Abstract

The Fourier phase spectrum plays a central role regarding where in an image contours occur, thereby defining the spatial relationship between those structures in the overall scene. Only a handful of studies have demonstrated psychophysically the relevance of the Fourier phase spectrum with respect to human visual processing, and none have demonstrated the relative amount of local cross-scale spatial phase alignment needed to perceptually extract meaningful structure from an image. We investigated the relative amount of spatial phase alignment needed for humans to perceptually match natural scene image structures at three different spatial frequencies [3, 6, and 12  cyclesperdegree (cpd)] as a function of the number of structures within the image (i.e., “structural sparseness”). The results showed that (1) the amount of spatial phase alignment needed to match structures depends on structural sparseness, with a bias for matching structures at 6cpd and (2) the ability to match partially phase-randomized images at a given spatial frequency is independent of structural sparseness at other spatial frequencies. The findings of the current study are discussed in terms of a network of feature integrators in the human visual system.

© 2007 Optical Society of America

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  44. D. J. Field, A. Hayes, and R. F. Hess, "The roles of polarity and symmetry in the perceptual grouping of contour fragments," Spatial Vis. 13, 51-66 (2000).
    [CrossRef]
  45. B. C. Hansen and R. F. Hess, "The role of spatial phase in texture segmentation and contour integration," J. Vision 6, 594-615 (2006).
    [CrossRef]
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    [CrossRef]
  47. D. H. Hubel and T. N. Wiesel, "Receptive fields, binocular interaction and functional architecture in the cat's visual cortex," J. Physiol. (London) 160, 106-154 (1962).
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    [CrossRef] [PubMed]
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  50. G. Krieger and C. Zetzsche, "Nonlinear image operators for the evaluation of local intrinsic dimensionality," IEEE Trans. Image Process. 5, 1026-1042 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]

2006 (3)

D. G. Pelli, C. W. Burns, B. Farell, and D. C. Moore-Page, "Feature detection and letter identification," Vision Res. 46, 4646-4674 (2006).
[CrossRef] [PubMed]

B. C. Hansen and R. F. Hess, "The role of spatial phase in texture segmentation and contour integration," J. Vision 6, 594-615 (2006).
[CrossRef]

F. A. Wichmann, D. I. Braun, and K. R. Gegenfurtner, "Phase noise and the classification of natural images," Vision Res. 46, 1520-1529 (2006).
[CrossRef]

2005 (4)

G. Felsen, J. Touryan, F. Han, and Y. Dan, "Cortical sensitivity to visual features in natural scenes," PLoS Biology 3, 1819-1828 (2005).
[CrossRef]

T. Ledgeway, R. F. Hess, and W. S. Geisler, "Grouping local orientation and direction signals to extract spatial contours: empirical tests of 'association field' models of contour integration," Vision Res. 45, 2511-2522 (2005).
[CrossRef] [PubMed]

E. Doi and M. S. Lewicki, "Relations between the statistical regularities of natural images and the response properties of the early visual system," in Proceedings of the Workshop of Special Interest Group of Pattern Recognition and Perception Model (SIG P&P) (Japanese Cognitive Science Society, Kyoto University, 2005), pp. 1-8.

B. C. Hansen and E. A. Essock, "Influence of scale and orientation on the visual perception of natural scenes," Visual Cogn. 12, 1199-1234 (2005).
[CrossRef]

2004 (2)

B. C. Hansen and E. A. Essock, "A horizontal bias in human visual processing of orientation and its correspondence to the structural components of natural scenes," J. Vision 4, 1044-1060 (2004).
[CrossRef]

Z. Wang and E. P. Simoncelli, "Local phase coherence and the perception of blur," in Advances in Neural Information Processing Systems, S.Thurn, L.Saul, and B.Schölkopf, eds. (MIT Press, 2004), pp. 1435-1442.

2003 (1)

B. C. Hansen, E. A. Essock, Y. Zheng, and J. K. DeFord, "Perceptual anisotropies in visual processing and their relation to natural image statistics," Network 14, 501-526 (2003).
[CrossRef] [PubMed]

2001 (4)

F. A. Wichmann and N. J. Hill, "The psychometric function: I. Fitting, sampling, and goodness of fit," Percept. Psychophys. 63, 1293-1313 (2001).
[CrossRef]

F. A. Wichmann and N. J. Hill, "The psychometric function: II. Bootstrap-based confidence intervals and sampling," Percept. Psychophys. 63, 1314-1329 (2001).
[CrossRef]

H. B. Barlow, "Redundancy reduction revisited," Network 12, 241-253 (2001).
[PubMed]

C. Chubb, L. Olzak, and A. Derrington, "Second-order processes in vision: introduction," J. Opt. Soc. Am. A 18, 2175-2178 (2001).
[CrossRef]

2000 (1)

D. J. Field, A. Hayes, and R. F. Hess, "The roles of polarity and symmetry in the perceptual grouping of contour fragments," Spatial Vis. 13, 51-66 (2000).
[CrossRef]

1999 (2)

P. Kovesi, "Image features from phase congruency," Videre 1, 1-26 (1999).

S. C. Dakin and R. F. Hess, "Contour integration and scale combination processes in visual edge detection," Spatial Vis. 12, 309-327 (1999).
[CrossRef]

1998 (3)

J. H. van Hateren and A. van der Schaaf, "Independent component filters of natural images compared with simple cells in primary visual cortex," Proc. R. Soc. London, Ser. B 265, 359-366 (1998).
[CrossRef]

M. J. Tarr and H. H. Bulthoff, "Image-based object recognition in man, monkey and machine," Cognition 67, 1-20 (1998).
[CrossRef] [PubMed]

