Abstract

To date a small palette of isotrigon textures have been available to study how the brain uses higher-order spatial correlation information. We introduce several hundred new isotrigon textures. Special modulation properties are illustrated that can be used to extract neural responses to higher-order spatial correlations. We also ask how many textures make an adequate training set and how representative individual examples are of their texture class. Human discrimination of 90 of these patterns was quantified. Modeling those responses shows that humanlike performance can be obtained providing a fourth-order classifier is used, although more than one mechanism is required.

© 2007 Optical Society of America

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  1. K. P. Purpura, J. D. Victor, and E. Katz, "Striate cortex extracts higher-order spatial correlations from visual textures," Phys. Rev. A 91, 8482-8486 (1994).
  2. M. O. Franz and B. Schölkopf, "Implicit Weiner series Part 1: cross-correlation vs. regression in reproducing kernel Hilbert spaces," Technical Report TR-114 (Max-Planck-Institut für biologische Kybernetik, 2003).
  3. M. Franz and B. Schölkopf, "Implicit Weiner series for higher-order image analysis," in Advances in Neural Information Processing Systems, 17: Proceedings of the 2004 Conference, L.K.Saul, Y.Weiss, and L.Bottou, eds., (MIT Press, 2005), pp. 1-8.
  4. C. Zetzsche, E. Barth, and B. Wegmann, "The importance of intrinsically two-dimensional image features in biological vision and picture coding," in Digital Images and Human Vision, A.Watson, ed. (MIT Press, 1993), pp. 109-138.
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    [CrossRef]
  6. B. A. Olshausen and D. J. Field, "Emergence of simple-cell receptive field properties by learning a sparse code for natural images," Nature 381, 607-609 (1996).
    [CrossRef] [PubMed]
  7. C. Zetzsche and U. Nuding, "Nonlinear and higher-order approaches to the encoding of natural scenes," Network 16, 191-221 (2005).
    [CrossRef]
  8. C. Zetzsche and F. Rohrbein, "Nonlinear and extra-classical receptive field properties and the statistics of natural scenes," Network 12, 331-350 (2001).
    [PubMed]
  9. G. Krieger and C. Zetzsche, "Nonlinear image operators for the evaluation of local intrinsic dimensionality," IEEE Trans. Image Process. 5, 1026-1042 (1996).
    [CrossRef] [PubMed]
  10. G. Krieger, I. Rentschler, G. Hauske, K. Schill, and C. Zetzsche, "Object and scene analysis by saccadic eye-movements: an investigation with higher-order statistics," Spatial Vis. 13, 201-214 (2000).
    [CrossRef]
  11. J. D. Victor and V. Zemon, "The human visual evoked potential: analysis of components due to elementary and complex aspects of form," Vision Res. 25, 1829-1842 (1985).
    [CrossRef] [PubMed]
  12. J. D. Victor and M. M. Conte, "Cortical interactions in texture processing: scale and dynamics," Visual Neurosci. 2, 297-313 (1989).
    [CrossRef]
  13. J. D. Victor and M. M. Conte, "Spatial organization of nonlinear interactions in form perception," Vision Res. 31, 1457-1488 (1991).
    [CrossRef] [PubMed]
  14. J. D. Victor, "Complex visual textures as a tool for studying the VEP," Vision Res. 25, 1811-1827 (1985).
    [CrossRef] [PubMed]
  15. J. D. Victor and M. M. Conte, "The role of high-order phase correlations in texture processing," Vision Res. 36, 1615-1631 (1996).
    [CrossRef] [PubMed]
  16. L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
    [CrossRef]
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    [CrossRef]
  18. L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, "Striate cortex in humans demonstrates the relationship between activation and variations in visual form," Exp. Brain Res. 130, 221-226 (2000).
    [CrossRef] [PubMed]
  19. T. Maddess and Y. Nagai, "Discriminating isotrigon textures," Vision Res. 41, 3837-3860 (2001).
    [CrossRef] [PubMed]
  20. J. D. Victor and M. M. Conte, "Visual working memory for image statistics," Vision Res. 44, 541-556 (2004).
    [CrossRef]
  21. C. W. Tyler, "Theory of texture discrimination based on higher-order perturbations in individual texture samples," Vision Res. 44, 2179-2186 (2004).
    [CrossRef] [PubMed]
  22. C. W. Tyler, "Beyond fourth-order texture discrimination: generation of extreme-order and statistically-balanced textures," Vision Res. 44, 2187-2199 (2004).
    [CrossRef] [PubMed]
  23. 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]
  24. A. Watson and A. Ahumada, "A standard model for foveal detection of spatial contrast," J. Vision 5, 717-740 (2005).
    [CrossRef]
  25. J. D. Victor, "Images, statistics and textures: implications of triple correlation uniqueness for texture statistics and the Julesz conjecture: comment," J. Opt. Soc. Am. A 11, 1680-1684 (1994).
    [CrossRef]
  26. C. Chubb and J. I. Yellott, "Every discrete, finite image is uniquely determined by its dipole histogram," Vision Res. 40, 485-492 (2000).
    [CrossRef] [PubMed]
  27. C. Chubb and J. I. Yellott, "Dipole statistics of discrete finite images: two visually motivated representation theorems," J. Opt. Soc. Am. A 19, 825-832 (2002).
    [CrossRef]
  28. 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]
  29. D. L. Ruderman, "Origins of scaling in natural images," Vision Res. 37, 3385-3398 (1997).
    [CrossRef]
  30. D. Ruderman and W. Bialek, "Statistics of natural images: scaling in the woods," Phys. Rev. Lett. 73, 814-817 (1994).
    [CrossRef] [PubMed]
  31. E. N. Gilbert, "Random colorings of a lattice on squares in the plane," SIAM J. Algebraic Discrete Methods 1, 152-159 (1980).
    [CrossRef]
  32. R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, 3rd ed. (Prentice Hall, 1992).
  33. D. Peterzell and D. Teller, "Individual differences in contrast sensitivity functions: the lowest spatial frequency channels," Vision Res. 36, 3077-3085 (1995).
    [CrossRef]
  34. J. D. Victor, C. Chubb, and M. M. Conte, "Interaction of luminance and higher-order statistics in texture discrimination," Vision Res. 45, 311-328 (2005).
    [CrossRef]
  35. J. D. Victor and M. M. Conte, "Motion mechanisms have only limited access to form information," Vision Res. 30, 289-301 (1990).
    [CrossRef] [PubMed]

