Abstract

We investigated how the detection of mirror symmetry depends on the distribution of contrast energy across spatial scales. Stimuli consisted of vertically symmetric noise patterns with fractal power spectra defined by 1/fβ slopes (-2β5). While overall rms contrast remained fixed at 25%, symmetry-detection thresholds were obtained by corrupting the signal with variable amounts of noise with identical spectral characteristics. A first experiment measured thresholds as a function of spectral slope, and performance was found to be substantially facilitated in images with power spectra that characterize natural scenes (1.2β3.2). In a second experiment, symmetry was removed from randomly chosen octave bands and replaced by noise with the same spectral profile. Results revealed that only in images with 1/f2 spectra does performance decrease by constant amounts across all frequency bands. Together, the results imply that symmetry mechanisms extract equal amounts of information from constant-octave frequency bands but lack the ability to whiten stimuli whose spectral slopes differ from those of natural scenes. Results are qualitatively well predicted by a multichannel model that (1) relies on spatial filters with equal-volume point-spread functions and constant-octave frequency bandwidths and (2) restricts the computation of symmetry to spatial regions whose dimensions are proportional to the filters’ spatial scale. These findings are also consistent with the notion that mechanisms that mediate the perception of form exploit the ability of early vision to reduce second-order redundancy in natural scenes.

© 1999 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. H. B. Barlow, B. C. Reeves, “The versatility and absolute efficiency of detecting mirror symmetry in random dot displays,” Vision Res. 19, 783–793 (1979).
    [CrossRef] [PubMed]
  8. S. C. Dakin, R. J. Watt, “Detection of bilateral symmetry using spatial filters,” Spatial Vision 8, 393–413 (1994).
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  30. D. J. Field, “Scale-invariance and self-similar ‘wavelet’ transforms: an analysis of natural scenes and mammalian visual systems,” in Wavelets, Fractals, and Fourier Transforms, M. Farge, J. C. R. Hunt, J. C. Vassilicos, eds. (Clarendon, Oxford, 1993), pp. 151–193.
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    [CrossRef]
  32. J. J. Atick, A. N. Redlich, “Towards a theory of early visual processing,” Neural Comput. 2, 308–320 (1990).
    [CrossRef]
  33. M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London, Ser. B 216, 427–59 (1982).
    [CrossRef]
  34. C. Blakemore, F. W. Campbell, “On the existence of neurones in the human vision system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).
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    [CrossRef] [PubMed]
  36. L. Croner, E. Kaplan, “Receptive fields of P and M ganglion cells across the primate retina,” Vision Res. 35, 7–24 (1995).
    [CrossRef] [PubMed]
  37. D. C. Knill, D. Field, D. Kersten, “Human discrimination of fractal images,” J. Opt. Soc. Am. A 7, 1113–1123 (1990).
    [CrossRef] [PubMed]
  38. Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the second-order statistics of natural and synthetic images,” Vision Res. 34, 541–554 (1994).
    [CrossRef] [PubMed]
  39. D. J. Tolhurst, Y. Tadmor, “Band-limited contrast in natural images explains the detectability of changes in the amplitude spectra,” Vision Res. 37, 3203–3215 (1997).
    [CrossRef]
  40. D. G. Pelli, “The VideoToolbox software for visual psychophysics: transforming numbers into movies,” Spatial Vision 10, 437–442 (1997).
    [CrossRef] [PubMed]
  41. A. B. Watson, D. G. Pelli, “QUEST: A Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
    [CrossRef] [PubMed]
  42. J. D. Victor, M. M. Conte, “The role of high-order phase correlations in texture processing,” Vision Res. 36, 1615–1631 (1996).
    [CrossRef] [PubMed]
  43. S. J. M. Rainville, F. A. A. Kingdom, “Is motion perception sensitive to local phase structures?” Invest. Ophthalmol. 38, S215 (1997).
  44. N. V. S. Graham, Visual Pattern Analyzers (Oxford U. Press, New York, 1989).
  45. N. Graham, J. G. Robson, “Summation of very close spatial frequencies: the importance of spatial probability summation,” Vision Res. 27, 815–826 (1987).
    [CrossRef]
  46. R. F. J. Quick, “A vector-magniture model of contrast detection,” Kybernetik 16, 65–67 (1974).
    [CrossRef]
  47. N. A. Macmillan, C. D. Creelman, Detection Theory: A User’s Guide (Cambridge U. Press, Cambridge, 1991).
  48. D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Peninsula, Los Altos, Calif., 1988).

