Abstract

The frontal image on the Shroud of Turin is shown to be consistent with a body shape covered with a naturally draping cloth in the sense that image shading can be derived from a single global mapping function of distance between these two surfaces. The visible image on the Shroud does not appear to be the work of an artist in an eye/brain/hand coordination sense nor does it appear to be the result of direct contact only, diffusion, radiation from a body shape or engraving, dabbing powder on a bas-relief, or electrostatic imaging. The visible image on the Shroud is probably not the result of a hot bas-relief impressed into cloth, but such a mechanism seems capable of accounting for the Shroud image’s distance correlation, resolution, and similar chemical structure. It does not simultaneously account for (1) the 3-D image residing on one side of the Shroud, (2) observed lateral image distortions (consistent with a draping cloth over a body shape), or (3) expected thermal perturbations associated with physically thick superimposed blood images. A complex mechanism involving more than one process may account for some of the Shroud image’s characteristics, but potential inconsistencies in shading continuity, cloth contact, lateral distortions, and pressure independence may exist.

© 1984 Optical Society of America

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References

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  1. I. Wilson, The Shroud of Turin (Doubleday, New York, 1978).
  2. E. J. Jumper, A. D. Adler, J. P. Jackson, S. F. Pellicori, J. H. Heller, J. R. Druzik, “A Comprehensive Examination of the Various Stains and Images on the Shroud of Turin,” in Archeological Chemistry (American Chemical Society, Washington D.C., 1983), Chap. 22.
  3. L. A. Schwalbe, R. N. Rogers, “Physics and Chemistry of the Shroud of Turin—A summary of the 1978 Investigation,” Anal. Chem. Acta 135, 3 (1982).
    [CrossRef]
  4. J. H. Heller, A. D. Adler, “A Chemical Investigation of the Shroud of Turin,” Can. Soc. Forensic Sci. J. 14, No. 3, 81 (1981).
  5. R. Gilbert, M. M. Gilbert, “Ultraviolet-Visible Reflectance and Fluorescence Spectra of the Shroud of Turin,” Appl. Opt. 19, 1930 (1980).
    [CrossRef] [PubMed]
  6. S. F. Pellicori, “Spectral Properties of the Shroud of Turin,” Appl. Opt. 19, 1913 (1980).
    [CrossRef] [PubMed]
  7. P. Kowaliski, Applied Photographic Theory, (Wiley, London, 1972).
  8. P. Vignon, The Shroud of Christ (1902; University Books, New Hyde Park, N.Y., 1970).
  9. Cloth prepared by Kate Edgerton; see Natl. Geogr. 157, 730 (1980).
  10. W. Ercoline et al., “Examination of the Turin Shroud for Image Distortions,” in Proceedings, 1982 IEEE Conference on Cybernetics & Society (IEEE, New York, 1982).
  11. J. E. Freund, Modern Elementary Statistics (Prentice-Hall, Englewood Cliffs, N.J., 1952).
  12. F. N. David, Tables of the Correlation Coefficient (Cambridge U.P., London, 1938), Chart II.
  13. N. Johnson, F. Leone, Statistics and Experimental Design Vol. 1 (Wiley, New York, 1977).
  14. LogE/ISI, Goleta, Calif.
  15. D. M. Stupin et al., “Quality Selection of Laser Fusion Target Microballoon,” at APS Division of Plasma Physics Meeting, San Francisco, 15 Nov. 1976.
  16. D. A. Rockwell, D. N. Garrett, “A New System for Accurate Laser Intensity Profile Analysis,” Proc. Soc. Photo-Opt. Instrum. Eng. 122 (Aug.1977.).
  17. A. Eoissonneau, J. K. Jegnum, “Regional Level of Wetlands Mapping for Northern Clay Section of Ontario,” in Proceedings, Third Canadian Symposium on Remote Sensing (Publisher, Location, 1975).
  18. Proceeding, First United States Conference of Research on the Shroud of Turin (Holy Shroud Guild, New York, 1977).
  19. Programmed by J. German, AF Weapons Lab.
  20. M. M. Mueller, reponse to “The Authentication of the Turin Shroud: An Issue in Archaeological Epistemology,” Curr. Anthropology 24, No. 3, 299 (June1983).
  21. J. Nickell, response to “The Authentication of the Turin Shroud: An Issue in Archaeological Epistemology,” Curr. Anthropol. 24, No. 3, 299 (June1983).
  22. R. Bucklin, “The Shroud of Turin: A Pathologist’s Viewpoint” Legal Med. (1982), pp. 33–39. Cyril Wecht, Ed.
  23. R. G. Gallagher, Information Theory and Reliable Communication (Wiley, New York, 1968).
  24. H. Thurston, “Relics, Authentic and Spurious,” The Month 156, 51 (1930).
  25. W. C. McCrone “Study of the Turin ‘Shroud’ I,” Microscope 28, No. 3 (1980).
  26. W. C. McCrone, “Light Microscopical Study of the Turin ‘Shroud’ II,” Microscope 28, No. 4 (1980).
  27. W. C. McCrone, “Microscopical Study of the Turin ‘Shroud’ III,” Microscope 29, 19 (1981).
  28. E. A. Wuenschel, “The Holy Shroud—Present State of the Question,” Am. Ecclesiastical Rev. CII, 465 (1940).
  29. D. Lloyd, Colorado Springs Police Department.
  30. T. Kruger, Boulder, Colo. Police Department.
  31. L. D. Harmon, “The Recognition of Faces,” Sci. Am. 229, No. 5, 70 (Nov.1973).
    [CrossRef]
  32. In Y. Le Grand, Light, Colour and Vision (Wiley, New York, 1957), p. 261.
  33. Cloth drape experiment at Biosteriometric Laboratory, Natl. Geogr.June1980.
  34. J. R. Howell, R. Siegel, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).
  35. With assistance of R. LaRue, U. Colorado.
  36. G. Ashe, “What Sort of Picture,” (Sindon, Turin, 1966), pp. 15–19.
  37. J. P. Jackson, P. E. Neilsen, “Role of Radiative Transport in the Propagation of Laser Supported Combustion Waves,” AIAA J. 12, 1499 (1974).
    [CrossRef]
  38. S. Pellicori, M. S. Evans, “The Shroud of Turin Through the Microscope” Archaeology 34, 34 (Jan./Feb.1981).
  39. A preliminary version of this paper appeared in J. Jackson et al., “Three Dimensional Characteristic of the Shroud Image,” Proceedings, 1982 Conference on Cybernetics and Society (IEEE, New York, 1982), pp. 559–575.

