Abstract

It is frequently assumed that the accuracy with which luminance gratings can be detected depends solely on the signal-to-noise ratio at the output of a single linear channel. Proportionality between threshold elevation and power spectral density is implicit in this assumption. I demonstrate that this proportionality does not hold for 1-cycle/degree gratings masked by low-pass noise with a 0.5-cycle/degree cutoff frequency. This implies that different channels can mediate detection, depending on the contrast of masking stimuli.

© 2000 Optical Society of America

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References

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  1. D. G. Pelli, “Effects of visual noise,” Ph.D. dissertation (University of Cambridge, Cambridge, UK, 1981).
  2. G. B. Henning, B. G. Hertz, J. L. Hinton, “Effects of different hypothetical detection mechanisms on the shape of spatial-frequency filters inferred from masking experiments: I. Noise masks,” J. Opt. Soc. Am. 71, 574–581 (1981).
    [CrossRef] [PubMed]
  3. J. A. Solomon, D. G. Pelli, “The visual filter mediating letter identification,” Nature 369, 395–397 (1994).
    [CrossRef] [PubMed]
  4. M. E. Perkins, M. S. Landy, “Nonadditivity of masking by narrow-band noises,” Vision Res. 31, 1053–1065 (1991).
    [CrossRef] [PubMed]
  5. M. A. Losada, K. T. Mullen, “Color and luminance spatial tuning estimated by noise masking in the absence of off-frequency looking,” J. Opt. Soc. Am. A 12, 250–260 (1995).
    [CrossRef]
  6. D. G. Pelli, “Channel properties revealed by noise masking,” Invest. Ophthalmol. Visual Sci. Suppl. 19, 44A (1980).
  7. A. E. Burgess, C. Li, C. K. Abbey, “Visual signal detectability with two noise components: anomalous masking effects,” J. Opt. Soc. Am. A 14, 2420–2442 (1997).
    [CrossRef]
  8. D. G. Pelli, L. Zhang, “Accurate control of contrast on microcomputer displays,” Vision Res. 31, 1337–1350 (1991).
    [CrossRef] [PubMed]
  9. A. B. Watson, J. A. Solomon, “Psychophysica: Mathematica notebooks for psychophysical experiments,” Spatial Vis. 10, 447–466 (1997).
    [CrossRef]
  10. A. B. Watson, D. G. Pelli, “QUEST: a Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
    [CrossRef] [PubMed]
  11. A pixel with luminance l has contrast (l-l0)/l0, where l0 is the expected or mean luminance. Each pixel’s power is its squared contrast. Note: Because target and mask are presented on alternate frames, the maximum power any pixel can have is 0.25.
  12. R. D. Patterson, “Auditory filter shapes derived with noise stimuli,” J. Acoust. Soc. Am. 59, 640–654 (1976).
    [CrossRef] [PubMed]
  13. J. Majaj, D. G. Pelli, P. Kurshan, M. Palomares, New York University, Department of Psychology, 6 Washington Place, New York, N.Y. 10003-6634 (personal communication, December1998).
  14. H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial-frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
    [CrossRef]
  15. C. F. Stromeyer, B. Julesz, “Spatial-frequency masking in vision: critical bands and spread of masking,” J. Opt. Soc. Am. 62, 1221–1232 (1972).
    [CrossRef] [PubMed]
  16. R. L. De Valois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in Macaque visual cortex,” Vision Res. 22, 545–559 (1982).
    [CrossRef] [PubMed]
  17. J. M. Valeton, A. B. Watson, “Contrast detection does not have a local spatial scale,” Invest. Ophthalmol. Visual Sci. 31, 428 (1990).
  18. When the axes in Fig. 7(a) are rotated so that the dashed line (which intersects the regression line where it intersects the mean threshold elevation) becomes the abscissa, the slope of the data points remains significant (p<0.04). For Figs. 7(b) and 7(c) the p values are approximately 0.5 and 0.4, respectively.
  19. J. Yao, “Predicting human performance by model observers,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1995).
  20. S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media/Cambridge U. Press, Cambridge, UK, 1996).
  21. D. G. Pelli, “The quantum efficiency of vision,” in Vision: Coding and Efficiency, C. B. Blakemore, ed. (Cambridge U. Press, Cambridge, UK, 1990).
  22. In another section of their paper, Losada and Mullen report absolute thresholds (where N=0, not shown in Fig. 3). These data were fitted simultaneously with the data in Fig. 3. If these data are excluded, the addition of parameters pi still does not improve the fit to subject MAL’s data, but it does reduce the RMS error of the fit to subject MJS’s data by 1%.

