Abstract

The observed density of a photographic record is dependent on the aperture of the illuminating and imaging optics. This effect, due to the scattering by the photographic emulsion, is analyzed using as few assumptions as possible. The results show a reasonable agreement with experiments on Callier effect found in the literature. In the realm of a study about noise and coherence in image processing, the influence of the Calier effect on image contrast is discussed. It is found in particular that contrast variations may be large (over a factor of 10 in some cases) and that the best contrast is obtained for a certin partially coherent illumination.

© 1978 Optical Society of America

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References

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  1. G. Häusler and A. W. Lohmann, “Hybrid Image Processing,” in Proceedings of the International Conference on Applications of Holography and Optical Data Processing (International Commission of Optics,. Jerusalem, 1976), paper MB-1.
  2. G. L. Rogers, “Non-coherent optical processing,” Opt. Las. Technol. 7, 153–162(1975).
    [Crossref]
  3. P. Chavel and S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,” J. Opt. Soc. Am. 66, 14–23(1976).
    [Crossref]
  4. J. W. Goodman, “Noise in Coherent Optical Information Processing,” in Optical Information Processing, edited by Yu. E. Nesterikhin, G. W. Stroke, and W. E. Kock (Plenum, New York, 1976), pp. 85–103.
    [Crossref]
  5. D. Tichenor and J. W. Goodman, “Practical Noise Limitations in Holographic Image Deblurring,” in Proceedings of the International Optical Computing Conference (I.E.E.E., Washington, D.C., 1975, I.E.E.E. Catalog No. 75 CH 0941-5C), pp. 82–84.
  6. R. S. Powers and J. W. Goodman, “Error rates in computer-generated holographic memories,” Appl. Opt. 14, 1690–1701(1975).
    [Crossref] [PubMed]
  7. W. M. W. Abney, “On the accuracy of the grease spot photometer for measuring the density of photographic plates, and a note on the sector photometer,” J. Soc. Chem. Ind. 9, 722–725(1890).
  8. W. M. W. Abney, “Grease spot photometer measures,” J. Soc. Chem. Ind. 10, 18–20(1891).
  9. F. Hurter and V. C. Driffield, “Reply to the preceding communication of Captain Abney, ‘On the accuracy of the grease spot photometer for measuring the density of photographic plates, and a note on the sector photometer’,” J. Soc. Chem. Ind. 9, 725(1890).
  10. F. Hurter and V. C. Driffield, “The sector and grease spot photometers and their results,” J. Soc. Chem. Ind. 10, 20–24(1891).
  11. F. Hurter and V. C. Driffield, “The sector and grease spot photometers,” J. Soc. Chem. Ind. 10, 98–100(1891).
  12. F. Hurter, “The sector and grease spot photometers,” J. Soc. Chem. Ind. 10, 318(1891).
  13. A. Callier, “Absorption und Diffusion des Lichtes in der entwickelten photographischen Platte, nach Messungen mit dem Martensschen Polarisationphotometer,” Z. Wiss. Photogr. Photophys. Photochem. 8, 257–272(1909).
  14. A. Callier, Photogr. J. 49(NS33), 200(1909).
  15. P. Vernier, “Sur l’action photographique des électrons—Application aux observations astronomiques,” Bull. Astron. 12, 84–127(1959).
  16. P. Vernier, “Etude de l’effet Callier et de sa relation avec la granularité des plaques photographiques,” C. R. Acad. Sci. (Paris) 246, 1527–1530(1958).
  17. P. Vernier, “Conditions for linearity between density and exposure for photographic plates,” J. Opt. Soc. Am. 59, 444–454(1969).
