Abstract

We propose a novel method to analyze a sequence of color images. A series of color images is examined in a four-dimensional space, which we call the temporal-color space, whose axes are the three color axes red, green, and blue and one temporal axis. The significance of the temporal-color space lies in its ability to represent the change of image color with time. A conventional color space analysis yields a histogram of the colors in an image, only for an instant of time. Conceptually, the two reflection components from the dichromatic-reflection model, the specular-reflection component and the body-reflection component, form two subspaces in temporal-color space. These two components can be extracted at each pixel in the image locally. Using this fact, we analyzed real color images and separated the two reflection components successfully. We did not make any assumptions about surface properties or the global distribution of surface normals. Finally, object shape was recovered.

© 1994 Optical Society of America

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References

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  1. R. M. Haralick, G. L. Kelly, “Pattern recognition with measurement space and spatial clustering for multiple images,” Proc. IEEE 57, 654–665 (1969).
    [CrossRef]
  2. S. A. Shafer, “Optical phenomena in computer vision,” presented at the meeting of the Canadian Society for Computational Studies of Intelligence, Ontario, Canada, May 1984.
  3. S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
    [CrossRef]
  4. G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlight in color images,” Int. J. Computer Vision 2, 7–32 (1988).
    [CrossRef]
  5. G. J. Klinker, S. A. Shafer, T. Kanade, “A physical approach to color image understanding,” Int. J. Computer Vision 4, 7–38 (1990).
    [CrossRef]
  6. C. L. Novak, “Anatomy of a histogram,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 599–605.
  7. L. B. Wolff, “Using polarization to separate reflection components,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 363–369.
    [CrossRef]
  8. S. N. Nayar, X. Fang, T. Boult, “Removal of specularities using color and polarization,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 583–589.
    [CrossRef]
  9. B. V. Funt, M. S. Drew, “Color space analysis of mutual illumination,”IEEE Trans. Pattern Anal. Machine Intell. 15, 1319–1326 (1993).
    [CrossRef]
  10. B. K. P. Horn, “Obtaining shape from shading information,” in Shape from Shading, B. K. P. Horn, M. J. Brooks, eds. (MIT Press, Cambridge, Mass., 1989).
  11. K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
    [CrossRef]
  12. R. J. Woodham, “Photometric stereo: a reflectance map technique for determining surface orientation from image intensity,” in Image Understanding Systems and Industrial Applications, R. Nevatia, ed., Proc. Soc. Photo-Opt. Instrum. Eng.155, 136–143 (1978).
    [CrossRef]
  13. S. K. Nayar, K. Ikeuchi, T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,”IEEE Trans. Robotics Automation 6, 418–431 (1990).
    [CrossRef]
  14. H. C. Lee, “Method for computing the scene-illuminant chromaticity from specular highlights,” J. Opt. Soc. Am. A 3, 1694–1699 (1986).
    [CrossRef] [PubMed]
  15. S. Tominaga, B. A. Wandell, “Standard surface-reflectance model and illuminant estimation,” J. Opt. Soc. Am. A 6, 576–584 (1989).
    [CrossRef]
  16. C. L. Novak, S. A. Shafer, R. G. Willson, “Obtaining accurate color images for machine vision research,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 54–68 (1990).
    [CrossRef]
  17. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  18. K. Torrance, E. Sparrow, “Theory for off-specular reflection from roughened surfaces,”J. Opt. Soc. Am. 57, 1105–1114 (1967).
    [CrossRef]
  19. S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: Physical and geometrical perspectives,”IEEE Trans. Pattern Anal. Machine Intell. 13, 611–634 (1991).
    [CrossRef]
  20. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr.160 (1977).
  21. H. C. Lee, E. J. Breneman, C. P. Schulte, “Modeling light reflection for computer color vision,”IEEE Trans. Pattern Anal. Machine Intell. 12, 402–409 (1990).
    [CrossRef]

1993

B. V. Funt, M. S. Drew, “Color space analysis of mutual illumination,”IEEE Trans. Pattern Anal. Machine Intell. 15, 1319–1326 (1993).
[CrossRef]

1991

S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: Physical and geometrical perspectives,”IEEE Trans. Pattern Anal. Machine Intell. 13, 611–634 (1991).
[CrossRef]

1990

G. J. Klinker, S. A. Shafer, T. Kanade, “A physical approach to color image understanding,” Int. J. Computer Vision 4, 7–38 (1990).
[CrossRef]

H. C. Lee, E. J. Breneman, C. P. Schulte, “Modeling light reflection for computer color vision,”IEEE Trans. Pattern Anal. Machine Intell. 12, 402–409 (1990).
[CrossRef]

S. K. Nayar, K. Ikeuchi, T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,”IEEE Trans. Robotics Automation 6, 418–431 (1990).
[CrossRef]

1989

1988

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlight in color images,” Int. J. Computer Vision 2, 7–32 (1988).
[CrossRef]

1986

1985

S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

1981

K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
[CrossRef]

1977

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr.160 (1977).

