Abstract

We argue that some aspects of human spatial vision, particularly for textured patterns and scenes, can be described in terms of demodulation and predictive coding. Such nonlinear processes encode a pattern into local phasors that represent it completely as a modulation, in phase and amplitude, of a prediction associated with the image structure in some region by its predominant undulation(s). The demodulation representation of a pattern is an anisotropic, second-order form of predictive coding, and it offers a particularly efficient way to analyze and encode textures, as it identifies and exploits their underlying redundancies. In addition, self-consistent domains of redundancy in image structure provide a basis for image segmentation. We first provide an algorithm for computing the three elements of a complete demodulation transform of any image, and we illustrate such decompositions for both natural and synthetic images. We then present psychophysical evidence from spatial masking experiments, as well as illustrations of perceptual organization, that suggest a possible role for such underlying representations in human vision. In psychophysical experiments employing masks with more than two oriented Fourier components, we find that peaks of threshold elevation occur at locations in the Fourier plane remote from the orientations and frequencies of the actual mask components. Rather, as would occur from demodulation, these peaks in the frequency plane are related to the vector difference frequencies between the actual masking components and their spectral centers of mass. We offer a neural interpretation of demodulation coding, and finally we demonstrate a practical application of this process in a system for automatic visual recognition of personal identity by demodulation of a facial feature.

© 1995 Optical Society of America

Full Article  |  PDF Article

Corrections

John G. Daugman and Cathryn J. Downing, "Demodulation, predictive coding, and spatial vision: erratum," J. Opt. Soc. Am. A 12, 2077-2077 (1995)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-12-9-2077

References

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  10. G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyzes spatial patterns in independent bands of spatial frequency,” Vision Res. 15, 887–897 (1975).
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  18. J. G. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression. Invited Paper,” IEEE Trans. Acoust. Speech Signal Process. 36, 1169–1179 (1988).
    [Crossref]
  19. A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
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  25. D. M. Green, An Introduction to Hearing (Erlbaum, Hillsdale, N.J., 1976).
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  27. L. O. Harvey, V. V. Doan, “Visual masking at different polar angles in the two-dimensional Fourier plane,” J. Opt. Soc. Am. A 7, 116–127 (1990).
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  28. D. Marr, Vision: A Computational Investigation into the Human Representation and Processing of Visual Information (Freeman, New York, 1982).
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    [Crossref] [PubMed]
  30. G. B. Wetherill, H. Levitt, “Sequential estimation of points on a psychometric function,” Brit. J. Math. Stat. Psychol. 18, 1–10 (1965).
    [Crossref]
  31. J. G. Daugman, R. E. Kronauer, Y. Y. Zeevi, “Perception of two-dimensional phase modulation and amplitude modulation signals in spatio-temporal bandlimited textures,” Perception 13, A16 (1984).
  32. D. J. Fleet, K. Langley, “Computational analysis of non-Fourier motion,” Vision Res. 34, 3057–3079 (1994).
    [Crossref] [PubMed]
  33. A. M. Derrington, D. R. Badcock, “Two-stage analysis of the motion of 2-D patterns: what is the first stage?” Vision Res. 32, 691–698 (1992).
    [Crossref] [PubMed]
  34. A. M. Derrington, G. B. Henning, “Some observations on the masking effects of two-dimensional stimuli,” Vision Res. 29, 241–246 (1989).
    [Crossref] [PubMed]
  35. H. R. Wilson, W. A. Richards, “Curvature and separation discrimination at texture boundaries,” J. Opt. Soc. Am. A 9, 1653–1662 (1992).
    [Crossref] [PubMed]
  36. A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
    [Crossref]
  37. D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
    [Crossref] [PubMed]
  38. J. G. Daugman, “High confidence visual recognition of persons by a test of statistical independence,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 1148–1161 (1993).
    [Crossref]
  39. J. G. Daugman, “Biometric personal identification system based on iris analysis,” U.S. patent5,291,560 (March1, 1994).
  40. S. Marčelja, “Mathematical description of the responses of simple cortical cells,” J. Opt. Soc. Am. 70, 1297–1300 (1980).
    [Crossref]
  41. J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 847–856 (1980).
    [Crossref] [PubMed]
  42. M. S. Landy, J. R. Bergen, “Texture segregation and orientation gradient,” Vision Res. 31, 679–691 (1991).
    [Crossref] [PubMed]
  43. T. Caelli, “Energy processing and coding factors in texture discrimination and image processing,” Percept. Psychophys. 34, 349–355 (1983).
    [Crossref] [PubMed]
  44. J. G. Daugman, “Uncertainty relation 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]
  45. E. Mach, “Über den Einfluss räumlich und zeitlich variierender Lichtreize auf die Gesichtswahrnehmung,” Sitzungsber. Math. Naturwiss. Kl. Kaiser. Akad. Wiss. 115, 633–648 (1906).
  46. F. Attneave, “Some informational aspects of visual perception,” Psycholog. Rev. 61, 183–193 (1954).
    [Crossref]
  47. H. B. Barlow, “The coding of sensory messages,” in Current Problems in Animal Behavior, W. Thorpe, L. Zangwill, eds. (Cambridge U. Press, Cambridge, 1961), pp. 331–360.
  48. D. Kersten, “Predictability and redundancy of natural images,” J. Opt. Soc. Am. A 4, 2395–2400 (1987).
    [Crossref] [PubMed]
  49. M. C. Morrone, D. C. Burr, L. Maffei, “Functional implications of cross-orientation inhibition of cortical visual cells. I. Neurophysiological evidence,” Proc. R. Soc. London Ser. B 216, 335–354 (1982).
    [Crossref]

