Abstract

The major cue to shape from texture is the compression of texture as a function of surface curvature [ J. Exp. Psychol. 13, 242 ( 1987); Vision Res. 33, 827 ( 1993)]. A number of computational models have been proposed in which compression is measured by detection of changes in the spatial-frequency spectrum [ Comput. Graphics Image Process. 5, 52 ( 1976)]. We propose that the visual system uses a strategy of characterizing the frequency spectrum by a simple set of measures and of tracking the changes in this characterization rather than determining changes in the shape of the actual spectra. Our evidence is based on a number of psychophysical demonstrations that use stimuli with specifically tailored frequency spectra, constructed from white noise filtered in the frequency domain. Our evidence suggests that the visual system determines the average peak frequency of the spectrum and uses this measure as its characterization. Changes in fp¯ are strongly correlated with the degree of surface curvature, and, over a range of stimuli, fp¯ takes account of the variance in local estimates of the frequency spectrum. One computes f¯p by determining the peak frequency at each spatial location and then averaging these frequency values over a local spatial region. We show that f¯p is related to the second-order moment but is more biologically plausible and shows superior ability to function in the presence of noise. As a test of this model, we have constructed a neural network architecture for computing shape from texture. Our model is limited to orthographically projected, homogeneous textures without in-surface rotation. The early stages of the model consist of multiple simple-cell units tuned to different orientations and spatial frequencies. We show that these simple cells are inadequate for the determination of compression but that the outputs of complex-cell-like units after normalization generate estimates of surface slant and tilt. The network shows qualitative agreement with human perception of shape from texture over a wide range of real and artificial stimuli.

© 1995 Optical Society of America

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  26. R. G. Szulborski, L. A. Palmer, “The two-dimensional spatial structure of nonlinear subunits in the receptive fields of complex cells,” Vision Res. 30, 249–254 (1990).
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  27. H. R. Wilson, W. A. Richards, “Curvature and separation discrimination at texture boundaries,” J. Opt. Soc. Am. A 9, 1653–1662 (1992).
    [Crossref] [PubMed]
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  31. S. R. Lehky, T. J. Sejnowski, R. Desimone, “Predicting responses of nonlinear neurons in monkey striate cortex to complex patterns,” J. Neurosci. 12, 3568–3581 (1992).
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  35. G. C. DeAngelis, J. G. Robson, I. Ohzawa, R. D. Freeman, “The organization of suppression in receptive fields of neurons in the cat’s visual cortex,” J. Neurophysiol. 68, 1440–1463 (1992).
  36. Z. F. Kisvarday, U. T. Eysel, “Functional and structural topography of horizontal inhibitory connections in cat visual cortex,” Eur. J. Neurosci. 5, 1558–1572 (1993).
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    [Crossref]
  39. P. Sajda, K. Sakai, S-C. Yen, L. H. Finkel, “NEXUS: a neural simulator for integrating top-down and bottom-up modeling,” in Neural Network Simulation Environments, J. Skrzypek, ed. (Kluwer, Norwell, Mass., 1994).
    [Crossref]
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  41. A. Blake, C. Marinos, “Shape from texture: estimation, isotropy and moment,” Artif. Intell. 45, 323–380 (1990).
    [Crossref]
  42. C. Blakemore, E. A. Tobin, “Lateral inhibition between orientation detectors in the cat’s visual cortex,” Exp. Brain Res. 15, 439–440 (1972).
    [Crossref]
  43. M. C. Morrone, D. C. Burr, L. Maffei, “Functional implications of cross-orientation inhibition of cortical visual cells,” Proc. R. Soc. London Ser. B 249, 335–354 (1982).
    [Crossref]
  44. A. M. Sillito, T. E. Salt, J. A. Kemp, “Modulatory and inhibitory processes in the visual cortex,” Vision Res. 25, 375–381 (1985).
    [Crossref] [PubMed]
  45. B. Julesz, “Textons, the elements of texture perception and their interactions,” Nature (London) 290, 91–97 (1981).
    [Crossref]
  46. R. Shapley, C. Enroth-Cugell, “Visual adaptation and retinal gain control,” Prog. Retinal Res. 3, 263–346 (1984).
    [Crossref]
  47. H. R. Wilson, “Nonlinear processes in visual pattern discrimination,” Proc. Natl. Acad. Sci. USA 90, 9785–9790 (1993).
    [Crossref] [PubMed]
  48. D. J. Heeger, “Nonlinear model of neural responses in cat visual cortex,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991).
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    [Crossref]
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1995 (1)

S. M. Courtney, L. H. Finkel, G. Buchsbaum, “Network simulations of retinal and cortical contributions to color constancy,” Vision Res. 35, 413–434 (1995).
[Crossref] [PubMed]

1993 (4)

Z. F. Kisvarday, U. T. Eysel, “Functional and structural topography of horizontal inhibitory connections in cat visual cortex,” Eur. J. Neurosci. 5, 1558–1572 (1993).
[Crossref] [PubMed]

B. G. Cumming, E. B. Johnston, A. J. Parker, “Effects of different texture cues on curved surfaces viewed stereoscopically,” Vision Res. 33, 827–838 (1993).
[Crossref] [PubMed]

J. S. Mansfield, A. J. Parker, “An orientation-tuned component in the contrast masking of stereopsis,” Vision Res. 33, 1535–1544 (1993).
[Crossref] [PubMed]

H. R. Wilson, “Nonlinear processes in visual pattern discrimination,” Proc. Natl. Acad. Sci. USA 90, 9785–9790 (1993).
[Crossref] [PubMed]

1992 (7)

S. Nishida, T. Sato, “Positive motion after-effect induced by bandpass-filtered random-dot kinematograms,” Vision Res. 32, 1635–1646 (1992).
[Crossref] [PubMed]

J. Gårding, “Shape from texture for smooth curved surfaces in perspective projection,” J. Math. Imag. Vis. 2, 329–352 (1992).
[Crossref]

P. Sajda, L. H. Finkel, “NEXUS: a simulation environment for large-scale neural systems,” Simulation 59, 358–364 (1992).
[Crossref]

G. C. DeAngelis, J. G. Robson, I. Ohzawa, R. D. Freeman, “The organization of suppression in receptive fields of neurons in the cat’s visual cortex,” J. Neurophysiol. 68, 1440–1463 (1992).

