Abstract

Previous analyses of natural image statistics have dealt mainly with their Fourier power spectra. Here we explore image statistics by examining responses to biologically motivated filters that are spatially localized and respond to first-order (luminance-defined) and second-order (contrast- or texture-defined) characteristics. We compare the distribution of natural image responses across filter parameters for first- and second-order information. We find that second-order information in natural scenes shows the same self-similarity previously described for first-order information but has substantially less orientational anisotropy. The magnitudes of the two kinds of information, as well as their mutual unsigned correlation, are much stronger for particular combinations of filter parameters in natural images but not in unstructured fractal images having the same power spectra.

© 2004 Optical Society of America

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  1. D. J. Field, “Relations between the statistics of natural images and the response profiles of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
    [CrossRef] [PubMed]
  2. D. J. Tolhurst, Y. Tadmor, T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229–232 (1992).
    [CrossRef] [PubMed]
  3. A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
    [CrossRef] [PubMed]
  4. D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3384 (1997).
    [CrossRef]
  5. D. J. Field, “What is the goal of sensory coding,” Neural Comput. 6, 559–601 (1994).
    [CrossRef]
  6. R. J. Baddeley, “Searching for filters with ‘interesting’ output distributions: an uninteresting direction to explore?” Network 7, 409–421 (1996).
    [CrossRef] [PubMed]
  7. A. J. Schofield, “What does second-order vision see in an image?” Perception 29, 1071–1086 (2000). An additional figure is available at http://www.perceptionweb.com/perc0900/schofield.html .
    [CrossRef]
  8. O. Schwartz, E. P. Simoncelli, “Natural signal statistics and sensory gain control,” Nat. Neurosci. 4, 819–825 (2001).
    [CrossRef] [PubMed]
  9. D. L. Ringach, “Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex,” J. Neurophysiol. 88, 455–463 (2002).
    [PubMed]
  10. E. P. Simoncelli, B. A. Olshausen, “Natural image statistics and neural representation,” Annu. Rev. Neurosci. 24, 1193–1216 (2001).
    [CrossRef] [PubMed]
  11. A. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661–674 (1984).
    [CrossRef]
  12. D. L. Ruderman, “Origins of scaling in natural images,” Vision Res. 37, 3385–3398 (1997).
    [CrossRef]
  13. M. G. A. Thomsom, “Beats, kurtosis and visual coding,” Network 12, 271–287 (2001).
    [CrossRef]
  14. D. L. Ruderman, “The statistics of natural images,” Network 5, 517–548 (1994).
    [CrossRef]
  15. A. J. Bell, T. J. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
    [CrossRef]
  16. B. A. Olshausen, D. J. Field, “Emergence of simple-cell receptive field properties by learning a sparse code for natural images,” Nature (London) 381, 607–609 (1996).
    [CrossRef]
  17. J. H. van Hateren, D. L. Ruderman, “Independent component analysis of natural image sequences yields spatio-temporal filters similar to simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 2315–2320 (1998).
    [CrossRef]
  18. A. Hyvärinen, P. O. Hoyer, “A two-layer sparse coding model learns simple and complex cell receptive fields and topography from natural images,” Vision Res. 41, 2413–2423 (2001).
    [CrossRef] [PubMed]
  19. B. A. Olshausen, D. J. Field, “Sparse coding with an overcomplete basis set: a strategy employed by V1?” Vision Res. 37, 3311–3325 (1996).
    [CrossRef]
  20. D. H. Hubel, T. N. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex,” J. Physiol. (London) 160, 106–154 (1962).
  21. F. W. Campbell, G. F. Cooper, C. Enroth-Cugell, “The spatial selectivity of the visual cells of the cat,” J. Physiol. (London) 203, 223–235 (1969).
  22. 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]
  23. L. A. Palmer, T. L. Davis, “Receptive-field structure in cat striate cortex,” J. Neurophysiol. 46, 260–276 (1981).
    [PubMed]
  24. D. L. Ringach, M. J. Hawkin, R. Shapley, “Dynamics of orientation tuning in primary visual cortex,” Nature (London) 387, 281–284 (1997).
    [CrossRef]
  25. F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).
  26. C. Blakemore, F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. (London) 203, 237–260 (1969).
  27. J. G. Daugman, “Uncertainty relations for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters,” J. Opt. Soc. Am. A 2, 1160–1169 (1985).
    [CrossRef] [PubMed]
  28. D. J. Field, D. J. Tolhurst, “The structure and symmetry of simple cell receptive-field profiles in the cat’s visual cortex,” Proc. R. Soc. London Ser. B 228, 379–400 (1986).
    [CrossRef]
  29. C. Chubb, G. Sperling, “Drift-balanced random stimuli: a general basis for studying non-Fourier motion perception,” J. Opt. Soc. Am. A 5, 1986–2007 (1988).
    [CrossRef] [PubMed]
  30. P. Cavanagh, G. Mather, “Motion: the long and short of it,” Spatial Vision 4, 103–129 (1989).
    [CrossRef] [PubMed]
  31. A. M. Derrington, O. I. Ukkonen, “Second-order motion discrimination by feature tracking,” Vision Res. 39, 1465–1475 (1999).
    [CrossRef] [PubMed]
  32. T. Ledgeway, A. T. Smith, “Evidence for separate motion-detecting mechanisms for first- and second-order motion in human vision,” Vision Res. 34, 2727–2740 (1994).
    [CrossRef] [PubMed]
  33. A. Sutter, G. Sperling, C. Chubb, “Measuring the spatial frequency selectivity of second-order texture mechanisms,” Vision Res. 35, 915–924 (1995).
    [CrossRef] [PubMed]
  34. A. T. Smith, T. Ledgeway, “Separate detection of moving luminance and contrast modulations: fact or artifact?” Vision Res. 37, 45–62 (1997).
    [CrossRef] [PubMed]
  35. A. J. Schofield, M. A. Georgeson, “Sensitivity to modulations of luminance and contrast in visual white noise: separate mechanisms with similar behaviour,” Vision Res. 39, 2697–2716 (1999).
    [CrossRef] [PubMed]
  36. L. M. Vaina, A. Cowey, D. K. Kennedy, “Perception of first- and second-order motion: separable neurological mechanisms?” Hum. Brain Mapp. 7, 67–77 (1999).
    [CrossRef] [PubMed]
  37. A. T. Smith, R. F. Hess, C. L. Baker, “Direction identification thresholds for second-order motion in central and peripheral vision,” J. Opt. Soc. Am. A 11, 506–514 (1994).
    [CrossRef]
  38. S. O. Dumoulin, C. L. Baker, R. F. Hess, A. C. Evans, “Cortical specialization for processing first- and second-order motion,” Cereb. Cortex 13, 1375–1385 (2003).
    [CrossRef] [PubMed]
  39. T. D. Albright, “Form-cue invariant motion processing in primate visual cortex,” Science 255, 1141–1143 (1992).
    [CrossRef] [PubMed]
  40. A. Chaudhuri, T. D. Albright, “Neuronal responses to edges defined by luminance vs. temporal texture in macaque area V1,” Visual Neurosci. 14, 949–962 (1997).
    [CrossRef]
  41. C. L. Baker, “Central neural mechanisms for detecting second-order motion,” Curr. Opin. Neurobiol. 9, 461–466 (1999).
    [CrossRef] [PubMed]
  42. 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.
  43. H. R. Wilson, V. P. Ferrera, C. Yo, “A psychophysically motivated model for two-dimensional motion perception,” Visual Neurosci. 9, 79–97 (1992).
    [CrossRef]
  44. M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
    [CrossRef] [PubMed]
  45. M. S. Landy, N. Graham, “Visual perception of texture,” in The Visual Neurosciences, L. M. Chalupa, J. S. Werner, eds. (MIT Press, Cambridge, Mass., 2003), pp. 1106–1118.
  46. N. Prins, F. A. A. Kingdom, “Detection and discrimination of texture modulations defined by orientation, spatial frequency, and contrast,” J. Opt. Soc. Am. A 20, 401–410 (2003).
    [CrossRef]
  47. C. L. Baker, I. Mareschal, “Processing of second-order stimuli in the visual cortex,” in Vision: From Neurons to Cognition, Progress in Brain Research, C. Casanova, M. Ptito, eds. (Elsevier Science, Amsterdam, 2001), Vol. 134, Chap. 12.
  48. N. Graham, A. Sutter, C. Venkatesan, “Spatial-frequency- and orientation-selectivity of simple and complex channels in region segregation,” Vision Res. 33, 1893–1911 (1993).
    [CrossRef] [PubMed]
  49. F. A. A. Kingdom, D. R. T. Keeble, “On the mechanism for scale invariance in orientation-defined textures,” Vision Res. 39, 1477–1489 (1999).
    [CrossRef] [PubMed]
  50. N. Graham, S. Wolfson, “A note about preferred orientations at the first and second stages of complex (second-order) texture channels,” J. Opt. Soc. Am. A 18, 2273–2281 (2001).
    [CrossRef]
  51. S. C. Dakin, I. Mareschal, “Sensitivity to contrast modulation depends on carrier spatial frequency and orientation,” Vision Res. 40, 311–329 (2000).
    [CrossRef] [PubMed]
  52. J. A. Solomon, G. Sperling, “Full-wave and half-wave rectification in second-order motion perception,” Vision Res. 34, 2239–2257 (1994).
    [CrossRef] [PubMed]
  53. N. Graham, A. Sutter, “Spatial summation in simple (Fourier) and complex (non-Fourier) channels in texture segregation,” Vision Res. 38, 231–257 (1998).
    [CrossRef] [PubMed]
  54. Y.-X. Zhou, C. L. Baker, “Spatial properties of envelope responses in Area 17 and 18 of the cat,” J. Neurophysiol. 75, 1038–1050 (1996).
    [PubMed]
  55. R. C. Gonzalez, R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 2001), pp. 199–205.
  56. I. Mareschal, C. L. Baker, “Cortical processing of second order motion,” Visual Neurosci. 16, 527–540 (1999).
    [CrossRef]
  57. K. V. Marida, Statistics of Directional Data (Academic, London, 1972).
  58. I. Mareschal, C. L. Baker, “Temporal and spatial response to second-order stimuli in cat A18,” J. Neurophysiol. 80, 2811–2823 (1998).
    [PubMed]
  59. D. C. Knill, D. J. Field, D. Kersten, “Human discrimination of fractal images,” J. Opt. Soc. Am. A 7, 1113–1123 (1990).
    [CrossRef] [PubMed]
  60. Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the 2nd-order statistics of natural and synthetic-images,” Vision Res. 34, 541–554 (1994).
    [CrossRef] [PubMed]
  61. S. J. M. Rainville, F. A. A. Kingdom, “Spatial-scale contribution to the detection of mirror symmetry in fractal noise,” J. Opt. Soc. Am. A 16, 2112–2122 (1999).
    [CrossRef]
  62. R. J. Baddeley, P. J. B. Hancock, “A statistical analysis of natural images matches psychophysically derived orientation tuning curves,” Proc. R. Soc. London Ser. B 246, 219–223 (1991).
    [CrossRef]
  63. E. A. Essock, J. K. DeFord, B. C. Hansen, M. J. Sinai, “Oblique stimuli are seen best (not worst!) in naturalistic broad-band stimuli: a horizontal effect,” Vision Res. 43, 1329–1335 (2003).
    [CrossRef] [PubMed]
  64. C. S. Furmanski, S. A. Engel, “An oblique effect in human primary visual cortex,” Nat. Neurosci. 3, 535–536 (2000).
    [CrossRef] [PubMed]
  65. D. Schluppeck, S. A. Engel, “Oblique effect in human MT+ follows pattern rather than component motion,” J. Vision 3, Abstract 282, p. 282a (2003).
  66. B. W. Li, M. R. Peterson, R. D. Freeman, “The oblique effect: a neural basis in the visual cortex,” J. Neurophysiol. 90, 204–217 (2003).
    [CrossRef] [PubMed]
  67. H. B. Barlow, D. J. Tolhurst, “Why do you have edge detectors?” in OSA Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 172.
  68. J. P. Jones, L. A. Palmer, “An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1233–1258 (1987).
    [PubMed]
  69. J. P. Jones, L. A. Palmer, “The two-dimensional spatial structure of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1187–1211 (1987).
    [PubMed]
  70. D. J. Tolhurst, I. D. Thompson, “On the variety of spatial frequency selectivity shown by neurons in area 17 of the cat,” Proc. R. Soc. London Ser. B 213, 183–199 (1981).
    [CrossRef]
  71. D. B. Hamilton, D. G. Albrecht, W. S. Geisler, “Visual cortical receptive fields in monkey and cat: spatial and temporal phase transfer function,” Vision Res. 29, 1285–1308 (1989).
    [CrossRef] [PubMed]
  72. Z.-L. Lu, G. Sperling, “The functional architecture of human visual motion perception,” Vision Res. 35, 2697–2722 (1995).
    [CrossRef] [PubMed]
  73. N. Graham, A. Sutter, “Normalization: contrast-gain control in simple (Fourier) and complex (non-Fourier) pathways of pattern vision,” Vision Res. 40, 2737–2761 (2000).
    [CrossRef] [PubMed]
  74. D. J. Heeger, “Normalization of cell responses in cat striate cortex,” Visual Neurosci. 9, 271–294 (1992).
  75. A. G. Leventhal, Y. Wang, M. T. Schmolesky, Y. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
    [CrossRef]
  76. I. Mareschal, C. L. Baker, “A cortical locus for the processing of contrast-defined contours,” Nat. Neurosci. 1, 150–154 (1998).
    [CrossRef]