    [Crossref]
  18. F. Hurter and V. C. Driffield, “Photo-chemical investigations and a new method of determination of the sensitiveness of photographic plates,” J. Soc. Chem. Ind. 9, 455–469(1890).
    [Crossref]
  19. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Chap. X.
  20. R. Görisch, “Untersuchungen über den Callier-Effekt I,” Z. Wiss. Photogr. Photophys. Photochem. 48, 85–102(1953).
  21. J. L. Tupper, “General sensitometry,” in The Theory of the Photographic Process, 3rd ed., edited by C. E. K. Mees and T. H. James (Macmillan, New York, 1966), pp. 409–436.
  22. P. Vernier, “Définition du contour du spot d’un densitomètre, effet Callier et granulation photographique,” Rev. Opt. 39, 505–515(1960).
  23. A. M. Koerner and C. Tuttle, “Experimental determination of photographic density,” J. Opt. Soc. Am. 27, 241–256(1937).
    [Crossref]
  24. C. Tuttle and J. W. McFarlane, “The measurement of density in variable density sound film,” J. Soc. Motion Pict. Telev. Eng. 15, 345–351(1930).
  25. H. J. McNicholas, “Absolute methods in reflectometry,” Natl. Bur. Stand. J. Res. 1, 29–73(1928).
  26. H. von Helmholtz, Handbuch der physiologischen Optik, 1st ed. (Voss, Leipzig, 1856), pp. 166–191.
  27. P. Croce and M. Marquet, “Détermination optique du “bruit de fond” photographique,” Opt. Acta 2, 107–108(1955).
    [Crossref]
  28. G. B. Brandt, “Coherent optical power spectra of photographic materials,” Appl. Opt. 9, 1424–1428(1970).
    [Crossref] [PubMed]
  29. K. Biedermann, “The scattered flux spectrum of photographic materials for holography,” Optik 31, 367–389(1970).
  30. E. N. Leith, “Photographic film as an element of a coherent optical system,” J. Photogr. Soc. Am. 6, 75–80(1962).
  31. C. B. Burckhardt, “Storage capacity of an optically formed spatial filter for character recognition,” Appl. Opt. 6, 1359–1366(1967).
    [Crossref] [PubMed]
  32. H. Stark, W. R. Bennett, and M. Arm, “Design considerations in power spectra measurements by diffraction of coherent light,” Appl. Opt. 8, 2165–2172(1969).
    [Crossref] [PubMed]
  33. H. Stark, “Some film-noise measurements by diffraction of coherent light,” Appl. Opt. 10, 333–337(1971).
    [Crossref] [PubMed]
  34. D. H. R. Vilkomerson, “Measurements of the noise spectral power density of photosensitive materials at high spatial frequencies,” Appl. Opt. 9, 2080–2087(1970).
    [Crossref] [PubMed]
  35. H. M. Smith, “Light scattering in photographic materials for holography,” Appl. Opt. 11, 26–32(1972).
    [Crossref] [PubMed]
  36. C. E. Thomas, “Film characteristics pertinent to coherent optical data processing systems,” Appl. Opt. 11, 1756–1765(1972).
    [Crossref] [PubMed]
  37. J. C. Dainty and R. Shaw, Image Science (Academic, London, 1974), pp. 68–115.
  38. C. Lanczos, Applied Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1956).
  39. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 887 and 916.
  40. S. Lowenthal and P. Chavel, “Noise Problems in Optical Image Processing,” in Proceedings of the International Conference on Applications of Holography and Optical Data Processing (International Commission of Optics, Jerusalem, 1976), paper TuA-1.
  41. R. E. Swing, “Conditions for microdensitometer linearity,” J. Opt. Soc. Am. 62, 199–207(1972).
    [Crossref]
  42. R. E. Kinzly, “Partially coherent imaging in a microdensitometer,” J. Opt. Soc. Am. 62, 386–394(1972).
    [Crossref]