S. C. Dakin and R. F. Hess, "Spatial-frequency tuning of visual contour integration," J. Opt. Soc. Am. A 15, 1486-1499 (1998).
[CrossRef]

1997 (2)

G. Krieger, C. Zetzsche, and E. Barth, "Higher-order statistics of natural images and their exploitation by operators selective to intrinsic dimensionality," in Proceedings of the IEEE Signal Processing Workshop on Higher Order Statistics (IEEE, 1997), pp. 147-151.
[CrossRef]

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

1996 (3)

R. Baddeley, "Searching for filters with "interesting" output distributions: an uninteresting direction to explore?" Network 7, 409-421 (1996).
[CrossRef] [PubMed]

A. van der Schaaf and J. H. van Hateren, "Modeling the power spectra of natural images: statistics and Information," Vision Res. 36, 2759-2770 (1996).
[CrossRef] [PubMed]

G. Krieger and C. Zetzsche, "Nonlinear image operators for the evaluation of local intrinsic dimensionality," IEEE Trans. Image Process. 5, 1026-1042 (1996).
[CrossRef] [PubMed]

1994 (1)

D. L. Ruderman and W. Bialek, "Statistics of natural images: scaling in the Woods," Phys. Rev. Lett. 73, 814-817 (1994).
[CrossRef] [PubMed]

1993 (2)

D. J. Field, A. Hayes, and R. F. Hess, "Contour integration by the human visual system: Evidence for a local association field," Vision Res. 33, 173-193 (1993).
[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: New Developments and New Applications, M.Farge, J.C. R.Hunt, and J.C.Vassilicos, eds. (Oxford U. Press, 1993).

1992 (2)

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

J. J. Atick and A. N. Redlich, "What does the retina know about natural scenes?" Neural Comput. 4, 196-210 (1992).
[CrossRef]

1989 (1)

J. G. Daugman, "Entropy reduction and decorrelation in visual coding by oriented neural receptive fields," IEEE Trans. Biomed. Eng. 36, 107-114 (1989).
[CrossRef] [PubMed]

1988 (1)

M. C. Morrone and D. C. Burr, "Feature detection in human vision: a phase dependent energy model," Proc. R. Soc. London, Ser. B 235, 221-245 (1988).
[CrossRef]

1987 (6)

M. C. Morrone and R. A. Owens, "Feature detection from local energy," Pattern Recogn. Lett. 6, 303-313 (1987).
[CrossRef]

J. P. Jones and 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 and L. A. Palmer, "An evaluation of the two-dimensional gabor filter models of simple receptive fields in cat striate cortex," J. Neurophysiol. 58, 1233-1258 (1987).
[PubMed]

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

G. J. Burton and I. R. Moorhead, "Color and spatial structure in natural scenes," Appl. Opt. 26, 157-170 (1987).
[CrossRef] [PubMed]

I. Biederman, "Recognition-by-components: a theory of human image understanding," Psychol. Rev. 94, 115-147 (1987).
[CrossRef] [PubMed]

1986 (2)

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

L. Olzak and J. P. Thomas, "Seeing spatial patterns," in Handbook of Perception and Human Performance: Sensory Processes and Perception, K.R.Boff, L.Kaufman, and J.P.Thomas, eds. (Wiley, 1986), Vol. 1.

1982 (3)

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

D. Marr, Vision: A Computational Investigation into the Human Representation and Processing of Visual Information (Freeman, 1982).
[PubMed]

L. N. Piotrowski and F. W. Campbell, "A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase," Perception 11, 337-346 (1982).
[CrossRef] [PubMed]

1981 (2)

A. V. Oppenheim and J. S. Lim, "The importance of phase in signals," Proc. IEEE 69, 529-541 (1981).
[CrossRef]

J. G. Robson and N. Graham, "Probability summation and regional variation in contrast sensitivity across the visual field," Vision Res. 21, 409-418 (1981).
[CrossRef]

1980 (2)

N. Graham, "Spatial-frequency channels in human vision: detecting edges without edge detectors," in Visual Coding and Adaptability, C.Harris, ed. (Erlbaum, 1980), pp. 215-252.

A. M. Treisman and G. Gelade, "A feature-integration theory of attention," Cogn. Psychol. 12, 97-136 (1980).
[CrossRef] [PubMed]

1979 (1)

B. W. Andrews and D. A. Pollen, "Relationship between spatial frequency selectivity and receptive field profile of simple cells," J. Physiol. (London) 287, 163-176 (1979).

1978 (2)

J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Spatial summation in the receptive fields of simple cells in the cat's striate cortex," J. Physiol. (London) 283, 53-77 (1978).

J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Receptive field organization of complex cells in the cat's striate cortex," J. Physiol. (London) 283, 79-99 (1978).

1962 (1)

D. H. Hubel and 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.

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

Andrews, B. W.

B. W. Andrews and D. A. Pollen, "Relationship between spatial frequency selectivity and receptive field profile of simple cells," J. Physiol. (London) 287, 163-176 (1979).

Atick, J. J.

J. J. Atick and A. N. Redlich, "What does the retina know about natural scenes?" Neural Comput. 4, 196-210 (1992).
[CrossRef]

Baddeley, R.

R. Baddeley, "Searching for filters with "interesting" output distributions: an uninteresting direction to explore?" Network 7, 409-421 (1996).
[CrossRef] [PubMed]

Barlow, H. B.

H. B. Barlow, "Redundancy reduction revisited," Network 12, 241-253 (2001).
[PubMed]

Barth, E.

