Abstract

A quantitative method for reconstruction and superresolution of red-blood-cell image holograms (interferograms) is described, with resolution improvement to 4% of the wavelength; the approach is based on the principles of coherent physical optics and is automated in the form of computer programs. The red-blood-cell phase information is reconstructed from the hologram; then the phase-distribution resolution is improved using symmetry, surface-continuous differentiability, and space-limited conditions.

© 1971 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 418.
  2. Y. C. Fung, Federation Am. Soc. Exptl. Biol. 25, 1761 (1966).
  3. Y. C. Fung and P. Tong, Biophys. J. 8, 175 (1968).
  4. Y. C. Fung, in Biomedical Sciences Instrumentation, Vol. 4, edited by R. D. Allison (Plenum, New York, 1968), p. 310.
  5. F. Walter, Blut Z. Blutforsch.  9, 297 (1963).
    [Crossref]
  6. O. Bryngdahl and A. W. Lohmann, J. Opt. Soc. Am. 58, 141 (1968).
    [Crossref]
  7. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123 (1962).
    [Crossref]
  8. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 54, 1295 (1964).
    [Crossref]
  9. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 1377 (1963).
    [Crossref]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  11. L. C. Martin, The Theory of the Microscope (American Elsevier, New York, 1966).
  12. E. Wolf, Opt. Commun. 1, 153 (1969); J. Opt. Soc. Am. 60, 18 (1970).
    [Crossref]
  13. W. H. Carter, J. Opt. Soc. Am. 60, 306 (1970).
    [Crossref]
  14. G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
    [Crossref]
  15. C. W. McCutchen, J. Opt. Soc. Am. 57, 1190 (1967).
    [Crossref] [PubMed]
  16. E. A. Evans, Opt. Commun. 2, 317 (1970).
    [Crossref]
  17. J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).
    [Crossref]
  18. C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966).
    [Crossref]
  19. B. R. Frieden, J. Opt. Soc. Am. 57, 1013 (1967).
    [Crossref]
  20. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw–Hill, New York, 1968).
  21. I. H. Barkdoll and B. L. McGlamery, in Proceedings of the 23rd National Conference of the Association of Computing Machinery (Brandon/Systems Press, Princeton, N. J., 1968).
  22. J. W. Cooley and J. W. Tukey, Math. Comp. 19, 297 (1965).
    [Crossref]
  23. A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill, New York, 1962).
  24. D. L. Marquardt, J. Soc. Indust. Appl. Math. 2, 431 (1963).
  25. R. Barer and S. Joseph, Quart. J. Microscop. Sci. 95, 399 (1954);Quart. J. Microscop. Sci. 96, 1 (1955).

1970 (2)

1969 (1)

E. Wolf, Opt. Commun. 1, 153 (1969); J. Opt. Soc. Am. 60, 18 (1970).
[Crossref]

1968 (2)

Y. C. Fung and P. Tong, Biophys. J. 8, 175 (1968).

O. Bryngdahl and A. W. Lohmann, J. Opt. Soc. Am. 58, 141 (1968).
[Crossref]

1967 (2)

1966 (2)

C. W. Barnes, J. Opt. Soc. Am. 56, 575 (1966).
[Crossref]

Y. C. Fung, Federation Am. Soc. Exptl. Biol. 25, 1761 (1966).

1965 (1)

J. W. Cooley and J. W. Tukey, Math. Comp. 19, 297 (1965).
[Crossref]

1964 (2)

1963 (3)

E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 1377 (1963).
[Crossref]

F. Walter, Blut Z. Blutforsch.  9, 297 (1963).
[Crossref]

D. L. Marquardt, J. Soc. Indust. Appl. Math. 2, 431 (1963).

1962 (1)

1954 (1)

R. Barer and S. Joseph, Quart. J. Microscop. Sci. 95, 399 (1954);Quart. J. Microscop. Sci. 96, 1 (1955).

1952 (1)

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[Crossref]

Barer, R.

R. Barer and S. Joseph, Quart. J. Microscop. Sci. 95, 399 (1954);Quart. J. Microscop. Sci. 96, 1 (1955).

Barkdoll, I. H.

I. H. Barkdoll and B. L. McGlamery, in Proceedings of the 23rd National Conference of the Association of Computing Machinery (Brandon/Systems Press, Princeton, N. J., 1968).

Barnes, C. W.

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 418.

Bryngdahl, O.

Carter, W. H.

Cooley, J. W.

J. W. Cooley and J. W. Tukey, Math. Comp. 19, 297 (1965).
[Crossref]

Evans, E. A.

E. A. Evans, Opt. Commun. 2, 317 (1970).
[Crossref]

Frieden, B. R.