2003 (5)

S. O. Dumoulin, C. L. Baker, R. F. Hess, A. C. Evans, “Cortical specialization for processing first- and second-order motion,” Cereb. Cortex 13, 1375–1385 (2003).
[CrossRef] [PubMed]

D. Schluppeck, S. A. Engel, “Oblique effect in human MT+ follows pattern rather than component motion,” J. Vision 3, Abstract 282, p. 282a (2003).

B. W. Li, M. R. Peterson, R. D. Freeman, “The oblique effect: a neural basis in the visual cortex,” J. Neurophysiol. 90, 204–217 (2003).
[CrossRef] [PubMed]

E. A. Essock, J. K. DeFord, B. C. Hansen, M. J. Sinai, “Oblique stimuli are seen best (not worst!) in naturalistic broad-band stimuli: a horizontal effect,” Vision Res. 43, 1329–1335 (2003).
[CrossRef] [PubMed]

N. Prins, F. A. A. Kingdom, “Detection and discrimination of texture modulations defined by orientation, spatial frequency, and contrast,” J. Opt. Soc. Am. A 20, 401–410 (2003).
[CrossRef]

2002 (1)

D. L. Ringach, “Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex,” J. Neurophysiol. 88, 455–463 (2002).
[PubMed]

2001 (5)

E. P. Simoncelli, B. A. Olshausen, “Natural image statistics and neural representation,” Annu. Rev. Neurosci. 24, 1193–1216 (2001).
[CrossRef] [PubMed]

O. Schwartz, E. P. Simoncelli, “Natural signal statistics and sensory gain control,” Nat. Neurosci. 4, 819–825 (2001).
[CrossRef] [PubMed]

M. G. A. Thomsom, “Beats, kurtosis and visual coding,” Network 12, 271–287 (2001).
[CrossRef]

A. Hyvärinen, P. O. Hoyer, “A two-layer sparse coding model learns simple and complex cell receptive fields and topography from natural images,” Vision Res. 41, 2413–2423 (2001).
[CrossRef] [PubMed]

N. Graham, S. Wolfson, “A note about preferred orientations at the first and second stages of complex (second-order) texture channels,” J. Opt. Soc. Am. A 18, 2273–2281 (2001).
[CrossRef]

2000 (4)

A. J. Schofield, “What does second-order vision see in an image?” Perception 29, 1071–1086 (2000). An additional figure is available at http://www.perceptionweb.com/perc0900/schofield.html .
[CrossRef]

C. S. Furmanski, S. A. Engel, “An oblique effect in human primary visual cortex,” Nat. Neurosci. 3, 535–536 (2000).
[CrossRef] [PubMed]

N. Graham, A. Sutter, “Normalization: contrast-gain control in simple (Fourier) and complex (non-Fourier) pathways of pattern vision,” Vision Res. 40, 2737–2761 (2000).
[CrossRef] [PubMed]

S. C. Dakin, I. Mareschal, “Sensitivity to contrast modulation depends on carrier spatial frequency and orientation,” Vision Res. 40, 311–329 (2000).
[CrossRef] [PubMed]

1999 (7)