1994 (2)

1993 (2)

J. G. Daugman, “High confidence visual recognition of persons by a test of statistical independence,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 1148–1161 (1993).
[Crossref]

J. G. Daugman, “Quadrature-phase simple-cell pairs are appropriately described in complex analytic form,” J. Opt. Soc. Am. A 10, 375–377 (1993).
[Crossref]

1992 (2)

A. M. Derrington, D. R. Badcock, “Two-stage analysis of the motion of 2-D patterns: what is the first stage?” Vision Res. 32, 691–698 (1992).
[Crossref] [PubMed]

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

1991 (2)

A. C. Bovik, “Analysis of multichannel narrow-band filters for image texture segmentation,” IEEE Trans. Signal Process. 39, 2025–2043 (1991).
[Crossref]

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

1990 (3)

L. O. Harvey, V. V. Doan, “Visual masking at different polar angles in the two-dimensional Fourier plane,” J. Opt. Soc. Am. A 7, 116–127 (1990).
[Crossref] [PubMed]

A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
[Crossref]

A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
[Crossref]

1989 (3)

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]

J. Nachmias, “Contrast-modulated maskers: test of a late nonlinearity hypothesis,” Vision Res. 29, 137–142 (1989).
[Crossref]

A. M. Derrington, G. B. Henning, “Some observations on the masking effects of two-dimensional stimuli,” Vision Res. 29, 241–246 (1989).
[Crossref] [PubMed]

1988 (2)

J. G. Daugman, “Pattern and motion vision without Laplacian zero-crossings,” J. Opt. Soc. Am. A 5, 1142–1148 (1988).
[Crossref] [PubMed]

J. G. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression. Invited Paper,” IEEE Trans. Acoust. Speech Signal Process. 36, 1169–1179 (1988).
[Crossref]

1987 (2)

A. M. Derrington, “Distortion products in geniculate X-cells: a physiological basis for masking by spatially modulated gratings,” Vision Res. 27, 1377–1387 (1987).
[Crossref]

D. Kersten, “Predictability and redundancy of natural images,” J. Opt. Soc. Am. A 4, 2395–2400 (1987).
[Crossref] [PubMed]

1986 (1)

A. M. Derrington, D. R. Badcock, “Detection of spatial beats: non-linearity or contrast-increment detection?” Vision Res. 26, 343–348 (1986).
[Crossref]

1985 (1)

1984 (2)

J. G. Daugman, R. E. Kronauer, Y. Y. Zeevi, “Perception of two-dimensional phase modulation and amplitude modulation signals in spatio-temporal bandlimited textures,” Perception 13, A16 (1984).

