F. Bicha, C. Simler, A. Dieterlen, C. Xu, B. Colicchio, S. Jacquey, and M.-P. Gramain, “Identification of acquisition parameters from the point spread function of a fluorescence microscope,” Opt. Commun. 196, 109–117 (2001).
[CrossRef]

A. Dieterlen, A. De Meyer, P. Kessler, B. Colicchio, and J. De Mey, “On the use of Zernike polynomials in PSF processing: 4-D wide field fast microscopy images deconvolution,” presented at the International Conference Focus on Microscopy 2005, Jena, Germany (March 20–23, 2005).

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]

Conchello, J.-A.

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]

De Mey, J.

A. Dieterlen, A. De Meyer, P. Kessler, B. Colicchio, and J. De Mey, “On the use of Zernike polynomials in PSF processing: 4-D wide field fast microscopy images deconvolution,” presented at the International Conference Focus on Microscopy 2005, Jena, Germany (March 20–23, 2005).

De Meyer, A.

A. De Meyer, “Contribution à l’amélioration des outils de restauration d’image et de caractérisation de l’instrument en microscopie 3D par fluorescence,” Ph.D. dissertation (Université de Mulhouse, 2008).

A. Dieterlen, A. De Meyer, P. Kessler, B. Colicchio, and J. De Mey, “On the use of Zernike polynomials in PSF processing: 4-D wide field fast microscopy images deconvolution,” presented at the International Conference Focus on Microscopy 2005, Jena, Germany (March 20–23, 2005).

de Monvel, J. B.

M. Von Tiedemann, A. Fridberger, M. Ulfendahl, I. Tomo, J. B. de Monvel, and J. B. De Monvel, “Image adaptive point-spread function estimation and deconvolution for in vivo confocal microscopy,” Microsc. Res. Tech. 69, 10–20 (2006).
[CrossRef] [PubMed]

M. Von Tiedemann, A. Fridberger, M. Ulfendahl, I. Tomo, J. B. de Monvel, and J. B. De Monvel, “Image adaptive point-spread function estimation and deconvolution for in vivo confocal microscopy,” Microsc. Res. Tech. 69, 10–20 (2006).
[CrossRef] [PubMed]

J. B. de Monvel, E. Scarfone, S. Le Calvez, and M. Ulfendahl, “Image-adaptive deconvolution for three-dimensional deep biological imaging,” Biophys. J. 85, 3991–4001 (2003).
[CrossRef] [PubMed]

Diaspro, A.

Dieterlen, A.

O. Haeberlé, F. Bicha, C. Simler, A. Dieterlen, C. Xu, B. Colicchio, S. Jacquey, and M.-P. Gramain, “Identification of acquisition parameters from the point spread function of a fluorescence microscope,” Opt. Commun. 196, 109–117 (2001).
[CrossRef]

A. Chomik, A. Dieterlen, C. Xu, O. Haeberle, J. J. Meyer, and S. Jacquey, “Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation,” J. Opt. 28, 225–233 (1997).
[CrossRef]

A. Dieterlen, A. De Meyer, P. Kessler, B. Colicchio, and J. De Mey, “On the use of Zernike polynomials in PSF processing: 4-D wide field fast microscopy images deconvolution,” presented at the International Conference Focus on Microscopy 2005, Jena, Germany (March 20–23, 2005).

Federici, F.

Fletcher, D. A.

Fridberger, A.

M. Von Tiedemann, A. Fridberger, M. Ulfendahl, I. Tomo, J. B. de Monvel, and J. B. De Monvel, “Image adaptive point-spread function estimation and deconvolution for in vivo confocal microscopy,” Microsc. Res. Tech. 69, 10–20 (2006).
[CrossRef] [PubMed]

Gibson, S. F.

Gramain, M.-P.

O. Haeberlé, F. Bicha, C. Simler, A. Dieterlen, C. Xu, B. Colicchio, S. Jacquey, and M.-P. Gramain, “Identification of acquisition parameters from the point spread function of a fluorescence microscope,” Opt. Commun. 196, 109–117 (2001).
[CrossRef]

Gu, J.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Pattern Recogn. 35, 2905–2911 (2002).
[CrossRef]

Gustafsson, M. G. L.

