Abstract

Over a range of high temporal and low spatial frequencies, counterphase flickering gratings evoke the so-called frequency-doubling illusion, in which the apparent brightness of the grating varies at twice its real spatial frequency. The form of the nonlinearity that causes this second-harmonic distortion of the visual response was determined by a cancellation technique. The harmonic distortion can be measured as a function of amplitude (or contrast) by adding to the flickering grating a real, nonflickering, double-frequency component with the amplitude and phase required to cancel the illusory second harmonic. Harmonic distortion curves obtained in this way imply that the nonlinearity is of the form |<i>s</i>| <sup><i>P</i></sup>, where <i>s</i> is the stimulus pattern (without its d<i>c</i> component) and <i>p</i> is close to 0.6. If <i>p</i> = 1, or if the absolute value is not taken, this expression predicts distortion curves that differ significantly from the experimental results. Hence neither rectification nor compression alone is sufficient to account for the second-harmonic distortion; both are required.

© 1981 Optical Society of America

PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. H. Kelly, "Frequency doubling in visual responses," J. Opt. Soc. Am. 56, 1628–1633 (1966).
  2. C. A. Burbeck and D. H. Kelly, "Retinal mechanisms inferred from measurements of threshold sensitivity versus suprathreshold orthogonal mask contrast," in Proceedings of Topical Meeting on Recent Advances in Vision (Optical Society of America, Washington, D.C., 1980), paper ThB4.
  3. B. G. Cleland, W. R. Levick, and K. J. Sanderson, "Properties of sustained and transient ganglion cells in the cat retine," J. Physiol. (London) 228, 649–680 (1973); see also C. Enroth-Cugell and J. G. Robson, "The contrast sensitivity of retinal ganglion cells," J. Physiol. (London) 187, 517–552 (1966).
  4. W. Richards and T. B. Felton, "Spatial frequency doubling: retinal or central?" Vision Res. 13, 2129–2137 (1973).
  5. C. W. Tyler, "Observations on spatial frequency doubling," Perception 3, 81–86 (1974).
  6. V. Virsu and P. Laurinen, "Long-lasting afterimages caused by neural adaptation," Vision Res. 17, 853–860 (1977).
  7. K. I. Naka and W. A. H. Rushton, "S-potentials from luminosity units in the retina of fish (cyprinidae)," J. Physiol. (London) 185, 587–599 (1966).
  8. D. H. Kelly and R. E. Savoie, "Theory of flicker and transient responses. III. An essential nonlinearity," J. Opt. Soc. Am. 68, 1481–1490 (1978).
  9. C. Rashbass, "The visibility of transient changes of luminance," J. Physiol. (London) 210, 165–186 (1970).
  10. See, for example, I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965). This integral is evaluated on p. 372 (Formula 3.631-9) in terms of the beta function B, where B(x, y) = [Γ(x)Γ(y)]/Γ(x + y).
  11. Regardless of the form of the nonlinearity, its odd part can create only odd harmonics and therefore can have no effect on the second harmonic. Conversely, the even part of the nonlinearity can create only even harmonics and therefore can have no effect on the fundamental. Thus, even if these two frequency components are responses of the same nonlinear transducer, they represent separate, independent, additive aspects of its behavior. This is true for any transducer function that can he expanded in a Taylor series, and it does not depend on the phase of the stimulus. It follows from the fact that odd powers of a sinusoidal input (sin2n-1 θ and cos2n-l θ) can always be expressed as a (finite) sum of odd harmonic terms, sin(2n - 2k - 1)θ and cos(2n - 2k - 1)θ, while even powers (sin2n θ and cos2n θ) can be expressed as a sum of even harmonic terms, in the latter case always of the form cos 2(n - k)θ, where K ranges from 0 to n - 1. See Ref. 10, pp. 25, 26.
  12. D. H. Kelly, "Visual nonlinearity measurement," J. Opt. Soc. Am. 71, 368A (1981).