2005 (3)

C. Zetzsche and U. Nuding, "Nonlinear and higher-order approaches to the encoding of natural scenes," Network 16, 191-221 (2005).
[CrossRef]

A. Watson and A. Ahumada, "A standard model for foveal detection of spatial contrast," J. Vision 5, 717-740 (2005).
[CrossRef]

J. D. Victor, C. Chubb, and M. M. Conte, "Interaction of luminance and higher-order statistics in texture discrimination," Vision Res. 45, 311-328 (2005).
[CrossRef]

2004 (4)

J. D. Victor and M. M. Conte, "Visual working memory for image statistics," Vision Res. 44, 541-556 (2004).
[CrossRef]

C. W. Tyler, "Theory of texture discrimination based on higher-order perturbations in individual texture samples," Vision Res. 44, 2179-2186 (2004).
[CrossRef] [PubMed]

C. W. Tyler, "Beyond fourth-order texture discrimination: generation of extreme-order and statistically-balanced textures," Vision Res. 44, 2187-2199 (2004).
[CrossRef] [PubMed]

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]

2002 (1)

2001 (2)

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

C. Zetzsche and F. Rohrbein, "Nonlinear and extra-classical receptive field properties and the statistics of natural scenes," Network 12, 331-350 (2001).
[PubMed]

2000 (3)

G. Krieger, I. Rentschler, G. Hauske, K. Schill, and C. Zetzsche, "Object and scene analysis by saccadic eye-movements: an investigation with higher-order statistics," Spatial Vis. 13, 201-214 (2000).
[CrossRef]

C. Chubb and J. I. Yellott, "Every discrete, finite image is uniquely determined by its dipole histogram," Vision Res. 40, 485-492 (2000).
[CrossRef] [PubMed]

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, "Striate cortex in humans demonstrates the relationship between activation and variations in visual form," Exp. Brain Res. 130, 221-226 (2000).
[CrossRef] [PubMed]

1999 (1)

1998 (2)

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[CrossRef]

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, J. M. Maisog, E. M. Daly, D. J. Mangot, R. E. Desmond, L. M. Optican, M. B. Schapiro, and J. W. VanMeter, "Cortical regions involved in visual texture perception: a fMRI study," Brain Res. Cognit. Brain Res. 7, 111-118 (1998).
[CrossRef]

1997 (1)

D. L. Ruderman, "Origins of scaling in natural images," Vision Res. 37, 3385-3398 (1997).
[CrossRef]

1996 (3)

J. D. Victor and M. M. Conte, "The role of high-order phase correlations in texture processing," Vision Res. 36, 1615-1631 (1996).
[CrossRef] [PubMed]

B. A. Olshausen and D. J. Field, "Emergence of simple-cell receptive field properties by learning a sparse code for natural images," Nature 381, 607-609 (1996).
[CrossRef] [PubMed]

G. Krieger and C. Zetzsche, "Nonlinear image operators for the evaluation of local intrinsic dimensionality," IEEE Trans. Image Process. 5, 1026-1042 (1996).
[CrossRef] [PubMed]

1995 (1)

D. Peterzell and D. Teller, "Individual differences in contrast sensitivity functions: the lowest spatial frequency channels," Vision Res. 36, 3077-3085 (1995).
[CrossRef]

1994 (3)

D. Ruderman and W. Bialek, "Statistics of natural images: scaling in the woods," Phys. Rev. Lett. 73, 814-817 (1994).
[CrossRef] [PubMed]

J. D. Victor, "Images, statistics and textures: implications of triple correlation uniqueness for texture statistics and the Julesz conjecture: comment," J. Opt. Soc. Am. A 11, 1680-1684 (1994).
[CrossRef]

K. P. Purpura, J. D. Victor, and E. Katz, "Striate cortex extracts higher-order spatial correlations from visual textures," Phys. Rev. A 91, 8482-8486 (1994).

1991 (1)

J. D. Victor and M. M. Conte, "Spatial organization of nonlinear interactions in form perception," Vision Res. 31, 1457-1488 (1991).
[CrossRef] [PubMed]

1990 (1)

J. D. Victor and M. M. Conte, "Motion mechanisms have only limited access to form information," Vision Res. 30, 289-301 (1990).
[CrossRef] [PubMed]

1989 (1)

J. D. Victor and M. M. Conte, "Cortical interactions in texture processing: scale and dynamics," Visual Neurosci. 2, 297-313 (1989).
[CrossRef]

1985 (2)

J. D. Victor and V. Zemon, "The human visual evoked potential: analysis of components due to elementary and complex aspects of form," Vision Res. 25, 1829-1842 (1985).
[CrossRef] [PubMed]

J. D. Victor, "Complex visual textures as a tool for studying the VEP," Vision Res. 25, 1811-1827 (1985).
[CrossRef] [PubMed]

1980 (1)

E. N. Gilbert, "Random colorings of a lattice on squares in the plane," SIAM J. Algebraic Discrete Methods 1, 152-159 (1980).
[CrossRef]

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]

Ahumada, A.