1997 (2)

A. E. Burgess, C. Li, C. K. Abbey, “Visual signal detectability with two noise components: anomalous masking effects,” J. Opt. Soc. Am. A 14, 2420–2442 (1997).
[CrossRef]

A. B. Watson, J. A. Solomon, “Psychophysica: Mathematica notebooks for psychophysical experiments,” Spatial Vis. 10, 447–466 (1997).
[CrossRef]

1995 (1)

1994 (1)

J. A. Solomon, D. G. Pelli, “The visual filter mediating letter identification,” Nature 369, 395–397 (1994).
[CrossRef] [PubMed]

1991 (2)

M. E. Perkins, M. S. Landy, “Nonadditivity of masking by narrow-band noises,” Vision Res. 31, 1053–1065 (1991).
[CrossRef] [PubMed]

D. G. Pelli, L. Zhang, “Accurate control of contrast on microcomputer displays,” Vision Res. 31, 1337–1350 (1991).
[CrossRef] [PubMed]

1990 (1)

J. M. Valeton, A. B. Watson, “Contrast detection does not have a local spatial scale,” Invest. Ophthalmol. Visual Sci. 31, 428 (1990).

1983 (2)

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial-frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef]

A. B. Watson, D. G. Pelli, “QUEST: a Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
[CrossRef] [PubMed]

1982 (1)

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

1981 (1)

1980 (1)

D. G. Pelli, “Channel properties revealed by noise masking,” Invest. Ophthalmol. Visual Sci. Suppl. 19, 44A (1980).

1976 (1)

R. D. Patterson, “Auditory filter shapes derived with noise stimuli,” J. Acoust. Soc. Am. 59, 640–654 (1976).
[CrossRef] [PubMed]

1972 (1)

Abbey, C. K.

Albrecht, D. G.

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

Burgess, A. E.

De Valois, R. L.

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

Henning, G. B.

Hertz, B. G.

Hinton, J. L.

Julesz, B.

Kurshan, P.

J. Majaj, D. G. Pelli, P. Kurshan, M. Palomares, New York University, Department of Psychology, 6 Washington Place, New York, N.Y. 10003-6634 (personal communication, December1998).

Landy, M. S.

M. E. Perkins, M. S. Landy, “Nonadditivity of masking by narrow-band noises,” Vision Res. 31, 1053–1065 (1991).
[CrossRef] [PubMed]

Li, C.

Losada, M. A.

Majaj, J.

J. Majaj, D. G. Pelli, P. Kurshan, M. Palomares, New York University, Department of Psychology, 6 Washington Place, New York, N.Y. 10003-6634 (personal communication, December1998).

McFarlane, D. K.

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial-frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef]

Mullen, K. T.

Palomares, M.

J. Majaj, D. G. Pelli, P. Kurshan, M. Palomares, New York University, Department of Psychology, 6 Washington Place, New York, N.Y. 10003-6634 (personal communication, December1998).

Patterson, R. D.

R. D. Patterson, “Auditory filter shapes derived with noise stimuli,” J. Acoust. Soc. Am. 59, 640–654 (1976).
[CrossRef] [PubMed]

Pelli, D. G.

