Abstract

A computational theory of color transparency for color images containing X junctions is described. This theory is based on physical models of color transparency under conditions of additive or subtractive color mixture that describe the relationship among four colors at an X junction. Algorithms are derived for recovering transmittance and surface reflectance functions of a transparent medium from a set of sensor responses at an X junction. The algorithms are based on the ability to describe surface reflectance and transmittance functions by using a linear combination of orthogonal basis functions. We also address algorithms for determination of depth ordering of overlapping surfaces and the type of color mixture by checking the physical realizability of recovered functions.

© 1999 Optical Society of America

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References

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  1. F. Metelli, “An algebraic development of the theory of perceptual transparency,” Ergonomics 13, 59–66 (1970).
    [CrossRef] [PubMed]
  2. F. Metelli, “The perception of transparency,” Sci. Am. 230(4), 90–98 (1974).
    [CrossRef] [PubMed]
  3. J. Beck, “Additive and subtractive color mixture in color transparency,” Percept. Psychophys. 23, 265–267 (1978).
    [CrossRef] [PubMed]
  4. J. Beck, K. Prazdny, R. Ivry, “The perception of transparency with achromatic colors,” Percept. Psychophys. 35, 407–422 (1984).
    [CrossRef] [PubMed]
  5. W. Gerbino, C. I. F. H. J. Stultiens, J. M. Troost, C. M. M. de Weert, “Transparent layer constancy,” J. Exp. Psychol. 16, 3–20 (1990).
  6. E. H. Adelson, P. Anandan, “Ordinal characteristics of transparency,” presented at the AAAI-90 Workshop on Qualitative Vision, Boston, Mass., July 20, 1990.
  7. B. L. Anderson, “A theory of illusory lightness and transparency in monocular and binocular images: the role of contour junctions,” Perception 26, 419–454 (1997).
    [CrossRef] [PubMed]
  8. M. H. Brill, “Physical and informational constraints on the perception of transparency and translucency,” Computer Vision Graph. Image Process. 28, 356–362 (1984).
    [CrossRef]
  9. J. Beck, “The perception of surface color,” Sci. Am. 232(2), 65–75 (1975).
  10. O. Da Pos, Trasparenze, Serie Alma Mater (ICONE s.r.l., Padova, Italy, 1989).
  11. M. H. Brill, “The perception of a colored translucent sheet on a background,” Color Res. Appl. 19, 34–36 (1994).
  12. F. Faul, “Chromatic scission in perceptual transparency,” Perception 25 (Supplement), 105 (1996).
  13. F. Faul, “Theoretische und experimentale Untersuchung chromatischer Determinanten perzeptueller Transparenz,” Ph.D. dissertation (Christian-Albrechts-Universität zu Kiel, Kiel, Germany, 1997).
  14. M. D’Zmura, P. Colantoni, K. Knoblauch, B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
    [CrossRef] [PubMed]
  15. V. J. Chen, M. D’Zmura, “Test of a convergence model for color transparency,” Perception 27, 595–608 (1998).
    [CrossRef]
  16. M. D’Zmura, O. Rinner, K. Gegenfurtner, “Colour and lightness of a surface seen behind a transparent filter,” Perception 27 (Supplement), 170 (1998).
  17. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
    [CrossRef]
  18. D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031–1040 (1964).
    [CrossRef]
  19. L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A 3, 1673–1683 (1986).
    [CrossRef] [PubMed]
  20. J. P. S. Parkkinen, J. Hallikainen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989).
    [CrossRef]
  21. G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
    [CrossRef]
  22. L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
    [CrossRef] [PubMed]
  23. M. D’Zmura, G. Iverson, “Color constancy. I. Basic theory of two-stage linear recovery of spectral descriptions of lights and surfaces,” J. Opt. Soc. Am. A 10, 2148–2165 (1993).
    [CrossRef]
  24. M. D’Zmura, G. Iverson, “Color constancy. II. Results for two-stage linear recovery of spectral descriptions of lights and surfaces,” J. Opt. Soc. Am. A 10, 2166–2180 (1993).
    [CrossRef]
  25. M. D’Zmura, G. Iverson, “Color constancy. III. General linear recovery of spectral descriptions of lights and surfaces,” J. Opt. Soc. Am. A 11, 2389–2400 (1994).
    [CrossRef]
  26. K. Takebe, S. Nakauchi, S. Usui, “A computational model for color constancy by separating reflectance and illuminant edges within a scene,” Neural Networks 9, 1405–1415 (1996).
    [CrossRef] [PubMed]
  27. G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).
  28. J. D. Foley, A. Van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics, Principles and Practice, 2nd ed. (Addison-Wesley, New York, 1990).
  29. F. Metelli, O. Da Pos, A. Cavedon, “Balanced and unbalanced, complete and partial transparency,” Percept. Psychophys. 38, 354–366 (1985).
    [CrossRef] [PubMed]
  30. M. Fukuda, S. C. Masin, “Test of balanced transparency,” Perception 23, 37–43 (1994).
    [CrossRef] [PubMed]
  31. M. Singh, D. D. Hoffman, “Part boundaries alter the perception of transparency,” Psychol. Sci. 9, 370–378 (1997).
    [CrossRef]

