Abstract

We present a computational algorithm for isotrigon texture discrimination. The aim of this method consists in discriminating isotrigon textures against a binary random background. The extension of the method to the problem of multitexture discrimination is considered as well. The method relies on the fact that the information content of time or space–frequency representations of signals, including images, can be readily analyzed by means of generalized entropy measures. In such a scenario, the Rényi entropy appears as an effective tool, given that Rényi measures can be used to provide information about a local neighborhood within an image. Localization is essential for comparing images on a pixel-by-pixel basis. Discrimination is performed through a local Rényi entropy measurement applied on a spatially oriented 1-D pseudo-Wigner distribution (PWD) of the test image. The PWD is normalized so that it may be interpreted as a probability distribution. Prior to the calculation of the texture’s PWD, a preprocessing filtering step replaces the original texture with its localized spatially oriented Allan variances. The anisotropic structure of the textures, as revealed by the Allan variances, turns out to be crucial later to attain a high discrimination by the extraction of Rényi entropy measures. The method has been empirically evaluated with a family of isotrigon textures embedded in a binary random background. The extension to the case of multiple isotrigon mosaics has also been considered. Discrimination results are compared with other existing methods.

© 2008 Optical Society of America

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  1. B. Julesz, E. N. Gilbert, and J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137-140 (1978).
    [CrossRef] [PubMed]
  2. T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vision Res. 44, 1093-1113 (2004).
    [CrossRef] [PubMed]
  3. K. P. Purpura, J. D. Victor, and E. Katz, “Striate cortex extracts higher-order spatial correlations from visual textures,” Proc. Natl. Acad. Sci. U.S.A. 91, 8482-8486 (1994).
    [CrossRef] [PubMed]
  4. T. Maddess and Y. Nagai, “Discriminating isotrigon textures,” Vision Res. 41, 3837-3860 (2001).
    [CrossRef] [PubMed]
  5. D. W. Allan, “Statistics of atomic frequency standard,” Proc. IEEE 54, 223-231 (1966).
    [CrossRef]
  6. R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis (Prentice Hall, 1992).
  7. J. L. Starck, E. Candes, and D. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670-684 (2002).
    [CrossRef]
  8. M. N. Do and M. Vetterli, “The contourlet transform: an efficient directional multiresolution image representation,” IEEE Trans. Image Process. 14, 2091-2106 (2005).
    [CrossRef] [PubMed]
  9. D. Donoho, “Wedgelets: nearly minimax estimation of edges,” Ann. Stat. 27, 859-867 (1999).
    [CrossRef]
  10. V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P. L. Dragotti, “Directionlets: anisotropic multidirectional representation with separable filtering,” IEEE Trans. Image Process. 15, 1916-1933 (2006).
    [CrossRef] [PubMed]
  11. A. Cumani, “Edge detection in multispectral images,” Comput. Vis. Graph. Image Process. 53, 40-51 (1991).
  12. L. D. Jacobson and H. Wechsler, “Joint spatial/spatial-frequency representation,” Signal Process. 14, 37-68 (1988).
    [CrossRef]
  13. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  14. L. Cohen, “Generalized phase-space distribution functions,” J. Math. Phys. 7, 781-786 (1966).
    [CrossRef]
  15. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part I. Continuous-time signals,” Philips J. Res. 35, 237-250 (1980).
  16. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part II. Discrete-time signals,” Philips J. Res. 35, 276-300 (1980).
