Abstract

To remove the axial sidelobes from 4Pi images, deconvolution forms an integral part of 4Pi microscopy. As a result of its high axial resolution, the 4Pi point spread function (PSF) is particularly susceptible to imperfect optical conditions within the sample. This is typically observed as a shift in the position of the maxima under the PSF envelope. A significantly varying phase shift renders deconvolution procedures based on a spatially invariant PSF essentially useless. We present a technique for computing the forward transformation in the case of a varying phase at a computational expense of the same order of magnitude as that of the shift invariant case, a method for the estimation of PSF phase from an acquired image, and a deconvolution procedure built on these techniques.

© 2006 Optical Society of America

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References

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  1. S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
    [CrossRef]
  2. P. E. Hänninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, "Two-photon excitation 4Pi confocal microscope: Enhanced axial resolution microscope for biological research," Appl. Phys. Lett. 66, 1698-1700 (1995).
    [CrossRef]
  3. 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, 36-48 (2001).
    [CrossRef]
  4. M. Schrader, M. Kozubek, S. W. Hell, and T. Wilson, "Optical transfer functions of 4Pi confocal microscopes: theory and experiment," Opt. Lett. 22, 436-438 (1997).
    [CrossRef] [PubMed]
  5. M. Schrader, S. W. Hell, and H. T. M. van der Voort, "Three-dimensional superresolution with a 4Pi-confocal microscope using image restoration," J. Appl. Phys. 84, 4033-4042 (1998).
    [CrossRef]
  6. 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, 211-217 (1998).
    [CrossRef]
  7. M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal imaging in fixed biological specimens," Biophys. J. 75, 1659-1668 (1998).
    [CrossRef] [PubMed]
  8. S. W. Hell, C. M. Blanca, and J. Bewersdorf, "Phase determination in interference-based superresolving microscopes through critical frequency analysis," Opt. Lett. 27, 888-890 (2002).
    [CrossRef]
  9. 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, 281-285 (2002).
    [CrossRef]
  10. W. H. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. Am. 62, 55-59 (1972).
    [CrossRef]
  11. S. M. Tan, "Aperture synthesis mapping and parameter estimation," Ph.D. dissertation (Mullard Radio Astronomy Observatory, Cavendish Laboratory, Cambridge, 1987).
  12. S. M. Tan, C. Fox, and G. K. Nicholls, "Physics 707 Inverse Problems (Course Notes)," http://www.math.auckland.ac.nz/∼phy707/.
  13. R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
    [CrossRef]
  14. R. Heintzmann, "Resolution Enhancement of Biological Light Microscopic Data," Ph.D. dissertation (University of Heidelberg 1999).
  15. C. Stockklausner and N. Klöcker, "Surface expression of inward rectifier potassium channels is controlled by selective Golgi export," J. Biol. Chem. 278, 17000-17005 (2003).
    [CrossRef] [PubMed]

2003 (1)

C. Stockklausner and N. Klöcker, "Surface expression of inward rectifier potassium channels is controlled by selective Golgi export," J. Biol. Chem. 278, 17000-17005 (2003).
[CrossRef] [PubMed]

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, 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, 888-890 (2002).
[CrossRef]

2001 (1)

1998 (3)

M. Schrader, S. W. Hell, and H. T. M. van der Voort, "Three-dimensional superresolution with a 4Pi-confocal microscope using image restoration," J. Appl. Phys. 84, 4033-4042 (1998).
[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, 211-217 (1998).
[CrossRef]

M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal imaging in fixed biological specimens," Biophys. J. 75, 1659-1668 (1998).
[CrossRef] [PubMed]

1997 (1)

1995 (1)

P. E. Hänninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, "Two-photon excitation 4Pi confocal microscope: Enhanced axial resolution microscope for biological research," Appl. Phys. Lett. 66, 1698-1700 (1995).
[CrossRef]

1994 (1)

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
[CrossRef]

1972 (1)

Bahlmann, K.

M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal imaging in fixed biological specimens," Biophys. J. 75, 1659-1668 (1998).
[CrossRef] [PubMed]

Barrett, R.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Berry, M.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Bewersdorf, J.

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, 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, 888-890 (2002).
[CrossRef]

Blanca, C. M.

S. W. Hell, C. M. Blanca, and J. Bewersdorf, "Phase determination in interference-based superresolving microscopes through critical frequency analysis," Opt. Lett. 27, 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, 281-285 (2002).
[CrossRef]

Chan, T. F.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Cremer, C.

P. E. Hänninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, "Two-photon excitation 4Pi confocal microscope: Enhanced axial resolution microscope for biological research," Appl. Phys. Lett. 66, 1698-1700 (1995).
[CrossRef]

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
[CrossRef]

Demmel, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

der Vorst, H. V.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Donato, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Dongarra, J.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Egner, A.

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, 211-217 (1998).
[CrossRef]

Eijkhout, V.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Fox, C.

S. M. Tan, C. Fox, and G. K. Nicholls, "Physics 707 Inverse Problems (Course Notes)," http://www.math.auckland.ac.nz/∼phy707/.