1976 (1)

1975 (2)

1972 (4)

1971 (1)

1970 (3)

1969 (2)

1967 (1)

1962 (1)

E. N. Leith, “Photographic film as an element of a coherent optical system,” J. Photogr. Soc. Am. 6, 75–80(1962).

1960 (1)

P. Vernier, “Définition du contour du spot d’un densitomètre, effet Callier et granulation photographique,” Rev. Opt. 39, 505–515(1960).

1959 (1)

P. Vernier, “Sur l’action photographique des électrons—Application aux observations astronomiques,” Bull. Astron. 12, 84–127(1959).

1958 (1)

P. Vernier, “Etude de l’effet Callier et de sa relation avec la granularité des plaques photographiques,” C. R. Acad. Sci. (Paris) 246, 1527–1530(1958).

1955 (1)

P. Croce and M. Marquet, “Détermination optique du “bruit de fond” photographique,” Opt. Acta 2, 107–108(1955).
[Crossref]

1953 (1)

R. Görisch, “Untersuchungen über den Callier-Effekt I,” Z. Wiss. Photogr. Photophys. Photochem. 48, 85–102(1953).

1937 (1)

1930 (1)

C. Tuttle and J. W. McFarlane, “The measurement of density in variable density sound film,” J. Soc. Motion Pict. Telev. Eng. 15, 345–351(1930).

1928 (1)

H. J. McNicholas, “Absolute methods in reflectometry,” Natl. Bur. Stand. J. Res. 1, 29–73(1928).

1909 (2)

A. Callier, “Absorption und Diffusion des Lichtes in der entwickelten photographischen Platte, nach Messungen mit dem Martensschen Polarisationphotometer,” Z. Wiss. Photogr. Photophys. Photochem. 8, 257–272(1909).

A. Callier, Photogr. J. 49(NS33), 200(1909).

1891 (4)

F. Hurter and V. C. Driffield, “The sector and grease spot photometers and their results,” J. Soc. Chem. Ind. 10, 20–24(1891).

F. Hurter and V. C. Driffield, “The sector and grease spot photometers,” J. Soc. Chem. Ind. 10, 98–100(1891).

F. Hurter, “The sector and grease spot photometers,” J. Soc. Chem. Ind. 10, 318(1891).

W. M. W. Abney, “Grease spot photometer measures,” J. Soc. Chem. Ind. 10, 18–20(1891).

1890 (3)

F. Hurter and V. C. Driffield, “Reply to the preceding communication of Captain Abney, ‘On the accuracy of the grease spot photometer for measuring the density of photographic plates, and a note on the sector photometer’,” J. Soc. Chem. Ind. 9, 725(1890).

W. M. W. Abney, “On the accuracy of the grease spot photometer for measuring the density of photographic plates, and a note on the sector photometer,” J. Soc. Chem. Ind. 9, 722–725(1890).

F. Hurter and V. C. Driffield, “Photo-chemical investigations and a new method of determination of the sensitiveness of photographic plates,” J. Soc. Chem. Ind. 9, 455–469(1890).
[Crossref]

Abney, W. M. W.

W. M. W. Abney, “Grease spot photometer measures,” J. Soc. Chem. Ind. 10, 18–20(1891).

W. M. W. Abney, “On the accuracy of the grease spot photometer for measuring the density of photographic plates, and a note on the sector photometer,” J. Soc. Chem. Ind. 9, 722–725(1890).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 887 and 916.

Arm, M.

Bennett, W. R.

Biedermann, K.

K. Biedermann, “The scattered flux spectrum of photographic materials for holography,” Optik 31, 367–389(1970).

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Chap. X.

Brandt, G. B.

Burckhardt, C. B.

Callier, A.

A. Callier, “Absorption und Diffusion des Lichtes in der entwickelten photographischen Platte, nach Messungen mit dem Martensschen Polarisationphotometer,” Z. Wiss. Photogr. Photophys. Photochem. 8, 257–272(1909).

A. Callier, Photogr. J. 49(NS33), 200(1909).

Chavel, P.

P. Chavel and S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,” J. Opt. Soc. Am. 66, 14–23(1976).
[Crossref]

S. Lowenthal and P. Chavel, “Noise Problems in Optical Image Processing,” in Proceedings of the International Conference on Applications of Holography and Optical Data Processing (International Commission of Optics, Jerusalem, 1976), paper TuA-1.

Croce, P.

P. Croce and M. Marquet, “Détermination optique du “bruit de fond” photographique,” Opt. Acta 2, 107–108(1955).
[Crossref]

Dainty, J. C.

J. C. Dainty and R. Shaw, Image Science (Academic, London, 1974), pp. 68–115.

Driffield, V. C.

F. Hurter and V. C. Driffield, “The sector and grease spot photometers,” J. Soc. Chem. Ind. 10, 98–100(1891).

F. Hurter and V. C. Driffield, “The sector and grease spot photometers and their results,” J. Soc. Chem. Ind. 10, 20–24(1891).

F. Hurter and V. C. Driffield, “Reply to the preceding communication of Captain Abney, ‘On the accuracy of the grease spot photometer for measuring the density of photographic plates, and a note on the sector photometer’,” J. Soc. Chem. Ind. 9, 725(1890).