S. R. Lehky, T. J. Sejnowski, R. Desimone, “Predicting responses of nonlinear neurons in monkey striate cortex to complex patterns,” J. Neurosci. 12, 3568–3581 (1992).
[PubMed]

H. R. Wilson, W. A. Richards, “Curvature and separation discrimination at texture boundaries,” J. Opt. Soc. Am. A 9, 1653–1662 (1992).
[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]

1991 (1)

M. R. Turner, G. L. Gerstein, R. Bajcsy, “Underestimation of visual texture slant by human observers: a model,” Biol. Cybern. 65, 215–226 (1991).
[Crossref] [PubMed]

1990 (4)

R. G. Szulborski, L. A. Palmer, “The two-dimensional spatial structure of nonlinear subunits in the receptive fields of complex cells,” Vision Res. 30, 249–254 (1990).
[Crossref] [PubMed]

A. Blake, C. Marinos, “Shape from texture: estimation, isotropy and moment,” Artif. Intell. 45, 323–380 (1990).
[Crossref]

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

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

1989 (2)

D. Blostein, N. Ahuja, “Shape from texture: integrating texture element extraction and surface estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 1233–1251 (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 (1)

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

1987 (1)

J. T. Todd, R. A. Akerstrom, “Perception of three-dimensional form from patterns of optical texture,” J. Exp. Psychol. 13, 242–255 (1987).

1985 (1)

A. M. Sillito, T. E. Salt, J. A. Kemp, “Modulatory and inhibitory processes in the visual cortex,” Vision Res. 25, 375–381 (1985).
[Crossref] [PubMed]

1984 (3)

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

K. Kanatani, “Detection of surface orientation and motion from texture by a stereological technique,” Artif. Intell. 23, 213–237 (1984).
[Crossref]

J. E. Cutting, R. T. Millard, “Three gradients and the perception of flat and curved surfaces,” J. Exp. Psychol. 113, 198–216 (1984).
[Crossref]

1983 (1)

K. A. Stevens, “Slant-tilt: the visual encoding of surface orientation,” Biol. Cybern. 46, 183–195 (1983).
[Crossref] [PubMed]

1982 (1)

M. C. Morrone, D. C. Burr, L. Maffei, “Functional implications of cross-orientation inhibition of cortical visual cells,” Proc. R. Soc. London Ser. B 249, 335–354 (1982).
[Crossref]

1981 (3)

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

A. P. Witkin, “Recovering surface shape and orientation from texture,” Artif. Intell. 17, 17–47 (1981).
[Crossref]

K. A. Stevens, “The information content of texture gradients,” Biol. Cybern. 42, 95–105 (1981).
[Crossref] [PubMed]

1980 (1)

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[Crossref]

1979 (1)

H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
[Crossref]

1978 (1)

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat striate cortex,” J. Physiol. (London) 283, 79–99 (1978).

1977 (1)

P. Hammond, D. M. MacKey, “Differential responsiveness of simple and complex cells in cat striate cortex to visual texture,” Exp. Brain Res. 30, 275–296 (1977).
[PubMed]

1976 (1)

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

1972 (1)

C. Blakemore, E. A. Tobin, “Lateral inhibition between orientation detectors in the cat’s visual cortex,” Exp. Brain Res. 15, 439–440 (1972).
[Crossref]

1969 (1)

M. L. Braunstein, J. W. Payne, “Perspective and form ratio as determinants of relative slant judgments,” J. Exp. Psychol. 81, 584–590 (1969).
[Crossref]

1968 (1)

D. H. Hubel, T. N. Wiesel, “Receptive fields and functional architecture of monkey striate cortex,” J. Physiol. (London) 195, 215–243 (1968).

Adelson, E. H.

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

Ahuja, N.

D. Blostein, N. Ahuja, “Shape from texture: integrating texture element extraction and surface estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 1233–1251 (1989).
[Crossref]

Akerstrom, R. A.

J. T. Todd, R. A. Akerstrom, “Perception of three-dimensional form from patterns of optical texture,” J. Exp. Psychol. 13, 242–255 (1987).

Anderson, A.

M. L. J. Crawford, A. Anderson, R. Blake, G. H. Jacobs, C. Neumeyer, “Interspecies comparisons in the understanding of human visual perception,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Bajcsy, R.

M. R. Turner, G. L. Gerstein, R. Bajcsy, “Underestimation of visual texture slant by human observers: a model,” Biol. Cybern. 65, 215–226 (1991).
[Crossref] [PubMed]

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

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.

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

H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
[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).

Blake, A.

A. Blake, C. Marinos, “Shape from texture: estimation, isotropy and moment,” Artif. Intell. 45, 323–380 (1990).
[Crossref]

Blake, R.

M. L. J. Crawford, A. Anderson, R. Blake, G. H. Jacobs, C. Neumeyer, “Interspecies comparisons in the understanding of human visual perception,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Blakemore, C.

C. Blakemore, E. A. Tobin, “Lateral inhibition between orientation detectors in the cat’s visual cortex,” Exp. Brain Res. 15, 439–440 (1972).
[Crossref]

Blostein, D.

D. Blostein, N. Ahuja, “Shape from texture: integrating texture element extraction and surface estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 1233–1251 (1989).
[Crossref]

Bovik, A. C.

B. J. Super, A. C. Bovik, “Shape-from-texture by wavelet-based measurement of local spectral moments,” in Proceedings of the Computer Vision and Pattern Recognition Meeting 1992 (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 296–300.
[Crossref]

Braunstein, M. L.

