Abstract

We suggest that intrinsic two-dimensional (i2D) features, computationally defined as the outputs of nonlinear operators that model the activity of end-stopped neurons, play a role in preattentive texture discrimination. We first show that for discriminable textures with identical power spectra the predictions of traditional models depend on the type of nonlinearity and fail for energy measures. We then argue that the concept of intrinsic dimensionality, and the existence of end-stopped neurons, can help us to understand the role of the nonlinearities. Furthermore, we show examples in which models without strong i2D selectivity fail to predict the correct ranking order of perceptual segregation. Our arguments regarding the importance of i2D features resemble the arguments of Julesz and co-workers regarding textons such as terminators and crossings. However, we provide a computational framework that identifies textons with the outputs of nonlinear operators that are selective to i2D features.

© 1998 Optical Society of America

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  1. B. Julesz, B. Kröse, “Features and spatial filters,” Nature 333, 302–303 (1988).
    [CrossRef] [PubMed]
  2. B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature 290, 91–97 (1981).
    [CrossRef] [PubMed]
  3. J. R. Bergen, E. H. Adelson, “Early vision and texture perception,” Nature 333, 363–364 (1988).
    [CrossRef] [PubMed]
  4. D. Wermser, C.-E. Liedtke, “Texture analysis using a model of the visual system,” in Proceedings of the Sixth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1982), pp. 1078–1080.
  5. J. M. Coggins, A. K. Jain, “A spatial filtering approach to texture analysis,” Pattern Recogn. Lett. 3, 195–203 (1985).
    [CrossRef]
  6. M. R. Turner, “Texture discrimination by Gabor functions,” Biol. Cybern. 55, 71–82 (1986).
    [PubMed]
  7. C. Zetzsche, W. Schönecker, “Orientation selective filters lead to entropy reduction in the processing of natural images,” Perception 16, 229 (1987).
  8. U. Kriegeskotten-Thiede, C. Zetzsche, “Local amplitude of filter outputs predicts the influence of micropattern spacing, orientation, and elongation on texture discrimination,” Perception 17, 398 (1988).
  9. I. Fogel, D. Sagi, “Gabor filters as texture discriminators,” Biol. Cybern. 61, 103–113 (1989).
    [CrossRef]
  10. A. Sutter, J. Beck, N. Graham, “Contrast and spatial variables in texture segregation: testing a simple spatial-frequency channels model,” Percept. Psychophys. 46, 312–332 (1989).
    [CrossRef] [PubMed]
  11. A. C. Bovik, M. Clark, W. S. Geisler, “Multi-channel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
    [CrossRef]
  12. J. Malik, P. Perona, “Preattentive texture discrimination with early vision mechanisms,” J. Opt. Soc. Am. A 7, 923–932 (1990).
    [CrossRef] [PubMed]
  13. M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
    [CrossRef] [PubMed]
  14. N. Graham, J. Beck, A. Sutter, “Nonlinear processes in spatial-frequency channel models of perceived texture segregation: effects of sign and amount of contrast,” Vision Res. 32, 719–743 (1992).
    [CrossRef] [PubMed]
  15. A. Gorea, T. V. Papathomas, “Double opponency as a generalized concept in texture segregation illustrated with stimuli defined by color, luminance, and orientation,” J. Opt. Soc. Am. A 10, 1450–1462 (1993).
    [CrossRef]
  16. T. M. Caelli, “Three processing characteristics of visual texture segmentation,” Spatial Vis. 1, 19–30 (1985).
    [CrossRef]
  17. T. M. Caelli, “An adaptive computational model for texture segmentation,” IEEE Trans. Semicond. Manuf. 18, 9–17 (1988).
  18. D. Carević, T. Caelli, “Application of partial modelling techniques for texture segmentation,” J. Opt. Soc. Am. A 14, 2924–2937 (1997).
    [CrossRef]
  19. H. C. Nothdurft, “Texton segregation by associated differences in global and local luminance distribution,” Proc. R. Soc. London Ser. B 239, 295–320 (1990).
    [CrossRef]
  20. B. Julesz, “Early vision and focal attention,” Review Mod. Phys. 63, 735–772 (1991).
    [CrossRef]
  21. J. R. Bergen, “Theories of visual texture perception,” in Vision and Visual Disfunction, D. Regan, ed. (Macmillan, New York, 1991), Vol. 10B, pp. 114–134.
  22. A. Dobbins, S. W. Zucker, M. S. Cynader, “Endstopped neurons in the visual cortex as a substrate for calculating curvature,” Nature 329, 438–441 (1987).
    [CrossRef] [PubMed]
  23. C. Zetzsche, “Statistical properties of the representation of natural images at different levels in the visual system,” Perception 17, 359 (1988).
  24. J. J. Koenderink, W. Richards, “Two-dimensional curvature operators,” J. Opt. Soc. Am. A 5, 1136–1141 (1988).
    [CrossRef]
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    [CrossRef] [PubMed]
  26. C. Zetzsche, E. Barth, “Fundamental limits of linear filters in the visual processing of two-dimensional signals,” Vision Res. 30, 1111–1117 (1990).
    [CrossRef] [PubMed]
  27. C. Zetzsche, E. Barth, “Image surface predicates and the neural encoding of two-dimensional signal variation,” in Human Vision and Electronic Imaging: Models, Methods, and Applications, B. Rogowitz, ed., Proc. SPIE1249, 160–177 (1990).
  28. F. Heitger, L. Rosenthaler, R. von der Heydt, E. Peterhans, O. Kübler, “Simulation of neural contour mechanisms: from simple to end-stopped cells,” Vision Res. 32, 63–981 (1992).
    [CrossRef]
  29. C. Zetzsche, E. Barth, “Detection of intrinsic signal dimensionality in images and optic flow fields,” Perception 20, 71 (1991).
  30. C. Zetzsche, E. Barth, 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, Cambridge, Mass., 1993), pp. 109–138.
  31. E. Barth, T. Caelli, C. Zetzsche, “Image encoding, labelling and reconstruction from differential geometry,” CVGIP: Graph. Models Image Process. 55, 428–446 (1993).
  32. G. Krieger, C. Zetzsche, E. Barth, “2D-detectors in biological vision: Volterra–Wiener kernels for end-stopped, dot-responsive, and motion-specific cells,” Perception 22 (Suppl.), 143 (1993).
  33. G. Krieger, C. Zetzsche, E. Barth, “Nonlinear image operators for the detection of local intrinsic dimensionality,” in Proceedings of the IEEE Workshop Nonlinear Signal and Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 182–185.
  34. G. Krieger, C. Zetzsche, “Nonlinear image operators for the evaluation of local intrinsic dimensionality,” Special issue on Nonlinear Image Processing, IEEE Trans. Image Process. 5, 1026–1042 (1996).
    [CrossRef]
  35. T. M. Caelli, M. Hübner, I. Rentschler, “On the discrimination of micropatterns and textures,” Hum. Neurobiol. 5, 129–136 (1986).
    [PubMed]
  36. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962).
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    [CrossRef] [PubMed]
  38. J. A. Solomon, A. B. Watson, “Cinematica: a system for calibrated, Macintosh-driven displays from within Mathematica,” Behav. Res. Methods Instrum. Comput. 28, 607–610 (1996).
    [CrossRef] [PubMed]
  39. I. Rentschler, B. Treutwein, “Loss of spatial phase relationships in extrafoveal vision,” Nature 313, 308–310 (1985).
    [CrossRef] [PubMed]
  40. I. Rentschler, M. Huebner, T. Caelli, “On the discrimination of compound Gabor signals and textures,” Vision Res. 28, 279–291 (1988).
    [CrossRef] [PubMed]
  41. S. A. Klein, C. W. Tyler, “Phase discrimination of compound gratings: generalized autocorrelation analysis,” J. Opt. Soc. Am. A 3, 868–879 (1986).
    [CrossRef] [PubMed]
  42. C. Zetzsche, B. Wegmann, “Coding properties of local amplitude and phase of two-dimensional filter outputs,” Perception 17, 396 (1988).
  43. T. M. Caelli, B. Julesz, E. Gilbert, “On perceptual analysers underlying visual texture discrimination. Part II,” Biol. Cybern. 29, 201–214 (1978).
    [CrossRef] [PubMed]
  44. D. S. Simmons, D. H. Foster, “Segmenting textures of curved-line elements,” in Artificial and Biological Vision Systems, G. A. Orban, H.-H. Nagel, eds. (Springer-Verlag, Berlin, 1992), pp. 324–349.
  45. B. S. Rubenstein, D. Sagi, “Preattentive texture segmentation: the role of line terminations, size, and filter wavelength,” Percept. Psychophys. 58, 489–509 (1996).
    [CrossRef] [PubMed]
  46. E. Barth, G. Krieger, I. Rentschler, B. Treutwein, “Receptor-horizontal cell interactions may induce endstopping,” Invest. Ophthalmol. Visual Sci. 37 (Suppl.), 1056 (1996).
  47. E. Barth, C. Zetzsche, “Endstopped operators based on iterated nonlinear center-surround inhibition,” in Human Vision and Electronic Imaging III, B. Rogowitz, ed., Proc. SPIE3299, pp. 41–53 (1998).
  48. E. Barth, M. Ferraro, C. Zetzsche, I. Rentschler, “Computational models for the topological selectivity in early and primitive vision systems,” OSA Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 186.
  49. K. Koffka, Principles of Gestalt Psychology (Harcourt, Brace, New York, 1935).
  50. B. Julesz, “Spatial nonlinearities in the instantaneous perception of textures with identical power spectra,” Philos. Trans. R. Soc. London Ser. B 290, 83–94 (1980).
    [CrossRef]
  51. J. Elder, S. Zucker, “The effect of contour closure on the rapid discrimination of two-dimensional shapes,” Vision Res. 33, 981–991 (1993).
    [CrossRef] [PubMed]
  52. J. Elder, S. Zucker, “A measure of closure,” Vision Res. 34, 3361–3369 (1994).
    [CrossRef] [PubMed]
  53. E. Barth, C. Zetzsche, I. Rentschler, “End-stopping may yield textons,” Perception 24 (Suppl.), 19 (1995).