1998 (2)

V. J. Chen, M. D’Zmura, “Test of a convergence model for color transparency,” Perception 27, 595–608 (1998).
[CrossRef]

M. D’Zmura, O. Rinner, K. Gegenfurtner, “Colour and lightness of a surface seen behind a transparent filter,” Perception 27 (Supplement), 170 (1998).

1997 (3)

M. D’Zmura, P. Colantoni, K. Knoblauch, B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
[CrossRef] [PubMed]

B. L. Anderson, “A theory of illusory lightness and transparency in monocular and binocular images: the role of contour junctions,” Perception 26, 419–454 (1997).
[CrossRef] [PubMed]

M. Singh, D. D. Hoffman, “Part boundaries alter the perception of transparency,” Psychol. Sci. 9, 370–378 (1997).
[CrossRef]

1996 (2)

K. Takebe, S. Nakauchi, S. Usui, “A computational model for color constancy by separating reflectance and illuminant edges within a scene,” Neural Networks 9, 1405–1415 (1996).
[CrossRef] [PubMed]

F. Faul, “Chromatic scission in perceptual transparency,” Perception 25 (Supplement), 105 (1996).

1994 (3)

M. H. Brill, “The perception of a colored translucent sheet on a background,” Color Res. Appl. 19, 34–36 (1994).

M. D’Zmura, G. Iverson, “Color constancy. III. General linear recovery of spectral descriptions of lights and surfaces,” J. Opt. Soc. Am. A 11, 2389–2400 (1994).
[CrossRef]

M. Fukuda, S. C. Masin, “Test of balanced transparency,” Perception 23, 37–43 (1994).
[CrossRef] [PubMed]

1993 (2)

1990 (1)

W. Gerbino, C. I. F. H. J. Stultiens, J. M. Troost, C. M. M. de Weert, “Transparent layer constancy,” J. Exp. Psychol. 16, 3–20 (1990).

1989 (1)

1986 (2)

1985 (1)

F. Metelli, O. Da Pos, A. Cavedon, “Balanced and unbalanced, complete and partial transparency,” Percept. Psychophys. 38, 354–366 (1985).
[CrossRef] [PubMed]

1984 (2)

M. H. Brill, “Physical and informational constraints on the perception of transparency and translucency,” Computer Vision Graph. Image Process. 28, 356–362 (1984).
[CrossRef]

J. Beck, K. Prazdny, R. Ivry, “The perception of transparency with achromatic colors,” Percept. Psychophys. 35, 407–422 (1984).
[CrossRef] [PubMed]

1980 (1)

G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

1978 (1)

J. Beck, “Additive and subtractive color mixture in color transparency,” Percept. Psychophys. 23, 265–267 (1978).
[CrossRef] [PubMed]

1975 (1)

J. Beck, “The perception of surface color,” Sci. Am. 232(2), 65–75 (1975).

1974 (1)

F. Metelli, “The perception of transparency,” Sci. Am. 230(4), 90–98 (1974).
[CrossRef] [PubMed]

1970 (1)

F. Metelli, “An algebraic development of the theory of perceptual transparency,” Ergonomics 13, 59–66 (1970).
[CrossRef] [PubMed]

1964 (2)

Adelson, E. H.

