Abstract

This study’s main result is to show that under the conditions imposed by the Maloney–Wandell color constancy algorithm, whereby illuminants are three dimensional and reflectances two dimensional (the 3–2 world), color constancy can be expressed in terms of a simple independent adjustment of the sensor responses (in other words, as a von Kries adaptation type of coefficient rule algorithm) as long as the sensor space is first transformed to a new basis. A consequence of this result is that any color constancy algorithm that makes 3–2 assumptions, such as the Maloney–Wandell subspace algorithm, Forsyth’s MWEXT, and the Funt–Drew lightness algorithm, must effectively calculate a simple von Kries-type scaling of sensor responses, i.e., a diagonal matrix. Our results are strong in the sense that no constraint is placed on the initial spectral sensitivities of the sensors. In addition to purely theoretical arguments, we present results from simulations of von Kries-type color constancy in which the spectra of real illuminants and reflectances along with the human-cone-sensitivity functions are used. The simulations demonstrate that when the cone sensor space is transformed to its new basis in the appropriate manner a diagonal matrix supports nearly optimal color constancy.

© 1994 Optical Society of America

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References

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  1. G. West, M. H. Brill, “Necessary and sufficient conditions for von Kries chromatic adaption to give colour constancy,”J. Math. Biol. 15, 249–258 (1982).
    [CrossRef]
  2. G. D. Finlayson, M. S. Drew, B. V. Funt, “Enhancing von Kries adaptation via sensor transformations,” in Human Vision, Visual Processing, and Digital Display IV, J. P. Allebach, B. E. Rogowitz, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1913, 473–484 (1993).
    [CrossRef]
  3. Adaptation made with linear combinations of the adapted cone functions is sometimes referred to as second-site adaptation (see, e.g., Ref. 9). This can be confusing, however, because second-site adaptation implies a second adaptation stage, whereas we use only a single adaptation stage, with the difference being that the adaptation is applied to sensors derived as linear transformations of the cone-sensitivity functions.
  4. E. H. Land, J. J. McCann, “Lightness and retinex theory,”J. Opt. Soc. Am. 61, 1–11 (1971).
    [CrossRef] [PubMed]
  5. 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]
  6. B. V. Funt, M. S. Drew, “Color constancy computation in near-Mondrian scenes using a finite dimensional linear model,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 544–549.
    [CrossRef]
  7. D. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vision 5, 5–36 (1990).
    [CrossRef]
  8. G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).
  9. M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1672 (1986).
    [CrossRef]
  10. G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
    [CrossRef]
  11. A. Blake, “Boundary conditions for lightness computation in Mondrian world,” Comput. Vision Graphics Image Process. 32, 314–327 (1985).
    [CrossRef]
  12. L. T. Maloney, “Computational approaches to color constancy,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1985).
  13. 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]
  14. J. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964).
  15. S. M. Newhall, D. Nickerson, D. B. Judd, “Final report of the OSA subcommittee on the spacing of the Munsell colors,”J. Opt. Soc. Am. 33, 385–418 (1943).
    [CrossRef]
  16. 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]
  17. D. H. Marimont, B. A. Wandell, A. B. Poirson, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913 (1992).
    [CrossRef] [PubMed]
  18. B. K. P. Horn, “Determining lightness from an image,” Comput. Vision Graphics Image Process. 3, 277–299 (1974).
    [CrossRef]
  19. A. Hurlbert, “Formal connections between lightness algorithms,” J. Opt. Soc. Am. A 3, 1684–1692 (1986).
    [CrossRef] [PubMed]
  20. B. V. Funt, G. D. Finlayson, “Color constant color indexing,” IEEE Pattern Anal. Mach. Intell. (to be published).
  21. M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,”J. Theor. Biol. 71, 473–478 (1978).
    [CrossRef] [PubMed]
  22. E. H. Land, “The retinex theory of color vision,” Sci. Am. 237(6), 108–129 (1977).
    [CrossRef] [PubMed]
  23. J. J. Vos, P. L. Walraven, “On the derivation of the foveal receptor primaries,” Vision Res. 11, 799–818 (1971).
    [CrossRef] [PubMed]
  24. D. H. Foster, R. S. Snelgar, “Initial analysis of opponent-colour interactions revealed in sharpened field sensitivities,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 303–312.
  25. W. Jaeger, H. Krastel, S. Braun, “An increment-threshold evaluation of mechanisms underlying colour constancy,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 545–552.
  26. H. G. Sperling, R. S. Harwerth, “Red–green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
    [CrossRef] [PubMed]
  27. M. Kalloniatis, R. S. Harwerth, “Spectral sensitivity and adaptation characteristics of cone mechanisms under white-light adaptation,” J. Opt. Soc. Am. A 7, 1912–1928 (1990).
    [CrossRef] [PubMed]
  28. A. B. Poirson, B. A. Wandell, “Task-dependent color discrimination,” J. Opt. Soc. Am. A 7, 776–782 (1990).
    [CrossRef] [PubMed]
  29. M. J. Vrhel, H. J. Trussell, “Physical device illumination correction,” in Device-Independent Color Imaging and Imaging Systems Integration, R. J. Motta, H. A. Berberian, eds., Proc. Soc. Photo-Opt. Instrum Eng.1909, 84–91 (1993).
    [CrossRef]

