Abstract

We introduce an image model for color texture based on spatial correlation functions defined both within and between color bands. We show that three-dimensional geometric transformations of a surface in the scene produce corresponding transformations in these correlation functions. From this analysis we derive invariants of color correlation functions that can be computed efficiently and that can be used for geometry-invariant recognition. We show experimentally that these invariants are effective for recognition in situations in which neither color distributions nor gray-scale texture is sufficient. After recognition the estimated color correlation functions can be used to recover the three-dimensional location and orientation of surfaces. The color-texture invariants can be used in the recognition of three-dimensional objects, the segmentation of images of three-dimensional scenes, and the searching of image data bases.

© 1994 Optical Society of America

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References

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  1. G. Healey, S. Shafer, L. Wolff, eds., Physics-Based Vision: Principles and Practice. COLOR (Jones & Bartlett, Boston, Mass., 1992).
  2. S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
    [CrossRef]
  3. G. Healey, “Using color for geometry-insensitive segmentation,” J. Opt. Soc. Am. A 6, 920–937 (1989).
    [CrossRef]
  4. G. Healey, “Segmenting images using normalized color,”IEEE Trans. Syst. Man Cybern. 22, 64–73 (1992).
    [CrossRef]
  5. G. J. Klinker, S. A. Shafer, T. Kanade, “A physical approach to color image understanding,” Int. J. Comput. Vision 4, 7–38 (1990).
    [CrossRef]
  6. C. L. Novak, S. A. Shafer, “Method for estimating scene parameters from color histograms,” J. Opt. Soc. Am. A 11, 3020–3036 (1994).
    [CrossRef]
  7. M. Daily, “Color image segmentation using Markov random fields,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 304–312.
    [CrossRef]
  8. D. Panjwani, G. Healey, “Unsupervised segmentation of textured color images using Markov random field models,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 776–777.
    [CrossRef]
  9. W. Wright, “A Markov random field approach to data fusion and color segmentation,” Image Vision Comput. 7, 144–150 (1989).
    [CrossRef]
  10. M. Nagao, T. Matsuyama, Y. Ikeda, “Region extraction and shape analysis in aerial photographs,” Comput. Vision Graphics Image Process. 10, 195–223 (1979).
    [CrossRef]
  11. M. Swain, D. Ballard, “Color indexing,” Int. J. Comput. Vision 7, 11–32 (1991).
    [CrossRef]
  12. B. Funt, G. Finlayson, “Color constant color indexing,” Tech. Rep. CSS/LCCR TR 91-09 (School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada, 1991).
  13. G. Healey, D. Slater, “Global color constancy: recognition of objects by use of illumination-invariant properties of color distributions,” J. Opt. Soc. Am. A 11, 3003–3010 (1994).
    [CrossRef]
  14. J. J. Gibson, “The perception of visual surfaces,” Am. J. Psychol. 63, 367–384 (1950).
    [CrossRef] [PubMed]
  15. A. P. Witkin, “Recovering surface shape and orientation from texture,” Artif. Intell. 17, 17–45 (1981).
    [CrossRef]
  16. J. Y. Jau, R. T. Chin, “Shape from texture using Wigner distribution,” Comput. Vision Graphics Image Process. 52, 248–263 (1990).
    [CrossRef]
  17. B. J. Super, A. C. Bovik, “Shape-from-texture by wavelet based measurement of local spectral moments,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 296–301.
  18. R. Bajscy, L. Lieberman, “Texture gradient as a depth cue,” Comput. Vision Graphics Image Process. 5, 52–67 (1976).
    [CrossRef]
  19. J. Krumm, S. Shafer, “Shape from periodic texture using the spectrogram,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 284–289.
  20. L. G. Brown, H. Shvaytser, “Surface orientation from projective foreshortening of isotropic texture autocorrelation,”IEEE Trans. Pattern Anal. Mach. Intell. 12, 584–588 (1990).
    [CrossRef]
  21. M. A. S. Patel, F. S. Cohen, “Local surface shape estimation of 3D textured surfaces using Gaussian Markov random fields and stereo windows,”IEEE Trans. Pattern Anal. Mach. Intell. (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1091–1098.
    [CrossRef]
  22. J. Malik, R. Rosenholtz, “A differential method for computing local shape-from-texture for planar and curved surfaces,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 267–273.
    [CrossRef]
  23. R. L. Kashyap, A. Khotanzad, “A model based method for rotation invariant texture classification,”IEEE Trans. Pattern Anal. Mach. Intell. 8, 472–481 (1986).
    [CrossRef]
  24. F. S. Cohen, Z. Fan, M. S. Patel, “Classification of rotated and scaled textured images using Gaussian Markov random field models,”IEEE Trans. Pattern Anal. Mach. Intell. 13, 192–202 (1991).
    [CrossRef]
  25. E. Polak, Computational Methods in Optimization: a Unified Approach (Academic, New York, 1971).
  26. G. Taubin, D. B. Cooper, “Object recognition based on moment (or algebraic) invariants,” in Geometric Invariance in Computer Vision, J. L. Mundy, A. Zisserman, eds. (MIT Press, Cambridge, Mass., 1992), Chap. 19, pp. 375–397).
  27. R. Duda, P. Hart, Pattern Classification and Scene Analysis (Wiley-Interscience, New York, 1973).

