Abstract

The visual representation of spatial patterns begins with a series of linear transformations: the stimulus is blurred by the optics, spatially sampled by the photoreceptor array, spatially pooled by the ganglion-cell receptive fields, and so forth. Models of human spatial-pattern vision commonly summarize the initial transformations by a single linear transformation that maps the stimulus into an array of sensor responses. Some components of the initial linear transformations (e.g., lens blurring, photoreceptor sampling) have been estimated empirically; others have not. A computable model must include some assumptions concerning the unknown components of the initial linear encoding. Even a modest sketch of the initial visual encoding requires the specification of a large number of sensors, making the calculations required for performance predictions quite large. We describe procedures for reducing the computational burden of current models of spatial vision that ensure that the simplifications are consistent with the predictions of the complete model. We also describe a method for using pattern-sensitivity measurements to estimate the initial linear transformation. The method is based on the assumption that detection performance is monotonic with the vector length of the sensor responses. We show how contrast-threshold data can be used to estimate the linear transformation needed to characterize threshold performance.

© 1988 Optical Society of America

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References

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    [CrossRef] [PubMed]
  21. In calculations for which the stimulus is represented before the optical blurring, a higher sampling frequency may be required to represent stimuli adequately. This may occur when sensor characterizations include both the ocular optics and the neural sensors. In our implementation of the Wilson–Gelb calculations, we used a spatial resolution of 120 samples per degree. Our simulations confirmed all their calculations.
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  23. The Wilson–Gelb sensor set7is not consistent, in this sense, with the sensor set described in Ref. 8.
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    [CrossRef] [PubMed]
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  29. These methods of selecting sensors deserve a careful review, but doing so would take us too far afield from the present analysis.
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    [CrossRef]
  31. N. Graham, J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
    [CrossRef] [PubMed]
  32. C. Rashbass, “The visibility of transient changes of luminance,” J. Physiol. London 210, 165–186 (1970).
    [PubMed]
  33. There will be a correspondingly simple structure in the general 2D case as well. A proper discussion of the structure of the resulting matrix, however, should take into account whether the 2D transformation is separable. The added complexity would take us far afield from our main point. The reader may consult various standard references [e.g., A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1982), Vol. 1, p. 137;W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), p. 203] for a review of some of the issues concerning the structure in 2D linear operators and their associated quadratic forms.
  34. Graham et al.30report their data in units of contrast normalized so that the threshold to each of the components alone is set near 1. Since there are four measurements along the axes and there are only two scale parameters, it is not possible for all the data along the axes to be scaled to 1.
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    [CrossRef] [PubMed]
  37. G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1982).
  38. M. A. Berkley, F. Kitterle, D. W. Watkins, “Grating visibility as a function of orientation and retinal eccentricity,” Vision Res. 15, 239–244 (1975).
    [CrossRef] [PubMed]
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    [CrossRef]
  42. M. W. Greenlee, S. Magnussen, “Higher-harmonic adaptation and the detection of squarewave gratings,” Vision Res. 27, 249–255 (1987).
    [CrossRef] [PubMed]
  43. G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyzes patterns in independent bands of spatial frequency,” Vision Res. 15, 887–899 (1975).
    [CrossRef]
  44. J. Nachmias, B. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1630 (1983).
    [CrossRef] [PubMed]
  45. J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat’s striate cortex,” J. Physiol. 283, 79–99 (1978).
  46. R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in the macaque visual cortex,” Vision Res. 22, 545–559 (1982).
    [CrossRef]
  47. G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).
  48. J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart, linpack Users Guide (SIAM Publications, Philadelphia, Pa., 1978).

1987 (2)

A. Ahumada, “Aliasing predictions from a spin-glass model for assigning cones to color sensitivity classes,” Invest. Opthalmol. 28, 137 (1987).

M. W. Greenlee, S. Magnussen, “Higher-harmonic adaptation and the detection of squarewave gratings,” Vision Res. 27, 249–255 (1987).
[CrossRef] [PubMed]

1985 (4)

1984 (5)

1983 (2)

J. Nachmias, B. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1630 (1983).
[CrossRef] [PubMed]

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature 302, 419–422 (1983).
[CrossRef] [PubMed]

1982 (2)

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in the macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

J. I. Yellott, “Spectral analysis of spatial sampling by photoreceptors: topological disorder prevents aliasing,” Vision Res. 22, 1205–1210 (1982).
[CrossRef] [PubMed]

1981 (1)

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

1979 (1)

The sensors used by Wilson and his colleagues have evolved over the years. Thus the set of sensors used in the computations by Wilson and Bergen [H. R. Wilson, J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–32 (1979)] are different from that used by Wilson and Gelb7; both of these sets are different from that used by Wilson and Regan.8
[CrossRef] [PubMed]

1978 (2)

N. Graham, J. G. Robson, J. Nachmias, “Grating summation of fovea and periphery,” Vision Res. 18, 815–825 (1978).
[CrossRef]

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat’s striate cortex,” J. Physiol. 283, 79–99 (1978).