1997

1996

G. Krieger, C. Zetzsche, “Nonlinear image operators for the evaluation of local intrinsic dimensionality,” Special issue on Nonlinear Image Processing, IEEE Trans. Image Process. 5, 1026–1042 (1996).
[CrossRef]

J. A. Solomon, A. B. Watson, “Cinematica: a system for calibrated, Macintosh-driven displays from within Mathematica,” Behav. Res. Methods Instrum. Comput. 28, 607–610 (1996).
[CrossRef] [PubMed]

B. S. Rubenstein, D. Sagi, “Preattentive texture segmentation: the role of line terminations, size, and filter wavelength,” Percept. Psychophys. 58, 489–509 (1996).
[CrossRef] [PubMed]

E. Barth, G. Krieger, I. Rentschler, B. Treutwein, “Receptor-horizontal cell interactions may induce endstopping,” Invest. Ophthalmol. Visual Sci. 37 (Suppl.), 1056 (1996).

1995

E. Barth, C. Zetzsche, I. Rentschler, “End-stopping may yield textons,” Perception 24 (Suppl.), 19 (1995).

1994

J. Elder, S. Zucker, “A measure of closure,” Vision Res. 34, 3361–3369 (1994).
[CrossRef] [PubMed]

1993

A. Gorea, T. V. Papathomas, “Double opponency as a generalized concept in texture segregation illustrated with stimuli defined by color, luminance, and orientation,” J. Opt. Soc. Am. A 10, 1450–1462 (1993).
[CrossRef]

J. Elder, S. Zucker, “The effect of contour closure on the rapid discrimination of two-dimensional shapes,” Vision Res. 33, 981–991 (1993).
[CrossRef] [PubMed]

E. Barth, T. Caelli, C. Zetzsche, “Image encoding, labelling and reconstruction from differential geometry,” CVGIP: Graph. Models Image Process. 55, 428–446 (1993).