1994 (2)

1992 (1)

G. Healey, “Segmenting images using normalized color,”IEEE Trans. Syst. Man Cybern. 22, 64–73 (1992).
[CrossRef]

1991 (2)

M. Swain, D. Ballard, “Color indexing,” Int. J. Comput. Vision 7, 11–32 (1991).
[CrossRef]

F. S. Cohen, Z. Fan, M. S. Patel, “Classification of rotated and scaled textured images using Gaussian Markov random field models,”IEEE Trans. Pattern Anal. Mach. Intell. 13, 192–202 (1991).
[CrossRef]

1990 (3)

L. G. Brown, H. Shvaytser, “Surface orientation from projective foreshortening of isotropic texture autocorrelation,”IEEE Trans. Pattern Anal. Mach. Intell. 12, 584–588 (1990).
[CrossRef]

J. Y. Jau, R. T. Chin, “Shape from texture using Wigner distribution,” Comput. Vision Graphics Image Process. 52, 248–263 (1990).
[CrossRef]

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

1989 (2)

G. Healey, “Using color for geometry-insensitive segmentation,” J. Opt. Soc. Am. A 6, 920–937 (1989).
[CrossRef]

W. Wright, “A Markov random field approach to data fusion and color segmentation,” Image Vision Comput. 7, 144–150 (1989).
[CrossRef]

1986 (1)

R. L. Kashyap, A. Khotanzad, “A model based method for rotation invariant texture classification,”IEEE Trans. Pattern Anal. Mach. Intell. 8, 472–481 (1986).
[CrossRef]

1985 (1)

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

1981 (1)

A. P. Witkin, “Recovering surface shape and orientation from texture,” Artif. Intell. 17, 17–45 (1981).
[CrossRef]

1979 (1)

M. Nagao, T. Matsuyama, Y. Ikeda, “Region extraction and shape analysis in aerial photographs,” Comput. Vision Graphics Image Process. 10, 195–223 (1979).
[CrossRef]

1976 (1)

R. Bajscy, L. Lieberman, “Texture gradient as a depth cue,” Comput. Vision Graphics Image Process. 5, 52–67 (1976).
[CrossRef]

1950 (1)

J. J. Gibson, “The perception of visual surfaces,” Am. J. Psychol. 63, 367–384 (1950).
[CrossRef] [PubMed]

Bajscy, R.

R. Bajscy, L. Lieberman, “Texture gradient as a depth cue,” Comput. Vision Graphics Image Process. 5, 52–67 (1976).
[CrossRef]

Ballard, D.

M. Swain, D. Ballard, “Color indexing,” Int. J. Comput. Vision 7, 11–32 (1991).
[CrossRef]

Bovik, A. C.

B. J. Super, A. C. Bovik, “Shape-from-texture by wavelet based measurement of local spectral moments,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 296–301.

Brown, L. G.

L. G. Brown, H. Shvaytser, “Surface orientation from projective foreshortening of isotropic texture autocorrelation,”IEEE Trans. Pattern Anal. Mach. Intell. 12, 584–588 (1990).
[CrossRef]

Chin, R. T.

J. Y. Jau, R. T. Chin, “Shape from texture using Wigner distribution,” Comput. Vision Graphics Image Process. 52, 248–263 (1990).
[CrossRef]

Cohen, F. S.

