Abstract

A new principle of tomographic optical microscope imaging has been developed that uses a computerized reconstruction algorithm and a transmission optical microscope. A conventional transmission microscope with a rotationally oblique illumination system provides projections of a thick specimen in various directions within the numerical aperture of the objective lens. The images obtained are combined to reconstruct a three-dimensional distribution of the sample by inverting the imaging system in a computer. The three-dimensional optical transfer function of this projection system is analyzed, and it is found that this system is strictly angularly band limited. For improving the spatial resolution we utilize a priori knowledge of the spatial extent of the object as the support constraint. Experimental results are presented to demonstrate the tomographic imaging capability of this principle.

© 1987 Optical Society of America

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  1. D. A. Agard, J. W. Sedat, “Three-dimensional architecture of a polytene nucleus,” Nature 302, 676–681 (1983).
    [Crossref] [PubMed]
  2. B. Wendroff, Theoretical Numerical Analysis (Academic, New York, 1966).
  3. S. Kawata, Y. Ichioka, “Iterative image restoration for linearly degraded images. I. Basis,”J. Opt. Soc. Am. 70, 762–772 (1980).
    [Crossref]
  4. A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Opt. 24, 194–200 (1985).
    [Crossref] [PubMed]
  5. A. W. Lohmann, “Three-dimensional properties of wavefield,” Optik 51, 105–117 (1978).
  6. L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 101.
  7. B. R. Frieden, “Optical transfer of the three-dimensional object,”J. Opt. Soc. Am. 57, 56–66 (1967).
    [Crossref]
  8. N. Streibl, “Fundamental restrictions for 3-D light distributions,” Optik 66, 341–354 (1984).
  9. N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1984).
    [Crossref]
  10. S. Kawata, Y. Touki, S. Minami, “Optical microscopic tomography,” in Inverse Optics II, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.558, 15–20 (1985).
    [Crossref]
  11. D. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,”IEEE Trans. Biomed. Eng. BME-19, 20–28 (1972).
    [Crossref]
  12. G. Groh, “Holographic tomography using a circular synthetic aperture,” Appl. Opt. 10, 2549–2550 (1971).
    [Crossref] [PubMed]
  13. L. T. Chang, B. Macdonald, V. Perez-Mendez, “Axial tomography and three dimensional image reconstruction,”IEEE Trans. Nucl. Sci. NS-23, 568–572 (1976).
    [Crossref]
  14. M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Cho, J. W. Arendt, G. R. Gindi, “Three-dimensional radiographic imaging with a restricted view angle,”J. Opt. Soc. Am. 69, 1323–1333 (1979).
    [Crossref]
  15. S. Kawata, J. Sklansky, “Elimination of nonpivotal plane images from x-ray motion-tomograms,”IEEE Trans. Med. Imaging MI-4, 153–159 (1985).
    [Crossref]
  16. T. Inouye, “Image reconstruction with limited angle projection data,”IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).
  17. K. C. Tam, V. Perez-Mendez, “Tomographical imaging with limited-angle input,”J. Opt. Soc. Am. 71, 582–592 (1981).
    [Crossref]
  18. A. N. Tikhonov, V. V. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).
  19. J. B. Abbiss, M. Defrise, C. De Mol, H. S. Dhadwal, “Regularized iterative and noniterative procedures for object restoration in the presence of noise: an error analysis,”J. Opt. Soc. Am. 73, 1470–1475 (1983).
    [Crossref]
  20. C. W. Helstrom, “Image restoration by the method of least squares,”J. Opt. Soc. Am. 57, 297–303 (1967).
    [Crossref]
  21. S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,”IEEE Trans. Med. Imaging MI-4, 65–71 (1985).
    [Crossref]
  22. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [Crossref]
  23. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [Crossref]
  24. P. F. C. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,”J. Theor. Biol. 36, 105–117 (1972).
    [Crossref] [PubMed]
  25. For example, B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer-Verlag, Berlin, 1975), Secs. 5.13.2 and 5.4.
  26. K. Minami, S. Kawata, S. Minami, “Superresolution of Fourier transform spectra by autoregressive model fitting with singular value decomposition,” Appl. Opt. 24, 162–167 (1985).
    [Crossref] [PubMed]
  27. For example, A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sensing GE-22, 3–13 (1984).
    [Crossref]

1985 (4)

A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Opt. 24, 194–200 (1985).
[Crossref] [PubMed]

S. Kawata, J. Sklansky, “Elimination of nonpivotal plane images from x-ray motion-tomograms,”IEEE Trans. Med. Imaging MI-4, 153–159 (1985).
[Crossref]

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,”IEEE Trans. Med. Imaging MI-4, 65–71 (1985).
[Crossref]

K. Minami, S. Kawata, S. Minami, “Superresolution of Fourier transform spectra by autoregressive model fitting with singular value decomposition,” Appl. Opt. 24, 162–167 (1985).
[Crossref] [PubMed]

