Abstract

A new psychophysical methodology is introduced, histogram contrast analysis, that allows one to measure stimulus transformations, f, used by the visual system to draw distinctions between different image regions. The method involves the discrimination of images constructed by selecting texture micropatterns randomly and independently (across locations) on the basis of a given micropattern histogram. Different components of f are measured by use of different component functions to modulate the micropattern histogram until the resulting textures are discriminable. When no discrimination threshold can be obtained for a given modulating component function, a second titration technique may be used to measure the contribution of that component to f. The method includes several strong tests of its own assumptions. An example is given of the method applied to visual textures composed of small, uniform squares with randomly chosen gray levels. In particular, for a fixed mean gray level μ and a fixed gray-level variance σ2, histogram contrast analysis is used to establish that the class S of all textures composed of small squares with jointly independent, identically distributed gray levels with mean μ and variance σ2 is perceptually elementary in the following sense: there exists a single, real-valued function fS of gray level, such that two textures I and J in S are discriminable only if the average value of fS applied to the gray levels in I is significantly different from the average value of fS applied to the gray levels in J. Finally, histogram contrast analysis is used to obtain a seventh-order polynomial approximation of fS.

© 1994 Optical Society of America

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References

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  1. G. T. Fechner, Elements of Psychophysics, D. H. Howes, E. C. Boring, eds., H. E. Adler, trans. (Holt, Reinhart & Winston, New York, 1966; originally published, 1860).
  2. E. Mach, The Analysis of Sensations and the Relation of the Physical to the Psychical (Dover, New York, 1959, with use of the fifth German edition revised and supplemented by S. Waterlow; first German edition 1886).
  3. H. B. Barlow, “Optic nerve impulses and Weber’s law,” Cold Spring Harbor Symp. Quant. Biol. 30, 539–546 (1965).
    [Crossref]
  4. H. B. Barlow, W. R. Levick, “Three factors limiting the reliable detection of light by retinal ganglion cells of the cat,”J. Physiol. 200, 1–24 (1969).
    [PubMed]
  5. R. Shapley, C. Enroth-Cugell, “Visual adaptation and retinal gain controls,” Prog. Retinal Res. 3, 263–346 (1984).
    [Crossref]
  6. J. Müller, “Of the senses,” in Visual Perception: the Nineteenth Century, W. N. Dember, ed. (Wiley, New York, 1964), pp. 35–69; recopied from the original English version published in Elements of Physiology, W. Baly, trans. (Taylor & Walton, London, 1942), Vol. 2.
  7. J. R. Bergen, “Theories of visual texture perception,” in Vision and Visual Dysfunction, D. Regan, ed. (Macmillan, New York, 1991), Vol. 10, pp. 114–134.
  8. J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Computer Vision Graphics Image Process. 37, 299–325 (1987).
    [Crossref]
  9. J. R. Bergen, E. H. Adelson, “Early vision and texture perception,” Nature (London) 333, 363–364 (1988).
    [Crossref]
  10. J. R. Bergen, M. S. Landy, “Computational modeling of visual texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 252–271.
  11. T. Caelli, “Three processing characteristics of visual texture segmentation,” Spatial Vision 1, 19–30 (1985).
    [Crossref] [PubMed]
  12. I. Fogel, D. Sagi, “Gabor filters as texture discriminator,” Biol. Cybern. 61, 103–113 (1989).
    [Crossref]
  13. N. Graham, “Complex channels, early local nonlinearities, and normalization in texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 273–290.
  14. M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
    [Crossref] [PubMed]
  15. J. Malik, P. Perona, “Preattentive texture discrimination with early vision mechanisms,” J. Opt. Soc. Am. A 7, 923–932 (1990).
    [Crossref] [PubMed]
  16. 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]
  17. H. Knutsson, G. H. Granlund, “Texture analysis using two-dimensional quadrature filters,” presented at the IEEE Computer Society Workshop on Computer Architecture for Pattern Analysis and Image Database Management, Pasadena, Calif., October 12–14, 1983.
  18. J. D. Victor, “Models for preattentive texture discrimination: Fourier analysis and local feature processing in a unified framework,” Spatial Vision 3, 263–280 (1988).
    [Crossref] [PubMed]
  19. R. L. De Valois, K. K. De Valois, Spatial Vision (Oxford U. Press, New York, 1988).
  20. N. Graham, Visual Pattern Analyzers (Oxford U. Press, New York, 1989).
    [Crossref]
  21. R. C. Fisher, Linear Algebra (Dickenson, Encino, Calif., 1970).
  22. U. W. Hochstrasser, “Orthogonal polynomials,” in Handbook of Mathematical Functions, Vol. 55 of Applied Mathematics Series, M. Abramowitz, I. A. Stegun, eds. (National Bureau of Standards, Washington, D.C., 1972), Chap. 22, pp. 771–802.
  23. D. H. Brainard, “Calibration of a computer controlled color monitor,” Color Res. Appl. 14, 23–43 (1989).
    [Crossref]
  24. J. B. Mulligan, L. S. Stone, “Halftoning method for the generation of motion stimuli,” J. Opt. Soc. Am. A 6, 1217–1227 (1989).
    [Crossref]
  25. L. T. Maloney, “Confidence intervals for the parameters of psychometric functions,” Percept. Psychophys. 47, 127–134 (1990).
    [Crossref] [PubMed]
  26. M. Eckstein, J. S. Whiting, J. P. Thomas, S. Shimozaki, “A novel temporal integration of intensity,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 17.
  27. C. Chubb, M. S. Landy, “Orthogonal distribution analysis: a systematic approach to the study of texture perception,” Invest. Ophthalmol. Vis. Sci. Suppl. 31, 561 (1990).
  28. C. Chubb, M. S. Landy, “Orthogonal distribution analysis: a new approach to the study of texture perception,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 291–301.
  29. J. Econopouly, C. Chubb, M. S. Landy, “Segregation of 1st-order noise textures,” Invest. Ophthalmol. Vis. Sci. Suppl. 32, 714 (1991).
  30. J. Econopouly, C. Chubb, M. S. Landy, “Segregation of 1st-order noise textures is perceptually one-dimensional,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 956 (1992).