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[CrossRef] [PubMed]

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Haeberle, O.

A. Chomik, A. Dieterlen, C. Xu, O. Haeberle, J. J. Meyer, and S. Jacquey, “Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation,” J. Opt. 28, 225–233 (1997).
[CrossRef]

Haeberlé, O.

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: conventional microscopy,” Opt. Commun. 216, 55–63 (2003).
[CrossRef]

O. Haeberlé, F. Bicha, C. Simler, A. Dieterlen, C. Xu, B. Colicchio, S. Jacquey, and M.-P. Gramain, “Identification of acquisition parameters from the point spread function of a fluorescence microscope,” Opt. Commun. 196, 109–117 (2001).
[CrossRef]

Hanser, B.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Hanser, B. M.

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[CrossRef] [PubMed]

Hesser, J.

N. Becherer, H. Jodicke, G. Schlosser, J. Hesser, F. Zeilfelder, and R. Manner, “On soft clipping of Zernike moments for deblurring and enhancement of optical point spread functions,” Proc. SPIE 6065, 60650C-11 (2006).
[CrossRef]

Jacquey, S.

O. Haeberlé, F. Bicha, C. Simler, A. Dieterlen, C. Xu, B. Colicchio, S. Jacquey, and M.-P. Gramain, “Identification of acquisition parameters from the point spread function of a fluorescence microscope,” Opt. Commun. 196, 109–117 (2001).
[CrossRef]

A. Chomik, A. Dieterlen, C. Xu, O. Haeberle, J. J. Meyer, and S. Jacquey, “Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation,” J. Opt. 28, 225–233 (1997).
[CrossRef]

Jodicke, H.

N. Becherer, H. Jodicke, G. Schlosser, J. Hesser, F. Zeilfelder, and R. Manner, “On soft clipping of Zernike moments for deblurring and enhancement of optical point spread functions,” Proc. SPIE 6065, 60650C-11 (2006).
[CrossRef]

Kam, Z.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98, 3790–3795 (2001).
[CrossRef] [PubMed]

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]

Kessler, P.

A. Dieterlen, A. De Meyer, P. Kessler, B. Colicchio, and J. De Mey, “On the use of Zernike polynomials in PSF processing: 4-D wide field fast microscopy images deconvolution,” presented at the International Conference Focus on Microscopy 2005, Jena, Germany (March 20–23, 2005).

Laczik, Z.

Lanni, F.

Le Calvez, S.

J. B. de Monvel, E. Scarfone, S. Le Calvez, and M. Ulfendahl, “Image-adaptive deconvolution for three-dimensional deep biological imaging,” Biophys. J. 85, 3991–4001 (2003).
[CrossRef] [PubMed]

Liao, S. X.

M. Pawlak and S. X. Liao, “On the recovery of a function on a circular domain,” IEEE Trans. Inf. Theory 48, 2736–2753 (2002).
[CrossRef]

S. X. Liao and M. Pawlak, “Image analysis with Zernike moment descriptors,” in 1997 IEEE Canadian Conference on Electrical and Computer Engineering (IEEE, 1997), Vol. 2, pp. 700–703.

Luo, L. M.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Pattern Recogn. 35, 2905–2911 (2002).
[CrossRef]

Manner, R.

N. Becherer, H. Jodicke, G. Schlosser, J. Hesser, F. Zeilfelder, and R. Manner, “On soft clipping of Zernike moments for deblurring and enhancement of optical point spread functions,” Proc. SPIE 6065, 60650C-11 (2006).
[CrossRef]

McGowan, J.

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]

J. G. McNally, C. Preza, J.-A. Conchello, and L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A 11, 1056–1067 (1994).
[CrossRef]

Meyer, J. J.

A. Chomik, A. Dieterlen, C. Xu, O. Haeberle, J. J. Meyer, and S. Jacquey, “Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation,” J. Opt. 28, 225–233 (1997).
[CrossRef]

Mohammed, A.-R.