  17. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part III. Relations with other time-frequency transformations,” Philips J. Res. 35, 372-389 (1980).
  18. K. H. Brenner, “A discrete version of the Wigner distribution function,” EURASIP J. Appl. Signal Process. 2005, 307-309.
  19. L. Stankovic, “A measure of some time-frequency distributions concentration,” Signal Process. 81, 623-631 (2001).
    [CrossRef]
  20. C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, 1949).
  21. N. Wiener, Cybernetics (Wiley, 1948).
  22. A. Rényi, “Some fundamental questions of information theory,” in Selected Papers of Alfréd Rényi, P.Turán, ed. (Akadémiai Kiadó, 1976), pp. 526-552 (1976). [Originally published in Magy. Tud. Akad. Mat. Fiz Tud. Oszt. Kozl., 10, 251-282 (1960)].
  23. W. J. Williams, M. L. Brown, and A. O. Hero, “Uncertainty, information and time-frequency distributions,” Proc. SPIE 1566, 144-156 (1991).
    [CrossRef]
  24. T. H. Sang and W. J. Williams, “Rényi information and signal dependent optimal kernel design.” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1995), Vol. 2, pp. 997-1000.
  25. P. Flandrin, R. G. Baraniuk, and O. Michel, “Time-frequency complexity and information,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1994), Vol. 3, pp. 329-332.
  26. R. Eisberg and R. Resnick, Quantum Physics (Wiley, 1974).
  27. G. Süßmann, “Uncertainty relation: from inequality to equality,” Z. Naturforsch. 52, 49-52 (1997).
  28. J. J. Wlodarz, “Entropy and Wigner distribution functions revisited,” Int. J. Theor. Phys. 42, 1075-1084 (2003).
    [CrossRef]
  29. J. Ville, “Théorie and applications de la Notion de Signal Analytique,” Cables Transm. 2A, 61-74 (1948).
  30. D. Dragoman, “Applications of the Wigner distribution function in signal processing,” EURASIP J. Appl. Signal Process. 10, 1520-1534 (2005).
  31. B. Yegnanarayana, P. Pavan Kumar, and S. Das, “One-dimensional Gabor filtering for texture edge detection,” Proceedings of Indian Conference on Computer Vision, Graphics and Image Processing (IEEE, 1998), pp. 231-237.
  32. MATLAB code for segmentation and classification of multitexture images is available from http://www.cse.iitk.ac.in/~amit/courses/768/00/rajrup/.
  33. P. de Rivaz and N. Kingsbury, “Complex wavelet features for fast texture image retrieval,” Proceedings of IEEE Conference on Image Processing (IEEE, 1999), pp. 25-28.
  34. M. E. Barilla, M. G. Forero, and M. Spann, “Color-based texture segmentation,” in Information Optics, 5th International Workshop, G.Cristóbal, B.Javidi, and S.Vallmitjana, eds. (AIP, 2006), pp. 401-409.
  35. G. Smith and I. Burns, “MeasTex image texture database and test suite” (1997). Available online at http://www.texturesynthesis.com/meastex/meastex.html.
  36. C. W. Tyler, “Stereoscopic tilt and size effects,” Perception 4, 287-192 (1975).
    [CrossRef]
  37. R. Schumer and L. Ganz, “Independent stereoscopic channels for different extents of spatial pooling,” Vision Res. 19, 1303-1314 (1979).
    [CrossRef] [PubMed]
  38. T. Maddess and Y. Nagai, “Lessons from biological processing of image texture,” in International Congress Series (Elsevier, 2004), Vol. 1269, pp. 26-29.
    [CrossRef]

2006 (1)

V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P. L. Dragotti, “Directionlets: anisotropic multidirectional representation with separable filtering,” IEEE Trans. Image Process. 15, 1916-1933 (2006).
[CrossRef] [PubMed]

2005 (2)

M. N. Do and M. Vetterli, “The contourlet transform: an efficient directional multiresolution image representation,” IEEE Trans. Image Process. 14, 2091-2106 (2005).
[CrossRef] [PubMed]

D. Dragoman, “Applications of the Wigner distribution function in signal processing,” EURASIP J. Appl. Signal Process. 10, 1520-1534 (2005).