F. Hurter and V. C. Driffield, “Photo-chemical investigations and a new method of determination of the sensitiveness of photographic plates,” J. Soc. Chem. Ind. 9, 455–469(1890).
[Crossref]

Goodman, J. W.

R. S. Powers and J. W. Goodman, “Error rates in computer-generated holographic memories,” Appl. Opt. 14, 1690–1701(1975).
[Crossref] [PubMed]

D. Tichenor and J. W. Goodman, “Practical Noise Limitations in Holographic Image Deblurring,” in Proceedings of the International Optical Computing Conference (I.E.E.E., Washington, D.C., 1975, I.E.E.E. Catalog No. 75 CH 0941-5C), pp. 82–84.

J. W. Goodman, “Noise in Coherent Optical Information Processing,” in Optical Information Processing, edited by Yu. E. Nesterikhin, G. W. Stroke, and W. E. Kock (Plenum, New York, 1976), pp. 85–103.
[Crossref]

Görisch, R.

R. Görisch, “Untersuchungen über den Callier-Effekt I,” Z. Wiss. Photogr. Photophys. Photochem. 48, 85–102(1953).

Häusler, G.

G. Häusler and A. W. Lohmann, “Hybrid Image Processing,” in Proceedings of the International Conference on Applications of Holography and Optical Data Processing (International Commission of Optics,. Jerusalem, 1976), paper MB-1.

Hurter, F.

F. Hurter and V. C. Driffield, “The sector and grease spot photometers and their results,” J. Soc. Chem. Ind. 10, 20–24(1891).

F. Hurter and V. C. Driffield, “The sector and grease spot photometers,” J. Soc. Chem. Ind. 10, 98–100(1891).

F. Hurter, “The sector and grease spot photometers,” J. Soc. Chem. Ind. 10, 318(1891).

F. Hurter and V. C. Driffield, “Photo-chemical investigations and a new method of determination of the sensitiveness of photographic plates,” J. Soc. Chem. Ind. 9, 455–469(1890).
[Crossref]

F. Hurter and V. C. Driffield, “Reply to the preceding communication of Captain Abney, ‘On the accuracy of the grease spot photometer for measuring the density of photographic plates, and a note on the sector photometer’,” J. Soc. Chem. Ind. 9, 725(1890).

Kinzly, R. E.

Koerner, A. M.

Lanczos, C.

C. Lanczos, Applied Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1956).

Leith, E. N.

E. N. Leith, “Photographic film as an element of a coherent optical system,” J. Photogr. Soc. Am. 6, 75–80(1962).

Lohmann, A. W.

G. Häusler and A. W. Lohmann, “Hybrid Image Processing,” in Proceedings of the International Conference on Applications of Holography and Optical Data Processing (International Commission of Optics,. Jerusalem, 1976), paper MB-1.

Lowenthal, S.

P. Chavel and S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,” J. Opt. Soc. Am. 66, 14–23(1976).
[Crossref]

S. Lowenthal and P. Chavel, “Noise Problems in Optical Image Processing,” in Proceedings of the International Conference on Applications of Holography and Optical Data Processing (International Commission of Optics, Jerusalem, 1976), paper TuA-1.

Marquet, M.

P. Croce and M. Marquet, “Détermination optique du “bruit de fond” photographique,” Opt. Acta 2, 107–108(1955).
[Crossref]

McFarlane, J. W.

C. Tuttle and J. W. McFarlane, “The measurement of density in variable density sound film,” J. Soc. Motion Pict. Telev. Eng. 15, 345–351(1930).

McNicholas, H. J.

H. J. McNicholas, “Absolute methods in reflectometry,” Natl. Bur. Stand. J. Res. 1, 29–73(1928).

Powers, R. S.

Rogers, G. L.

G. L. Rogers, “Non-coherent optical processing,” Opt. Las. Technol. 7, 153–162(1975).
[Crossref]

Shaw, R.

J. C. Dainty and R. Shaw, Image Science (Academic, London, 1974), pp. 68–115.

Smith, H. M.

Stark, H.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 887 and 916.

Swing, R. E.

Thomas, C. E.

Tichenor, D.

D. Tichenor and J. W. Goodman, “Practical Noise Limitations in Holographic Image Deblurring,” in Proceedings of the International Optical Computing Conference (I.E.E.E., Washington, D.C., 1975, I.E.E.E. Catalog No. 75 CH 0941-5C), pp. 82–84.