I. Mareschal, C. L. Baker, “Cortical processing of second order motion,” Visual Neurosci. 16, 527–540 (1999).
[CrossRef]

S. J. M. Rainville, F. A. A. Kingdom, “Spatial-scale contribution to the detection of mirror symmetry in fractal noise,” J. Opt. Soc. Am. A 16, 2112–2122 (1999).
[CrossRef]

C. L. Baker, “Central neural mechanisms for detecting second-order motion,” Curr. Opin. Neurobiol. 9, 461–466 (1999).
[CrossRef] [PubMed]

F. A. A. Kingdom, D. R. T. Keeble, “On the mechanism for scale invariance in orientation-defined textures,” Vision Res. 39, 1477–1489 (1999).
[CrossRef] [PubMed]

A. J. Schofield, M. A. Georgeson, “Sensitivity to modulations of luminance and contrast in visual white noise: separate mechanisms with similar behaviour,” Vision Res. 39, 2697–2716 (1999).
[CrossRef] [PubMed]

L. M. Vaina, A. Cowey, D. K. Kennedy, “Perception of first- and second-order motion: separable neurological mechanisms?” Hum. Brain Mapp. 7, 67–77 (1999).
[CrossRef] [PubMed]

A. M. Derrington, O. I. Ukkonen, “Second-order motion discrimination by feature tracking,” Vision Res. 39, 1465–1475 (1999).
[CrossRef] [PubMed]

1998 (5)

J. H. van Hateren, D. L. Ruderman, “Independent component analysis of natural image sequences yields spatio-temporal filters similar to simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 2315–2320 (1998).
[CrossRef]

I. Mareschal, C. L. Baker, “Temporal and spatial response to second-order stimuli in cat A18,” J. Neurophysiol. 80, 2811–2823 (1998).
[PubMed]

N. Graham, A. Sutter, “Spatial summation in simple (Fourier) and complex (non-Fourier) channels in texture segregation,” Vision Res. 38, 231–257 (1998).
[CrossRef] [PubMed]

A. G. Leventhal, Y. Wang, M. T. Schmolesky, Y. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
[CrossRef]

I. Mareschal, C. L. Baker, “A cortical locus for the processing of contrast-defined contours,” Nat. Neurosci. 1, 150–154 (1998).
[CrossRef]

1997 (6)

A. Chaudhuri, T. D. Albright, “Neuronal responses to edges defined by luminance vs. temporal texture in macaque area V1,” Visual Neurosci. 14, 949–962 (1997).
[CrossRef]

D. L. Ringach, M. J. Hawkin, R. Shapley, “Dynamics of orientation tuning in primary visual cortex,” Nature (London) 387, 281–284 (1997).
[CrossRef]

A. T. Smith, T. Ledgeway, “Separate detection of moving luminance and contrast modulations: fact or artifact?” Vision Res. 37, 45–62 (1997).
[CrossRef] [PubMed]

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3384 (1997).
[CrossRef]

A. J. Bell, T. J. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
[CrossRef]

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

1996 (5)

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

B. A. Olshausen, D. J. Field, “Sparse coding with an overcomplete basis set: a strategy employed by V1?” Vision Res. 37, 3311–3325 (1996).
[CrossRef]

A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

R. J. Baddeley, “Searching for filters with ‘interesting’ output distributions: an uninteresting direction to explore?” Network 7, 409–421 (1996).
[CrossRef] [PubMed]

Y.-X. Zhou, C. L. Baker, “Spatial properties of envelope responses in Area 17 and 18 of the cat,” J. Neurophysiol. 75, 1038–1050 (1996).
[PubMed]

1995 (2)

Z.-L. Lu, G. Sperling, “The functional architecture of human visual motion perception,” Vision Res. 35, 2697–2722 (1995).
[CrossRef] [PubMed]

A. Sutter, G. Sperling, C. Chubb, “Measuring the spatial frequency selectivity of second-order texture mechanisms,” Vision Res. 35, 915–924 (1995).
[CrossRef] [PubMed]

1994 (6)

T. Ledgeway, A. T. Smith, “Evidence for separate motion-detecting mechanisms for first- and second-order motion in human vision,” Vision Res. 34, 2727–2740 (1994).
[CrossRef] [PubMed]

D. J. Field, “What is the goal of sensory coding,” Neural Comput. 6, 559–601 (1994).
[CrossRef]

D. L. Ruderman, “The statistics of natural images,” Network 5, 517–548 (1994).
[CrossRef]

J. A. Solomon, G. Sperling, “Full-wave and half-wave rectification in second-order motion perception,” Vision Res. 34, 2239–2257 (1994).
[CrossRef] [PubMed]

Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the 2nd-order statistics of natural and synthetic-images,” Vision Res. 34, 541–554 (1994).
[CrossRef] [PubMed]

A. T. Smith, R. F. Hess, C. L. Baker, “Direction identification thresholds for second-order motion in central and peripheral vision,” J. Opt. Soc. Am. A 11, 506–514 (1994).
[CrossRef]

1993 (1)

N. Graham, A. Sutter, C. Venkatesan, “Spatial-frequency- and orientation-selectivity of simple and complex channels in region segregation,” Vision Res. 33, 1893–1911 (1993).
[CrossRef] [PubMed]

1992 (4)

H. R. Wilson, V. P. Ferrera, C. Yo, “A psychophysically motivated model for two-dimensional motion perception,” Visual Neurosci. 9, 79–97 (1992).
[CrossRef]

D. J. Heeger, “Normalization of cell responses in cat striate cortex,” Visual Neurosci. 9, 271–294 (1992).

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

T. D. Albright, “Form-cue invariant motion processing in primate visual cortex,” Science 255, 1141–1143 (1992).
[CrossRef] [PubMed]

1991 (2)

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

R. J. Baddeley, P. J. B. Hancock, “A statistical analysis of natural images matches psychophysically derived orientation tuning curves,” Proc. R. Soc. London Ser. B 246, 219–223 (1991).
[CrossRef]

1990 (1)

1989 (2)

D. B. Hamilton, D. G. Albrecht, W. S. Geisler, “Visual cortical receptive fields in monkey and cat: spatial and temporal phase transfer function,” Vision Res. 29, 1285–1308 (1989).
[CrossRef] [PubMed]

P. Cavanagh, G. Mather, “Motion: the long and short of it,” Spatial Vision 4, 103–129 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (3)

D. J. Field, “Relations between the statistics of natural images and the response profiles of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
[CrossRef] [PubMed]

J. P. Jones, L. A. Palmer, “An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1233–1258 (1987).
[PubMed]

J. P. Jones, L. A. Palmer, “The two-dimensional spatial structure of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1187–1211 (1987).
[PubMed]

1986 (1)

D. J. Field, D. J. Tolhurst, “The structure and symmetry of simple cell receptive-field profiles in the cat’s visual cortex,” Proc. R. Soc. London Ser. B 228, 379–400 (1986).
[CrossRef]

1985 (1)

1984 (1)

A. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661–674 (1984).
[CrossRef]

1982 (1)

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]

1981 (2)

L. A. Palmer, T. L. Davis, “Receptive-field structure in cat striate cortex,” J. Neurophysiol. 46, 260–276 (1981).
[PubMed]

D. J. Tolhurst, I. D. Thompson, “On the variety of spatial frequency selectivity shown by neurons in area 17 of the cat,” Proc. R. Soc. London Ser. B 213, 183–199 (1981).
[CrossRef]

1969 (2)

F. W. Campbell, G. F. Cooper, C. Enroth-Cugell, “The spatial selectivity of the visual cells of the cat,” J. Physiol. (London) 203, 223–235 (1969).

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

1968 (1)

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

1962 (1)

D. H. Hubel, T. N. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex,” J. Physiol. (London) 160, 106–154 (1962).

Albrecht, D. G.

D. B. Hamilton, D. G. Albrecht, W. S. Geisler, “Visual cortical receptive fields in monkey and cat: spatial and temporal phase transfer function,” Vision Res. 29, 1285–1308 (1989).
[CrossRef] [PubMed]

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]

Albright, T. D.

A. Chaudhuri, T. D. Albright, “Neuronal responses to edges defined by luminance vs. temporal texture in macaque area V1,” Visual Neurosci. 14, 949–962 (1997).
[CrossRef]

T. D. Albright, “Form-cue invariant motion processing in primate visual cortex,” Science 255, 1141–1143 (1992).
[CrossRef] [PubMed]

Baddeley, R. J.