F. S. Cohen, Z. Fan, M. S. Patel, “Classification of rotated and scaled textured images using Gaussian Markov random field models,”IEEE Trans. Pattern Anal. Mach. Intell. 13, 192–202 (1991).
[CrossRef]

M. A. S. Patel, F. S. Cohen, “Local surface shape estimation of 3D textured surfaces using Gaussian Markov random fields and stereo windows,”IEEE Trans. Pattern Anal. Mach. Intell. (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1091–1098.
[CrossRef]

Cooper, D. B.

G. Taubin, D. B. Cooper, “Object recognition based on moment (or algebraic) invariants,” in Geometric Invariance in Computer Vision, J. L. Mundy, A. Zisserman, eds. (MIT Press, Cambridge, Mass., 1992), Chap. 19, pp. 375–397).

Daily, M.

M. Daily, “Color image segmentation using Markov random fields,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 304–312.
[CrossRef]

Duda, R.

R. Duda, P. Hart, Pattern Classification and Scene Analysis (Wiley-Interscience, New York, 1973).

Fan, Z.

F. S. Cohen, Z. Fan, M. S. Patel, “Classification of rotated and scaled textured images using Gaussian Markov random field models,”IEEE Trans. Pattern Anal. Mach. Intell. 13, 192–202 (1991).
[CrossRef]

Finlayson, G.

B. Funt, G. Finlayson, “Color constant color indexing,” Tech. Rep. CSS/LCCR TR 91-09 (School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada, 1991).

Funt, B.

B. Funt, G. Finlayson, “Color constant color indexing,” Tech. Rep. CSS/LCCR TR 91-09 (School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada, 1991).

Gibson, J. J.

J. J. Gibson, “The perception of visual surfaces,” Am. J. Psychol. 63, 367–384 (1950).
[CrossRef] [PubMed]

Hart, P.

R. Duda, P. Hart, Pattern Classification and Scene Analysis (Wiley-Interscience, New York, 1973).

Healey, G.

G. Healey, D. Slater, “Global color constancy: recognition of objects by use of illumination-invariant properties of color distributions,” J. Opt. Soc. Am. A 11, 3003–3010 (1994).
[CrossRef]

G. Healey, “Segmenting images using normalized color,”IEEE Trans. Syst. Man Cybern. 22, 64–73 (1992).
[CrossRef]

G. Healey, “Using color for geometry-insensitive segmentation,” J. Opt. Soc. Am. A 6, 920–937 (1989).
[CrossRef]

D. Panjwani, G. Healey, “Unsupervised segmentation of textured color images using Markov random field models,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 776–777.
[CrossRef]

Ikeda, Y.

M. Nagao, T. Matsuyama, Y. Ikeda, “Region extraction and shape analysis in aerial photographs,” Comput. Vision Graphics Image Process. 10, 195–223 (1979).
[CrossRef]

Jau, J. Y.

J. Y. Jau, R. T. Chin, “Shape from texture using Wigner distribution,” Comput. Vision Graphics Image Process. 52, 248–263 (1990).
[CrossRef]

Kanade, T.

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

Kashyap, R. L.

R. L. Kashyap, A. Khotanzad, “A model based method for rotation invariant texture classification,”IEEE Trans. Pattern Anal. Mach. Intell. 8, 472–481 (1986).
[CrossRef]

Khotanzad, A.

R. L. Kashyap, A. Khotanzad, “A model based method for rotation invariant texture classification,”IEEE Trans. Pattern Anal. Mach. Intell. 8, 472–481 (1986).
[CrossRef]

Klinker, G. J.

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

Krumm, J.

J. Krumm, S. Shafer, “Shape from periodic texture using the spectrogram,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 284–289.

Lieberman, L.

R. Bajscy, L. Lieberman, “Texture gradient as a depth cue,” Comput. Vision Graphics Image Process. 5, 52–67 (1976).
[CrossRef]

Malik, J.

J. Malik, R. Rosenholtz, “A differential method for computing local shape-from-texture for planar and curved surfaces,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 267–273.
[CrossRef]

Matsuyama, T.

M. Nagao, T. Matsuyama, Y. Ikeda, “Region extraction and shape analysis in aerial photographs,” Comput. Vision Graphics Image Process. 10, 195–223 (1979).
[CrossRef]

Nagao, M.

M. Nagao, T. Matsuyama, Y. Ikeda, “Region extraction and shape analysis in aerial photographs,” Comput. Vision Graphics Image Process. 10, 195–223 (1979).
[CrossRef]

Novak, C. L.