1983 (2)

M. M. Mueller, reponse to “The Authentication of the Turin Shroud: An Issue in Archaeological Epistemology,” Curr. Anthropology 24, No. 3, 299 (June1983).

J. Nickell, response to “The Authentication of the Turin Shroud: An Issue in Archaeological Epistemology,” Curr. Anthropol. 24, No. 3, 299 (June1983).

1982 (1)

L. A. Schwalbe, R. N. Rogers, “Physics and Chemistry of the Shroud of Turin—A summary of the 1978 Investigation,” Anal. Chem. Acta 135, 3 (1982).
[CrossRef]

1981 (3)

J. H. Heller, A. D. Adler, “A Chemical Investigation of the Shroud of Turin,” Can. Soc. Forensic Sci. J. 14, No. 3, 81 (1981).

S. Pellicori, M. S. Evans, “The Shroud of Turin Through the Microscope” Archaeology 34, 34 (Jan./Feb.1981).

W. C. McCrone, “Microscopical Study of the Turin ‘Shroud’ III,” Microscope 29, 19 (1981).

1980 (5)

W. C. McCrone “Study of the Turin ‘Shroud’ I,” Microscope 28, No. 3 (1980).

W. C. McCrone, “Light Microscopical Study of the Turin ‘Shroud’ II,” Microscope 28, No. 4 (1980).

Cloth prepared by Kate Edgerton; see Natl. Geogr. 157, 730 (1980).

S. F. Pellicori, “Spectral Properties of the Shroud of Turin,” Appl. Opt. 19, 1913 (1980).
[CrossRef] [PubMed]

R. Gilbert, M. M. Gilbert, “Ultraviolet-Visible Reflectance and Fluorescence Spectra of the Shroud of Turin,” Appl. Opt. 19, 1930 (1980).
[CrossRef] [PubMed]

1977 (1)

D. A. Rockwell, D. N. Garrett, “A New System for Accurate Laser Intensity Profile Analysis,” Proc. Soc. Photo-Opt. Instrum. Eng. 122 (Aug.1977.).