1969

R. M. Haralick, G. L. Kelly, “Pattern recognition with measurement space and spatial clustering for multiple images,” Proc. IEEE 57, 654–665 (1969).
[CrossRef]

1967

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Boult, T.

S. N. Nayar, X. Fang, T. Boult, “Removal of specularities using color and polarization,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 583–589.
[CrossRef]

Breneman, E. J.

H. C. Lee, E. J. Breneman, C. P. Schulte, “Modeling light reflection for computer color vision,”IEEE Trans. Pattern Anal. Machine Intell. 12, 402–409 (1990).
[CrossRef]

Drew, M. S.

B. V. Funt, M. S. Drew, “Color space analysis of mutual illumination,”IEEE Trans. Pattern Anal. Machine Intell. 15, 1319–1326 (1993).
[CrossRef]

Fang, X.

S. N. Nayar, X. Fang, T. Boult, “Removal of specularities using color and polarization,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 583–589.
[CrossRef]

Funt, B. V.

B. V. Funt, M. S. Drew, “Color space analysis of mutual illumination,”IEEE Trans. Pattern Anal. Machine Intell. 15, 1319–1326 (1993).
[CrossRef]

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr.160 (1977).

Haralick, R. M.

R. M. Haralick, G. L. Kelly, “Pattern recognition with measurement space and spatial clustering for multiple images,” Proc. IEEE 57, 654–665 (1969).
[CrossRef]

Horn, B. K. P.

K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
[CrossRef]

B. K. P. Horn, “Obtaining shape from shading information,” in Shape from Shading, B. K. P. Horn, M. J. Brooks, eds. (MIT Press, Cambridge, Mass., 1989).

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr.160 (1977).

Ikeuchi, K.

S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: Physical and geometrical perspectives,”IEEE Trans. Pattern Anal. Machine Intell. 13, 611–634 (1991).
[CrossRef]

S. K. Nayar, K. Ikeuchi, T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,”IEEE Trans. Robotics Automation 6, 418–431 (1990).
[CrossRef]

K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
[CrossRef]

Kanade, T.

S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: Physical and geometrical perspectives,”IEEE Trans. Pattern Anal. Machine Intell. 13, 611–634 (1991).
[CrossRef]

S. K. Nayar, K. Ikeuchi, T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,”IEEE Trans. Robotics Automation 6, 418–431 (1990).
[CrossRef]

G. J. Klinker, S. A. Shafer, T. Kanade, “A physical approach to color image understanding,” Int. J. Computer Vision 4, 7–38 (1990).
[CrossRef]

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlight in color images,” Int. J. Computer Vision 2, 7–32 (1988).
[CrossRef]

Kelly, G. L.

R. M. Haralick, G. L. Kelly, “Pattern recognition with measurement space and spatial clustering for multiple images,” Proc. IEEE 57, 654–665 (1969).
[CrossRef]

Klinker, G. J.

G. J. Klinker, S. A. Shafer, T. Kanade, “A physical approach to color image understanding,” Int. J. Computer Vision 4, 7–38 (1990).
[CrossRef]

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlight in color images,” Int. J. Computer Vision 2, 7–32 (1988).
[CrossRef]

Lee, H. C.

H. C. Lee, E. J. Breneman, C. P. Schulte, “Modeling light reflection for computer color vision,”IEEE Trans. Pattern Anal. Machine Intell. 12, 402–409 (1990).
[CrossRef]

H. C. Lee, “Method for computing the scene-illuminant chromaticity from specular highlights,” J. Opt. Soc. Am. A 3, 1694–1699 (1986).
[CrossRef] [PubMed]

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr.160 (1977).

Nayar, S. K.

S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: Physical and geometrical perspectives,”IEEE Trans. Pattern Anal. Machine Intell. 13, 611–634 (1991).
[CrossRef]

S. K. Nayar, K. Ikeuchi, T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,”IEEE Trans. Robotics Automation 6, 418–431 (1990).
[CrossRef]

Nayar, S. N.