1998 (3)

S. C. Dakin, A. M. Herbert, “The spatial region of integration for visual symmetry detection,” Proc. R. Soc. London, Ser. B 265, 659–664 (1998).
[CrossRef]

S. J. M. Rainville, F. A. A. Kingdom, “Does oblique structure support the detection of mirror symmetry?” Invest. Ophthalmol. 39, S170 (1998).

S. J. M. Rainville, F. A. A. Kingdom, “From spatial filters to mirror symmetry: new findings and new model,” Perception 27, 58a (1998).

1997 (5)

S. C. Dakin, R. F. Hess, “The spatial mechanisms mediating symmetry perception,” Vision Res. 37, 2915–2930 (1997).
[CrossRef]

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3383 (1997).
[CrossRef]

D. J. Tolhurst, Y. Tadmor, “Band-limited contrast in natural images explains the detectability of changes in the amplitude spectra,” Vision Res. 37, 3203–3215 (1997).
[CrossRef]

D. G. Pelli, “The VideoToolbox software for visual psychophysics: transforming numbers into movies,” Spatial Vision 10, 437–442 (1997).
[CrossRef] [PubMed]

S. J. M. Rainville, F. A. A. Kingdom, “Is motion perception sensitive to local phase structures?” Invest. Ophthalmol. 38, S215 (1997).

1996 (2)

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

P. Wenderoth, “The effects of the contrast polarity of dot-pair partners on the detection of bilateral symmetry,” Perception 25, 757–771 (1996).
[CrossRef] [PubMed]

1995 (4)

F. Labonte, Y. Shapira, P. Cohen, J. Faubert, “A model for global symmetry detection in dense images,” Spatial Vision 9, 33–55 (1995).
[CrossRef] [PubMed]

C. W. Tyler, L. Hardage, R. T. Miller, “Multiple mechanisms for the detection of mirror symmetry,” Spatial Vision 9, 79–100 (1995).
[CrossRef] [PubMed]

L. Croner, E. Kaplan, “Receptive fields of P and M ganglion cells across the primate retina,” Vision Res. 35, 7–24 (1995).
[CrossRef] [PubMed]

N. Brady, D. J. Field, “What’s constant in contrast constancy? The effects of scaling on the perceived contrast of bandpass patterns,” Vision Res. 35, 739–756 (1995).
[CrossRef] [PubMed]

1994 (4)

P. Wenderoth, “The salience of vertical symmetry,” Perception 23, 221–236 (1994).
[CrossRef] [PubMed]

D. L. Ruderman, W. Bialeck, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the second-order statistics of natural and synthetic images,” Vision Res. 34, 541–554 (1994).
[CrossRef] [PubMed]

S. C. Dakin, R. J. Watt, “Detection of bilateral symmetry using spatial filters,” Spatial Vision 8, 393–413 (1994).
[CrossRef] [PubMed]

1993 (2)

J. Wagemans, L. Van Gool, V. Swinnen, J. Van Horebeek, “Higher-order structure in regularity detection,” Vision Res. 33, 1067–1088 (1993).
[CrossRef] [PubMed]

U. Koeppl, “Local orientation versus local position as determinants of perceived symmetry,” Perception 22, 111 (1993).

1992 (1)

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthal. Physiol. Opt. 12, 229–232 (1992).
[CrossRef]

1990 (2)

J. J. Atick, A. N. Redlich, “Towards a theory of early visual processing,” Neural Comput. 2, 308–320 (1990).
[CrossRef]

D. C. Knill, D. Field, D. Kersten, “Human discrimination of fractal images,” J. Opt. Soc. Am. A 7, 1113–1123 (1990).
[CrossRef] [PubMed]