1974 (1)

J. P. Jackson, P. E. Neilsen, “Role of Radiative Transport in the Propagation of Laser Supported Combustion Waves,” AIAA J. 12, 1499 (1974).
[CrossRef]

1973 (1)

L. D. Harmon, “The Recognition of Faces,” Sci. Am. 229, No. 5, 70 (Nov.1973).
[CrossRef]

1940 (1)

E. A. Wuenschel, “The Holy Shroud—Present State of the Question,” Am. Ecclesiastical Rev. CII, 465 (1940).

1930 (1)

H. Thurston, “Relics, Authentic and Spurious,” The Month 156, 51 (1930).

Adler, A. D.

J. H. Heller, A. D. Adler, “A Chemical Investigation of the Shroud of Turin,” Can. Soc. Forensic Sci. J. 14, No. 3, 81 (1981).

E. J. Jumper, A. D. Adler, J. P. Jackson, S. F. Pellicori, J. H. Heller, J. R. Druzik, “A Comprehensive Examination of the Various Stains and Images on the Shroud of Turin,” in Archeological Chemistry (American Chemical Society, Washington D.C., 1983), Chap. 22.

Ashe, G.

G. Ashe, “What Sort of Picture,” (Sindon, Turin, 1966), pp. 15–19.

Bucklin, R.

R. Bucklin, “The Shroud of Turin: A Pathologist’s Viewpoint” Legal Med. (1982), pp. 33–39. Cyril Wecht, Ed.

David, F. N.

F. N. David, Tables of the Correlation Coefficient (Cambridge U.P., London, 1938), Chart II.

Druzik, J. R.

E. J. Jumper, A. D. Adler, J. P. Jackson, S. F. Pellicori, J. H. Heller, J. R. Druzik, “A Comprehensive Examination of the Various Stains and Images on the Shroud of Turin,” in Archeological Chemistry (American Chemical Society, Washington D.C., 1983), Chap. 22.

Edgerton, Kate

Cloth prepared by Kate Edgerton; see Natl. Geogr. 157, 730 (1980).

Eoissonneau, A.

A. Eoissonneau, J. K. Jegnum, “Regional Level of Wetlands Mapping for Northern Clay Section of Ontario,” in Proceedings, Third Canadian Symposium on Remote Sensing (Publisher, Location, 1975).

Ercoline, W.

W. Ercoline et al., “Examination of the Turin Shroud for Image Distortions,” in Proceedings, 1982 IEEE Conference on Cybernetics & Society (IEEE, New York, 1982).

Evans, M. S.

S. Pellicori, M. S. Evans, “The Shroud of Turin Through the Microscope” Archaeology 34, 34 (Jan./Feb.1981).

Freund, J. E.

J. E. Freund, Modern Elementary Statistics (Prentice-Hall, Englewood Cliffs, N.J., 1952).

Gallagher, R. G.

R. G. Gallagher, Information Theory and Reliable Communication (Wiley, New York, 1968).

Garrett, D. N.

D. A. Rockwell, D. N. Garrett, “A New System for Accurate Laser Intensity Profile Analysis,” Proc. Soc. Photo-Opt. Instrum. Eng. 122 (Aug.1977.).

German, J.

Programmed by J. German, AF Weapons Lab.

Gilbert, M. M.

Gilbert, R.

Harmon, L. D.

L. D. Harmon, “The Recognition of Faces,” Sci. Am. 229, No. 5, 70 (Nov.1973).
[CrossRef]

Heller, J. H.

J. H. Heller, A. D. Adler, “A Chemical Investigation of the Shroud of Turin,” Can. Soc. Forensic Sci. J. 14, No. 3, 81 (1981).

E. J. Jumper, A. D. Adler, J. P. Jackson, S. F. Pellicori, J. H. Heller, J. R. Druzik, “A Comprehensive Examination of the Various Stains and Images on the Shroud of Turin,” in Archeological Chemistry (American Chemical Society, Washington D.C., 1983), Chap. 22.

Howell, J. R.

J. R. Howell, R. Siegel, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).

Jackson, J.