S. N. Nayar, X. Fang, T. Boult, “Removal of specularities using color and polarization,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 583–589.
[CrossRef]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr.160 (1977).

Novak, C. L.

C. L. Novak, “Anatomy of a histogram,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 599–605.

C. L. Novak, S. A. Shafer, R. G. Willson, “Obtaining accurate color images for machine vision research,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 54–68 (1990).
[CrossRef]

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr.160 (1977).

Schulte, C. P.

H. C. Lee, E. J. Breneman, C. P. Schulte, “Modeling light reflection for computer color vision,”IEEE Trans. Pattern Anal. Machine Intell. 12, 402–409 (1990).
[CrossRef]

Shafer, S. A.

G. J. Klinker, S. A. Shafer, T. Kanade, “A physical approach to color image understanding,” Int. J. Computer Vision 4, 7–38 (1990).
[CrossRef]

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlight in color images,” Int. J. Computer Vision 2, 7–32 (1988).
[CrossRef]

S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

C. L. Novak, S. A. Shafer, R. G. Willson, “Obtaining accurate color images for machine vision research,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 54–68 (1990).
[CrossRef]

S. A. Shafer, “Optical phenomena in computer vision,” presented at the meeting of the Canadian Society for Computational Studies of Intelligence, Ontario, Canada, May 1984.

Sparrow, E.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Tominaga, S.

Torrance, K.

Wandell, B. A.

Willson, R. G.

C. L. Novak, S. A. Shafer, R. G. Willson, “Obtaining accurate color images for machine vision research,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 54–68 (1990).
[CrossRef]

Wolff, L. B.

L. B. Wolff, “Using polarization to separate reflection components,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 363–369.
[CrossRef]

Woodham, R. J.

R. J. Woodham, “Photometric stereo: a reflectance map technique for determining surface orientation from image intensity,” in Image Understanding Systems and Industrial Applications, R. Nevatia, ed., Proc. Soc. Photo-Opt. Instrum. Eng.155, 136–143 (1978).
[CrossRef]

Artif. Intell.

K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
[CrossRef]

Color Res. Appl.

S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell.

B. V. Funt, M. S. Drew, “Color space analysis of mutual illumination,”IEEE Trans. Pattern Anal. Machine Intell. 15, 1319–1326 (1993).
[CrossRef]

S. K. Nayar, K. Ikeuchi, T. Kanade, “Surface reflection: Physical and geometrical perspectives,”IEEE Trans. Pattern Anal. Machine Intell. 13, 611–634 (1991).
[CrossRef]

H. C. Lee, E. J. Breneman, C. P. Schulte, “Modeling light reflection for computer color vision,”IEEE Trans. Pattern Anal. Machine Intell. 12, 402–409 (1990).
[CrossRef]

IEEE Trans. Robotics Automation

S. K. Nayar, K. Ikeuchi, T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,”IEEE Trans. Robotics Automation 6, 418–431 (1990).
[CrossRef]

Int. J. Computer Vision

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlight in color images,” Int. J. Computer Vision 2, 7–32 (1988).
[CrossRef]

G. J. Klinker, S. A. Shafer, T. Kanade, “A physical approach to color image understanding,” Int. J. Computer Vision 4, 7–38 (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Natl. Bur. Stand. (U.S.) Monogr.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” Natl. Bur. Stand. (U.S.) Monogr.160 (1977).

Proc. IEEE

R. M. Haralick, G. L. Kelly, “Pattern recognition with measurement space and spatial clustering for multiple images,” Proc. IEEE 57, 654–665 (1969).
[CrossRef]

Other

S. A. Shafer, “Optical phenomena in computer vision,” presented at the meeting of the Canadian Society for Computational Studies of Intelligence, Ontario, Canada, May 1984.

B. K. P. Horn, “Obtaining shape from shading information,” in Shape from Shading, B. K. P. Horn, M. J. Brooks, eds. (MIT Press, Cambridge, Mass., 1989).