G. Krieger, C. Zetzsche, E. Barth, “2D-detectors in biological vision: Volterra–Wiener kernels for end-stopped, dot-responsive, and motion-specific cells,” Perception 22 (Suppl.), 143 (1993).

1992

F. Heitger, L. Rosenthaler, R. von der Heydt, E. Peterhans, O. Kübler, “Simulation of neural contour mechanisms: from simple to end-stopped cells,” Vision Res. 32, 63–981 (1992).
[CrossRef]

N. Graham, J. Beck, A. Sutter, “Nonlinear processes in spatial-frequency channel models of perceived texture segregation: effects of sign and amount of contrast,” Vision Res. 32, 719–743 (1992).
[CrossRef] [PubMed]

1991

M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
[CrossRef] [PubMed]

B. Julesz, “Early vision and focal attention,” Review Mod. Phys. 63, 735–772 (1991).
[CrossRef]

C. Zetzsche, E. Barth, “Detection of intrinsic signal dimensionality in images and optic flow fields,” Perception 20, 71 (1991).

1990

H. C. Nothdurft, “Texton segregation by associated differences in global and local luminance distribution,” Proc. R. Soc. London Ser. B 239, 295–320 (1990).
[CrossRef]

C. Zetzsche, E. Barth, “Fundamental limits of linear filters in the visual processing of two-dimensional signals,” Vision Res. 30, 1111–1117 (1990).
[CrossRef] [PubMed]

A. C. Bovik, M. Clark, W. S. Geisler, “Multi-channel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
[CrossRef]

J. Malik, P. Perona, “Preattentive texture discrimination with early vision mechanisms,” J. Opt. Soc. Am. A 7, 923–932 (1990).
[CrossRef] [PubMed]

1989

H. R. Wilson, W. A. Richards, “Mechanisms of contour curvature discrimination,” J. Opt. Soc. Am. A 6, 106–115 (1989).
[CrossRef] [PubMed]

I. Fogel, D. Sagi, “Gabor filters as texture discriminators,” Biol. Cybern. 61, 103–113 (1989).
[CrossRef]

A. Sutter, J. Beck, N. Graham, “Contrast and spatial variables in texture segregation: testing a simple spatial-frequency channels model,” Percept. Psychophys. 46, 312–332 (1989).
[CrossRef] [PubMed]

1988

B. Julesz, B. Kröse, “Features and spatial filters,” Nature 333, 302–303 (1988).
[CrossRef] [PubMed]

J. R. Bergen, E. H. Adelson, “Early vision and texture perception,” Nature 333, 363–364 (1988).
[CrossRef] [PubMed]

U. Kriegeskotten-Thiede, C. Zetzsche, “Local amplitude of filter outputs predicts the influence of micropattern spacing, orientation, and elongation on texture discrimination,” Perception 17, 398 (1988).

T. M. Caelli, “An adaptive computational model for texture segmentation,” IEEE Trans. Semicond. Manuf. 18, 9–17 (1988).

I. Rentschler, M. Huebner, T. Caelli, “On the discrimination of compound Gabor signals and textures,” Vision Res. 28, 279–291 (1988).
[CrossRef] [PubMed]

C. Zetzsche, B. Wegmann, “Coding properties of local amplitude and phase of two-dimensional filter outputs,” Perception 17, 396 (1988).

C. Zetzsche, “Statistical properties of the representation of natural images at different levels in the visual system,” Perception 17, 359 (1988).

J. J. Koenderink, W. Richards, “Two-dimensional curvature operators,” J. Opt. Soc. Am. A 5, 1136–1141 (1988).
[CrossRef]

1987

A. Dobbins, S. W. Zucker, M. S. Cynader, “Endstopped neurons in the visual cortex as a substrate for calculating curvature,” Nature 329, 438–441 (1987).
[CrossRef] [PubMed]

C. Zetzsche, W. Schönecker, “Orientation selective filters lead to entropy reduction in the processing of natural images,” Perception 16, 229 (1987).

1986

M. R. Turner, “Texture discrimination by Gabor functions,” Biol. Cybern. 55, 71–82 (1986).
[PubMed]

T. M. Caelli, M. Hübner, I. Rentschler, “On the discrimination of micropatterns and textures,” Hum. Neurobiol. 5, 129–136 (1986).
[PubMed]

S. A. Klein, C. W. Tyler, “Phase discrimination of compound gratings: generalized autocorrelation analysis,” J. Opt. Soc. Am. A 3, 868–879 (1986).
[CrossRef] [PubMed]

1985

I. Rentschler, B. Treutwein, “Loss of spatial phase relationships in extrafoveal vision,” Nature 313, 308–310 (1985).
[CrossRef] [PubMed]

J. M. Coggins, A. K. Jain, “A spatial filtering approach to texture analysis,” Pattern Recogn. Lett. 3, 195–203 (1985).
[CrossRef]

T. M. Caelli, “Three processing characteristics of visual texture segmentation,” Spatial Vis. 1, 19–30 (1985).
[CrossRef]

1981

B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature 290, 91–97 (1981).
[CrossRef] [PubMed]

1980

B. Julesz, “Spatial nonlinearities in the instantaneous perception of textures with identical power spectra,” Philos. Trans. R. Soc. London Ser. B 290, 83–94 (1980).
[CrossRef]

1978

T. M. Caelli, B. Julesz, E. Gilbert, “On perceptual analysers underlying visual texture discrimination. Part II,” Biol. Cybern. 29, 201–214 (1978).
[CrossRef] [PubMed]

1975

B. Julesz, “Experiments in the visual perception of texture,” Sci. Am. 232, 34–43 (1975).
[CrossRef] [PubMed]

Adelson, E. H.

J. R. Bergen, E. H. Adelson, “Early vision and texture perception,” Nature 333, 363–364 (1988).
[CrossRef] [PubMed]

Barth, E.

E. Barth, G. Krieger, I. Rentschler, B. Treutwein, “Receptor-horizontal cell interactions may induce endstopping,” Invest. Ophthalmol. Visual Sci. 37 (Suppl.), 1056 (1996).

