Abstract

Under appropriate conditions zero-crossings of a bandpass signal are very rich in information. The authors examine here the relevance of this result to the early stages of visual information processing, where zero-crossings in the output of independent spatial-frequency-tuned channels may contain sufficient information for much of the subsequent processing.

© 1979 Optical Society of America

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References

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  1. D. Marr, “Early processing of visual information,” Phil. Trans. R. Soc. Lond. B 275, 483–524 (1976), p. 488.
  2. F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–556 (1968).
  3. H. R. Wilson and S. C. Giese, “Threshold visibility of frequency gradient patterns,” Vision Res. 17, 1177–1190 (1977).
    [Crossref]
  4. 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).
  5. D. Marr and T. Poggio, “Theory of human stereopsis,” J. Opt. Soc. Am. 67, 1400 (1977).
  6. D. Marr and T. Poggio, “A computational theory of human stereo vision,” Proc. Roy. Soc. London B, (in press).
  7. B. F. Logan, “Information in the zero-crossings of bandpass signals,” Bell Sys. Techn. J. 56, 487–510 (1977).
    [Crossref]
  8. M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
    [Crossref] [PubMed]
  9. J. J. Kulikowski and P. E. King-Smith, “Spatial arrangement of line, edge and grating detectors revealed by subthreshold summation,” Vision Res. 13, 1455–1478 (1973).
    [Crossref] [PubMed]
  10. R. M. Shapley and D. J. Tolhurst, “Edge detectors in human vision,” J. Physiol. (London) 229, 165–183 (1973).
  11. G. E. Legge, “Sustained and transient mechanisms in human vision: temporal and spatial properties,” Vision Res. 18, 69–81 (1978).
    [Crossref] [PubMed]
  12. J. D. Cowan, “Some remarks on channel bandwidths for visual contrast detection,” Neurosciences Res. Prog. Bull,  15, 492–517 (1977), Fig. A 12.

1978 (1)

G. E. Legge, “Sustained and transient mechanisms in human vision: temporal and spatial properties,” Vision Res. 18, 69–81 (1978).
[Crossref] [PubMed]

1977 (4)

J. D. Cowan, “Some remarks on channel bandwidths for visual contrast detection,” Neurosciences Res. Prog. Bull,  15, 492–517 (1977), Fig. A 12.

H. R. Wilson and S. C. Giese, “Threshold visibility of frequency gradient patterns,” Vision Res. 17, 1177–1190 (1977).
[Crossref]

D. Marr and T. Poggio, “Theory of human stereopsis,” J. Opt. Soc. Am. 67, 1400 (1977).

B. F. Logan, “Information in the zero-crossings of bandpass signals,” Bell Sys. Techn. J. 56, 487–510 (1977).
[Crossref]

1976 (1)

D. Marr, “Early processing of visual information,” Phil. Trans. R. Soc. Lond. B 275, 483–524 (1976), p. 488.

1973 (2)

J. J. Kulikowski and P. E. King-Smith, “Spatial arrangement of line, edge and grating detectors revealed by subthreshold summation,” Vision Res. 13, 1455–1478 (1973).
[Crossref] [PubMed]

R. M. Shapley and D. J. Tolhurst, “Edge detectors in human vision,” J. Physiol. (London) 229, 165–183 (1973).

1971 (1)

1968 (1)

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–556 (1968).

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).

Campbell, F. W.

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–556 (1968).

Cowan, J. D.

J. D. Cowan, “Some remarks on channel bandwidths for visual contrast detection,” Neurosciences Res. Prog. Bull,  15, 492–517 (1977), Fig. A 12.

Giese, S. C.

H. R. Wilson and S. C. Giese, “Threshold visibility of frequency gradient patterns,” Vision Res. 17, 1177–1190 (1977).
[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).

King-Smith, P. E.

J. J. Kulikowski and P. E. King-Smith, “Spatial arrangement of line, edge and grating detectors revealed by subthreshold summation,” Vision Res. 13, 1455–1478 (1973).
[Crossref] [PubMed]

Kulikowski, J. J.

J. J. Kulikowski and P. E. King-Smith, “Spatial arrangement of line, edge and grating detectors revealed by subthreshold summation,” Vision Res. 13, 1455–1478 (1973).
[Crossref] [PubMed]

Legge, G. E.

G. E. Legge, “Sustained and transient mechanisms in human vision: temporal and spatial properties,” Vision Res. 18, 69–81 (1978).
[Crossref] [PubMed]

Logan, B. F.

B. F. Logan, “Information in the zero-crossings of bandpass signals,” Bell Sys. Techn. J. 56, 487–510 (1977).
[Crossref]

Marr, D.