J. G. Daugman, “Spatial visual channels in the Fourier plane,” Vision Res. 24, 891–910 (1984).
[Crossref] [PubMed]

1983 (2)

J. Nachmias, B. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1629 (1983).
[Crossref] [PubMed]

T. Caelli, “Energy processing and coding factors in texture discrimination and image processing,” Percept. Psychophys. 34, 349–355 (1983).
[Crossref] [PubMed]

1982 (2)

M. C. Morrone, D. C. Burr, L. Maffei, “Functional implications of cross-orientation inhibition of cortical visual cells. I. Neurophysiological evidence,” Proc. R. Soc. London Ser. B 216, 335–354 (1982).
[Crossref]

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc R. Soc. London Ser. B 216, 427–459 (1982).
[Crossref]

1981 (2)

D. G. Albrecht, R. L. DeValois, “Striate cortex responses to periodic patterns with and without the fundamental harmonics,” J. Physiol. 319, 497–514 (1981).
[PubMed]

D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
[Crossref] [PubMed]

1980 (2)

S. Marčelja, “Mathematical description of the responses of simple cortical cells,” J. Opt. Soc. Am. 70, 1297–1300 (1980).
[Crossref]

J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 847–856 (1980).
[Crossref] [PubMed]

1975 (1)

G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyzes spatial patterns in independent bands of spatial frequency,” Vision Res. 15, 887–897 (1975).
[Crossref]

1973 (1)

G. J. Burton, “Evidence for non-linear response processes in the human visual system from measurements of the thresholds of spatial beat frequencies,” Vision Res. 13, 1211–1225 (1973).
[Crossref] [PubMed]

1965 (1)

G. B. Wetherill, H. Levitt, “Sequential estimation of points on a psychometric function,” Brit. J. Math. Stat. Psychol. 18, 1–10 (1965).
[Crossref]

1954 (1)

F. Attneave, “Some informational aspects of visual perception,” Psycholog. Rev. 61, 183–193 (1954).
[Crossref]

1952 (2)

C. W. Harrison, “Experiments with linear prediction in television,” Bell Syst. Tech. J. 31, 764–783 (1952).

B. M. Oliver, “Efficient coding,” Bell Syst. Tech. J. 31, 724–750 (1952).

1928 (1)

R. V. L. Hartley, “Transmission of information,” Bell Syst. Tech. J. 7, 535–563 (1928).

1906 (1)

E. Mach, “Über den Einfluss räumlich und zeitlich variierender Lichtreize auf die Gesichtswahrnehmung,” Sitzungsber. Math. Naturwiss. Kl. Kaiser. Akad. Wiss. 115, 633–648 (1906).

1841 (1)

A. Seebeck, “Beohachtungen über einige Bedingungen der Entstehung von Tönen,” Ann. Phys. Chem. 53, 417–436 (1841).

Albrecht, D. G.

D. G. Albrecht, R. L. DeValois, “Striate cortex responses to periodic patterns with and without the fundamental harmonics,” J. Physiol. 319, 497–514 (1981).
[PubMed]

Attneave, F.

F. Attneave, “Some informational aspects of visual perception,” Psycholog. Rev. 61, 183–193 (1954).
[Crossref]

Badcock, D. R.

A. M. Derrington, D. R. Badcock, “Two-stage analysis of the motion of 2-D patterns: what is the first stage?” Vision Res. 32, 691–698 (1992).
[Crossref] [PubMed]

A. M. Derrington, D. R. Badcock, “Detection of spatial beats: non-linearity or contrast-increment detection?” Vision Res. 26, 343–348 (1986).
[Crossref]

Barlow, H. B.