Fung, Y. C.

Y. C. Fung and P. Tong, Biophys. J. 8, 175 (1968).

Y. C. Fung, Federation Am. Soc. Exptl. Biol. 25, 1761 (1966).

Y. C. Fung, in Biomedical Sciences Instrumentation, Vol. 4, edited by R. D. Allison (Plenum, New York, 1968), p. 310.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Harris, J. L.

Joseph, S.

R. Barer and S. Joseph, Quart. J. Microscop. Sci. 95, 399 (1954);Quart. J. Microscop. Sci. 96, 1 (1955).

Leith, E. N.

Lohmann, A. W.

Marquardt, D. L.

D. L. Marquardt, J. Soc. Indust. Appl. Math. 2, 431 (1963).

Martin, L. C.

L. C. Martin, The Theory of the Microscope (American Elsevier, New York, 1966).

McCutchen, C. W.

McGlamery, B. L.

I. H. Barkdoll and B. L. McGlamery, in Proceedings of the 23rd National Conference of the Association of Computing Machinery (Brandon/Systems Press, Princeton, N. J., 1968).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw–Hill, New York, 1968).

A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill, New York, 1962).

Tong, P.

Y. C. Fung and P. Tong, Biophys. J. 8, 175 (1968).

Toraldo di Francia, G.

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[Crossref]

Tukey, J. W.

J. W. Cooley and J. W. Tukey, Math. Comp. 19, 297 (1965).
[Crossref]

Upatnieks, J.

Walter, F.

F. Walter, Blut Z. Blutforsch.  9, 297 (1963).
[Crossref]

Wolf, E.

E. Wolf, Opt. Commun. 1, 153 (1969); J. Opt. Soc. Am. 60, 18 (1970).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 418.

Biophys. J. (1)

Y. C. Fung and P. Tong, Biophys. J. 8, 175 (1968).

Blut Z. Blutforsch (1)

F. Walter, Blut Z. Blutforsch.  9, 297 (1963).
[Crossref]

Federation Am. Soc. Exptl. Biol. (1)

Y. C. Fung, Federation Am. Soc. Exptl. Biol. 25, 1761 (1966).

J. Opt. Soc. Am. (9)

J. Soc. Indust. Appl. Math. (1)

D. L. Marquardt, J. Soc. Indust. Appl. Math. 2, 431 (1963).

Math. Comp. (1)

J. W. Cooley and J. W. Tukey, Math. Comp. 19, 297 (1965).
[Crossref]

Nuovo Cimento Suppl. (1)

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[Crossref]

Opt. Commun. (2)

E. A. Evans, Opt. Commun. 2, 317 (1970).
[Crossref]

E. Wolf, Opt. Commun. 1, 153 (1969); J. Opt. Soc. Am. 60, 18 (1970).
[Crossref]

Quart. J. Microscop. Sci. (1)

R. Barer and S. Joseph, Quart. J. Microscop. Sci. 95, 399 (1954);Quart. J. Microscop. Sci. 96, 1 (1955).

Other (7)

A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill, New York, 1962).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw–Hill, New York, 1968).

I. H. Barkdoll and B. L. McGlamery, in Proceedings of the 23rd National Conference of the Association of Computing Machinery (Brandon/Systems Press, Princeton, N. J., 1968).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

L. C. Martin, The Theory of the Microscope (American Elsevier, New York, 1966).

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965), p. 418.

Y. C. Fung, in Biomedical Sciences Instrumentation, Vol. 4, edited by R. D. Allison (Plenum, New York, 1968), p. 310.

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Figures (11)

Fig. 1
Fig. 1

Image hologram (interferogram) of a red blood cell in the live state using 0.546-μm illumination.

Fig. 2
Fig. 2

Schematic of Leitz interference microscope as an image-hologram construction device.

Fig. 3
Fig. 3

Spatial-frequency-plane map of the Fourier-transformed image-hologram irradiance distribution.

Fig. 4
Fig. 4

Illustration of the presumed red-blood-cell biconcave cross section.

Fig. 5
Fig. 5

The image phase for X cuts through the cell center and on either side of the cell are shown as reconstructed by computation from Fig. 1; corrected for phase-plane tilt. The abscissa is in sampling units (4.89 sampling units per micrometer).

Fig. 6
Fig. 6

The image phase for Y cuts through the cell center and on either side of the cell are shown as reconstructed by computation from Fig. 1; corrected for phase-plane tilt. The abscissa is in sampling units (4.89 sampling units per micrometer).

Fig. 7
Fig. 7

The image phase for cuts through the cell center and on either side of the cell are shown as reconstructed by computation, using a νc=7 artificial-aperture function. The abscissa is in sampling units (4.89 sampling units per micrometer).