Giese, G.

M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal imaging in fixed biological specimens," Biophys. J. 75, 1659-1668 (1998).
[CrossRef] [PubMed]

Hänninen, P. E.

P. E. Hänninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, "Two-photon excitation 4Pi confocal microscope: Enhanced axial resolution microscope for biological research," Appl. Phys. Lett. 66, 1698-1700 (1995).
[CrossRef]

Heintzmann, R.

R. Heintzmann, "Resolution Enhancement of Biological Light Microscopic Data," Ph.D. dissertation (University of Heidelberg 1999).

Hell, S. W.

S. W. Hell, C. M. Blanca, and J. Bewersdorf, "Phase determination in interference-based superresolving microscopes through critical frequency analysis," Opt. Lett. 27, 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, 281-285 (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, 36-48 (2001).
[CrossRef]

M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal imaging in fixed biological specimens," Biophys. J. 75, 1659-1668 (1998).
[CrossRef] [PubMed]

M. Schrader, S. W. Hell, and H. T. M. van der Voort, "Three-dimensional superresolution with a 4Pi-confocal microscope using image restoration," J. Appl. Phys. 84, 4033-4042 (1998).
[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, 211-217 (1998).
[CrossRef]

M. Schrader, M. Kozubek, S. W. Hell, and T. Wilson, "Optical transfer functions of 4Pi confocal microscopes: theory and experiment," Opt. Lett. 22, 436-438 (1997).
[CrossRef] [PubMed]

P. E. Hänninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, "Two-photon excitation 4Pi confocal microscope: Enhanced axial resolution microscope for biological research," Appl. Phys. Lett. 66, 1698-1700 (1995).
[CrossRef]

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
[CrossRef]

Klöcker, N.

C. Stockklausner and N. Klöcker, "Surface expression of inward rectifier potassium channels is controlled by selective Golgi export," J. Biol. Chem. 278, 17000-17005 (2003).
[CrossRef] [PubMed]

Kozubek, M.

Lindek, S.

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
[CrossRef]

Nagorni, M.

Nicholls, G. K.

S. M. Tan, C. Fox, and G. K. Nicholls, "Physics 707 Inverse Problems (Course Notes)," http://www.math.auckland.ac.nz/∼phy707/.

Pozo, R.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Richardson, W. H.

Romine, C.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

Salo, J.

P. E. Hänninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, "Two-photon excitation 4Pi confocal microscope: Enhanced axial resolution microscope for biological research," Appl. Phys. Lett. 66, 1698-1700 (1995).
[CrossRef]

Schrader, M.

M. Schrader, S. W. Hell, and H. T. M. van der Voort, "Three-dimensional superresolution with a 4Pi-confocal microscope using image restoration," J. Appl. Phys. 84, 4033-4042 (1998).
[CrossRef]

M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal imaging in fixed biological specimens," Biophys. J. 75, 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, 211-217 (1998).
[CrossRef]

M. Schrader, M. Kozubek, S. W. Hell, and T. Wilson, "Optical transfer functions of 4Pi confocal microscopes: theory and experiment," Opt. Lett. 22, 436-438 (1997).
[CrossRef] [PubMed]

Soini, E.

P. E. Hänninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, "Two-photon excitation 4Pi confocal microscope: Enhanced axial resolution microscope for biological research," Appl. Phys. Lett. 66, 1698-1700 (1995).
[CrossRef]

Stelzer, E. H. K.

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
[CrossRef]

Stockklausner, C.

C. Stockklausner and N. Klöcker, "Surface expression of inward rectifier potassium channels is controlled by selective Golgi export," J. Biol. Chem. 278, 17000-17005 (2003).
[CrossRef] [PubMed]

Tan, S. M.

S. M. Tan, "Aperture synthesis mapping and parameter estimation," Ph.D. dissertation (Mullard Radio Astronomy Observatory, Cavendish Laboratory, Cambridge, 1987).

S. M. Tan, C. Fox, and G. K. Nicholls, "Physics 707 Inverse Problems (Course Notes)," http://www.math.auckland.ac.nz/∼phy707/.

van der Voort, H. T. M.

M. Schrader, S. W. Hell, and H. T. M. van der Voort, "Three-dimensional superresolution with a 4Pi-confocal microscope using image restoration," J. Appl. Phys. 84, 4033-4042 (1998).
[CrossRef]

Wilson, T.