E. H. Adelson, P. Anandan, “Ordinal characteristics of transparency,” presented at the AAAI-90 Workshop on Qualitative Vision, Boston, Mass., July 20, 1990.

Anandan, P.

E. H. Adelson, P. Anandan, “Ordinal characteristics of transparency,” presented at the AAAI-90 Workshop on Qualitative Vision, Boston, Mass., July 20, 1990.

Anderson, B. L.

B. L. Anderson, “A theory of illusory lightness and transparency in monocular and binocular images: the role of contour junctions,” Perception 26, 419–454 (1997).
[CrossRef] [PubMed]

Beck, J.

J. Beck, K. Prazdny, R. Ivry, “The perception of transparency with achromatic colors,” Percept. Psychophys. 35, 407–422 (1984).
[CrossRef] [PubMed]

J. Beck, “Additive and subtractive color mixture in color transparency,” Percept. Psychophys. 23, 265–267 (1978).
[CrossRef] [PubMed]

J. Beck, “The perception of surface color,” Sci. Am. 232(2), 65–75 (1975).

Brill, M. H.

M. H. Brill, “The perception of a colored translucent sheet on a background,” Color Res. Appl. 19, 34–36 (1994).

M. H. Brill, “Physical and informational constraints on the perception of transparency and translucency,” Computer Vision Graph. Image Process. 28, 356–362 (1984).
[CrossRef]

Buchsbaum, G.

G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

Cavedon, A.

F. Metelli, O. Da Pos, A. Cavedon, “Balanced and unbalanced, complete and partial transparency,” Percept. Psychophys. 38, 354–366 (1985).
[CrossRef] [PubMed]

Chen, V. J.

V. J. Chen, M. D’Zmura, “Test of a convergence model for color transparency,” Perception 27, 595–608 (1998).
[CrossRef]

Cohen, J.

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
[CrossRef]

Colantoni, P.

M. D’Zmura, P. Colantoni, K. Knoblauch, B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
[CrossRef] [PubMed]

D’Zmura, M.

Da Pos, O.

F. Metelli, O. Da Pos, A. Cavedon, “Balanced and unbalanced, complete and partial transparency,” Percept. Psychophys. 38, 354–366 (1985).
[CrossRef] [PubMed]

O. Da Pos, Trasparenze, Serie Alma Mater (ICONE s.r.l., Padova, Italy, 1989).

de Weert, C. M. M.

W. Gerbino, C. I. F. H. J. Stultiens, J. M. Troost, C. M. M. de Weert, “Transparent layer constancy,” J. Exp. Psychol. 16, 3–20 (1990).

Faul, F.

F. Faul, “Chromatic scission in perceptual transparency,” Perception 25 (Supplement), 105 (1996).

F. Faul, “Theoretische und experimentale Untersuchung chromatischer Determinanten perzeptueller Transparenz,” Ph.D. dissertation (Christian-Albrechts-Universität zu Kiel, Kiel, Germany, 1997).

Feiner, S. K.

J. D. Foley, A. Van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics, Principles and Practice, 2nd ed. (Addison-Wesley, New York, 1990).

Foley, J. D.

J. D. Foley, A. Van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics, Principles and Practice, 2nd ed. (Addison-Wesley, New York, 1990).

Fukuda, M.

M. Fukuda, S. C. Masin, “Test of balanced transparency,” Perception 23, 37–43 (1994).
[CrossRef] [PubMed]

Gegenfurtner, K.

M. D’Zmura, O. Rinner, K. Gegenfurtner, “Colour and lightness of a surface seen behind a transparent filter,” Perception 27 (Supplement), 170 (1998).

