Abstract

Physical models indicate that, in general, reflectance is a complicated function of wavelength and geometry. An analysis of general reflectance models, however, shows that approximate reflectance models exist that preserve much of the structure of the more-detailed models. In particular, I show from general models that Shafer’s dichromatic reflection model [ Color Res. Appl. 10, 210 ( 1985)] is a reasonable approximation for a large class of inhomogeneous dielectrics. I also show that a unichromatic reflection model is a useful approximation for metals. The approximate color-reflectance model is the basis for two algorithms that use color information. The first algorithm uses normalized color to classify surfaces according to material composition and is insensitive to geometrical variation in the scene. The second algorithm is used to identify metal and dielectric materials from their images. Experimental results are presented.

© 1989 Optical Society of America

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References

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  1. S. Shafer, “Using color to separate reflection components,” Color Research and Application 10, 210–218 (1985); also available as (University of Rochester, Rochester, N.Y., 1984).
    [CrossRef]
  2. B. K. P. Horn, “Obtaining shape from shading information,” in The Psychology of Computer Vision, P. Winston, ed. (McGraw-Hill, New York, 1975), pp. 115–155.
  3. V. Weisskopf, “How light interacts with matter,” Sci. Am. 219, 60–71 (1968).
    [CrossRef]
  4. G. Klinker, S. Shafer, T. Kanade, “Using a color reflection model to separate highlights from object color,” in Proceedings of the First International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 145–150.
  5. G. Klinker, S. Shafer, T. Kanade, “Image segmentation and reflection analysis through color,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 838–853.
  6. H. C. Lee, “Method for computing the scene-illuminant chromaticity from specular highlights,” J. Opt. Soc. Am. A 3, 1694–1699 (1986).
    [CrossRef] [PubMed]
  7. J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media. 1: theory,” Appl. Opt. 12, 1811–1815 (1973).
    [CrossRef] [PubMed]
  8. K. Torrance, E. Sparrow, “Theory for off-specular reflection from roughened surfaces,”J. Opt. Soc. Am. 57, 1105–1114 (1967).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).
  10. R. S. Hunter, The Measurement of Appearance (Wiley, New York, 1975).
  11. W. Egan, T. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).
  12. B. K. P. Horn, Robot Vision (McGraw-Hill, New York, 1986).
  13. R. Siegel, J. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1981).
  14. G. Healey, W. E. Blanz, “Identifying metal surfaces in color images,” to be published in Optics, Electro-Optics, and Sensors, Proc. Soc. Photo-Opt. Instrum. Eng. (1988).
  15. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Macmillan, New York, 1963).
  16. R. Cook, K. Torrance, “A reflectance model for computer graphics,” Comput. Graphics 15, 307–316 (1981).
    [CrossRef]
  17. N. Mott, H. Jones, The Theory of the Properties of Metals and Alloys (Oxford U. Press, Oxford, 1958).
  18. J. Chambers, Computational Methods for Data Analysis (Wiley, New York, 1977).
  19. K. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1978).
  20. D. Gray, ed., American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1972).
  21. E. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).
  22. J. Weaver, C. Krafka, Physics Data: Optical Properties of Metals (Fachinformationszentrum, Karlsruhe1981).
  23. E. Sparrow, R. Cess, Radiation Heat Transfer (Hemisphere, Washington, D.C., 1978).
  24. H. C. Lee, E. Breneman, C. Schulte, “An experimental study of a color reflection model,” (Eastman Kodak, Rochester, N.Y., 1986).
  25. M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1672 (1986).
    [CrossRef]
  26. G. Healey, T. O. Binford, “Color algorithms for a general vision system,” in Proceedings of the Tenth International Joint Conference on Artificial Intelligence, J. McDermott, ed. (Morgan Kaufmann, Los Altos, Calif., 1987), pp. 759–762.
  27. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Techn. Physik. 12, 593 (1931).
  28. S. Orchard, “Reflection and transmission of light by diffusing suspensions,”J. Opt. Soc. Am. 59, 1584–1597 (1969).
    [CrossRef]
  29. J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media. 2: experiment,” Appl. Opt. 12, 1816–1823 (1973).
    [CrossRef] [PubMed]
  30. D. Nickerson, “Spectrophotometric data for a collection of munsell samples,” (U.S. Department of Agriculture, Washington, D.C., 1957).
  31. G. Healey, T. O. Binford, “A color metric for computer vision,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 10–17.
  32. R. Ohlander, K. Price, D. Reddy, “Picture segmentation using a recursive region splitting method,” Comput. Graphics Image Process 8, 313–333 (1978).
    [CrossRef]
  33. Y. Ohta, T. Kanade, T. Sakai, “Color information for region segmentation,” Comput. Graphics Image Process. 13, 222–241 (1980).
    [CrossRef]
  34. T. Sumanaweera, G. Healey, B. U. Lee, T. O. Binford, J. Ponce, “Image segmentation using geometrical and physical constraints,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 1091–1099.
  35. J. Canny, “A computational approach to edge detection,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 679–697 (1986).
    [CrossRef]
  36. B. K. P. Horn, R. Sjoberg, “Calculating the reflectance map,” Appl. Opt. 18, 1770–1779 (1979).
    [CrossRef] [PubMed]
  37. E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978).
  38. B. K. P. Horn, “Exact reproduction of colored images,” Comput. Vision Graphics Image Process. 26, 135–167 (1984).
    [CrossRef]
  39. B. Wandell, “The synthesis and analysis of color images,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2–13 (1987).
    [CrossRef]
  40. G. Porter, J. Mundy, “Automatic visual inspection of metal surfaces,” in Techniques and Applications of Image Understanding, J. J. Pearson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.281, 176–181 (1981).
    [CrossRef]
  41. G. Healey, T. O. Binford, “Predicting material classes,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 1140–1146.