Appl. Phys. Lett. (2)

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
[CrossRef]

P. E. Hänninen, S. W. Hell, J. Salo, E. Soini, and C. Cremer, "Two-photon excitation 4Pi confocal microscope: Enhanced axial resolution microscope for biological research," Appl. Phys. Lett. 66, 1698-1700 (1995).
[CrossRef]

Biophys. J. (1)

M. Schrader, K. Bahlmann, G. Giese, and S. W. Hell, "4Pi-Confocal imaging in fixed biological specimens," Biophys. J. 75, 1659-1668 (1998).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

M. Schrader, S. W. Hell, and H. T. M. van der Voort, "Three-dimensional superresolution with a 4Pi-confocal microscope using image restoration," J. Appl. Phys. 84, 4033-4042 (1998).
[CrossRef]

J. Biol. Chem. (1)

C. Stockklausner and N. Klöcker, "Surface expression of inward rectifier potassium channels is controlled by selective Golgi export," J. Biol. Chem. 278, 17000-17005 (2003).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (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, 211-217 (1998).
[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, 281-285 (2002).
[CrossRef]

Opt. Lett. (2)

Other (4)

S. M. Tan, "Aperture synthesis mapping and parameter estimation," Ph.D. dissertation (Mullard Radio Astronomy Observatory, Cavendish Laboratory, Cambridge, 1987).

S. M. Tan, C. Fox, and G. K. Nicholls, "Physics 707 Inverse Problems (Course Notes)," http://www.math.auckland.ac.nz/∼phy707/.

R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd ed. (SIAM, 1994).
[CrossRef]

R. Heintzmann, "Resolution Enhancement of Biological Light Microscopic Data," Ph.D. dissertation (University of Heidelberg 1999).

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Figures (12)

Fig. 1
Fig. 1

(Color online) Schematic representation of a z-axis profile through a two photon 4Pi A PSF showing the effect of a varying phase. This is seen as a shift of the interference fringes with respect to the maximum of the envelope.

Fig. 2
Fig. 2

Line scan along z axis through simulated test image showing the principles of the phase estimation procedure. A cross correlation is performed between (A) the raw data and the PSF envelope (i.e., the appropriate confocal PSF) giving (B) the profile d env . From the peaks in this image, the centers of all objects are estimated (as shown by the vertical lines). The raw data are additionally correlated with I env cos ( 4 πnz λ ex ) to extract the fringe position [as shown in profile (C)]. To verify that the reference point belongs to an isolated object, the relative heights of the peaks in the low pass filtered image (D) are analyzed. In the displayed case, the second object from the left would be rejected. Note that the cross correlations∕filtering were carried out on the xz image, resulting in a better signal to noise ratio than would be expected from the 1D filtering of the data shown.

Fig. 3
Fig. 3

Simulated test pattern.

Fig. 4
Fig. 4

Simulated phase map.

Fig. 5
Fig. 5

Two photon 4Pi A image generated from test pattern and phase map with Poisson noise.

Fig. 6
Fig. 6

(Color online) Reconstructed phase map showing automatically selected control points (crosses).

Fig. 7
Fig. 7

Reconstructed image.

Fig. 8
Fig. 8

(Color online) Line profiles through the test object, simulated image, and deconvolution result.

Fig. 9
Fig. 9

xz slice at y = 2 through raw Kir 2.1 ion channel data after Gaussian blur for noise reduction. Control points for phase estimation were manually selected at points in three different y slices. The location of these control points is indicated on the image. The dynamic range was compressed to improve contrast by saturating the bright spot.

Fig. 10
Fig. 10

xz slice at y = 2 through Kir 2.1 ion channel image reconstructed with our ML algorithm and the phase set to zero over the whole image. This represents what would be obtained from a conventional ML deconvolution algorithm. The dynamic range was compressed to improve contrast by saturating the bright spot.

Fig. 11
Fig. 11

xz slice at y = 2 through Kir 2.1 ion channel image reconstructed with our ML algorithm and a phase map interpolated from 20 control points selected on the cell membrane. The dynamic range was compressed to improve contrast by saturating the bright spot.

Fig. 12
Fig. 12

(Color online) Line profiles taken vertically through the raw data and deconvolution results with and without phase correction.

Tables (1)

Tables Icon

Table 1 Correlation of Simulated Data and Reconstructions With and Without Phase Compensation to the Test Pattern Used to Generate the Simulated Data a

Equations (6)

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

I PSF = I env ( x , y , z ) I fringes ( x , y , z ) .
I PSF I env ( r ) cos 4 [ 2 πnz λ ex + ϕ ( r ) ] .
d ( r ) = - f ( s ) I PSF ( r - s , r ) d s x d s y d s z ,
I PSF I env 1 4 [ 3 2 + 2 cos ( 2 ϕ ) cos ( 4 πnz λ ex ) - 2 sin ( 2 ϕ ) sin ( 4 πnz λ ex ) + 1 2 cos ( 4 ϕ ) cos ( 8 πnz λ ex ) - 1 2 sin ( 4 ϕ ) sin ( 8 πnz λ ex ) ] .
d 1 4 ( 3 2 I env f + 2 cos ( 2 ϕ ) { [ cos ( 4 πnz λ ex ) I env ] f } 2 sin ( 2 ϕ ) { [ sin ( 4 πnz λ ex ) I env ] f } + 1 2 cos ( 4 ϕ ) { [ cos ( 8 πnz λ ex ) I env ] f } 1 2 sin ( 4 ϕ ) { [ sin ( 8 πnz λ ex ) I env ] f } ) ,
arg f ^ min [ W ( d - H f ^ ) 2 + λ 2 L ( f ^ - f def ) 2 ]

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