M. L. Braunstein, J. W. Payne, “Perspective and form ratio as determinants of relative slant judgments,” J. Exp. Psychol. 81, 584–590 (1969).
[Crossref]

Brown, L. G.

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

Buchsbaum, G.

S. M. Courtney, L. H. Finkel, G. Buchsbaum, “Network simulations of retinal and cortical contributions to color constancy,” Vision Res. 35, 413–434 (1995).
[Crossref] [PubMed]

Bülthoff, H. H.

H. H. Bülthoff, “Shape from X: psychophysics and computation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991).

Burr, D. C.

M. C. Morrone, D. C. Burr, L. Maffei, “Functional implications of cross-orientation inhibition of cortical visual cells,” Proc. R. Soc. London Ser. B 249, 335–354 (1982).
[Crossref]

Courtney, S. M.

S. M. Courtney, L. H. Finkel, G. Buchsbaum, “Network simulations of retinal and cortical contributions to color constancy,” Vision Res. 35, 413–434 (1995).
[Crossref] [PubMed]

Crawford, M. L. J.

M. L. J. Crawford, A. Anderson, R. Blake, G. H. Jacobs, C. Neumeyer, “Interspecies comparisons in the understanding of human visual perception,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Cumming, B. G.

B. G. Cumming, E. B. Johnston, A. J. Parker, “Effects of different texture cues on curved surfaces viewed stereoscopically,” Vision Res. 33, 827–838 (1993).
[Crossref] [PubMed]

Cutting, J. E.

J. E. Cutting, R. T. Millard, “Three gradients and the perception of flat and curved surfaces,” J. Exp. Psychol. 113, 198–216 (1984).
[Crossref]

DeAngelis, G. C.

G. C. DeAngelis, J. G. Robson, I. Ohzawa, R. D. Freeman, “The organization of suppression in receptive fields of neurons in the cat’s visual cortex,” J. Neurophysiol. 68, 1440–1463 (1992).

Desimone, R.

S. R. Lehky, T. J. Sejnowski, R. Desimone, “Predicting responses of nonlinear neurons in monkey striate cortex to complex patterns,” J. Neurosci. 12, 3568–3581 (1992).
[PubMed]

DeValois, R.

H. R. Wilson, D. Levi, L. Maffei, J. Rovamo, R. DeValois, “The perception of form in visual perception,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Enroth-Cugell, C.

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

J. Walraven, C. Enroth-Cugell, “The control of visual sensitivity: receptoral and postreceptoral processes,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Eysel, U. T.

Z. F. Kisvarday, U. T. Eysel, “Functional and structural topography of horizontal inhibitory connections in cat visual cortex,” Eur. J. Neurosci. 5, 1558–1572 (1993).
[Crossref] [PubMed]

Finkel, L. H.

S. M. Courtney, L. H. Finkel, G. Buchsbaum, “Network simulations of retinal and cortical contributions to color constancy,” Vision Res. 35, 413–434 (1995).
[Crossref] [PubMed]

P. Sajda, L. H. Finkel, “NEXUS: a simulation environment for large-scale neural systems,” Simulation 59, 358–364 (1992).
[Crossref]

P. Sajda, K. Sakai, L. H. Finkel, “NEXUS: a tool for simulating large-scale hybrid neural networks,” in Proceedings of the Summer Computer Simulation Conference 1992Society for Computer Simulation, San Diego, Calif., 1992), pp. 72–76.

P. Sajda, K. Sakai, S-C. Yen, L. H. Finkel, “NEXUS: a neural simulator for integrating top-down and bottom-up modeling,” in Neural Network Simulation Environments, J. Skrzypek, ed. (Kluwer, Norwell, Mass., 1994).
[Crossref]

Freeman, R. D.

G. C. DeAngelis, J. G. Robson, I. Ohzawa, R. D. Freeman, “The organization of suppression in receptive fields of neurons in the cat’s visual cortex,” J. Neurophysiol. 68, 1440–1463 (1992).

Gårding, J.

J. Gårding, “Shape from texture for smooth curved surfaces in perspective projection,” J. Math. Imag. Vis. 2, 329–352 (1992).
[Crossref]

Gerstein, G. L.

M. R. Turner, G. L. Gerstein, R. Bajcsy, “Underestimation of visual texture slant by human observers: a model,” Biol. Cybern. 65, 215–226 (1991).
[Crossref] [PubMed]

Gibson, J. J.

J. J. Gibson, The Perception of the Visual World (Houghton Mifflin, Boston, Mass., 1950).

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]

Hammond, P.

P. Hammond, D. M. MacKey, “Differential responsiveness of simple and complex cells in cat striate cortex to visual texture,” Exp. Brain Res. 30, 275–296 (1977).
[PubMed]

Heeger, D. J.

D. J. Heeger, “Nonlinear model of neural responses in cat visual cortex,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991).

Hildreth, E.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[Crossref]

Hubel, D. H.

D. H. Hubel, T. N. Wiesel, “Receptive fields and functional architecture of monkey striate cortex,” J. Physiol. (London) 195, 215–243 (1968).

Jacobs, G. H.

M. L. J. Crawford, A. Anderson, R. Blake, G. H. Jacobs, C. Neumeyer, “Interspecies comparisons in the understanding of human visual perception,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Johnston, E. B.

B. G. Cumming, E. B. Johnston, A. J. Parker, “Effects of different texture cues on curved surfaces viewed stereoscopically,” Vision Res. 33, 827–838 (1993).
[Crossref] [PubMed]

Julesz, B.

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

Kanatani, K.

K. Kanatani, “Detection of surface orientation and motion from texture by a stereological technique,” Artif. Intell. 23, 213–237 (1984).
[Crossref]

Kemp, J. A.

A. M. Sillito, T. E. Salt, J. A. Kemp, “Modulatory and inhibitory processes in the visual cortex,” Vision Res. 25, 375–381 (1985).
[Crossref] [PubMed]

Kisvarday, Z. F.