A preliminary version of this paper appeared in J. Jackson et al., “Three Dimensional Characteristic of the Shroud Image,” Proceedings, 1982 Conference on Cybernetics and Society (IEEE, New York, 1982), pp. 559–575.

Jackson, J. P.

J. P. Jackson, P. E. Neilsen, “Role of Radiative Transport in the Propagation of Laser Supported Combustion Waves,” AIAA J. 12, 1499 (1974).
[CrossRef]

E. J. Jumper, A. D. Adler, J. P. Jackson, S. F. Pellicori, J. H. Heller, J. R. Druzik, “A Comprehensive Examination of the Various Stains and Images on the Shroud of Turin,” in Archeological Chemistry (American Chemical Society, Washington D.C., 1983), Chap. 22.

Jegnum, J. K.

A. Eoissonneau, J. K. Jegnum, “Regional Level of Wetlands Mapping for Northern Clay Section of Ontario,” in Proceedings, Third Canadian Symposium on Remote Sensing (Publisher, Location, 1975).

Johnson, N.

N. Johnson, F. Leone, Statistics and Experimental Design Vol. 1 (Wiley, New York, 1977).

Jumper, E. J.

E. J. Jumper, A. D. Adler, J. P. Jackson, S. F. Pellicori, J. H. Heller, J. R. Druzik, “A Comprehensive Examination of the Various Stains and Images on the Shroud of Turin,” in Archeological Chemistry (American Chemical Society, Washington D.C., 1983), Chap. 22.

Kowaliski, P.

P. Kowaliski, Applied Photographic Theory, (Wiley, London, 1972).

Kruger, T.

T. Kruger, Boulder, Colo. Police Department.

LaRue, R.

With assistance of R. LaRue, U. Colorado.

Le Grand, Y.

In Y. Le Grand, Light, Colour and Vision (Wiley, New York, 1957), p. 261.

Leone, F.

N. Johnson, F. Leone, Statistics and Experimental Design Vol. 1 (Wiley, New York, 1977).

Lloyd, D.

D. Lloyd, Colorado Springs Police Department.

McCrone, W. C.

W. C. McCrone, “Microscopical Study of the Turin ‘Shroud’ III,” Microscope 29, 19 (1981).

W. C. McCrone “Study of the Turin ‘Shroud’ I,” Microscope 28, No. 3 (1980).

W. C. McCrone, “Light Microscopical Study of the Turin ‘Shroud’ II,” Microscope 28, No. 4 (1980).

Mueller, M. M.

M. M. Mueller, reponse to “The Authentication of the Turin Shroud: An Issue in Archaeological Epistemology,” Curr. Anthropology 24, No. 3, 299 (June1983).

Neilsen, P. E.

J. P. Jackson, P. E. Neilsen, “Role of Radiative Transport in the Propagation of Laser Supported Combustion Waves,” AIAA J. 12, 1499 (1974).
[CrossRef]

Nickell, J.

J. Nickell, response to “The Authentication of the Turin Shroud: An Issue in Archaeological Epistemology,” Curr. Anthropol. 24, No. 3, 299 (June1983).

Pellicori, S.

S. Pellicori, M. S. Evans, “The Shroud of Turin Through the Microscope” Archaeology 34, 34 (Jan./Feb.1981).

Pellicori, S. F.

S. F. Pellicori, “Spectral Properties of the Shroud of Turin,” Appl. Opt. 19, 1913 (1980).
[CrossRef] [PubMed]

E. J. Jumper, A. D. Adler, J. P. Jackson, S. F. Pellicori, J. H. Heller, J. R. Druzik, “A Comprehensive Examination of the Various Stains and Images on the Shroud of Turin,” in Archeological Chemistry (American Chemical Society, Washington D.C., 1983), Chap. 22.

Rockwell, D. A.

D. A. Rockwell, D. N. Garrett, “A New System for Accurate Laser Intensity Profile Analysis,” Proc. Soc. Photo-Opt. Instrum. Eng. 122 (Aug.1977.).

Rogers, R. N.

L. A. Schwalbe, R. N. Rogers, “Physics and Chemistry of the Shroud of Turin—A summary of the 1978 Investigation,” Anal. Chem. Acta 135, 3 (1982).
[CrossRef]

Schwalbe, L. A.