R. J. Baddeley, “Searching for filters with ‘interesting’ output distributions: an uninteresting direction to explore?” Network 7, 409–421 (1996).
[CrossRef] [PubMed]

R. J. Baddeley, P. J. B. Hancock, “A statistical analysis of natural images matches psychophysically derived orientation tuning curves,” Proc. R. Soc. London Ser. B 246, 219–223 (1991).
[CrossRef]

Baker, C. L.

S. O. Dumoulin, C. L. Baker, R. F. Hess, A. C. Evans, “Cortical specialization for processing first- and second-order motion,” Cereb. Cortex 13, 1375–1385 (2003).
[CrossRef] [PubMed]

I. Mareschal, C. L. Baker, “Cortical processing of second order motion,” Visual Neurosci. 16, 527–540 (1999).
[CrossRef]

C. L. Baker, “Central neural mechanisms for detecting second-order motion,” Curr. Opin. Neurobiol. 9, 461–466 (1999).
[CrossRef] [PubMed]

I. Mareschal, C. L. Baker, “A cortical locus for the processing of contrast-defined contours,” Nat. Neurosci. 1, 150–154 (1998).
[CrossRef]

I. Mareschal, C. L. Baker, “Temporal and spatial response to second-order stimuli in cat A18,” J. Neurophysiol. 80, 2811–2823 (1998).
[PubMed]

Y.-X. Zhou, C. L. Baker, “Spatial properties of envelope responses in Area 17 and 18 of the cat,” J. Neurophysiol. 75, 1038–1050 (1996).
[PubMed]

A. T. Smith, R. F. Hess, C. L. Baker, “Direction identification thresholds for second-order motion in central and peripheral vision,” J. Opt. Soc. Am. A 11, 506–514 (1994).
[CrossRef]

C. L. Baker, I. Mareschal, “Processing of second-order stimuli in the visual cortex,” in Vision: From Neurons to Cognition, Progress in Brain Research, C. Casanova, M. Ptito, eds. (Elsevier Science, Amsterdam, 2001), Vol. 134, Chap. 12.

Barlow, H. B.

H. B. Barlow, D. J. Tolhurst, “Why do you have edge detectors?” in OSA Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 172.

Bell, A. J.

A. J. Bell, T. J. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
[CrossRef]

Bergen, J. R.

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

Blakemore, C.

C. Blakemore, F. W. Campbell, “On the existence of neurones in the human visual 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, 3367–3384 (1997).
[CrossRef]

Campbell, F. W.

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

F. W. Campbell, G. F. Cooper, C. Enroth-Cugell, “The spatial selectivity of the visual cells of the cat,” J. Physiol. (London) 203, 223–235 (1969).

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

Cavanagh, P.

P. Cavanagh, G. Mather, “Motion: the long and short of it,” Spatial Vision 4, 103–129 (1989).
[CrossRef] [PubMed]

Chao, T.

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

Chaudhuri, A.

A. Chaudhuri, T. D. Albright, “Neuronal responses to edges defined by luminance vs. temporal texture in macaque area V1,” Visual Neurosci. 14, 949–962 (1997).
[CrossRef]

Chubb, C.

A. Sutter, G. Sperling, C. Chubb, “Measuring the spatial frequency selectivity of second-order texture mechanisms,” Vision Res. 35, 915–924 (1995).
[CrossRef] [PubMed]

C. Chubb, G. Sperling, “Drift-balanced random stimuli: a general basis for studying non-Fourier motion perception,” J. Opt. Soc. Am. A 5, 1986–2007 (1988).
[CrossRef] [PubMed]

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.

Cooper, G. F.

F. W. Campbell, G. F. Cooper, C. Enroth-Cugell, “The spatial selectivity of the visual cells of the cat,” J. Physiol. (London) 203, 223–235 (1969).

Cowey, A.

L. M. Vaina, A. Cowey, D. K. Kennedy, “Perception of first- and second-order motion: separable neurological mechanisms?” Hum. Brain Mapp. 7, 67–77 (1999).
[CrossRef] [PubMed]

Dakin, S. C.

S. C. Dakin, I. Mareschal, “Sensitivity to contrast modulation depends on carrier spatial frequency and orientation,” Vision Res. 40, 311–329 (2000).
[CrossRef] [PubMed]

Daugman, J. G.

Davis, T. L.

L. A. Palmer, T. L. Davis, “Receptive-field structure in cat striate cortex,” J. Neurophysiol. 46, 260–276 (1981).
[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]

DeFord, J. K.

E. A. Essock, J. K. DeFord, B. C. Hansen, M. J. Sinai, “Oblique stimuli are seen best (not worst!) in naturalistic broad-band stimuli: a horizontal effect,” Vision Res. 43, 1329–1335 (2003).
[CrossRef] [PubMed]

Derrington, A. M.

A. M. Derrington, O. I. Ukkonen, “Second-order motion discrimination by feature tracking,” Vision Res. 39, 1465–1475 (1999).
[CrossRef] [PubMed]

Dumoulin, S. O.

S. O. Dumoulin, C. L. Baker, R. F. Hess, A. C. Evans, “Cortical specialization for processing first- and second-order motion,” Cereb. Cortex 13, 1375–1385 (2003).
[CrossRef] [PubMed]

Engel, S. A.

D. Schluppeck, S. A. Engel, “Oblique effect in human MT+ follows pattern rather than component motion,” J. Vision 3, Abstract 282, p. 282a (2003).

C. S. Furmanski, S. A. Engel, “An oblique effect in human primary visual cortex,” Nat. Neurosci. 3, 535–536 (2000).
[CrossRef] [PubMed]

Enroth-Cugell, C.

F. W. Campbell, G. F. Cooper, C. Enroth-Cugell, “The spatial selectivity of the visual cells of the cat,” J. Physiol. (London) 203, 223–235 (1969).

Essock, E. A.

E. A. Essock, J. K. DeFord, B. C. Hansen, M. J. Sinai, “Oblique stimuli are seen best (not worst!) in naturalistic broad-band stimuli: a horizontal effect,” Vision Res. 43, 1329–1335 (2003).
[CrossRef] [PubMed]

Evans, A. C.

S. O. Dumoulin, C. L. Baker, R. F. Hess, A. C. Evans, “Cortical specialization for processing first- and second-order motion,” Cereb. Cortex 13, 1375–1385 (2003).
[CrossRef] [PubMed]

Ferrera, V. P.

H. R. Wilson, V. P. Ferrera, C. Yo, “A psychophysically motivated model for two-dimensional motion perception,” Visual Neurosci. 9, 79–97 (1992).
[CrossRef]

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, 3367–3384 (1997).
[CrossRef]

B. A. Olshausen, D. J. Field, “Sparse coding with an overcomplete basis set: a strategy employed by V1?” Vision Res. 37, 3311–3325 (1996).
[CrossRef]

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

D. J. Field, “What is the goal of sensory coding,” Neural Comput. 6, 559–601 (1994).
[CrossRef]

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

D. J. Field, “Relations between the statistics of natural images and the response profiles of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
[CrossRef] [PubMed]

D. J. Field, D. J. Tolhurst, “The structure and symmetry of simple cell receptive-field profiles in the cat’s visual cortex,” Proc. R. Soc. London Ser. B 228, 379–400 (1986).
[CrossRef]

Freeman, R. D.

B. W. Li, M. R. Peterson, R. D. Freeman, “The oblique effect: a neural basis in the visual cortex,” J. Neurophysiol. 90, 204–217 (2003).
[CrossRef] [PubMed]

Furmanski, C. S.

C. S. Furmanski, S. A. Engel, “An oblique effect in human primary visual cortex,” Nat. Neurosci. 3, 535–536 (2000).
[CrossRef] [PubMed]

Geisler, W. S.

D. B. Hamilton, D. G. Albrecht, W. S. Geisler, “Visual cortical receptive fields in monkey and cat: spatial and temporal phase transfer function,” Vision Res. 29, 1285–1308 (1989).
[CrossRef] [PubMed]

Georgeson, M. A.

A. J. Schofield, M. A. Georgeson, “Sensitivity to modulations of luminance and contrast in visual white noise: separate mechanisms with similar behaviour,” Vision Res. 39, 2697–2716 (1999).
[CrossRef] [PubMed]

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 2001), pp. 199–205.

Graham, N.

