Abstract

An approach to achieve superresolution in confocal scanning microscopy by using the singular-system approach to the inverse problem was recently proposed. It consists of using a specially designed mask that performs the task of data inversion by means of all-optical processing. We discuss an approximate binary form of such a mask that permits its practical manufacture for use in incoherent confocal microscopy. The performance characteristics of such an approximate mask are compared with those of an exact mask and with those of a conventional confocal scanning microscope. Although the resolution of the approximate microscope is slightly inferior to the exact one, they are both still advances over the conventional confocal one (the improvement of resolution being 65% and 70%, respectively).

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 1, 59–64 (1993).
    [CrossRef]
  2. M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks.” Inverse Probl. 8, 1–23 (1992).
    [CrossRef]
  3. M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of super-resolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
    [CrossRef]
  4. M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
    [CrossRef]
  5. M. Bertero, P. Boccacci, R. E. Davies, E. R. Pike, “Superresolution in confocal scanning microscopy: III. The case of circular pupils,” Inverse Probl. 7, 655–674 (1991).
    [CrossRef]
  6. E. R. Pike, “Inverse problems in confocal microscopy,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, Bristol, 1992), pp. 164–179.
  7. M. Bertero, P. Boccacci, “Computation of the singular system for a class of integral operators related to data inversion in confocal microscopy,” Inverse Probl. 5, 935–957 (1989).
    [CrossRef]
  8. T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

1993 (1)

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 1, 59–64 (1993).
[CrossRef]

1992 (2)

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks.” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of super-resolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

1991 (1)

M. Bertero, P. Boccacci, R. E. Davies, E. R. Pike, “Superresolution in confocal scanning microscopy: III. The case of circular pupils,” Inverse Probl. 7, 655–674 (1991).
[CrossRef]

1989 (2)

M. Bertero, P. Boccacci, “Computation of the singular system for a class of integral operators related to data inversion in confocal microscopy,” Inverse Probl. 5, 935–957 (1989).
[CrossRef]

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

Bertero, M.

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 1, 59–64 (1993).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks.” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of super-resolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, E. R. Pike, “Superresolution in confocal scanning microscopy: III. The case of circular pupils,” Inverse Probl. 7, 655–674 (1991).
[CrossRef]

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

M. Bertero, P. Boccacci, “Computation of the singular system for a class of integral operators related to data inversion in confocal microscopy,” Inverse Probl. 5, 935–957 (1989).
[CrossRef]

Boccacci, P.

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks.” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, E. R. Pike, “Superresolution in confocal scanning microscopy: III. The case of circular pupils,” Inverse Probl. 7, 655–674 (1991).
[CrossRef]

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

M. Bertero, P. Boccacci, “Computation of the singular system for a class of integral operators related to data inversion in confocal microscopy,” Inverse Probl. 5, 935–957 (1989).
[CrossRef]

Brakenhoff, G. J.

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 1, 59–64 (1993).
[CrossRef]

Davies, R. E.

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 1, 59–64 (1993).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of super-resolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks.” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, E. R. Pike, “Superresolution in confocal scanning microscopy: III. The case of circular pupils,” Inverse Probl. 7, 655–674 (1991).
[CrossRef]

De Mol, C.

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

Defrise, M.

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

Jiang, S. H.

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of super-resolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

Malfanti, F.

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks.” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

Pike, E. R.

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 1, 59–64 (1993).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks.” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of super-resolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, E. R. Pike, “Superresolution in confocal scanning microscopy: III. The case of circular pupils,” Inverse Probl. 7, 655–674 (1991).
[CrossRef]

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

E. R. Pike, “Inverse problems in confocal microscopy,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, Bristol, 1992), pp. 164–179.

Sheppard, C.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Walker, J. G.

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 1, 59–64 (1993).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of super-resolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks.” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

Wilson, T.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Young, M. R.