Tupper, J. L.

J. L. Tupper, “General sensitometry,” in The Theory of the Photographic Process, 3rd ed., edited by C. E. K. Mees and T. H. James (Macmillan, New York, 1966), pp. 409–436.

Tuttle, C.

A. M. Koerner and C. Tuttle, “Experimental determination of photographic density,” J. Opt. Soc. Am. 27, 241–256(1937).
[Crossref]

C. Tuttle and J. W. McFarlane, “The measurement of density in variable density sound film,” J. Soc. Motion Pict. Telev. Eng. 15, 345–351(1930).

Vernier, P.

P. Vernier, “Conditions for linearity between density and exposure for photographic plates,” J. Opt. Soc. Am. 59, 444–454(1969).
[Crossref]

P. Vernier, “Définition du contour du spot d’un densitomètre, effet Callier et granulation photographique,” Rev. Opt. 39, 505–515(1960).

P. Vernier, “Sur l’action photographique des électrons—Application aux observations astronomiques,” Bull. Astron. 12, 84–127(1959).

P. Vernier, “Etude de l’effet Callier et de sa relation avec la granularité des plaques photographiques,” C. R. Acad. Sci. (Paris) 246, 1527–1530(1958).

Vilkomerson, D. H. R.

von Helmholtz, H.

H. von Helmholtz, Handbuch der physiologischen Optik, 1st ed. (Voss, Leipzig, 1856), pp. 166–191.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Chap. X.

Appl. Opt. (8)

Bull. Astron. (1)

P. Vernier, “Sur l’action photographique des électrons—Application aux observations astronomiques,” Bull. Astron. 12, 84–127(1959).

C. R. Acad. Sci. (Paris) (1)

P. Vernier, “Etude de l’effet Callier et de sa relation avec la granularité des plaques photographiques,” C. R. Acad. Sci. (Paris) 246, 1527–1530(1958).

J. Opt. Soc. Am. (5)

J. Photogr. Soc. Am. (1)

E. N. Leith, “Photographic film as an element of a coherent optical system,” J. Photogr. Soc. Am. 6, 75–80(1962).

J. Soc. Chem. Ind. (7)

F. Hurter and V. C. Driffield, “Photo-chemical investigations and a new method of determination of the sensitiveness of photographic plates,” J. Soc. Chem. Ind. 9, 455–469(1890).
[Crossref]

W. M. W. Abney, “On the accuracy of the grease spot photometer for measuring the density of photographic plates, and a note on the sector photometer,” J. Soc. Chem. Ind. 9, 722–725(1890).

W. M. W. Abney, “Grease spot photometer measures,” J. Soc. Chem. Ind. 10, 18–20(1891).

F. Hurter and V. C. Driffield, “Reply to the preceding communication of Captain Abney, ‘On the accuracy of the grease spot photometer for measuring the density of photographic plates, and a note on the sector photometer’,” J. Soc. Chem. Ind. 9, 725(1890).

F. Hurter and V. C. Driffield, “The sector and grease spot photometers and their results,” J. Soc. Chem. Ind. 10, 20–24(1891).

F. Hurter and V. C. Driffield, “The sector and grease spot photometers,” J. Soc. Chem. Ind. 10, 98–100(1891).

F. Hurter, “The sector and grease spot photometers,” J. Soc. Chem. Ind. 10, 318(1891).

J. Soc. Motion Pict. Telev. Eng. (1)

C. Tuttle and J. W. McFarlane, “The measurement of density in variable density sound film,” J. Soc. Motion Pict. Telev. Eng. 15, 345–351(1930).

Natl. Bur. Stand. J. Res. (1)

H. J. McNicholas, “Absolute methods in reflectometry,” Natl. Bur. Stand. J. Res. 1, 29–73(1928).

Opt. Acta (1)

P. Croce and M. Marquet, “Détermination optique du “bruit de fond” photographique,” Opt. Acta 2, 107–108(1955).
[Crossref]

Opt. Las. Technol. (1)

G. L. Rogers, “Non-coherent optical processing,” Opt. Las. Technol. 7, 153–162(1975).
[Crossref]

Optik (1)

K. Biedermann, “The scattered flux spectrum of photographic materials for holography,” Optik 31, 367–389(1970).

Photogr. J. (1)

A. Callier, Photogr. J. 49(NS33), 200(1909).

Rev. Opt. (1)

P. Vernier, “Définition du contour du spot d’un densitomètre, effet Callier et granulation photographique,” Rev. Opt. 39, 505–515(1960).