G. Krieger, C. Zetzsche, and E. Barth, "Higher-order statistics of natural images and their exploitation by operators selective to intrinsic dimensionality," in Proceedings of the IEEE Signal Processing Workshop on Higher Order Statistics (IEEE, 1997), pp. 147-151.
[CrossRef]

Bialek, W.

D. L. Ruderman and W. Bialek, "Statistics of natural images: scaling in the Woods," Phys. Rev. Lett. 73, 814-817 (1994).
[CrossRef] [PubMed]

Biederman, I.

I. Biederman, "Recognition-by-components: a theory of human image understanding," Psychol. Rev. 94, 115-147 (1987).
[CrossRef] [PubMed]

Brady, N.

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

Braun, D. I.

F. A. Wichmann, D. I. Braun, and K. R. Gegenfurtner, "Phase noise and the classification of natural images," Vision Res. 46, 1520-1529 (2006).
[CrossRef]

Bulthoff, H. H.

M. J. Tarr and H. H. Bulthoff, "Image-based object recognition in man, monkey and machine," Cognition 67, 1-20 (1998).
[CrossRef] [PubMed]

Burns, C. W.

D. G. Pelli, C. W. Burns, B. Farell, and D. C. Moore-Page, "Feature detection and letter identification," Vision Res. 46, 4646-4674 (2006).
[CrossRef] [PubMed]

Burr, D. C.

M. C. Morrone and D. C. Burr, "Feature detection in human vision: a phase dependent energy model," Proc. R. Soc. London, Ser. B 235, 221-245 (1988).
[CrossRef]

Burton, G. J.

Campbell, F. W.

L. N. Piotrowski and F. W. Campbell, "A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase," Perception 11, 337-346 (1982).
[CrossRef] [PubMed]

Chao, T.

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

Chubb, C.

Dakin, S. C.

S. C. Dakin and R. F. Hess, "Contour integration and scale combination processes in visual edge detection," Spatial Vis. 12, 309-327 (1999).
[CrossRef]

S. C. Dakin and R. F. Hess, "Spatial-frequency tuning of visual contour integration," J. Opt. Soc. Am. A 15, 1486-1499 (1998).
[CrossRef]

Dan, Y.

G. Felsen, J. Touryan, F. Han, and Y. Dan, "Cortical sensitivity to visual features in natural scenes," PLoS Biology 3, 1819-1828 (2005).
[CrossRef]

Daugman, J. G.

J. G. Daugman, "Entropy reduction and decorrelation in visual coding by oriented neural receptive fields," IEEE Trans. Biomed. Eng. 36, 107-114 (1989).
[CrossRef] [PubMed]

De Valois, R. L.

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

DeFord, J. K.

B. C. Hansen, E. A. Essock, Y. Zheng, and J. K. DeFord, "Perceptual anisotropies in visual processing and their relation to natural image statistics," Network 14, 501-526 (2003).
[CrossRef] [PubMed]

Derrington, A.

Doi, E.

E. Doi and M. S. Lewicki, "Relations between the statistical regularities of natural images and the response properties of the early visual system," in Proceedings of the Workshop of Special Interest Group of Pattern Recognition and Perception Model (SIG P&P) (Japanese Cognitive Science Society, Kyoto University, 2005), pp. 1-8.

Essock, E. A.

B. C. Hansen and E. A. Essock, "Influence of scale and orientation on the visual perception of natural scenes," Visual Cogn. 12, 1199-1234 (2005).
[CrossRef]

B. C. Hansen and E. A. Essock, "A horizontal bias in human visual processing of orientation and its correspondence to the structural components of natural scenes," J. Vision 4, 1044-1060 (2004).
[CrossRef]

B. C. Hansen, E. A. Essock, Y. Zheng, and J. K. DeFord, "Perceptual anisotropies in visual processing and their relation to natural image statistics," Network 14, 501-526 (2003).
[CrossRef] [PubMed]

Farell, B.

D. G. Pelli, C. W. Burns, B. Farell, and D. C. Moore-Page, "Feature detection and letter identification," Vision Res. 46, 4646-4674 (2006).
[CrossRef] [PubMed]

Felsen, G.

G. Felsen, J. Touryan, F. Han, and Y. Dan, "Cortical sensitivity to visual features in natural scenes," PLoS Biology 3, 1819-1828 (2005).
[CrossRef]

Field, D. J.

D. J. Field, A. Hayes, and R. F. Hess, "The roles of polarity and symmetry in the perceptual grouping of contour fragments," Spatial Vis. 13, 51-66 (2000).
[CrossRef]

D. J. Field and 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. Field, A. Hayes, and R. F. Hess, "Contour integration by the human visual system: Evidence for a local association field," Vision Res. 33, 173-193 (1993).
[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: New Developments and New Applications, M.Farge, J.C. R.Hunt, and J.C.Vassilicos, eds. (Oxford U. Press, 1993).

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

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

Gegenfurtner, K. R.

F. A. Wichmann, D. I. Braun, and K. R. Gegenfurtner, "Phase noise and the classification of natural images," Vision Res. 46, 1520-1529 (2006).
[CrossRef]

Geisler, W. S.

T. Ledgeway, R. F. Hess, and W. S. Geisler, "Grouping local orientation and direction signals to extract spatial contours: empirical tests of 'association field' models of contour integration," Vision Res. 45, 2511-2522 (2005).
[CrossRef] [PubMed]

Gelade, G.

A. M. Treisman and G. Gelade, "A feature-integration theory of attention," Cogn. Psychol. 12, 97-136 (1980).
[CrossRef] [PubMed]

Graham, N.