A.-R. Mohammed and J. Yang, “Practical fast computation of Zernike moments,” J. Comp. Sci. Technol. 17, 181–188 (2002).
[CrossRef]

Mukundan, R.

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation invariants of Zernike moments,” Pattern Recogn. 36, 1765–1773(2003).
[CrossRef]

Nagy, J. G.

J. G. Nagy and D. P. O’Leary, “Fast iterative image restoration with a spatially-varying PSF,” Proc. SPIE 3162, 388–399 (1997).
[CrossRef]

Neelamani, R.

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

Nehorai, A.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006).
[CrossRef]

O’Leary, D. P.

J. G. Nagy and D. P. O’Leary, “Fast iterative image restoration with a spatially-varying PSF,” Proc. SPIE 3162, 388–399 (1997).
[CrossRef]

Pawlak, M.

M. Pawlak and S. X. Liao, “On the recovery of a function on a circular domain,” IEEE Trans. Inf. Theory 48, 2736–2753 (2002).
[CrossRef]

S. X. Liao and M. Pawlak, “Image analysis with Zernike moment descriptors,” in 1997 IEEE Canadian Conference on Electrical and Computer Engineering (IEEE, 1997), Vol. 2, pp. 700–703.

Preza, C.

Raveendran, P.

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation invariants of Zernike moments,” Pattern Recogn. 36, 1765–1773(2003).
[CrossRef]

Robello, M.

Sarder, P.

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006).
[CrossRef]

Scarfone, E.

J. B. de Monvel, E. Scarfone, S. Le Calvez, and M. Ulfendahl, “Image-adaptive deconvolution for three-dimensional deep biological imaging,” Biophys. J. 85, 3991–4001 (2003).
[CrossRef] [PubMed]

Schlosser, G.

N. Becherer, H. Jodicke, G. Schlosser, J. Hesser, F. Zeilfelder, and R. Manner, “On soft clipping of Zernike moments for deblurring and enhancement of optical point spread functions,” Proc. SPIE 6065, 60650C-11 (2006).
[CrossRef]

Sedat, J. W.

M. Arigovindan, J. Shaevitz, J. McGowan, J. W. Sedat, and D. A. Agard, “A parallel product-convolution approach for representing the depth varying point spread functions in 3D widefield microscopy based on principalcomponent analysis,” Opt. Express 18, 6461–6476 (2010).
[CrossRef] [PubMed]

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[CrossRef] [PubMed]

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Shaevitz, J.

Shaevitz, J. W.

Shu, H. Z.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Pattern Recogn. 35, 2905–2911 (2002).
[CrossRef]

Simler, C.

O. Haeberlé, F. Bicha, C. Simler, A. Dieterlen, C. Xu, B. Colicchio, S. Jacquey, and M.-P. Gramain, “Identification of acquisition parameters from the point spread function of a fluorescence microscope,” Opt. Commun. 196, 109–117 (2001).
[CrossRef]

Teague, M. R.

Thomas, L. J.

Tomo, I.

M. Von Tiedemann, A. Fridberger, M. Ulfendahl, I. Tomo, J. B. de Monvel, and J. B. De Monvel, “Image adaptive point-spread function estimation and deconvolution for in vivo confocal microscopy,” Microsc. Res. Tech. 69, 10–20 (2006).
[CrossRef] [PubMed]

Török, P.

Toumoulin, C.

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Pattern Recogn. 35, 2905–2911 (2002).
[CrossRef]

Ulfendahl, M.

M. Von Tiedemann, A. Fridberger, M. Ulfendahl, I. Tomo, J. B. de Monvel, and J. B. De Monvel, “Image adaptive point-spread function estimation and deconvolution for in vivo confocal microscopy,” Microsc. Res. Tech. 69, 10–20 (2006).
[CrossRef] [PubMed]

J. B. de Monvel, E. Scarfone, S. Le Calvez, and M. Ulfendahl, “Image-adaptive deconvolution for three-dimensional deep biological imaging,” Biophys. J. 85, 3991–4001 (2003).
[CrossRef] [PubMed]

Unser, M.