Gerbino, W.

W. Gerbino, C. I. F. H. J. Stultiens, J. M. Troost, C. M. M. de Weert, “Transparent layer constancy,” J. Exp. Psychol. 16, 3–20 (1990).

Hallikainen, J.

Hoffman, D. D.

M. Singh, D. D. Hoffman, “Part boundaries alter the perception of transparency,” Psychol. Sci. 9, 370–378 (1997).
[CrossRef]

Hughes, J. F.

J. D. Foley, A. Van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics, Principles and Practice, 2nd ed. (Addison-Wesley, New York, 1990).

Iverson, G.

Ivry, R.

J. Beck, K. Prazdny, R. Ivry, “The perception of transparency with achromatic colors,” Percept. Psychophys. 35, 407–422 (1984).
[CrossRef] [PubMed]

Jaaskelainen, T.

Judd, D. B.

Knoblauch, K.

M. D’Zmura, P. Colantoni, K. Knoblauch, B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
[CrossRef] [PubMed]

Laget, B.

M. D’Zmura, P. Colantoni, K. Knoblauch, B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
[CrossRef] [PubMed]

MacAdam, D. L.

Maloney, L. T.

Masin, S. C.

M. Fukuda, S. C. Masin, “Test of balanced transparency,” Perception 23, 37–43 (1994).
[CrossRef] [PubMed]

Metelli, F.

F. Metelli, O. Da Pos, A. Cavedon, “Balanced and unbalanced, complete and partial transparency,” Percept. Psychophys. 38, 354–366 (1985).
[CrossRef] [PubMed]

F. Metelli, “The perception of transparency,” Sci. Am. 230(4), 90–98 (1974).
[CrossRef] [PubMed]

F. Metelli, “An algebraic development of the theory of perceptual transparency,” Ergonomics 13, 59–66 (1970).
[CrossRef] [PubMed]

Nakauchi, S.

K. Takebe, S. Nakauchi, S. Usui, “A computational model for color constancy by separating reflectance and illuminant edges within a scene,” Neural Networks 9, 1405–1415 (1996).
[CrossRef] [PubMed]

Parkkinen, J. P. S.

Prazdny, K.

J. Beck, K. Prazdny, R. Ivry, “The perception of transparency with achromatic colors,” Percept. Psychophys. 35, 407–422 (1984).
[CrossRef] [PubMed]

Rinner, O.

M. D’Zmura, O. Rinner, K. Gegenfurtner, “Colour and lightness of a surface seen behind a transparent filter,” Perception 27 (Supplement), 170 (1998).

Singh, M.

M. Singh, D. D. Hoffman, “Part boundaries alter the perception of transparency,” Psychol. Sci. 9, 370–378 (1997).
[CrossRef]

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

Stultiens, C. I. F. H. J.

W. Gerbino, C. I. F. H. J. Stultiens, J. M. Troost, C. M. M. de Weert, “Transparent layer constancy,” J. Exp. Psychol. 16, 3–20 (1990).

Takebe, K.

K. Takebe, S. Nakauchi, S. Usui, “A computational model for color constancy by separating reflectance and illuminant edges within a scene,” Neural Networks 9, 1405–1415 (1996).
[CrossRef] [PubMed]

Troost, J. M.

W. Gerbino, C. I. F. H. J. Stultiens, J. M. Troost, C. M. M. de Weert, “Transparent layer constancy,” J. Exp. Psychol. 16, 3–20 (1990).

Usui, S.

K. Takebe, S. Nakauchi, S. Usui, “A computational model for color constancy by separating reflectance and illuminant edges within a scene,” Neural Networks 9, 1405–1415 (1996).
[CrossRef] [PubMed]

Van Dam, A.

J. D. Foley, A. Van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics, Principles and Practice, 2nd ed. (Addison-Wesley, New York, 1990).

Wandell, B. A.

Wyszecki, G.