1992 (1)

J. Econopouly, C. Chubb, M. S. Landy, “Segregation of 1st-order noise textures is perceptually one-dimensional,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 956 (1992).

1991 (2)

J. Econopouly, C. Chubb, M. S. Landy, “Segregation of 1st-order noise textures,” Invest. Ophthalmol. Vis. Sci. Suppl. 32, 714 (1991).

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

1990 (3)

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

L. T. Maloney, “Confidence intervals for the parameters of psychometric functions,” Percept. Psychophys. 47, 127–134 (1990).
[Crossref] [PubMed]

C. Chubb, M. S. Landy, “Orthogonal distribution analysis: a systematic approach to the study of texture perception,” Invest. Ophthalmol. Vis. Sci. Suppl. 31, 561 (1990).

1989 (4)

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

D. H. Brainard, “Calibration of a computer controlled color monitor,” Color Res. Appl. 14, 23–43 (1989).
[Crossref]

J. B. Mulligan, L. S. Stone, “Halftoning method for the generation of motion stimuli,” J. Opt. Soc. Am. A 6, 1217–1227 (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 (2)

J. D. Victor, “Models for preattentive texture discrimination: Fourier analysis and local feature processing in a unified framework,” Spatial Vision 3, 263–280 (1988).
[Crossref] [PubMed]

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

1987 (1)

J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Computer Vision Graphics Image Process. 37, 299–325 (1987).
[Crossref]

1985 (1)

T. Caelli, “Three processing characteristics of visual texture segmentation,” Spatial Vision 1, 19–30 (1985).
[Crossref] [PubMed]

1984 (1)

R. Shapley, C. Enroth-Cugell, “Visual adaptation and retinal gain controls,” Prog. Retinal Res. 3, 263–346 (1984).
[Crossref]

1969 (1)

H. B. Barlow, W. R. Levick, “Three factors limiting the reliable detection of light by retinal ganglion cells of the cat,”J. Physiol. 200, 1–24 (1969).
[PubMed]

1965 (1)

H. B. Barlow, “Optic nerve impulses and Weber’s law,” Cold Spring Harbor Symp. Quant. Biol. 30, 539–546 (1965).
[Crossref]

Adelson, E. H.

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

Barlow, H. B.