1987 (1)

B. Wandell, “The synthesis and analysis of color images,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2–13 (1987).
[CrossRef]

1986 (3)

1985 (1)

S. Shafer, “Using color to separate reflection components,” Color Research and Application 10, 210–218 (1985); also available as (University of Rochester, Rochester, N.Y., 1984).
[CrossRef]

1984 (1)

B. K. P. Horn, “Exact reproduction of colored images,” Comput. Vision Graphics Image Process. 26, 135–167 (1984).
[CrossRef]

1981 (1)

R. Cook, K. Torrance, “A reflectance model for computer graphics,” Comput. Graphics 15, 307–316 (1981).
[CrossRef]

1980 (1)

Y. Ohta, T. Kanade, T. Sakai, “Color information for region segmentation,” Comput. Graphics Image Process. 13, 222–241 (1980).
[CrossRef]

1979 (1)

1978 (1)

R. Ohlander, K. Price, D. Reddy, “Picture segmentation using a recursive region splitting method,” Comput. Graphics Image Process 8, 313–333 (1978).
[CrossRef]

1973 (2)

1969 (1)

1968 (1)

V. Weisskopf, “How light interacts with matter,” Sci. Am. 219, 60–71 (1968).
[CrossRef]

1967 (1)

1931 (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Techn. Physik. 12, 593 (1931).

Atkinson, K.

K. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1978).

Beckmann, P.

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

Binford, T. O.

G. Healey, T. O. Binford, “Color algorithms for a general vision system,” in Proceedings of the Tenth International Joint Conference on Artificial Intelligence, J. McDermott, ed. (Morgan Kaufmann, Los Altos, Calif., 1987), pp. 759–762.

G. Healey, T. O. Binford, “A color metric for computer vision,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 10–17.

T. Sumanaweera, G. Healey, B. U. Lee, T. O. Binford, J. Ponce, “Image segmentation using geometrical and physical constraints,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 1091–1099.

G. Healey, T. O. Binford, “Predicting material classes,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 1140–1146.

Blanz, W. E.

G. Healey, W. E. Blanz, “Identifying metal surfaces in color images,” to be published in Optics, Electro-Optics, and Sensors, Proc. Soc. Photo-Opt. Instrum. Eng. (1988).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Breneman, E.

H. C. Lee, E. Breneman, C. Schulte, “An experimental study of a color reflection model,” (Eastman Kodak, Rochester, N.Y., 1986).

Canny, J.

J. Canny, “A computational approach to edge detection,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 679–697 (1986).
[CrossRef]

Cess, R.

E. Sparrow, R. Cess, Radiation Heat Transfer (Hemisphere, Washington, D.C., 1978).

Chambers, J.

J. Chambers, Computational Methods for Data Analysis (Wiley, New York, 1977).

Cook, R.

R. Cook, K. Torrance, “A reflectance model for computer graphics,” Comput. Graphics 15, 307–316 (1981).
[CrossRef]

D’Zmura, M.