L. A. Schwalbe, R. N. Rogers, “Physics and Chemistry of the Shroud of Turin—A summary of the 1978 Investigation,” Anal. Chem. Acta 135, 3 (1982).
[CrossRef]

Siegel, R.

J. R. Howell, R. Siegel, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).

Stupin, D. M.

D. M. Stupin et al., “Quality Selection of Laser Fusion Target Microballoon,” at APS Division of Plasma Physics Meeting, San Francisco, 15 Nov. 1976.

Thurston, H.

H. Thurston, “Relics, Authentic and Spurious,” The Month 156, 51 (1930).

Vignon, P.

P. Vignon, The Shroud of Christ (1902; University Books, New Hyde Park, N.Y., 1970).

Wilson, I.

I. Wilson, The Shroud of Turin (Doubleday, New York, 1978).

Wuenschel, E. A.

E. A. Wuenschel, “The Holy Shroud—Present State of the Question,” Am. Ecclesiastical Rev. CII, 465 (1940).

AIAA J. (1)

J. P. Jackson, P. E. Neilsen, “Role of Radiative Transport in the Propagation of Laser Supported Combustion Waves,” AIAA J. 12, 1499 (1974).
[CrossRef]

Am. Ecclesiastical Rev. CII (1)

E. A. Wuenschel, “The Holy Shroud—Present State of the Question,” Am. Ecclesiastical Rev. CII, 465 (1940).

Anal. Chem. Acta (1)

L. A. Schwalbe, R. N. Rogers, “Physics and Chemistry of the Shroud of Turin—A summary of the 1978 Investigation,” Anal. Chem. Acta 135, 3 (1982).
[CrossRef]

Appl. Opt. (2)

Archaeology (1)

S. Pellicori, M. S. Evans, “The Shroud of Turin Through the Microscope” Archaeology 34, 34 (Jan./Feb.1981).

Can. Soc. Forensic Sci. J. (1)

J. H. Heller, A. D. Adler, “A Chemical Investigation of the Shroud of Turin,” Can. Soc. Forensic Sci. J. 14, No. 3, 81 (1981).

Curr. Anthropol. (1)

J. Nickell, response to “The Authentication of the Turin Shroud: An Issue in Archaeological Epistemology,” Curr. Anthropol. 24, No. 3, 299 (June1983).

Curr. Anthropology (1)

M. M. Mueller, reponse to “The Authentication of the Turin Shroud: An Issue in Archaeological Epistemology,” Curr. Anthropology 24, No. 3, 299 (June1983).

Microscope (3)

W. C. McCrone “Study of the Turin ‘Shroud’ I,” Microscope 28, No. 3 (1980).

W. C. McCrone, “Light Microscopical Study of the Turin ‘Shroud’ II,” Microscope 28, No. 4 (1980).

W. C. McCrone, “Microscopical Study of the Turin ‘Shroud’ III,” Microscope 29, 19 (1981).

Natl. Geogr. (1)

Cloth prepared by Kate Edgerton; see Natl. Geogr. 157, 730 (1980).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. A. Rockwell, D. N. Garrett, “A New System for Accurate Laser Intensity Profile Analysis,” Proc. Soc. Photo-Opt. Instrum. Eng. 122 (Aug.1977.).

Sci. Am. (1)

L. D. Harmon, “The Recognition of Faces,” Sci. Am. 229, No. 5, 70 (Nov.1973).
[CrossRef]

The Month (1)

H. Thurston, “Relics, Authentic and Spurious,” The Month 156, 51 (1930).

Other (23)

R. Bucklin, “The Shroud of Turin: A Pathologist’s Viewpoint” Legal Med. (1982), pp. 33–39. Cyril Wecht, Ed.

R. G. Gallagher, Information Theory and Reliable Communication (Wiley, New York, 1968).

P. Kowaliski, Applied Photographic Theory, (Wiley, London, 1972).

P. Vignon, The Shroud of Christ (1902; University Books, New Hyde Park, N.Y., 1970).

A. Eoissonneau, J. K. Jegnum, “Regional Level of Wetlands Mapping for Northern Clay Section of Ontario,” in Proceedings, Third Canadian Symposium on Remote Sensing (Publisher, Location, 1975).

Proceeding, First United States Conference of Research on the Shroud of Turin (Holy Shroud Guild, New York, 1977).