1984 (3)

For example, A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sensing GE-22, 3–13 (1984).
[Crossref]

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

N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1984).
[Crossref]

1983 (2)

1981 (1)

1980 (1)

1979 (2)

1978 (1)

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

1976 (1)

L. T. Chang, B. Macdonald, V. Perez-Mendez, “Axial tomography and three dimensional image reconstruction,”IEEE Trans. Nucl. Sci. NS-23, 568–572 (1976).
[Crossref]

1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

1974 (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

1972 (2)

D. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,”IEEE Trans. Biomed. Eng. BME-19, 20–28 (1972).
[Crossref]

P. F. C. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,”J. Theor. Biol. 36, 105–117 (1972).
[Crossref] [PubMed]

1971 (1)

1967 (2)

Abbiss, J. B.

Agard, D. A.

D. A. Agard, J. W. Sedat, “Three-dimensional architecture of a polytene nucleus,” Nature 302, 676–681 (1983).
[Crossref] [PubMed]

Arendt, J. W.

Arsenin, V. V.

A. N. Tikhonov, V. V. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Barrett, H. H.

Bille, J.

Chang, L. T.

L. T. Chang, B. Macdonald, V. Perez-Mendez, “Axial tomography and three dimensional image reconstruction,”IEEE Trans. Nucl. Sci. NS-23, 568–572 (1976).
[Crossref]

Chiu, M. Y.

Cho, C.

De Mol, C.

Defrise, M.

Devaney, A. J.

For example, A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sensing GE-22, 3–13 (1984).
[Crossref]

Dhadwal, H. S.

Erhardt, A.

Frieden, B. R.

B. R. Frieden, “Optical transfer of the three-dimensional object,”J. Opt. Soc. Am. 57, 56–66 (1967).
[Crossref]

For example, B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer-Verlag, Berlin, 1975), Secs. 5.13.2 and 5.4.

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

Gilbert, P. F. C.

P. F. C. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,”J. Theor. Biol. 36, 105–117 (1972).
[Crossref] [PubMed]

Gindi, G. R.

Grant, D.

D. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,”IEEE Trans. Biomed. Eng. BME-19, 20–28 (1972).
[Crossref]

Groh, G.

Helstrom, C. W.

Ichioka, Y.

Inouye, T.

T. Inouye, “Image reconstruction with limited angle projection data,”IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).

Kawata, S.

S. Kawata, J. Sklansky, “Elimination of nonpivotal plane images from x-ray motion-tomograms,”IEEE Trans. Med. Imaging MI-4, 153–159 (1985).
[Crossref]

K. Minami, S. Kawata, S. Minami, “Superresolution of Fourier transform spectra by autoregressive model fitting with singular value decomposition,” Appl. Opt. 24, 162–167 (1985).
[Crossref] [PubMed]

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,”IEEE Trans. Med. Imaging MI-4, 65–71 (1985).
[Crossref]

S. Kawata, Y. Ichioka, “Iterative image restoration for linearly degraded images. I. Basis,”J. Opt. Soc. Am. 70, 762–772 (1980).
[Crossref]

S. Kawata, Y. Touki, S. Minami, “Optical microscopic tomography,” in Inverse Optics II, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.558, 15–20 (1985).
[Crossref]

Komitowski, D.

Lohmann, A. W.

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

Macdonald, B.

L. T. Chang, B. Macdonald, V. Perez-Mendez, “Axial tomography and three dimensional image reconstruction,”IEEE Trans. Nucl. Sci. NS-23, 568–572 (1976).
[Crossref]

Mertz, L.

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

Minami, K.

Minami, S.

K. Minami, S. Kawata, S. Minami, “Superresolution of Fourier transform spectra by autoregressive model fitting with singular value decomposition,” Appl. Opt. 24, 162–167 (1985).
[Crossref] [PubMed]

S. Kawata, Y. Touki, S. Minami, “Optical microscopic tomography,” in Inverse Optics II, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.558, 15–20 (1985).
[Crossref]

Nalcioglu, O.

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,”IEEE Trans. Med. Imaging MI-4, 65–71 (1985).
[Crossref]

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

Perez-Mendez, V.

K. C. Tam, V. Perez-Mendez, “Tomographical imaging with limited-angle input,”J. Opt. Soc. Am. 71, 582–592 (1981).
[Crossref]

L. T. Chang, B. Macdonald, V. Perez-Mendez, “Axial tomography and three dimensional image reconstruction,”IEEE Trans. Nucl. Sci. NS-23, 568–572 (1976).
[Crossref]

Sedat, J. W.

D. A. Agard, J. W. Sedat, “Three-dimensional architecture of a polytene nucleus,” Nature 302, 676–681 (1983).
[Crossref] [PubMed]

Simpson, R. G.

Sklansky, J.

