Abstract

In transmission microscopy, many objects are three dimensional, that is, they are thicker than the depth of focus of the imaging system. The three-dimensional (3-D) image-intensity distribution consists of a series of two-dimensional images (optical slices) with different parts of the object in focus. First, we deal with the fundamental limitations of 3-D imaging with classical optical systems. Second, a transfer theory of 3-D image formation is derived that relates the 3-D object (complex index of refraction) to the 3-D image intensity distribution in first-order Born approximation. This theory applies to weak objects that do not scatter much light. Since, in a microscope, the illumination is neither coherent nor completely incoherent, a theory for partially coherent light is needed, but in this case the object phase distribution and the absorptive parts of the object play different roles. Finally, some experimental results are presented.

© 1985 Optical Society of America

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References

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  1. L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 101.
  2. R. Frieden, “Optical transfer of the three-dimensional object,”J. Opt. Soc. Am. 57, 56–66 (1967).
    [CrossRef]
  3. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  4. R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
    [CrossRef]
  5. A. F. Fercher, H. Bartelt, H. Becker, E. Wiltschko, “Image formation by inversion of scattered field data, experiments,” Appl. Opt. 18, 2427–2439 (1976).
    [CrossRef]
  6. G. Zinser, A. Ehrhardt, J. Bille, “Erzeugung und Rekonstruktion dreidimensionaler lichtmikroskopischer Bilder,” presented at Fifth Deutsche Arbeitsgemeinschaft Mustererkennung Symposium, Karlsruhe, Germany, 1983; G. Zinser, “Erzeugung und Analyse digitaler lichtmikroshopischer Bilder histologischer Präparate bie hoher Ortsauflösung,” dissertation (University of Heidelberg, Heidelberg, Federal Republic of Germany, 1982).
  7. G. Häusler, E. Körner, “Simple focusing criterion,” Appl. Opt. 23, 2468–2469 (1984); “Acquisition and use of three-dimensional object information in image processing,” presented at the Thirteenth Meeting of the International Commission for Optics, Sapporo, Japan, 1984.
    [CrossRef] [PubMed]
  8. E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikrosk. Anat. Entwicklungsmech. 9, 413–459 (1873).
    [CrossRef]
  9. A. W. Lohmann: “Three-dimensional properties of wavefields,” Optik 51, 105–117 (1978).
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1968).
  11. R. B. Rall, Computational Solutions of Nonlinear Operator Equations (Wiley, New York, 1969).
  12. N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).
  13. N. Streibl, “Depth transfer by an imaging system,” Opt. Acta (to be published).

1984 (2)

1979 (1)

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

1978 (1)

A. W. Lohmann: “Three-dimensional properties of wavefields,” Optik 51, 105–117 (1978).

1976 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1967 (1)

1873 (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikrosk. Anat. Entwicklungsmech. 9, 413–459 (1873).
[CrossRef]

Abbe, E.

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikrosk. Anat. Entwicklungsmech. 9, 413–459 (1873).
[CrossRef]

Bartelt, H.

Becker, H.

Bille, J.

G. Zinser, A. Ehrhardt, J. Bille, “Erzeugung und Rekonstruktion dreidimensionaler lichtmikroskopischer Bilder,” presented at Fifth Deutsche Arbeitsgemeinschaft Mustererkennung Symposium, Karlsruhe, Germany, 1983; G. Zinser, “Erzeugung und Analyse digitaler lichtmikroshopischer Bilder histologischer Präparate bie hoher Ortsauflösung,” dissertation (University of Heidelberg, Heidelberg, Federal Republic of Germany, 1982).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1968).

Ehrhardt, A.

G. Zinser, A. Ehrhardt, J. Bille, “Erzeugung und Rekonstruktion dreidimensionaler lichtmikroskopischer Bilder,” presented at Fifth Deutsche Arbeitsgemeinschaft Mustererkennung Symposium, Karlsruhe, Germany, 1983; G. Zinser, “Erzeugung und Analyse digitaler lichtmikroshopischer Bilder histologischer Präparate bie hoher Ortsauflösung,” dissertation (University of Heidelberg, Heidelberg, Federal Republic of Germany, 1982).