H. B. Barlow, W. R. Levick, “Three factors limiting the reliable detection of light by retinal ganglion cells of the cat,”J. Physiol. 200, 1–24 (1969).
[PubMed]

H. B. Barlow, “Optic nerve impulses and Weber’s law,” Cold Spring Harbor Symp. Quant. Biol. 30, 539–546 (1965).
[Crossref]

Beck, J.

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]

J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Computer Vision Graphics Image Process. 37, 299–325 (1987).
[Crossref]

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 (London) 333, 363–364 (1988).
[Crossref]

J. R. Bergen, M. S. Landy, “Computational modeling of visual texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 252–271.

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

Brainard, D. H.

D. H. Brainard, “Calibration of a computer controlled color monitor,” Color Res. Appl. 14, 23–43 (1989).
[Crossref]

Caelli, T.

T. Caelli, “Three processing characteristics of visual texture segmentation,” Spatial Vision 1, 19–30 (1985).
[Crossref] [PubMed]

Chubb, C.

J. Econopouly, C. Chubb, M. S. Landy, “Segregation of 1st-order noise textures is perceptually one-dimensional,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 956 (1992).

J. Econopouly, C. Chubb, M. S. Landy, “Segregation of 1st-order noise textures,” Invest. Ophthalmol. Vis. Sci. Suppl. 32, 714 (1991).

C. Chubb, M. S. Landy, “Orthogonal distribution analysis: a systematic approach to the study of texture perception,” Invest. Ophthalmol. Vis. Sci. Suppl. 31, 561 (1990).

C. Chubb, M. S. Landy, “Orthogonal distribution analysis: a new approach to the study of texture perception,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 291–301.

De Valois, K. K.

R. L. De Valois, K. K. De Valois, Spatial Vision (Oxford U. Press, New York, 1988).

De Valois, R. L.

R. L. De Valois, K. K. De Valois, Spatial Vision (Oxford U. Press, New York, 1988).

Eckstein, M.

M. Eckstein, J. S. Whiting, J. P. Thomas, S. Shimozaki, “A novel temporal integration of intensity,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 17.

Econopouly, J.

J. Econopouly, C. Chubb, M. S. Landy, “Segregation of 1st-order noise textures is perceptually one-dimensional,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 956 (1992).

J. Econopouly, C. Chubb, M. S. Landy, “Segregation of 1st-order noise textures,” Invest. Ophthalmol. Vis. Sci. Suppl. 32, 714 (1991).

Enroth-Cugell, C.

R. Shapley, C. Enroth-Cugell, “Visual adaptation and retinal gain controls,” Prog. Retinal Res. 3, 263–346 (1984).
[Crossref]

Fechner, G. T.

G. T. Fechner, Elements of Psychophysics, D. H. Howes, E. C. Boring, eds., H. E. Adler, trans. (Holt, Reinhart & Winston, New York, 1966; originally published, 1860).

Fisher, R. C.

R. C. Fisher, Linear Algebra (Dickenson, Encino, Calif., 1970).

Fogel, I.

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

Graham, N.

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]

N. Graham, “Complex channels, early local nonlinearities, and normalization in texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 273–290.

N. Graham, Visual Pattern Analyzers (Oxford U. Press, New York, 1989).
[Crossref]

Granlund, G. H.

H. Knutsson, G. H. Granlund, “Texture analysis using two-dimensional quadrature filters,” presented at the IEEE Computer Society Workshop on Computer Architecture for Pattern Analysis and Image Database Management, Pasadena, Calif., October 12–14, 1983.

Hochstrasser, U. W.

U. W. Hochstrasser, “Orthogonal polynomials,” in Handbook of Mathematical Functions, Vol. 55 of Applied Mathematics Series, M. Abramowitz, I. A. Stegun, eds. (National Bureau of Standards, Washington, D.C., 1972), Chap. 22, pp. 771–802.

Ivry, R.

J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Computer Vision Graphics Image Process. 37, 299–325 (1987).
[Crossref]

Knutsson, H.

H. Knutsson, G. H. Granlund, “Texture analysis using two-dimensional quadrature filters,” presented at the IEEE Computer Society Workshop on Computer Architecture for Pattern Analysis and Image Database Management, Pasadena, Calif., October 12–14, 1983.

Landy, M. S.

J. Econopouly, C. Chubb, M. S. Landy, “Segregation of 1st-order noise textures is perceptually one-dimensional,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 956 (1992).