A. Watson and A. Ahumada, "A standard model for foveal detection of spatial contrast," J. Vision 5, 717-740 (2005).
[CrossRef]

Alexander, G. E.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[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]

Azari, N. P.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[CrossRef]

Barth, E.

C. Zetzsche, E. Barth, and B. Wegmann, "The importance of intrinsically two-dimensional image features in biological vision and picture coding," in Digital Images and Human Vision, A.Watson, ed. (MIT Press, 1993), pp. 109-138.

Beason-Held, L. L.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, "Striate cortex in humans demonstrates the relationship between activation and variations in visual form," Exp. Brain Res. 130, 221-226 (2000).
[CrossRef] [PubMed]

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[CrossRef]

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, J. M. Maisog, E. M. Daly, D. J. Mangot, R. E. Desmond, L. M. Optican, M. B. Schapiro, and J. W. VanMeter, "Cortical regions involved in visual texture perception: a fMRI study," Brain Res. Cognit. Brain Res. 7, 111-118 (1998).
[CrossRef]

Bialek, W.

D. Ruderman and W. Bialek, "Statistics of natural images: scaling in the woods," Phys. Rev. Lett. 73, 814-817 (1994).
[CrossRef] [PubMed]

Chubb, C.

J. D. Victor, C. Chubb, and M. M. Conte, "Interaction of luminance and higher-order statistics in texture discrimination," Vision Res. 45, 311-328 (2005).
[CrossRef]

C. Chubb and J. I. Yellott, "Dipole statistics of discrete finite images: two visually motivated representation theorems," J. Opt. Soc. Am. A 19, 825-832 (2002).
[CrossRef]

C. Chubb and J. I. Yellott, "Every discrete, finite image is uniquely determined by its dipole histogram," Vision Res. 40, 485-492 (2000).
[CrossRef] [PubMed]

Conte, M. M.

J. D. Victor, C. Chubb, and M. M. Conte, "Interaction of luminance and higher-order statistics in texture discrimination," Vision Res. 45, 311-328 (2005).
[CrossRef]

J. D. Victor and M. M. Conte, "Visual working memory for image statistics," Vision Res. 44, 541-556 (2004).
[CrossRef]

J. D. Victor and M. M. Conte, "The role of high-order phase correlations in texture processing," Vision Res. 36, 1615-1631 (1996).
[CrossRef] [PubMed]

J. D. Victor and M. M. Conte, "Spatial organization of nonlinear interactions in form perception," Vision Res. 31, 1457-1488 (1991).
[CrossRef] [PubMed]

J. D. Victor and M. M. Conte, "Motion mechanisms have only limited access to form information," Vision Res. 30, 289-301 (1990).
[CrossRef] [PubMed]

J. D. Victor and M. M. Conte, "Cortical interactions in texture processing: scale and dynamics," Visual Neurosci. 2, 297-313 (1989).
[CrossRef]

Daly, E. M.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, "Striate cortex in humans demonstrates the relationship between activation and variations in visual form," Exp. Brain Res. 130, 221-226 (2000).
[CrossRef] [PubMed]

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, J. M. Maisog, E. M. Daly, D. J. Mangot, R. E. Desmond, L. M. Optican, M. B. Schapiro, and J. W. VanMeter, "Cortical regions involved in visual texture perception: a fMRI study," Brain Res. Cognit. Brain Res. 7, 111-118 (1998).
[CrossRef]

Desmond, R. E.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, "Striate cortex in humans demonstrates the relationship between activation and variations in visual form," Exp. Brain Res. 130, 221-226 (2000).
[CrossRef] [PubMed]

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, J. M. Maisog, E. M. Daly, D. J. Mangot, R. E. Desmond, L. M. Optican, M. B. Schapiro, and J. W. VanMeter, "Cortical regions involved in visual texture perception: a fMRI study," Brain Res. Cognit. Brain Res. 7, 111-118 (1998).
[CrossRef]

Field, D. J.

B. A. Olshausen and D. J. Field, "Emergence of simple-cell receptive field properties by learning a sparse code for natural images," Nature 381, 607-609 (1996).
[CrossRef] [PubMed]

Franz, M.

M. Franz and B. Schölkopf, "Implicit Weiner series for higher-order image analysis," in Advances in Neural Information Processing Systems, 17: Proceedings of the 2004 Conference, L.K.Saul, Y.Weiss, and L.Bottou, eds., (MIT Press, 2005), pp. 1-8.

Franz, M. O.

M. O. Franz and B. Schölkopf, "Implicit Weiner series Part 1: cross-correlation vs. regression in reproducing kernel Hilbert spaces," Technical Report TR-114 (Max-Planck-Institut für biologische Kybernetik, 2003).

Gilbert, E. N.

E. N. Gilbert, "Random colorings of a lattice on squares in the plane," SIAM J. Algebraic Discrete Methods 1, 152-159 (1980).
[CrossRef]

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]

Grady, C. L.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[CrossRef]

Hauske, G.

G. Krieger, I. Rentschler, G. Hauske, K. Schill, and C. Zetzsche, "Object and scene analysis by saccadic eye-movements: an investigation with higher-order statistics," Spatial Vis. 13, 201-214 (2000).
[CrossRef]

Horwitz, B.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[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, 3rd ed. (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," Phys. Rev. A 91, 8482-8486 (1994).