Fercher, A. F.

Frieden, R.

Häusler, G.

Kaveh, M.

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

Körner, E.

Lohmann, A. W.

A. W. Lohmann: “Three-dimensional properties of wavefields,” Optik 51, 105–117 (1978).

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 101.

Mueller, R. K.

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

Rall, R. B.

R. B. Rall, Computational Solutions of Nonlinear Operator Equations (Wiley, New York, 1969).

Streibl, N.

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).

N. Streibl, “Depth transfer by an imaging system,” Opt. Acta (to be published).

Wade, G.

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

Wiltschko, E.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1968).

Zinser, G.

G. Zinser, A. Ehrhardt, J. Bille, “Erzeugung und Rekonstruktion dreidimensionaler lichtmikroskopischer Bilder,” presented at Fifth Deutsche Arbeitsgemeinschaft Mustererkennung Symposium, Karlsruhe, Germany, 1983; G. Zinser, “Erzeugung und Analyse digitaler lichtmikroshopischer Bilder histologischer Präparate bie hoher Ortsauflösung,” dissertation (University of Heidelberg, Heidelberg, Federal Republic of Germany, 1982).

Appl. Opt. (2)

Arch. Mikrosk. Anat. Entwicklungsmech. (1)

E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Arch. Mikrosk. Anat. Entwicklungsmech. 9, 413–459 (1873).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Optik (2)

A. W. Lohmann: “Three-dimensional properties of wavefields,” Optik 51, 105–117 (1978).

N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).

Proc. IEEE (1)

R. K. Mueller, M. Kaveh, G. Wade, “Reconstructive tomography and applications to ultrasonics,” Proc. IEEE 67, 567–587 (1979).
[CrossRef]

Other (5)

G. Zinser, A. Ehrhardt, J. Bille, “Erzeugung und Rekonstruktion dreidimensionaler lichtmikroskopischer Bilder,” presented at Fifth Deutsche Arbeitsgemeinschaft Mustererkennung Symposium, Karlsruhe, Germany, 1983; G. Zinser, “Erzeugung und Analyse digitaler lichtmikroshopischer Bilder histologischer Präparate bie hoher Ortsauflösung,” dissertation (University of Heidelberg, Heidelberg, Federal Republic of Germany, 1982).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1968).

R. B. Rall, Computational Solutions of Nonlinear Operator Equations (Wiley, New York, 1969).

N. Streibl, “Depth transfer by an imaging system,” Opt. Acta (to be published).

L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 101.

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Figures (7)

Fig. 1
Fig. 1

3-D imaging of a partially coherently transilluminated object by a telecentric system.

Fig. 2
Fig. 2

Shape of the 3-D aperture depends on the size of the illuminating source ρs compared with the size of the pupil of the imaging system ρp.

Fig. 3
Fig. 3

3-D OTF’s for the absorptive parts and for the phase distribution for different radii of the source ρs and the pupil ρp: a, b, ρs/ρp = 0.1 almost coherent; c, d, ρs/ρp = 0.72 partially coherent; e, f, ρs/ρp = 0.95 still partially coherent but approaching incoherent illumination (ρs/ρp ≫ 2).

Fig. 4
Fig. 4

Sections through a 3-D data cube (256 × 256 × 50 volume elements) recorded with a microscope (objective Pl100/1.32) and a digital TV-frame store with a sampling rate of δx = 0.1 μm, δy = 0.07 μm, δz = 0.5 μm. The object consists of Urothel cells with clearly visible kernels. a, Lateral section z = 20δz; b, lateral section z = 30δz; c, longitudinal section y = 80δy.

Fig. 5
Fig. 5

Central cross section through the real part of the 3-D Fourier spectrum of a microscopic image (128 × 128 × 32 volume elements obtained by FFT). The border of the doughnut-shaped 3-D aperture is visible. The bipolar signal is shown in a bias (gray background).