Panjwani, D.

D. Panjwani, G. Healey, “Unsupervised segmentation of textured color images using Markov random field models,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 776–777.
[CrossRef]

Patel, M. A. S.

M. A. S. Patel, F. S. Cohen, “Local surface shape estimation of 3D textured surfaces using Gaussian Markov random fields and stereo windows,”IEEE Trans. Pattern Anal. Mach. Intell. (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1091–1098.
[CrossRef]

Patel, M. S.

F. S. Cohen, Z. Fan, M. S. Patel, “Classification of rotated and scaled textured images using Gaussian Markov random field models,”IEEE Trans. Pattern Anal. Mach. Intell. 13, 192–202 (1991).
[CrossRef]

Polak, E.

E. Polak, Computational Methods in Optimization: a Unified Approach (Academic, New York, 1971).

Rosenholtz, R.

J. Malik, R. Rosenholtz, “A differential method for computing local shape-from-texture for planar and curved surfaces,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 267–273.
[CrossRef]

Shafer, S.

J. Krumm, S. Shafer, “Shape from periodic texture using the spectrogram,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 284–289.

Shafer, S. A.

C. L. Novak, S. A. Shafer, “Method for estimating scene parameters from color histograms,” J. Opt. Soc. Am. A 11, 3020–3036 (1994).
[CrossRef]

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

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

Shvaytser, H.

L. G. Brown, H. Shvaytser, “Surface orientation from projective foreshortening of isotropic texture autocorrelation,”IEEE Trans. Pattern Anal. Mach. Intell. 12, 584–588 (1990).
[CrossRef]

Slater, D.

Super, B. J.

B. J. Super, A. C. Bovik, “Shape-from-texture by wavelet based measurement of local spectral moments,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 296–301.

Swain, M.

M. Swain, D. Ballard, “Color indexing,” Int. J. Comput. Vision 7, 11–32 (1991).
[CrossRef]

Taubin, G.

G. Taubin, D. B. Cooper, “Object recognition based on moment (or algebraic) invariants,” in Geometric Invariance in Computer Vision, J. L. Mundy, A. Zisserman, eds. (MIT Press, Cambridge, Mass., 1992), Chap. 19, pp. 375–397).

Witkin, A. P.

A. P. Witkin, “Recovering surface shape and orientation from texture,” Artif. Intell. 17, 17–45 (1981).
[CrossRef]

Wright, W.

W. Wright, “A Markov random field approach to data fusion and color segmentation,” Image Vision Comput. 7, 144–150 (1989).
[CrossRef]

Am. J. Psychol. (1)

J. J. Gibson, “The perception of visual surfaces,” Am. J. Psychol. 63, 367–384 (1950).
[CrossRef] [PubMed]

Artif. Intell. (1)

A. P. Witkin, “Recovering surface shape and orientation from texture,” Artif. Intell. 17, 17–45 (1981).
[CrossRef]

Color Res. Appl. (1)

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

Comput. Vision Graphics Image Process. (3)

J. Y. Jau, R. T. Chin, “Shape from texture using Wigner distribution,” Comput. Vision Graphics Image Process. 52, 248–263 (1990).
[CrossRef]

M. Nagao, T. Matsuyama, Y. Ikeda, “Region extraction and shape analysis in aerial photographs,” Comput. Vision Graphics Image Process. 10, 195–223 (1979).
[CrossRef]

R. Bajscy, L. Lieberman, “Texture gradient as a depth cue,” Comput. Vision Graphics Image Process. 5, 52–67 (1976).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

R. L. Kashyap, A. Khotanzad, “A model based method for rotation invariant texture classification,”IEEE Trans. Pattern Anal. Mach. Intell. 8, 472–481 (1986).
[CrossRef]

F. S. Cohen, Z. Fan, M. S. Patel, “Classification of rotated and scaled textured images using Gaussian Markov random field models,”IEEE Trans. Pattern Anal. Mach. Intell. 13, 192–202 (1991).
[CrossRef]

L. G. Brown, H. Shvaytser, “Surface orientation from projective foreshortening of isotropic texture autocorrelation,”IEEE Trans. Pattern Anal. Mach. Intell. 12, 584–588 (1990).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

G. Healey, “Segmenting images using normalized color,”IEEE Trans. Syst. Man Cybern. 22, 64–73 (1992).
[CrossRef]

Image Vision Comput. (1)