Z. Wiss. Photogr. Photophys. Photochem. (2)

R. Görisch, “Untersuchungen über den Callier-Effekt I,” Z. Wiss. Photogr. Photophys. Photochem. 48, 85–102(1953).

A. Callier, “Absorption und Diffusion des Lichtes in der entwickelten photographischen Platte, nach Messungen mit dem Martensschen Polarisationphotometer,” Z. Wiss. Photogr. Photophys. Photochem. 8, 257–272(1909).

Other (10)

G. Häusler and A. W. Lohmann, “Hybrid Image Processing,” in Proceedings of the International Conference on Applications of Holography and Optical Data Processing (International Commission of Optics,. Jerusalem, 1976), paper MB-1.

J. W. Goodman, “Noise in Coherent Optical Information Processing,” in Optical Information Processing, edited by Yu. E. Nesterikhin, G. W. Stroke, and W. E. Kock (Plenum, New York, 1976), pp. 85–103.
[Crossref]

D. Tichenor and J. W. Goodman, “Practical Noise Limitations in Holographic Image Deblurring,” in Proceedings of the International Optical Computing Conference (I.E.E.E., Washington, D.C., 1975, I.E.E.E. Catalog No. 75 CH 0941-5C), pp. 82–84.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Chap. X.

J. L. Tupper, “General sensitometry,” in The Theory of the Photographic Process, 3rd ed., edited by C. E. K. Mees and T. H. James (Macmillan, New York, 1966), pp. 409–436.

H. von Helmholtz, Handbuch der physiologischen Optik, 1st ed. (Voss, Leipzig, 1856), pp. 166–191.

J. C. Dainty and R. Shaw, Image Science (Academic, London, 1974), pp. 68–115.

C. Lanczos, Applied Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1956).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 887 and 916.

S. Lowenthal and P. Chavel, “Noise Problems in Optical Image Processing,” in Proceedings of the International Conference on Applications of Holography and Optical Data Processing (International Commission of Optics, Jerusalem, 1976), paper TuA-1.

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Figures (12)

FIG. 1
FIG. 1

Schematic diagram of the setup. The photographic emulsion located in the object plane Π0 is imaged on the image plane Π by the lens L located in the pupil plane P. The condenser C placed against the object images the plane ∑ of the source of luminance s ˜(Ω) on the pupil plane.

FIG. 2
FIG. 2

A thick photographic emulsion illuminated by an inclined plane wave.

FIG. 3
FIG. 3

The image is due to the parts of the nondiffuse light and of the diffuse light which are transmitted through the pupil. This figure shows the nondiffuse component and the diffuse light spectrum ϕ ˜ corresponding to the central point of the source.

FIG. 4
FIG. 4

Geometrical interpretation of the weighting function w(Ω) of Eq. (14).

FIG. 5
FIG. 5

Diffuse light spectrum ϕ ˜ and weighting function w in the case of equal source and pupil radii.

FIG. 6
FIG. 6

Behavior of the weighting function w(Ω) with fixed source radius Rs = λdΩs, for 5 different values of the pupil radius Rp = λdΩp, the parameter indicated is the ratio Rp/Rs = Ωps.

FIG. 7
FIG. 7

Callier effect: variations of the measured intensity transmittance with the source radius Rs = λdΩs for fixed pupil radius Rp = λdΩp (or conversely); the parameter is the ratio Ωp/λ, which varies from 0 to 1 by steps of 0.1. (a) fine grain, (b) medium grain.

FIG. 8
FIG. 8

Callier effect: variations of the density with source radius Rs = λdΩs and pupil radius Rp = λdΩp. The solid has a sharp ridge for Ωs = Ωp corresponding to a maximum on the lines of constant Ωs and of constant Ωp. The plane Ωs = Ωp is a plane of symmetry. The curves drawn correspond to values of Ωs/λ and of Ωp/λ varying from 0 to 1 by steps of 0.1. (a) fine grain; (b) medium grain.

FIG. 9
FIG. 9

Callier effect: experimental results obtained by Vernier15 on “maximum resolution” electron plates. The notations are the same as in Fig. 8.

