Abstract

In part I of this series we proposed a new principle of tomographic imaging using an optical microscope devised with a support-constraint reconstruction algorithm [ J. Opt. Soc. Am. A 4, 292 ( 1987)]. In this paper we describe a new reconstruction algorithm with a nonnegative constraint that utilizes a priori knowledge of nonnegativity of absorbance or optical density of three-dimensional sample distribution. This method drastically improves the longitudinal resolution in tomography reconstruction, which was the most serious problem with the developed microscope tomography principle. Compared with conventional nonnegative-constrained iterative reconstruction algorithms, the algorithm developed here essentially reduces computation-time and memory-capacity requirements because it includes conjugate directional searching and gradient projection based on the Kuhn–Tucker condition, guaranteeing the convergence of the algorithm within a finite (the smallest) number of iterations. Results of experiments with a biological specimen, using the proposed algorithm, demonstrate an evident improvement of longitudinal resolution with a computation time only three times longer than that of the fastest nonconstrained algorithm.

© 1988 Optical Society of America

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References

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  1. Y. Maruyama, K. Iwata, R. Nagata, “Measurement of the refractive index distribution in the interior of a solid object from multi-directional interferograms,” Jpn. J. Appl. Phys. 16, 1171–1176 (1977).
    [CrossRef]
  2. G. N. Vishnyakov, G. G. Levin, “Optical tomography of phase objects,” Opt. Spectrosc. (USSR) 53, 434–437 (1982).
  3. B. W. Stuck, “A new proposal for estimating the spatial concentration of certain types of air pollutants,” J. Opt. Soc. Am. 67, 668–678 (1977).
    [CrossRef]
  4. K. E. Bennet, G. W. Faris, R. L. Byer, “Experimental optical fan beam tomography,” Appl. Opt. 23, 2678–2685 (1984).
    [CrossRef]
  5. G. W. Faris, R. L. Byer, “Quantitative optical tomographic imaging,” Opt. Lett. 11, 413–415 (1986).
    [CrossRef] [PubMed]
  6. B. R. Myers, M. A. Levine, “Two-dimensional spectral line emission reconstruction,” Rev. Sci. Instrum. 49, 610–616 (1978).
    [CrossRef]
  7. T. Abe, Y. Mitsunaga, H. Koga, “Photoelastic computer tomography: a novel measurement method for axial residual stress profile in optical fibers,” J. Opt. Soc. Am. A 3, 133–138 (1986).
    [CrossRef]
  8. S. Kawata, O. Nakamura, S. Minami, “Optical microscope tomography. I. Support constraint,” J. Opt. Soc. Am. A 4, 292–297 (1987).
    [CrossRef]
  9. S. Kawata, S. Minami, “The principle and applications of optical microscope tomography,” Acta Histochem. Cytochem. 19, 73–81 (1986).
    [CrossRef]
  10. B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer-Verlag, Berlin, 1975), Sees. 5.13.2 and 5.14.
  11. P. A. Jansson, “Modern constrained nonlinear methods,” in Deconvolution, P. A. Jansson, ed. (Academic, New York, 1984), Chap. 4.
  12. P. A. Jansson, R. H. Hunt, E. K. Plyler, “Resolution enhancement of spectra,” J. Opt. Soc. Am. 60, 596–599 (1970).
    [CrossRef]
  13. R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  14. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—Theory,” IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
    [CrossRef]
  15. M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—Applications and numerical results,” IEEE Trans. Med. Image. MI-1, 95–101 (1982).
    [CrossRef]
  16. K. C. Tam, “The use of multispectral imaging in limited-angle reconstruction,” IEEE Trans Nucl. Sci. NS-29, 512–515 (1982).
    [CrossRef]
  17. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  18. C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974), Chap. 23.
  19. S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imag. MI-4, 65–71 (1985).
    [CrossRef]
  20. S. Kawata, O. Nakamura, S. Minami, “Constrained resolution enhancement in optical microscopic tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 1753–1756 (1986).
  21. D. G. Luenberger, Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1973).
  22. The relation between the least-squares (regularized) iteration and the Gerchberg–Papoulis method is described in J. B. Abbis, 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),and S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imag. MI-4, 65–71 (1985).
    [CrossRef]

1987 (1)

1986 (4)

S. Kawata, S. Minami, “The principle and applications of optical microscope tomography,” Acta Histochem. Cytochem. 19, 73–81 (1986).
[CrossRef]

G. W. Faris, R. L. Byer, “Quantitative optical tomographic imaging,” Opt. Lett. 11, 413–415 (1986).
[CrossRef] [PubMed]

T. Abe, Y. Mitsunaga, H. Koga, “Photoelastic computer tomography: a novel measurement method for axial residual stress profile in optical fibers,” J. Opt. Soc. Am. A 3, 133–138 (1986).
[CrossRef]

S. Kawata, O. Nakamura, S. Minami, “Constrained resolution enhancement in optical microscopic tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 1753–1756 (1986).