1981

D. H. Kelly, "Visual nonlinearity measurement," J. Opt. Soc. Am. 71, 368A (1981).

1978

1977

V. Virsu and P. Laurinen, "Long-lasting afterimages caused by neural adaptation," Vision Res. 17, 853–860 (1977).

1974

C. W. Tyler, "Observations on spatial frequency doubling," Perception 3, 81–86 (1974).

1973

B. G. Cleland, W. R. Levick, and K. J. Sanderson, "Properties of sustained and transient ganglion cells in the cat retine," J. Physiol. (London) 228, 649–680 (1973); see also C. Enroth-Cugell and J. G. Robson, "The contrast sensitivity of retinal ganglion cells," J. Physiol. (London) 187, 517–552 (1966).

W. Richards and T. B. Felton, "Spatial frequency doubling: retinal or central?" Vision Res. 13, 2129–2137 (1973).

1970

C. Rashbass, "The visibility of transient changes of luminance," J. Physiol. (London) 210, 165–186 (1970).

1966

D. H. Kelly, "Frequency doubling in visual responses," J. Opt. Soc. Am. 56, 1628–1633 (1966).

K. I. Naka and W. A. H. Rushton, "S-potentials from luminosity units in the retina of fish (cyprinidae)," J. Physiol. (London) 185, 587–599 (1966).

Burbeck, C. A.

C. A. Burbeck and D. H. Kelly, "Retinal mechanisms inferred from measurements of threshold sensitivity versus suprathreshold orthogonal mask contrast," in Proceedings of Topical Meeting on Recent Advances in Vision (Optical Society of America, Washington, D.C., 1980), paper ThB4.

Cleland, B. G.

B. G. Cleland, W. R. Levick, and K. J. Sanderson, "Properties of sustained and transient ganglion cells in the cat retine," J. Physiol. (London) 228, 649–680 (1973); see also C. Enroth-Cugell and J. G. Robson, "The contrast sensitivity of retinal ganglion cells," J. Physiol. (London) 187, 517–552 (1966).

Felton, T. B.

W. Richards and T. B. Felton, "Spatial frequency doubling: retinal or central?" Vision Res. 13, 2129–2137 (1973).

Gradshteyn, I. S.

See, for example, I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965). This integral is evaluated on p. 372 (Formula 3.631-9) in terms of the beta function B, where B(x, y) = [Γ(x)Γ(y)]/Γ(x + y).

Kelly, D. H.

D. H. Kelly, "Visual nonlinearity measurement," J. Opt. Soc. Am. 71, 368A (1981).

D. H. Kelly and R. E. Savoie, "Theory of flicker and transient responses. III. An essential nonlinearity," J. Opt. Soc. Am. 68, 1481–1490 (1978).

D. H. Kelly, "Frequency doubling in visual responses," J. Opt. Soc. Am. 56, 1628–1633 (1966).

C. A. Burbeck and D. H. Kelly, "Retinal mechanisms inferred from measurements of threshold sensitivity versus suprathreshold orthogonal mask contrast," in Proceedings of Topical Meeting on Recent Advances in Vision (Optical Society of America, Washington, D.C., 1980), paper ThB4.

Laurinen, P.

V. Virsu and P. Laurinen, "Long-lasting afterimages caused by neural adaptation," Vision Res. 17, 853–860 (1977).

Levick, W. R.

B. G. Cleland, W. R. Levick, and K. J. Sanderson, "Properties of sustained and transient ganglion cells in the cat retine," J. Physiol. (London) 228, 649–680 (1973); see also C. Enroth-Cugell and J. G. Robson, "The contrast sensitivity of retinal ganglion cells," J. Physiol. (London) 187, 517–552 (1966).

Naka, K. I.

K. I. Naka and W. A. H. Rushton, "S-potentials from luminosity units in the retina of fish (cyprinidae)," J. Physiol. (London) 185, 587–599 (1966).

Rashbass, C.