FIG. 10
FIG. 10

As a consequence of the Callier effect, the image contrast C of a weakly modulated object depends on the scattering by the emulsion and hence, on the source radius Rs = λdΩs and on the pupil radius Rp = λdΩp. For given Ωs, the contrast is maximum for Ωs = Ωp, and conversely. The overall maximum is obtained for very small Ωs and Ωp (coherent illumination and very low resolution). The curves drawn correspond to values of Ωs/λ and of Ωp/λ varying from 0 to 1 by steps of 0.1. (a) fine grain; (b) medium grain. The C coordinate is in arbitrary units.

FIG. 11
FIG. 11

The weighting function w(Ω) for two values Ωs1 and Ωs2 of Ωss1 < Ωs2 < Ωp).

FIG. 12
FIG. 12

Explaining Eq. (A10).

Equations (46)

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E 0 exp ( i k · r - i ω t ) ,
[ τ ] [ E 0 ] exp ( i k · r - i ω t ) ,
I ( r ) = Σ Π Π s ˜ ( Ω ) τ ( r 1 ) τ * ( r 2 ) p ( r - r 1 ) p * ( r - r 2 ) × exp [ 2 i π Ω · ( r 1 - r 2 ) ] d r 1 d r 2 d Ω ,
τ ( r ) = τ 0 ( r ) + X ( r )
τ ( r 1 ) τ * ( r 2 ) = τ 0 ( r 1 ) τ 0 * ( r 2 ) + ϕ ( r 1 , r 2 ) .
T ( r ) = I ( r ) with emulsion I ( r ) without emulsion = A B
A = Σ Π Π s ˜ ( Ω ) τ 0 ( r 1 ) τ 0 * ( r 2 ) p ( r - r 1 ) p * ( r - r 2 ) × exp [ 2 i π Ω · ( r 1 - r 2 ) ] d r 1 d r 2 d Ω + Σ Π Π s ˜ ( Ω ) ϕ ( r 1 , r 2 ) p ( r - r 1 ) p * ( r - r 2 ) × exp [ 2 i π Ω · ( r 1 - r 2 ) ] d r 1 d r 2 d Ω
B = Σ Π Π s ˜ ( Ω ) p ( r 1 ) p * ( r 2 ) × exp [ 2 i π Ω · ( r 2 - r 1 ) ] d r 1 d r 2 d Ω .
T = τ 0 2 + H
H = 2 ϕ ˜ ( Ω ) [ s ˜ ( Ω ) p ˜ ( Ω ) 2 ] d Ω 2 s ˜ ( Ω ) p ˜ ( Ω ) 2 d Ω
T = τ 0 2 + 2 ϕ ˜ ( Ω ) p ˜ ( Ω ) 2 d Ω
T = τ 0 2 + σ 2 = τ 2 ,
T = τ 0 2 + ϕ ( 0 ) A c 2 2 s ˜ ( Ω ) d Ω 2 p ˜ ( Ω ) 2 d Ω 2 s ˜ ( Ω ) p ˜ ( Ω ) 2 d Ω .
T = τ 0 2 + σ 2 / N ,
N = image resolution cell area correlation area of object transmittance .
N = coherence area of object illumination correlation area of object transmittance .
s ˜ ( Ω ) = L C ( Ω s ) ,
C ( Ω s ) = 1 , for Ω < Ω s = 0 , for Ω > Ω s .
p ˜ ( Ω ) = C ( Ω p ) .
T = τ 0 2 + 2 ϕ ˜ ( Ω ) w ( Ω ) d Ω ,