J. G. Robson and N. Graham, "Probability summation and regional variation in contrast sensitivity across the visual field," Vision Res. 21, 409-418 (1981).
[CrossRef]

N. Graham, "Spatial-frequency channels in human vision: detecting edges without edge detectors," in Visual Coding and Adaptability, C.Harris, ed. (Erlbaum, 1980), pp. 215-252.

Han, F.

G. Felsen, J. Touryan, F. Han, and Y. Dan, "Cortical sensitivity to visual features in natural scenes," PLoS Biology 3, 1819-1828 (2005).
[CrossRef]

Hansen, B. C.

B. C. Hansen and R. F. Hess, "The role of spatial phase in texture segmentation and contour integration," J. Vision 6, 594-615 (2006).
[CrossRef]

B. C. Hansen and E. A. Essock, "Influence of scale and orientation on the visual perception of natural scenes," Visual Cogn. 12, 1199-1234 (2005).
[CrossRef]

B. C. Hansen and E. A. Essock, "A horizontal bias in human visual processing of orientation and its correspondence to the structural components of natural scenes," J. Vision 4, 1044-1060 (2004).
[CrossRef]

B. C. Hansen, E. A. Essock, Y. Zheng, and J. K. DeFord, "Perceptual anisotropies in visual processing and their relation to natural image statistics," Network 14, 501-526 (2003).
[CrossRef] [PubMed]

Hayes, A.

D. J. Field, A. Hayes, and R. F. Hess, "The roles of polarity and symmetry in the perceptual grouping of contour fragments," Spatial Vis. 13, 51-66 (2000).
[CrossRef]

D. J. Field, A. Hayes, and R. F. Hess, "Contour integration by the human visual system: Evidence for a local association field," Vision Res. 33, 173-193 (1993).
[CrossRef] [PubMed]

Hess, R. F.

B. C. Hansen and R. F. Hess, "The role of spatial phase in texture segmentation and contour integration," J. Vision 6, 594-615 (2006).
[CrossRef]

T. Ledgeway, R. F. Hess, and W. S. Geisler, "Grouping local orientation and direction signals to extract spatial contours: empirical tests of 'association field' models of contour integration," Vision Res. 45, 2511-2522 (2005).
[CrossRef] [PubMed]

D. J. Field, A. Hayes, and R. F. Hess, "The roles of polarity and symmetry in the perceptual grouping of contour fragments," Spatial Vis. 13, 51-66 (2000).
[CrossRef]

S. C. Dakin and R. F. Hess, "Contour integration and scale combination processes in visual edge detection," Spatial Vis. 12, 309-327 (1999).
[CrossRef]

S. C. Dakin and R. F. Hess, "Spatial-frequency tuning of visual contour integration," J. Opt. Soc. Am. A 15, 1486-1499 (1998).
[CrossRef]

D. J. Field, A. Hayes, and R. F. Hess, "Contour integration by the human visual system: Evidence for a local association field," Vision Res. 33, 173-193 (1993).
[CrossRef] [PubMed]

Hill, N. J.

F. A. Wichmann and N. J. Hill, "The psychometric function: II. Bootstrap-based confidence intervals and sampling," Percept. Psychophys. 63, 1314-1329 (2001).
[CrossRef]

F. A. Wichmann and N. J. Hill, "The psychometric function: I. Fitting, sampling, and goodness of fit," Percept. Psychophys. 63, 1293-1313 (2001).
[CrossRef]

Hubel, D. H.

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

Jones, J. P.

J. P. Jones and 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 and L. A. Palmer, "An evaluation of the two-dimensional gabor filter models of simple receptive fields in cat striate cortex," J. Neurophysiol. 58, 1233-1258 (1987).
[PubMed]

Kovesi, P.

P. Kovesi, "Image features from phase congruency," Videre 1, 1-26 (1999).

Krieger, G.

G. Krieger, C. Zetzsche, and E. Barth, "Higher-order statistics of natural images and their exploitation by operators selective to intrinsic dimensionality," in Proceedings of the IEEE Signal Processing Workshop on Higher Order Statistics (IEEE, 1997), pp. 147-151.
[CrossRef]

G. Krieger and C. Zetzsche, "Nonlinear image operators for the evaluation of local intrinsic dimensionality," IEEE Trans. Image Process. 5, 1026-1042 (1996).
[CrossRef] [PubMed]

Ledgeway, T.

T. Ledgeway, R. F. Hess, and W. S. Geisler, "Grouping local orientation and direction signals to extract spatial contours: empirical tests of 'association field' models of contour integration," Vision Res. 45, 2511-2522 (2005).
[CrossRef] [PubMed]

Lewicki, M. S.

E. Doi and M. S. Lewicki, "Relations between the statistical regularities of natural images and the response properties of the early visual system," in Proceedings of the Workshop of Special Interest Group of Pattern Recognition and Perception Model (SIG P&P) (Japanese Cognitive Science Society, Kyoto University, 2005), pp. 1-8.

Lim, J. S.

A. V. Oppenheim and J. S. Lim, "The importance of phase in signals," Proc. IEEE 69, 529-541 (1981).
[CrossRef]

Marr, D.

D. Marr, Vision: A Computational Investigation into the Human Representation and Processing of Visual Information (Freeman, 1982).
[PubMed]

Moore-Page, D. C.

D. G. Pelli, C. W. Burns, B. Farell, and D. C. Moore-Page, "Feature detection and letter identification," Vision Res. 46, 4646-4674 (2006).
[CrossRef] [PubMed]

Moorhead, I. R.