J. Econopouly, C. Chubb, M. S. Landy, “Segregation of 1st-order noise textures,” Invest. Ophthalmol. Vis. Sci. Suppl. 32, 714 (1991).

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

C. Chubb, M. S. Landy, “Orthogonal distribution analysis: a systematic approach to the study of texture perception,” Invest. Ophthalmol. Vis. Sci. Suppl. 31, 561 (1990).

C. Chubb, M. S. Landy, “Orthogonal distribution analysis: a new approach to the study of texture perception,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 291–301.

J. R. Bergen, M. S. Landy, “Computational modeling of visual texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 252–271.

Levick, W. R.

H. B. Barlow, W. R. Levick, “Three factors limiting the reliable detection of light by retinal ganglion cells of the cat,”J. Physiol. 200, 1–24 (1969).
[PubMed]

Mach, E.

E. Mach, The Analysis of Sensations and the Relation of the Physical to the Psychical (Dover, New York, 1959, with use of the fifth German edition revised and supplemented by S. Waterlow; first German edition 1886).

Malik, J.

Maloney, L. T.

L. T. Maloney, “Confidence intervals for the parameters of psychometric functions,” Percept. Psychophys. 47, 127–134 (1990).
[Crossref] [PubMed]

Müller, J.

J. Müller, “Of the senses,” in Visual Perception: the Nineteenth Century, W. N. Dember, ed. (Wiley, New York, 1964), pp. 35–69; recopied from the original English version published in Elements of Physiology, W. Baly, trans. (Taylor & Walton, London, 1942), Vol. 2.

Mulligan, J. B.

Perona, P.

Sagi, D.

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

Shapley, R.

R. Shapley, C. Enroth-Cugell, “Visual adaptation and retinal gain controls,” Prog. Retinal Res. 3, 263–346 (1984).
[Crossref]

Shimozaki, S.

M. Eckstein, J. S. Whiting, J. P. Thomas, S. Shimozaki, “A novel temporal integration of intensity,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 17.

Stone, L. S.

Sutter, A.

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]

J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Computer Vision Graphics Image Process. 37, 299–325 (1987).
[Crossref]

Thomas, J. P.

M. Eckstein, J. S. Whiting, J. P. Thomas, S. Shimozaki, “A novel temporal integration of intensity,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 17.

Victor, J. D.

J. D. Victor, “Models for preattentive texture discrimination: Fourier analysis and local feature processing in a unified framework,” Spatial Vision 3, 263–280 (1988).
[Crossref] [PubMed]

Whiting, J. S.

M. Eckstein, J. S. Whiting, J. P. Thomas, S. Shimozaki, “A novel temporal integration of intensity,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 17.

Biol. Cybern. (1)

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

Cold Spring Harbor Symp. Quant. Biol. (1)

H. B. Barlow, “Optic nerve impulses and Weber’s law,” Cold Spring Harbor Symp. Quant. Biol. 30, 539–546 (1965).
[Crossref]

Color Res. Appl. (1)

D. H. Brainard, “Calibration of a computer controlled color monitor,” Color Res. Appl. 14, 23–43 (1989).
[Crossref]

Computer Vision Graphics Image Process. (1)

J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Computer Vision Graphics Image Process. 37, 299–325 (1987).
[Crossref]

Invest. Ophthalmol. Vis. Sci. Suppl. (3)

C. Chubb, M. S. Landy, “Orthogonal distribution analysis: a systematic approach to the study of texture perception,” Invest. Ophthalmol. Vis. Sci. Suppl. 31, 561 (1990).

J. Econopouly, C. Chubb, M. S. Landy, “Segregation of 1st-order noise textures,” Invest. Ophthalmol. Vis. Sci. Suppl. 32, 714 (1991).

J. Econopouly, C. Chubb, M. S. Landy, “Segregation of 1st-order noise textures is perceptually one-dimensional,” Invest. Ophthalmol. Vis. Sci. Suppl. 33, 956 (1992).