1994

1992

1990

1986

1985

A. Blake, “Boundary conditions for lightness computation in Mondrian world,” Comput. Vision Graphics Image Process. 32, 314–327 (1985).
[CrossRef]

1982

G. West, M. H. Brill, “Necessary and sufficient conditions for von Kries chromatic adaption to give colour constancy,”J. Math. Biol. 15, 249–258 (1982).
[CrossRef]

1978

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,”J. Theor. Biol. 71, 473–478 (1978).
[CrossRef] [PubMed]

1977

E. H. Land, “The retinex theory of color vision,” Sci. Am. 237(6), 108–129 (1977).
[CrossRef] [PubMed]

1974

B. K. P. Horn, “Determining lightness from an image,” Comput. Vision Graphics Image Process. 3, 277–299 (1974).
[CrossRef]

1971

J. J. Vos, P. L. Walraven, “On the derivation of the foveal receptor primaries,” Vision Res. 11, 799–818 (1971).
[CrossRef] [PubMed]

H. G. Sperling, R. S. Harwerth, “Red–green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
[CrossRef] [PubMed]

E. H. Land, J. J. McCann, “Lightness and retinex theory,”J. Opt. Soc. Am. 61, 1–11 (1971).
[CrossRef] [PubMed]

1964

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]

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

1943

Blake, A.

A. Blake, “Boundary conditions for lightness computation in Mondrian world,” Comput. Vision Graphics Image Process. 32, 314–327 (1985).
[CrossRef]

Braun, S.

W. Jaeger, H. Krastel, S. Braun, “An increment-threshold evaluation of mechanisms underlying colour constancy,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 545–552.

Brill, M. H.

G. West, M. H. Brill, “Necessary and sufficient conditions for von Kries chromatic adaption to give colour constancy,”J. Math. Biol. 15, 249–258 (1982).
[CrossRef]

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,”J. Theor. Biol. 71, 473–478 (1978).
[CrossRef] [PubMed]

Cohen, J.

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

D’Zmura, M.

Drew, M. S.

G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
[CrossRef]

B. V. Funt, M. S. Drew, “Color constancy computation in near-Mondrian scenes using a finite dimensional linear model,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 544–549.
[CrossRef]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Enhancing von Kries adaptation via sensor transformations,” in Human Vision, Visual Processing, and Digital Display IV, J. P. Allebach, B. E. Rogowitz, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1913, 473–484 (1993).
[CrossRef]

Finlayson, G. D.

G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
[CrossRef]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Enhancing von Kries adaptation via sensor transformations,” in Human Vision, Visual Processing, and Digital Display IV, J. P. Allebach, B. E. Rogowitz, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1913, 473–484 (1993).
[CrossRef]

B. V. Funt, G. D. Finlayson, “Color constant color indexing,” IEEE Pattern Anal. Mach. Intell. (to be published).

Forsyth, D.

D. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vision 5, 5–36 (1990).
[CrossRef]

Foster, D. H.

D. H. Foster, R. S. Snelgar, “Initial analysis of opponent-colour interactions revealed in sharpened field sensitivities,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 303–312.

Funt, B. V.

G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
[CrossRef]

B. V. Funt, G. D. Finlayson, “Color constant color indexing,” IEEE Pattern Anal. Mach. Intell. (to be published).

B. V. Funt, M. S. Drew, “Color constancy computation in near-Mondrian scenes using a finite dimensional linear model,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 544–549.
[CrossRef]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Enhancing von Kries adaptation via sensor transformations,” in Human Vision, Visual Processing, and Digital Display IV, J. P. Allebach, B. E. Rogowitz, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1913, 473–484 (1993).
[CrossRef]

Harwerth, R. S.

M. Kalloniatis, R. S. Harwerth, “Spectral sensitivity and adaptation characteristics of cone mechanisms under white-light adaptation,” J. Opt. Soc. Am. A 7, 1912–1928 (1990).
[CrossRef] [PubMed]

H. G. Sperling, R. S. Harwerth, “Red–green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
[CrossRef] [PubMed]

Horn, B. K. P.