1985 (1)

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

1984 (1)

1983 (1)

1982 (5)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—Theory,” IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—Applications and numerical results,” IEEE Trans. Med. Image. MI-1, 95–101 (1982).
[CrossRef]

K. C. Tam, “The use of multispectral imaging in limited-angle reconstruction,” IEEE Trans Nucl. Sci. NS-29, 512–515 (1982).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

G. N. Vishnyakov, G. G. Levin, “Optical tomography of phase objects,” Opt. Spectrosc. (USSR) 53, 434–437 (1982).

1978 (1)

B. R. Myers, M. A. Levine, “Two-dimensional spectral line emission reconstruction,” Rev. Sci. Instrum. 49, 610–616 (1978).
[CrossRef]

1977 (2)

B. W. Stuck, “A new proposal for estimating the spatial concentration of certain types of air pollutants,” J. Opt. Soc. Am. 67, 668–678 (1977).
[CrossRef]

Y. Maruyama, K. Iwata, R. Nagata, “Measurement of the refractive index distribution in the interior of a solid object from multi-directional interferograms,” Jpn. J. Appl. Phys. 16, 1171–1176 (1977).
[CrossRef]

1974 (1)

R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1970 (1)

Abbis, J. B.

Abe, T.

Bennet, K. E.

Byer, R. L.

De Mol, C.

Defrise, M.

Dhadwal, H. S.

Faris, G. W.

Fienup, J. R.

Frieden, B. R.

B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer-Verlag, Berlin, 1975), Sees. 5.13.2 and 5.14.

Gerchberg, R. W.

R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Hanson, R. J.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974), Chap. 23.

Hunt, R. H.

Iwata, K.

Y. Maruyama, K. Iwata, R. Nagata, “Measurement of the refractive index distribution in the interior of a solid object from multi-directional interferograms,” Jpn. J. Appl. Phys. 16, 1171–1176 (1977).
[CrossRef]

Jansson, P. A.

P. A. Jansson, R. H. Hunt, E. K. Plyler, “Resolution enhancement of spectra,” J. Opt. Soc. Am. 60, 596–599 (1970).
[CrossRef]

P. A. Jansson, “Modern constrained nonlinear methods,” in Deconvolution, P. A. Jansson, ed. (Academic, New York, 1984), Chap. 4.

Kawata, S.

S. Kawata, O. Nakamura, S. Minami, “Optical microscope tomography. I. Support constraint,” J. Opt. Soc. Am. A 4, 292–297 (1987).
[CrossRef]

S. Kawata, S. Minami, “The principle and applications of optical microscope tomography,” Acta Histochem. Cytochem. 19, 73–81 (1986).
[CrossRef]

S. Kawata, O. Nakamura, S. Minami, “Constrained resolution enhancement in optical microscopic tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 1753–1756 (1986).

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

Koga, H.

Lawson, C. L.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974), Chap. 23.

Levin, G. G.

G. N. Vishnyakov, G. G. Levin, “Optical tomography of phase objects,” Opt. Spectrosc. (USSR) 53, 434–437 (1982).

Levine, M. A.

B. R. Myers, M. A. Levine, “Two-dimensional spectral line emission reconstruction,” Rev. Sci. Instrum. 49, 610–616 (1978).
[CrossRef]

Luenberger, D. G.

D. G. Luenberger, Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1973).

Maruyama, Y.

Y. Maruyama, K. Iwata, R. Nagata, “Measurement of the refractive index distribution in the interior of a solid object from multi-directional interferograms,” Jpn. J. Appl. Phys. 16, 1171–1176 (1977).
[CrossRef]

Minami, S.

S. Kawata, O. Nakamura, S. Minami, “Optical microscope tomography. I. Support constraint,” J. Opt. Soc. Am. A 4, 292–297 (1987).
[CrossRef]

S. Kawata, S. Minami, “The principle and applications of optical microscope tomography,” Acta Histochem. Cytochem. 19, 73–81 (1986).
[CrossRef]

S. Kawata, O. Nakamura, S. Minami, “Constrained resolution enhancement in optical microscopic tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 1753–1756 (1986).

Mitsunaga, Y.

Myers, B. R.

B. R. Myers, M. A. Levine, “Two-dimensional spectral line emission reconstruction,” Rev. Sci. Instrum. 49, 610–616 (1978).
[CrossRef]

Nagata, R.

Y. Maruyama, K. Iwata, R. Nagata, “Measurement of the refractive index distribution in the interior of a solid object from multi-directional interferograms,” Jpn. J. Appl. Phys. 16, 1171–1176 (1977).
[CrossRef]

Nakamura, O.

S. Kawata, O. Nakamura, S. Minami, “Optical microscope tomography. I. Support constraint,” J. Opt. Soc. Am. A 4, 292–297 (1987).
[CrossRef]

S. Kawata, O. Nakamura, S. Minami, “Constrained resolution enhancement in optical microscopic tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 1753–1756 (1986).

Nalcioglu, O.

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

Plyler, E. K.

Sezan, M. I.