Egan, W.

W. Egan, T. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

Healey, G.

G. Healey, W. E. Blanz, “Identifying metal surfaces in color images,” to be published in Optics, Electro-Optics, and Sensors, Proc. Soc. Photo-Opt. Instrum. Eng. (1988).

G. Healey, T. O. Binford, “Color algorithms for a general vision system,” in Proceedings of the Tenth International Joint Conference on Artificial Intelligence, J. McDermott, ed. (Morgan Kaufmann, Los Altos, Calif., 1987), pp. 759–762.

T. Sumanaweera, G. Healey, B. U. Lee, T. O. Binford, J. Ponce, “Image segmentation using geometrical and physical constraints,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 1091–1099.

G. Healey, T. O. Binford, “A color metric for computer vision,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 10–17.

G. Healey, T. O. Binford, “Predicting material classes,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 1140–1146.

Hilgeman, T.

W. Egan, T. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

Horn, B. K. P.

B. K. P. Horn, “Exact reproduction of colored images,” Comput. Vision Graphics Image Process. 26, 135–167 (1984).
[CrossRef]

B. K. P. Horn, R. Sjoberg, “Calculating the reflectance map,” Appl. Opt. 18, 1770–1779 (1979).
[CrossRef] [PubMed]

B. K. P. Horn, Robot Vision (McGraw-Hill, New York, 1986).

B. K. P. Horn, “Obtaining shape from shading information,” in The Psychology of Computer Vision, P. Winston, ed. (McGraw-Hill, New York, 1975), pp. 115–155.

Howell, J.

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

Hunter, R. S.

R. S. Hunter, The Measurement of Appearance (Wiley, New York, 1975).

Jones, H.

N. Mott, H. Jones, The Theory of the Properties of Metals and Alloys (Oxford U. Press, Oxford, 1958).

Kanade, T.

Y. Ohta, T. Kanade, T. Sakai, “Color information for region segmentation,” Comput. Graphics Image Process. 13, 222–241 (1980).
[CrossRef]

G. Klinker, S. Shafer, T. Kanade, “Image segmentation and reflection analysis through color,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 838–853.

G. Klinker, S. Shafer, T. Kanade, “Using a color reflection model to separate highlights from object color,” in Proceedings of the First International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 145–150.

Klinker, G.

G. Klinker, S. Shafer, T. Kanade, “Image segmentation and reflection analysis through color,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 838–853.

G. Klinker, S. Shafer, T. Kanade, “Using a color reflection model to separate highlights from object color,” in Proceedings of the First International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 145–150.

Krafka, C.

J. Weaver, C. Krafka, Physics Data: Optical Properties of Metals (Fachinformationszentrum, Karlsruhe1981).

Kreyszig, E.

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978).

Kubelka, P.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Techn. Physik. 12, 593 (1931).

Lee, B. U.

T. Sumanaweera, G. Healey, B. U. Lee, T. O. Binford, J. Ponce, “Image segmentation using geometrical and physical constraints,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 1091–1099.

Lee, H. C.

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

H. C. Lee, E. Breneman, C. Schulte, “An experimental study of a color reflection model,” (Eastman Kodak, Rochester, N.Y., 1986).

Lennie, P.

Mott, N.

N. Mott, H. Jones, The Theory of the Properties of Metals and Alloys (Oxford U. Press, Oxford, 1958).

Mundy, J.

G. Porter, J. Mundy, “Automatic visual inspection of metal surfaces,” in Techniques and Applications of Image Understanding, J. J. Pearson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.281, 176–181 (1981).
[CrossRef]

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Techn. Physik. 12, 593 (1931).

Nickerson, D.

D. Nickerson, “Spectrophotometric data for a collection of munsell samples,” (U.S. Department of Agriculture, Washington, D.C., 1957).

Ohlander, R.

R. Ohlander, K. Price, D. Reddy, “Picture segmentation using a recursive region splitting method,” Comput. Graphics Image Process 8, 313–333 (1978).
[CrossRef]

Ohta, Y.

Y. Ohta, T. Kanade, T. Sakai, “Color information for region segmentation,” Comput. Graphics Image Process. 13, 222–241 (1980).
[CrossRef]

Orchard, S.

Ponce, J.