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

J. Physiol. (1)

H. B. Barlow, W. R. Levick, “Three factors limiting the reliable detection of light by retinal ganglion cells of the cat,”J. Physiol. 200, 1–24 (1969).
[PubMed]

Nature (London) (1)

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

Percept. Psychophys. (2)

L. T. Maloney, “Confidence intervals for the parameters of psychometric functions,” Percept. Psychophys. 47, 127–134 (1990).
[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]

Prog. Retinal Res. (1)

R. Shapley, C. Enroth-Cugell, “Visual adaptation and retinal gain controls,” Prog. Retinal Res. 3, 263–346 (1984).
[Crossref]

Spatial Vision (2)

T. Caelli, “Three processing characteristics of visual texture segmentation,” Spatial Vision 1, 19–30 (1985).
[Crossref] [PubMed]

J. D. Victor, “Models for preattentive texture discrimination: Fourier analysis and local feature processing in a unified framework,” Spatial Vision 3, 263–280 (1988).
[Crossref] [PubMed]

Vision Res. (1)

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

Other (13)

N. Graham, “Complex channels, early local nonlinearities, and normalization in texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 273–290.

R. L. De Valois, K. K. De Valois, Spatial Vision (Oxford U. Press, New York, 1988).

N. Graham, Visual Pattern Analyzers (Oxford U. Press, New York, 1989).
[Crossref]

R. C. Fisher, Linear Algebra (Dickenson, Encino, Calif., 1970).

U. W. Hochstrasser, “Orthogonal polynomials,” in Handbook of Mathematical Functions, Vol. 55 of Applied Mathematics Series, M. Abramowitz, I. A. Stegun, eds. (National Bureau of Standards, Washington, D.C., 1972), Chap. 22, pp. 771–802.

J. Müller, “Of the senses,” in Visual Perception: the Nineteenth Century, W. N. Dember, ed. (Wiley, New York, 1964), pp. 35–69; recopied from the original English version published in Elements of Physiology, W. Baly, trans. (Taylor & Walton, London, 1942), Vol. 2.

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

J. R. Bergen, M. S. Landy, “Computational modeling of visual texture segregation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 252–271.

G. T. Fechner, Elements of Psychophysics, D. H. Howes, E. C. Boring, eds., H. E. Adler, trans. (Holt, Reinhart & Winston, New York, 1966; originally published, 1860).

E. Mach, The Analysis of Sensations and the Relation of the Physical to the Psychical (Dover, New York, 1959, with use of the fifth German edition revised and supplemented by S. Waterlow; first German edition 1886).

H. Knutsson, G. H. Granlund, “Texture analysis using two-dimensional quadrature filters,” presented at the IEEE Computer Society Workshop on Computer Architecture for Pattern Analysis and Image Database Management, Pasadena, Calif., October 12–14, 1983.

C. Chubb, M. S. Landy, “Orthogonal distribution analysis: a new approach to the study of texture perception,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991), pp. 291–301.

M. Eckstein, J. S. Whiting, J. P. Thomas, S. Shimozaki, “A novel temporal integration of intensity,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 17.

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Figures (10)

Fig. 1
Fig. 1

IID textures for a micropattern set Γ of small gray squares of various luminances. (a) the micropattern set Γ; (b) IU for U, the uniform distribution (in this and subsequent panels the texel distribution is shown in an inset); (c) IU+λ1; (d) IUλ1; (e) IU+λ2;(f) IUλ2; (g) IU+λ3;(h) IUλ3; (i) IU+λ4; (j) IUλ4. The set of modulators λi is defined in Subsection 3.A. The visible distinctions between (g) and (h) and between (i) and (j) are for texture pairs with equal mean and variance. Hence such distinctions cannot be drawn simply on the basis of average luminance or contrast.

Fig. 2
Fig. 2

(a) Landy–Bergen10,14 model, which is an example of the back-pocket model of texture segregation. The input image is processed by a number of linear filters varying in dominant orientation and spatial frequency. Within a given scale each oriented filter output is squared (computing a texture energy), orthogonal orientations are put in opponency, and these opponent responses are divided by the sum of all four oriented energies as a form of contrast gain control. The intent is to convert an input textural difference into a difference in intensity of response. Finally, an edge detector is applied to localize such intensity edges. (b) For the textures illustrated in Figs. 1(g)–1(j), this much simpler model may suffice. It corresponds to a single channel of the model in (a) with no up-front linear filtering and a single pointwise nonlinearity.

Fig. 3
Fig. 3

Histogram contrast threshold. The figure illustrates a series of IID texture pairs with increasing amplitude of modulation of the probability distributions used to generate the textures. Here a third-order polynomial λ3 was used as the modulator of the uniform distribution U [as in Figs. 1(g) and 1(h)]. The histogram contrast threshold is the amount of modulation that results in threshold segregation performance (e.g., 75% correct on a forced-choice discrimination task).