1975 (2)

G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyzes patterns in independent bands of spatial frequency,” Vision Res. 15, 887–899 (1975).
[CrossRef]

M. A. Berkley, F. Kitterle, D. W. Watkins, “Grating visibility as a function of orientation and retinal eccentricity,” Vision Res. 15, 239–244 (1975).
[CrossRef] [PubMed]

1974 (1)

R. F. Quick, “A vector magnitude model of contrast detection,” Kybernetik 16, 65–67 (1974).
[CrossRef]

1973 (1)

J. Nachmias, R. Sansbury, A. Vassilev, A. Weber, “Adaptation to square-wave gratings: in search of the elusive third harmonic,” Vision Res. 13, 1335–1342 (1973).
[CrossRef]

1971 (2)

N. Graham, J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[CrossRef] [PubMed]

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

1970 (1)

C. Rashbass, “The visibility of transient changes of luminance,” J. Physiol. London 210, 165–186 (1970).
[PubMed]

1968 (1)

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

1958 (2)

1956 (1)

1954 (1)

Ahumada, A.

A. Ahumada, “Aliasing predictions from a spin-glass model for assigning cones to color sensitivity classes,” Invest. Opthalmol. 28, 137 (1987).

Ahumada, A. J.

Albrecht, D. G.

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in the macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

Barlow, H. B.

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature 302, 419–422 (1983).
[CrossRef] [PubMed]

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Bergen, J. R.

The sensors used by Wilson and his colleagues have evolved over the years. Thus the set of sensors used in the computations by Wilson and Bergen [H. R. Wilson, J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–32 (1979)] are different from that used by Wilson and Gelb7; both of these sets are different from that used by Wilson and Regan.8
[CrossRef] [PubMed]

Berkley, M. A.

M. A. Berkley, F. Kitterle, D. W. Watkins, “Grating visibility as a function of orientation and retinal eccentricity,” Vision Res. 15, 239–244 (1975).
[CrossRef] [PubMed]

Broadbent, D. E.

G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyzes patterns in independent bands of spatial frequency,” Vision Res. 15, 887–899 (1975).
[CrossRef]

Bunch, J. R.

J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart, linpack Users Guide (SIAM Publications, Philadelphia, Pa., 1978).

Burgess, A. E.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Campbell, F. W.

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

Chandler, J. P.

J. P. Chandler, “stepit,” Quantum Chemistry Program Exchange (Department of Chemistry, Indiana University, Bloomington, Ind., 1965).

Cornsweet, T. N.

J. I. Yellott, B. A. Wandell, T. N. Cornsweet, “The beginnings of visual perception: the retinal image and its initial encoding,” in Handbook of Physiology: The Nervous System, I. Darien-Smith, ed. (Easton, New York, 1984), Vol. 1, pp. 257–316.

deLange, H.

DeValois, R. L.

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in the macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

Dongarra, J.

J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart, linpack Users Guide (SIAM Publications, Philadelphia, Pa., 1978).

Geisler, W. S.

W. S. Geisler, “Ideal observer analysis of visual discrimination,” in Frontiers of Visual Science, The Committee on Vision, National Academy of Sciences, ed. (National Academy Press, Washington, D.C., 1987), pp. 17–31.

Gelb, D. J.

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).

Graham, N.

N. Graham, J. G. Robson, J. Nachmias, “Grating summation of fovea and periphery,” Vision Res. 18, 815–825 (1978).
[CrossRef]

N. Graham, J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[CrossRef] [PubMed]

Greenlee, M. W.

M. W. Greenlee, S. Magnussen, “Higher-harmonic adaptation and the detection of squarewave gratings,” Vision Res. 27, 249–255 (1987).
[CrossRef] [PubMed]

Henning, G. B.

G. B. Henning, B. G. Hertz, D. E. Broadbent, “Some experiments bearing on the hypothesis that the visual system analyzes patterns in independent bands of spatial frequency,” Vision Res. 15, 887–899 (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 patterns in independent bands of spatial frequency,” Vision Res. 15, 887–899 (1975).
[CrossRef]

Jennings, R. J.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Kak, A. C.