Fig. 8
Fig. 8

A radial section of the red-cell-image phase (using the νc=7 artificial-aperture function) from Fig. 7. The abscissa is in sampling units (4.89 sampling units per micrometer).

Fig. 9
Fig. 9

The sum-of-square errors as a function of the trial radius, illustrating the sharp minimum. (This figure is for the object search using Fig. 8 data and R0, C0, C2, C3 trial coefficients.) The abscissa is in sampling units (4.89 sampling units per micrometer).

Fig. 10
Fig. 10

A radial section of the best-trial-image phase (using νc=7 artificial-aperture function) found by computation (to be compared with Fig. 8). The abscissa is in sampling units (4.89 sampling units per micrometer).

Fig. 11
Fig. 11

The optimum-object-phase radial section corresponding to Fig. 10 image phase, which represents the superresolved object from the red-cell data, Fig. 8. The abscissa is in sampling units (4.89 sampling units per micrometer).

Equations (35)

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U H ( X , Y , Z , t ) = U R ( X , Y , Z , t ) + U ¯ I ( X , Y , Z , t ) ,
U R ( X , Y , Z , t ) = exp ( 2 π i Z cos θ λ - i ω t ) e - π i α Y ,
α ( 2 sin θ ) / λ .
U ¯ I ( X , Y , Z , t ) = exp ( 2 π i Z cos θ λ - i ω t ) × e π i α Y U I ( X , Y , Z ) ,
U I ( X , Y , Z ) = 1 M - h ( X - X ˜ 0 , Y - Y ˜ 0 , Z , Z 0 ) × U 0 ( - X ˜ 0 M , - Y ˜ 0 M ) d X ˜ 0 d Y ˜ 0 .
U 0 ( X 0 , Y 0 ) = A 0 ( X 0 , Y 0 ) exp [ i φ 0 ( X 0 , Y 0 ) ] .
U I = h * U 0 / M
F ( U I ) = F ( h ) F ( U 0 ) / M .
I H = U H U H * .
I H = e - π i α Y + e π i α Y U I ( X , Y ) 2 , I H = 1 + U I ( X , Y ) U I * ( X , Y ) + e 2 π i α Y U I ( X , Y ) + e - 2 π i α Y U I * ( X , Y ) .
φ 0 ( X , Y ) = ( 2 π / λ ) Δ η D ( X , Y ) .
Î H ( ν , μ ) = δ ( ν , μ ) + Û I 2 ( ν , μ ) + Û I ( ν , μ - α ) + Û I * ( ν , μ + α ) ,
δ ( ν , μ ) = lim N sin ( N π ν ) sin ( N π μ ) π 2 ν μ .
α 3 2 B Y .
α M A X = 2 / λ 3 2 B Y ,
B Y 4 3 λ .
α B Y / 2.
Î N ( ν , μ ) = Û I ( ν , μ ) .
F - 1 ( Î N ) = U 0 ( X , Y ) = A 0 ( X , Y ) exp [ i φ 0 ( X , Y ) ] .
U 0 ( X , Y ) = f ( X , Y ) ,
U I = h * U 0 / M .
= U I - U I .
2 = - U I - U I 2 d X d Y .
2 = 1 4 π 2 - Û I - Û I 2 d ν d μ .
2 = 1 4 π 2 M 2 Σ ν μ Û 0 - Û 0 2 d ν d μ ,
Σ ν μ the limitation ( ν 2 + μ 2 ) 1 2 ν c .
D ( X ) = ( 1 - X n ) m ( C 0 + C 1 X + C 2 X 2 + ) .
d D / d X = - m n X n - 1 ( 1 - X n ) m - 1 ( C 0 + C 1 X + C 2 X 2 + ) + ( 1 - X n ) m ( C 1 + 2 C 2 X + ) .
D ( X ) = ( 1 - X 2 ) 1 2 ( C 0 + C 2 X 2 + C 3 X 3 + ) .
U 0 ( r ) = exp [ i φ 0 ( r ) ] , U I = h * U 0 / M ,
φ 0 ( r ) = ( 2 π / λ ) Δ η D ( r / R 0 ) .
B Y / 2 = μ M - α .
Reconstructed U I = U I exp { 2 π i [ ( δ α X ) X + ( δ α Y ) Y ] } .
λ 2 ν e 2 < 0.6 ( λ / D ¯ ) ,
D ( X ) = ( 1 - X 2 ) 1 2 ( C 0 + C 2 X 2 + C 2 X 3 + C 4 X 4 + C 5 X 5 ) 1 2 ,