Fig. 6
Fig. 6

Comparison between a, the OTF of the absorptive part (ρs = ρp) and b, its product with the 3-D filter [Eq. (33)].

Fig. 7
Fig. 7

Result of the approximate deconvolution of the object of Fig. 4 with the 3-D filter [Eq. (33)]: a, lateral section z = 20δz; b, lateral section z = 30δz.

Equations (35)

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ρ D = ρ s + ρ 0 ,
μ D = μ s + μ 0 ρ p .
μ s ρ s .
μ 0 = μ D - μ s ρ s + ρ p .
η 0 = [ λ - 2 - ( μ 0 + μ s ) 2 ] 1 / 2 - ( λ - 2 - μ s 2 ) 1 / 2 .
[ 2 + n 2 ( r ) k 2 ] u ( r ) = 0.
V ( r ) = k 2 ( 1 - n 2 ( r ) ) ,
( 2 + k 2 ) u ( r ) = V ( r ) u ( r ) .
J ( r 1 ; r 2 ) = u ( r 1 ) u * ( r 2 ) .
( 1 2 + k 2 ) J ( r 1 ; r 2 ) = V ( r 1 ) J ( r 1 ; r 2 ) ,
( 2 2 + k 2 ) J ( r 1 ; r 2 ) = V * ( r 2 ) J ( r 1 ; r 2 ) ,
( j 2 + k 2 ) J IN ( r 1 ; r 2 ) = 0             j = 1 , 2.
( 2 + k 2 ) G ( r - r ) = δ ( r - r ) .
J ( r 1 ; r 2 ) = J IN ( r 1 ; r 2 ) + [ G ( r 1 - r 1 ) V ( r 1 ) δ ( r 2 - r 2 ) + δ ( r 1 - r 1 ) G * ( r 2 - r 2 ) V * ( r 2 ) - G ( r 1 - r 1 ) V ( r 1 ) G * ( r 2 - r 2 ) V * ( r 2 ) ] × J ( r 1 ; r 2 ) d 3 r 1 d 3 r 2 .
J ( r 1 ; r 2 ) = J IN ( r 1 ; r 2 ) + [ G ( r 1 - r 1 ) V ( r 1 ) δ ( r 2 - r 2 ) + δ ( r 1 - r 1 ) G * ( r 2 - r 2 ) V * ( r 2 ) ] × J I N ( r 1 ; r 2 ) d 3 r 1 d 3 r 2 .
J IN ( x 1 , z 1 ; x 2 , z 2 ) = S ˜ ( μ ) exp { 2 π i [ ( z 1 - z 2 ) × ( λ - 2 - μ 2 ) 1 / 2 + ( x 1 - x 2 ) μ ] } d 2 μ ,
H = G ( r 1 - r 1 ) V ( r 1 ) V ( r 1 ) J IN ( r 1 ; r 2 ) d 3 r 1 .
G ˜ ( μ , z ) = 1 4 π i ( λ - 2 - μ 2 ) 1 / 2 exp [ 2 π i z ( λ - 2 - μ 2 ) 1 / 2 ] .
H = V ( x 1 , z 1 ) S ˜ ( μ 2 ) 4 π i ( λ - 2 - μ 1 2 ) 1 / 2 exp { 2 π i [ ( z 1 - z 2 ) × ( λ - 2 - μ 2 2 ) 1 / 2 - ( x 1 - x 2 ) μ 2 + z 1 - z 1 ( λ - 2 - μ 1 2 ) 1 / 2 + ( x 1 - x 1 ) μ 1 ] } d 2 μ 1 d 2 μ 2 d 2 x 1 d z 1 .