Krasuski, J. S.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, "Striate cortex in humans demonstrates the relationship between activation and variations in visual form," Exp. Brain Res. 130, 221-226 (2000).
[CrossRef] [PubMed]

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, J. M. Maisog, E. M. Daly, D. J. Mangot, R. E. Desmond, L. M. Optican, M. B. Schapiro, and J. W. VanMeter, "Cortical regions involved in visual texture perception: a fMRI study," Brain Res. Cognit. Brain Res. 7, 111-118 (1998).
[CrossRef]

Krieger, G.

G. Krieger, I. Rentschler, G. Hauske, K. Schill, and C. Zetzsche, "Object and scene analysis by saccadic eye-movements: an investigation with higher-order statistics," Spatial Vis. 13, 201-214 (2000).
[CrossRef]

C. Zetzsche, G. Krieger, and B. Wegmann, "The atoms of vision: Cartesian or polar?" J. Opt. Soc. Am. A 16, 1554-1565 (1999).
[CrossRef]

G. Krieger and C. Zetzsche, "Nonlinear image operators for the evaluation of local intrinsic dimensionality," IEEE Trans. Image Process. 5, 1026-1042 (1996).
[CrossRef] [PubMed]

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]

Maisog, J. M.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, J. M. Maisog, E. M. Daly, D. J. Mangot, R. E. Desmond, L. M. Optican, M. B. Schapiro, and J. W. VanMeter, "Cortical regions involved in visual texture perception: a fMRI study," Brain Res. Cognit. Brain Res. 7, 111-118 (1998).
[CrossRef]

Mangot, D. J.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, "Striate cortex in humans demonstrates the relationship between activation and variations in visual form," Exp. Brain Res. 130, 221-226 (2000).
[CrossRef] [PubMed]

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, J. M. Maisog, E. M. Daly, D. J. Mangot, R. E. Desmond, L. M. Optican, M. B. Schapiro, and J. W. VanMeter, "Cortical regions involved in visual texture perception: a fMRI study," Brain Res. Cognit. Brain Res. 7, 111-118 (1998).
[CrossRef]

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[CrossRef]

Mentis, M. J.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[CrossRef]

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]

Nuding, U.

C. Zetzsche and U. Nuding, "Nonlinear and higher-order approaches to the encoding of natural scenes," Network 16, 191-221 (2005).
[CrossRef]

Olshausen, B. A.

B. A. Olshausen and D. J. Field, "Emergence of simple-cell receptive field properties by learning a sparse code for natural images," Nature 381, 607-609 (1996).
[CrossRef] [PubMed]

Optican, L. M.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, "Striate cortex in humans demonstrates the relationship between activation and variations in visual form," Exp. Brain Res. 130, 221-226 (2000).
[CrossRef] [PubMed]

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[CrossRef]

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, J. M. Maisog, E. M. Daly, D. J. Mangot, R. E. Desmond, L. M. Optican, M. B. Schapiro, and J. W. VanMeter, "Cortical regions involved in visual texture perception: a fMRI study," Brain Res. Cognit. Brain Res. 7, 111-118 (1998).
[CrossRef]

Peterzell, D.

D. Peterzell and D. Teller, "Individual differences in contrast sensitivity functions: the lowest spatial frequency channels," Vision Res. 36, 3077-3085 (1995).
[CrossRef]

Purpura, K. P.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, "Striate cortex in humans demonstrates the relationship between activation and variations in visual form," Exp. Brain Res. 130, 221-226 (2000).
[CrossRef] [PubMed]

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, J. M. Maisog, E. M. Daly, D. J. Mangot, R. E. Desmond, L. M. Optican, M. B. Schapiro, and J. W. VanMeter, "Cortical regions involved in visual texture perception: a fMRI study," Brain Res. Cognit. Brain Res. 7, 111-118 (1998).
[CrossRef]

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[CrossRef]

K. P. Purpura, J. D. Victor, and E. Katz, "Striate cortex extracts higher-order spatial correlations from visual textures," Phys. Rev. A 91, 8482-8486 (1994).

Rapoport, S. I.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, "Striate cortex in humans demonstrates the relationship between activation and variations in visual form," Exp. Brain Res. 130, 221-226 (2000).
[CrossRef] [PubMed]

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[CrossRef]

Rentschler, I.

G. Krieger, I. Rentschler, G. Hauske, K. Schill, and C. Zetzsche, "Object and scene analysis by saccadic eye-movements: an investigation with higher-order statistics," Spatial Vis. 13, 201-214 (2000).
[CrossRef]

Rohrbein, F.

C. Zetzsche and F. Rohrbein, "Nonlinear and extra-classical receptive field properties and the statistics of natural scenes," Network 12, 331-350 (2001).
[PubMed]

Ruderman, D.

D. Ruderman and W. Bialek, "Statistics of natural images: scaling in the woods," Phys. Rev. Lett. 73, 814-817 (1994).
[CrossRef] [PubMed]

Ruderman, D. L.

D. L. Ruderman, "Origins of scaling in natural images," Vision Res. 37, 3385-3398 (1997).
[CrossRef]

Schapiro, M. B.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[CrossRef]

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, J. M. Maisog, E. M. Daly, D. J. Mangot, R. E. Desmond, L. M. Optican, M. B. Schapiro, and J. W. VanMeter, "Cortical regions involved in visual texture perception: a fMRI study," Brain Res. Cognit. Brain Res. 7, 111-118 (1998).
[CrossRef]

Schill, K.

G. Krieger, I. Rentschler, G. Hauske, K. Schill, and C. Zetzsche, "Object and scene analysis by saccadic eye-movements: an investigation with higher-order statistics," Spatial Vis. 13, 201-214 (2000).
[CrossRef]

Schölkopf, B.