N. Graham, S. Wolfson, “A note about preferred orientations at the first and second stages of complex (second-order) texture channels,” J. Opt. Soc. Am. A 18, 2273–2281 (2001).
[CrossRef]

N. Graham, A. Sutter, “Normalization: contrast-gain control in simple (Fourier) and complex (non-Fourier) pathways of pattern vision,” Vision Res. 40, 2737–2761 (2000).
[CrossRef] [PubMed]

N. Graham, A. Sutter, “Spatial summation in simple (Fourier) and complex (non-Fourier) channels in texture segregation,” Vision Res. 38, 231–257 (1998).
[CrossRef] [PubMed]

N. Graham, A. Sutter, C. Venkatesan, “Spatial-frequency- and orientation-selectivity of simple and complex channels in region segregation,” Vision Res. 33, 1893–1911 (1993).
[CrossRef] [PubMed]

M. S. Landy, N. Graham, “Visual perception of texture,” in The Visual Neurosciences, L. M. Chalupa, J. S. Werner, eds. (MIT Press, Cambridge, Mass., 2003), pp. 1106–1118.

Hamilton, D. B.

D. B. Hamilton, D. G. Albrecht, W. S. Geisler, “Visual cortical receptive fields in monkey and cat: spatial and temporal phase transfer function,” Vision Res. 29, 1285–1308 (1989).
[CrossRef] [PubMed]

Hancock, P. J. B.

R. J. Baddeley, P. J. B. Hancock, “A statistical analysis of natural images matches psychophysically derived orientation tuning curves,” Proc. R. Soc. London Ser. B 246, 219–223 (1991).
[CrossRef]

Hansen, B. C.

E. A. Essock, J. K. DeFord, B. C. Hansen, M. J. Sinai, “Oblique stimuli are seen best (not worst!) in naturalistic broad-band stimuli: a horizontal effect,” Vision Res. 43, 1329–1335 (2003).
[CrossRef] [PubMed]

Hawkin, M. J.

D. L. Ringach, M. J. Hawkin, R. Shapley, “Dynamics of orientation tuning in primary visual cortex,” Nature (London) 387, 281–284 (1997).
[CrossRef]

Heeger, D. J.

D. J. Heeger, “Normalization of cell responses in cat striate cortex,” Visual Neurosci. 9, 271–294 (1992).

Hess, R. F.

S. O. Dumoulin, C. L. Baker, R. F. Hess, A. C. Evans, “Cortical specialization for processing first- and second-order motion,” Cereb. Cortex 13, 1375–1385 (2003).
[CrossRef] [PubMed]

A. T. Smith, R. F. Hess, C. L. Baker, “Direction identification thresholds for second-order motion in central and peripheral vision,” J. Opt. Soc. Am. A 11, 506–514 (1994).
[CrossRef]

Hoyer, P. O.

A. Hyvärinen, P. O. Hoyer, “A two-layer sparse coding model learns simple and complex cell receptive fields and topography from natural images,” Vision Res. 41, 2413–2423 (2001).
[CrossRef] [PubMed]

Hubel, D. H.

D. H. Hubel, T. N. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex,” J. Physiol. (London) 160, 106–154 (1962).

Hyvärinen, A.

A. Hyvärinen, P. O. Hoyer, “A two-layer sparse coding model learns simple and complex cell receptive fields and topography from natural images,” Vision Res. 41, 2413–2423 (2001).
[CrossRef] [PubMed]

Jones, J. P.

J. P. Jones, L. A. Palmer, “The two-dimensional spatial structure of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1187–1211 (1987).
[PubMed]

J. P. Jones, L. A. Palmer, “An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1233–1258 (1987).
[PubMed]

Keeble, D. R. T.

F. A. A. Kingdom, D. R. T. Keeble, “On the mechanism for scale invariance in orientation-defined textures,” Vision Res. 39, 1477–1489 (1999).
[CrossRef] [PubMed]

Kennedy, D. K.

L. M. Vaina, A. Cowey, D. K. Kennedy, “Perception of first- and second-order motion: separable neurological mechanisms?” Hum. Brain Mapp. 7, 67–77 (1999).
[CrossRef] [PubMed]

Kersten, D.

Kingdom, F. A. A.

Knill, D. C.

Landy, M. S.

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 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. S. Landy, N. Graham, “Visual perception of texture,” in The Visual Neurosciences, L. M. Chalupa, J. S. Werner, eds. (MIT Press, Cambridge, Mass., 2003), pp. 1106–1118.

Ledgeway, T.

A. T. Smith, T. Ledgeway, “Separate detection of moving luminance and contrast modulations: fact or artifact?” Vision Res. 37, 45–62 (1997).
[CrossRef] [PubMed]

T. Ledgeway, A. T. Smith, “Evidence for separate motion-detecting mechanisms for first- and second-order motion in human vision,” Vision Res. 34, 2727–2740 (1994).
[CrossRef] [PubMed]

Leventhal, A. G.

A. G. Leventhal, Y. Wang, M. T. Schmolesky, Y. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
[CrossRef]

Li, B. W.

B. W. Li, M. R. Peterson, R. D. Freeman, “The oblique effect: a neural basis in the visual cortex,” J. Neurophysiol. 90, 204–217 (2003).
[CrossRef] [PubMed]

Lu, Z.-L.

Z.-L. Lu, G. Sperling, “The functional architecture of human visual motion perception,” Vision Res. 35, 2697–2722 (1995).
[CrossRef] [PubMed]

Mareschal, I.

S. C. Dakin, I. Mareschal, “Sensitivity to contrast modulation depends on carrier spatial frequency and orientation,” Vision Res. 40, 311–329 (2000).
[CrossRef] [PubMed]

I. Mareschal, C. L. Baker, “Cortical processing of second order motion,” Visual Neurosci. 16, 527–540 (1999).
[CrossRef]

I. Mareschal, C. L. Baker, “A cortical locus for the processing of contrast-defined contours,” Nat. Neurosci. 1, 150–154 (1998).
[CrossRef]

I. Mareschal, C. L. Baker, “Temporal and spatial response to second-order stimuli in cat A18,” J. Neurophysiol. 80, 2811–2823 (1998).
[PubMed]

C. L. Baker, I. Mareschal, “Processing of second-order stimuli in the visual cortex,” in Vision: From Neurons to Cognition, Progress in Brain Research, C. Casanova, M. Ptito, eds. (Elsevier Science, Amsterdam, 2001), Vol. 134, Chap. 12.

Marida, K. V.

K. V. Marida, Statistics of Directional Data (Academic, London, 1972).

Mather, G.

P. Cavanagh, G. Mather, “Motion: the long and short of it,” Spatial Vision 4, 103–129 (1989).
[CrossRef] [PubMed]

Olshausen, B. A.

E. P. Simoncelli, B. A. Olshausen, “Natural image statistics and neural representation,” Annu. Rev. Neurosci. 24, 1193–1216 (2001).
[CrossRef] [PubMed]

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

B. A. Olshausen, D. J. Field, “Sparse coding with an overcomplete basis set: a strategy employed by V1?” Vision Res. 37, 3311–3325 (1996).
[CrossRef]

Palmer, L. A.

J. P. Jones, L. A. Palmer, “The two-dimensional spatial structure of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1187–1211 (1987).
[PubMed]

J. P. Jones, L. A. Palmer, “An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1233–1258 (1987).
[PubMed]

L. A. Palmer, T. L. Davis, “Receptive-field structure in cat striate cortex,” J. Neurophysiol. 46, 260–276 (1981).
[PubMed]

Pentland, A.

A. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661–674 (1984).
[CrossRef]

Peterson, M. R.

B. W. Li, M. R. Peterson, R. D. Freeman, “The oblique effect: a neural basis in the visual cortex,” J. Neurophysiol. 90, 204–217 (2003).
[CrossRef] [PubMed]

Prins, N.

Rainville, S. J. M.

Ringach, D. L.

D. L. Ringach, “Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex,” J. Neurophysiol. 88, 455–463 (2002).
[PubMed]

D. L. Ringach, M. J. Hawkin, R. Shapley, “Dynamics of orientation tuning in primary visual cortex,” Nature (London) 387, 281–284 (1997).
[CrossRef]

Robson, J. G.

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.

J. H. van Hateren, D. L. Ruderman, “Independent component analysis of natural image sequences yields spatio-temporal filters similar to simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 2315–2320 (1998).
[CrossRef]

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

D. L. Ruderman, “The statistics of natural images,” Network 5, 517–548 (1994).
[CrossRef]

Schluppeck, D.