2004 (1)

T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vision Res. 44, 1093-1113 (2004).
[CrossRef] [PubMed]

2003 (1)

J. J. Wlodarz, “Entropy and Wigner distribution functions revisited,” Int. J. Theor. Phys. 42, 1075-1084 (2003).
[CrossRef]

2002 (1)

J. L. Starck, E. Candes, and D. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670-684 (2002).
[CrossRef]

2001 (2)

T. Maddess and Y. Nagai, “Discriminating isotrigon textures,” Vision Res. 41, 3837-3860 (2001).
[CrossRef] [PubMed]

L. Stankovic, “A measure of some time-frequency distributions concentration,” Signal Process. 81, 623-631 (2001).
[CrossRef]

1999 (1)

D. Donoho, “Wedgelets: nearly minimax estimation of edges,” Ann. Stat. 27, 859-867 (1999).
[CrossRef]

1997 (1)

G. Süßmann, “Uncertainty relation: from inequality to equality,” Z. Naturforsch. 52, 49-52 (1997).

1994 (1)

K. P. Purpura, J. D. Victor, and E. Katz, “Striate cortex extracts higher-order spatial correlations from visual textures,” Proc. Natl. Acad. Sci. U.S.A. 91, 8482-8486 (1994).
[CrossRef] [PubMed]

1991 (2)

W. J. Williams, M. L. Brown, and A. O. Hero, “Uncertainty, information and time-frequency distributions,” Proc. SPIE 1566, 144-156 (1991).
[CrossRef]

A. Cumani, “Edge detection in multispectral images,” Comput. Vis. Graph. Image Process. 53, 40-51 (1991).

1988 (1)

L. D. Jacobson and H. Wechsler, “Joint spatial/spatial-frequency representation,” Signal Process. 14, 37-68 (1988).
[CrossRef]

1980 (3)

T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part I. Continuous-time signals,” Philips J. Res. 35, 237-250 (1980).

T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part II. Discrete-time signals,” Philips J. Res. 35, 276-300 (1980).

T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part III. Relations with other time-frequency transformations,” Philips J. Res. 35, 372-389 (1980).

1979 (1)

R. Schumer and L. Ganz, “Independent stereoscopic channels for different extents of spatial pooling,” Vision Res. 19, 1303-1314 (1979).
[CrossRef] [PubMed]

1978 (1)

B. Julesz, E. N. Gilbert, and J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137-140 (1978).
[CrossRef] [PubMed]

1975 (1)

C. W. Tyler, “Stereoscopic tilt and size effects,” Perception 4, 287-192 (1975).
[CrossRef]

1966 (2)

L. Cohen, “Generalized phase-space distribution functions,” J. Math. Phys. 7, 781-786 (1966).
[CrossRef]

D. W. Allan, “Statistics of atomic frequency standard,” Proc. IEEE 54, 223-231 (1966).
[CrossRef]

1948 (1)

J. Ville, “Théorie and applications de la Notion de Signal Analytique,” Cables Transm. 2A, 61-74 (1948).

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Allan, D. W.

D. W. Allan, “Statistics of atomic frequency standard,” Proc. IEEE 54, 223-231 (1966).
[CrossRef]

Ankiewcz, A.

T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vision Res. 44, 1093-1113 (2004).
[CrossRef] [PubMed]

Baraniuk, R. G.

P. Flandrin, R. G. Baraniuk, and O. Michel, “Time-frequency complexity and information,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1994), Vol. 3, pp. 329-332.

Barilla, M. E.

M. E. Barilla, M. G. Forero, and M. Spann, “Color-based texture segmentation,” in Information Optics, 5th International Workshop, G.Cristóbal, B.Javidi, and S.Vallmitjana, eds. (AIP, 2006), pp. 401-409.

Beferull-Lozano, B.

V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P. L. Dragotti, “Directionlets: anisotropic multidirectional representation with separable filtering,” IEEE Trans. Image Process. 15, 1916-1933 (2006).
[CrossRef] [PubMed]

Brenner, K. H.

K. H. Brenner, “A discrete version of the Wigner distribution function,” EURASIP J. Appl. Signal Process. 2005, 307-309.

Brown, M. L.

W. J. Williams, M. L. Brown, and A. O. Hero, “Uncertainty, information and time-frequency distributions,” Proc. SPIE 1566, 144-156 (1991).
[CrossRef]

Burns, I.

G. Smith and I. Burns, “MeasTex image texture database and test suite” (1997). Available online at http://www.texturesynthesis.com/meastex/meastex.html.

Candes, E.

J. L. Starck, E. Candes, and D. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670-684 (2002).
[CrossRef]

Claasen, T. A. C. M.