Fig. 4
Fig. 4

Method of titration. Points in this plane represent IID texture pairs that are generated by mixing two maximal modulators ϕ and ρ of the base distribution p. A point (x, y) in this plane corresponds to the discrimination of Ip++ from Ipxρyϕ. If the measure used by the subject for this space of textures is one dimensional, then the locus of points corresponding to a given performance level is a straight line. The point (0, Aϕ) corresponds to a direct measurement of the histogram contrast threshold of ϕ. Here Aρ is not directly measurable. Instead, we perform experiments along the two indicated 45-deg lines, which correspond to a mixing of ϕ with ±ρ to determine the proportions π+ and π of ±ρ that must be mixed with ϕ for threshold performance to be achieved. A line fitted through these three empirically measured points is then extrapolated to its x intercept. The absolute value of the x intercept is an estimate of Aρ. A negative titration-line slope indicates that Aϕ and Aρ have the same sign, and a positive slope indicates opposite signs.

Fig. 5
Fig. 5

Back-pocket model and IID textures. (a) An IID texture Ip and its gray-level histogram, (b) the impulse response of a typical linear filter used in the modeling of early vision, (c) the resulting filtered texture and its intensity histogram. Note that the filtered texture-intensity histogram is approximately Gaussian and thus is determined only by the mean and the variance of Ip (and by the characteristics of the filter).

Fig. 6
Fig. 6

Spatial filtering that preserves information about the moments of IID texture histograms other than mean and variance. (a) Patches of IID texture IUλ3 and IU+λ3. The bar graphs enclosed within the patches of texture are the intensity histograms of the patches. These histograms approximate the texel distributions IUλ3 and IU+λ3 (which have equal mean and variance). (b) The receptive field F of a linear filter capable of preserving information about the higher moments (other than mean and variance) of the histograms of IID textures of Γ. This receptive field has an excitatory center exactly the size of a micropattern of Γ and an annular inhibitory surround covering a region equal in size to the ring of texels surrounding a given texel. If the excitatory center of a neuron with such a receptive field is aligned with a texel τ, then the output of that neuron will be dominated by the intensity of τ. (c) The result of convolving the IID texture field shown in (a) with receptive field F. Enclosed bar graphs are intensity histograms of left-hand and right-hand filtered textures, F * IUλ3 and F * IU+λ3. (d) The intensity histogram (enlarged) of the left-hand patch of filtered texture, F * IUλ3. (e) The intensity histogram (enlarged) of the right-hand patch of filtered texture, F * IU+λ3. (f) The difference between the two histograms shown in (d) and (e). Note that the form of the modulator λ3 is preserved quite well in the difference between the histograms of the filtered textures. Thus the particular isotropic bandpass filter whose receptive field is shown in (b) does a good job of preserving the histogram information in the original IID texture to which it is applied.

Fig. 7
Fig. 7

Example stimuli: (a) a vertical IID texture square wave modulating between IU+λ3 and IUλ3, (b) a horizontal IID texture square wave modulating between IU+λ4 and IUλ4. The black border was not present in the experiments; rather, the stimulus was surrounded by a gray field with the same luminance as the mean luminance of the textures.

Fig. 8
Fig. 8

Titration results for subject JE. Panels correspond to different titrations of each of λ3, λ4, and λ5 with each of λ6 and λ7. The number of trials per point and other statistics of interest are tabulated in Table 3. Intervals around the data points are 95% nonparametric confidence intervals. A more comprehensive statistic, p, refers to the ability of a straight line to model the data as a whole. The intervals around the x intercepts are standard deviations. (See text for details.)

Fig. 9
Fig. 9

Titration results for subject CC. Format as in Fig. 8.

Fig. 10
Fig. 10

Reconstructions of fS for subjects (a) JE and (b) CC. These reconstructions include neither of components λ1 and λ2; these components cannot be measured by use of IID textures with equal mean and variance because measurement for such textures is no longer one dimensional (that is, the space of all IID textures is not perceptually elementary).