1987 (3)

1983 (3)

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef] [PubMed]

A. B. Watson, D. G. Pelli, “QUEST: A Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
[CrossRef] [PubMed]

B. Jenkins, “Component processes in the perception of bilaterally symmetric dot textures,” Percept. Psychophys. 34, 433–440 (1983).
[CrossRef] [PubMed]

1982 (3)

R. L. De Valois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef] [PubMed]

B. Jenkins, “Redundancy in the perception of bilateral symmetry in dot textures,” Percept. Psychophys. 32, 171–177 (1982).
[CrossRef] [PubMed]

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London, Ser. B 216, 427–59 (1982).
[CrossRef]

1979 (2)

B. Julesz, J. Chang, “Symmetry perception and spatial-frequency channels,” Perception 8, 711–718 (1979).
[CrossRef] [PubMed]

H. B. Barlow, B. C. Reeves, “The versatility and absolute efficiency of detecting mirror symmetry in random dot displays,” Vision Res. 19, 783–793 (1979).
[CrossRef] [PubMed]

1975 (1)

V. G. Bruce, M. J. Morgan, “Violations of symmetry and repetition in visual patterns,” Perception 4, 239–249 (1975).
[CrossRef]

1974 (1)

R. F. J. Quick, “A vector-magniture model of contrast detection,” Kybernetik 16, 65–67 (1974).
[CrossRef]

1969 (1)

C. Blakemore, F. W. Campbell, “On the existence of neurones in the human vision system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

1968 (2)

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

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Albrecht, D. G.

R. L. De Valois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef] [PubMed]

Atick, J. J.

J. J. Atick, A. N. Redlich, “Towards a theory of early visual processing,” Neural Comput. 2, 308–320 (1990).
[CrossRef]

Barlow, H. B.

H. B. Barlow, B. C. Reeves, “The versatility and absolute efficiency of detecting mirror symmetry in random dot displays,” Vision Res. 19, 783–793 (1979).
[CrossRef] [PubMed]

Bialeck, W.

D. L. Ruderman, W. Bialeck, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

Blakemore, C.

C. Blakemore, F. W. Campbell, “On the existence of neurones in the human vision system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

Brady, N.

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3383 (1997).
[CrossRef]

N. Brady, D. J. Field, “What’s constant in contrast constancy? The effects of scaling on the perceived contrast of bandpass patterns,” Vision Res. 35, 739–756 (1995).
[CrossRef] [PubMed]

Bruce, V. G.

V. G. Bruce, M. J. Morgan, “Violations of symmetry and repetition in visual patterns,” Perception 4, 239–249 (1975).
[CrossRef]

Burton, G. J.

Campbell, F. W.

C. Blakemore, F. W. Campbell, “On the existence of neurones in the human vision system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Carlin, P.

P. Carlin, “On symmetry in visual perception,” Ph.D dissertation (University of Stirling, Stirling, Scotland, 1996).

Chang, J.

B. Julesz, J. Chang, “Symmetry perception and spatial-frequency channels,” Perception 8, 711–718 (1979).
[CrossRef] [PubMed]

Chao, T.

D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthal. Physiol. Opt. 12, 229–232 (1992).
[CrossRef]

Cohen, P.

F. Labonte, Y. Shapira, P. Cohen, J. Faubert, “A model for global symmetry detection in dense images,” Spatial Vision 9, 33–55 (1995).
[CrossRef] [PubMed]

Conte, M. M.

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

Creelman, C. D.

N. A. Macmillan, C. D. Creelman, Detection Theory: A User’s Guide (Cambridge U. Press, Cambridge, 1991).

Croner, L.

L. Croner, E. Kaplan, “Receptive fields of P and M ganglion cells across the primate retina,” Vision Res. 35, 7–24 (1995).
[CrossRef] [PubMed]

Dakin, S. C.