T. Sumanaweera, G. Healey, B. U. Lee, T. O. Binford, J. Ponce, “Image segmentation using geometrical and physical constraints,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 1091–1099.

Porter, G.

G. Porter, J. Mundy, “Automatic visual inspection of metal surfaces,” in Techniques and Applications of Image Understanding, J. J. Pearson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.281, 176–181 (1981).
[CrossRef]

Price, K.

R. Ohlander, K. Price, D. Reddy, “Picture segmentation using a recursive region splitting method,” Comput. Graphics Image Process 8, 313–333 (1978).
[CrossRef]

Reddy, D.

R. Ohlander, K. Price, D. Reddy, “Picture segmentation using a recursive region splitting method,” Comput. Graphics Image Process 8, 313–333 (1978).
[CrossRef]

Reichman, J.

Sakai, T.

Y. Ohta, T. Kanade, T. Sakai, “Color information for region segmentation,” Comput. Graphics Image Process. 13, 222–241 (1980).
[CrossRef]

Schulte, C.

H. C. Lee, E. Breneman, C. Schulte, “An experimental study of a color reflection model,” (Eastman Kodak, Rochester, N.Y., 1986).

Shafer, S.

S. Shafer, “Using color to separate reflection components,” Color Research and Application 10, 210–218 (1985); also available as (University of Rochester, Rochester, N.Y., 1984).
[CrossRef]

G. Klinker, S. Shafer, T. Kanade, “Image segmentation and reflection analysis through color,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 838–853.

G. Klinker, S. Shafer, T. Kanade, “Using a color reflection model to separate highlights from object color,” in Proceedings of the First International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 145–150.

Siegel, R.

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

Sjoberg, R.

Sparrow, E.

Spizzichino, A.

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

Sumanaweera, T.

T. Sumanaweera, G. Healey, B. U. Lee, T. O. Binford, J. Ponce, “Image segmentation using geometrical and physical constraints,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 1091–1099.

Torrance, K.

R. Cook, K. Torrance, “A reflectance model for computer graphics,” Comput. Graphics 15, 307–316 (1981).
[CrossRef]

K. Torrance, E. Sparrow, “Theory for off-specular reflection from roughened surfaces,”J. Opt. Soc. Am. 57, 1105–1114 (1967).
[CrossRef]

Wandell, B.

B. Wandell, “The synthesis and analysis of color images,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2–13 (1987).
[CrossRef]

Weaver, J.

J. Weaver, C. Krafka, Physics Data: Optical Properties of Metals (Fachinformationszentrum, Karlsruhe1981).

Weisskopf, V.

V. Weisskopf, “How light interacts with matter,” Sci. Am. 219, 60–71 (1968).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Appl. Opt. (3)

Color Research and Application (1)

S. Shafer, “Using color to separate reflection components,” Color Research and Application 10, 210–218 (1985); also available as (University of Rochester, Rochester, N.Y., 1984).
[CrossRef]

Comput. Graphics (1)

R. Cook, K. Torrance, “A reflectance model for computer graphics,” Comput. Graphics 15, 307–316 (1981).
[CrossRef]

Comput. Graphics Image Process (1)

R. Ohlander, K. Price, D. Reddy, “Picture segmentation using a recursive region splitting method,” Comput. Graphics Image Process 8, 313–333 (1978).
[CrossRef]

Comput. Graphics Image Process. (1)

Y. Ohta, T. Kanade, T. Sakai, “Color information for region segmentation,” Comput. Graphics Image Process. 13, 222–241 (1980).
[CrossRef]

Comput. Vision Graphics Image Process. (1)

B. K. P. Horn, “Exact reproduction of colored images,” Comput. Vision Graphics Image Process. 26, 135–167 (1984).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

B. Wandell, “The synthesis and analysis of color images,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-9, 2–13 (1987).
[CrossRef]

J. Canny, “A computational approach to edge detection,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-8, 679–697 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Sci. Am. (1)

V. Weisskopf, “How light interacts with matter,” Sci. Am. 219, 60–71 (1968).
[CrossRef]

Z. Techn. Physik. (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Techn. Physik. 12, 593 (1931).

Other (25)

G. Healey, T. O. Binford, “Color algorithms for a general vision system,” in Proceedings of the Tenth International Joint Conference on Artificial Intelligence, J. McDermott, ed. (Morgan Kaufmann, Los Altos, Calif., 1987), pp. 759–762.