D. B. Judd, D. L. MacAdam, G. Wyszecki, “Spectral distribution of typical daylight as a function of correlated color temperature,” J. Opt. Soc. Am. 54, 1031–1040 (1964).
[CrossRef]

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

Color Res. Appl. (1)

M. H. Brill, “The perception of a colored translucent sheet on a background,” Color Res. Appl. 19, 34–36 (1994).

Computer Vision Graph. Image Process. (1)

M. H. Brill, “Physical and informational constraints on the perception of transparency and translucency,” Computer Vision Graph. Image Process. 28, 356–362 (1984).
[CrossRef]

Ergonomics (1)

F. Metelli, “An algebraic development of the theory of perceptual transparency,” Ergonomics 13, 59–66 (1970).
[CrossRef] [PubMed]

J. Exp. Psychol. (1)

W. Gerbino, C. I. F. H. J. Stultiens, J. M. Troost, C. M. M. de Weert, “Transparent layer constancy,” J. Exp. Psychol. 16, 3–20 (1990).

J. Franklin Inst. (1)

G. Buchsbaum, “A spatial processor model for object colour perception,” J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Neural Networks (1)

K. Takebe, S. Nakauchi, S. Usui, “A computational model for color constancy by separating reflectance and illuminant edges within a scene,” Neural Networks 9, 1405–1415 (1996).
[CrossRef] [PubMed]

Percept. Psychophys. (3)

F. Metelli, O. Da Pos, A. Cavedon, “Balanced and unbalanced, complete and partial transparency,” Percept. Psychophys. 38, 354–366 (1985).
[CrossRef] [PubMed]

J. Beck, “Additive and subtractive color mixture in color transparency,” Percept. Psychophys. 23, 265–267 (1978).
[CrossRef] [PubMed]

J. Beck, K. Prazdny, R. Ivry, “The perception of transparency with achromatic colors,” Percept. Psychophys. 35, 407–422 (1984).
[CrossRef] [PubMed]

Perception (6)

F. Faul, “Chromatic scission in perceptual transparency,” Perception 25 (Supplement), 105 (1996).

M. D’Zmura, P. Colantoni, K. Knoblauch, B. Laget, “Color transparency,” Perception 26, 471–492 (1997).
[CrossRef] [PubMed]

V. J. Chen, M. D’Zmura, “Test of a convergence model for color transparency,” Perception 27, 595–608 (1998).
[CrossRef]

M. D’Zmura, O. Rinner, K. Gegenfurtner, “Colour and lightness of a surface seen behind a transparent filter,” Perception 27 (Supplement), 170 (1998).

M. Fukuda, S. C. Masin, “Test of balanced transparency,” Perception 23, 37–43 (1994).
[CrossRef] [PubMed]

B. L. Anderson, “A theory of illusory lightness and transparency in monocular and binocular images: the role of contour junctions,” Perception 26, 419–454 (1997).
[CrossRef] [PubMed]

Psychol. Sci. (1)

M. Singh, D. D. Hoffman, “Part boundaries alter the perception of transparency,” Psychol. Sci. 9, 370–378 (1997).
[CrossRef]

Psychon. Sci. (1)

J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
[CrossRef]

Sci. Am. (2)

F. Metelli, “The perception of transparency,” Sci. Am. 230(4), 90–98 (1974).
[CrossRef] [PubMed]

J. Beck, “The perception of surface color,” Sci. Am. 232(2), 65–75 (1975).

Other (5)

O. Da Pos, Trasparenze, Serie Alma Mater (ICONE s.r.l., Padova, Italy, 1989).

E. H. Adelson, P. Anandan, “Ordinal characteristics of transparency,” presented at the AAAI-90 Workshop on Qualitative Vision, Boston, Mass., July 20, 1990.

F. Faul, “Theoretische und experimentale Untersuchung chromatischer Determinanten perzeptueller Transparenz,” Ph.D. dissertation (Christian-Albrechts-Universität zu Kiel, Kiel, Germany, 1997).

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).