B. K. P. Horn, “Determining lightness from an image,” Comput. Vision Graphics Image Process. 3, 277–299 (1974).
[CrossRef]

Hurlbert, A.

Jaeger, W.

W. Jaeger, H. Krastel, S. Braun, “An increment-threshold evaluation of mechanisms underlying colour constancy,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 545–552.

Judd, D. B.

Kalloniatis, M.

Krastel, H.

W. Jaeger, H. Krastel, S. Braun, “An increment-threshold evaluation of mechanisms underlying colour constancy,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 545–552.

Land, E. H.

E. H. Land, “The retinex theory of color vision,” Sci. Am. 237(6), 108–129 (1977).
[CrossRef] [PubMed]

E. H. Land, J. J. McCann, “Lightness and retinex theory,”J. Opt. Soc. Am. 61, 1–11 (1971).
[CrossRef] [PubMed]

Lennie, P.

MacAdam, D. L.

Maloney, L. T.

Marimont, D. H.

McCann, J. J.

Newhall, S. M.

Nickerson, D.

Poirson, A. B.

Snelgar, R. S.

D. H. Foster, R. S. Snelgar, “Initial analysis of opponent-colour interactions revealed in sharpened field sensitivities,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 303–312.

Sperling, H. G.

H. G. Sperling, R. S. Harwerth, “Red–green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
[CrossRef] [PubMed]

Stiles, W. S.

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

Trussell, H. J.

M. J. Vrhel, H. J. Trussell, “Physical device illumination correction,” in Device-Independent Color Imaging and Imaging Systems Integration, R. J. Motta, H. A. Berberian, eds., Proc. Soc. Photo-Opt. Instrum Eng.1909, 84–91 (1993).
[CrossRef]

Vos, J. J.

J. J. Vos, P. L. Walraven, “On the derivation of the foveal receptor primaries,” Vision Res. 11, 799–818 (1971).
[CrossRef] [PubMed]

Vrhel, M. J.

M. J. Vrhel, H. J. Trussell, “Physical device illumination correction,” in Device-Independent Color Imaging and Imaging Systems Integration, R. J. Motta, H. A. Berberian, eds., Proc. Soc. Photo-Opt. Instrum Eng.1909, 84–91 (1993).
[CrossRef]

Walraven, P. L.

J. J. Vos, P. L. Walraven, “On the derivation of the foveal receptor primaries,” Vision Res. 11, 799–818 (1971).
[CrossRef] [PubMed]

Wandell, B. A.

West, G.

G. West, M. H. Brill, “Necessary and sufficient conditions for von Kries chromatic adaption to give colour constancy,”J. Math. Biol. 15, 249–258 (1982).
[CrossRef]

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 Formulas, 2nd ed. (Wiley, New York, 1982).

Comput. Vision Graphics Image Process.

A. Blake, “Boundary conditions for lightness computation in Mondrian world,” Comput. Vision Graphics Image Process. 32, 314–327 (1985).
[CrossRef]

B. K. P. Horn, “Determining lightness from an image,” Comput. Vision Graphics Image Process. 3, 277–299 (1974).
[CrossRef]

Int. J. Comput. Vision

D. Forsyth, “A novel algorithm for color constancy,” Int. J. Comput. Vision 5, 5–36 (1990).
[CrossRef]

J. Math. Biol.

G. West, M. H. Brill, “Necessary and sufficient conditions for von Kries chromatic adaption to give colour constancy,”J. Math. Biol. 15, 249–258 (1982).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Theor. Biol.

M. H. Brill, “A device performing illuminant-invariant assessment of chromatic relations,”J. Theor. Biol. 71, 473–478 (1978).
[CrossRef] [PubMed]

Psychon. Sci.

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

Sci. Am.

E. H. Land, “The retinex theory of color vision,” Sci. Am. 237(6), 108–129 (1977).
[CrossRef] [PubMed]

Science

H. G. Sperling, R. S. Harwerth, “Red–green cone interactions in the increment-threshold spectral sensitivity of primates,” Science 172, 180–184 (1971).
[CrossRef] [PubMed]

Vision Res.

J. J. Vos, P. L. Walraven, “On the derivation of the foveal receptor primaries,” Vision Res. 11, 799–818 (1971).
[CrossRef] [PubMed]

Other

D. H. Foster, R. S. Snelgar, “Initial analysis of opponent-colour interactions revealed in sharpened field sensitivities,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 303–312.

W. Jaeger, H. Krastel, S. Braun, “An increment-threshold evaluation of mechanisms underlying colour constancy,” in Colour Vision: Physiology and Psychophysics, J. D. Mollon, L. T. Sharpe, eds. (Academic, New York, 1983), pp. 545–552.