Z. F. Kisvarday, U. T. Eysel, “Functional and structural topography of horizontal inhibitory connections in cat visual cortex,” Eur. J. Neurosci. 5, 1558–1572 (1993).
[Crossref] [PubMed]

Krumm, J.

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

Landy, M. S.

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).

Lehky, S. R.

S. R. Lehky, T. J. Sejnowski, R. Desimone, “Predicting responses of nonlinear neurons in monkey striate cortex to complex patterns,” J. Neurosci. 12, 3568–3581 (1992).
[PubMed]

Lennie, P.

P. Lennie, C. Trevarthen, D. Van Essen, H. Wassle, “Parallel processing of visual information,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Levi, D.

H. R. Wilson, D. Levi, L. Maffei, J. Rovamo, R. DeValois, “The perception of form in visual perception,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Lieberman, L.

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

MacKey, D. M.

P. Hammond, D. M. MacKey, “Differential responsiveness of simple and complex cells in cat striate cortex to visual texture,” Exp. Brain Res. 30, 275–296 (1977).
[PubMed]

Maffei, L.

M. C. Morrone, D. C. Burr, L. Maffei, “Functional implications of cross-orientation inhibition of cortical visual cells,” Proc. R. Soc. London Ser. B 249, 335–354 (1982).
[Crossref]

H. R. Wilson, D. Levi, L. Maffei, J. Rovamo, R. DeValois, “The perception of form in visual perception,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Malik, J.

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

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

Mansfield, J. S.

J. S. Mansfield, A. J. Parker, “An orientation-tuned component in the contrast masking of stereopsis,” Vision Res. 33, 1535–1544 (1993).
[Crossref] [PubMed]

Marinos, C.

A. Blake, C. Marinos, “Shape from texture: estimation, isotropy and moment,” Artif. Intell. 45, 323–380 (1990).
[Crossref]

Marr, D.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[Crossref]

Millard, R. T.

J. E. Cutting, R. T. Millard, “Three gradients and the perception of flat and curved surfaces,” J. Exp. Psychol. 113, 198–216 (1984).
[Crossref]

Morrone, M. C.

M. C. Morrone, D. C. Burr, L. Maffei, “Functional implications of cross-orientation inhibition of cortical visual cells,” Proc. R. Soc. London Ser. B 249, 335–354 (1982).
[Crossref]

Movshon, J. A.

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat striate cortex,” J. Physiol. (London) 283, 79–99 (1978).

Neumeyer, C.

M. L. J. Crawford, A. Anderson, R. Blake, G. H. Jacobs, C. Neumeyer, “Interspecies comparisons in the understanding of human visual perception,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Nishida, S.

S. Nishida, T. Sato, “Positive motion after-effect induced by bandpass-filtered random-dot kinematograms,” Vision Res. 32, 1635–1646 (1992).
[Crossref] [PubMed]

Ohzawa, I.

G. C. DeAngelis, J. G. Robson, I. Ohzawa, R. D. Freeman, “The organization of suppression in receptive fields of neurons in the cat’s visual cortex,” J. Neurophysiol. 68, 1440–1463 (1992).

Palmer, L. A.

R. G. Szulborski, L. A. Palmer, “The two-dimensional spatial structure of nonlinear subunits in the receptive fields of complex cells,” Vision Res. 30, 249–254 (1990).
[Crossref] [PubMed]

Parker, A. J.

J. S. Mansfield, A. J. Parker, “An orientation-tuned component in the contrast masking of stereopsis,” Vision Res. 33, 1535–1544 (1993).
[Crossref] [PubMed]

B. G. Cumming, E. B. Johnston, A. J. Parker, “Effects of different texture cues on curved surfaces viewed stereoscopically,” Vision Res. 33, 827–838 (1993).
[Crossref] [PubMed]

Payne, J. W.

M. L. Braunstein, J. W. Payne, “Perspective and form ratio as determinants of relative slant judgments,” J. Exp. Psychol. 81, 584–590 (1969).
[Crossref]

Perona, P.

Richards, W. A.

Robson, J. G.

G. C. DeAngelis, J. G. Robson, I. Ohzawa, R. D. Freeman, “The organization of suppression in receptive fields of neurons in the cat’s visual cortex,” J. Neurophysiol. 68, 1440–1463 (1992).

Rosenholtz, R.

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

Rovamo, J.

H. R. Wilson, D. Levi, L. Maffei, J. Rovamo, R. DeValois, “The perception of form in visual perception,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Sajda, P.

P. Sajda, L. H. Finkel, “NEXUS: a simulation environment for large-scale neural systems,” Simulation 59, 358–364 (1992).
[Crossref]

P. Sajda, K. Sakai, L. H. Finkel, “NEXUS: a tool for simulating large-scale hybrid neural networks,” in Proceedings of the Summer Computer Simulation Conference 1992Society for Computer Simulation, San Diego, Calif., 1992), pp. 72–76.

P. Sajda, K. Sakai, S-C. Yen, L. H. Finkel, “NEXUS: a neural simulator for integrating top-down and bottom-up modeling,” in Neural Network Simulation Environments, J. Skrzypek, ed. (Kluwer, Norwell, Mass., 1994).
[Crossref]

Sakai, K.

P. Sajda, K. Sakai, S-C. Yen, L. H. Finkel, “NEXUS: a neural simulator for integrating top-down and bottom-up modeling,” in Neural Network Simulation Environments, J. Skrzypek, ed. (Kluwer, Norwell, Mass., 1994).
[Crossref]

P. Sajda, K. Sakai, L. H. Finkel, “NEXUS: a tool for simulating large-scale hybrid neural networks,” in Proceedings of the Summer Computer Simulation Conference 1992Society for Computer Simulation, San Diego, Calif., 1992), pp. 72–76.

Salt, T. E.

A. M. Sillito, T. E. Salt, J. A. Kemp, “Modulatory and inhibitory processes in the visual cortex,” Vision Res. 25, 375–381 (1985).
[Crossref] [PubMed]

Sato, T.