Morrone, M. C.

M. C. Morrone and D. C. Burr, "Feature detection in human vision: a phase dependent energy model," Proc. R. Soc. London, Ser. B 235, 221-245 (1988).
[CrossRef]

M. C. Morrone and R. A. Owens, "Feature detection from local energy," Pattern Recogn. Lett. 6, 303-313 (1987).
[CrossRef]

Movshon, J. A.

J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Receptive field organization of complex cells in the cat's striate cortex," J. Physiol. (London) 283, 79-99 (1978).

J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Spatial summation in the receptive fields of simple cells in the cat's striate cortex," J. Physiol. (London) 283, 53-77 (1978).

Olzak, L.

C. Chubb, L. Olzak, and A. Derrington, "Second-order processes in vision: introduction," J. Opt. Soc. Am. A 18, 2175-2178 (2001).
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L. Olzak and J. P. Thomas, "Seeing spatial patterns," in Handbook of Perception and Human Performance: Sensory Processes and Perception, K.R.Boff, L.Kaufman, and J.P.Thomas, eds. (Wiley, 1986), Vol. 1.

Oppenheim, A. V.

A. V. Oppenheim and J. S. Lim, "The importance of phase in signals," Proc. IEEE 69, 529-541 (1981).
[CrossRef]

Owens, R. A.

M. C. Morrone and R. A. Owens, "Feature detection from local energy," Pattern Recogn. Lett. 6, 303-313 (1987).
[CrossRef]

Palmer, L. A.

J. P. Jones and 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 and L. A. Palmer, "An evaluation of the two-dimensional gabor filter models of simple receptive fields in cat striate cortex," J. Neurophysiol. 58, 1233-1258 (1987).
[PubMed]

Pelli, D. G.

D. G. Pelli, C. W. Burns, B. Farell, and D. C. Moore-Page, "Feature detection and letter identification," Vision Res. 46, 4646-4674 (2006).
[CrossRef] [PubMed]

Piotrowski, L. N.

L. N. Piotrowski and F. W. Campbell, "A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase," Perception 11, 337-346 (1982).
[CrossRef] [PubMed]

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B. W. Andrews and D. A. Pollen, "Relationship between spatial frequency selectivity and receptive field profile of simple cells," J. Physiol. (London) 287, 163-176 (1979).

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J. J. Atick and A. N. Redlich, "What does the retina know about natural scenes?" Neural Comput. 4, 196-210 (1992).
[CrossRef]

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J. G. Robson and N. Graham, "Probability summation and regional variation in contrast sensitivity across the visual field," Vision Res. 21, 409-418 (1981).
[CrossRef]

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D. L. Ruderman and W. Bialek, "Statistics of natural images: scaling in the Woods," Phys. Rev. Lett. 73, 814-817 (1994).
[CrossRef] [PubMed]

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Z. Wang and E. P. Simoncelli, "Local phase coherence and the perception of blur," in Advances in Neural Information Processing Systems, S.Thurn, L.Saul, and B.Schölkopf, eds. (MIT Press, 2004), pp. 1435-1442.

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D. J. Tolhurst, Y. Tadmor, and T. Chao, "Amplitude spectra of natural images," Ophthalmic Physiol. Opt. 12, 229-232 (1992).
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M. J. Tarr and H. H. Bulthoff, "Image-based object recognition in man, monkey and machine," Cognition 67, 1-20 (1998).
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L. Olzak and J. P. Thomas, "Seeing spatial patterns," in Handbook of Perception and Human Performance: Sensory Processes and Perception, K.R.Boff, L.Kaufman, and J.P.Thomas, eds. (Wiley, 1986), Vol. 1.

Thompson, I. D.

J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Spatial summation in the receptive fields of simple cells in the cat's striate cortex," J. Physiol. (London) 283, 53-77 (1978).

J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Receptive field organization of complex cells in the cat's striate cortex," J. Physiol. (London) 283, 79-99 (1978).

Thorell, L. G.

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

Tolhurst, D. J.

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

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

J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Receptive field organization of complex cells in the cat's striate cortex," J. Physiol. (London) 283, 79-99 (1978).

J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Spatial summation in the receptive fields of simple cells in the cat's striate cortex," J. Physiol. (London) 283, 53-77 (1978).

Touryan, J.

G. Felsen, J. Touryan, F. Han, and Y. Dan, "Cortical sensitivity to visual features in natural scenes," PLoS Biology 3, 1819-1828 (2005).
[CrossRef]

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A. M. Treisman and G. Gelade, "A feature-integration theory of attention," Cogn. Psychol. 12, 97-136 (1980).
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J. H. van Hateren and A. van der Schaaf, "Independent component filters of natural images compared with simple cells in primary visual cortex," Proc. R. Soc. London, Ser. B 265, 359-366 (1998).
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A. van der Schaaf and J. H. van Hateren, "Modeling the power spectra of natural images: statistics and Information," Vision Res. 36, 2759-2770 (1996).
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van Hateren, J. H.

J. H. van Hateren and A. van der Schaaf, "Independent component filters of natural images compared with simple cells in primary visual cortex," Proc. R. Soc. London, Ser. B 265, 359-366 (1998).
[CrossRef]

A. van der Schaaf and J. H. van Hateren, "Modeling the power spectra of natural images: statistics and Information," Vision Res. 36, 2759-2770 (1996).
[CrossRef] [PubMed]

Wang, Z.

Z. Wang and E. P. Simoncelli, "Local phase coherence and the perception of blur," in Advances in Neural Information Processing Systems, S.Thurn, L.Saul, and B.Schölkopf, eds. (MIT Press, 2004), pp. 1435-1442.