M. J. Vrhel, H. J. Trussell, “Physical device illumination correction,” in Device-Independent Color Imaging and Imaging Systems Integration, R. J. Motta, H. A. Berberian, eds., Proc. Soc. Photo-Opt. Instrum Eng.1909, 84–91 (1993).
[CrossRef]

L. T. Maloney, “Computational approaches to color constancy,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1985).

B. V. Funt, G. D. Finlayson, “Color constant color indexing,” IEEE Pattern Anal. Mach. Intell. (to be published).

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

B. V. Funt, M. S. Drew, “Color constancy computation in near-Mondrian scenes using a finite dimensional linear model,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 544–549.
[CrossRef]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Enhancing von Kries adaptation via sensor transformations,” in Human Vision, Visual Processing, and Digital Display IV, J. P. Allebach, B. E. Rogowitz, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1913, 473–484 (1993).
[CrossRef]

Adaptation made with linear combinations of the adapted cone functions is sometimes referred to as second-site adaptation (see, e.g., Ref. 9). This can be confusing, however, because second-site adaptation implies a second adaptation stage, whereas we use only a single adaptation stage, with the difference being that the adaptation is applied to sensors derived as linear transformations of the cone-sensitivity functions.

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Figures (3)

Fig. 1
Fig. 1

Result of sensor transformation T. Solid curves, Vos–Walraven cone fundamentals; dashed curves, transformed sensors.

Fig. 2
Fig. 2

Comparison of transformed sensors derived under 3–2 and 2–3 model assumptions. Solid curves, sensors derived assuming a 3–2 world; dashed curves, sensors derived assuming a 2–3 world.

Fig. 3
Fig. 3

Cumulative histograms showing improved performance of generalized diagonal color constancy. Dashed curves, simple diagonal color constancy; dotted curves, generalized diagonal color constancy; solid curves, optimal (nondiagonal) color constancy.

Equations (42)

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

d = D p             ( simple diagonal constancy ) ,
T d = D T p             ( generalized diagonal constancy ) .
p k X = ω C X ( λ ) R k ( λ ) d λ             ( color observation ) ,
S ( λ ) i = 1 d S S i ( λ ) σ i ,
E ( λ ) j = 1 d S E j ( λ ) j ,
p = Λ ( ) σ ,
p = Ω ( σ ) ,
d = Q p ,
d M = [ Λ ( ) ] - 1 Λ ( ) σ             ( Maloney ' s descriptor ) ,
d F = Λ ( c ) [ Λ ( ) ] - 1 Λ ( ) σ             ( Forsyth ' s descriptor ) .
D i j p i , x = p j , x .
p i , c = D e c p i , e ,             p j , c = D e c p j , e             ( same surfaces ) ,
p i , e = D i j p j , e .
D e c p i , e = D e c D i j p j , e .
D e c p i , e = D i j D e c p j , e .
p i , c = D i j p j , c .
Ω ( 1 ) ( 1 , 0 ) T ,             Ω ( 2 ) ( 0 , 1 ) T ,
Ω ( σ ) = σ 1 Ω ( 1 ) + σ 2 Ω ( 2 ) .
p = σ 1 Ω ( 1 ) + σ 2 Ω ( 2 ) .
Ω ( 2 ) = M Ω ( 1 ) ,
M = Ω ( 2 ) [ Ω ( 1 ) ] - 1 .
p = [ σ 1 + σ 2 M ] Ω ( 1 ) ,
M = T - 1 D T .
= T - 1 T .
T p = [ σ 1 + σ 2 D ] T Ω ( 1 ) .
T p i = [ σ 1 i + σ 2 i D ] T p s ,
T p i = [ σ 1 i + σ 2 j D ] T p s .
T p i = D i j T p j ,
D i j = [ σ 1 i + σ 2 i D ] [ σ 1 j + σ 2 j D ] - 1 .
[ D n ] t V = [ 0 0 ] .
v 1 = [ n 2 - n 1 0 ] t ,
v 2 = [ 0 n 3 - n 2 ] t .
D 11 n 1 n 2 - D 22 n 2 n 1 = 0 , D 22 n 2 n 3 - D 33 n 3 n 2 = 0.
D 1 V o = V c A 1 ,
D 2 V o = V c A 2 ,
D 2 [ D 1 ] - 1 V c A 1 = V c A 2 .
[ α D 11 + β D 12 ] v 1 o = α v 1 c + β v 2 c .
p i = D i w p w .
d i w = [ diag ( p w ) ] - 1 p i ,
ND = 100 * d i w , e - d i w , c d i w , c ,
d = D p ,             D k k = d k p k .
d = T - 1 D T p .

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