S. Nishida, T. Sato, “Positive motion after-effect induced by bandpass-filtered random-dot kinematograms,” Vision Res. 32, 1635–1646 (1992).
[Crossref] [PubMed]

Sejnowski, T. J.

S. R. Lehky, T. J. Sejnowski, R. Desimone, “Predicting responses of nonlinear neurons in monkey striate cortex to complex patterns,” J. Neurosci. 12, 3568–3581 (1992).
[PubMed]

Shafer, S. A.

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

Shapley, R.

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

Shvayster, H.

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

Sillito, A. M.

A. M. Sillito, T. E. Salt, J. A. Kemp, “Modulatory and inhibitory processes in the visual cortex,” Vision Res. 25, 375–381 (1985).
[Crossref] [PubMed]

Stevens, K. A.

K. A. Stevens, “Slant-tilt: the visual encoding of surface orientation,” Biol. Cybern. 46, 183–195 (1983).
[Crossref] [PubMed]

K. A. Stevens, “The information content of texture gradients,” Biol. Cybern. 42, 95–105 (1981).
[Crossref] [PubMed]

Super, B. J.

B. J. Super, A. C. Bovik, “Shape-from-texture by wavelet-based measurement of local spectral moments,” in Proceedings of the Computer Vision and Pattern Recognition Meeting 1992 (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 296–300.
[Crossref]

Sutter, A.

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]

Szulborski, R. G.

R. G. Szulborski, L. A. Palmer, “The two-dimensional spatial structure of nonlinear subunits in the receptive fields of complex cells,” Vision Res. 30, 249–254 (1990).
[Crossref] [PubMed]

Thompson, I. D.

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat striate cortex,” J. Physiol. (London) 283, 79–99 (1978).

Tobin, E. A.

C. Blakemore, E. A. Tobin, “Lateral inhibition between orientation detectors in the cat’s visual cortex,” Exp. Brain Res. 15, 439–440 (1972).
[Crossref]

Todd, J. T.

J. T. Todd, R. A. Akerstrom, “Perception of three-dimensional form from patterns of optical texture,” J. Exp. Psychol. 13, 242–255 (1987).

Tolhurst, D. J.

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat striate cortex,” J. Physiol. (London) 283, 79–99 (1978).

Trevarthen, C.

P. Lennie, C. Trevarthen, D. Van Essen, H. Wassle, “Parallel processing of visual information,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Turner, M. R.

M. R. Turner, G. L. Gerstein, R. Bajcsy, “Underestimation of visual texture slant by human observers: a model,” Biol. Cybern. 65, 215–226 (1991).
[Crossref] [PubMed]

Van Essen, D.

P. Lennie, C. Trevarthen, D. Van Essen, H. Wassle, “Parallel processing of visual information,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Walraven, J.

J. Walraven, C. Enroth-Cugell, “The control of visual sensitivity: receptoral and postreceptoral processes,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Wassle, H.

P. Lennie, C. Trevarthen, D. Van Essen, H. Wassle, “Parallel processing of visual information,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Wiesel, T. N.

D. H. Hubel, T. N. Wiesel, “Receptive fields and functional architecture of monkey striate cortex,” J. Physiol. (London) 195, 215–243 (1968).

Wilson, H. R.

H. R. Wilson, “Nonlinear processes in visual pattern discrimination,” Proc. Natl. Acad. Sci. USA 90, 9785–9790 (1993).
[Crossref] [PubMed]

H. R. Wilson, W. A. Richards, “Curvature and separation discrimination at texture boundaries,” J. Opt. Soc. Am. A 9, 1653–1662 (1992).
[Crossref] [PubMed]

H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
[Crossref]

H. R. Wilson, D. Levi, L. Maffei, J. Rovamo, R. DeValois, “The perception of form in visual perception,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

Witkin, A. P.

A. P. Witkin, “Recovering surface shape and orientation from texture,” Artif. Intell. 17, 17–47 (1981).
[Crossref]

Yen, S-C.

P. Sajda, K. Sakai, S-C. Yen, L. H. Finkel, “NEXUS: a neural simulator for integrating top-down and bottom-up modeling,” in Neural Network Simulation Environments, J. Skrzypek, ed. (Kluwer, Norwell, Mass., 1994).
[Crossref]

Artif. Intell. (3)

K. Kanatani, “Detection of surface orientation and motion from texture by a stereological technique,” Artif. Intell. 23, 213–237 (1984).
[Crossref]

A. P. Witkin, “Recovering surface shape and orientation from texture,” Artif. Intell. 17, 17–47 (1981).
[Crossref]

A. Blake, C. Marinos, “Shape from texture: estimation, isotropy and moment,” Artif. Intell. 45, 323–380 (1990).
[Crossref]

Biol. Cybern. (3)

K. A. Stevens, “The information content of texture gradients,” Biol. Cybern. 42, 95–105 (1981).
[Crossref] [PubMed]

K. A. Stevens, “Slant-tilt: the visual encoding of surface orientation,” Biol. Cybern. 46, 183–195 (1983).
[Crossref] [PubMed]

M. R. Turner, G. L. Gerstein, R. Bajcsy, “Underestimation of visual texture slant by human observers: a model,” Biol. Cybern. 65, 215–226 (1991).
[Crossref] [PubMed]

Comput. Graphics Image Process. (1)

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

Eur. J. Neurosci. (1)

Z. F. Kisvarday, U. T. Eysel, “Functional and structural topography of horizontal inhibitory connections in cat visual cortex,” Eur. J. Neurosci. 5, 1558–1572 (1993).
[Crossref] [PubMed]

Exp. Brain Res. (2)

C. Blakemore, E. A. Tobin, “Lateral inhibition between orientation detectors in the cat’s visual cortex,” Exp. Brain Res. 15, 439–440 (1972).
[Crossref]

P. Hammond, D. M. MacKey, “Differential responsiveness of simple and complex cells in cat striate cortex to visual texture,” Exp. Brain Res. 30, 275–296 (1977).
[PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

D. Blostein, N. Ahuja, “Shape from texture: integrating texture element extraction and surface estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 1233–1251 (1989).
[Crossref]

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

J. Exp. Psychol. (3)

J. T. Todd, R. A. Akerstrom, “Perception of three-dimensional form from patterns of optical texture,” J. Exp. Psychol. 13, 242–255 (1987).

