Abstract

A reconstruction procedure based on linear system theory has been developed for 3-D light-microscopic images. Inverse filtering with the 3-D optical transfer function was used for image reconstruction. The procedure allows a significant improvement in spatial resolution in the image planes perpendicular to the optical axis.

© 1985 Optical Society of America

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References

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  1. H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 231, 91 (1955).
    [CrossRef]
  2. P. A. Stokseth, “Properties of a Defocused Optical System,” J. Opt. Soc. Am. 59, 1314 (1969).
    [CrossRef]
  3. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).
  4. J. W. Cooley, J. W. Tukey, “An Algorithm for Machine Computation of Complex Fourier Series,” Math. Computation 19, 297 (1965).
    [CrossRef]
  5. D. A. Agard, J. W. Sedat, “Threedimensional Architecture of a Polythene Nucleus,” Nature London 302, 676 (1983).
    [CrossRef] [PubMed]
  6. D. Mathog, M. Hochstrasser, Y. Gruenbaum, H. Saumweber, J. Sedat, “Characteristic Folding Pattern of Polythene Chromosomes in Drosophila Salivary Gland Nuclei,” Nature London 308, 414 (1984).
    [CrossRef] [PubMed]
  7. H. G. Zimmer, Geometrische Optik (Springer, Berlin, 1967).
    [CrossRef]

1984 (1)

D. Mathog, M. Hochstrasser, Y. Gruenbaum, H. Saumweber, J. Sedat, “Characteristic Folding Pattern of Polythene Chromosomes in Drosophila Salivary Gland Nuclei,” Nature London 308, 414 (1984).
[CrossRef] [PubMed]

1983 (1)

D. A. Agard, J. W. Sedat, “Threedimensional Architecture of a Polythene Nucleus,” Nature London 302, 676 (1983).
[CrossRef] [PubMed]

1969 (1)

1965 (1)

J. W. Cooley, J. W. Tukey, “An Algorithm for Machine Computation of Complex Fourier Series,” Math. Computation 19, 297 (1965).
[CrossRef]

1955 (1)

H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 231, 91 (1955).
[CrossRef]

Agard, D. A.

D. A. Agard, J. W. Sedat, “Threedimensional Architecture of a Polythene Nucleus,” Nature London 302, 676 (1983).
[CrossRef] [PubMed]

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An Algorithm for Machine Computation of Complex Fourier Series,” Math. Computation 19, 297 (1965).
[CrossRef]

Gruenbaum, Y.

D. Mathog, M. Hochstrasser, Y. Gruenbaum, H. Saumweber, J. Sedat, “Characteristic Folding Pattern of Polythene Chromosomes in Drosophila Salivary Gland Nuclei,” Nature London 308, 414 (1984).
[CrossRef] [PubMed]

Hochstrasser, M.

D. Mathog, M. Hochstrasser, Y. Gruenbaum, H. Saumweber, J. Sedat, “Characteristic Folding Pattern of Polythene Chromosomes in Drosophila Salivary Gland Nuclei,” Nature London 308, 414 (1984).
[CrossRef] [PubMed]

Hopkins, H. H.

H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 231, 91 (1955).
[CrossRef]

Mathog, D.

D. Mathog, M. Hochstrasser, Y. Gruenbaum, H. Saumweber, J. Sedat, “Characteristic Folding Pattern of Polythene Chromosomes in Drosophila Salivary Gland Nuclei,” Nature London 308, 414 (1984).
[CrossRef] [PubMed]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

Saumweber, H.

D. Mathog, M. Hochstrasser, Y. Gruenbaum, H. Saumweber, J. Sedat, “Characteristic Folding Pattern of Polythene Chromosomes in Drosophila Salivary Gland Nuclei,” Nature London 308, 414 (1984).
[CrossRef] [PubMed]

Sedat, J.

D. Mathog, M. Hochstrasser, Y. Gruenbaum, H. Saumweber, J. Sedat, “Characteristic Folding Pattern of Polythene Chromosomes in Drosophila Salivary Gland Nuclei,” Nature London 308, 414 (1984).
[CrossRef] [PubMed]

Sedat, J. W.