D. Schluppeck, S. A. Engel, “Oblique effect in human MT+ follows pattern rather than component motion,” J. Vision 3, Abstract 282, p. 282a (2003).

Schmolesky, M. T.

A. G. Leventhal, Y. Wang, M. T. Schmolesky, Y. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
[CrossRef]

Schofield, A. J.

A. J. Schofield, “What does second-order vision see in an image?” Perception 29, 1071–1086 (2000). An additional figure is available at http://www.perceptionweb.com/perc0900/schofield.html .
[CrossRef]

A. J. Schofield, M. A. Georgeson, “Sensitivity to modulations of luminance and contrast in visual white noise: separate mechanisms with similar behaviour,” Vision Res. 39, 2697–2716 (1999).
[CrossRef] [PubMed]

Schwartz, O.

O. Schwartz, E. P. Simoncelli, “Natural signal statistics and sensory gain control,” Nat. Neurosci. 4, 819–825 (2001).
[CrossRef] [PubMed]

Sejnowski, T. J.

A. J. Bell, T. J. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
[CrossRef]

Shapley, R.

D. L. Ringach, M. J. Hawkin, R. Shapley, “Dynamics of orientation tuning in primary visual cortex,” Nature (London) 387, 281–284 (1997).
[CrossRef]

Simoncelli, E. P.

E. P. Simoncelli, B. A. Olshausen, “Natural image statistics and neural representation,” Annu. Rev. Neurosci. 24, 1193–1216 (2001).
[CrossRef] [PubMed]

O. Schwartz, E. P. Simoncelli, “Natural signal statistics and sensory gain control,” Nat. Neurosci. 4, 819–825 (2001).
[CrossRef] [PubMed]

Sinai, M. J.

E. A. Essock, J. K. DeFord, B. C. Hansen, M. J. Sinai, “Oblique stimuli are seen best (not worst!) in naturalistic broad-band stimuli: a horizontal effect,” Vision Res. 43, 1329–1335 (2003).
[CrossRef] [PubMed]

Smith, A. T.

A. T. Smith, T. Ledgeway, “Separate detection of moving luminance and contrast modulations: fact or artifact?” Vision Res. 37, 45–62 (1997).
[CrossRef] [PubMed]

T. Ledgeway, A. T. Smith, “Evidence for separate motion-detecting mechanisms for first- and second-order motion in human vision,” Vision Res. 34, 2727–2740 (1994).
[CrossRef] [PubMed]

A. T. Smith, R. F. Hess, C. L. Baker, “Direction identification thresholds for second-order motion in central and peripheral vision,” J. Opt. Soc. Am. A 11, 506–514 (1994).
[CrossRef]

Solomon, J. A.

J. A. Solomon, G. Sperling, “Full-wave and half-wave rectification in second-order motion perception,” Vision Res. 34, 2239–2257 (1994).
[CrossRef] [PubMed]

Sperling, G.

Z.-L. Lu, G. Sperling, “The functional architecture of human visual motion perception,” Vision Res. 35, 2697–2722 (1995).
[CrossRef] [PubMed]

A. Sutter, G. Sperling, C. Chubb, “Measuring the spatial frequency selectivity of second-order texture mechanisms,” Vision Res. 35, 915–924 (1995).
[CrossRef] [PubMed]

J. A. Solomon, G. Sperling, “Full-wave and half-wave rectification in second-order motion perception,” Vision Res. 34, 2239–2257 (1994).
[CrossRef] [PubMed]

C. Chubb, G. Sperling, “Drift-balanced random stimuli: a general basis for studying non-Fourier motion perception,” J. Opt. Soc. Am. A 5, 1986–2007 (1988).
[CrossRef] [PubMed]

Sutter, A.

N. Graham, A. Sutter, “Normalization: contrast-gain control in simple (Fourier) and complex (non-Fourier) pathways of pattern vision,” Vision Res. 40, 2737–2761 (2000).
[CrossRef] [PubMed]

N. Graham, A. Sutter, “Spatial summation in simple (Fourier) and complex (non-Fourier) channels in texture segregation,” Vision Res. 38, 231–257 (1998).
[CrossRef] [PubMed]

A. Sutter, G. Sperling, C. Chubb, “Measuring the spatial frequency selectivity of second-order texture mechanisms,” Vision Res. 35, 915–924 (1995).
[CrossRef] [PubMed]

N. Graham, A. Sutter, C. Venkatesan, “Spatial-frequency- and orientation-selectivity of simple and complex channels in region segregation,” Vision Res. 33, 1893–1911 (1993).
[CrossRef] [PubMed]

Tadmor, Y.

Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the 2nd-order statistics of natural and synthetic-images,” Vision Res. 34, 541–554 (1994).
[CrossRef] [PubMed]

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

Thompson, I. D.

D. J. Tolhurst, I. D. Thompson, “On the variety of spatial frequency selectivity shown by neurons in area 17 of the cat,” Proc. R. Soc. London Ser. B 213, 183–199 (1981).
[CrossRef]

Thomsom, M. G. A.

M. G. A. Thomsom, “Beats, kurtosis and visual coding,” Network 12, 271–287 (2001).
[CrossRef]

Thorell, L. 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]

Tolhurst, D. J.

Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the 2nd-order statistics of natural and synthetic-images,” Vision Res. 34, 541–554 (1994).
[CrossRef] [PubMed]

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

D. J. Field, D. J. Tolhurst, “The structure and symmetry of simple cell receptive-field profiles in the cat’s visual cortex,” Proc. R. Soc. London Ser. B 228, 379–400 (1986).
[CrossRef]

D. J. Tolhurst, I. D. Thompson, “On the variety of spatial frequency selectivity shown by neurons in area 17 of the cat,” Proc. R. Soc. London Ser. B 213, 183–199 (1981).
[CrossRef]

H. B. Barlow, D. J. Tolhurst, “Why do you have edge detectors?” in OSA Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 172.

Ukkonen, O. I.

A. M. Derrington, O. I. Ukkonen, “Second-order motion discrimination by feature tracking,” Vision Res. 39, 1465–1475 (1999).
[CrossRef] [PubMed]

Vaina, L. M.

L. M. Vaina, A. Cowey, D. K. Kennedy, “Perception of first- and second-order motion: separable neurological mechanisms?” Hum. Brain Mapp. 7, 67–77 (1999).
[CrossRef] [PubMed]

van der Schaaf, A.

A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

van Hateren, J. H.

J. H. van Hateren, D. L. Ruderman, “Independent component analysis of natural image sequences yields spatio-temporal filters similar to simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 2315–2320 (1998).
[CrossRef]

A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

Venkatesan, C.

N. Graham, A. Sutter, C. Venkatesan, “Spatial-frequency- and orientation-selectivity of simple and complex channels in region segregation,” Vision Res. 33, 1893–1911 (1993).
[CrossRef] [PubMed]

Wang, Y.

A. G. Leventhal, Y. Wang, M. T. Schmolesky, Y. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
[CrossRef]

Wiesel, T. N.

D. H. Hubel, T. N. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex,” J. Physiol. (London) 160, 106–154 (1962).

Wilson, H. R.

H. R. Wilson, V. P. Ferrera, C. Yo, “A psychophysically motivated model for two-dimensional motion perception,” Visual Neurosci. 9, 79–97 (1992).
[CrossRef]

Wolfson, S.

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 2001), pp. 199–205.

Yo, C.

H. R. Wilson, V. P. Ferrera, C. Yo, “A psychophysically motivated model for two-dimensional motion perception,” Visual Neurosci. 9, 79–97 (1992).
[CrossRef]

Zhou, Y.

A. G. Leventhal, Y. Wang, M. T. Schmolesky, Y. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
[CrossRef]

Zhou, Y.-X.