M. O. Franz and B. Schölkopf, "Implicit Weiner series Part 1: cross-correlation vs. regression in reproducing kernel Hilbert spaces," Technical Report TR-114 (Max-Planck-Institut für biologische Kybernetik, 2003).

M. Franz and B. Schölkopf, "Implicit Weiner series for higher-order image analysis," in Advances in Neural Information Processing Systems, 17: Proceedings of the 2004 Conference, L.K.Saul, Y.Weiss, and L.Bottou, eds., (MIT Press, 2005), pp. 1-8.

Teller, D.

D. Peterzell and D. Teller, "Individual differences in contrast sensitivity functions: the lowest spatial frequency channels," Vision Res. 36, 3077-3085 (1995).
[CrossRef]

Tyler, C. W.

C. W. Tyler, "Theory of texture discrimination based on higher-order perturbations in individual texture samples," Vision Res. 44, 2179-2186 (2004).
[CrossRef] [PubMed]

C. W. Tyler, "Beyond fourth-order texture discrimination: generation of extreme-order and statistically-balanced textures," Vision Res. 44, 2187-2199 (2004).
[CrossRef] [PubMed]

Van Meter, J. W.

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[CrossRef]

VanMeter, J. W.

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, "Striate cortex in humans demonstrates the relationship between activation and variations in visual form," Exp. Brain Res. 130, 221-226 (2000).
[CrossRef] [PubMed]

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, J. M. Maisog, E. M. Daly, D. J. Mangot, R. E. Desmond, L. M. Optican, M. B. Schapiro, and J. W. VanMeter, "Cortical regions involved in visual texture perception: a fMRI study," Brain Res. Cognit. Brain Res. 7, 111-118 (1998).
[CrossRef]

Victor, J. D.

J. D. Victor, C. Chubb, and M. M. Conte, "Interaction of luminance and higher-order statistics in texture discrimination," Vision Res. 45, 311-328 (2005).
[CrossRef]

J. D. Victor and M. M. Conte, "Visual working memory for image statistics," Vision Res. 44, 541-556 (2004).
[CrossRef]

J. D. Victor and M. M. Conte, "The role of high-order phase correlations in texture processing," Vision Res. 36, 1615-1631 (1996).
[CrossRef] [PubMed]

K. P. Purpura, J. D. Victor, and E. Katz, "Striate cortex extracts higher-order spatial correlations from visual textures," Phys. Rev. A 91, 8482-8486 (1994).

J. D. Victor, "Images, statistics and textures: implications of triple correlation uniqueness for texture statistics and the Julesz conjecture: comment," J. Opt. Soc. Am. A 11, 1680-1684 (1994).
[CrossRef]

J. D. Victor and M. M. Conte, "Spatial organization of nonlinear interactions in form perception," Vision Res. 31, 1457-1488 (1991).
[CrossRef] [PubMed]

J. D. Victor and M. M. Conte, "Motion mechanisms have only limited access to form information," Vision Res. 30, 289-301 (1990).
[CrossRef] [PubMed]

J. D. Victor and M. M. Conte, "Cortical interactions in texture processing: scale and dynamics," Visual Neurosci. 2, 297-313 (1989).
[CrossRef]

J. D. Victor, "Complex visual textures as a tool for studying the VEP," Vision Res. 25, 1811-1827 (1985).
[CrossRef] [PubMed]

J. D. Victor and V. Zemon, "The human visual evoked potential: analysis of components due to elementary and complex aspects of form," Vision Res. 25, 1829-1842 (1985).
[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]

Watson, A.

A. Watson and A. Ahumada, "A standard model for foveal detection of spatial contrast," J. Vision 5, 717-740 (2005).
[CrossRef]

Wegmann, B.

C. Zetzsche, G. Krieger, and B. Wegmann, "The atoms of vision: Cartesian or polar?" J. Opt. Soc. Am. A 16, 1554-1565 (1999).
[CrossRef]

C. Zetzsche, E. Barth, and B. Wegmann, "The importance of intrinsically two-dimensional image features in biological vision and picture coding," in Digital Images and Human Vision, A.Watson, ed. (MIT Press, 1993), pp. 109-138.

Wichern, D. W.

R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, 3rd ed. (Prentice Hall, 1992).

Yellott, J. I.

C. Chubb and J. I. Yellott, "Dipole statistics of discrete finite images: two visually motivated representation theorems," J. Opt. Soc. Am. A 19, 825-832 (2002).
[CrossRef]

C. Chubb and J. I. Yellott, "Every discrete, finite image is uniquely determined by its dipole histogram," Vision Res. 40, 485-492 (2000).
[CrossRef] [PubMed]

Zemon, V.

J. D. Victor and V. Zemon, "The human visual evoked potential: analysis of components due to elementary and complex aspects of form," Vision Res. 25, 1829-1842 (1985).
[CrossRef] [PubMed]

Zetzsche, C.

C. Zetzsche and U. Nuding, "Nonlinear and higher-order approaches to the encoding of natural scenes," Network 16, 191-221 (2005).
[CrossRef]

C. Zetzsche and F. Rohrbein, "Nonlinear and extra-classical receptive field properties and the statistics of natural scenes," Network 12, 331-350 (2001).
[PubMed]

G. Krieger, I. Rentschler, G. Hauske, K. Schill, and C. Zetzsche, "Object and scene analysis by saccadic eye-movements: an investigation with higher-order statistics," Spatial Vis. 13, 201-214 (2000).
[CrossRef]

C. Zetzsche, G. Krieger, and B. Wegmann, "The atoms of vision: Cartesian or polar?" J. Opt. Soc. Am. A 16, 1554-1565 (1999).
[CrossRef]

G. Krieger and C. Zetzsche, "Nonlinear image operators for the evaluation of local intrinsic dimensionality," IEEE Trans. Image Process. 5, 1026-1042 (1996).
[CrossRef] [PubMed]

C. Zetzsche, E. Barth, and B. Wegmann, "The importance of intrinsically two-dimensional image features in biological vision and picture coding," in Digital Images and Human Vision, A.Watson, ed. (MIT Press, 1993), pp. 109-138.