H = S ˜ ( μ 2 ) 4 π i ( λ - 2 - μ 1 2 ) 1 / 2 exp [ 2 π i ( x 1 · μ 1 - x 2 · μ 2 ) ] × V ( x 1 , z 1 ) exp { - 2 π i [ x 1 · ( μ 1 - μ 2 ) + z 1 ( λ - 2 - μ 1 2 ) 1 / 2 - ( λ - 2 - μ 2 2 ) 1 / 2 ] } × d 2 x 1 d z 1 d 2 μ 1 d 2 μ 2 .
J ˜ OBJ ( μ 1 ; μ 2 ) = S ˜ ( μ 1 ) δ ( μ 1 - μ 2 ) + S ˜ ( μ 2 ) 4 π i ( λ - 2 - μ 1 2 ) 1 / 2 × V ¯ [ μ 1 - μ 2 , ( λ - 2 - μ 1 2 ) 1 / 2 - ( λ - 2 - μ 2 2 ) 1 / 2 ] - S ˜ ( μ 1 ) 4 π i ( λ - 2 - μ 2 2 ) 1 / 2 V ¯ * ( μ 2 - μ 1 ) , ( λ - 2 - μ 2 2 ) 1 / 2 - ( λ - 2 - μ 1 2 ) 1 / 2 ,
V ( x , z ) = P ( x , z ) + i A ( x , z ) .
J ˜ OBJ ( μ 1 ; μ 2 ) = S ˜ ( μ 1 ) δ ( μ 1 - μ 2 ) + λ 4 π i [ S ˜ ( μ 2 ) - S ˜ ( μ 1 ) ] P ¯ [ μ 1 - μ 2 , λ 2 ( μ 2 2 - μ 1 2 ) ] + λ 4 π [ S ˜ ( μ 2 ) + S ˜ ( μ 1 ) ] A ˜ [ μ 1 - μ 2 , λ 2 ( μ 2 2 - μ 1 2 ) ] .
J ˜ IMG ( μ 1 ; μ 2 ) = p ˜ ( μ 1 ) J ˜ OBJ ( μ 1 ; μ 2 ) p ˜ * ( μ 2 ) .
E ¯ IMG ( μ , η ) = J ˜ IMG ( μ + ½ μ ; μ - ½ μ ) × δ ( η + λ μ · μ ) d 2 μ .
E ¯ IMG ( μ , η ) = B δ ( μ , η ) + P ¯ ( μ , η ) T ¯ p ( μ , η ) + A ¯ ( μ , η ) T ¯ A ( μ , η ) ,
B = S ˜ ( μ ) p ˜ ( μ ) 2 d 2 μ .
T ¯ A ( μ , η ) = λ 4 π p ˜ ( μ + ½ μ ) [ S ˜ ( μ + ½ μ ) + S ˜ ( μ - ½ μ ) ] p ˜ * ( μ - ½ μ ) δ ( η + λ μ · μ ) d 2 μ ,
T ¯ p ( μ , η ) = i λ 4 π p ˜ ( μ + ½ μ ) [ S ˜ ( μ + ½ μ ) - S ˜ ( μ - ½ μ ) ] p ˜ * ( μ - ½ μ ) δ ( η + λ μ · μ ) d 2 μ .
E IMG ( r ) = B + P ( r ) * * * T P ( r ) + A ( r ) * * * T A ( r ) .
p ˜ ( μ ) = { 1 if μ < ρ p 0 otherwise ,
S ˜ ( μ ) = { 1 if μ < ρ S 0 otherwise .
T ¯ A ( ρ , η ) = 1 2 π ρ Re { [ 1 2 ( ρ p 2 + ρ S 2 ) - 1 4 ρ 2 - ( η λ ρ ) 2 - | η λ - 1 2 ( ρ p 2 - ρ S 2 ) | ] 1 / 2 + [ 1 2 ( ρ p 2 + ρ S 2 ) - 1 4 ρ 2 - ( η λ ρ ) 2 - | η λ + 1 2 ( ρ p 2 - ρ S 2 ) | ] 1 / 2 } .
T ¯ p ( ρ , η ) = i 2 π ρ Re [ ( ) 1 / 2 - ( ) 1 / 2 ] ,
F ¯ ( ρ , η ) ~ 1 ρ ( ρ 2 + η 2 ) .

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