Y.-X. Zhou, C. L. Baker, “Spatial properties of envelope responses in Area 17 and 18 of the cat,” J. Neurophysiol. 75, 1038–1050 (1996).
[PubMed]

Annu. Rev. Neurosci. (1)

E. P. Simoncelli, B. A. Olshausen, “Natural image statistics and neural representation,” Annu. Rev. Neurosci. 24, 1193–1216 (2001).
[CrossRef] [PubMed]

Cereb. Cortex (1)

S. O. Dumoulin, C. L. Baker, R. F. Hess, A. C. Evans, “Cortical specialization for processing first- and second-order motion,” Cereb. Cortex 13, 1375–1385 (2003).
[CrossRef] [PubMed]

Curr. Opin. Neurobiol. (1)

C. L. Baker, “Central neural mechanisms for detecting second-order motion,” Curr. Opin. Neurobiol. 9, 461–466 (1999).
[CrossRef] [PubMed]

Hum. Brain Mapp. (1)

L. M. Vaina, A. Cowey, D. K. Kennedy, “Perception of first- and second-order motion: separable neurological mechanisms?” Hum. Brain Mapp. 7, 67–77 (1999).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

A. Pentland, “Fractal-based description of natural scenes,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661–674 (1984).
[CrossRef]

J. Neurophysiol. (7)

D. L. Ringach, “Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex,” J. Neurophysiol. 88, 455–463 (2002).
[PubMed]

L. A. Palmer, T. L. Davis, “Receptive-field structure in cat striate cortex,” J. Neurophysiol. 46, 260–276 (1981).
[PubMed]

Y.-X. Zhou, C. L. Baker, “Spatial properties of envelope responses in Area 17 and 18 of the cat,” J. Neurophysiol. 75, 1038–1050 (1996).
[PubMed]

I. Mareschal, C. L. Baker, “Temporal and spatial response to second-order stimuli in cat A18,” J. Neurophysiol. 80, 2811–2823 (1998).
[PubMed]

J. P. Jones, L. A. Palmer, “An evaluation of the two-dimensional Gabor filter model of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1233–1258 (1987).
[PubMed]

J. P. Jones, L. A. Palmer, “The two-dimensional spatial structure of simple receptive fields in cat striate cortex,” J. Neurophysiol. 58, 1187–1211 (1987).
[PubMed]

B. W. Li, M. R. Peterson, R. D. Freeman, “The oblique effect: a neural basis in the visual cortex,” J. Neurophysiol. 90, 204–217 (2003).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (8)

J. Physiol. (London) (4)

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

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

D. H. Hubel, T. N. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex,” J. Physiol. (London) 160, 106–154 (1962).

F. W. Campbell, G. F. Cooper, C. Enroth-Cugell, “The spatial selectivity of the visual cells of the cat,” J. Physiol. (London) 203, 223–235 (1969).

J. Vision (1)

D. Schluppeck, S. A. Engel, “Oblique effect in human MT+ follows pattern rather than component motion,” J. Vision 3, Abstract 282, p. 282a (2003).

Nat. Neurosci. (3)

I. Mareschal, C. L. Baker, “A cortical locus for the processing of contrast-defined contours,” Nat. Neurosci. 1, 150–154 (1998).
[CrossRef]

C. S. Furmanski, S. A. Engel, “An oblique effect in human primary visual cortex,” Nat. Neurosci. 3, 535–536 (2000).
[CrossRef] [PubMed]

O. Schwartz, E. P. Simoncelli, “Natural signal statistics and sensory gain control,” Nat. Neurosci. 4, 819–825 (2001).
[CrossRef] [PubMed]

Nature (London) (2)

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

D. L. Ringach, M. J. Hawkin, R. Shapley, “Dynamics of orientation tuning in primary visual cortex,” Nature (London) 387, 281–284 (1997).
[CrossRef]

Network (3)

M. G. A. Thomsom, “Beats, kurtosis and visual coding,” Network 12, 271–287 (2001).
[CrossRef]

D. L. Ruderman, “The statistics of natural images,” Network 5, 517–548 (1994).
[CrossRef]

R. J. Baddeley, “Searching for filters with ‘interesting’ output distributions: an uninteresting direction to explore?” Network 7, 409–421 (1996).
[CrossRef] [PubMed]

Neural Comput. (1)

D. J. Field, “What is the goal of sensory coding,” Neural Comput. 6, 559–601 (1994).
[CrossRef]

Ophthalmic Physiol. Opt. (1)

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

Perception (1)

A. J. Schofield, “What does second-order vision see in an image?” Perception 29, 1071–1086 (2000). An additional figure is available at http://www.perceptionweb.com/perc0900/schofield.html .
[CrossRef]

Proc. R. Soc. London Ser. B (4)

J. H. van Hateren, D. L. Ruderman, “Independent component analysis of natural image sequences yields spatio-temporal filters similar to simple cells in primary visual cortex,” Proc. R. Soc. London Ser. B 265, 2315–2320 (1998).
[CrossRef]

D. J. Tolhurst, I. D. Thompson, “On the variety of spatial frequency selectivity shown by neurons in area 17 of the cat,” Proc. R. Soc. London Ser. B 213, 183–199 (1981).
[CrossRef]

D. J. Field, D. J. Tolhurst, “The structure and symmetry of simple cell receptive-field profiles in the cat’s visual cortex,” Proc. R. Soc. London Ser. B 228, 379–400 (1986).
[CrossRef]

R. J. Baddeley, P. J. B. Hancock, “A statistical analysis of natural images matches psychophysically derived orientation tuning curves,” Proc. R. Soc. London Ser. B 246, 219–223 (1991).
[CrossRef]

Science (1)

T. D. Albright, “Form-cue invariant motion processing in primate visual cortex,” Science 255, 1141–1143 (1992).
[CrossRef] [PubMed]

Spatial Vision (1)

P. Cavanagh, G. Mather, “Motion: the long and short of it,” Spatial Vision 4, 103–129 (1989).
[CrossRef] [PubMed]

Vision Res. (23)

A. M. Derrington, O. I. Ukkonen, “Second-order motion discrimination by feature tracking,” Vision Res. 39, 1465–1475 (1999).
[CrossRef] [PubMed]

T. Ledgeway, A. T. Smith, “Evidence for separate motion-detecting mechanisms for first- and second-order motion in human vision,” Vision Res. 34, 2727–2740 (1994).
[CrossRef] [PubMed]

A. Sutter, G. Sperling, C. Chubb, “Measuring the spatial frequency selectivity of second-order texture mechanisms,” Vision Res. 35, 915–924 (1995).
[CrossRef] [PubMed]

A. T. Smith, T. Ledgeway, “Separate detection of moving luminance and contrast modulations: fact or artifact?” Vision Res. 37, 45–62 (1997).
[CrossRef] [PubMed]

A. J. Schofield, M. A. Georgeson, “Sensitivity to modulations of luminance and contrast in visual white noise: separate mechanisms with similar behaviour,” Vision Res. 39, 2697–2716 (1999).
[CrossRef] [PubMed]

A. Hyvärinen, P. O. Hoyer, “A two-layer sparse coding model learns simple and complex cell receptive fields and topography from natural images,” Vision Res. 41, 2413–2423 (2001).
[CrossRef] [PubMed]

B. A. Olshausen, D. J. Field, “Sparse coding with an overcomplete basis set: a strategy employed by V1?” Vision Res. 37, 3311–3325 (1996).
[CrossRef]

A. J. Bell, T. J. Sejnowski, “The ‘independent components’ of natural scenes are edge filters,” Vision Res. 37, 3327–3338 (1997).
[CrossRef]

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]

A. van der Schaaf, J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759–2770 (1996).
[CrossRef] [PubMed]

D. J. Field, N. Brady, “Visual sensitivity, blur and the sources of variability in the amplitude spectra of natural scenes,” Vision Res. 37, 3367–3384 (1997).
[CrossRef]

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

E. A. Essock, J. K. DeFord, B. C. Hansen, M. J. Sinai, “Oblique stimuli are seen best (not worst!) in naturalistic broad-band stimuli: a horizontal effect,” Vision Res. 43, 1329–1335 (2003).
[CrossRef] [PubMed]

Y. Tadmor, D. J. Tolhurst, “Discrimination of changes in the 2nd-order statistics of natural and synthetic-images,” Vision Res. 34, 541–554 (1994).
[CrossRef] [PubMed]

N. Graham, A. Sutter, C. Venkatesan, “Spatial-frequency- and orientation-selectivity of simple and complex channels in region segregation,” Vision Res. 33, 1893–1911 (1993).
[CrossRef] [PubMed]

F. A. A. Kingdom, D. R. T. Keeble, “On the mechanism for scale invariance in orientation-defined textures,” Vision Res. 39, 1477–1489 (1999).
[CrossRef] [PubMed]

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

S. C. Dakin, I. Mareschal, “Sensitivity to contrast modulation depends on carrier spatial frequency and orientation,” Vision Res. 40, 311–329 (2000).
[CrossRef] [PubMed]

J. A. Solomon, G. Sperling, “Full-wave and half-wave rectification in second-order motion perception,” Vision Res. 34, 2239–2257 (1994).
[CrossRef] [PubMed]