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]

Brain Res. Cognit. Brain Res. (1)

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, J. M. Maisog, E. M. Daly, D. J. Mangot, R. E. Desmond, L. M. Optican, M. B. Schapiro, and J. W. VanMeter, "Cortical regions involved in visual texture perception: a fMRI study," Brain Res. Cognit. Brain Res. 7, 111-118 (1998).
[CrossRef]

Exp. Brain Res. (1)

L. L. Beason-Held, K. P. Purpura, J. S. Krasuski, R. E. Desmond, D. J. Mangot, E. M. Daly, L. M. Optican, S. I. Rapoport, and J. W. VanMeter, "Striate cortex in humans demonstrates the relationship between activation and variations in visual form," Exp. Brain Res. 130, 221-226 (2000).
[CrossRef] [PubMed]

IEEE Trans. Image Process. (1)

G. Krieger and C. Zetzsche, "Nonlinear image operators for the evaluation of local intrinsic dimensionality," IEEE Trans. Image Process. 5, 1026-1042 (1996).
[CrossRef] [PubMed]

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

J. Vision (1)

A. Watson and A. Ahumada, "A standard model for foveal detection of spatial contrast," J. Vision 5, 717-740 (2005).
[CrossRef]

Nature (1)

B. A. Olshausen and D. J. Field, "Emergence of simple-cell receptive field properties by learning a sparse code for natural images," Nature 381, 607-609 (1996).
[CrossRef] [PubMed]

Network (2)

C. Zetzsche and U. Nuding, "Nonlinear and higher-order approaches to the encoding of natural scenes," Network 16, 191-221 (2005).
[CrossRef]

C. Zetzsche and F. Rohrbein, "Nonlinear and extra-classical receptive field properties and the statistics of natural scenes," Network 12, 331-350 (2001).
[PubMed]

Phys. Rev. A (1)

K. P. Purpura, J. D. Victor, and E. Katz, "Striate cortex extracts higher-order spatial correlations from visual textures," Phys. Rev. A 91, 8482-8486 (1994).

Phys. Rev. Lett. (1)

D. Ruderman and W. Bialek, "Statistics of natural images: scaling in the woods," Phys. Rev. Lett. 73, 814-817 (1994).
[CrossRef] [PubMed]

SIAM J. Algebraic Discrete Methods (1)

E. N. Gilbert, "Random colorings of a lattice on squares in the plane," SIAM J. Algebraic Discrete Methods 1, 152-159 (1980).
[CrossRef]

Spatial Vis. (1)

G. Krieger, I. Rentschler, G. Hauske, K. Schill, and C. Zetzsche, "Object and scene analysis by saccadic eye-movements: an investigation with higher-order statistics," Spatial Vis. 13, 201-214 (2000).
[CrossRef]

Vision Res. (14)

J. D. Victor and V. Zemon, "The human visual evoked potential: analysis of components due to elementary and complex aspects of form," Vision Res. 25, 1829-1842 (1985).
[CrossRef] [PubMed]

J. D. Victor and M. M. Conte, "Spatial organization of nonlinear interactions in form perception," Vision Res. 31, 1457-1488 (1991).
[CrossRef] [PubMed]

J. D. Victor, "Complex visual textures as a tool for studying the VEP," Vision Res. 25, 1811-1827 (1985).
[CrossRef] [PubMed]

J. D. Victor and M. M. Conte, "The role of high-order phase correlations in texture processing," Vision Res. 36, 1615-1631 (1996).
[CrossRef] [PubMed]

D. L. Ruderman, "Origins of scaling in natural images," Vision Res. 37, 3385-3398 (1997).
[CrossRef]

D. Peterzell and D. Teller, "Individual differences in contrast sensitivity functions: the lowest spatial frequency channels," Vision Res. 36, 3077-3085 (1995).
[CrossRef]

J. D. Victor, C. Chubb, and M. M. Conte, "Interaction of luminance and higher-order statistics in texture discrimination," Vision Res. 45, 311-328 (2005).
[CrossRef]

J. D. Victor and M. M. Conte, "Motion mechanisms have only limited access to form information," Vision Res. 30, 289-301 (1990).
[CrossRef] [PubMed]

C. Chubb and J. I. Yellott, "Every discrete, finite image is uniquely determined by its dipole histogram," Vision Res. 40, 485-492 (2000).
[CrossRef] [PubMed]

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

J. D. Victor and M. M. Conte, "Visual working memory for image statistics," Vision Res. 44, 541-556 (2004).
[CrossRef]

C. W. Tyler, "Theory of texture discrimination based on higher-order perturbations in individual texture samples," Vision Res. 44, 2179-2186 (2004).
[CrossRef] [PubMed]

C. W. Tyler, "Beyond fourth-order texture discrimination: generation of extreme-order and statistically-balanced textures," Vision Res. 44, 2187-2199 (2004).
[CrossRef] [PubMed]

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]

Visual Neurosci. (2)

L. L. Beason-Held, K. P. Purpura, J. W. Van Meter, N. P. Azari, D. J. Mangot, L. M. Optican, M. J. Mentis, G. E. Alexander, C. L. Grady, B. Horwitz, S. I. Rapoport, and M. B. Schapiro, "PET reveals occipitotemporal pathway activation during elementary form perception in humans," Visual Neurosci. 15, 503-510 (1998).
[CrossRef]

J. D. Victor and M. M. Conte, "Cortical interactions in texture processing: scale and dynamics," Visual Neurosci. 2, 297-313 (1989).
[CrossRef]

Other (4)

M. O. Franz and B. Schölkopf, "Implicit Weiner series Part 1: cross-correlation vs. regression in reproducing kernel Hilbert spaces," Technical Report TR-114 (Max-Planck-Institut für biologische Kybernetik, 2003).