G. Klinker, S. Shafer, T. Kanade, “Using a color reflection model to separate highlights from object color,” in Proceedings of the First International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 145–150.

G. Klinker, S. Shafer, T. Kanade, “Image segmentation and reflection analysis through color,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 838–853.

B. K. P. Horn, “Obtaining shape from shading information,” in The Psychology of Computer Vision, P. Winston, ed. (McGraw-Hill, New York, 1975), pp. 115–155.

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978).

T. Sumanaweera, G. Healey, B. U. Lee, T. O. Binford, J. Ponce, “Image segmentation using geometrical and physical constraints,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 1091–1099.

D. Nickerson, “Spectrophotometric data for a collection of munsell samples,” (U.S. Department of Agriculture, Washington, D.C., 1957).

G. Healey, T. O. Binford, “A color metric for computer vision,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 10–17.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

R. S. Hunter, The Measurement of Appearance (Wiley, New York, 1975).

W. Egan, T. Hilgeman, Optical Properties of Inhomogeneous Materials (Academic, New York, 1979).

B. K. P. Horn, Robot Vision (McGraw-Hill, New York, 1986).

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

G. Healey, W. E. Blanz, “Identifying metal surfaces in color images,” to be published in Optics, Electro-Optics, and Sensors, Proc. Soc. Photo-Opt. Instrum. Eng. (1988).

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

N. Mott, H. Jones, The Theory of the Properties of Metals and Alloys (Oxford U. Press, Oxford, 1958).

J. Chambers, Computational Methods for Data Analysis (Wiley, New York, 1977).

K. Atkinson, An Introduction to Numerical Analysis (Wiley, New York, 1978).

D. Gray, ed., American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, New York, 1972).

E. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).

J. Weaver, C. Krafka, Physics Data: Optical Properties of Metals (Fachinformationszentrum, Karlsruhe1981).

E. Sparrow, R. Cess, Radiation Heat Transfer (Hemisphere, Washington, D.C., 1978).

H. C. Lee, E. Breneman, C. Schulte, “An experimental study of a color reflection model,” (Eastman Kodak, Rochester, N.Y., 1986).

G. Porter, J. Mundy, “Automatic visual inspection of metal surfaces,” in Techniques and Applications of Image Understanding, J. J. Pearson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.281, 176–181 (1981).
[CrossRef]

G. Healey, T. O. Binford, “Predicting material classes,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1988), pp. 1140–1146.

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Figures (11)

Fig. 1
Fig. 1

Surface reflection.

Fig. 2
Fig. 2

Scattering by colorant particles.

Fig. 3
Fig. 3

Reflection geometry.

Fig. 4
Fig. 4

(a) F1(θl) for copper. (b) F2*(λ) for copper. (c)–(f) Unichromatic fits for copper for the following angles θl: (c) 0°, (d) 30°, (e) 60°, (f) 80°.

Fig. 5
Fig. 5

(a) F1(θl) for aluminum. (b) F2*(λ) for aluminum. (c)–(f) Unichromatic fits for aluminum for the following angles θl: (c) 0°, (d) 30°, (e) 60°, (f) 80°.

Fig. 6
Fig. 6

(a) F1(θl) for sample 19. (b) F2(λ) for sample 19. (c)–(f) RB fits for sample 19 for the following angles θl: (c) 0°, (d) 30°, (e) 60°, (f) 80°.

Fig. 7
Fig. 7

(a) F1(θl) for sample 41. (b) F2(λ) for sample 41. (c)–(f) RB fits for sample 41 for the following angles θl: (c) 0°,(d) 30°, (e) 60°, (f) 80°.

Fig. 8
Fig. 8

Image of a valve.

Fig. 9
Fig. 9

Result of edge detection.

Fig. 10
Fig. 10

Prominent regions.

Fig. 11
Fig. 11

Regions in normalized color space.