W. Wright, “A Markov random field approach to data fusion and color segmentation,” Image Vision Comput. 7, 144–150 (1989).
[CrossRef]

Int. J. Comput. Vision (2)

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

M. Swain, D. Ballard, “Color indexing,” Int. J. Comput. Vision 7, 11–32 (1991).
[CrossRef]

J. Opt. Soc. Am. A (3)

Other (11)

M. A. S. Patel, F. S. Cohen, “Local surface shape estimation of 3D textured surfaces using Gaussian Markov random fields and stereo windows,”IEEE Trans. Pattern Anal. Mach. Intell. (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1091–1098.
[CrossRef]

J. Malik, R. Rosenholtz, “A differential method for computing local shape-from-texture for planar and curved surfaces,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 267–273.
[CrossRef]

E. Polak, Computational Methods in Optimization: a Unified Approach (Academic, New York, 1971).

G. Taubin, D. B. Cooper, “Object recognition based on moment (or algebraic) invariants,” in Geometric Invariance in Computer Vision, J. L. Mundy, A. Zisserman, eds. (MIT Press, Cambridge, Mass., 1992), Chap. 19, pp. 375–397).

R. Duda, P. Hart, Pattern Classification and Scene Analysis (Wiley-Interscience, New York, 1973).

G. Healey, S. Shafer, L. Wolff, eds., Physics-Based Vision: Principles and Practice. COLOR (Jones & Bartlett, Boston, Mass., 1992).

M. Daily, “Color image segmentation using Markov random fields,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 304–312.
[CrossRef]

D. Panjwani, G. Healey, “Unsupervised segmentation of textured color images using Markov random field models,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 776–777.
[CrossRef]

B. Funt, G. Finlayson, “Color constant color indexing,” Tech. Rep. CSS/LCCR TR 91-09 (School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada, 1991).

J. Krumm, S. Shafer, “Shape from periodic texture using the spectrogram,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 284–289.

B. J. Super, A. C. Bovik, “Shape-from-texture by wavelet based measurement of local spectral moments,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 296–301.

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Figures (30)

Fig. 1
Fig. 1

Gaussian sphere.

Fig. 2
Fig. 2

Color stripe texture.

Fig. 3
Fig. 3

Intensity image of Fig. 2.

Fig. 4
Fig. 4

Different texture.

Fig. 5
Fig. 5

Color correlations of Fig. 2.

Fig. 6
Fig. 6

R( n ¯).

Fig. 7
Fig. 7

R′( n ¯).

Fig. 8
Fig. 8

R(L n ¯).

Fig. 9
Fig. 9

R′(L n ¯).

Fig. 10
Fig. 10

Histograms of two textures.

Fig. 11
Fig. 11

Sweater.

Fig. 12
Fig. 12

Correlation function of Fig. 11.

Fig. 13
Fig. 13

Rotated Sweater.

Fig. 14
Fig. 14

Correlation function of Fig. 13.

Fig. 15
Fig. 15

Carpet.

Fig. 16
Fig. 16

Correlation function of Fig. 15.

Fig. 17
Fig. 17

Carpet at different orientation.

Fig. 18
Fig. 18

Correlation function of Fig. 17.

Fig. 19
Fig. 19

Grand Canyon.

Fig. 20
Fig. 20

Correlation function of Fig. 19.

Fig. 21
Fig. 21

Different view of Grand Canyon.

Fig. 22
Fig. 22

Correlation function of Fig. 21.

Fig. 23
Fig. 23

Two sides of a box.

Fig. 24
Fig. 24

Correlation function of left-hand side of box.

Fig. 25
Fig. 25

Correlation function of right-hand side of box.

Fig. 27
Fig. 27

Correlation function of Fig. 26.

Fig. 28
Fig. 28

Tie at different angle.

Fig. 29
Fig. 29

Correlation function of Fig. 28.

Fig. 30
Fig. 30

Plane geometry.

