Abstract

A method of observing 3-D phase structures through a microscope incorporating computer reconstruction is discussed. This microscope is equipped with an annular pupil in illumination optics, but no phase shifter is included in the imaging optics. The sample stage is longitudinally (z-axial) scanned to collect a focus image series. The 3-D phase transfer function is derived and computer-plotted based on Streibl’s 3-D image transfer theory under the first-order Born approximation and the mutual intensity propagation theorem. Experimental results of 3-D phase reconstruction are shown with cultured tobacco cells by Helstrom’s inverse filtering of the transfer function to a series of focused images.

© 1990 Optical Society of America

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References

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  1. B. Kachar, “Asymmetric Illumination Contrast: A Method of Image Formation for Video Light Microscopy,” Science 227, 766–768 (1985).
    [Crossref] [PubMed]
  2. S. Inoue, Video Microscopy (Plenum, New York, 1986), pp. 410–412.
  3. T. S. McKechnie, “The Effect of Condenser Obstruction on the Two-Point Resolution of a Microscope,” Opt. Acta 19, 729–737 (1972).
    [Crossref]
  4. N. Streibl, “Fundamental Restrictions for 3-D Light Distributions,” Optik 66, 341–354 (1984).
  5. A. Takahashi, I. Nemoto, “Inverse Problem in Microscopy,” Shingakugihou MBE88-58 (1988) (in Japanese).
  6. E. Wolf, “Three-Dimensional Structure Determination of Semi-Transparent Objects from Holographic Data,” Opt. Commun. 1, 153–156 (1969).
    [Crossref]
  7. N. Streibl, “Three-Dimensional Imaging by a Microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [Crossref]
  8. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  9. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), p. 294.
  10. C. W. Helstrom, “Image Restoration by the Method of Least Squares,” J. Opt. Soc. Am. 57, 297–303 (1967).
    [Crossref]

1988 (1)

A. Takahashi, I. Nemoto, “Inverse Problem in Microscopy,” Shingakugihou MBE88-58 (1988) (in Japanese).

1985 (2)

B. Kachar, “Asymmetric Illumination Contrast: A Method of Image Formation for Video Light Microscopy,” Science 227, 766–768 (1985).
[Crossref] [PubMed]

N. Streibl, “Three-Dimensional Imaging by a Microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
[Crossref]

1984 (1)

N. Streibl, “Fundamental Restrictions for 3-D Light Distributions,” Optik 66, 341–354 (1984).

1972 (1)

T. S. McKechnie, “The Effect of Condenser Obstruction on the Two-Point Resolution of a Microscope,” Opt. Acta 19, 729–737 (1972).
[Crossref]

1969 (1)

E. Wolf, “Three-Dimensional Structure Determination of Semi-Transparent Objects from Holographic Data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

1967 (1)

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Helstrom, C. W.

Inoue, S.

S. Inoue, Video Microscopy (Plenum, New York, 1986), pp. 410–412.

Kachar, B.

B. Kachar, “Asymmetric Illumination Contrast: A Method of Image Formation for Video Light Microscopy,” Science 227, 766–768 (1985).
[Crossref] [PubMed]

McKechnie, T. S.

T. S. McKechnie, “The Effect of Condenser Obstruction on the Two-Point Resolution of a Microscope,” Opt. Acta 19, 729–737 (1972).
[Crossref]

Nemoto, I.

A. Takahashi, I. Nemoto, “Inverse Problem in Microscopy,” Shingakugihou MBE88-58 (1988) (in Japanese).

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), p. 294.

Streibl, N.

N. Streibl, “Three-Dimensional Imaging by a Microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
[Crossref]

N. Streibl, “Fundamental Restrictions for 3-D Light Distributions,” Optik 66, 341–354 (1984).

Takahashi, A.

A. Takahashi, I. Nemoto, “Inverse Problem in Microscopy,” Shingakugihou MBE88-58 (1988) (in Japanese).

Wolf, E.

E. Wolf, “Three-Dimensional Structure Determination of Semi-Transparent Objects from Holographic Data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

T. S. McKechnie, “The Effect of Condenser Obstruction on the Two-Point Resolution of a Microscope,” Opt. Acta 19, 729–737 (1972).
[Crossref]

Opt. Commun. (1)

E. Wolf, “Three-Dimensional Structure Determination of Semi-Transparent Objects from Holographic Data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

Optik (1)

N. Streibl, “Fundamental Restrictions for 3-D Light Distributions,” Optik 66, 341–354 (1984).

Science (1)

B. Kachar, “Asymmetric Illumination Contrast: A Method of Image Formation for Video Light Microscopy,” Science 227, 766–768 (1985).
[Crossref] [PubMed]

Shingakugihou MBE88-58 (1)

A. Takahashi, I. Nemoto, “Inverse Problem in Microscopy,” Shingakugihou MBE88-58 (1988) (in Japanese).

Other (3)

S. Inoue, Video Microscopy (Plenum, New York, 1986), pp. 410–412.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), p. 294.