J. D. Foley, A. Van Dam, S. K. Feiner, J. F. Hughes, Computer Graphics, Principles and Practice, 2nd ed. (Addison-Wesley, New York, 1990).

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Figures (16)

Fig. 1
Fig. 1

Image formed by juxtaposing two squares. The left square F is assumed to be transparent. The right square B and the background surface A are opaque. a(λ), b(λ), p(λ), and q(λ) are surface reflectances of four colors surrounding an X junction in the image. The transparent surface F is characterized by surface reflectance fe(λ) for additive color mixture and by surface reflectance f(λ) and transmittance t(λ) for subtractive color mixture.

Fig. 2
Fig. 2

Pattern of light reflection and transmission through the transparent filter when subtractive color mixture occurs.

Fig. 3
Fig. 3

Pattern of light reflection and transmission within the filter medium.

Fig. 4
Fig. 4

Evaluation of the model of a filter’s surface reflectance. (a) Measured transmittance (thick solid curve) and surface reflectance (thin solid curve) of a filter. A reflectance curve fitted by the model (dashed curve) is also drawn, although it overlaps the measurement. (b) Measured and model-fitted reflectances plotted on a different scale from that of (a). ERefm is the error between measured and model-fitted reflectances.

Fig. 5
Fig. 5

Evaluation of the model of subtractive color mixture: (a) transmittance and reflectance functions of transparent filter and reflectance of a background paper and (b) measured reflectances and curves fitted by models with and without inner reflection (I.R.).

Fig. 6
Fig. 6

Possible correspondences between colors and regions A, B, P, and Q. XY means that region X is the same surface as Y but that Y is overlapped by the filter. The corresponding filter region is shown in gray. (a) and (c) correspond to a square-shaped filter, while (b) and (d) correspond to an irregular-shaped filter.

Fig. 7
Fig. 7

Color settings of the stimulus in the numerical experiment.

Fig. 8
Fig. 8

Relationship of four colors at the X junction in CIE 1931 XYZ space.

Fig. 9
Fig. 9

True and recovered surface reflectance and estimated value of α and Eσ(fe) for four color correspondences when additive color mixture was used both for generating an image and for recovering functions. Dots show true functions, and solid curves show recovered functions.

Fig. 10
Fig. 10

True and recovered squared transmittance and surface reflectance of the transparent medium and estimated value of β and ERefm for four color combinations when subtractive color mixture was used both for generating an image and for recovering functions. Dots and open circles show true squared transmittance and reflectance functions, respectively, and solid and dotted–dashed curves show the recovered squared transmittance and surface reflectance, respectively.

Fig. 11
Fig. 11

Surface reflectance and α recovered by the additive color mixture model by using a set of sensor responses generated by the subtractive color mixture. Symbols are the same as those in Fig. 9.

Fig. 12
Fig. 12

Recovered squared transmittance (solid curves) and surface reflectance (dotted–dashed curves) estimated by a subtractive color mixture model by using a set of sensor responses generated by additive color mixture. The recovered value of β and the error are also shown. Dots and open circles show the surface reflectance of the overlapping surface fe(λ) and the calculated reflectance 0.05[1+fe2(λ)], which is not used in image generation.

Fig. 13
Fig. 13

Contour plots of Ctrs and Cref as a function of the CIE 1931 (x, y) chromaticity of the overlapping region Q. (x, y) chromaticities of the background (A), right (B), and left (P) regions are (0.35, 0.35), (0.45, 0.35), and (0.25, 0.35), respectively. Y is 20 for all colors. Plots (a) and (c) and plots (b) and (d) are the results when color correspondences shown in Figs. 6(a) and 6(c) are hypothesized. Contour lines are drawn only for values smaller than 10 for both Ctrs and Cref. Darker colors correspond to smaller values, which means that constraint violations are smaller.

Fig. 14
Fig. 14

Contour plots of Ctrs and Cref as a function of the CIE 1931 (x, y) chromaticity of the overlapping color when the background color is blue: (x, y)=(0.25, 0.25). Other chromaticities are the same as those in Fig. 13.