M. L. Braunstein, J. W. Payne, “Perspective and form ratio as determinants of relative slant judgments,” J. Exp. Psychol. 81, 584–590 (1969).
[Crossref]

J. E. Cutting, R. T. Millard, “Three gradients and the perception of flat and curved surfaces,” J. Exp. Psychol. 113, 198–216 (1984).
[Crossref]

J. Math. Imag. Vis. (1)

J. Gårding, “Shape from texture for smooth curved surfaces in perspective projection,” J. Math. Imag. Vis. 2, 329–352 (1992).
[Crossref]

J. Neurophysiol. (1)

G. C. DeAngelis, J. G. Robson, I. Ohzawa, R. D. Freeman, “The organization of suppression in receptive fields of neurons in the cat’s visual cortex,” J. Neurophysiol. 68, 1440–1463 (1992).

J. Neurosci. (1)

S. R. Lehky, T. J. Sejnowski, R. Desimone, “Predicting responses of nonlinear neurons in monkey striate cortex to complex patterns,” J. Neurosci. 12, 3568–3581 (1992).
[PubMed]

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

J. Physiol. (London) (2)

D. H. Hubel, T. N. Wiesel, “Receptive fields and functional architecture of monkey striate cortex,” J. Physiol. (London) 195, 215–243 (1968).

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat striate cortex,” J. Physiol. (London) 283, 79–99 (1978).

Nature (London) (2)

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

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

Percept. Psychophys. (1)

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]

Proc. Natl. Acad. Sci. USA (1)

H. R. Wilson, “Nonlinear processes in visual pattern discrimination,” Proc. Natl. Acad. Sci. USA 90, 9785–9790 (1993).
[Crossref] [PubMed]

Proc. R. Soc. London Ser. B (2)

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).
[Crossref]

M. C. Morrone, D. C. Burr, L. Maffei, “Functional implications of cross-orientation inhibition of cortical visual cells,” Proc. R. Soc. London Ser. B 249, 335–354 (1982).
[Crossref]

Prog. Retinal Res. (1)

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

Simulation (1)

P. Sajda, L. H. Finkel, “NEXUS: a simulation environment for large-scale neural systems,” Simulation 59, 358–364 (1992).
[Crossref]

Vision Res. (8)

A. M. Sillito, T. E. Salt, J. A. Kemp, “Modulatory and inhibitory processes in the visual cortex,” Vision Res. 25, 375–381 (1985).
[Crossref] [PubMed]

R. G. Szulborski, L. A. Palmer, “The two-dimensional spatial structure of nonlinear subunits in the receptive fields of complex cells,” Vision Res. 30, 249–254 (1990).
[Crossref] [PubMed]

H. R. Wilson, J. R. Bergen, “A four mechanism model for spatial vision,” Vision Res. 19, 19–32 (1979).
[Crossref]

J. S. Mansfield, A. J. Parker, “An orientation-tuned component in the contrast masking of stereopsis,” Vision Res. 33, 1535–1544 (1993).
[Crossref] [PubMed]

S. Nishida, T. Sato, “Positive motion after-effect induced by bandpass-filtered random-dot kinematograms,” Vision Res. 32, 1635–1646 (1992).
[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]

S. M. Courtney, L. H. Finkel, G. Buchsbaum, “Network simulations of retinal and cortical contributions to color constancy,” Vision Res. 35, 413–434 (1995).
[Crossref] [PubMed]

B. G. Cumming, E. B. Johnston, A. J. Parker, “Effects of different texture cues on curved surfaces viewed stereoscopically,” Vision Res. 33, 827–838 (1993).
[Crossref] [PubMed]

Other (13)

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

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

B. J. Super, A. C. Bovik, “Shape-from-texture by wavelet-based measurement of local spectral moments,” in Proceedings of the Computer Vision and Pattern Recognition Meeting 1992 (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 296–300.
[Crossref]

J. J. Gibson, The Perception of the Visual World (Houghton Mifflin, Boston, Mass., 1950).

P. Lennie, C. Trevarthen, D. Van Essen, H. Wassle, “Parallel processing of visual information,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

J. Walraven, C. Enroth-Cugell, “The control of visual sensitivity: receptoral and postreceptoral processes,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

H. R. Wilson, D. Levi, L. Maffei, J. Rovamo, R. DeValois, “The perception of form in visual perception,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

M. L. J. Crawford, A. Anderson, R. Blake, G. H. Jacobs, C. Neumeyer, “Interspecies comparisons in the understanding of human visual perception,” in Visual Perception, L. Spillmann, J. S. Werner, eds. (Academic, San Diego, Calif., 1990).

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).

P. Sajda, K. Sakai, S-C. Yen, L. H. Finkel, “NEXUS: a neural simulator for integrating top-down and bottom-up modeling,” in Neural Network Simulation Environments, J. Skrzypek, ed. (Kluwer, Norwell, Mass., 1994).
[Crossref]

H. H. Bülthoff, “Shape from X: psychophysics and computation,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991).

P. Sajda, K. Sakai, L. H. Finkel, “NEXUS: a tool for simulating large-scale hybrid neural networks,” in Proceedings of the Summer Computer Simulation Conference 1992Society for Computer Simulation, San Diego, Calif., 1992), pp. 72–76.

D. J. Heeger, “Nonlinear model of neural responses in cat visual cortex,” in Computational Models of Visual Processing, M. S. Landy, J. A. Movshon, eds. (MIT Press, Cambridge, Mass., 1991).