M. Franz and B. Schölkopf, "Implicit Weiner series for higher-order image analysis," in Advances in Neural Information Processing Systems, 17: Proceedings of the 2004 Conference, L.K.Saul, Y.Weiss, and L.Bottou, eds., (MIT Press, 2005), pp. 1-8.

C. Zetzsche, E. Barth, and B. Wegmann, "The importance of intrinsically two-dimensional image features in biological vision and picture coding," in Digital Images and Human Vision, A.Watson, ed. (MIT Press, 1993), pp. 109-138.

R. A. Johnson and D. W. Wichern, Applied Multivariate Statistical Analysis, 3rd ed. (Prentice Hall, 1992).

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Figures (12)

Fig. 1
Fig. 1

Examples of textures generated with the Box glider and the four groups of six rules for combining the inputs during the recursion process. The 24 rules are given in a succinct modulo 3 form in Table 1. In each case the patterns were generated from the same matrix of evenly distributed binary noise. Although some of the patterns look similar, their minitexture content is quite disparate (Table 4, Appendix A).

Fig. 2
Fig. 2

M rule Box textures generated with ternary seed noise. Table 4, Appendix A, shows that only the M 0 , group 1 patterns are not changed from the case of using a binary seed. As shown in Table 4, the I textures are the same for both types of initial noise, each having the correct number and characteristic set of 243 minitextures.

Fig. 3
Fig. 3

Textures generated from the group 1 M rules by using eight other gilders. The Corners glider is given in Eq. (3), and the others can be found in Ref. [19].

Fig. 4
Fig. 4

Minitexture spectra example. All the figures are for the Box M 1 textures of the type shown in A, which is 33 × 33 pixels in size. All texture examples were generated from ternary noise. B, Number of minitextures (mtx) found in each of 150 examples, each 33 × 33 pixels; the mean ± SD is 210.5 ± 16.81 . Note that the maximum number is 243 minitextures. C, Mean ± SD cumulative number of minitextures found; all possible minitextures are observed by an average of four examples (mean, solid curve; ± 1 SD, dotted curves). To compute the SD the 150 examples were divided into 10 sets of 15, and the cumulative function was computed 10 times. The horizontal dashed line indicates 243, the maximum number of observable minitextures in any Box texture. D, Raw minitexture spectrum, showing all 3 9 = 19,683 possible bins, 243 of which are filled. The ordinate shows the mean minitexture count. E, 243 nonzero bins of D concatenated (black curve) displayed with ± 1 SD. Overall each example is highly representative of the parent texture type from which it is drawn. Minitexture spectra for all glider types were similarly flat. Nonisotrigon patterns can have very uneven spectra.

Fig. 5
Fig. 5

Illustration of the special matrix modulation properties of textures created from M rules. For all four rule groups there is an increment matrix, A, which is a regular tessellation of a unit cell (Table 2), the unit cell for group 0 rules being shown at top right. Left column, titled Original, three textures created from one (ternary) seed noise matrix by applying rules M 0 , M 1 , M 2 , and the Box glider. Middle column, titled mod 3 ( M ± A ) , textures obtained by incrementing (or decrementing) by A as shown on the ordinate labels. Right column, (with the unit cell at the top) shows that there is no difference between the ordinals and the increment derived textures. It is worth noting that, in modulating, the patterns commonly undergo local orientation changes.

Fig. 6
Fig. 6

Convergence functions exploring how many examples are necessary to obtain mean 2CFs and 3CFs that do not differ from zero for textures and noise. For each texture type 200 examples were created, and these were blocked into 8 sets of 25 examples. The 2CF and 3CF of each were computed, and then the mean and SD for each of the 8 sets of 25. The ratio of the resulting functions was computed pointwise, yielding a t value for each coefficient in the original CF. This was repeated for the first 3 , 4 , 5 , 6 25 textures in each set of 25. The maximum t statistic was then found in each case. The process was repeated for the ternary seed noise matrices used to generate each texture. A, D, Plots of the maximum t value for binary (black and white) Box textures (squares) for the 2CFs (A) and 3CFs (D), and for the random noise (circles). The textures actually proceeded more rapidly to 0 than the random textures. B, E, Similar plots for Group 0, M 1 Box textures, perhaps the most similar to the binary Box textures. These (squares) converged at a rate that was not appreciably different than for the random ternary noise (circles). The gray regions are the envelopes enclosing the convergence functions for M 1 textures for the all gliders shown in Fig. 3. C, F, Convergence function for 2V3L textures shown in Fig. 7. Error bars are SD.

Fig. 7
Fig. 7

Examples of 2V3L isotrigon textures. We examined textures created from the 4 rules (Table 5) and 21 gliders defined on a 3 × 3 pixel domain. The four gliders shown in the top row gave reasonably structured looking patterns. Note that these representations of the gliders are greatly magnified, as their block pixels are in fact the size of the pixels in the texture examples; gray pixels denote inputs ( a , b ) , and the white pixels denote the location where the output, f, should be inserted during recursive generation of the textures. Interestingly, many 3VL3 rules generate a texture very similar to that of the three-level triangles of glider 3, rule 1, here but none of those 3V3L textures is isotrigon,[23] (Fig. 9).