T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part II. Discrete-time signals,” Philips J. Res. 35, 276-300 (1980).

T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part III. Relations with other time-frequency transformations,” Philips J. Res. 35, 372-389 (1980).

T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part I. Continuous-time signals,” Philips J. Res. 35, 237-250 (1980).

Cohen, L.

L. Cohen, “Generalized phase-space distribution functions,” J. Math. Phys. 7, 781-786 (1966).
[CrossRef]

Cumani, A.

A. Cumani, “Edge detection in multispectral images,” Comput. Vis. Graph. Image Process. 53, 40-51 (1991).

Das, S.

B. Yegnanarayana, P. Pavan Kumar, and S. Das, “One-dimensional Gabor filtering for texture edge detection,” Proceedings of Indian Conference on Computer Vision, Graphics and Image Processing (IEEE, 1998), pp. 231-237.

de Rivaz, P.

P. de Rivaz and N. Kingsbury, “Complex wavelet features for fast texture image retrieval,” Proceedings of IEEE Conference on Image Processing (IEEE, 1999), pp. 25-28.

Do, M. N.

M. N. Do and M. Vetterli, “The contourlet transform: an efficient directional multiresolution image representation,” IEEE Trans. Image Process. 14, 2091-2106 (2005).
[CrossRef] [PubMed]

Donoho, D.

J. L. Starck, E. Candes, and D. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670-684 (2002).
[CrossRef]

D. Donoho, “Wedgelets: nearly minimax estimation of edges,” Ann. Stat. 27, 859-867 (1999).
[CrossRef]

Dragoman, D.

D. Dragoman, “Applications of the Wigner distribution function in signal processing,” EURASIP J. Appl. Signal Process. 10, 1520-1534 (2005).

Dragotti, P. L.

V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P. L. Dragotti, “Directionlets: anisotropic multidirectional representation with separable filtering,” IEEE Trans. Image Process. 15, 1916-1933 (2006).
[CrossRef] [PubMed]

Eisberg, R.

R. Eisberg and R. Resnick, Quantum Physics (Wiley, 1974).

Flandrin, P.

P. Flandrin, R. G. Baraniuk, and O. Michel, “Time-frequency complexity and information,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1994), Vol. 3, pp. 329-332.

Forero, M. G.

M. E. Barilla, M. G. Forero, and M. Spann, “Color-based texture segmentation,” in Information Optics, 5th International Workshop, G.Cristóbal, B.Javidi, and S.Vallmitjana, eds. (AIP, 2006), pp. 401-409.

Ganz, L.

R. Schumer and L. Ganz, “Independent stereoscopic channels for different extents of spatial pooling,” Vision Res. 19, 1303-1314 (1979).
[CrossRef] [PubMed]

Gilbert, E. N.

B. Julesz, E. N. Gilbert, and J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137-140 (1978).
[CrossRef] [PubMed]

Hero, A. O.

W. J. Williams, M. L. Brown, and A. O. Hero, “Uncertainty, information and time-frequency distributions,” Proc. SPIE 1566, 144-156 (1991).
[CrossRef]

Jacobson, L. D.

L. D. Jacobson and H. Wechsler, “Joint spatial/spatial-frequency representation,” Signal Process. 14, 37-68 (1988).
[CrossRef]

James, A. C.

T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vision Res. 44, 1093-1113 (2004).
[CrossRef] [PubMed]

Johnson, R. A.

R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis (Prentice Hall, 1992).

Julesz, B.

B. Julesz, E. N. Gilbert, and J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137-140 (1978).
[CrossRef] [PubMed]

Katz, E.

K. P. Purpura, J. D. Victor, and E. Katz, “Striate cortex extracts higher-order spatial correlations from visual textures,” Proc. Natl. Acad. Sci. U.S.A. 91, 8482-8486 (1994).
[CrossRef] [PubMed]

Kingsbury, N.

P. de Rivaz and N. Kingsbury, “Complex wavelet features for fast texture image retrieval,” Proceedings of IEEE Conference on Image Processing (IEEE, 1999), pp. 25-28.

Maddess, T.