N. Graham, A. Sutter, “Spatial summation in simple (Fourier) and complex (non-Fourier) channels in texture segregation,” Vision Res. 38, 231–257 (1998).
[CrossRef] [PubMed]

D. B. Hamilton, D. G. Albrecht, W. S. Geisler, “Visual cortical receptive fields in monkey and cat: spatial and temporal phase transfer function,” Vision Res. 29, 1285–1308 (1989).
[CrossRef] [PubMed]

Z.-L. Lu, G. Sperling, “The functional architecture of human visual motion perception,” Vision Res. 35, 2697–2722 (1995).
[CrossRef] [PubMed]

N. Graham, A. Sutter, “Normalization: contrast-gain control in simple (Fourier) and complex (non-Fourier) pathways of pattern vision,” Vision Res. 40, 2737–2761 (2000).
[CrossRef] [PubMed]

Visual Neurosci. (5)

D. J. Heeger, “Normalization of cell responses in cat striate cortex,” Visual Neurosci. 9, 271–294 (1992).

A. G. Leventhal, Y. Wang, M. T. Schmolesky, Y. Zhou, “Neural correlates of boundary perception,” Visual Neurosci. 15, 1107–1118 (1998).
[CrossRef]

I. Mareschal, C. L. Baker, “Cortical processing of second order motion,” Visual Neurosci. 16, 527–540 (1999).
[CrossRef]

A. Chaudhuri, T. D. Albright, “Neuronal responses to edges defined by luminance vs. temporal texture in macaque area V1,” Visual Neurosci. 14, 949–962 (1997).
[CrossRef]

H. R. Wilson, V. P. Ferrera, C. Yo, “A psychophysically motivated model for two-dimensional motion perception,” Visual Neurosci. 9, 79–97 (1992).
[CrossRef]

Other (6)

H. B. Barlow, D. J. Tolhurst, “Why do you have edge detectors?” in OSA Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 172.

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.

K. V. Marida, Statistics of Directional Data (Academic, London, 1972).

R. C. Gonzalez, R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 2001), pp. 199–205.

M. S. Landy, N. Graham, “Visual perception of texture,” in The Visual Neurosciences, L. M. Chalupa, J. S. Werner, eds. (MIT Press, Cambridge, Mass., 2003), pp. 1106–1118.

C. L. Baker, I. Mareschal, “Processing of second-order stimuli in the visual cortex,” in Vision: From Neurons to Cognition, Progress in Brain Research, C. Casanova, M. Ptito, eds. (Elsevier Science, Amsterdam, 2001), Vol. 134, Chap. 12.

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Figures (8)

Fig. 1
Fig. 1

Example of (top) first-order model and (bottom) filter–rectify–filter (FRF) model. The first-order model convolves the image with a single Gabor, to detect luminance variations. The second-order model convolves the image with a high-spatial-frequency Gabor filter (F1), whose full-wave rectified response is then convolved with a second Gabor (F2) having a lower spatial frequency to detect texture variations. In this example the first-order (F0) and second-order (F2) filters (shown at magnified scale) have the same orientation, phase, and spatial wavelength. The first-order filtering detects the luminance change in the boat hulls and the dock, while the second-order filter detects the texture and contrast changes at the same locations. For this pair of filters, the first- and second-order response images have a correlation of 0.44.

Fig. 2
Fig. 2

Consequences of signed versus unsigned correlation for a cartoon synthetic image depicting three surfaces with two occlusion boundaries. Spatial images are at the top, and pixel profiles are below. A, Luminance (first-order) component of test image, with luminance changes (from left to right) of high→medium→low. B, Contrast modulation (second-order) component of test image, a sine-wave grating “texture” with contrast changes of highlowhigh; the dashed curve indicates the modulation envelope. C, Test image with mixed-attribute changes, formed as a combination of components in A and B. Note that luminance and contrast changes are spatially coincident and are of the same sign at the left boundary but are of opposite sign at the right boundary. Consequently a simple correlation of the raw “signed” responses would give a net correlation coefficient R=0. However, a correlation of “unsigned” (full-wave rectified) responses would give R=1, successfully capturing the spatially coincident changes in first- and second-order information.

Fig. 3
Fig. 3

Effect of spatial frequency on filter responses. A., Examples of images and corresponding plots of the spatial-frequency responses for B., first-order and C., second-order filtered images; plotted values are averages across other filter parameters. Since the filter gains are scaled with spatial frequency to compensate for the expected 1/f rolloff (see text), the first-order responses are nearly flat. In each case the equivalent rolloff slope (if the 1/f compensation had not been applied) is derived from a best-fitting line, and indicated by α. Spectral slopes for the natural images are similar to those for the artificial fractal (although some variability occurs, since individual natural scenes rarely have a perfect amplitude spectrum of 1/f), with uncorrected (solid) and corrected (dashed) images producing similar responses.

Fig. 4
Fig. 4

Effect of orientation on filter responses for two example images. Responses are shown as polar plots, with spatial wavelength λ (reciprocal of spatial frequency) as a parameter, and averaged across other filter parameters. A., Noise image with 1/f amplitude spectrum like that of natural images. B., First-order responses are approximately the same for all orientations and spatial wavelengths. C., Responses to the FRF model, shown at the same scale as B., are much smaller. D., Same as C. but at magnified scale to show that responses are uniform with orientation and spatial frequency. E., Example of a natural image whose responses were typical. F., First-order filter responses (same scale as B.) show an orientational bias toward the horizontal and, at high spatial frequencies, a pronounced orientational anisotropy in favor of vertical and horizontal. G., Responses to the FRF model (same scale as C.). H., Same as G. but at the same magnified scale as D., showing small but systematic orientational bias toward the horizontal (probably due to foreshortening; see Section 4). Also note that the second-order responses are substantially larger to a natural image (H.) than to 1/f noise (D.).

Fig. 5
Fig. 5

Correlation between first-order (λF0) and second-order (λF2) responses as a function of spatial frequency, averaged across the ensemble of natural scenes (n=8) and averaged across other filter parameters satisfying the constraint that the late-stage (λF2) to early-stage (λF1) FRF spatial-frequency ratio is 8:1. Results when early- and late-stage filters were in-phase (black) and quadrature-phase (gray) show no significant difference. However, the correlations are slightly greater when the filter orientations are parallel (A) than when orthogonal (B), although both are significantly greater than the response correlations from 1/f noise (gray dotted curves). Error bars represent standard deviation.

Fig. 6
Fig. 6

Correlation between first- and second-order responses as a function of difference in orientation (θF2-θF0), averaged across other filter parameters; the example shown here is for wavelength ratio λF2/λF0 of 4:1 (optimal ratio; see Fig. 5A). The correlations in natural images show the strongest response when the early- and late-stage filters are parallel in orientation. As in Fig. 5, note that the correlation for natural images (black curve) is significantly greater than that for 1/f noise (gray curve). Error bars represent standrad deviation.

Fig. 7
Fig. 7

Plot of second-order response amplitude for varying parameters of early- and late-stage Gabor filters (F1, F2) in the FRF model. A, B, Effect of varying the spatial wavelengths (λF1, λF2), averaged across orientations. The filters are either in-phase (A) or in quadrature-phase (B). Hatching indicates excluded filter combinations (see the text). C, D, Effect of varying filter orientations while averaging across spatial wavelengths. The strongest responses occur for early and late filters having the same orientation, particularly vertical (90°). Results are shown for filters either in-phase (C), or in quadrature-phase (D). The gray-scale color bar indicates strength of response, with the arrow indicating average response for 1/f noise.

Fig. 8
Fig. 8

Correlation between first- and second-order responses as a function of varying parameters of early- and late-stages (F1, F2) of FRF for a fixed wavelength ratio λF2/λF0 of 4:1. A, Correlation as a function of ratio of spatial scales between early- and late-stage filters, showing a maximum at a ratio of 8:1. B, Correlation as a function of relative orientation of early- versus late-stage filters, showing roughly equal correlation at all orientation differences. The average correlation is ∼0.2, but particular filter combinations give correlations of >0.6. Error bars represent standard deviation.

Equations (4)

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

gλθΦσ(x, y)=A exp[(x2+y2)/2σ2]cos[2π(x/λ)+Φ],
x=x cos θ+y sin θ,y=-x sin θ+y cos θ,
V=1-abskRkexp(i2θk)kRk,
R=XY(aXY×bXY)[XY(aXY×aXY)×XY(bXY×bXY)]1/2,

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