There will be a correspondingly simple structure in the general 2D case as well. A proper discussion of the structure of the resulting matrix, however, should take into account whether the 2D transformation is separable. The added complexity would take us far afield from our main point. The reader may consult various standard references [e.g., A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1982), Vol. 1, p. 137;W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), p. 203] for a review of some of the issues concerning the structure in 2D linear operators and their associated quadratic forms.

Kelly, D. H.

Kitterle, F.

M. A. Berkley, F. Kitterle, D. W. Watkins, “Grating visibility as a function of orientation and retinal eccentricity,” Vision Res. 15, 239–244 (1975).
[CrossRef] [PubMed]

Klein, S. A.

Levi, D. M.

Magnussen, S.

M. W. Greenlee, S. Magnussen, “Higher-harmonic adaptation and the detection of squarewave gratings,” Vision Res. 27, 249–255 (1987).
[CrossRef] [PubMed]

Maloney, L. T.

L. T. Maloney, B. A. Wandell, “A model of a single visual channel’s response to weak test lights,” Vision Res. 24, 633–640 (1984).
[CrossRef]

Marr, D.

D. Marr, Vision (Freeman, San Francisco, 1982).

Moler, C. B.

J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart, linpack Users Guide (SIAM Publications, Philadelphia, Pa., 1978).

Movshon, J. A.

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat’s striate cortex,” J. Physiol. 283, 79–99 (1978).

Nachmias, J.

J. Nachmias, B. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1630 (1983).
[CrossRef] [PubMed]

N. Graham, J. G. Robson, J. Nachmias, “Grating summation of fovea and periphery,” Vision Res. 18, 815–825 (1978).
[CrossRef]

J. Nachmias, R. Sansbury, A. Vassilev, A. Weber, “Adaptation to square-wave gratings: in search of the elusive third harmonic,” Vision Res. 13, 1335–1342 (1973).
[CrossRef]

N. Graham, J. Nachmias, “Detection of grating patterns containing two spatial frequencies: a comparison of single-channel and multiple-channel models,” Vision Res. 11, 251–259 (1971).
[CrossRef] [PubMed]

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

Nielsen, K. R. K.

Olzak, L. A.

L. A. Olzak, J. P. Thomas, “Seeing spatial patterns,” in Handbook of Perception and Human Performance, J. P. Thomas, ed. (Wiley, New York, 1986), pp. 7-1, 7-53.

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

Quick, R. F.

R. F. Quick, “A vector magnitude model of contrast detection,” Kybernetik 16, 65–67 (1974).
[CrossRef]

Rashbass, C.

C. Rashbass, “The visibility of transient changes of luminance,” J. Physiol. London 210, 165–186 (1970).
[PubMed]

Regan, D.

Robson, J. G.

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature 302, 419–422 (1983).
[CrossRef] [PubMed]

N. Graham, J. G. Robson, J. Nachmias, “Grating summation of fovea and periphery,” Vision Res. 18, 815–825 (1978).
[CrossRef]

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

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

Rogowitz, B. E.

J. Nachmias, B. E. Rogowitz, “Masking by spatially-modulated gratings,” Vision Res. 23, 1621–1630 (1983).
[CrossRef] [PubMed]

Rosenfeld, A.

There will be a correspondingly simple structure in the general 2D case as well. A proper discussion of the structure of the resulting matrix, however, should take into account whether the 2D transformation is separable. The added complexity would take us far afield from our main point. The reader may consult various standard references [e.g., A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1982), Vol. 1, p. 137;W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), p. 203] for a review of some of the issues concerning the structure in 2D linear operators and their associated quadratic forms.

Sachs, M. B.

Sansbury, R.

J. Nachmias, R. Sansbury, A. Vassilev, A. Weber, “Adaptation to square-wave gratings: in search of the elusive third harmonic,” Vision Res. 13, 1335–1342 (1973).
[CrossRef]

Schade, O. H.

Stewart, G. W.

J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart, linpack Users Guide (SIAM Publications, Philadelphia, Pa., 1978).

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1982).

Thomas, J. P.

L. A. Olzak, J. P. Thomas, “Seeing spatial patterns,” in Handbook of Perception and Human Performance, J. P. Thomas, ed. (Wiley, New York, 1986), pp. 7-1, 7-53.

Thompson, I. D.

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat’s striate cortex,” J. Physiol. 283, 79–99 (1978).