C. Vonesch and M. Unser, “A fast thresholded landweber algorithm for wavelet-regularized multidimensional deconvolution,” IEEE Trans. Image Process. 17, 539–549 (2008).
[CrossRef] [PubMed]

F. Aguet, D. Van de ville, and M. Unser, “An accurate PSF model with few parameters for axially shift-variant deconvolution,” in IEEE International Symposium on Biomedical Imaging: From Nano to Macro (IEEE, 2008), pp. 157–160.
[CrossRef]

Van de ville, D.

F. Aguet, D. Van de ville, and M. Unser, “An accurate PSF model with few parameters for axially shift-variant deconvolution,” in IEEE International Symposium on Biomedical Imaging: From Nano to Macro (IEEE, 2008), pp. 157–160.
[CrossRef]

Varga, P.

Von Tiedemann, M.

M. Von Tiedemann, A. Fridberger, M. Ulfendahl, I. Tomo, J. B. de Monvel, and J. B. De Monvel, “Image adaptive point-spread function estimation and deconvolution for in vivo confocal microscopy,” Microsc. Res. Tech. 69, 10–20 (2006).
[CrossRef] [PubMed]

Vonesch, C.

C. Vonesch and M. Unser, “A fast thresholded landweber algorithm for wavelet-regularized multidimensional deconvolution,” IEEE Trans. Image Process. 17, 539–549 (2008).
[CrossRef] [PubMed]

Xu, C.

O. Haeberlé, F. Bicha, C. Simler, A. Dieterlen, C. Xu, B. Colicchio, S. Jacquey, and M.-P. Gramain, “Identification of acquisition parameters from the point spread function of a fluorescence microscope,” Opt. Commun. 196, 109–117 (2001).
[CrossRef]

A. Chomik, A. Dieterlen, C. Xu, O. Haeberle, J. J. Meyer, and S. Jacquey, “Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation,” J. Opt. 28, 225–233 (1997).
[CrossRef]

Yang, J.

A.-R. Mohammed and J. Yang, “Practical fast computation of Zernike moments,” J. Comp. Sci. Technol. 17, 181–188 (2002).
[CrossRef]

Zeilfelder, F.

N. Becherer, H. Jodicke, G. Schlosser, J. Hesser, F. Zeilfelder, and R. Manner, “On soft clipping of Zernike moments for deblurring and enhancement of optical point spread functions,” Proc. SPIE 6065, 60650C-11 (2006).
[CrossRef]

Appl. Opt. (1)

Biophys. J. (1)

J. B. de Monvel, E. Scarfone, S. Le Calvez, and M. Ulfendahl, “Image-adaptive deconvolution for three-dimensional deep biological imaging,” Biophys. J. 85, 3991–4001 (2003).
[CrossRef] [PubMed]

IEEE Signal Process. Mag. (1)

P. Sarder and A. Nehorai, “Deconvolution methods for 3-D fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006).
[CrossRef]

IEEE Trans. Image Process. (1)

C. Vonesch and M. Unser, “A fast thresholded landweber algorithm for wavelet-regularized multidimensional deconvolution,” IEEE Trans. Image Process. 17, 539–549 (2008).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

M. Pawlak and S. X. Liao, “On the recovery of a function on a circular domain,” IEEE Trans. Inf. Theory 48, 2736–2753 (2002).
[CrossRef]

IEEE Trans. Signal Process. (1)

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

J. Comp. Sci. Technol. (1)

A.-R. Mohammed and J. Yang, “Practical fast computation of Zernike moments,” J. Comp. Sci. Technol. 17, 181–188 (2002).
[CrossRef]

J. Microsc. (1)

B. M. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Phase-retrieved pupil functions in wide-field fluorescence microscopy,” J. Microsc. 216, 32–48 (2004).
[CrossRef] [PubMed]