Tables (5)

Tables Icon

Table 1 Electromagnetic Properties of Matter

Tables Icon

Table 2 Conductivity of Materials

Tables Icon

Table 3 Quality of Unichromatic Fit

Tables Icon

Table 4 Quality of Dichromatic Fita

Tables Icon

Table 5 Computed ED Values for Aluminum and Plastic

Equations (73)

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

R ( θ l , θ v , θ p , λ ) = R S ( θ l , θ v , θ p , λ ) + R B ( θ l , θ v , θ p , λ ) ,
I ( θ l , θ v , θ p , λ ) = R ( θ l , θ v , θ p , λ ) L ( λ ) ,
R ( g , λ ) = { M S ( g ) C S ( λ ) metal M S ( g ) C S ( λ ) + M B ( g ) C B ( λ ) inhomogeneous dielectric ,
I ( x ) = I ( 0 ) exp ( - 2 ω K 0 x / c ) ,
n 2 ( λ ) = μ c 2 2 { 1 + [ 1 + ( λ σ 2 π c ) 2 ] 1 / 2 } ,
K 0 2 ( λ ) = μ c 2 2 { - 1 + [ 1 + ( λ σ 2 π c ) 2 ] 1 / 2 } .
F ( θ l , λ ) = 0.5 ( R + R ) ,
R ( θ l , λ ) = a 2 + b 2 - 2 a cos θ l + cos 2 θ l a 2 + b 2 + 2 a cos θ l + cos 2 θ l ,
R ( θ l , λ ) = R a 2 + b 2 - 2 a sin θ l tan θ l + sin 2 θ l tan 2 θ l a 2 + b 2 + 2 a sin θ l tan θ l + sin 2 θ l tan 2 θ l ,
a = ( 0.5 ) 1 / 2 ( { [ n 2 ( λ ) - K 0 2 ( λ ) - sin 2 θ l ] 2 + 4 n 2 ( λ ) K 0 2 ( λ ) } 1 / 2 + [ n 2 ( λ ) - K 0 2 ( λ ) - sin 2 θ l ] ) 1 / 2 ,
b = ( 0.5 ) 1 / 2 ( { [ n 2 ( λ ) - K 0 2 ( λ ) - sin 2 θ l ] 2 + 4 n 2 ( λ ) K 0 2 ( λ ) } 1 / 2 - [ n 2 ( λ ) - K 0 2 ( λ ) - sin 2 θ l ] ) 1 / 2 .
R S ( θ l , θ v , θ p , λ ) = F ( γ , λ ) P ( α ) G ( θ l , θ v , θ p ) N ^ · V ^ ,
G = min { 1 , 2 ( N ^ · H ^ ) ( N ^ · V ^ ) ( V ^ · H ^ ) , 2 ( N ^ · H ^ ) ( N ^ · L ^ ) ( V ^ · H ^ ) } .
P ( α ) = 1 m 2 cos 4 α exp ( - tan 2 α m 2 ) ,
I ( g , λ ) = F ( g , λ ) P ( g ) G ( g ) L ( λ ) N ^ · V ^ ( g ) ,
I ˜ ( g , λ ) = I 1 ( g ) I 2 ( λ ) .
I 1 ( g ) = F 1 ( g ) P ( g ) G ( g ) N ^ · V ^ ( g ) ,
I 2 ( λ ) = F 2 ( λ ) L ( λ ) ,
F ˜ ( g , λ ) = F 1 ( g ) F 2 ( λ )
θ l λ [ F ¯ ( θ l , λ ) - F 1 ( θ l ) F 2 ( λ ) ] 2 d λ d θ l ,
E = 1 i n 1 j m [ F ¯ ( i , j ) - F 1 ( i ) F 2 ( j ) ] 2 .
F ¯ = U D V .
F 1 * ( i ) = σ 1 u i 1 ,
F 2 * ( j ) = v 1 j .
I 1 ( g ) = F 1 ( g ) P ( g ) G ( g ) N ^ · V ^ ( g ) ,
F 1 ( g ) = F 1 * ( g ) F ( g )
I 2 ( λ ) = F 2 * ( λ ) L ( λ ) ,
F ¯ ( i , j ) - F 1 * ( i ) F 2 * ( j ) 2 = 2 j m ( σ j u i j ) 2 .
E = 1 i n 2 j m σ j 2 u i j 2 = 2 j m σ j 2 ,
E n = 1 n 2 j m σ j 2 .
F 1 = σ 1 2 / ( 1 i m σ i 2 ) .
n = ( μ c 2 ) 1 / 2 ,
K 0 = 0.