Fig. 15
Fig. 15

Example of perceived depth ordering depending on the background’s luminance. In this example the right square seems to be overlapping when the background is black, while the left square is perceived to be overlapping when the background is white.

Fig. 16
Fig. 16

Ctrs and Cref as a function of the luminance of a background. (x, y) chromaticities of regions A, B, P, and Q are (0.33, 0.33), (0.30, 0.30), (0.40, 0.40), and (0.35, 0.35), respectively. Luminance (Y) of regions B, P, and Q is 15, 75, and 30. Y of the background (A) is varied from 10 to 80.

Tables (1)

Tables Icon

Table 1 Sensory Responses in the Numerical Experiments

Equations (47)

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

p(λ)=αa(λ)+(1-α)fe(λ),
q(λ)=αb(λ)+(1-α)fe(λ),
p(λ)=f(λ)+t2(λ)a(λ)+t2(λ)a2(λ)f(λ)+
=f(λ)+t2(λ)a(λ)1-f(λ)a(λ),
q(λ)=f(λ)+t2(λ)b(λ)1-f(λ)b(λ).
t(λ)=PtPi=(1-β)21-[βθ(λ)]2 θ(λ),
t(λ)=(1-β)2θ(λ).
f(λ)=PrPi=β+β(1-β)2θ2(λ)=β+βt2(λ)(1-β)2.
f(λ)=β[1.0+t2(λ)].
ERefm=1N i=1N|f(λi)-fˆ(λi)|,
ri=λRi(λ)E(λ)S(λ),
r=REs,
rp=RE[αa+(1-α)fe]=αrα+(1-α)rfe,
rq=αrb+(1-α)rfe,
rp=RE(f+Ma)=rf+ra,
[M]k,k=t2(λk)1-f(λk)a(λk).
rp=rf+RE(tsq)a,
a(λ)=i=13σiSi(λ)+m(λ),
a=Bsσ+m,
tsq=Btτ+n,
r¯p=(REBs)[ασa+(1-α)σfe],
r¯q=(REBs)[ασb+(1-α)σfe],
α=(r¯a-r¯b)-1(r¯p-r¯q),
σfe(p)=(REBs)-1 r¯p-αr¯a1-α,
σfe(q)=(REBs)-1 r¯q-αr¯b1-α,
runderlay=1α [r-(1-α)rfe],
σunderlay=(REBs)-1r¯underlay.
funderlay=Bsσunderlay+m.
r¯p=r¯f+Λ(σa)τ+la,
r¯q=r¯f+Λ(σb)τ+lb,
[Λ(σ)]i,j=λRi(λ)E(λ)k=13σkSk(λ)+m(λ)Tj(λ),
[l]i=λRi(λ)E(λ)k=13σkSk(λ)+m(λ)n(λ).
σa=(REBs)-1r¯a,
σb=(REBs)-1r¯b.
τ=[Λ(σa)-Λ(σb)]-1[(r¯p-la)-(r¯q-lb)].
σf=(REBs)-1[(r¯p-la)-Λ(σa)τ]=(REBs)-1[(r¯q-lb)-Λ(σb)τ].
β=(1+tsq)-1f,
r¯=r¯f+Λ(τ)σunderlay+h,
[Λ(τ)]i,j=λRi(λ)E(λ)k=13τkTk(λ)+n(λ)Sj(λ),
[h]i=λRi(λ)E(λ)k=13τkTk(λ)+n(λ)m(λ).
σunderlay=Λ(τ)-1(r¯-r¯f-h).
Eσ(fe)=σfe(p)-σfe(q)
rp=rf+RE(tsq)arf+kra,
Ctrs=1N i=1N|t(λi)-Clp[t(λi)]|,
Cref=1N i=1N|f(λi)-Clp[f(λi)]|,
Clp[x]=1,x>1x,0x10,x<0.
rq=α(rb-ra)+rpifleftsquareisoverlappingα(rp-ra)+rbifrightsquareisoverlapping.

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