E. Barth, C. Zetzsche, I. Rentschler, “End-stopping may yield textons,” Perception 24 (Suppl.), 19 (1995).

E. Barth, T. Caelli, C. Zetzsche, “Image encoding, labelling and reconstruction from differential geometry,” CVGIP: Graph. Models Image Process. 55, 428–446 (1993).

G. Krieger, C. Zetzsche, E. Barth, “2D-detectors in biological vision: Volterra–Wiener kernels for end-stopped, dot-responsive, and motion-specific cells,” Perception 22 (Suppl.), 143 (1993).

C. Zetzsche, E. Barth, “Detection of intrinsic signal dimensionality in images and optic flow fields,” Perception 20, 71 (1991).

C. Zetzsche, E. Barth, “Fundamental limits of linear filters in the visual processing of two-dimensional signals,” Vision Res. 30, 1111–1117 (1990).
[CrossRef] [PubMed]

E. Barth, C. Zetzsche, “Endstopped operators based on iterated nonlinear center-surround inhibition,” in Human Vision and Electronic Imaging III, B. Rogowitz, ed., Proc. SPIE3299, pp. 41–53 (1998).

C. Zetzsche, E. Barth, 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, Cambridge, Mass., 1993), pp. 109–138.

C. Zetzsche, E. Barth, “Image surface predicates and the neural encoding of two-dimensional signal variation,” in Human Vision and Electronic Imaging: Models, Methods, and Applications, B. Rogowitz, ed., Proc. SPIE1249, 160–177 (1990).

G. Krieger, C. Zetzsche, E. Barth, “Nonlinear image operators for the detection of local intrinsic dimensionality,” in Proceedings of the IEEE Workshop Nonlinear Signal and Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 182–185.

E. Barth, M. Ferraro, C. Zetzsche, I. Rentschler, “Computational models for the topological selectivity in early and primitive vision systems,” OSA Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 186.

Beck, J.

N. Graham, J. Beck, A. Sutter, “Nonlinear processes in spatial-frequency channel models of perceived texture segregation: effects of sign and amount of contrast,” Vision Res. 32, 719–743 (1992).
[CrossRef] [PubMed]

A. Sutter, J. Beck, N. Graham, “Contrast and spatial variables in texture segregation: testing a simple spatial-frequency channels model,” Percept. Psychophys. 46, 312–332 (1989).
[CrossRef] [PubMed]

Bergen, J. R.

M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
[CrossRef] [PubMed]

J. R. Bergen, E. H. Adelson, “Early vision and texture perception,” Nature 333, 363–364 (1988).
[CrossRef] [PubMed]

J. R. Bergen, “Theories of visual texture perception,” in Vision and Visual Disfunction, D. Regan, ed. (Macmillan, New York, 1991), Vol. 10B, pp. 114–134.

Bovik, A. C.

A. C. Bovik, M. Clark, W. S. Geisler, “Multi-channel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
[CrossRef]

Caelli, T.

D. Carević, T. Caelli, “Application of partial modelling techniques for texture segmentation,” J. Opt. Soc. Am. A 14, 2924–2937 (1997).
[CrossRef]

E. Barth, T. Caelli, C. Zetzsche, “Image encoding, labelling and reconstruction from differential geometry,” CVGIP: Graph. Models Image Process. 55, 428–446 (1993).

I. Rentschler, M. Huebner, T. Caelli, “On the discrimination of compound Gabor signals and textures,” Vision Res. 28, 279–291 (1988).
[CrossRef] [PubMed]

Caelli, T. M.

T. M. Caelli, “An adaptive computational model for texture segmentation,” IEEE Trans. Semicond. Manuf. 18, 9–17 (1988).

T. M. Caelli, M. Hübner, I. Rentschler, “On the discrimination of micropatterns and textures,” Hum. Neurobiol. 5, 129–136 (1986).
[PubMed]

T. M. Caelli, “Three processing characteristics of visual texture segmentation,” Spatial Vis. 1, 19–30 (1985).
[CrossRef]

T. M. Caelli, B. Julesz, E. Gilbert, “On perceptual analysers underlying visual texture discrimination. Part II,” Biol. Cybern. 29, 201–214 (1978).
[CrossRef] [PubMed]

Carevic, D.

Clark, M.

A. C. Bovik, M. Clark, W. S. Geisler, “Multi-channel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
[CrossRef]

Coggins, J. M.

J. M. Coggins, A. K. Jain, “A spatial filtering approach to texture analysis,” Pattern Recogn. Lett. 3, 195–203 (1985).
[CrossRef]

Cynader, M. S.

A. Dobbins, S. W. Zucker, M. S. Cynader, “Endstopped neurons in the visual cortex as a substrate for calculating curvature,” Nature 329, 438–441 (1987).
[CrossRef] [PubMed]

Dobbins, A.

A. Dobbins, S. W. Zucker, M. S. Cynader, “Endstopped neurons in the visual cortex as a substrate for calculating curvature,” Nature 329, 438–441 (1987).
[CrossRef] [PubMed]

Elder, J.

J. Elder, S. Zucker, “A measure of closure,” Vision Res. 34, 3361–3369 (1994).
[CrossRef] [PubMed]

J. Elder, S. Zucker, “The effect of contour closure on the rapid discrimination of two-dimensional shapes,” Vision Res. 33, 981–991 (1993).
[CrossRef] [PubMed]

Ferraro, M.

E. Barth, M. Ferraro, C. Zetzsche, I. Rentschler, “Computational models for the topological selectivity in early and primitive vision systems,” OSA Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 186.

Fogel, I.

I. Fogel, D. Sagi, “Gabor filters as texture discriminators,” Biol. Cybern. 61, 103–113 (1989).
[CrossRef]

Foster, D. H.

D. S. Simmons, D. H. Foster, “Segmenting textures of curved-line elements,” in Artificial and Biological Vision Systems, G. A. Orban, H.-H. Nagel, eds. (Springer-Verlag, Berlin, 1992), pp. 324–349.