Wichmann, F. A.

F. A. Wichmann, D. I. Braun, and K. R. Gegenfurtner, "Phase noise and the classification of natural images," Vision Res. 46, 1520-1529 (2006).
[CrossRef]

F. A. Wichmann and N. J. Hill, "The psychometric function: II. Bootstrap-based confidence intervals and sampling," Percept. Psychophys. 63, 1314-1329 (2001).
[CrossRef]

F. A. Wichmann and N. J. Hill, "The psychometric function: I. Fitting, sampling, and goodness of fit," Percept. Psychophys. 63, 1293-1313 (2001).
[CrossRef]

Wiesel, T. N.

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

Zetzsche, C.

G. Krieger, C. Zetzsche, and E. Barth, "Higher-order statistics of natural images and their exploitation by operators selective to intrinsic dimensionality," in Proceedings of the IEEE Signal Processing Workshop on Higher Order Statistics (IEEE, 1997), pp. 147-151.
[CrossRef]

G. Krieger and C. Zetzsche, "Nonlinear image operators for the evaluation of local intrinsic dimensionality," IEEE Trans. Image Process. 5, 1026-1042 (1996).
[CrossRef] [PubMed]

Zheng, Y.

B. C. Hansen, E. A. Essock, Y. Zheng, and J. K. DeFord, "Perceptual anisotropies in visual processing and their relation to natural image statistics," Network 14, 501-526 (2003).
[CrossRef] [PubMed]

Appl. Opt. (1)

Cogn. Psychol. (1)

A. M. Treisman and G. Gelade, "A feature-integration theory of attention," Cogn. Psychol. 12, 97-136 (1980).
[CrossRef] [PubMed]

Cognition (1)

M. J. Tarr and H. H. Bulthoff, "Image-based object recognition in man, monkey and machine," Cognition 67, 1-20 (1998).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng. (1)

J. G. Daugman, "Entropy reduction and decorrelation in visual coding by oriented neural receptive fields," IEEE Trans. Biomed. Eng. 36, 107-114 (1989).
[CrossRef] [PubMed]

IEEE Trans. Image Process. (1)

G. Krieger and C. Zetzsche, "Nonlinear image operators for the evaluation of local intrinsic dimensionality," IEEE Trans. Image Process. 5, 1026-1042 (1996).
[CrossRef] [PubMed]

J. Neurophysiol. (2)

J. P. Jones and 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 and L. A. Palmer, "An evaluation of the two-dimensional gabor filter models of simple receptive fields in cat striate cortex," J. Neurophysiol. 58, 1233-1258 (1987).
[PubMed]

J. Opt. Soc. Am. A (3)

J. Physiol. (London) (4)

J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Receptive field organization of complex cells in the cat's striate cortex," J. Physiol. (London) 283, 79-99 (1978).

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

J. A. Movshon, I. D. Thompson, and D. J. Tolhurst, "Spatial summation in the receptive fields of simple cells in the cat's striate cortex," J. Physiol. (London) 283, 53-77 (1978).

B. W. Andrews and D. A. Pollen, "Relationship between spatial frequency selectivity and receptive field profile of simple cells," J. Physiol. (London) 287, 163-176 (1979).

J. Vision (2)

B. C. Hansen and R. F. Hess, "The role of spatial phase in texture segmentation and contour integration," J. Vision 6, 594-615 (2006).
[CrossRef]

B. C. Hansen and E. A. Essock, "A horizontal bias in human visual processing of orientation and its correspondence to the structural components of natural scenes," J. Vision 4, 1044-1060 (2004).
[CrossRef]

Network (3)

R. Baddeley, "Searching for filters with "interesting" output distributions: an uninteresting direction to explore?" Network 7, 409-421 (1996).
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H. B. Barlow, "Redundancy reduction revisited," Network 12, 241-253 (2001).
[PubMed]

B. C. Hansen, E. A. Essock, Y. Zheng, and J. K. DeFord, "Perceptual anisotropies in visual processing and their relation to natural image statistics," Network 14, 501-526 (2003).
[CrossRef] [PubMed]

Neural Comput. (1)

J. J. Atick and A. N. Redlich, "What does the retina know about natural scenes?" Neural Comput. 4, 196-210 (1992).
[CrossRef]

Ophthalmic Physiol. Opt. (1)

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

Pattern Recogn. Lett. (1)

M. C. Morrone and R. A. Owens, "Feature detection from local energy," Pattern Recogn. Lett. 6, 303-313 (1987).
[CrossRef]

Percept. Psychophys. (2)

F. A. Wichmann and N. J. Hill, "The psychometric function: I. Fitting, sampling, and goodness of fit," Percept. Psychophys. 63, 1293-1313 (2001).
[CrossRef]

F. A. Wichmann and N. J. Hill, "The psychometric function: II. Bootstrap-based confidence intervals and sampling," Percept. Psychophys. 63, 1314-1329 (2001).
[CrossRef]

Perception (1)

L. N. Piotrowski and F. W. Campbell, "A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase," Perception 11, 337-346 (1982).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

D. L. Ruderman and W. Bialek, "Statistics of natural images: scaling in the Woods," Phys. Rev. Lett. 73, 814-817 (1994).
[CrossRef] [PubMed]

PLoS Biology (1)

G. Felsen, J. Touryan, F. Han, and Y. Dan, "Cortical sensitivity to visual features in natural scenes," PLoS Biology 3, 1819-1828 (2005).
[CrossRef]

Proc. IEEE (1)