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Figures (19)

Fig. 1
Fig. 1

(A) Stimulus generated from white noise. (B) Schematic illustration of the construction of the stimulus shown in (A). The cylinder is approximated by a 56-sided polygon, and the projection of each surface is constructed by applying different filters to the same white-noise pattern in the Fourier domain. In the remainder of the paper elliptic cylinders with specified eccentricities seen from the direction of the long axis (perpendicular to the rotational axis) are used, depending on the purpose of the experiments. For this figure and Figs. 2, 4, and 5 below, an eccentricity of 1.5 is used. The frequency component of a white-noise pattern [the leftmost pattern in (B)] is multiplied by rectangular filters (the second column). The top filter is used for generating the frontal part of the cylinder, and the bottom filter is used for the periphery of the cylinder. The U axis corresponds to horizontal orientation in the space domain. The real parts of the generated frequency components are shown in the third column. The imaginary parts are processed identically, so that phase information is not modified. The frontal part of the cylinder includes only lower-frequency components, and the periphery of the cylinder includes high-frequency components in the horizontal orientation. The magnitude of the dc component is truncated for presentation purposes.

Fig. 2
Fig. 2

Vertical cylinders constructed from (A) a two-dimensional (2-D) sinusoidal pattern and (B) a white-noise pattern. The illustrations at the bottom show the ranges of the frequency distributions of these stimuli in the horizontal orientation. As one moves from the center to the periphery, in (A) the spectrum shifts to the higher frequency and in (B) the spectrum expands to the higher frequency. We can perceive a similar 3-D cylinder in both cases, although the variance of the frequency distributions is very different.

Fig. 3
Fig. 3

(A) and (B) are images with different frequency distributions over space; thus they are perceived to have different 3-D shapes. In (A) we perceive a vertical cylinder that flattens into a plane, whereas in (B) we see a surface with a wide flat center and sharply curved edges. The cross sections of (A) and (B) computed from their mean frequencies are shown next to these stimuli. The frequency distribution of surface (C) is the summation of those of (A) and (B), as shown in (E). We perceive a third kind of 3-D shape, a regular cylinder, in (C). (D) is the image constructed from a simple frequency distribution that is so controlled that the mean frequency is identical to that of (C). The computed cross sections of (C) and (D) are also shown next to these stimuli. (E) shows the frequency distributions of surfaces (A)–(D) in the Fourier domain at the position indicated by vertical arrows in (A) and (F). Only gray regions have nonzero components. Horizontal arrows next to the gray circles show the direction of shift as one moves to the periphery, and the length of the arrows shows the rate of the shift. (F) shows the mean frequencies in the horizontal orientation of (A)–(D) as a function of horizontal position X. The origin of X corresponds to the center of the images. We perceive the same 3-D shape in (C) and (D). This suggests that the visual system characterizes the spectrum and does not track individual components.

Fig. 4
Fig. 4

Two stimuli with substantial peak frequency components: (A) one with shifting strong peaks but fixed mean frequency, (B) one with fixed strong peaks but shifting mean frequency. A 3-D cylinder is visible in (A) but not in (B). The magnitude of the peak component varies between 4 and 8 times that of the rest of the components, and the peak or the mean frequency changes by the same ratio as that of real cylinders. The plots at the bottom show filters’ cross sections in horizontal orientation at the center and the periphery of the cylinder.

Fig. 5
Fig. 5

Stimuli with relatively weak peak frequency components: (A) stimulus whose peak frequency changes from low to high along the horizontal orientation but whose mean frequency does not change, (B) similarly constructed stimulus whose mean frequency changes but whose peak frequency does not change. The plots at the bottom show filters’ cross sections in horizontal orientation at the center and the periphery of the cylinder. The peak frequency in (A) and the mean frequency in (B) change in the same fashion as that of a real cylinder. A 3-D cylinder is visible in (B) but not in (A).

Fig. 6
Fig. 6

(A) and (B) show the distribution of local peak frequencies in the horizontal orientation measured across a 15 × 15-pixel region near the center of the stimuli shown in Figs. 4(A) and 5(B). Lighter colors indicate higher frequencies. (C) and (D) show the measured local peak frequencies, mean frequencies, and average peak frequencies in the horizontal orientation for points lying along a horizontal scan line in Figs. 4(A) and 5(B). Frequencies are normalized by the largest value. X = 0 corresponds to the center of the cylinder, and X = 1 corresponds to the periphery. In (C) the average peak frequency closely approximates the peak frequency, whereas in (D) the average peak frequency closely approximates the mean frequency. Note that there is a considerable variance in local peak in (D) but not in (C).

Fig. 7
Fig. 7

Measured average peak frequency in the horizontal orientation, f p ¯, of stimuli shown in Figs. 2, 4, and 5. X = 0 corresponds to the center of the cylinder, and X = 1 corresponds to the edge. Stimuli in which we can perceive 3-D depth [Figs. 2(B), 4(A), and 5(B)] have a large difference in f p ¯ between the center and the periphery. Stimuli in which we do not perceive 3-D depth [Figs. 4(B) and 5(A)] have a small difference or no difference in f p ¯. The f p ¯ have been normalized by the largest value.

Fig. 8
Fig. 8

Stimuli with controlled average peak frequency. The ratio of the average peak frequency in the horizontal orientation between the center and the periphery (fp/fc) is (A) 1, (B) 2, and (C) 4. We perceive greater curvature from stimuli with larger ratios.

Fig. 9
Fig. 9

Rank of perceived 3-D depth as a function of the ratio of average peak frequency in the horizontal orientation, f p ¯, between the center and the periphery of the cylinder. Circles show the mean value, and the lengths of the bars show the standard deviation. There is a monotonically increasing relation between the observed depth and the ratio of f p ¯.