S. C. Dakin, A. M. Herbert, “The spatial region of integration for visual symmetry detection,” Proc. R. Soc. London, Ser. B 265, 659–664 (1998).
[CrossRef]

S. C. Dakin, R. F. Hess, “The spatial mechanisms mediating symmetry perception,” Vision Res. 37, 2915–2930 (1997).
[CrossRef]

S. C. Dakin, R. J. Watt, “Detection of bilateral symmetry using spatial filters,” Spatial Vision 8, 393–413 (1994).
[CrossRef] [PubMed]

De Valois, R. L.

R. L. De Valois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef] [PubMed]

Dubs, A.

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London, Ser. B 216, 427–59 (1982).
[CrossRef]

Faubert, J.

F. Labonte, Y. Shapira, P. Cohen, J. Faubert, “A model for global symmetry detection in dense images,” Spatial Vision 9, 33–55 (1995).
[CrossRef] [PubMed]

Field, D.

Field, D. J.

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3383 (1997).
[CrossRef]

N. Brady, D. J. Field, “What’s constant in contrast constancy? The effects of scaling on the perceived contrast of bandpass patterns,” Vision Res. 35, 739–756 (1995).
[CrossRef] [PubMed]

D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–9234 (1987).
[CrossRef] [PubMed]

D. J. Field, “Scale-invariance and self-similar ‘wavelet’ transforms: an analysis of natural scenes and mammalian visual systems,” in Wavelets, Fractals, and Fourier Transforms, M. Farge, J. C. R. Hunt, J. C. Vassilicos, eds. (Clarendon, Oxford, 1993), pp. 151–193.

Graham, N.

N. Graham, J. G. Robson, “Summation of very close spatial frequencies: the importance of spatial probability summation,” Vision Res. 27, 815–826 (1987).
[CrossRef]

Graham, N. V. S.

N. V. S. Graham, Visual Pattern Analyzers (Oxford U. Press, New York, 1989).

Green, D. M.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Peninsula, Los Altos, Calif., 1988).

Hardage, L.

C. W. Tyler, L. Hardage, R. T. Miller, “Multiple mechanisms for the detection of mirror symmetry,” Spatial Vision 9, 79–100 (1995).
[CrossRef] [PubMed]

C. W. Tyler, L. Hardage, “Mirror symmetry detection: predominance of second-order pattern processing throughout the visual field,” in Human Symmetry Perception and Its Computational Analysis, C. W. Tyler, ed. (VSP, Utrecht, The Netherlands, 1996).

Herbert, A. M.

S. C. Dakin, A. M. Herbert, “The spatial region of integration for visual symmetry detection,” Proc. R. Soc. London, Ser. B 265, 659–664 (1998).
[CrossRef]

Hess, R. F.

S. C. Dakin, R. F. Hess, “The spatial mechanisms mediating symmetry perception,” Vision Res. 37, 2915–2930 (1997).
[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).

Jenkins, B.

B. Jenkins, “Component processes in the perception of bilaterally symmetric dot textures,” Percept. Psychophys. 34, 433–440 (1983).
[CrossRef] [PubMed]

B. Jenkins, “Redundancy in the perception of bilateral symmetry in dot textures,” Percept. Psychophys. 32, 171–177 (1982).
[CrossRef] [PubMed]

Julesz, B.

B. Julesz, J. Chang, “Symmetry perception and spatial-frequency channels,” Perception 8, 711–718 (1979).
[CrossRef] [PubMed]

Kaplan, E.

L. Croner, E. Kaplan, “Receptive fields of P and M ganglion cells across the primate retina,” Vision Res. 35, 7–24 (1995).
[CrossRef] [PubMed]

Kersten, D.

Kingdom, F. A. A.

S. J. M. Rainville, F. A. A. Kingdom, “Does oblique structure support the detection of mirror symmetry?” Invest. Ophthalmol. 39, S170 (1998).

S. J. M. Rainville, F. A. A. Kingdom, “From spatial filters to mirror symmetry: new findings and new model,” Perception 27, 58a (1998).