A. V. Oppenheim and J. S. Lim, "The importance of phase in signals," Proc. IEEE 69, 529-541 (1981).
[CrossRef]

Proc. R. Soc. London, Ser. B (3)

M. C. Morrone and D. C. Burr, "Feature detection in human vision: a phase dependent energy model," Proc. R. Soc. London, Ser. B 235, 221-245 (1988).
[CrossRef]

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

J. H. van Hateren and A. van der Schaaf, "Independent component filters of natural images compared with simple cells in primary visual cortex," Proc. R. Soc. London, Ser. B 265, 359-366 (1998).
[CrossRef]

Psychol. Rev. (1)

I. Biederman, "Recognition-by-components: a theory of human image understanding," Psychol. Rev. 94, 115-147 (1987).
[CrossRef] [PubMed]

Spatial Vis. (2)

D. J. Field, A. Hayes, and R. F. Hess, "The roles of polarity and symmetry in the perceptual grouping of contour fragments," Spatial Vis. 13, 51-66 (2000).
[CrossRef]

S. C. Dakin and R. F. Hess, "Contour integration and scale combination processes in visual edge detection," Spatial Vis. 12, 309-327 (1999).
[CrossRef]

Videre (1)

P. Kovesi, "Image features from phase congruency," Videre 1, 1-26 (1999).

Vision Res. (8)

J. G. Robson and N. Graham, "Probability summation and regional variation in contrast sensitivity across the visual field," Vision Res. 21, 409-418 (1981).
[CrossRef]

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

F. A. Wichmann, D. I. Braun, and K. R. Gegenfurtner, "Phase noise and the classification of natural images," Vision Res. 46, 1520-1529 (2006).
[CrossRef]

A. van der Schaaf and J. H. van Hateren, "Modeling the power spectra of natural images: statistics and Information," Vision Res. 36, 2759-2770 (1996).
[CrossRef] [PubMed]

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

D. G. Pelli, C. W. Burns, B. Farell, and D. C. Moore-Page, "Feature detection and letter identification," Vision Res. 46, 4646-4674 (2006).
[CrossRef] [PubMed]

D. J. Field, A. Hayes, and R. F. Hess, "Contour integration by the human visual system: Evidence for a local association field," Vision Res. 33, 173-193 (1993).
[CrossRef] [PubMed]

T. Ledgeway, R. F. Hess, and W. S. Geisler, "Grouping local orientation and direction signals to extract spatial contours: empirical tests of 'association field' models of contour integration," Vision Res. 45, 2511-2522 (2005).
[CrossRef] [PubMed]

Visual Cogn. (1)

B. C. Hansen and E. A. Essock, "Influence of scale and orientation on the visual perception of natural scenes," Visual Cogn. 12, 1199-1234 (2005).
[CrossRef]

Other (7)

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: New Developments and New Applications, M.Farge, J.C. R.Hunt, and J.C.Vassilicos, eds. (Oxford U. Press, 1993).

E. Doi and M. S. Lewicki, "Relations between the statistical regularities of natural images and the response properties of the early visual system," in Proceedings of the Workshop of Special Interest Group of Pattern Recognition and Perception Model (SIG P&P) (Japanese Cognitive Science Society, Kyoto University, 2005), pp. 1-8.

D. Marr, Vision: A Computational Investigation into the Human Representation and Processing of Visual Information (Freeman, 1982).
[PubMed]

L. Olzak and J. P. Thomas, "Seeing spatial patterns," in Handbook of Perception and Human Performance: Sensory Processes and Perception, K.R.Boff, L.Kaufman, and J.P.Thomas, eds. (Wiley, 1986), Vol. 1.

Z. Wang and E. P. Simoncelli, "Local phase coherence and the perception of blur," in Advances in Neural Information Processing Systems, S.Thurn, L.Saul, and B.Schölkopf, eds. (MIT Press, 2004), pp. 1435-1442.

N. Graham, "Spatial-frequency channels in human vision: detecting edges without edge detectors," in Visual Coding and Adaptability, C.Harris, ed. (Erlbaum, 1980), pp. 215-252.

G. Krieger, C. Zetzsche, and E. Barth, "Higher-order statistics of natural images and their exploitation by operators selective to intrinsic dimensionality," in Proceedings of the IEEE Signal Processing Workshop on Higher Order Statistics (IEEE, 1997), pp. 147-151.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

(a) Top left, illustration of an edge, below which is an illustration of a series of sinusoidal waveforms that have arrival phase convergence at the central position, which, over a full series of sinusoidal waveforms of increasingly higher spatial frequencies (not shown), will sum up to the edge shown above. The next four illustrations show how the saliency of the edge is corrupted as the alignment of the sinusoidal waveforms is increasingly perturbed. (b) 2D illustration of how the saliency of the contours in a given 2D natural image can be disrupted as a function of increasing (left to right) phase randomization.

Fig. 2
Fig. 2

Example stimuli from Experiment 1. Each column contains three different natural images filtered with PP FILT set to a bandwidth of 1 (left column) or 2 (right column) octaves. For each row PP FILT was set at a different central spatial frequency at (top to bottom) 3, 6, and 12 cpd , respectively.

Fig. 3
Fig. 3

Illustration of structural sparseness for vertically oriented image content. Top row, two natural scene images dominated by vertical structure, with the image on the left possessing fewer overall vertical structures than the image on the right. Middle row, filter response images from a 1-octave log-Gabor filter selective for vertical orientations (15° orientation bandwidth, full width at half-height). Notice that filter responses are more prominent (i.e., less sparse) in the image on the right compared with the response image on the left. Bottom row, pixel value histograms of the filter response images. On the abscissa are the gray-scale pixel values, and on the ordinate is the frequency of occurrence of each gray-scale value in each corresponding filter response image (note that the scales of the two ordinates are not identical). Notice that the distribution is much more peaked on the left than on the right, indicating that image is more sparse with respect to structures at the example spatial frequency range. Refer to the text for further details.