C. Rashbass, "The visibility of transient changes of luminance," J. Physiol. (London) 210, 165–186 (1970).

Richards, W.

W. Richards and T. B. Felton, "Spatial frequency doubling: retinal or central?" Vision Res. 13, 2129–2137 (1973).

Rushton, W. A. H.

K. I. Naka and W. A. H. Rushton, "S-potentials from luminosity units in the retina of fish (cyprinidae)," J. Physiol. (London) 185, 587–599 (1966).

Ryzhik, I. M.

See, for example, I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965). This integral is evaluated on p. 372 (Formula 3.631-9) in terms of the beta function B, where B(x, y) = [Γ(x)Γ(y)]/Γ(x + y).

Sanderson, K. J.

B. G. Cleland, W. R. Levick, and K. J. Sanderson, "Properties of sustained and transient ganglion cells in the cat retine," J. Physiol. (London) 228, 649–680 (1973); see also C. Enroth-Cugell and J. G. Robson, "The contrast sensitivity of retinal ganglion cells," J. Physiol. (London) 187, 517–552 (1966).

Savoie, R. E.

Tyler, C. W.

C. W. Tyler, "Observations on spatial frequency doubling," Perception 3, 81–86 (1974).

Virsu, V.

V. Virsu and P. Laurinen, "Long-lasting afterimages caused by neural adaptation," Vision Res. 17, 853–860 (1977).

J. Opt. Soc. Am.

J. Physiol.

K. I. Naka and W. A. H. Rushton, "S-potentials from luminosity units in the retina of fish (cyprinidae)," J. Physiol. (London) 185, 587–599 (1966).

C. Rashbass, "The visibility of transient changes of luminance," J. Physiol. (London) 210, 165–186 (1970).

J. Physiol. London

B. G. Cleland, W. R. Levick, and K. J. Sanderson, "Properties of sustained and transient ganglion cells in the cat retine," J. Physiol. (London) 228, 649–680 (1973); see also C. Enroth-Cugell and J. G. Robson, "The contrast sensitivity of retinal ganglion cells," J. Physiol. (London) 187, 517–552 (1966).

Perception

C. W. Tyler, "Observations on spatial frequency doubling," Perception 3, 81–86 (1974).

Vision Res.

V. Virsu and P. Laurinen, "Long-lasting afterimages caused by neural adaptation," Vision Res. 17, 853–860 (1977).

W. Richards and T. B. Felton, "Spatial frequency doubling: retinal or central?" Vision Res. 13, 2129–2137 (1973).

Other

See, for example, I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965). This integral is evaluated on p. 372 (Formula 3.631-9) in terms of the beta function B, where B(x, y) = [Γ(x)Γ(y)]/Γ(x + y).

Regardless of the form of the nonlinearity, its odd part can create only odd harmonics and therefore can have no effect on the second harmonic. Conversely, the even part of the nonlinearity can create only even harmonics and therefore can have no effect on the fundamental. Thus, even if these two frequency components are responses of the same nonlinear transducer, they represent separate, independent, additive aspects of its behavior. This is true for any transducer function that can he expanded in a Taylor series, and it does not depend on the phase of the stimulus. It follows from the fact that odd powers of a sinusoidal input (sin2n-1 θ and cos2n-l θ) can always be expressed as a (finite) sum of odd harmonic terms, sin(2n - 2k - 1)θ and cos(2n - 2k - 1)θ, while even powers (sin2n θ and cos2n θ) can be expressed as a sum of even harmonic terms, in the latter case always of the form cos 2(n - k)θ, where K ranges from 0 to n - 1. See Ref. 10, pp. 25, 26.

C. A. Burbeck and D. H. Kelly, "Retinal mechanisms inferred from measurements of threshold sensitivity versus suprathreshold orthogonal mask contrast," in Proceedings of Topical Meeting on Recent Advances in Vision (Optical Society of America, Washington, D.C., 1980), paper ThB4.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.