C. L. Novak, “Anatomy of a histogram,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 599–605.

L. B. Wolff, “Using polarization to separate reflection components,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 363–369.
[CrossRef]

S. N. Nayar, X. Fang, T. Boult, “Removal of specularities using color and polarization,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 583–589.
[CrossRef]

C. L. Novak, S. A. Shafer, R. G. Willson, “Obtaining accurate color images for machine vision research,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 54–68 (1990).
[CrossRef]

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

R. J. Woodham, “Photometric stereo: a reflectance map technique for determining surface orientation from image intensity,” in Image Understanding Systems and Industrial Applications, R. Nevatia, ed., Proc. Soc. Photo-Opt. Instrum. Eng.155, 136–143 (1978).
[CrossRef]

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Figures (32)

Fig. 1
Fig. 1

T shape in the RGB color space (synthesized data).

Fig. 2
Fig. 2

Viewer-centered coordinate system. n is a surface normal, and θs and θn denote the angle between the viewing direction and the light source direction and the angle between the viewing direction and the surface normal, respectively.

Fig. 3
Fig. 3

Iθs space.

Fig. 4
Fig. 4

Temporal-color space (synthesized data).

Fig. 5
Fig. 5

Measurement at one pixel (synthesized data).

Fig. 6
Fig. 6

Estimation of illuminant color in the xy chromaticity diagram. The three pixels of different colors are manually selected in the image (see Fig. 7).

Fig. 7
Fig. 7

Multicolored object. The three pixels of different colors are manually selected in the image.

Fig. 8
Fig. 8

Estimation of the color vector KLT.

Fig. 9
Fig. 9

Geometry matrix D (synthesized data): ●, column 1; +, column 2.

Fig. 10
Fig. 10

Body-reflection plane and the specular-reflection plane (synthesized data).

Fig. 11
Fig. 11

Geometry of the experimental setup.

Fig. 12
Fig. 12

Geometry of the extended light source.

Fig. 13
Fig. 13

Green shiny plastic cylinder.

Fig. 14
Fig. 14

Measured intensities in the temporal-color space.

Fig. 15
Fig. 15

Decomposed two reflection components: ○, surface reflection; +, body reflection.

Fig. 16
Fig. 16

Loci of two reflection components in the temporal-color space.

Fig. 17
Fig. 17

Body-reflection plane (+) and specular-reflection plane

Fig. 18
Fig. 18

Result of fitting.

Fig. 19
Fig. 19

Green matte plastic cylinder.

Fig. 20
Fig. 20

Measured intensities in the temporal-color space.

Fig. 21
Fig. 21

Two decomposed reflection components.

Fig. 22
Fig. 22

Result of fitting.

Fig. 23
Fig. 23

Aluminum triangular prism.

Fig. 24
Fig. 24

Locus of the intensity in the temporal-color space.

Fig. 25
Fig. 25

Two decomposed reflection components.

Fig. 26
Fig. 26

Purple plastic cylinder.

Fig. 27
Fig. 27

Needle map.

Fig. 28
Fig. 28

Recovered object shape.

Fig. 29
Fig. 29

Body-reflection image.

Fig. 30
Fig. 30

Specular-reflection image.

Fig. 31
Fig. 31

Hybrid reflectance model.

Fig. 32
Fig. 32

Two reflection components under an extended light source.

Equations (34)