T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vision Res. 44, 1093-1113 (2004).
[CrossRef] [PubMed]

T. Maddess and Y. Nagai, “Discriminating isotrigon textures,” Vision Res. 41, 3837-3860 (2001).
[CrossRef] [PubMed]

T. Maddess and Y. Nagai, “Lessons from biological processing of image texture,” in International Congress Series (Elsevier, 2004), Vol. 1269, pp. 26-29.
[CrossRef]

Mecklenbräuker, W. F. G.

T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part II. Discrete-time signals,” Philips J. Res. 35, 276-300 (1980).

T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part I. Continuous-time signals,” Philips J. Res. 35, 237-250 (1980).

T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution--a tool for time-frequency analysis. Part III. Relations with other time-frequency transformations,” Philips J. Res. 35, 372-389 (1980).

Michel, O.

P. Flandrin, R. G. Baraniuk, and O. Michel, “Time-frequency complexity and information,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1994), Vol. 3, pp. 329-332.

Nagai, Y.

T. Maddess, Y. Nagai, A. C. James, and A. Ankiewcz, “Binary and ternary textures containing higher-order spatial correlations,” Vision Res. 44, 1093-1113 (2004).
[CrossRef] [PubMed]

T. Maddess and Y. Nagai, “Discriminating isotrigon textures,” Vision Res. 41, 3837-3860 (2001).
[CrossRef] [PubMed]

T. Maddess and Y. Nagai, “Lessons from biological processing of image texture,” in International Congress Series (Elsevier, 2004), Vol. 1269, pp. 26-29.
[CrossRef]

Pavan Kumar, P.

B. Yegnanarayana, P. Pavan Kumar, and S. Das, “One-dimensional Gabor filtering for texture edge detection,” Proceedings of Indian Conference on Computer Vision, Graphics and Image Processing (IEEE, 1998), pp. 231-237.

Purpura, K. P.

K. P. Purpura, J. D. Victor, and E. Katz, “Striate cortex extracts higher-order spatial correlations from visual textures,” Proc. Natl. Acad. Sci. U.S.A. 91, 8482-8486 (1994).
[CrossRef] [PubMed]

Rényi, A.

A. Rényi, “Some fundamental questions of information theory,” in Selected Papers of Alfréd Rényi, P.Turán, ed. (Akadémiai Kiadó, 1976), pp. 526-552 (1976). [Originally published in Magy. Tud. Akad. Mat. Fiz Tud. Oszt. Kozl., 10, 251-282 (1960)].

Resnick, R.

R. Eisberg and R. Resnick, Quantum Physics (Wiley, 1974).

Sang, T. H.

T. H. Sang and W. J. Williams, “Rényi information and signal dependent optimal kernel design.” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1995), Vol. 2, pp. 997-1000.

Schumer, R.

R. Schumer and L. Ganz, “Independent stereoscopic channels for different extents of spatial pooling,” Vision Res. 19, 1303-1314 (1979).
[CrossRef] [PubMed]

Shannon, C. E.

C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, 1949).

Smith, G.

G. Smith and I. Burns, “MeasTex image texture database and test suite” (1997). Available online at http://www.texturesynthesis.com/meastex/meastex.html.

Spann, M.

M. E. Barilla, M. G. Forero, and M. Spann, “Color-based texture segmentation,” in Information Optics, 5th International Workshop, G.Cristóbal, B.Javidi, and S.Vallmitjana, eds. (AIP, 2006), pp. 401-409.

Stankovic, L.

L. Stankovic, “A measure of some time-frequency distributions concentration,” Signal Process. 81, 623-631 (2001).
[CrossRef]

Starck, J. L.

J. L. Starck, E. Candes, and D. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670-684 (2002).
[CrossRef]

Süßmann, G.

G. Süßmann, “Uncertainty relation: from inequality to equality,” Z. Naturforsch. 52, 49-52 (1997).

Tyler, C. W.

C. W. Tyler, “Stereoscopic tilt and size effects,” Perception 4, 287-192 (1975).
[CrossRef]

Velisavljevic, V.

V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P. L. Dragotti, “Directionlets: anisotropic multidirectional representation with separable filtering,” IEEE Trans. Image Process. 15, 1916-1933 (2006).
[CrossRef] [PubMed]

Vetterli, M.