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 1, 59–64 (1993).
[CrossRef]

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of super-resolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

Inverse Probl. (4)

M. Bertero, P. Boccacci, R. E. Davies, F. Malfanti, E. R. Pike, J. G. Walker, “Superresolution in confocal scanning microscopy: IV. Theory of data inversion by the use of optical masks.” Inverse Probl. 8, 1–23 (1992).
[CrossRef]

M. Bertero, P. Boccacci, M. Defrise, C. De Mol, E. R. Pike, “Superresolution in confocal scanning microscopy: II. The incoherent case,” Inverse Probl. 5, 441–461 (1989).
[CrossRef]

M. Bertero, P. Boccacci, R. E. Davies, E. R. Pike, “Superresolution in confocal scanning microscopy: III. The case of circular pupils,” Inverse Probl. 7, 655–674 (1991).
[CrossRef]

M. Bertero, P. Boccacci, “Computation of the singular system for a class of integral operators related to data inversion in confocal microscopy,” Inverse Probl. 5, 935–957 (1989).
[CrossRef]

J. Microsc. (1)

M. R. Young, S. H. Jiang, R. E. Davies, J. G. Walker, E. R. Pike, M. Bertero, “Experimental confirmation of super-resolution in coherent confocal scanning microscopy using optical masks,” J. Microsc. 165, 131–138 (1992).
[CrossRef]

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

J. G. Walker, E. R. Pike, R. E. Davies, M. R. Young, G. J. Brakenhoff, M. Bertero, “Superresolving scanning optical microscopy using holographic optical processing,” J. Opt. Soc. Am. A 1, 59–64 (1993).
[CrossRef]

Other (2)

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

E. R. Pike, “Inverse problems in confocal microscopy,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, Bristol, 1992), pp. 164–179.

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 (5)

Fig. 1
Fig. 1

Arbitrarily normalized image-plane mask M(ρ) for incoherent imaging that we constructed from the first five singular functions.

Fig. 2
Fig. 2

Incoherent confocal microscope setup with the elliptical annular mask that we inserted at 45° in the image plane: L’s, lenses; M, masks.

Fig. 3
Fig. 3

Approximate binary mask design Ma(ρ) corresponding to the exact mask that is shown in Fig. 1.

Fig. 4
Fig. 4

PSF’s of the exact mask T(ρ) (circles) and approximate binary mask Ta(ρ) (triangles) compared with the conventional confocal microscope t(ρ) (solid curve). Both masks show super-resolution inasmuch as their profiles are significantly narrower than that of the confocal microscope.

Fig. 5
Fig. 5

Transfer functions of the exact mask T ^(ω) (circles) and approximate mask T ^ a(ω) (triangles) compared with the confocal microscope t ^(ω) (solid curve). Both masks display considerably stronger frequency response in the upper half of the spectrum.

Equations (13)

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

g ( x ) = ( A f ) ( x ) = S 2 ( x - y ) S 1 ( y ) f ( y ) d y ,
A u k = α k v k ,             A * v k = α k u k ,
( A * g ) ( y ) = S 1 ( y ) S 2 ( x - y ) g ( x ) d x .
f ˜ ( 0 ) = k = 1 K ( g , v k ) α k u k ( 0 ) .
( g , v k ) = g ( x ) v k ( x ) d x .
M ( x ) = k = 1 K 1 α k u k ( 0 ) v k ( x ) ,
f ˜ ( 0 ) = g ( x ) M ( x ) d x = ( g , M ) .
S 1 ( x ) = S 2 ( x ) = [ 2 J 1 ( π ρ ) π ρ ] 2 ,             ρ = x .
f ˜ ( 0 ) = ( m , g ) = ( m , A f ) = ( A * m , f ) ,
f ˜ ( 0 ) = T ( y ) f ( y ) d y .
f ˜ ( z ) = T ( y ) f ( z - y ) d y = ( T * f ) ( z ) .
T ( y ) = S 1 ( y ) S 2 ( x - y ) m ( x ) d x .
t ( y ) = [ 2 J 1 ( π ρ ) π ρ ] 4 ,             ρ = x .

Metrics