H. B. Barlow, “The coding of sensory messages,” in Current Problems in Animal Behavior, W. Thorpe, L. Zangwill, eds. (Cambridge U. Press, Cambridge, 1961), pp. 331–360.

Bergen, J. R.

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

Bovik, A. C.

A. C. Bovik, “Analysis of multichannel narrow-band filters for image texture segmentation,” IEEE Trans. Signal Process. 39, 2025–2043 (1991).
[Crossref]

A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
[Crossref]

A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
[Crossref]

Broadbent, D. E.

G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyzes spatial patterns in independent bands of spatial frequency,” Vision Res. 15, 887–897 (1975).
[Crossref]

Burr, D. C.

M. C. Morrone, D. C. Burr, L. Maffei, “Functional implications of cross-orientation inhibition of cortical visual cells. I. Neurophysiological evidence,” Proc. R. Soc. London Ser. B 216, 335–354 (1982).
[Crossref]

Burton, G. J.

G. J. Burton, “Evidence for non-linear response processes in the human visual system from measurements of the thresholds of spatial beat frequencies,” Vision Res. 13, 1211–1225 (1973).
[Crossref] [PubMed]

Caelli, T.

T. Caelli, “Energy processing and coding factors in texture discrimination and image processing,” Percept. Psychophys. 34, 349–355 (1983).
[Crossref] [PubMed]

Clark, M.

A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
[Crossref]

A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
[Crossref]

Daugman, J. G.

J. G. Daugman, “High confidence visual recognition of persons by a test of statistical independence,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 1148–1161 (1993).
[Crossref]

J. G. Daugman, “Quadrature-phase simple-cell pairs are appropriately described in complex analytic form,” J. Opt. Soc. Am. A 10, 375–377 (1993).
[Crossref]

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]

J. G. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression. Invited Paper,” IEEE Trans. Acoust. Speech Signal Process. 36, 1169–1179 (1988).
[Crossref]

J. G. Daugman, “Pattern and motion vision without Laplacian zero-crossings,” J. Opt. Soc. Am. A 5, 1142–1148 (1988).
[Crossref] [PubMed]

J. G. Daugman, “Uncertainty relation 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]

J. G. Daugman, “Spatial visual channels in the Fourier plane,” Vision Res. 24, 891–910 (1984).
[Crossref] [PubMed]

J. G. Daugman, R. E. Kronauer, Y. Y. Zeevi, “Perception of two-dimensional phase modulation and amplitude modulation signals in spatio-temporal bandlimited textures,” Perception 13, A16 (1984).

J. G. Daugman, “Two-dimensional spectral analysis of cortical receptive field profiles,” Vision Res. 20, 847–856 (1980).
[Crossref] [PubMed]

J. G. Daugman, “Biometric personal identification system based on iris analysis,” U.S. patent5,291,560 (March1, 1994).

J. G. Daugman, “Image analysis and compact coding by oriented 2D Gabor primitives,” in Image Understanding and the Man–Machine Interface, J. J. Pearson, E. Barrett, eds., Proc. Soc. Photo-Opt. Instrum. Eng.758, 19–30 (1987).
[Crossref]

Derrington, A. M.

A. M. Derrington, D. R. Badcock, “Two-stage analysis of the motion of 2-D patterns: what is the first stage?” Vision Res. 32, 691–698 (1992).
[Crossref] [PubMed]

A. M. Derrington, G. B. Henning, “Some observations on the masking effects of two-dimensional stimuli,” Vision Res. 29, 241–246 (1989).
[Crossref] [PubMed]

A. M. Derrington, “Distortion products in geniculate X-cells: a physiological basis for masking by spatially modulated gratings,” Vision Res. 27, 1377–1387 (1987).
[Crossref]

A. M. Derrington, D. R. Badcock, “Detection of spatial beats: non-linearity or contrast-increment detection?” Vision Res. 26, 343–348 (1986).
[Crossref]

DeValois, R. L.