V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P. L. Dragotti, “Directionlets: anisotropic multidirectional representation with separable filtering,” IEEE Trans. Image Process. 15, 1916-1933 (2006).
[CrossRef] [PubMed]

M. N. Do and M. Vetterli, “The contourlet transform: an efficient directional multiresolution image representation,” IEEE Trans. Image Process. 14, 2091-2106 (2005).
[CrossRef] [PubMed]

Victor, J. D.

K. P. Purpura, J. D. Victor, and E. Katz, “Striate cortex extracts higher-order spatial correlations from visual textures,” Proc. Natl. Acad. Sci. U.S.A. 91, 8482-8486 (1994).
[CrossRef] [PubMed]

B. Julesz, E. N. Gilbert, and J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137-140 (1978).
[CrossRef] [PubMed]

Ville, J.

J. Ville, “Théorie and applications de la Notion de Signal Analytique,” Cables Transm. 2A, 61-74 (1948).

Weaver, W.

C. E. Shannon and W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, 1949).

Wechsler, H.

L. D. Jacobson and H. Wechsler, “Joint spatial/spatial-frequency representation,” Signal Process. 14, 37-68 (1988).
[CrossRef]

Wichern, D. W.

R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis (Prentice Hall, 1992).

Wiener, N.

N. Wiener, Cybernetics (Wiley, 1948).

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Williams, W. J.

W. J. Williams, M. L. Brown, and A. O. Hero, “Uncertainty, information and time-frequency distributions,” Proc. SPIE 1566, 144-156 (1991).
[CrossRef]

T. H. Sang and W. J. Williams, “Rényi information and signal dependent optimal kernel design.” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1995), Vol. 2, pp. 997-1000.

Wlodarz, J. J.

J. J. Wlodarz, “Entropy and Wigner distribution functions revisited,” Int. J. Theor. Phys. 42, 1075-1084 (2003).
[CrossRef]

Yegnanarayana, B.

B. Yegnanarayana, P. Pavan Kumar, and S. Das, “One-dimensional Gabor filtering for texture edge detection,” Proceedings of Indian Conference on Computer Vision, Graphics and Image Processing (IEEE, 1998), pp. 231-237.

Ann. Stat. (1)

D. Donoho, “Wedgelets: nearly minimax estimation of edges,” Ann. Stat. 27, 859-867 (1999).
[CrossRef]

Biol. Cybern. (1)

B. Julesz, E. N. Gilbert, and J. D. Victor, “Visual discrimination of textures with identical third-order statistics,” Biol. Cybern. 31, 137-140 (1978).
[CrossRef] [PubMed]

Cables Transm. (1)

J. Ville, “Théorie and applications de la Notion de Signal Analytique,” Cables Transm. 2A, 61-74 (1948).

Comput. Vis. Graph. Image Process. (1)

A. Cumani, “Edge detection in multispectral images,” Comput. Vis. Graph. Image Process. 53, 40-51 (1991).

EURASIP J. Appl. Signal Process. (2)

D. Dragoman, “Applications of the Wigner distribution function in signal processing,” EURASIP J. Appl. Signal Process. 10, 1520-1534 (2005).

K. H. Brenner, “A discrete version of the Wigner distribution function,” EURASIP J. Appl. Signal Process. 2005, 307-309.

IEEE Trans. Image Process. (3)

V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and P. L. Dragotti, “Directionlets: anisotropic multidirectional representation with separable filtering,” IEEE Trans. Image Process. 15, 1916-1933 (2006).
[CrossRef] [PubMed]

J. L. Starck, E. Candes, and D. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11, 670-684 (2002).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

A. Particular sample of an isotrigon texture (even zigzag) of 512 × 512   pixels . B. Allan variance of the given texture, using a horizontal window analysis (note that the result reveals an anisotropic nature with a privileged direction tilted 45°). C. Binary noise. D. Allan variance of the binary noise calculated in the same horizontal scheme (anisotropy is not detectable, as expected, indicating that binary noise is isotropic and all directionalities are equivalent). Variance calculations have been obtained by applying Eq. (2) for N = 8   pixels .