Geisler, W. S.

A. C. Bovik, M. Clark, W. S. Geisler, “Multi-channel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
[CrossRef]

Gilbert, E.

T. M. Caelli, B. Julesz, E. Gilbert, “On perceptual analysers underlying visual texture discrimination. Part II,” Biol. Cybern. 29, 201–214 (1978).
[CrossRef] [PubMed]

Gorea, A.

Graham, N.

N. Graham, J. Beck, A. Sutter, “Nonlinear processes in spatial-frequency channel models of perceived texture segregation: effects of sign and amount of contrast,” Vision Res. 32, 719–743 (1992).
[CrossRef] [PubMed]

A. Sutter, J. Beck, N. Graham, “Contrast and spatial variables in texture segregation: testing a simple spatial-frequency channels model,” Percept. Psychophys. 46, 312–332 (1989).
[CrossRef] [PubMed]

Heitger, F.

F. Heitger, L. Rosenthaler, R. von der Heydt, E. Peterhans, O. Kübler, “Simulation of neural contour mechanisms: from simple to end-stopped cells,” Vision Res. 32, 63–981 (1992).
[CrossRef]

Hübner, M.

T. M. Caelli, M. Hübner, I. Rentschler, “On the discrimination of micropatterns and textures,” Hum. Neurobiol. 5, 129–136 (1986).
[PubMed]

Huebner, M.

I. Rentschler, M. Huebner, T. Caelli, “On the discrimination of compound Gabor signals and textures,” Vision Res. 28, 279–291 (1988).
[CrossRef] [PubMed]

Jain, A. K.

J. M. Coggins, A. K. Jain, “A spatial filtering approach to texture analysis,” Pattern Recogn. Lett. 3, 195–203 (1985).
[CrossRef]

Julesz, B.

B. Julesz, “Early vision and focal attention,” Review Mod. Phys. 63, 735–772 (1991).
[CrossRef]

B. Julesz, B. Kröse, “Features and spatial filters,” Nature 333, 302–303 (1988).
[CrossRef] [PubMed]

B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature 290, 91–97 (1981).
[CrossRef] [PubMed]

B. Julesz, “Spatial nonlinearities in the instantaneous perception of textures with identical power spectra,” Philos. Trans. R. Soc. London Ser. B 290, 83–94 (1980).
[CrossRef]

T. M. Caelli, B. Julesz, E. Gilbert, “On perceptual analysers underlying visual texture discrimination. Part II,” Biol. Cybern. 29, 201–214 (1978).
[CrossRef] [PubMed]

B. Julesz, “Experiments in the visual perception of texture,” Sci. Am. 232, 34–43 (1975).
[CrossRef] [PubMed]

Klein, S. A.

Koenderink, J. J.

Koffka, K.

K. Koffka, Principles of Gestalt Psychology (Harcourt, Brace, New York, 1935).

Krieger, G.

G. Krieger, C. Zetzsche, “Nonlinear image operators for the evaluation of local intrinsic dimensionality,” Special issue on Nonlinear Image Processing, IEEE Trans. Image Process. 5, 1026–1042 (1996).
[CrossRef]

E. Barth, G. Krieger, I. Rentschler, B. Treutwein, “Receptor-horizontal cell interactions may induce endstopping,” Invest. Ophthalmol. Visual Sci. 37 (Suppl.), 1056 (1996).

G. Krieger, C. Zetzsche, E. Barth, “2D-detectors in biological vision: Volterra–Wiener kernels for end-stopped, dot-responsive, and motion-specific cells,” Perception 22 (Suppl.), 143 (1993).

G. Krieger, C. Zetzsche, E. Barth, “Nonlinear image operators for the detection of local intrinsic dimensionality,” in Proceedings of the IEEE Workshop Nonlinear Signal and Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 182–185.

Kriegeskotten-Thiede, U.

U. Kriegeskotten-Thiede, C. Zetzsche, “Local amplitude of filter outputs predicts the influence of micropattern spacing, orientation, and elongation on texture discrimination,” Perception 17, 398 (1988).

Kröse, B.

B. Julesz, B. Kröse, “Features and spatial filters,” Nature 333, 302–303 (1988).
[CrossRef] [PubMed]

Kübler, O.

F. Heitger, L. Rosenthaler, R. von der Heydt, E. Peterhans, O. Kübler, “Simulation of neural contour mechanisms: from simple to end-stopped cells,” Vision Res. 32, 63–981 (1992).
[CrossRef]

Landy, M. S.

M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
[CrossRef] [PubMed]

Liedtke, C.-E.

D. Wermser, C.-E. Liedtke, “Texture analysis using a model of the visual system,” in Proceedings of the Sixth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1982), pp. 1078–1080.

Malik, J.

Nothdurft, H. C.

H. C. Nothdurft, “Texton segregation by associated differences in global and local luminance distribution,” Proc. R. Soc. London Ser. B 239, 295–320 (1990).
[CrossRef]

Papathomas, T. V.

Papoulis, A.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962).

Perona, P.

Peterhans, E.

F. Heitger, L. Rosenthaler, R. von der Heydt, E. Peterhans, O. Kübler, “Simulation of neural contour mechanisms: from simple to end-stopped cells,” Vision Res. 32, 63–981 (1992).
[CrossRef]

Rentschler, I.

E. Barth, G. Krieger, I. Rentschler, B. Treutwein, “Receptor-horizontal cell interactions may induce endstopping,” Invest. Ophthalmol. Visual Sci. 37 (Suppl.), 1056 (1996).

E. Barth, C. Zetzsche, I. Rentschler, “End-stopping may yield textons,” Perception 24 (Suppl.), 19 (1995).

I. Rentschler, M. Huebner, T. Caelli, “On the discrimination of compound Gabor signals and textures,” Vision Res. 28, 279–291 (1988).
[CrossRef] [PubMed]

T. M. Caelli, M. Hübner, I. Rentschler, “On the discrimination of micropatterns and textures,” Hum. Neurobiol. 5, 129–136 (1986).
[PubMed]

I. Rentschler, B. Treutwein, “Loss of spatial phase relationships in extrafoveal vision,” Nature 313, 308–310 (1985).
[CrossRef] [PubMed]

E. Barth, M. Ferraro, C. Zetzsche, I. Rentschler, “Computational models for the topological selectivity in early and primitive vision systems,” OSA Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 186.