D. G. Albrecht, R. L. DeValois, “Striate cortex responses to periodic patterns with and without the fundamental harmonics,” J. Physiol. 319, 497–514 (1981).
[PubMed]

Doan, V. V.

Dubs, A.

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc R. Soc. London Ser. B 216, 427–459 (1982).
[Crossref]

Fleet, D. J.

D. J. Fleet, K. Langley, “Computational analysis of non-Fourier motion,” Vision Res. 34, 3057–3079 (1994).
[Crossref] [PubMed]

Gallagher, R.

R. Gallagher, Information Theory and Reliable Communication (Wiley, New York, 1968).

Geisler, W. S.

A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
[Crossref]

A. C. Bovik, M. Clark, W. S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 55–73 (1990).
[Crossref]

Ghiglia, D. C.

Green, D. M.

D. M. Green, An Introduction to Hearing (Erlbaum, Hillsdale, N.J., 1976).

Harrison, C. W.

C. W. Harrison, “Experiments with linear prediction in television,” Bell Syst. Tech. J. 31, 764–783 (1952).

Hartley, R. V. L.

R. V. L. Hartley, “Transmission of information,” Bell Syst. Tech. J. 7, 535–563 (1928).

Harvey, L. O.

Henning, G. B.

A. M. Derrington, G. B. Henning, “Some observations on the masking effects of two-dimensional stimuli,” Vision Res. 29, 241–246 (1989).
[Crossref] [PubMed]

G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyzes spatial patterns in independent bands of spatial frequency,” Vision Res. 15, 887–897 (1975).
[Crossref]

Hertz, B. G.

G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyzes spatial patterns in independent bands of spatial frequency,” Vision Res. 15, 887–897 (1975).
[Crossref]

Kersten, D.

Kronauer, R. E.

J. G. Daugman, R. E. Kronauer, Y. Y. Zeevi, “Perception of two-dimensional phase modulation and amplitude modulation signals in spatio-temporal bandlimited textures,” Perception 13, A16 (1984).

Landy, M. S.

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

Langley, K.

D. J. Fleet, K. Langley, “Computational analysis of non-Fourier motion,” Vision Res. 34, 3057–3079 (1994).
[Crossref] [PubMed]

Laughlin, S. B.

M. V. Srinivasan, S. B. Laughlin, A. Dubs, “Predictive coding: a fresh view of inhibition in the retina,” Proc R. Soc. London Ser. B 216, 427–459 (1982).
[Crossref]

Levitt, H.

G. B. Wetherill, H. Levitt, “Sequential estimation of points on a psychometric function,” Brit. J. Math. Stat. Psychol. 18, 1–10 (1965).
[Crossref]

Mach, E.

E. Mach, “Über den Einfluss räumlich und zeitlich variierender Lichtreize auf die Gesichtswahrnehmung,” Sitzungsber. Math. Naturwiss. Kl. Kaiser. Akad. Wiss. 115, 633–648 (1906).

Maffei, L.

M. C. Morrone, D. C. Burr, L. Maffei, “Functional implications of cross-orientation inhibition of cortical visual cells. I. Neurophysiological evidence,” Proc. R. Soc. London Ser. B 216, 335–354 (1982).
[Crossref]

Marcelja, S.

Marr, D.

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

Morrone, M. C.

M. C. Morrone, D. C. Burr, L. Maffei, “Functional implications of cross-orientation inhibition of cortical visual cells. I. Neurophysiological evidence,” Proc. R. Soc. London Ser. B 216, 335–354 (1982).
[Crossref]

Nachmias, J.

J. Nachmias, “Contrast-modulated maskers: test of a late nonlinearity hypothesis,” Vision Res. 29, 137–142 (1989).
[Crossref]

J. Nachmias, B. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1629 (1983).
[Crossref] [PubMed]

Oliver, B. M.