Thorell, L. G.

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in the macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

Tolhurst, D. J.

J. A. Movshon, I. D. Thompson, D. J. Tolhurst, “Receptive field organization of complex cells in the cat’s striate cortex,” J. Physiol. 283, 79–99 (1978).

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).

Vassilev, A.

J. Nachmias, R. Sansbury, A. Vassilev, A. Weber, “Adaptation to square-wave gratings: in search of the elusive third harmonic,” Vision Res. 13, 1335–1342 (1973).
[CrossRef]

Wagner, R. F.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Wandell, B. A.

B. A. Wandell, “Color measurement and discrimination,” J. Opt. Soc. Am. A 2, 62–71 (1985).
[CrossRef] [PubMed]

L. T. Maloney, B. A. Wandell, “A model of a single visual channel’s response to weak test lights,” Vision Res. 24, 633–640 (1984).
[CrossRef]

J. I. Yellott, B. A. Wandell, T. N. Cornsweet, “The beginnings of visual perception: the retinal image and its initial encoding,” in Handbook of Physiology: The Nervous System, I. Darien-Smith, ed. (Easton, New York, 1984), Vol. 1, pp. 257–316.

Watkins, D. W.

M. A. Berkley, F. Kitterle, D. W. Watkins, “Grating visibility as a function of orientation and retinal eccentricity,” Vision Res. 15, 239–244 (1975).
[CrossRef] [PubMed]

Watson, A. B.

K. R. K. Nielsen, A. B. Watson, A. J. Ahumada, “Application of a computable model of human spatial vision to phase discrimination,” J. Opt. Soc. Am. A 2, 1600–1606 (1985).
[CrossRef] [PubMed]

A. J. Ahumada, A. B. Watson, “Equivalent-noise model for contrast detection and discrimination,” J. Opt. Soc. Am. A 2, 1133–1139 (1985).
[CrossRef] [PubMed]

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature 302, 419–422 (1983).
[CrossRef] [PubMed]

A. B. Watson, “The ideal observer concept as a modeling tool,” in Frontiers of Visual Science,The Committee on Vision, National Academy of Sciences, ed. (National Academy Press, Washington, D.C., 1987), pp. 32–37.

A. B. Watson, “Detection and recognition of simple spatial forms,” in Physical and Biological Processing of Images, A. C. Slade, ed. (Springer-Verlag, Berlin, 1983), pp. 100–114.
[CrossRef]

Weber, A.

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The sensors used by Wilson and his colleagues have evolved over the years. Thus the set of sensors used in the computations by Wilson and Bergen [H. R. Wilson, J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–32 (1979)] are different from that used by Wilson and Gelb7; both of these sets are different from that used by Wilson and Regan.8
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[CrossRef]

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[CrossRef] [PubMed]

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Other (17)

J. I. Yellott, B. A. Wandell, T. N. Cornsweet, “The beginnings of visual perception: the retinal image and its initial encoding,” in Handbook of Physiology: The Nervous System, I. Darien-Smith, ed. (Easton, New York, 1984), Vol. 1, pp. 257–316.

These methods of selecting sensors deserve a careful review, but doing so would take us too far afield from the present analysis.

The Wilson–Gelb sensor set7is not consistent, in this sense, with the sensor set described in Ref. 8.

In some simulations, the effects of noise are modeled by adding a random variable to each of the sensor responses. If noise is added to each sensor output, then we can never be sure that any sensor response is always zero but only that its mean response is zero. Such a sensor contributes only noise. When the total number of sensors is reduced by incorporating stimulus characteristics into the S matrix, then the properties of the noise added to the remaining sensors may have to be adjusted to include the effects of noise that have been deleted from the sensors with a zero-mean response.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

G. Wyszecki, W. S. Stiles, Color Science (Wiley, New York, 1982).

There will be a correspondingly simple structure in the general 2D case as well. A proper discussion of the structure of the resulting matrix, however, should take into account whether the 2D transformation is separable. The added complexity would take us far afield from our main point. The reader may consult various standard references [e.g., A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1982), Vol. 1, p. 137;W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), p. 203] for a review of some of the issues concerning the structure in 2D linear operators and their associated quadratic forms.

Graham et al.30report their data in units of contrast normalized so that the threshold to each of the components alone is set near 1. Since there are four measurements along the axes and there are only two scale parameters, it is not possible for all the data along the axes to be scaled to 1.

J. P. Chandler, “stepit,” Quantum Chemistry Program Exchange (Department of Chemistry, Indiana University, Bloomington, Ind., 1965).