J. Opt. (1)

A. Chomik, A. Dieterlen, C. Xu, O. Haeberle, J. J. Meyer, and S. Jacquey, “Quantification in optical sectioning microscopy: a comparison of some deconvolution algorithms in view of 3D image segmentation,” J. Opt. 28, 225–233 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Methods (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]

Microsc. Res. Tech. (1)

M. Von Tiedemann, A. Fridberger, M. Ulfendahl, I. Tomo, J. B. de Monvel, and J. B. De Monvel, “Image adaptive point-spread function estimation and deconvolution for in vivo confocal microscopy,” Microsc. Res. Tech. 69, 10–20 (2006).
[CrossRef] [PubMed]

Opt. Commun. (2)

O. Haeberlé, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: conventional microscopy,” Opt. Commun. 216, 55–63 (2003).
[CrossRef]

O. Haeberlé, F. Bicha, C. Simler, A. Dieterlen, C. Xu, B. Colicchio, S. Jacquey, and M.-P. Gramain, “Identification of acquisition parameters from the point spread function of a fluorescence microscope,” Opt. Commun. 196, 109–117 (2001).
[CrossRef]

Opt. Express (1)

Pattern Recogn. (2)

J. Gu, H. Z. Shu, C. Toumoulin, and L. M. Luo, “A novel algorithm for fast computation of Zernike moments,” Pattern Recogn. 35, 2905–2911 (2002).
[CrossRef]

C.-W. Chong, P. Raveendran, and R. Mukundan, “Translation invariants of Zernike moments,” Pattern Recogn. 36, 1765–1773(2003).
[CrossRef]

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

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, and J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. U.S.A. 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Proc. SPIE (3)

J. G. Nagy and D. P. O’Leary, “Fast iterative image restoration with a spatially-varying PSF,” Proc. SPIE 3162, 388–399 (1997).
[CrossRef]

C. Preza and J.-A. Conchello, “Image estimation accounting for point-spread function depth-variation in three-dimensional fluorescence microscopy,” Proc. SPIE 4964, 135–142 (2003).
[CrossRef]

N. Becherer, H. Jodicke, G. Schlosser, J. Hesser, F. Zeilfelder, and R. Manner, “On soft clipping of Zernike moments for deblurring and enhancement of optical point spread functions,” Proc. SPIE 6065, 60650C-11 (2006).
[CrossRef]

Other (4)

A. Dieterlen, A. De Meyer, P. Kessler, B. Colicchio, and J. De Mey, “On the use of Zernike polynomials in PSF processing: 4-D wide field fast microscopy images deconvolution,” presented at the International Conference Focus on Microscopy 2005, Jena, Germany (March 20–23, 2005).

A. De Meyer, “Contribution à l’amélioration des outils de restauration d’image et de caractérisation de l’instrument en microscopie 3D par fluorescence,” Ph.D. dissertation (Université de Mulhouse, 2008).

S. X. Liao and M. Pawlak, “Image analysis with Zernike moment descriptors,” in 1997 IEEE Canadian Conference on Electrical and Computer Engineering (IEEE, 1997), Vol. 2, pp. 700–703.

F. Aguet, D. Van de ville, and M. Unser, “An accurate PSF model with few parameters for axially shift-variant deconvolution,” in IEEE International Symposium on Biomedical Imaging: From Nano to Macro (IEEE, 2008), pp. 157–160.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Mapping transform between the Cartesian coordinates of an image to the polar coordinates inside a unit circle.

Fig. 2
Fig. 2

3D PSF representation—all slices are centered the same way in the unit circle (the center is calculated using the Airy disk).

Fig. 3
Fig. 3

Four Zernike moments variations over the 17th plane and the 32nd, along the depth ( 0 μm 15.75 μm ).

Fig. 4
Fig. 4

Original (calculated) PSF at 0.5 μm of depth and the reconstructed one using pseudo-3D Zernike with an order of 45. The contrast was deliberately modified in order to highlight low intensity structures.