B. M. Oliver, “Efficient coding,” Bell Syst. Tech. J. 31, 724–750 (1952).

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, Signals and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Pollen, D. A.

D. A. Pollen, S. F. Ronner, “Phase relationships between adjacent simple cells in the visual cortex,” Science 212, 1409–1411 (1981).
[Crossref] [PubMed]

Richards, W. A.

Rogowitz, B.

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Figures (20)

Fig. 1
Fig. 1

Illustration of the demodulation transform for a synthetic six-component texture and its reconstruction from the transform’s AM and PM parts.

Fig. 2
Fig. 2

Illustration of the demodulation transform for a synthetic (degenerate) six-component texture and its reconstruction from the transform’s AM and PM parts.

Fig. 3
Fig. 3

Illustration of the demodulation transform for a synthetic five-component texture and its reconstruction from the transform’s AM and PM parts.

Fig. 4
Fig. 4

Illustration of the demodulation transform for a plaid pattern and its reconstruction from the transform’s AM and PM parts. The AM component of the transform predicts the perceived beat pattern’s spatial frequency, orientation, and drift velocity, as described in the psychophysical experiments in Section 5.

Fig. 5
Fig. 5

Illustration of the demodulation transform for a three-component texture in a pure-AM configuration and its reconstruction. Such stimuli were used as the masker patterns in the psychophysical experiments described in Section 6 below. The AM component of the demodulation transform of such masker patterns predicts the spatial frequency and orientation at which peaks of threshold elevation occur in the Fourier plane.

Fig. 6
Fig. 6

Parallelogram rule for calculating 2D beat frequencies in the case of textures containing only two Fourier components (plaids). The vector difference frequency for each two-component plaid at the left is shown at the right, and this corresponds to the periodicity of the phasor AM component A(x, y) defined in Eq. (12) or (14) as illustrated in Fig. 4. This vector difference frequency corresponds to the perceived beat spatial frequency and orientation.

Fig. 7
Fig. 7

Demodulation transform of a pattern that induces illusory contours. The S-shaped contour perceived in the original stimulus corresponds to a ridge that is explicit in the AM component of the pattern’s demodulation transform.

Fig. 8
Fig. 8

Observers’ settings for spatial frequency to match the beat component perceived in moving plaids, plotted against the spatial frequency of the AM component of the plaid’s demodulation transform as defined in Eq. (12). Observers HW, JD, and RF.

Fig. 9
Fig. 9

Observers’ settings for orientation to match the beat component perceived in moving plaids, plotted against the orientation of the AM component of the plaid’s demodulation transform as defined in Eq. (12). Observers HW, JD, and RF.

Fig. 10
Fig. 10

Observers’ settings for velocity to match the beat component perceived in moving plaids, plotted against the velocity of the AM component of the plaid’s demodulation transform as defined in Eq. (12). Observers HW, JD, and RF.

Fig. 11
Fig. 11

General 2D spectral configuration for a three-component pure-AM stimulus. Filled symbols represent actual spectral components, which must lie on a parallelogram as indicated, and open circles represent the perceived but nonspectral (phantom) AM component. This component can always be computed (as in Subsection 2.A) by the demodulation transform algorithm. Numerical values are indicated for one such sample configuration.

Fig. 12
Fig. 12

Threshold elevation factors for detection of a horizontal test grating of various spatial frequencies. The masks were three-component textures as illustrated in Fig. 5, comprising a vertical carrier plus two oblique sidebands. An arrow indicates the spatial frequency of the horizontal AM phasor component in the demodulation representation of each mask. Even though no such component was spectrally present in the mask, this AM demodulation component correctly predicts the location of the peak horizontal masking effect for observers cjd and jgd in both experiments.