D. A. Agard, J. W. Sedat, “Threedimensional Architecture of a Polythene Nucleus,” Nature London 302, 676 (1983).
[CrossRef] [PubMed]

Stokseth, P. A.

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An Algorithm for Machine Computation of Complex Fourier Series,” Math. Computation 19, 297 (1965).
[CrossRef]

Zimmer, H. G.

H. G. Zimmer, Geometrische Optik (Springer, Berlin, 1967).
[CrossRef]

J. Opt. Soc. Am. (1)

Math. Computation (1)

J. W. Cooley, J. W. Tukey, “An Algorithm for Machine Computation of Complex Fourier Series,” Math. Computation 19, 297 (1965).
[CrossRef]

Nature London (2)

D. A. Agard, J. W. Sedat, “Threedimensional Architecture of a Polythene Nucleus,” Nature London 302, 676 (1983).
[CrossRef] [PubMed]

D. Mathog, M. Hochstrasser, Y. Gruenbaum, H. Saumweber, J. Sedat, “Characteristic Folding Pattern of Polythene Chromosomes in Drosophila Salivary Gland Nuclei,” Nature London 308, 414 (1984).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 231, 91 (1955).
[CrossRef]

Other (2)

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

H. G. Zimmer, Geometrische Optik (Springer, Berlin, 1967).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Two-dimensional radial section of the 3-D optical transfer function. The color code is based on a logarithmic scale: white corresponds to the value 1, black to values ≤10−3. The spatial frequencies are normalized to the incoherent cutoff frequency q0. The calculation of OTF(q,s) for this figure is based on magnification of the objective, 100×; N.A., 1.3; wavelength of light, 0.53 μm.

Fig. 2
Fig. 2

(a) One-dimensional section of the 3-D optical transfer function on the q axis (s = 0); (b) parallel to the s axis at q = 1/2. (c) Values of the 3-D optical transfer function at w = 0 (solid line) compared with the 2-D optical transfer function for zero defocus (dashed line). (d) The inverse optical transfer function (dashed line) and the inverse effective optical transfer function vs the normalized radial frequency q. All the calculations are based on objective magnification 100×, N.A. 1.3, light wavelength 0.53 μm.

Fig. 3
Fig. 3

Nucleus of a cell from the mucosa of the large intestine: diameter of object, 8 μm; objective, 100×; N.A. = 1.3, oil; total focal shift parallel to the optical axis, 16 μm; number of averaged images, 3; factor of oversampling and subsequent reduction, 3; sample spacing perpendicular to the optical axis, 0.125 μm (after reduction); sample spacing parallel to the optical axis, 0.25 μm; preprocessing steps, none; postprocessing steps, none. (a) and (b) Planes perpendicular to the optical axis. The original images show an optical dense region at coordinates x = 38, y = 28 which has a clearly visible fine structure, as can be seen in the reconstructed image, (c) and (d) Planes parallel to the optical axis. The optical dense region is located at (c) coordinates y = 28, z = 18 and at (d) coordinates x = 38, x = 18. The de-focused projections in the original images are apparent, as well as the enhancement of the spatial resolution perpendicular to the optical axis. (d) A 1-D section for y = 28, z = 18. The optical density at x = 32 − x = 40 in the original image shows no indication of a gap, while in the reconstructed image two maxima are visible. This shows that the reconstruction procedure does not merely amplify already existing maxima and minima in grey values but removes defocused projections in the 3-D image.