Fig. 4
Fig. 4

Illustration of the different stages of the log-Gabor filter construction process in the Fourier domain. Left to right, 2D polar coordinate reference map depicting the coordinate system (i.e., f and θ) and the axes utilized to construct the log-Gabor filter (i.e., the radius, or spatial frequency axis, f, and the angular arc, or orientation axis, θ); radial log-Gaussian filter component; theta Gaussian component; combination of the radial log-Gaussian and theta Gaussian components to give the log-Gabor filter; and an example of this filter in the spatial domain that has been assigned an even-symmetric local absolute phase angle. Note that this example gives only the real component.

Fig. 5
Fig. 5

Example natural images that have been assigned a SSM value. Smaller SSMs indicate higher degrees of overall structural sparseness (i.e., images contain relatively fewer edges/contours), while larger SSMs indicate lower degrees of structural sparseness (i.e., images contain relatively more edges/contours).

Fig. 6
Fig. 6

Example intervals from a given experiment session trial. Left (test interval), test image that has been filtered with PP FILT ( bandwidth = 1.2 octaves). Right (4AFC interval), three nonphase-filtered distracter images and one nonphase-filtered target image, which has a light-gray border (note that the test interval preceded the 4AFC interval). The task of the observer was to indicate which of the four-alternative images corresponded to the preceding test image. In this example, taken from Experiment 1, the three distracter stimuli are matched with respect to the type of content contained in the target/test image (i.e., carpentered structure).

Fig. 7
Fig. 7

Examples of natural scene images from the three different content-type categories. Left column, two images from the “natural only” category. Middle column, two images from the “natural-carpentered mixed” category. Right column, two images from the “carpentered only” category.

Fig. 8
Fig. 8

Results from Experiment 1. On the ordinate is the averaged phase alignment bandwidth threshold. On the abscissa is the averaged LSSM value for all orientations for each of the three central frequencies investigated in Experiment 1. For each of the five structural sparseness bins, refer to the text for further details.

Fig. 9
Fig. 9

Results from Experiment 1. The layout of each panel is identical to that in Fig. 8. (a) Data from images in the carpentered category. (b) Data from images in the natural-carpentered category. (c) Data from images in the natural category. Refer to the text for further details.

Fig. 10
Fig. 10

Examples of non- PP FILT -filtered stimuli created for Experiment 2. On the far left is an example phase spectrum illustrating the 20 different spatial frequency and orientation ranges carrying the spatial structure by which the natural image set was ranked (note that since the Fourier transform is odd-symmetric, each pair of corresponding segments counts once). The three spatial examples to the right of the example phase spectrum are examples of stimuli used in Experiment 2, ranked in order of increasing SSM. See text for further details.

Fig. 11
Fig. 11

Data from Experiment 2. The layout of each panel is identical to that in Fig. 8. Refer to the text for further details.

Fig. 12
Fig. 12

Data from Experiment 3. The layout of each panel is identical to that in Fig. 8. (a) Data from the condition where the stimuli contained a fixed amount of structural sparseness (SSM value of 0.09) for structures in the 3 and 12 cpd range and where the structures in the 6 cpd range were allowed to vary. (b) Data from the condition where the stimuli contained a fixed amount of structural sparseness (SSM value of 0.11) for structures in the 3 and 12 cpd range, and where the structures in the 6 cpd range were allowed to vary. Note that in both insets, the threshold phase alignment bandwidth needed to match natural image structures at 6 cpd increases with structural sparseness of the image structures near 6 cpd . That is, the ability to match the 6 cpd image structure was not influenced by the amount of structures present at spatial frequencies outside that range. Refer to the text for further details.

Equations (14)

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H ( u , v ) = 1 X Y x = 1 X y = 1 Y I ( x , y ) exp [ j 2 π ( u x X + v y Y ) ]
A ( u , v ) = Re ( u , v ) 2 + Im ( u , v ) 2 ,
Φ ( u , v ) = tan 1 ( Im ( u , v ) Re ( u , v ) ) ,
I A ( f i , θ j ) = 1 f i α .
PP FILT ( f i , θ j ) = { Φ ( f i , θ j ) f L f i f H rand ( [ π , π ] ) elsewhere } ,
I μ = 1 X Y x = 1 X y = 1 Y I FILT ( x , y ) ,
N I ( x , y ) = I FILT ( x , y ) I μ 1 ,
I SD = 1 X Y 1 x = 1 X y = 1 Y N I ( x , y ) 2 ,
I FILT ( x , y ) = { [ I SD I rms N I ( x , y ) ] 128 } + 128 ,
LG ( f i , θ j ) = exp { [ log ( R ( f i , θ j ) F peak ) 2 2 log ( σ 1 F peak ) 2 ] } exp { [ Θ ( f i , θ j ) 2 2 σ 2 2 ] } ,
F T ( x i , y j ) = { 1 F R ( x i , y j ) > + F r SD F R ( x i , y j ) < F r SD 0 elsewhere } ,
LSSM i = x = 1 X y = 1 Y F T i ( x , y ) X Y .
CP Φ r ( f i , θ j ) = N Φ r SF Φ r O ( f i , θ j ) ,
f SF L < f i < f SF H θ O 1 < θ j < θ O 2 ,

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