Fig. 13
Fig. 13

Threshold elevations for a horizontal test grating in the presence of one, two, or all three of the components used in the AM masking experiments summarized in Figs. 5 and 12. The icons beneath each panel signify both the orientation(s) and the number of mask components that were present. Bars sharing a common filled symbol overhead represent the only conditions in which threshold differences are not statistically significant. This pattern of results is predicted by the demodulation phasor AM component of each mask. The results show that the effectiveness of multicomponent masks does not stem from the separate action of their individual components.

Fig. 14
Fig. 14

Quadrature demodulator network whose function approximates (within a particular passband) the extraction of demodulation phasor AM and PM components. The phasor diagram at the bottom indicates how some basic operations of complex algebra are implemented by this neural model, all of whose elements are real valued, in resolving the AM and PM components of image projections onto the receptive fields. From Daugman.17

Fig. 15
Fig. 15

Illustration of how the complexity of the demodulation transform’s PM component is related to the predictive power of the carrier. The regions of the collage are defined by different primary orientations of correlation, but in this example the demodulation carrier was forced to be vertical. The resulting PM complexity is lowest when the vertical carrier most closely matches the correlation moments of the texture and thus best predicts its redundant structure. This clearly occurs in the T-shaped (vertically correlated) region of the original texture.

Fig. 16
Fig. 16

Demodulation transform of a natural wilderness scene and its exact reconstruction from the AM and PM components shown.

Fig. 17
Fig. 17

Demodulation transform of a natural stone texture and its exact reconstruction from the AM and PM components shown.

Fig. 18
Fig. 18

Application of AMPM texture demodulation in automatic visual recognition of personal identity, using the texture visible in the iris. A multicarrier, 256-byte iris code constructed from quantization of the phasor PM component is shown inscribed in the corner. Computing this code from a video image of the eye can establish its owner’s identity in less than 1 s with extremely high confidence.

Fig. 19
Fig. 19

Phase-quadrant quantization of the demodulation phasor PM component to only two bits for constructing the identifying iris code.

Fig. 20
Fig. 20

Performance histograms for the automatic personal identification system based on AMPM demodulation of visible iris texture. Measured Hamming distances between the multicarrier iris codes as illustrated in Fig. 18 tallied both for images comparing different eyes (Imposters, black histogram) and for different images of the same eye (Authentics, white histogram). The resulting decision task has d′ = 8.41 formal decidability. Solid curves are Eq. (31).

Equations (31)