Fig. 4
Fig. 4

Shell of Radiolaria Aulonia hexagonia: diameter of object, 60 μm; objective, 10×; N.A. = 0.25; total focal shift parallel to the optical axis, 128 μm; number of averaged images, 5; factor of oversampling and subsequent reduction, 1; sample spacing perpendicular to the optical axis, 0.5 μm; sample spacing parallel to the optical axis, 1.0 μm; preprocessing steps, Hanning window; postprocessing steps, inverse Hanning window. (a)–(d) Planes perpendicular to the optical axis; (e) plane parallel to the optical axis. All the sections show an enhanced spatial resolution perpendicular to the optical axis, while parallel to the optical axis, due to the information loss because of the band limitation of the 3-D optical transfer function, principally no equivalent enhancement of the spatial resolution can be achieved.

Fig. 5
Fig. 5

Test object, crossing bristles of an insect wing: diameter of object, 1 μm; objective, 100×; N.A. = 1.3; oil; total focal shift parallel to the optical axis, 16 μm; number of averaged images, 3; factor of oversampling and subsequent reduction, 3; sample spacing perpendicular to the optical axis, 0.125 μm (after reduction); sample spacing parallel to the optical axis, 0.25 μm; preprocessing steps, Hanning window; postprocessing steps, inverse Hanning window. (a) and (b) Planes perpendicular to the optical axis; (c) plane parallel to the optical axis. The reconstruction of a test object with known 3-D geometry ensures that the structures in the reconstructed images are not artifacts but fine structures of the object.

Fig. 6
Fig. 6

(a) Defocused optical system on the image side: A denotes the radius of the exit pupil; r′ the focal length; z′ the focal shift on the optical axis; α′ the aperture angle. The maximum phase error is given by the difference of path lengths of the rays at the edge of the aperture as a result of a focal shift. (b) Object and image side of an optical system for positive and negative focal shifts.

Tables (1)

Tables Icon

Table I Cutoff Frequencies q0 and w0/q0 and Maximum Sample Spacings δxy and δZ Perpendicular and Parallel to the Optical Axis for Various Objectives

Equations (22)

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g ( x , y , z ) psf ( x , y , z ) .
G ( u , υ , w ) = F ( u , υ , w ) OTF ( u , υ , w ) .
F ( u , υ , w ) = G ( u , υ , w ) / OTF ( u , υ , w )
σ = ( N . A . ) 2 z / 2 .
OTF ( q ; z ) = 2 f ( q ) J 1 [ 2 π h ( q ) q 0 z ] 2 π h ( q ) q 0 z ,
16 μ m for an objective 100 × / N . A . 1.3 200 μ m for an objective 25 × / N . A . 0.65 1000 μ m for an objective 10 × / N . A . 0.25 .
OTF ( q , s ) = { 1 q = 0 , s = 0 , 2 π 0 z 0 q 0 f ( q ) h ( q ) [ 1 s 2 h 2 ( q ) ] 1 / 2 0 < q 1 , | s | h ( q ) , elsewhere ,
OTF eff 1 ( q , s ) = { OTF 1 ( q , s ) if OTF 1 ( q , s ) 100 , OTF 1 ( q , s ) cos [ p ( q , s ) ] if 100 OTF 1 ( q , s ) 1000 , 0 elsewhere ,
( r + z ) 2 = ( r + σ ) 2 + z 2 2 ( r + σ ) z cos ( π α ) ,
σ = r z cos α + ( r + z ) ( 1 ε ) 1 / 2 ,
ε = z 2 sin 2 α ( r + z ) 2
n sin α = M sin α ,
N . A . = n sin α ,
1 cos α 1 2 sin 2 α
σ 1 2 N . A . 2 z M 2 ( 1 z r + z ) ,
N . A . 2 z = sin α sin β z ,
β = arctan ( A r + z ) A = r tan α [ Fig . 6 ( b ) ] .
z = i = 1 5 a i M i + 1 z i
= z M 2 [ 1 + f ( z ) ]
σ = 1 2 N . A . 2 z [ 1 + f ( z ) ] [ 1 g ( z ) ] ,
f ( z ) = ( a 1 1 ) + i = 2 5 a i M i 1 z i 1 , g ( z ) = i = 1 5 a i M i + 1 z i r + i = 1 5 a i M i + 1 z i
σ = ( N . A . ) 2 z / 2 .

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