Tables (2)

Tables Icon

Table 1 Classification of Unknown Texturesa

Tables Icon

Table 2 Classification of Unknown Textures by Intensity Images Onlya

Equations (34)

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

[ x y ] = g × [ x y ] ,
Q ( x , y , λ ) = r ( x , y , z , λ ) l ( x , y , z , λ ) .
I ( x , y , λ ) = Q ( x , y , λ ) * O ( x , y , λ ) ,
I r ( x , y ) = I ( x , y , λ ) F r ( λ ) d λ .
I r ( a , b ) = [ I r ( x , y ) * S ( x , y ) ] x = a , y = b ,
R i j ( n , m ) = E { [ I i ( a , b ) - I ¯ i ] [ I j ( a + n , b + m ) - I ¯ j ] } ,
x ^ = M 1 x ¯ + D ,
R i j ( n ¯ , θ , ϕ , α , k ) = E ( { [ I ( x ^ , λ ) F i ( λ ) d λ ] * S ( x ¯ ) } × { [ I ( x ^ + n ¯ , λ ) F j ( λ ) d λ ] * S ( x ¯ ) } x = a , y = b ) - I ¯ i I ¯ j ,
R i j ( n ¯ , θ , ϕ , α , k ) = R i j N ( M 1 n ¯ ) ,
R i j ( M 2 - 1 n ¯ ) = R i j ( n ¯ ) ,
m n [ R i j ( M 2 - 1 n ¯ ) - R i j ( n ¯ ) ] 2 = 0
i j { m n [ R i j estimate ( M 2 - 1 n ¯ ) - R i j 0 ( n ¯ ) ] 2 }
R ˜ i j ( n ¯ ) ϕ ( n ¯ ) d n ¯ ,
A i j α = 1 R i j ( n ¯ - R ¯ i j ) α R ˜ i j ( n ¯ ) d n ¯ ,
F ( A i j ) = M w F ( A i j ) ,
( 3 , 0 ) < ( 2 , 1 ) < ( 1 , 2 ) < ( 0 , 3 ) .
X [ 3 ] ( n , m ) = ( 1 6 n 3     1 2 n 2 m     1 2 n m 2     1 6 m 3 ) t .
X [ 2 , 3 ] ( n , m ) = [ 1 12 n 5 1 4 n 4 m 1 4 n 3 m 2 1 12 n 2 m 3 1 12 n 4 m 1 4 n 3 m 2 1 4 n 2 m 3 1 12 n m 4 1 12 n 3 m 2 1 4 n 2 m 3 1 4 n m 4 1 12 m 5 ] .
A [ d ] = 1 R X [ d ] ( n ¯ - R ¯ ) R ˜ ( n ¯ ) d n ¯ ,
A [ k , l ] = 1 R X [ k , l ] ( n ¯ - R ¯ ) R ˜ ( n ¯ ) d n ¯ .
L A [ 1 , 1 ] L t = I .
1 R n m n d - q m q R ˜ ( n , m )
1 R [ n > 0 m > 0 n d - q m q ( n , m ) + n > 0 m > 0 n d - q ( - m ) q × R ˜ ( n , - m ) + n > 0 m > 0 ( - n ) d - q m q R ˜ ( - n , m ) + n > 0 m > 0 ( - n ) d - q ( - m ) q R ˜ ( - n , - m ) ] .
[ x 1 y 1 ] = [ cos α - sin α sin α cos α ] [ x y ] .
[ x 2 y 2 ] = [ 1 + [ cos ( ϕ ) - 1 ] cos 2 θ [ cos ( ϕ ) - 1 ] cos θ sin θ [ cos ( ϕ ) - 1 ] cos θ sin θ [ cos ( ϕ ) - 1 ] sin 2 θ ] × [ x 1 y 1 ] .
[ x ^ y ^ ] = k ( [ x 2 y 2 ] + g [ d x d y ] ) .
x ^ = M 1 x ¯ + D ,
M 1 = k × [ 1 + [ cos ( ϕ ) - 1 ] cos 2 θ [ cos ( ϕ ) - 1 ] cos θ sin θ [ cos ( ϕ ) - 1 ] cos θ sin θ [ cos ( ϕ ) - 1 ] sin 2 θ ] × [ cos α - sin α sin α cos α ] ,
D = k g [ d x d y ] .
[ cos ( π / 2 - σ ) - sin ( π / 2 - σ ) sin ( π / 2 - σ ) cos ( π / 2 - σ ) ]
β = cos ( γ - δ ) cos ( γ ) .
[ β 0 0 1 ] .
[ cos α - p sin α ( sin α ) / p cos α ] ,
M 2 = k [ sin σ cos σ - cos σ sin σ ] [ cos α - p sin α ( sin α ) / p cos α ] × [ β 0 0 1 ] [ sin σ - cos σ cos σ sin σ ] ,

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