Richards, W.

Richards, W. A.

Rosenthaler, L.

F. Heitger, L. Rosenthaler, R. von der Heydt, E. Peterhans, O. Kübler, “Simulation of neural contour mechanisms: from simple to end-stopped cells,” Vision Res. 32, 63–981 (1992).
[CrossRef]

Rubenstein, B. S.

B. S. Rubenstein, D. Sagi, “Preattentive texture segmentation: the role of line terminations, size, and filter wavelength,” Percept. Psychophys. 58, 489–509 (1996).
[CrossRef] [PubMed]

Sagi, D.

B. S. Rubenstein, D. Sagi, “Preattentive texture segmentation: the role of line terminations, size, and filter wavelength,” Percept. Psychophys. 58, 489–509 (1996).
[CrossRef] [PubMed]

I. Fogel, D. Sagi, “Gabor filters as texture discriminators,” Biol. Cybern. 61, 103–113 (1989).
[CrossRef]

Schönecker, W.

C. Zetzsche, W. Schönecker, “Orientation selective filters lead to entropy reduction in the processing of natural images,” Perception 16, 229 (1987).

Simmons, D. S.

D. S. Simmons, D. H. Foster, “Segmenting textures of curved-line elements,” in Artificial and Biological Vision Systems, G. A. Orban, H.-H. Nagel, eds. (Springer-Verlag, Berlin, 1992), pp. 324–349.

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J. A. Solomon, A. B. Watson, “Cinematica: a system for calibrated, Macintosh-driven displays from within Mathematica,” Behav. Res. Methods Instrum. Comput. 28, 607–610 (1996).
[CrossRef] [PubMed]

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N. Graham, J. Beck, A. Sutter, “Nonlinear processes in spatial-frequency channel models of perceived texture segregation: effects of sign and amount of contrast,” Vision Res. 32, 719–743 (1992).
[CrossRef] [PubMed]

A. Sutter, J. Beck, N. Graham, “Contrast and spatial variables in texture segregation: testing a simple spatial-frequency channels model,” Percept. Psychophys. 46, 312–332 (1989).
[CrossRef] [PubMed]

Treutwein, B.

E. Barth, G. Krieger, I. Rentschler, B. Treutwein, “Receptor-horizontal cell interactions may induce endstopping,” Invest. Ophthalmol. Visual Sci. 37 (Suppl.), 1056 (1996).

I. Rentschler, B. Treutwein, “Loss of spatial phase relationships in extrafoveal vision,” Nature 313, 308–310 (1985).
[CrossRef] [PubMed]

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M. R. Turner, “Texture discrimination by Gabor functions,” Biol. Cybern. 55, 71–82 (1986).
[PubMed]

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von der Heydt, R.

F. Heitger, L. Rosenthaler, R. von der Heydt, E. Peterhans, O. Kübler, “Simulation of neural contour mechanisms: from simple to end-stopped cells,” Vision Res. 32, 63–981 (1992).
[CrossRef]

Watson, A. B.

J. A. Solomon, A. B. Watson, “Cinematica: a system for calibrated, Macintosh-driven displays from within Mathematica,” Behav. Res. Methods Instrum. Comput. 28, 607–610 (1996).
[CrossRef] [PubMed]

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C. Zetzsche, B. Wegmann, “Coding properties of local amplitude and phase of two-dimensional filter outputs,” Perception 17, 396 (1988).

C. Zetzsche, E. Barth, 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, Cambridge, Mass., 1993), pp. 109–138.

Wermser, D.

D. Wermser, C.-E. Liedtke, “Texture analysis using a model of the visual system,” in Proceedings of the Sixth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1982), pp. 1078–1080.

Wilson, H. R.

Zetzsche, C.

G. Krieger, C. Zetzsche, “Nonlinear image operators for the evaluation of local intrinsic dimensionality,” Special issue on Nonlinear Image Processing, IEEE Trans. Image Process. 5, 1026–1042 (1996).
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E. Barth, C. Zetzsche, I. Rentschler, “End-stopping may yield textons,” Perception 24 (Suppl.), 19 (1995).

E. Barth, T. Caelli, C. Zetzsche, “Image encoding, labelling and reconstruction from differential geometry,” CVGIP: Graph. Models Image Process. 55, 428–446 (1993).

G. Krieger, C. Zetzsche, E. Barth, “2D-detectors in biological vision: Volterra–Wiener kernels for end-stopped, dot-responsive, and motion-specific cells,” Perception 22 (Suppl.), 143 (1993).

C. Zetzsche, E. Barth, “Detection of intrinsic signal dimensionality in images and optic flow fields,” Perception 20, 71 (1991).

C. Zetzsche, E. Barth, “Fundamental limits of linear filters in the visual processing of two-dimensional signals,” Vision Res. 30, 1111–1117 (1990).
[CrossRef] [PubMed]

U. Kriegeskotten-Thiede, C. Zetzsche, “Local amplitude of filter outputs predicts the influence of micropattern spacing, orientation, and elongation on texture discrimination,” Perception 17, 398 (1988).

C. Zetzsche, “Statistical properties of the representation of natural images at different levels in the visual system,” Perception 17, 359 (1988).

C. Zetzsche, B. Wegmann, “Coding properties of local amplitude and phase of two-dimensional filter outputs,” Perception 17, 396 (1988).

C. Zetzsche, W. Schönecker, “Orientation selective filters lead to entropy reduction in the processing of natural images,” Perception 16, 229 (1987).

C. Zetzsche, E. Barth, “Image surface predicates and the neural encoding of two-dimensional signal variation,” in Human Vision and Electronic Imaging: Models, Methods, and Applications, B. Rogowitz, ed., Proc. SPIE1249, 160–177 (1990).