In calculations for which the stimulus is represented before the optical blurring, a higher sampling frequency may be required to represent stimuli adequately. This may occur when sensor characterizations include both the ocular optics and the neural sensors. In our implementation of the Wilson–Gelb calculations, we used a spatial resolution of 120 samples per degree. Our simulations confirmed all their calculations.

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A. B. Watson, “The ideal observer concept as a modeling tool,” in Frontiers of Visual Science,The Committee on Vision, National Academy of Sciences, ed. (National Academy Press, Washington, D.C., 1987), pp. 32–37.

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A. B. Watson, “Detection and recognition of simple spatial forms,” in Physical and Biological Processing of Images, A. C. Slade, ed. (Springer-Verlag, Berlin, 1983), pp. 100–114.
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Figures (6)

Fig. 1
Fig. 1

(a) Linear sensor functions of 3 of the 18 Wilson–Gelb sensors, (b) Expanded representation of the matrix product r = Sî describing the initial linear transformation. Each row of S is a sensor spatial sensitivity.

Fig. 2
Fig. 2

Reduction of the computational size of the initial linear transformation the representation of the stimulus with respect to an efficient set of bases. P is the number of sensors; N is the number of degrees of freedom in the original stimulus. The dimension of the vector w depends on the stimulus basis chosen for the experiment. In the example in the text, the vector’s dimension is 64.

Fig. 3
Fig. 3

Two stimuli that are indiscriminable to the Wilson–Gelb sensors. The difference between these two stimuli is within the null space of the Wilson–Gelb sensor set. The zero point marks the central fovea, so that one stimulus falls to the left of the central fovea and the other falls to the right.

Fig. 4
Fig. 4

Single-sensor analysis. N measurements of sensitivity to shifted delta functions permit an estimate of the values of a single-sensor model.

Fig. 5
Fig. 5

Quadratic model analysis, (a) N measurements of sensitivity to shifted delta functions permit an estimate of the diagonal entries of the Q matrix, (b) N(N + 1)/2 measurements of shifted delta functions and sums of shifted delta functions are required to estimate all of the parameters of the symmetric Q matrix. Shifted delta functions yield estimates along the diagonal. Once these are known, sums of shifted delta functions yield estimates of the off-diagonal entries.

Fig 6
Fig 6

The test-mixture thresholds of the two observers in two conditions from Ref. 30 are plotted along with the best-fitting ellipse. The ellipse was derived by estimating the quadratic matrix Q. The data fall near the ellipse, but the deviations are systematic. Data on the axes fall inside the predicted ellipse, whereas the points near the identity fall outside the ellipse. Data points plotted in the first quadrant refer to test mixtures in the peaks-added phase. Data points plotted in the fourth quadrant refer to test mixtures in the peaks-subtracted phase.

Tables (1)

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Table 1 Symbols and Their Definitions

Equations (24)

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î = R w .
α r = S α î = ( S R ) α w .
α î = B α w ,
Π N [ α f ( x ) ] = α Π N [ f ( x ) ]
Π N [ f ( x ) + g ( x ) ] = Π N [ f ( x ) ] + Π N [ g ( x ) ] .
Π N [ f ( x ) g ( x ) ] = Π N [ f ( x ) ] Π N [ g ( x ) ] .
cos [ 2 π x N ( f + Δ ) ] cos [ 2 π x N ( f Δ ) ] = 2 sin ( 2 π x N f ) sin ( 2 π x N Δ ) .
P ( correct ) = F [ ( α r ) 2 ] .
α a r a = ± α b r b ,
α a r a ± α b r b = { 2 α a r a 0 .
l 2 = α î t S t S α î .
P ( correct ) = F [ ( α a l a ) 2 ]
l a = 1 α a .
P ( correct ) = F ( α î t Q α î ) .
Q = S 1 t S 1 = S 2 t S 2 ,
F ( α δ l t Q α δ i ) = F ( α 2 q i i ) = P ( correct ) = 0.75 ,
( δ i + δ j ) t Q ( δ i + δ j ) = q i i + 2 q i j + q j j .
q i j = 1 2 ( 1 α i j 2 q i i q j j ) .
w l t Q w i = 1.0 , i = 1 , , 10 .
Q = [ 0.83 0.088 0.088 0.862 ] .
S 1 t S 1 = S 2 t S 2
S 1 = U 1 D 1 V 1 t
S 2 = U 2 D 2 V 2 t .
S 1 t S 1 = V 1 ( D 1 ) 2 V 1 t ,

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