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Z ( x , y ) = A ( x , y ) exp [ i ϕ ( x , y ) ]
S ( x , y ) = k = N N α k exp [ i ( u k x + ν k y ) ]
μ c = k = 1 N α k μ k k = 1 N α k ,
μ c = k = 1 N α k ν k k = 1 N ν k .
C ( x , y ) = exp [ i ( μ c x + ν c y ) ] ,
Z ( x , y ) = k = 1 N α k exp [ i ( Δ μ k x + Δ ν k y ) ] ,
Z ( x , y ) C ( x , y ) = { k = 1 N α k exp [ i ( Δ μ k x + Δ ν k y ) ] } × exp [ i ( μ c x + ν c y ) ] = k = 1 N α k exp [ i ( μ k x + ν k y ) ] ,
Z * ( x , y ) C * ( x , y ) = { k = 1 N α k * exp [ i ( Δ μ k x + Δ ν k y ) ] } × exp [ i ( μ c x + ν c y ) ] = k = 1 N α k * exp [ i ( μ k x + ν k y ) ] = k = N 1 α k exp [ i ( μ k x + ν k y ) ] ,
S ( x , y ) = Z ( x , y ) C ( x , y ) + Z * ( x , y ) C * ( x , y ) ,
S ( x , y ) = 2 Re [ Z ( x , y ) C ( x , y ) ] .
Z ( x , y ) = A ( x , y ) exp [ i ϕ ( x , y ) ] ,
A ( x , y ) = { [ k = 1 N α k cos ( Δ μ k x + Δ ν k y + θ k ) ] 2 + [ k = 1 N α k sin ( Δ μ k x + Δ ν k y + θ k ) ] 2 } 1 / 2 ,
ϕ ( x , y ) = tan 1 [ k = 1 N α k sin ( Δ μ k x + Δ ν k y + θ k ) k = 1 N α k cos ( Δ μ k x + Δ ν k y + θ k ) ] .
A ( x , y ) = { m = 1 N n = 1 N α m α n cos [ ( μ m μ n ) x + ( ν m ν n ) y + ( θ m θ n ) ] } 1 / 2 .
S ( x , y ) = i 2 { exp [ i ( μ 1 x + ν 1 y ) ] exp [ i ( μ 1 x + ν 1 y ) ] } + i 2 { exp [ i ( μ 2 x + ν 2 y ) ] exp [ i ( μ 2 x + ν 2 y ) ] } .
C ( x , y ) = exp { i [ ( μ 1 + μ 2 2 x + ( ν 1 + ν 2 2 ) y ] } .
Z ( x , y ) = i 2 exp { i [ ( μ 1 μ 2 2 ) x + ( ν 1 ν 2 2 ) y ] } i 2 exp { i [ ( μ 2 μ 1 2 ) x + ( ν 2 ν 1 2 ) y ] } = i cos [ ( μ 1 μ 2 ) x / 2 + ( ν 1 ν 2 ) y / 2 ] ,
A ( x , y ) = { cos 2 [ ( μ 1 μ 2 ) x / 2 + ( ν 1 ν 2 ) y / 2 ] } 1 / 2 = | cos [ ( μ 1 μ 2 ) x / 2 + ( ν 1 ν 2 ) y / 2 ] | = { ½ + ½ cos [ ( μ 1 μ 2 ) x + ( ν 1 ν 2 ) y ] } ½ .
f 3 = ½ [ f 1 2 + f 2 2 + 2 f 1 f 2 cos ( θ 1 θ 2 ) ] 1 / 2 ,
f 4 = ½ [ f 1 2 + f 2 2 2 f 1 f 2 cos ( θ 1 θ 2 ) ] 1 / 2 ,
θ 3 = tan 1 [ f 1 sin ( θ 1 ) + f 2 sin ( θ 2 ) f 1 cos ( θ 1 ) + f 2 cos ( θ 2 ) ] ,
θ 4 = tan 1 [ f 1 sin ( θ 1 ) f 2 sin ( θ 2 ) f 1 cos ( θ 1 ) f 2 cos ( θ 2 ) ] ,
ω 3 = ½ ( ω 1 + ω 2 ) ,
ω 4 = ½ ( ω 1 ω 2 ) .
S ( x , y ) = 2 cos ( μ c x + ν c y ) + cos [ ( μ c + Δ μ ) x + ( ν c + Δ ν ) y ] + cos [ ( μ c Δ μ ) x + ( ν c Δ ν ) y ] ,
S ( x , y ) = exp [ i ( μ c x + ν c y ) ] + exp [ i ( μ c x + ν c y ) ] + ½ exp { i [ ( μ c + Δ μ ) x + ( ν c + Δ ν ) y ] } + ½ exp { i [ ( μ c + Δ μ ) x + ( ν c + Δ ν ) y ] } + ½ exp { i [ ( μ c Δ μ ) x + ( ν c Δ ν ) y ] } + ½ exp { i [ ( μ c Δ μ ) x + ( ν c Δ ν ) y ] } .
C ( x , y ) = exp [ i ( μ c x + ν c y ) ] ,
Z ( x , y ) = 1 + ½ exp [ i ( x Δ μ + y Δ ν ) ] + ½ exp [ i ( x Δ μ y Δ ν ) ] = 1 + cos ( x Δ μ + y Δ ν ) .
A ( x , y ) = 1 + cos ( x Δ μ + y Δ ν ) ,
ϕ ( x , y ) = 0 .
f ( x ) = N ! m ! ( N m ) ! p m q ( N m ) .

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