Abstract

We find a simple analytic expression for the inverse of an operator related to the problem of data reduction in confocal scanning microscopy. Potential applications of this result to the practical scanning-microscope problem are outlined.

© 1987 Optical Society of America

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References

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  1. M. Bertero, C. De Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging. IV. The case of uncertain localization or non-uniform illumination of the object,” Opt. Acta 31, 923–946 (1984).
    [Crossref]
  2. M. Bertero, P. Brianzi, E. R. Pike, “Superresolution in confocal scanning microscopy,” Inverse Probl. 3, 195–212 (1987).
    [Crossref]
  3. T. Wilson, C. Sheppard, Theory and Practice of Scanning Microscopy (Academic, London, 1984).
  4. F. Gori, G. Guattari, “Signal restoration for linear systems with weighted inputs,” Inverse Probl. 1, 67–85 (1985).
    [Crossref]

1987 (1)

M. Bertero, P. Brianzi, E. R. Pike, “Superresolution in confocal scanning microscopy,” Inverse Probl. 3, 195–212 (1987).
[Crossref]

1985 (1)

F. Gori, G. Guattari, “Signal restoration for linear systems with weighted inputs,” Inverse Probl. 1, 67–85 (1985).
[Crossref]

1984 (1)

M. Bertero, C. De Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging. IV. The case of uncertain localization or non-uniform illumination of the object,” Opt. Acta 31, 923–946 (1984).
[Crossref]

Bertero, M.

M. Bertero, P. Brianzi, E. R. Pike, “Superresolution in confocal scanning microscopy,” Inverse Probl. 3, 195–212 (1987).
[Crossref]

M. Bertero, C. De Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging. IV. The case of uncertain localization or non-uniform illumination of the object,” Opt. Acta 31, 923–946 (1984).
[Crossref]

Brianzi, P.

M. Bertero, P. Brianzi, E. R. Pike, “Superresolution in confocal scanning microscopy,” Inverse Probl. 3, 195–212 (1987).
[Crossref]

De Mol, C.

M. Bertero, C. De Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging. IV. The case of uncertain localization or non-uniform illumination of the object,” Opt. Acta 31, 923–946 (1984).
[Crossref]

Gori, F.

F. Gori, G. Guattari, “Signal restoration for linear systems with weighted inputs,” Inverse Probl. 1, 67–85 (1985).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, “Signal restoration for linear systems with weighted inputs,” Inverse Probl. 1, 67–85 (1985).
[Crossref]

Pike, E. R.

M. Bertero, P. Brianzi, E. R. Pike, “Superresolution in confocal scanning microscopy,” Inverse Probl. 3, 195–212 (1987).
[Crossref]

M. Bertero, C. De Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging. IV. The case of uncertain localization or non-uniform illumination of the object,” Opt. Acta 31, 923–946 (1984).
[Crossref]

Sheppard, C.

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

Walker, J. G.

M. Bertero, C. De Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging. IV. The case of uncertain localization or non-uniform illumination of the object,” Opt. Acta 31, 923–946 (1984).
[Crossref]

Wilson, T.

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

Inverse Probl. (2)

M. Bertero, P. Brianzi, E. R. Pike, “Superresolution in confocal scanning microscopy,” Inverse Probl. 3, 195–212 (1987).
[Crossref]

F. Gori, G. Guattari, “Signal restoration for linear systems with weighted inputs,” Inverse Probl. 1, 67–85 (1985).
[Crossref]

Opt. Acta (1)

M. Bertero, C. De Mol, E. R. Pike, J. G. Walker, “Resolution in diffraction-limited imaging. IV. The case of uncertain localization or non-uniform illumination of the object,” Opt. Acta 31, 923–946 (1984).
[Crossref]

Other (1)

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

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Equations (27)

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g ( x ) = - + sinc ( x - y ) sinc ( y ) f ( y ) d y ,
f ( y ) = m = - + f ( y m ) sinc [ 2 ( y - y m ) ] ,
y 0 = 0 ,             y m = sgn ( m ) ( m - ½ ) ,             m = ± 1 , ± 2 , .
f ( y ) = k = 0 + 1 α k ( g , v k ) u k ( y ) .
g ( x ) = n = - + g ( n ) sinc ( x - n ) ,
g ( n ) = m = - + A n m f ( y m ) ,
A n o = ½ δ n o ,             n = 0 , ± 1 , ,
A n m = 1 2 π 2 ( - 1 ) n + 1 y m ( n - y m ) , m = ± 1 , ± 2 , , n = 0 , ± 1 , .
( A - 1 ) o n = 2 ( - 1 ) n ,             n = 0 , ± 1 , ,
( A - 1 ) m n = ( - 1 ) n + 1 2 n n - y m , m = ± 1 , ± 2 , , n = 0 , ± 1 , .
( A - 1 ) m n = k - 0 + 1 α k u k ( y m ) v k ( n ) ,
ϕ ( z ) = [ tan ( z 2 ) - 2 z ] - 1 ,
R 2 j = 2 β 2 j 2 ( 8 + β 2 j 2 ) - 1 .
( A - 1 ) o n = 8 ( - 1 ) n j = 0 + R 2 j β 2 j 2 - 4 π 2 n 2 ,
( A - 1 ) m n = 2 ( - 1 ) n + 1 n y m n 2 - y m 2 + 8 ( - 1 ) n + 1 × j = 0 + β 2 2 R 2 j ( β 2 j 2 - 4 π 2 n 2 ) ( β 2 j 2 - 4 π 2 y m 2 ) ,             m 0.
ϕ ( z ) ( z 2 - 4 π 2 n 2 ) - 1
z 2 ϕ ( z ) ( z 2 - 4 π 2 n 2 ) - 1 ( z 2 - 4 π 2 y m 2 ) - 1
j = 0 + R 2 j β 2 j 2 - 4 π 2 n 2 = ¼ ,
j = 0 + β 2 j 2 R 2 j ( β 2 j 2 - 4 π 2 n 2 ) ( β 2 j 2 - 4 π 2 y m 2 ) = n 2 4 ( n 2 - y m 2 ) ,
f ( 0 ) = 2 n = - + ( - 1 ) n g ( n ) .
cos ( π x ) = n = - + ( - 1 ) n sinc ( x - n ) .
2 n = - + ( - 1 ) n g ( n ) = 2 n = - + ( - 1 ) n × - + sinc ( n - y ) sinc ( y ) f ( y ) d y = 2 - + sinc ( y ) cos ( π y ) f ( y ) d y = 2 - + sinc ( 2 y ) f ( y ) d y = f ( 0 ) ,
f ( 0 ) = n = - N N t n g ( n ) ,
t n = k = 0 2 M 1 α N , k v N , k ( n ) u N , k ( 0 ) ,
f ( 0 , 0 ) = 4 p , q = - + ( - 1 ) p + 1 g ( p , q ) .
f ( 0 , 0 ) = p , q t p , q g ( p , q ) ,
t p , q = 4 ( - 1 ) p + q + η p , q

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