Fig. 5
Fig. 5

Original (calculated) PSF at 10 μm of depth and the reconstructed one using pseudo-3D Zernike with 45 orders. The contrast was deliberately modified in order to highlight low intensity structures.

Fig. 6
Fig. 6

Correlation coefficient as a function of PSF depth, between the interpolated PSF (using six PSF, noise-free and a SNR of 30, 20 dB) at 0, 3, 6, 9, 12, and 15.75 μm ) and the calculated one.

Fig. 7
Fig. 7

Interpolated PSF at 7 μm using six known PSFs, from left to right: noise-free, 20 dB Gaussian noise (the originals are given as comparison references).

Fig. 8
Fig. 8

Extracted PSF at a relative depth of (a) 4 μm and (b) 25.75 μm .

Fig. 9
Fig. 9

Interpolated PSF at a relative depth of (b) 18.25 μm compared to (a) the measured one at the same position .

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

Φ p q = ς Ψ p q ( x , y ) f ( x , y ) d x d y , p , q = 0 , 1 , 2 , 3 .
A p q = 1 + p π 0 1 π π f ( ρ , θ ) [ V p q ( ρ , θ ) ] ρ d ρ d θ { p = 0 , 1 , 2 , q Z p | q | is even , | q | < p .
V p q ( ρ , θ ) = R p q ( ρ ) e i q θ ,
R p q ( ρ ) = s = 0 ( p | q | ) / 2 ( 1 ) s ( p s ) ! s ! ( p + | q | 2 s ) ! ( p | q | 2 s ) ! ρ p 2 s .
A p q = τ ( N , p ) x = 0 N 1 y = 0 N 1 f ( ρ , θ ) V ( ρ , θ ) { ρ = x 2 + y 2 1 } .
f ( ρ , θ ) = p = 0 q = 0 p A p q V p q ( ρ , θ ) ,
f ͡ ( ρ , θ ) = p = 0 p max q = 0 p A p q V p q ( ρ , θ ) .
f ͡ ( ρ , θ ) = p = 0 N max q > 0 p ( C p q cos q θ + S p q sin q θ ) R p q ( ρ ) + C p 0 2 R p 0 ( ρ ) ,
C p q = 2 Re ( A p q ) = 2 p + 2 π 0 1 π π f ( ρ , θ ) R p q ( ρ ) cos ( q θ ) ρ d ρ d θ S p q = 2 Im ( A p q ) = 2 p 2 π 0 1 π π f ( ρ , θ ) R p q ( ρ ) sin ( q θ ) ρ d ρ d θ .
C p q n = 2 Re ( A p q n ) = 2 p + 2 π x y f ( ρ , θ ) n R ( ρ ) cos ( q θ ) , S p q n = 2 Im ( A p q n ) = 2 p 2 π x y f ( ρ , θ ) n R ( ρ ) sin ( q θ ) ,
f ͡ n ( ρ , θ ) = p = 0 P max q > 0 p ( C p q n cos ( q θ ) + S p q n sin ( q θ ) ) R p q ( ρ ) + C p 0 n 2 R p 0 ( ρ ) ,
Z k = [ B 00 k B p q k ] = [ A 00 1 A p q 1 A 00 n A p q n ] .
A p q n = 1 2 C p q n .
f ͡ n ( ρ , θ ) = p = 1 N max q > 0 p C p q n cos ( q θ ) R p q ( ρ ) + C p 0 n 2 R p 0 ( ρ ) .
V a n ( i ) = M Z ( a , n , i ) { 1 i K } .
a = { ( p 1 2 + 1 ) ( p 1 2 + 2 ) + q 2 + 1 p   is   even ( p 2 2 + 1 ) ( p 2 2 + 2 ) + p 1 2 + q 2 + 2 p   is   odd .
M Z ( a , n , k ) Poly   Fit V Z ( a , n ) = ( p 00 p a 0 p 0 n p a n ) ,
r p = i = 0 i < N ( X i X ¯ ) ( Y i Y ¯ ) i = 0 i < N ( X i X ¯ ) 2 i = 0 i < N ( Y i Y ¯ ) 2 .

Metrics