S. J. M. Rainville, F. A. A. Kingdom, “Is motion perception sensitive to local phase structures?” Invest. Ophthalmol. 38, S215 (1997).

Knill, D. C.

Koeppl, U.

U. Koeppl, “Local orientation versus local position as determinants of perceived symmetry,” Perception 22, 111 (1993).

Labonte, F.

F. Labonte, Y. Shapira, P. Cohen, J. Faubert, “A model for global symmetry detection in dense images,” Spatial Vision 9, 33–55 (1995).
[CrossRef] [PubMed]

Laughlin, S. B.

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc. R. Soc. London, Ser. B 216, 427–59 (1982).
[CrossRef]

Macmillan, N. A.

N. A. Macmillan, C. D. Creelman, Detection Theory: A User’s Guide (Cambridge U. Press, Cambridge, 1991).

McFarlane, D. K.

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef] [PubMed]

Miller, R. T.

C. W. Tyler, L. Hardage, R. T. Miller, “Multiple mechanisms for the detection of mirror symmetry,” Spatial Vision 9, 79–100 (1995).
[CrossRef] [PubMed]

Moorhead, I. R.

Morgan, M. J.

V. G. Bruce, M. J. Morgan, “Violations of symmetry and repetition in visual patterns,” Perception 4, 239–249 (1975).
[CrossRef]

Oomes, S.

S. Oomes, “Human visual perception of spatial structure: symmetry, orientation, and attitude,” Ph.D dissertation (Catholic University of Nijmegen, Nijmegen, The Netherlands, 1998).

Pelli, D. G.

D. G. Pelli, “The VideoToolbox software for visual psychophysics: transforming numbers into movies,” Spatial Vision 10, 437–442 (1997).
[CrossRef] [PubMed]

A. B. Watson, D. G. Pelli, “QUEST: A Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
[CrossRef] [PubMed]

Phillips, G. C.

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef] [PubMed]

Quick, R. F. J.

R. F. J. Quick, “A vector-magniture model of contrast detection,” Kybernetik 16, 65–67 (1974).
[CrossRef]

Rainville, S. J. M.

S. J. M. Rainville, F. A. A. Kingdom, “Does oblique structure support the detection of mirror symmetry?” Invest. Ophthalmol. 39, S170 (1998).

S. J. M. Rainville, F. A. A. Kingdom, “From spatial filters to mirror symmetry: new findings and new model,” Perception 27, 58a (1998).

S. J. M. Rainville, F. A. A. Kingdom, “Is motion perception sensitive to local phase structures?” Invest. Ophthalmol. 38, S215 (1997).

Redlich, A. N.

J. J. Atick, A. N. Redlich, “Towards a theory of early visual processing,” Neural Comput. 2, 308–320 (1990).
[CrossRef]

Reeves, B. C.

H. B. Barlow, B. C. Reeves, “The versatility and absolute efficiency of detecting mirror symmetry in random dot displays,” Vision Res. 19, 783–793 (1979).
[CrossRef] [PubMed]

Robson, J. G.

N. Graham, J. G. Robson, “Summation of very close spatial frequencies: the importance of spatial probability summation,” Vision Res. 27, 815–826 (1987).
[CrossRef]

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Ruderman, D. L.

D. L. Ruderman, W. Bialeck, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814–817 (1994).
[CrossRef] [PubMed]

Shapira, Y.

F. Labonte, Y. Shapira, P. Cohen, J. Faubert, “A model for global symmetry detection in dense images,” Spatial Vision 9, 33–55 (1995).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the second-order statistics of natural and synthetic images,” Vision Res. 34, 541–554 (1994).
[CrossRef] [PubMed]

D. J. Tolhurst, Y. Tadmor, “Band-limited contrast in natural images explains the detectability of changes in the amplitude spectra,” Vision Res. 37, 3203–3215 (1997).
[CrossRef]

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Other (9)

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P. Carlin, “On symmetry in visual perception,” Ph.D dissertation (University of Stirling, Stirling, Scotland, 1996).