R S ( θ l = 0 ° , λ ) = ( n - 1 ) 2 ( n + 1 ) 2 .
R B ( θ l , λ ) = ( 1 - R S ) C ( θ l , λ ) ( 1 - r i ) [ R ( λ ) - D ( θ l ) ] 2 [ 1 - r i R ( λ ) ] cos θ l ,
r i = 1 - 1.4399 - 0.7099 n + 0.3319 n 2 - 0.0636 n 3 n 2 ,
w ( λ ) = σ ( λ ) [ α ( λ ) + σ ( λ ) ] .
R ( λ ) = 2 - w ( λ ) - 2 [ 1 - w ( λ ) ] 1 / 2 w ( λ ) .
C ( θ l , λ ) = w ( λ ) cos θ l ( 2 cos θ l + 1 ) 1 - 4 [ 1 - w ( λ ) ] cos 2 θ l ,
D ( θ l ) = 2 cos θ l - 1 2 cos θ l + 1 .
R ( g , λ ) = R S ( g , λ ) + R B ( g , λ )
= M S ( g ) C S ( λ ) + M B ( g ) C B ( λ ) ,
R S ( g , λ ) = M S ( g ) C S ( λ ) .
R B ( g , λ ) = M B ( g ) C B ( λ ) .
E = 1 i n 1 j m [ R ¯ B ( i , j ) - F 1 F 2 ( j ) ] 2 .
R ¯ B = U D V ,
E n = 1 n 2 j m σ j 2 ,
F 1 = σ 1 2 / ( 1 i m σ i 2 ) .
R ( g , λ ) = { M S ( g ) C S ( λ ) metal M S ( g ) C S ( λ ) + M B ( g ) C B ( λ ) inhomogeneous dielectric ,
E ( g , λ ) = { L ( λ ) M S ( g ) C S ( λ ) metal L ( λ ) M S ( g ) C S ( λ ) + L ( λ ) M B ( g ) C B ( λ ) inhomogeneous dielectric .
I ( x , y , λ ) = E ( g , λ ) ,
I ( x , y , λ ) = L ( λ ) M ( g ) C ( λ ) ,
I 0 ( x 0 , y 0 , λ ) = L ( λ ) M ( g 0 ) C ( λ ) ,
I 1 ( x 1 , y 1 , λ ) = L ( λ ) M ( g 1 ) C ( λ ) ,
I ¯ ( x , y , λ ) = I ( x , y , λ ) I ( x , y , λ ) ,
I ¯ 0 ( x 0 , y 0 , λ ) = L ( λ ) M ( g 0 ) C ( λ ) L ( λ ) M ( g 0 ) C ( λ ) = L ( λ ) C ( λ ) L ( λ ) C ( λ ) ,
I ¯ 1 ( x 1 , y 1 , λ ) = L ( λ ) M ( g 1 ) C ( λ ) L ( λ ) M ( g 1 ) C ( λ ) = L ( λ ) C ( λ ) L ( λ ) C ( λ ) .
I ( x , y , λ ) = a x y f ( λ )
I ( x , y , λ ) = b x y g 1 ( λ ) + c x y g 2 ( λ )
S i = λ I ( x , y , λ ) s i ( λ ) d λ             ( 1 i N ) ,
[ S 1 S N ] = a x y [ α 1 α N ] ,
α i = λ f ( λ ) s i ( λ ) d λ .
[ S 1 S N ] = b x y [ β 1 β N ] + c x y [ γ 1 γ N ] ,
β i = λ g 1 ( λ ) s i ( λ ) d λ ,             γ i = λ g 2 ( λ ) s i ( λ ) d λ .
1 j N [ S ¯ ( i , j ) ] 2 = 1.
S ¯ = U D V ,
S ¯ i ( j ) = 1 j N σ k u i k v k .
S ¯ i * = 1 k D σ k u i k v k .
S ¯ i ( j ) - S ¯ i * 2 = ( D + 1 ) j N ( σ j u i j ) 2 ,
E D = 1 P 1 i P ( D + 1 ) j N ( σ j u i j ) 2 = 1 P ( D + 1 ) j N σ j 2 .
F ¯ ( i , j ) = 1 j m ( σ j u i j ) v j ,
F ¯ ( i , j ) - F 1 * ( i ) F 2 * ( j ) 2 = [ 1 j m ( σ j u i j ) v j ] - σ 1 u i 1 v 1 2 = 2 j m ( σ j u i j ) v j 2 = 2 j m ( σ j u i j ) v j 2 ,
F ¯ ( i , j ) - F 1 * ( i ) F 2 * ( j ) 2 = 2 j m ( σ j u i j ) 2 .

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