Tables (4)

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Table 1 Histogram Contrast Thresholds for Subject JE

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Table 2 Histogram Contrast Thresholds for Subject CC

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Table 3 Titrations for Subject JE

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Table 4 Titrations for Subject CC

Equations (64)

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m ( the greater value of v ) - m ( the lesser value of v ) threshold
C S ( x , y ) = S ( x , y ) - local average S ( x , y ) local average S ( x , y ) ,
C S ( x , y ) = S ( x , y ) - μ μ .
C S ( x , y ) = v - μ μ .
w - v v = threshold ,
f · g = ω Ω f ( ω ) g ( ω ) .
E [ f ( X p ) ] = ω Ω f ( ω ) p ( ω ) = f · p ,
var [ f ( X p ) ] = E [ f 2 ( X p ) ] - E 2 [ f ( X p ) ] = f 2 · p - ( f · p ) 2 ,
J g = - - g ( x , y ) J ( x , y ) d x d y .
1 g = - - g ( x , y ) d x d y
g g = - - g ( x , y ) 2 d x d y
TED ( J ) = T ( J ) g excit - T ( J ) g inhib .
d T ( ω ) = - - [ T ( ω alone ) ] ( x , y ) d x d y             ( ω Ω ) ,
TED ( J ) = d T J g excit - d T J g inhib ,
q = b B α a b .
q = b B α b b .
f = i = 0 N - 1 W i ϕ i .
W i = f · ϕ i ϕ i · ϕ i             ( i = 0 , 1 , , N - 1 ) .
TED ( J ) = T ( J ) g excit - T ( J ) g inhib
TED ( J p , q ) = T ( I p ) g excit - T ( I q ) g inhib = d T I p g excit - d T I q g inhib .
d ( J p , q ) = E [ TED ( J p , q ) ] { var [ TED ( J p , q ) ] } 1 / 2 = E [ d T I p g excit - d T I q g inhib ] ( var [ d T I p g excit - d T I q g inhib ] ) 1 / 2 .
d S ( p , q ) = E [ f I p g excit - f I q g inhib ] ( var [ f I p g excit - f I q g inhib ] ) 1 / 2 .
E [ f I p + ϕ g excit ] = K 1 E [ f ( any texel of I p + ϕ ) ] = K 1 E [ f ( X p + ϕ ) ] ,
var [ f I p + ϕ g excit ] = K 2 var [ f ( any texel of I p + ϕ ) ] = K 2 var [ f ( X p + ϕ ) ] .
E [ f I p + ϕ g inhib ] = K 1 E [ f ( X p - ϕ ) ] ,
var [ f I p + ϕ g inhib ] = K 2 var [ f ( X p - ϕ ) ] .
d S ( p + ϕ , p - ϕ ) = E [ f I p + ϕ g excit - f I p - ϕ g inhib ] ( var [ f I p + ϕ g excit - f I p - ϕ g inhib ] ) 1 / 2 = K E [ f ( X p + ϕ ) - f ( X p - ϕ ) ] { var [ f ( X p + ϕ ) - f ( X p - ϕ ) ] } 1 / 2
K = K 1 / K 2 .
E [ f ( X p + ϕ ) - f ( X p - ϕ ) ] { var [ f ( X p + ϕ ) - f ( X p - ϕ ) ] } 1 / 2 = R .
( f · ϕ ) 2 = R 2 var [ f ( X p ) ] R 2 + 2 .
var [ f ( X p + ϕ ) - f ( X p - ϕ ) ] = 2 var [ f ( X p ) ] - 2 ( f · ϕ ) 2