Fig. 10
Fig. 10

Measured second-order moment m2,0 and average peak frequency in the horizontal orientation, f p ¯, of Fig. 4. m2,0 and f p ¯ are normalized by their largest values. X = 0 corresponds to the center of the cylinder, and X = 1 corresponds to the periphery. Since the moments take into account all the frequency components, m2,0 of Fig. 4(B) increases as one moves from the center to the periphery, but m2,0 does not change in Fig. 4(A). On the other hand, f p ¯ changes in Fig. 4(A) but not in Fig. 4(B). Thus f p ¯ agrees with the human perception of Fig. 4, whereas the moment m2,0 does not.

Fig. 11
Fig. 11

(A) Vertical cylinder with small circles on its surface and (B) the same cylinder superimposed with a white-noise pattern. The power of the frequency components corresponding to the white noise is up to 25% of that of the peak components corresponding to the circles.

Fig. 12
Fig. 12

Measured second-order moment m2,0 and average peak frequency f p ¯ in the horizontal orientation of the cylinders shown in Fig. 11. m2,0 and f p ¯ are normalized by their largest values. X = 0 corresponds to the center, and X = 1 corresponds to the periphery. In Fig. 11(B) m2,0 is affected by the white-noise components, so that the moments of the center and the periphery are almost the same. f p ¯ is not significantly affected by the white noise.

Fig. 13
Fig. 13

Block diagram of the model. Dotted rectangles V (vertical), H (horizontal), R (45°), and L (135°) indicate orientation channels. The contents of H, R, and L are identical to those of V. An input image is decomposed into 72 channels (nine frequencies, four orientations, and On/Off) and gradually converges to generate a depth map at the bottom.

Fig. 14
Fig. 14

Responses of simple cells and complex cells in the model to three stimuli (left-most column). Images are a sphere composed of randomly placed circles (top row), a video image of a straw basket (middle row), and a video image of a cantaloupe (bottom row). Determined peak frequency distributions are shown in the two rightmost columns. Lighter colors indicate higher spatial frequencies. Any gradient-based model of shape from texture would be confounded by the noisy frequency distribution extracted by simple cells. The complex-cell distribution shows a more orderly gradient from low frequencies to higher frequencies as the surface slants.

Fig. 15
Fig. 15

Stimuli composed of (A) three planes and (B) two planes. In both stimuli the bottommost regions are perceived as frontoparallel, and the planes above these regions look slanted, although the patterns of the center region of (A) and the lower region of (B) are identical.

Fig. 16
Fig. 16

Stimuli used to test simulated depth versus eccentricity: generated regular ellipsoids with eccentricities of (A) 1, (B) 2, and (C) 4, viewed from the long axes. Small circles are placed randomly but do not overlap.

Fig. 17
Fig. 17

Network computed depth of ellipsoids with different eccentricities (filled squares). For ellipsoids with regular texture changes we see a linear relation between real depth and that determined by the model, which agrees with psychophysical experiments2,40 showing similar human performance for such stimuli (circles). The computed depth of pseudoellipsoids with irregular texture changes is far smaller than that of ellipsoids with regular texture change regardless of eccentricity (diamonds and open squares). Note that the scale of computed depth is linear but arbitrary.

Fig. 18
Fig. 18

Generated pseudoellipsoids with irregular texture changes: (A) no compression and regular density change, (B) randomly oriented regular compression, and (C) panorientational regular compression.

Fig. 19
Fig. 19

Images (first and third rows) and their 3-D depth (second and fourth rows) as determined by the model: artificial images of a shaded sphere (top left) and a torus (bottom left), real images of a basket (top center), a part of a cantaloupe (top right), and an avocado (bottom right), and the white-noise stimulus shown in Fig. 1 (bottom center).

Equations (15)

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f ¯ p ( x 0 , y 0 ) = 1 m I ( x 0 , y 0 ) f prob [ f p ( x , y ) = f ] f = 1 m I ( x 0 , y 0 ) f S x , y ( f ) S x , y ( f ) f f ¯ ,
m 2 , 0 = u 2 M ( u , v ) d u d v ,
f ¯ p = R u N ( u , v ) d u d v ,
apf o ( x 0 , y 0 ) = ave x , y I ( x 0 , y 0 ) { pfreq o [ ccell o , f ( x , y ) ] } ,
[ F o ( x , y ) - F min o ] / F min o ,
tan [ S o ( x , y ) ] = tan { cos - 1 [ F min o F o ( x , y ) ] } = { [ F o ( x , y ) F min o ] 2 - 1 } 1 / 2 .
O o , f 1 ( x 0 , y 0 ) = max x , y I 1 ( x 0 , y 0 ) { QLT [ ( I * M o , f ) + ( x , y ) ] } ,             QLT [ - ( I * M o , f ) - ( x , y ) ] } ,
QLT ( p ) = { 0 p < threshold min max p > threshold max c 1 p + c 2 otherwise .
O o 2 ( x 0 , y 0 ) = ave x , y I 2 ( x 0 , y 0 ) { pfreq o [ O o , f 1 ( x , y ) ] } ,
O o , f 1 ( x , y ) = max f [ O o , f 1 ( x , y ) ] .
O o 3 ( x 0 , y 0 ) = O o 2 ( x 0 , y 0 ) min x , y I ( x 0 , y 0 ) [ O o 2 ( x , y ) - 1.
O o 4 ( x 0 , y 0 ) = { 0 S o 1 < 0 O o 3 S o 1 0 S o 1 1 O o 3 1 < S o 1 ,
S o 1 ( x 0 , y 0 ) = C 1 { 1 - x , y I 3 ( x 0 , y 0 ) [ O o ¯ 3 ( x , y ) ] x , y I 3 ( x 0 , y 0 ) [ O o 3 ( x , y ) ] } + C 2 .
O 5 ( x 0 , y 0 ) = O 5 ( i s , j s ) + max o [ O o 4 ( x 0 , y 0 ) ] ,
i , j I 4 ( i 0 , j 0 ) o O o 4 ( x 0 + i , y 0 + j ) = min i 0 , j 0 [ i , j I 4 ( i 0 , j 0 ) o O o 4 ( x 0 + i , y 0 + j ) ] ,

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