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Figures (6)

Fig. 1
Fig. 1

Optical diagram of a phase imaging system under oblique illumination composed of Kohler illumination and telecentric imaging systems.

Fig. 2
Fig. 2

Computer plot of 3-D PTF under oblique illumination. The value of the PTF is positive for ζ < 0 and is negative for ζ > 0; the inside of the two shells is hollow.

Fig. 3
Fig. 3

(a) Computer plot of 3-D PTF under annular illumination; (b) perspective plot of the PTF value of ρ(= 1/r) and ζ(= 1/z).

Fig. 4
Fig. 4

Twelve images of cultured tobacco cells observed with an annular illumination microscope, each serially focused from (a) top to (l) bottom with 5-μm spacing. The wavelength for illumination is 550 nm.

Fig. 5
Fig. 5

Twelve cross sections of the power spectrum of observed images: (a) including ζ-axis and (b)–(l) parallel to (a) with a distance of 0.04 μm−1 between two planes.

Fig. 6
Fig. 6

Twelve reconstructed phase images; plane positions correspond to those shown in Fig. 4. Areas marked by squares are significantly different from those in the observations in Fig. 4.

Equations (11)

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S ˜ ( ξ , η ) = δ ( ξ - ρ s , η ) .
V p ( r , z ) = k 2 [ 1 - n 2 ( r , z ) ] .
E ˜ ˜ IMG ( ξ , η , ζ ) = B δ ( ξ , η , ζ ) + V ˜ ˜ p ( ξ , η , ζ ) T ˜ ˜ p ( ξ , η , ζ ) ,
T ˜ ˜ p ( ξ , η , ζ ) = j λ 4 π p ˜ ( ξ + 1 2 ξ , η + 1 2 η ) p ˜ * ( ξ - 1 2 ξ , η - 1 2 η ) × [ S ˜ ( ξ + 1 2 ξ , η + 1 2 η ) - S ˜ ( ξ - 1 2 ξ , η - 1 2 η ) ] × δ { ζ - [ λ - 2 - ( ξ + 1 2 ξ ) 2 - ( η + 1 2 η ) 2 ] 1 / 2 + [ λ - 2 - ( ξ - 1 2 ξ ) 2 - ( η - 1 2 η ) 2 ] 1 / 2 } d ξ d η ,
p ˜ ( ξ , η ) = p ˜ ( ρ ) = { 1 if ρ ρ p , 0 if ρ > ρ p ,
B = S ˜ ( ξ , η ) p ˜ ( ξ , η ) d ξ d η .
T ˜ ˜ p ( ξ , η , ζ ) = j λ 4 π p ˜ * ( ρ s - ξ , - η ) δ { ζ - ( λ - 2 - ρ s 2 ) 1 / 2 + [ λ - 2 - ( ρ s - ξ ) 2 - η 2 ] 1 / 2 } - j λ 4 π p ˜ ( ρ s + ξ , η ) δ { ζ + ( λ - 2 + ρ s 2 ) 1 / 2 - [ λ - 2 - ( ρ s + ξ ) 2 - η 2 ] 1 / 2 } .
ρ s = ρ p .
T ˜ ˜ p ( ρ , ζ ) = j λ 4 π [ p ˜ * [ - ρ ± ( ρ s 2 - ρ 2 ) 1 / 2 , ρ ] δ ( ζ - ( λ - 2 - ρ s 2 ) 1 / 2 + { λ - 2 - [ - ρ ± ( ρ s 2 - ρ 2 ) 1 / 2 ] 2 - ρ 2 } 1 / 2 ) - p ˜ [ ρ ± ( ρ s 2 - ρ 2 ) 1 / 2 , ρ ] × δ ( ζ + ( λ - 2 - ρ s ) 1 / 2 - { λ - 2 - [ ρ ± ( ρ s 2 - ρ 2 ) 1 / 2 ] 2 - ρ 2 } 1 / 2 ) ] d ρ ,
S ˜ ( ρ ) = δ ( ρ - ρ s )
H ˜ ˜ ( ρ , ζ ) = T ˜ ˜ p * ( ρ , ζ ) T ˜ ˜ p ( ρ , ζ ) T ˜ ˜ p * ( ρ , ζ ) + α ,

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