G. Krieger, C. Zetzsche, E. Barth, “Nonlinear image operators for the detection of local intrinsic dimensionality,” in Proceedings of the IEEE Workshop Nonlinear Signal and Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 182–185.

E. Barth, M. Ferraro, C. Zetzsche, I. Rentschler, “Computational models for the topological selectivity in early and primitive vision systems,” OSA Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 186.

E. Barth, C. Zetzsche, “Endstopped operators based on iterated nonlinear center-surround inhibition,” in Human Vision and Electronic Imaging III, B. Rogowitz, ed., Proc. SPIE3299, pp. 41–53 (1998).

C. Zetzsche, E. Barth, 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, Cambridge, Mass., 1993), pp. 109–138.

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J. Elder, S. Zucker, “A measure of closure,” Vision Res. 34, 3361–3369 (1994).
[CrossRef] [PubMed]

J. Elder, S. Zucker, “The effect of contour closure on the rapid discrimination of two-dimensional shapes,” Vision Res. 33, 981–991 (1993).
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A. Dobbins, S. W. Zucker, M. S. Cynader, “Endstopped neurons in the visual cortex as a substrate for calculating curvature,” Nature 329, 438–441 (1987).
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J. A. Solomon, A. B. Watson, “Cinematica: a system for calibrated, Macintosh-driven displays from within Mathematica,” Behav. Res. Methods Instrum. Comput. 28, 607–610 (1996).
[CrossRef] [PubMed]

Biol. Cybern.

M. R. Turner, “Texture discrimination by Gabor functions,” Biol. Cybern. 55, 71–82 (1986).
[PubMed]

I. Fogel, D. Sagi, “Gabor filters as texture discriminators,” Biol. Cybern. 61, 103–113 (1989).
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E. Barth, T. Caelli, C. Zetzsche, “Image encoding, labelling and reconstruction from differential geometry,” CVGIP: Graph. Models Image Process. 55, 428–446 (1993).

Hum. Neurobiol.

T. M. Caelli, M. Hübner, I. Rentschler, “On the discrimination of micropatterns and textures,” Hum. Neurobiol. 5, 129–136 (1986).
[PubMed]

IEEE Trans. Image Process

G. Krieger, C. Zetzsche, “Nonlinear image operators for the evaluation of local intrinsic dimensionality,” Special issue on Nonlinear Image Processing, IEEE Trans. Image Process. 5, 1026–1042 (1996).
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A. C. Bovik, M. Clark, W. S. Geisler, “Multi-channel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
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IEEE Trans. Semicond. Manuf.

T. M. Caelli, “An adaptive computational model for texture segmentation,” IEEE Trans. Semicond. Manuf. 18, 9–17 (1988).

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E. Barth, G. Krieger, I. Rentschler, B. Treutwein, “Receptor-horizontal cell interactions may induce endstopping,” Invest. Ophthalmol. Visual Sci. 37 (Suppl.), 1056 (1996).

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B. Julesz, “Textons, the elements of texture perception, and their interactions,” Nature 290, 91–97 (1981).
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J. R. Bergen, E. H. Adelson, “Early vision and texture perception,” Nature 333, 363–364 (1988).
[CrossRef] [PubMed]

I. Rentschler, B. Treutwein, “Loss of spatial phase relationships in extrafoveal vision,” Nature 313, 308–310 (1985).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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J. M. Coggins, A. K. Jain, “A spatial filtering approach to texture analysis,” Pattern Recogn. Lett. 3, 195–203 (1985).
[CrossRef]

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A. Sutter, J. Beck, N. Graham, “Contrast and spatial variables in texture segregation: testing a simple spatial-frequency channels model,” Percept. Psychophys. 46, 312–332 (1989).
[CrossRef] [PubMed]

B. S. Rubenstein, D. Sagi, “Preattentive texture segmentation: the role of line terminations, size, and filter wavelength,” Percept. Psychophys. 58, 489–509 (1996).
[CrossRef] [PubMed]

Perception

E. Barth, C. Zetzsche, I. Rentschler, “End-stopping may yield textons,” Perception 24 (Suppl.), 19 (1995).

C. Zetzsche, W. Schönecker, “Orientation selective filters lead to entropy reduction in the processing of natural images,” Perception 16, 229 (1987).

U. Kriegeskotten-Thiede, C. Zetzsche, “Local amplitude of filter outputs predicts the influence of micropattern spacing, orientation, and elongation on texture discrimination,” Perception 17, 398 (1988).

C. Zetzsche, “Statistical properties of the representation of natural images at different levels in the visual system,” Perception 17, 359 (1988).

G. Krieger, C. Zetzsche, E. Barth, “2D-detectors in biological vision: Volterra–Wiener kernels for end-stopped, dot-responsive, and motion-specific cells,” Perception 22 (Suppl.), 143 (1993).

C. Zetzsche, E. Barth, “Detection of intrinsic signal dimensionality in images and optic flow fields,” Perception 20, 71 (1991).

C. Zetzsche, B. Wegmann, “Coding properties of local amplitude and phase of two-dimensional filter outputs,” Perception 17, 396 (1988).

Philos. Trans. R. Soc. London Ser. B

B. Julesz, “Spatial nonlinearities in the instantaneous perception of textures with identical power spectra,” Philos. Trans. R. Soc. London Ser. B 290, 83–94 (1980).
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M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
[CrossRef] [PubMed]

N. Graham, J. Beck, A. Sutter, “Nonlinear processes in spatial-frequency channel models of perceived texture segregation: effects of sign and amount of contrast,” Vision Res. 32, 719–743 (1992).
[CrossRef] [PubMed]

I. Rentschler, M. Huebner, T. Caelli, “On the discrimination of compound Gabor signals and textures,” Vision Res. 28, 279–291 (1988).
[CrossRef] [PubMed]

F. Heitger, L. Rosenthaler, R. von der Heydt, E. Peterhans, O. Kübler, “Simulation of neural contour mechanisms: from simple to end-stopped cells,” Vision Res. 32, 63–981 (1992).
[CrossRef]

C. Zetzsche, E. Barth, “Fundamental limits of linear filters in the visual processing of two-dimensional signals,” Vision Res. 30, 1111–1117 (1990).
[CrossRef] [PubMed]

J. Elder, S. Zucker, “The effect of contour closure on the rapid discrimination of two-dimensional shapes,” Vision Res. 33, 981–991 (1993).
[CrossRef] [PubMed]

J. Elder, S. Zucker, “A measure of closure,” Vision Res. 34, 3361–3369 (1994).
[CrossRef] [PubMed]

Other

D. S. Simmons, D. H. Foster, “Segmenting textures of curved-line elements,” in Artificial and Biological Vision Systems, G. A. Orban, H.-H. Nagel, eds. (Springer-Verlag, Berlin, 1992), pp. 324–349.