D. Marr and T. Poggio, “Theory of human stereopsis,” J. Opt. Soc. Am. 67, 1400 (1977).

D. Marr, “Early processing of visual information,” Phil. Trans. R. Soc. Lond. B 275, 483–524 (1976), p. 488.

D. Marr and T. Poggio, “A computational theory of human stereo vision,” Proc. Roy. Soc. London B, (in press).

Nachmias, J.

Poggio, T.

D. Marr and T. Poggio, “Theory of human stereopsis,” J. Opt. Soc. Am. 67, 1400 (1977).

D. Marr and T. Poggio, “A computational theory of human stereo vision,” Proc. Roy. Soc. London B, (in press).

Robson, J. G.

M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
[Crossref] [PubMed]

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–556 (1968).

Sachs, M. B.

Shapley, R. M.

R. M. Shapley and D. J. Tolhurst, “Edge detectors in human vision,” J. Physiol. (London) 229, 165–183 (1973).

Tolhurst, D. J.

R. M. Shapley and D. J. Tolhurst, “Edge detectors in human vision,” J. Physiol. (London) 229, 165–183 (1973).

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).

Wilson, H. R.

H. R. Wilson and S. C. Giese, “Threshold visibility of frequency gradient patterns,” Vision Res. 17, 1177–1190 (1977).
[Crossref]

Bell Sys. Techn. J. (1)

B. F. Logan, “Information in the zero-crossings of bandpass signals,” Bell Sys. Techn. J. 56, 487–510 (1977).
[Crossref]

J. Opt. Soc. Am. (2)

J. Physiol. (London) (3)

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–556 (1968).

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).

R. M. Shapley and D. J. Tolhurst, “Edge detectors in human vision,” J. Physiol. (London) 229, 165–183 (1973).

Neurosciences Res. Prog. Bull (1)

J. D. Cowan, “Some remarks on channel bandwidths for visual contrast detection,” Neurosciences Res. Prog. Bull,  15, 492–517 (1977), Fig. A 12.

Phil. Trans. R. Soc. Lond. B (1)

D. Marr, “Early processing of visual information,” Phil. Trans. R. Soc. Lond. B 275, 483–524 (1976), p. 488.

Vision Res. (3)

H. R. Wilson and S. C. Giese, “Threshold visibility of frequency gradient patterns,” Vision Res. 17, 1177–1190 (1977).
[Crossref]

J. J. Kulikowski and P. E. King-Smith, “Spatial arrangement of line, edge and grating detectors revealed by subthreshold summation,” Vision Res. 13, 1455–1478 (1973).
[Crossref] [PubMed]

G. E. Legge, “Sustained and transient mechanisms in human vision: temporal and spatial properties,” Vision Res. 18, 69–81 (1978).
[Crossref] [PubMed]

Other (1)

D. Marr and T. Poggio, “A computational theory of human stereo vision,” Proc. Roy. Soc. London B, (in press).

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Figures (2)

FIG. 1
FIG. 1

The meaning of Logan’s theorem. (a) shows a stochastic band-limited gaussian signal f(x), and (c) exhibits the result fb(x) of filtering (a) through an ideal one-octave bandpass filter. The modulus of its transfer function is shown in (b). Since (c) has a bandwidth of one octave, and it has no zeroes in common with its Hilbert transform, Logan’s theorem tells us that (c) is determined, up to a multiplicative constant, by its zero-crossings alone. The aspect of Logan’s result that is important for this article is that under the right conditions, zero-crossings alone are very rich in information.

FIG. 2
FIG. 2

(a) On the left are shown short bar-shaped masks at the vertical and horizontal orientations, and on the right, the amplitude of their (idealized) transfer functions. The bandwidth shown here is one octave, the maximum value for which Logan’s theorem applies. (In practice, an ideal one-octave bandwidth requires side-lobes in the “receptive field”). If for each mask, zero-crossings are found along scan-lines lying perpendicular to the mask’s orientation, these zero-crossings contain full information about that part of the image whose spectrum falls within the shaded region (on the right) of the Fourier plane. The remaining regions of the Fourier plane can be covered by similar masks of different sizes. Provided that there is sufficient overlap in the Fourier domain, information from different masks can in principle be combined to give the original image up to a single scaling factor. (b) If the masks are more elongated, the support of their Fourier transform will become smaller. In order to cover the Fourier plane a set of such masks of several orientations will be required. Figure 2(b) shows an elongated mask, whose cross section is the difference of two gaussians, together with its Fourier transform. Interestingly, if one uses masks constructed from the difference of two gaussian curves,3 their Fourier transforms behave like ω2 for values of ω that are small compared to σ. In other words, they approximate a second derivative operator.