var [ f ( X p + ϕ ) - f ( X p - ϕ ) ] = var [ f ( X p + ϕ ) ] + var [ f ( X p - ϕ ) ] = f 2 · ( p + ϕ ) - [ f · ( p + ϕ ) ] 2 + f 2 · ( p - ϕ ) - [ f · ( p - ϕ ) ] 2 = f 2 · p + f 2 · ϕ - [ ( f · p ) 2 + 2 ( f · p ) ( f · ϕ ) + ( f · ϕ ) 2 ] + f 2 · p - f 2 · ϕ - [ ( f · p ) 2 - 2 ( f · p ) ( f · ϕ ) + ( f · ϕ ) 2 ] = 2 f 2 · p - 2 [ ( f · p ) 2 + ( f · ϕ ) 2 ] = 2 var [ f ( X p ) ] - 2 ( f · ϕ ) 2 .
E [ f ( X p + ϕ ) - f ( X p - ϕ ) ] = E [ f ( X p + ϕ ) ] - E [ f ( X p - ϕ ) ] = f · p + f · ϕ - ( f · p - f · ϕ ) = 2 ( f · ϕ ) .
2 f · ϕ { 2 var [ f ( X p ) ] - 2 ( f · ϕ ) 2 } 1 / 2 = R .
d S ( p + A ϕ , p - A ϕ ) = E [ f I p + A ϕ g excit - f I p - A ϕ g inhib ] ( var [ f I p + A ϕ g excit - f I p - A ϕ g inhib ] ) 1 / 2 = R α .
E [ f ( X p + A ϕ ) - f ( X p - A ϕ ) ] { var [ f ( X p + A ϕ ) - f ( X p - A ϕ ) ] } 1 / 2 = R α K .
( f · A ϕ ) 2 = R α 2 var [ f ( X p ) ] R α 2 + 2 K 2 .
f · ϕ = A - 1 Ψ α ( p )
Ψ α ( p ) = [ R α 2 var [ f ( X p ) ] R α 2 + 2 K 2 ] 1 / 2 .
f · ϕ i = Ψ ( p ) A i ,             i = 1 , 2 , , N ,
f · ϕ i = ± Ψ ( p ) A i             i = 1 , 2 , , N .
f S = i = 1 N f · ϕ i ϕ i · ϕ i ϕ i .
( a )             f S · ( A i ϕ i + A j ϕ j ) = 2 Ψ ( p ) , ( b )             f S · ( A i ϕ i - A j ϕ j ) = 0 ,
( a )             f S · ( A i ϕ i + A j ϕ j ) = 0 , ( b )             f S · ( A i ϕ i - A j ϕ j ) = 2 Ψ ( p ) .
I p + ϕ = I q ,             I p - ϕ = I r .
f S · ( x ρ + y ϕ ) = A - 1 Ψ ( p )
f S · ( x ρ + y ϕ ) = ± A - 1 Ψ ( p ) ,
y = ± A Ψ ( p ) - f S · ρ x f S · ϕ .
y = Ψ ( p ) - f S · ρ x f S · ϕ ,
y = - Ψ ( p ) - f S · ρ x f S · ϕ .
f S · ρ = Ψ ( p ) / x 0 .
I p + π ρ + ( 1 - π ) ϕ ( τ ) = { I p + ρ ( τ ) with probability π I p + ϕ ( τ ) otherwise .
ζ 0 ( v ) = 1 , ζ 1 ( v ) = v , ζ 2 ( v ) = v 2 .
f S = i = 3 7 f S · λ i λ i · λ i λ i .
E [ g I U + ϕ ] = ( h + f S ) · ( U + ϕ ) = ( h + f S ) · U + f S · ϕ ,
E [ g I U - ϕ ] = ( h + f S ) · ( U - ϕ ) = ( h + f S ) · U - f S · ϕ .
ζ 1 ( i ) = i ,             ξ 2 ( i ) = i 2 ,
E [ X p ] = i = 0 16 i p ( i ) = ζ 1 · p ,
var [ X p ] = E [ X p 2 ] - E 2 [ X p ] = i = 0 16 i 2 p ( i ) - [ i = 0 16 i p ( i ) ] 2 = ζ 2 · p - [ ζ 1 · p ] 2 .
ζ 1 · ϕ = ζ 1 · ( q - p ) = ζ 1 · q - ζ 1 · p = E [ X q ] - E [ X p ] .
ζ 2 · ϕ = ζ 2 · ( q - p ) = ζ 2 · q - ζ 2 · p = E [ X q 2 ] - E [ X p 2 ] .
E [ X q 2 ] - E 2 [ X q ] - ( E [ X p 2 ] - E 2 [ X p ] ) = var [ X p ] - var [ X q ] = 0 ,
E [ X p + ϕ ] = ξ 1 · ( p + ϕ ) = ζ 1 · p + ζ 1 · ϕ = ζ 1 · p = E [ X p ] ,
var [ X p + ϕ ] = ζ 2 · ( p + ϕ ) - [ ζ 1 · ( p + ϕ ) ] 2 = ζ 2 · p + ζ 2 · ϕ - ( ζ 1 · p + ζ 1 · ϕ ) 2 = ζ 2 · p - ( ζ 1 · p ) 2 = var [ X p ] .

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