w ( 0 ) = 1.
Ω = Ω s + Ω p .
Ω s = Ω p .
[ D ( Ω s = Ω p + ɛ , Ω p ) - D ( Ω s = Ω p - ɛ , Ω p ) ] / ɛ
D ( 0 , 0 ) = 2 D ( , )
ϕ ˜ ( Ω ) = A exp ( - Ω 2 / W 2 ) .
σ 2 = ϕ ( 0 ) = 2 ϕ ˜ ( Ω ) d Ω = τ 0 ( 1 - τ 0 ) ,
A = τ 0 ( 1 - τ 0 ) / W 2 .
T ( r ) = τ 0 2 + τ 0 m * ( r ) + τ 0 * m ( r ) + H
C = ( τ 0 m * + τ 0 * m ) / ( τ 0 2 + H ) .
H ( r ) = s ˜ ( Ω ) d Ω ϕ 0 { τ ( r 1 ) } p ( r - r 1 ) 2 d r 1 s ˜ ( Ω ) p ( r - r 1 ) p * ( r - r 2 ) exp [ 2 i π Ω · ( r 1 - r 2 ) ] d r 1 d r 2 d Ω
H ( Ω s , Ω p ) = 2 Π 0 + ϕ ˜ ( Ω ) w ( Ω ) Ω d Ω .
H ( Ω s 2 , Ω p ) - H ( Ω s 1 , Ω p ) = 2 Π Ω p - Ω s 2 Ω 0 ϕ ˜ ( Ω ) { w 2 ( Ω ) - w 1 ( Ω ) } Ω d Ω + 2 Π Ω 0 Ω p + Ω s 2 ϕ ˜ ( Ω ) { w 2 ( Ω ) - w 1 ( Ω ) } Ω d Ω .
- 2 Π 0 ϕ ˜ ( Ω 0 ) { w 2 ( Ω ) - w 1 ( Ω ) } Ω d Ω ,
H ( Ω s 2 , Ω p ) - H ( Ω s 1 , Ω p ) = 2 Π Ω p - Ω s 2 Ω 0 { ϕ ˜ ( Ω ) - ϕ ˜ ( Ω 0 ) } { w 2 ( Ω ) - w 1 ( Ω ) } Ω d Ω + 2 Π Ω 0 Ω p + Ω s 2 { ϕ ˜ ( Ω ) - ϕ ˜ ( Ω 0 ) } { w 2 ( Ω ) - w 1 ( Ω ) } Ω d Ω .
1 ,             i f Ω < Ω s - Ω p w ( Ω ) = Ω p 2 Arc cos ( X 1 / Ω p ) + Ω s 2 Arc cos ( X 2 / Ω s ) - Y Ω Π Inf ( Ω s 2 , Ω p 2 ) ,             if Ω s - Ω p < Ω < Ω s + Ω p 0 ,             if Ω > Ω s + Ω p
X 1 = ( Ω p 2 - Ω s 2 + Ω 2 ) / 2 Ω , X 2 = ( Ω s 2 - Ω p 2 + Ω 2 ) / 2 Ω , Y = ( 1 / Ω ) ( 2 Ω 2 Ω p 2 + 2 Ω 2 Ω s 2 + 2 Ω p 2 Ω s 2 - Ω 4 - Ω p 4 - Ω s 4 ) 1 / 2 .
d H d Ω s ( Ω s , Ω p ) = 2 Π 0 + ϕ ˜ ( Ω ) d w d Ω s Ω d Ω .
d w d Ω s = - 2 π Ω s 3 [ Ω 2 Arc cos ( X 1 / Ω p ) - Y Ω ] .
d w / d Ω s = ( 2 Ω s / π Ω p 2 ) Arc cos ( X 2 / Ω s )
d w / d Ω s = 0.
d H d Ω s ( Ω p + , Ω p ) = 4 Ω p 0 2 Ω p ϕ ˜ ( Ω ) Arc cos Ω 2 Ω p Ω d Ω .
d H d Ω s ( Ω p - , Ω p ) = - 4 Ω p 0 2 Ω p ϕ ˜ ( Ω ) { Arc cos Ω Ω p - 2 Ω 2 Ω p [ 1 - ( Ω 2 Ω p ) 2 ] 1 / 2 } Ω d Ω = - 16 Ω p 0 1 ϕ ˜ ( 2 Ω p t ) [ Arc cos t - 2 t ( 1 - t 2 ) 1 / 2 ] t d t .
0 1 [ Arc cos t - 2 t ( 1 - t 2 ) 1 / 2 ] t d t = 0.
( d H / d Ω s ) ( Ω p - , Ω p ) 0 ,
d H d Ω s ( Ω p - , Ω p ) + d H d Ω s ( Ω p + , Ω p ) = 4 Ω p 0 2 Ω p ϕ ˜ ( Ω ) 2 Ω 2 Ω p [ 1 - ( Ω 2 Ω p ) 2 ] 1 / 2 Ω d Ω .