Fig. 2
Fig. 2

Example of detection of a test texture (even zigzag) of 512 × 512   pixels . A. Texture embedded in a binary random noise background. B. Result of the application of the Allan variance to A with an horizontal orientation ( N = 8   pixels ) . C. Entropy map. D. DM after applying the Rényi entropy over a PWD of B tilted 45° and a window of N = 8   pixels .

Fig. 3
Fig. 3

Example of detection of a isotrigon texture (even triangle) by squaring of the differences of Allan variances. A. Target isotrigon texture embedded in a binary random noise background. B. Squaring of Allan variances ( AV 45 AV 0 ) 2 obtained from A ( N = 8   pixels ) . C. Entropy map. D. DM after applying the Rényi entropy, using a PWD tilted 90° and a window of N = 8   pixels .

Fig. 4
Fig. 4

Example of ROC for the odd cross texture versus random noise. The highest simultaneous quality is selected to determine the threshold values, i.e., R i values in Eq. (9). The experiment indicated by a black circle provides the best threshold values to discriminate this texture from binary noise, using the directionalities indicated in Table 2 for this texture.

Fig. 5
Fig. 5

Comparison of the results of this method for nine families of even isotrigon textures versus the most relevant QDA results given by Maddess and Nagai [4].

Fig. 6
Fig. 6

A. Image consisting of five isotrigon textures. B. Segmentation of region “0” (even oblong). C. Segmentation of region “1” (even triangle). D. Segmentation of region “2” (even zigzag). E. Segmentation of region “3” (even cross). F. Segmentation of region “4” (even box). Regions “2, 3, and 4” can be segmented straightforwardly by using the directionalities indicated in Table 2. Region “0” has the same entropy for the parameters given in Table 2 as region “4” but has been isolated by difference with this one. Region “1” has been obtained by eliminating the other four, as no suitable directionality was observed. G. DM after previous single results. H. DM by a complex wavelets method [33, 34]. Figure 6h courtesy of M. G. Forero.

Fig. 7
Fig. 7

A. Image consisting of five isotrigon textures. B. Segmentation of region “0” (even El). C. Segmentation of region “1” (even oblong). D. Segmentation of region “2” (even wye). E. Segmentation of region “3” (even tee). F. Segmentation of region “4” (even zigzag). All regions can be segmented straightforwardly by using the directionalities indicated in Table 2. Only regions “1” and “4” have been perfectly identified. Regions “0, 2, 3, and 4” have been recognized with some degree of uncertainty. G. DM after previous single results. H. DM by a complex wavelets method [33, 34]. Figure 7h courtesy of M. G. Forero.

Tables (3)

Tables Icon

Table 1 Possible Parameter Values in Eq. (2)

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Table 2 Discrimination Performance a

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Table 3 Percentage of Correct Decisions [PCD, Eq. (8)] of Isotrigon Even Texture Pairs Corresponding to Three Different Segmentation Methods

Equations (9)

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AV [ n ] = 1 2 ( N 1 ) k = N 2 N 2 1 ( x [ n + k + 1 ] x [ n + k ] ) 2 .
AV γ , λ [ p , q ] = 1 2 ( N 1 ) 2 k = N 2 N 2 1 l = N 2 N 2 1 ( x [ p + k + γ , q + l + λ ] x [ p + k , q + l ] ) 2 .
W [ n , k ] = 2 m = N 2 N 2 1 z [ n + m ] z * [ n m ] e 2 i ( 2 π m N ) k .
R α = 1 1 α log 2 ( n k P α [ n , k ] ) ,
R 3 [ n ] = 1 2 log 2 ( k W ̆ 3 [ n , k ] ) .
W ̆ [ n , k ] = W [ n , k ] W * [ n , k ] k ( W [ n , k ] W * [ n , k ] ) .
D i [ n ] = R i R T [ n ] N , i { 1 , 2 } .
DM [ n ] = arg i max ( D i [ n ] ) , i { 1 , 2 } .
PCD = 100 × number of accurate pixels total number of pixels .

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