C. W. Tyler, L. Hardage, “Mirror symmetry detection: predominance of second-order pattern processing throughout the visual field,” in Human Symmetry Perception and Its Computational Analysis, C. W. Tyler, ed. (VSP, Utrecht, The Netherlands, 1996).

S. Oomes, “Human visual perception of spatial structure: symmetry, orientation, and attitude,” Ph.D dissertation (Catholic University of Nijmegen, Nijmegen, The Netherlands, 1998).

J. Wagemans, L. Van Gool, J. Van Horebeek, “Orientation selective channels in symmetry detection: effects of cooperation and attention,” in Channels in the Visual Nervous System: Neurophysiology, Psychophysics and Models, B. E. Blum, ed. (Freund, London, 1991), pp. 425–445.

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Figures (6)

Fig. 1
Fig. 1

Examples of symmetric broadband noise patterns used in experiment 1. Each row shows three images of equal rms contrast but that vary in the slope of their power spectra: a–d, 1/f0; e–h, 1/f2; and i–l, 1/f4. Each column shows four patterns with decreasing amounts of mirror symmetry: a, e, i, 1.0; b, f, j, 0.67; d, h, k, 0.33; and e, i, l, 0.0.

Fig. 2
Fig. 2

Results from experiment 1. Symmetry-detection thresholds are shown as a function of spectral slope for observers SR and FK. Error bars indicate ±1 standard-deviation estimates.

Fig. 3
Fig. 3

Examples of stimuli used in experiment 2. Left column: Patterns are identical to those of experiment 1, except that information in one of four possible octave frequency bands (rows) is phase randomized. Middle column, symmetric (i.e., nonrandomized) component. Right column, phase-randomized components. Patterns from the second and third columns are added together to produce patterns in the first columns. Insets, complementary band–reject and band–pass filter pairs in the Fourier domain.

Fig. 4
Fig. 4

Results from Experiment 2. Performance is shown for two observers (columns) and four spectral slopes (rows). Graphs plot percent-correct performance as a function of the center frequency of the phase-randomized band. Spectral profiles are indicated along the rows. Dashed lines, performance with symmetry present in all four bands. Error bars, ±1 binomial standard-deviation estimates.

Fig. 5
Fig. 5

A, Schematic representation of four bandpass channels (a–d) viewing an image of finite dimensions. Individual circles represent the spatial extent of filters that constitute each channel. Circles shaded in gray represent filters recruited by an integration whose dimensions are proportional to the filter scale. B, Output from four channels (a–d) viewing a perfectly symmetric 1/f2 noise pattern. Channels are spatially limited by a Gaussian region of integration, and underlying filters have constant-octave frequency bandwidths and constant-volume point-spread functions.

Fig. 6
Fig. 6

Modeling results. A, Simulation for experiment 1. Symmetry-detection thresholds are plotted as a function of spectral slope. Results should be compared with human performance (Fig. 2). B, Simulation results for experiment 2. Model performance (d) plotted as a function of the center frequency of the randomized octave band. Dashed line indicate reference performance when no bands are phase randomized. Performance is shown for power spectra of 1/f-2, 1/f0, 1/f2, and 1/f4.

Tables (1)

Tables Icon

Table 1 Spatial-Frequency Parameters of Notch Filters Used in Experiment 2 a

Equations (9)

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

ξ=σs2σs2+σn2,
H(f )=0ifflowf<fhigh1otherwise,
Hn(u, v)=exp-12 ln(f/fn)ln(fnσ)2,
wn(x, y)=exp-(x-x0)22σn2exp-(y-y0)22σn2,
Cn(x, y)=wn(x, y)[hn(x, y)I(x, y)],
Sn=yx=1X/2-1|Cn(x, y)Cn(-x, y)|yx=1X/2-1Cn(x, y)2yx=1X/2-1Cn(-x, y)21/2.
LEDn=[Cn(x, y)-C¯n]2σn2,
αn=LEDnnLEDn.
S=n(αnSn)pNp.

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