E. Barth, C. Zetzsche, “Endstopped operators based on iterated nonlinear center-surround inhibition,” in Human Vision and Electronic Imaging III, B. Rogowitz, ed., Proc. SPIE3299, pp. 41–53 (1998).

E. Barth, M. Ferraro, C. Zetzsche, I. Rentschler, “Computational models for the topological selectivity in early and primitive vision systems,” OSA Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 186.

K. Koffka, Principles of Gestalt Psychology (Harcourt, Brace, New York, 1935).

C. Zetzsche, E. Barth, “Image surface predicates and the neural encoding of two-dimensional signal variation,” in Human Vision and Electronic Imaging: Models, Methods, and Applications, B. Rogowitz, ed., Proc. SPIE1249, 160–177 (1990).

C. Zetzsche, E. Barth, 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, Cambridge, Mass., 1993), pp. 109–138.

G. Krieger, C. Zetzsche, E. Barth, “Nonlinear image operators for the detection of local intrinsic dimensionality,” in Proceedings of the IEEE Workshop Nonlinear Signal and Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 182–185.

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962).

J. R. Bergen, “Theories of visual texture perception,” in Vision and Visual Disfunction, D. Regan, ed. (Macmillan, New York, 1991), Vol. 10B, pp. 114–134.

D. Wermser, C.-E. Liedtke, “Texture analysis using a model of the visual system,” in Proceedings of the Sixth International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1982), pp. 1078–1080.

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Figures (9)

Fig. 1
Fig. 1

Micropatterns with the same total line length. Bending leads to corners, breaking leads to line ends, and different relative positions of line segments can yield different types of junctions.

Fig. 2
Fig. 2

Different relative positions of line segments can lead to differences in size. The two disks drawn the same size for all micropatterns cover the micropatterns to different extents.

Fig. 3
Fig. 3

Arrow–Z (TaTz) and arrow–triangle (TaTt) texture pairs. Patterns are computer-generated raster images (1024×1024 pixels) with lines 70 pixels long (with one exception) and 2 pixels wide, randomly rotated by use of an affine transform with B-splines interpolation. (c), Lines in the upper right quadrant are 66-pixels long.

Fig. 4
Fig. 4

Representations of single micropatterns (arrow and triangle) at logarithmically sampled scales from 0.5 to 8 c/micropattern (Laplacian top rows, CEV bottom rows). Note that the CEV operator responds only to i2D features and that it responds differently to the tip and the line ends of the arrow compared with the corners of the triangle. Also note that Laplacian and CEV outputs become more similar at coarse scales.

Fig. 5
Fig. 5

Predicted segregation, for the texture pairs shown in Fig. 3, for four different operators. The predictions for the different texture pairs in Fig. 3 are plotted with thick lines and diamonds [Fig. 3(a), textures with equal power spectra which segregate well] dashed lines and stars [Fig. 3(b), textures that segregate less] and thin lines and squares [Fig. 3(c), a texture pair that does not segregate]. The differences that are due to the type of nonlinearity are discussed in Subsection 2.B. Table 2 (below) gives a qualitative interpretation of the predictions compared with the data presented in Section 4.

Fig. 6
Fig. 6

Power spectra for the arrow-, triangle-, and Z-textures (see text).

Fig. 7
Fig. 7

Responses of different i1D operators to the input patterns shown at the left: Laplacian followed by one-way rectification, Laplacian followed by squaring, local amplitude, and local energy (from left to right). Local amplitude is computed as i4(ei2+oi2) and local amplitude as i4(ei2+oi2)1/2, where e and o are functions of (x, y) denoting outputs of oriented even and odd filters and i indexes orientation. Details of the implementation are irrelevant for the point we make here, which is simply that the four operators have different selectivities for i2D features relative to i1D features.

Fig. 8
Fig. 8

Original in-square texture pairs are shown in (a). The lines are 82 pixels long and 2 pixels wide unless different values are given below for (b) and (c). The procedures for generating the textures are as in Fig. 3. In (b) the lines of the target texture Toc are 92 pixels long and in (c) the lines of the background texture are 90 pixels long. The values for line length have been chosen so as to approximately equalize the predictions of the Laplacian (Fig. 9) for the two pairs in (b) and (c) and raise them to a level comparable to the one obtained for (a).

Fig. 9
Fig. 9

Predicted segregation for the texture pairs shown in Fig. 8. Plotting conventions are as in Fig. 5, i.e., the predictions for the different texture pairs in Fig. 8 are plotted with thick lines and diamonds [Fig. 8(a)] dashed lines and stars [Fig. 8(b)] and thin lines and squares [Fig. 8(c)] (see text and Table 2).

Tables (2)

Tables Icon

Table 1 Experimental Results

Tables Icon

Table 2 Summary of Simulation Results

Equations (7)

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

G=14(lxx+lyy)2-14(lxx-lyy)2-lxy2=(2l)2-2,
d1=2l+;d2=2l-.
CEV=d2+-d1-,
xy[f(x, y)t1(x, y)]2dxdy
=uv|F(u, v)×T1(u, v)|2dudv=uv|F(u, v)|2×|T1(u, v)|2dudv
=uv|F(u, v)|2×|T2(u, v)|2dudv
=uv|F(u, v)×T2(u, v)|2dudv=xy[f(x, y)t2(x, y)]2dxdy,

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