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—Applications and numerical results,” IEEE Trans. Med. Image. MI-1, 95–101 (1982).
[CrossRef]

Stark, H.

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—Applications and numerical results,” IEEE Trans. Med. Image. MI-1, 95–101 (1982).
[CrossRef]

Stuck, B. W.

Tam, K. C.

K. C. Tam, “The use of multispectral imaging in limited-angle reconstruction,” IEEE Trans Nucl. Sci. NS-29, 512–515 (1982).
[CrossRef]

Vishnyakov, G. N.

G. N. Vishnyakov, G. G. Levin, “Optical tomography of phase objects,” Opt. Spectrosc. (USSR) 53, 434–437 (1982).

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—Theory,” IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—Theory,” IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

Acta Histochem. Cytochem. (1)

S. Kawata, S. Minami, “The principle and applications of optical microscope tomography,” Acta Histochem. Cytochem. 19, 73–81 (1986).
[CrossRef]

Appl. Opt. (2)

IEEE Trans Nucl. Sci. (1)

K. C. Tam, “The use of multispectral imaging in limited-angle reconstruction,” IEEE Trans Nucl. Sci. NS-29, 512–515 (1982).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

S. Kawata, O. Nakamura, S. Minami, “Constrained resolution enhancement in optical microscopic tomography,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 1753–1756 (1986).

IEEE Trans. Med. Imag. (2)

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

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: part 1—Theory,” IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

IEEE Trans. Med. Image. (1)

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections: part 2—Applications and numerical results,” IEEE Trans. Med. Image. MI-1, 95–101 (1982).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Jpn. J. Appl. Phys. (1)

Y. Maruyama, K. Iwata, R. Nagata, “Measurement of the refractive index distribution in the interior of a solid object from multi-directional interferograms,” Jpn. J. Appl. Phys. 16, 1171–1176 (1977).
[CrossRef]

Opt. Acta (1)

R. W. Gerchberg, “Superresolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (USSR) (1)

G. N. Vishnyakov, G. G. Levin, “Optical tomography of phase objects,” Opt. Spectrosc. (USSR) 53, 434–437 (1982).

Rev. Sci. Instrum. (1)

B. R. Myers, M. A. Levine, “Two-dimensional spectral line emission reconstruction,” Rev. Sci. Instrum. 49, 610–616 (1978).
[CrossRef]

Other (4)

B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer-Verlag, Berlin, 1975), Sees. 5.13.2 and 5.14.

P. A. Jansson, “Modern constrained nonlinear methods,” in Deconvolution, P. A. Jansson, ed. (Academic, New York, 1984), Chap. 4.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974), Chap. 23.

D. G. Luenberger, Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1973).

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Figures (7)

Fig. 1
Fig. 1

Explanatory graphs of a contour map of the squared error e of the two variates o1 and o2 by an iterative procedure including (a) the steepest-descent method started with ô and (b) the proposed method.

Fig. 2
Fig. 2

Block diagram of the developed tomographic optical microscope system.8,9 CCD, Charge-coupled device; GPIB, general-purpose interface bus.

Fig. 3
Fig. 3

Four of twelve projections of Spirogyra by the developed system. Each image comprises 64 × 64 pixels with a resolution of 2.3 μm × 1.8 μm.

Fig. 4
Fig. 4

Geometry of projections in the experiment.

Fig. 5
Fig. 5

(a) Resultant 3-D distribution of Spirogyra reconstructed by the conjugate-gradient method with only a thickness constraint. The longitudinal spacing between the slices is 5.4 μm. (b) Longitudinal cut of the reconstructed object along the line shown in the upper right-hand slice in (a).

Fig. 6
Fig. 6

Distribution pattern of negative values in Fig. 5(a).

Fig. 7
Fig. 7

(a) Resultant 3-D distribution of Spirogyra reconstructed by the proposed nonnegative constrained method, (b) Longitudinal cut of the reconstructed object along the line shown in the upper right-hand slice in (a).

Equations (17)

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

p = H o + n ,
e = ¯ p H T ô ̂ ¯ 2 + γ ¯ T ô ̂ 2 .
ô ̂ = [ T t ( H t H + γ I ) T ] 1 T t H t p .
e = ¯ p H T o * ¯ 2 + γ ¯ T o * ¯ 2 minimum ,
o * 0 ,
o * 0 ,
e / o * 0 ,
( o * ) t ( e / o * ) = 0 ,
e / o * = 2 ( T t ( H t H + γ I ) T o * T t H t p ) .
o k + 1 = o k + 1 / 2 α k ( e k / o k ) ;
if o j k + 1 < 0 , then set o j k + 1 = 0 for all j ;
e k = p H T o k 2 + γ T o k 2
e k / o k = 2 ( T t ( H t H + γ I ) T o k T t H t p ) .
α k = e k / o k 2 / H e k / o k 2 .
d j j = { 0 if e k / o j k > 0 and o j k = 0 , for all j 1 otherwise .
o = ( D t T t ( H t H + γ I ) T D ) 1 D t T t H t p
o j k + 1 = 0 for all j .

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