Programmed by J. German, AF Weapons Lab.

W. Ercoline et al., “Examination of the Turin Shroud for Image Distortions,” in Proceedings, 1982 IEEE Conference on Cybernetics & Society (IEEE, New York, 1982).

J. E. Freund, Modern Elementary Statistics (Prentice-Hall, Englewood Cliffs, N.J., 1952).

F. N. David, Tables of the Correlation Coefficient (Cambridge U.P., London, 1938), Chart II.

N. Johnson, F. Leone, Statistics and Experimental Design Vol. 1 (Wiley, New York, 1977).

LogE/ISI, Goleta, Calif.

D. M. Stupin et al., “Quality Selection of Laser Fusion Target Microballoon,” at APS Division of Plasma Physics Meeting, San Francisco, 15 Nov. 1976.

In Y. Le Grand, Light, Colour and Vision (Wiley, New York, 1957), p. 261.

Cloth drape experiment at Biosteriometric Laboratory, Natl. Geogr.June1980.

J. R. Howell, R. Siegel, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).

With assistance of R. LaRue, U. Colorado.

G. Ashe, “What Sort of Picture,” (Sindon, Turin, 1966), pp. 15–19.

A preliminary version of this paper appeared in J. Jackson et al., “Three Dimensional Characteristic of the Shroud Image,” Proceedings, 1982 Conference on Cybernetics and Society (IEEE, New York, 1982), pp. 559–575.

I. Wilson, The Shroud of Turin (Doubleday, New York, 1978).

E. J. Jumper, A. D. Adler, J. P. Jackson, S. F. Pellicori, J. H. Heller, J. R. Druzik, “A Comprehensive Examination of the Various Stains and Images on the Shroud of Turin,” in Archeological Chemistry (American Chemical Society, Washington D.C., 1983), Chap. 22.

D. Lloyd, Colorado Springs Police Department.

T. Kruger, Boulder, Colo. Police Department.

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Figures (27)

Figure 1
Figure 1

Shroud Image.

Figure 2
Figure 2

Shroud Correlation.

Figure 3
Figure 3

A. Albedo Face.

Figure 4
Figure 4

A. Albedo Correlation.

Figure 5
Figure 5

A. Albedo VP-8.

Figure 6
Figure 6

A. VP-8 of Shroud Frontal Image.

Figure 7
Figure 7

A. Shroud Face (negative).

Figure 8
Figure 8

A. Cardboard Model of Shroud VP-8.

Figure 9
Figure 9

Deformed VP-8 Reference Shroud.

Figure 10
Figure 10

Line Cloth Draping Over Derived Image.

Figure 11
Figure 11

A. Image of Phosphorescent Face Relative to Curved Surface.

Figure 12
Figure 12

A. (Artist A)—Rigorous.

Figure 13
Figure 13

A. (Artist B)—Rigorous.

Figure 14
Figure 14

A. (Artist A)—Freehand.

Figure 15
Figure 15

A. (Artist B)—Freehand.

Figure 16
Figure 16

VP-8 of (Artist A)—

Figure 17
Figure 17

VP-8 of (Artist B)—

Figure 18
Figure 18

A. “The Veil of Veronica,” Domenico Fetti, 1589–1623(VP-8).

Figure 19
Figure 19

Direct Contact.

Figure 20
Figure 20

Diffusion.

Figure 21
Figure 21

Radiation.

Figure 22
Figure 22

A. Electrostatic Image.

Figure 23
Figure 23

A. Multimechanism Hybrid Image.

Figure 24
Figure 24

Medallion.

Figure 25
Figure 25

A. Bas Relief on Dry Linen.

Figure 26
Figure 26

Powder Technique.

Figure 27
Figure 27

Engraving. Moderate Heating—No Oxide Formation.

Equations (6)

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r ( x , y , λ ) = ( a + b λ ) [ 1 + c ( λ - λ ) h ( x , y ) ] ,
I ( x , y ) = f [ P ( x , y ) ]
r 2 = 1 - S I · D 2 S I 2 ,
n 16 ( 1 - ρ 2 ) 2 ρ 2 [ d ( ρ 2 ) / ρ 2 ] 2 .
I h ( x , y ) = I s ( x , y )
t = ( d / d ) 2 t ,

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