S. Kawata, J. Sklansky, “Elimination of nonpivotal plane images from x-ray motion-tomograms,”IEEE Trans. Med. Imaging MI-4, 153–159 (1985).
[Crossref]

Streibl, N.

N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1984).
[Crossref]

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

Tam, K. C.

Tikhonov, A. N.

A. N. Tikhonov, V. V. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

Touki, Y.

S. Kawata, Y. Touki, S. Minami, “Optical microscopic tomography,” in Inverse Optics II, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.558, 15–20 (1985).
[Crossref]

Wendroff, B.

B. Wendroff, Theoretical Numerical Analysis (Academic, New York, 1966).

Zinser, G.

Appl. Opt. (3)

IEEE Trans. Biomed. Eng. (1)

D. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,”IEEE Trans. Biomed. Eng. BME-19, 20–28 (1972).
[Crossref]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,”IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[Crossref]

IEEE Trans. Geosci. Remote Sensing (1)

For example, A. J. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sensing GE-22, 3–13 (1984).
[Crossref]

IEEE Trans. Med. Imaging (2)

S. Kawata, J. Sklansky, “Elimination of nonpivotal plane images from x-ray motion-tomograms,”IEEE Trans. Med. Imaging MI-4, 153–159 (1985).
[Crossref]

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,”IEEE Trans. Med. Imaging MI-4, 65–71 (1985).
[Crossref]

IEEE Trans. Nucl. Sci. (2)

T. Inouye, “Image reconstruction with limited angle projection data,”IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).

L. T. Chang, B. Macdonald, V. Perez-Mendez, “Axial tomography and three dimensional image reconstruction,”IEEE Trans. Nucl. Sci. NS-23, 568–572 (1976).
[Crossref]

J. Opt. Soc. Am. (6)

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

J. Theor. Biol. (1)

P. F. C. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,”J. Theor. Biol. 36, 105–117 (1972).
[Crossref] [PubMed]

Nature (1)

D. A. Agard, J. W. Sedat, “Three-dimensional architecture of a polytene nucleus,” Nature 302, 676–681 (1983).
[Crossref] [PubMed]

Opt. Acta (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[Crossref]

Optik (2)

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

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

Other (5)

S. Kawata, Y. Touki, S. Minami, “Optical microscopic tomography,” in Inverse Optics II, A. J. Devaney, ed., Proc. Soc. Photo-Opt. Instrum. Eng.558, 15–20 (1985).
[Crossref]

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

B. Wendroff, Theoretical Numerical Analysis (Academic, New York, 1966).

For example, B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer-Verlag, Berlin, 1975), Secs. 5.13.2 and 5.4.

A. N. Tikhonov, V. V. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C., 1977).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the basic idea of optical microscope tomography: (a) reflection and fluorescence systems, (b) transmission system.

Fig. 2
Fig. 2

3-D OTF’s of the absorptive parts (upper plots) and the phase parts (lower parts) for the imaging by (a) nearly coherent illumination (ρ = 0.05), (b) oblique illumination (ρ = 0.1), and (c) incoherent illumination (ρ = 1.0), respectively, where ρ is the ratio of the N.A. for the condenser to that for the objective. (a) and (c) are rotationally symmetric around η axes.

Fig. 3
Fig. 3

Geometry of projection for (a) limited-view x-ray CT and (b) proposed optical microscope tomography. In both systems, the viewing angular range is limited in 3-D space; that for optical microscope tomography is a cone in 3-D space, while that for x-ray CT is a fan in 2-D space. Therefore the information attainable by optical microscope tomography could be much more than limited-view x-ray CT for the same angular range.

Fig. 4
Fig. 4

Block diagram of the developed optical microscope system.

Fig. 5
Fig. 5

Projectional 12 views of Spirogyra observed by the developed optical microscope system. The angle θ shown in Fig. 1(b) was 18°. Angular interval of off-axis pupil between images, every 30°. Each image was digitized into 64 × 64 pixels.

Fig. 6
Fig. 6

Resultant images of 6 neighboring sections of Spirogyra of 12 sections reconstructed from projectional data shown in Fig. 5. The section-to-section distance is 5.4 μm.

Equations (12)

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

p = H o + n ,
e = p - H o ^ 2 + γ o ^ 2
H t p = ( H t H + γ I ) o ^ ,
o ^ = T o ^ ^
T t H t p = T t ( H t H + γ ) T o ^ ^ ,
o k + 1 = o k + α k r k ,
r k = q k - β k - 1 r k - 1 ,
q k + 1 = q k - α k T t ( H t H + γ I ) T r k ,
α k = r k t q k r k t T t ( H t H + γ I ) T r k ,
β k - 1 = q k t T t ( H t H + γ I ) T r k - 1 r k - 1 t T t ( H t H + γ I ) T r k - 1 ,
o 0 = 0
r 0 = q 0 = T t H t p .

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