Fig. 8
Fig. 8

Examples of three-input, five-level isotrigon textures (3V5L) showing a range of behaviors. B, D were made with the Box glider, C the Oblong glider, and D with Zigzag.

Fig. 9
Fig. 9

Human discrimination of 90 isotrigon patterns from random ternary noise patterns. The test contained 16 trials for each of 3 texture sizes: 8 × 8 , 16 × 16 , and 32 × 32 pixels. All patterns were presented at the center of a display surrounded by random ternary noise. Test patterns were made by using three of the four possible rule groups (Table 1). The abscissa marks the gliders used: Bx, Box; Co, Corners; Cr, Cross; Ob, Oblong; Zi, Zigzag. The gliders repeat for each of the six rules per group, from left to right: M 0 , M 1 , M 2 , I 0 , I 1 , I 2 ; vertical dotted lines demark the Box textures. The dotted horizontal line in each panel marks chance performance ( p = 0.5 ) . Representative error bars (SE) for one each of the stimulus sizes are presented in each. Note that performance for 8 × 8 textures is similar to 16 × 16 and often better than for 32 × 32 , apparently due to a practice effect.

Fig. 10
Fig. 10

Accuracy functions from discriminant models based upon the variance in the responses of four receptive field types (see text) in response to textures versus ternary noise, i.e., the same comparison as in Fig. 9. Row A, Group 0 textures, B, Group 1; C, for Group 3; otherwise the same model form was used for all textures. The four receptive field types used in the model are shown in the inset at top right. The receptive field sizes can be gaged from the fact that the pixels in the two oriented versions are the size of the pixels in the textures. Two types of variance, the standard variance (SV) and the Allan variance (AV) were used. The dotted (AV) and dashed-dot (SV) lines hovering below 70% correct are the outputs of models where the best linear discriminant classifier operated on the variance measures. Note that none of these second-order models is able to discriminate the isotrigon textures efficiently. Thus, none of these second-order models performed like humans. In models where the best quadratic classifier (based on weighted products of the variances, hence fourth order) was used, performance for M textures was reminiscent of that of the human subjects, being high for Box and Oblong, but low for Cross. For the I textures, accuracy less clearly reflected that of humans.

Fig. 11
Fig. 11

Example receiver operator characteristic (ROC) plot. The plot was produced with the Box M 2 point in Fig. 10 for SV. The plot shows the mean ROC plot ± SE from bootstrap estimates ( N = 50 ) . The solid dot is the mean accuracy derived from 50 bootstrap estimates of the accuracy. This process was repeated to derive each point in Figs. 10, 12.

Fig. 12
Fig. 12

Model outputs where the best single quadratic classifier model was obtained for one texture versus random comparison and was then applied to all other texture versus random comparisons. Thus the models assume one fixed mechanism for all textures rather than one mechanism with adjustable decision weights as in Fig. 10. In all cases the training texture is marked with a dot. The SV (solid lines) and AV (dashed) were applied to the RF outputs as the inputs to the quadratic classifier. Linear classifiers were also computed in all cases, but all those accuracies were below 0.65 and so are not shown. The receptive fields for each model are illustrated in the small inset panel in the top right of each I rule accuracy plot. Notice that scale is slightly smaller for the receptive fields of D, the largest of which are 4 × 4 pixels. A and B compare accuracies for the same four RFs (see text) applied to group 1 and 2 textures, respectively. C, results for the models of Fig. 10; the low accuracies for Cross textures were actually observed for some subjects, which contributed to the low p values for those textures in Fig. 9. D, same as C, but data from two larger receptive fields were added, smoothing out the large variations. For E and F the training was done on I textures (dot), which improved accuracy only for that I texture and occasionally (F) gave poor performance for the M textures. Overall a single, simple, texture discrimination mechanism looks improbable. Figure 10 indicates that even a single mechanism that is reprogrammable with new weights in the classifier stage is unlikely to be sufficiently complex to emulate human performance on all textures.

Tables (6)

Tables Icon

Table 1 Modulo 3 Discrete Equation Form of Rules for Generating Isotrigon Textures for the Three-Input One-Output Case for Three Gray Levels a

Tables Icon

Table 2 Increment Matrix Unit Cells

Tables Icon

Table 3 Group 0 Discrete Equation [Eq. (5)] Coefficients

Tables Icon

Table 4 Number of Minitextures Shared between the Texture Types Shown in Fig. 1

Tables Icon

Table 5 Examples of Substitution Rules (Fig. 7)

Tables Icon

Table 6 Equivalent Substitution Rules (Fig. 7)

Equations (11)

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C 2 , f ( h , ν ) = 1 N 1 m 1 n I ( x , y ) I ( x + h , y + ν ) Δ x Δ y .
C 3 , f ( h 1 , ν 1 , h 2 , ν 2 ) = 1 N I ( x , y ) I ( x + h 1 , y + ν 1 ) I ( x + h 2 , y + ν 2 ) Δ x Δ y .
Corners = [ a n b n n n c n f ] .
f ( a , b , c ) = α = 0 n β = 0 n γ = 0 n χ α β γ a α b β c γ ,
f ( a , b , c ) = x 000 + x 100 a + x 010 b + x 001 c + x 110 a b + x 011 b c + x 101 a c + x 111 a b c ,
A B C D
f ( a , b ) = α = 0 2 β = 0 2 χ α β γ a α b β .
Group 0 = [ 0 0 0 2 ]
Group 1 = [ 0 0 0 0 1 2 0 2 1 ]
Group 2 = [ 0 0 0 1 0 2 ]
Group 3 = [ 0 0 0 0 1 2 ]

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