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

I = s ( λ ) h ( λ ) d λ .
I R = τ R ( λ ) s ( λ ) h ( λ ) d λ , I G = τ G ( λ ) s ( λ ) h ( λ ) d λ , I B = τ B ( λ ) s ( λ ) h ( λ ) d λ ,
C = [ I R I G I B ] = [ τ R ( λ ) s ( λ ) h ( λ ) d λ τ G ( λ ) s ( λ ) h ( λ ) d λ τ B ( λ ) s ( λ ) h ( λ ) d λ ] .
I ( θ s ) = g ( θ s ) s ( λ ) h ( λ ) d λ ,
p [ θ s , C ( θ s ) ] ,
C ( θ s ) = [ I R ( θ s ) I G ( θ s ) I B ( θ s ) ] = [ g ( θ s ) τ R ( λ ) s ( λ ) h ( λ ) d λ g ( θ s ) τ G ( λ ) s ( λ ) h ( λ ) d λ g ( θ s ) τ B ( λ ) s ( λ ) h ( λ ) d λ ] .
I ( θ s ) = I body ( θ s ) + I specular ( θ s ) = K L L ( θ s - θ n ) cos ( θ s - θ n ) + K s L ( θ s - θ n ) δ ( 2 θ n - θ s ) ,
K L = λ s ( λ ) c ( λ ) c L ( λ ) d λ , K S = c s λ s ( λ ) c ( λ ) d λ ,
I ( θ s ) = K L cos ( θ s - θ n ) + K S L ( θ s - 2 θ n ) .
K L = [ k L R k L G k L B ] = [ λ τ R ( λ ) s ( λ ) c ( λ ) c L ( λ ) d λ λ τ G ( λ ) s ( λ ) c ( λ ) c L ( λ ) d λ λ τ B ( λ ) s ( λ ) c ( λ ) c L ( λ ) d λ ] ,
K S = [ k S R k S G k S B ] = [ c S λ τ R ( λ ) s ( λ ) c ( λ ) d λ c S λ τ G ( λ ) s ( λ ) c ( λ ) d λ c S λ τ B ( λ ) s ( λ ) c ( λ ) d λ ] .
I = [ I R I G I B ] = [ cos ( θ s 1 - θ n ) L ( θ s 1 - 2 θ n ) cos ( θ s 2 - θ n ) L ( θ s 2 - 2 θ n ) cos ( θ s m - θ n ) L ( θ s m - 2 θ n ) ] [ k L R k L G k L B k S R k S G k S B ] = [ D L D S ] [ K L T K S T ] D K ,
I = [ I R I G I B ] = [ L ( θ s 1 - 2 θ n ) L ( θ s 2 - 2 θ n ) L ( θ s m - 2 θ n ) ] [ k S R k S G k s B ] = D S K S T
D = I K + ,
β = cos - 1 K S T · w i K S T w i .
I body = D L K L T ,
I specular = D S K S T .
A 1 cos ( θ s - A 2 ) + A 3 ,
B 1 exp [ - ( θ s - B 2 ) 2 B 3 2 ] .
K S = ( 1 3 , 1 3 , 1 3 ) .
I = I L + I S ,
f r ( θ i , θ r , λ ) = c L ( λ ) g L ( θ i , θ r ) + c S ( λ ) g S ( θ i , θ r ) ,
f r ( θ i , θ r , λ ) = c L ( λ ) g L ( θ i , θ r ) + c S g S ( θ i , θ r ) .
g L ( θ i , θ r ) = max ( 0 , cos θ i ) , g S ( θ i , θ r ) = δ ( θ i - θ r ) .
f r ( θ i , θ r , λ ) = c L ( λ ) cos θ i + c S δ ( θ i - θ r ) .
L i ( θ i , λ ) = c ( λ ) L ( θ i ) ,
I ( θ i , θ r ) = λ s ( λ ) L i ( θ i , λ ) f r ( θ i , θ r , λ ) d λ = λ s ( λ ) c ( λ ) L ( θ i ) [ c L cos θ i + c S δ ( θ i - θ r ) ] d λ = L ( θ i ) cos θ i λ s ( λ ) c ( λ ) c L ( λ ) d λ + L ( θ i ) δ ( θ i - θ r ) c S λ s ( λ ) c ( λ ) d λ ,
I ( θ s ) = K L L ( θ s - θ n ) cos ( θ s - θ n ) + K s L ( θ s - θ n ) δ ( 2 θ n - θ s ) ,
K L = λ s ( λ ) c ( λ ) c L ( λ ) d λ , K S = c s λ s ( λ ) c ( λ ) d λ .
L ( θ i ) = L ( θ - θ s ) = C J [ ( R + H ) cos ( θ - θ s ) - R ] { [ R + H - R cos ( θ - θ s ) ] 2 + [ R sin ( θ - θ s ) ] 2 } 3 / 2 .
α = cos - 1 R R + H .
I ( θ s ) = ( θ s - α ) ( θ s + α ) L ( θ , θ ) cos ( θ - θ n ) × λ τ ( λ ) s ( λ ) c ( λ ) c L ( λ ) d λ d θ + ( θ s - α ) ( θ s + α ) L ( θ , θ s ) δ ( θ - 2 θ n ) × c S λ τ ( λ ) s ( λ ) c ( λ ) d λ d θ = λ τ ( λ ) s ( λ ) c ( λ ) c L ( λ ) d λ cos ( θ s - θ n ) + c S λ τ ( λ ) s ( λ ) c ( λ ) d λ L ( θ s , 2 θ n ) .
I ( θ s ) = K L cos ( θ s - θ n ) + K S L ( θ s - 2 θ n ) ,
K L = λ τ ( λ ) s ( λ ) c ( λ ) c L ( λ ) d λ , K S = c S λ τ ( λ ) s ( λ ) c ( λ ) d λ .

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