J. A. Solomon, D. G. Pelli, “The visual filter mediating letter identification,” Nature 369, 395–397 (1994).
[CrossRef] [PubMed]

D. G. Pelli, L. Zhang, “Accurate control of contrast on microcomputer displays,” Vision Res. 31, 1337–1350 (1991).
[CrossRef] [PubMed]

A. B. Watson, D. G. Pelli, “QUEST: a Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
[CrossRef] [PubMed]

D. G. Pelli, “Channel properties revealed by noise masking,” Invest. Ophthalmol. Visual Sci. Suppl. 19, 44A (1980).

D. G. Pelli, “Effects of visual noise,” Ph.D. dissertation (University of Cambridge, Cambridge, UK, 1981).

J. Majaj, D. G. Pelli, P. Kurshan, M. Palomares, New York University, Department of Psychology, 6 Washington Place, New York, N.Y. 10003-6634 (personal communication, December1998).

D. G. Pelli, “The quantum efficiency of vision,” in Vision: Coding and Efficiency, C. B. Blakemore, ed. (Cambridge U. Press, Cambridge, UK, 1990).

Perkins, M. E.

M. E. Perkins, M. S. Landy, “Nonadditivity of masking by narrow-band noises,” Vision Res. 31, 1053–1065 (1991).
[CrossRef] [PubMed]

Phillips, G. C.

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial-frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef]

Solomon, J. A.

A. B. Watson, J. A. Solomon, “Psychophysica: Mathematica notebooks for psychophysical experiments,” Spatial Vis. 10, 447–466 (1997).
[CrossRef]

J. A. Solomon, D. G. Pelli, “The visual filter mediating letter identification,” Nature 369, 395–397 (1994).
[CrossRef] [PubMed]

Stromeyer, C. F.

Thorell, L. G.

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

Valeton, J. M.

J. M. Valeton, A. B. Watson, “Contrast detection does not have a local spatial scale,” Invest. Ophthalmol. Visual Sci. 31, 428 (1990).

Watson, A. B.

A. B. Watson, J. A. Solomon, “Psychophysica: Mathematica notebooks for psychophysical experiments,” Spatial Vis. 10, 447–466 (1997).
[CrossRef]

J. M. Valeton, A. B. Watson, “Contrast detection does not have a local spatial scale,” Invest. Ophthalmol. Visual Sci. 31, 428 (1990).

A. B. Watson, D. G. Pelli, “QUEST: a Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
[CrossRef] [PubMed]

Wilson, H. R.

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial-frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef]

Wolfram, S.

S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media/Cambridge U. Press, Cambridge, UK, 1996).

Yao, J.

J. Yao, “Predicting human performance by model observers,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1995).

Zhang, L.

D. G. Pelli, L. Zhang, “Accurate control of contrast on microcomputer displays,” Vision Res. 31, 1337–1350 (1991).
[CrossRef] [PubMed]

Invest. Ophthalmol. Visual Sci. (1)

J. M. Valeton, A. B. Watson, “Contrast detection does not have a local spatial scale,” Invest. Ophthalmol. Visual Sci. 31, 428 (1990).

Invest. Ophthalmol. Visual Sci. Suppl. (1)

D. G. Pelli, “Channel properties revealed by noise masking,” Invest. Ophthalmol. Visual Sci. Suppl. 19, 44A (1980).

J. Acoust. Soc. Am. (1)

R. D. Patterson, “Auditory filter shapes derived with noise stimuli,” J. Acoust. Soc. Am. 59, 640–654 (1976).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Nature (1)

J. A. Solomon, D. G. Pelli, “The visual filter mediating letter identification,” Nature 369, 395–397 (1994).
[CrossRef] [PubMed]

Percept. Psychophys. (1)

A. B. Watson, D. G. Pelli, “QUEST: a Bayesian adaptive psychometric method,” Percept. Psychophys. 33, 113–120 (1983).
[CrossRef] [PubMed]

Spatial Vis. (1)

A. B. Watson, J. A. Solomon, “Psychophysica: Mathematica notebooks for psychophysical experiments,” Spatial Vis. 10, 447–466 (1997).
[CrossRef]

Vision Res. (4)

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial-frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef]

D. G. Pelli, L. Zhang, “Accurate control of contrast on microcomputer displays,” Vision Res. 31, 1337–1350 (1991).
[CrossRef] [PubMed]

M. E. Perkins, M. S. Landy, “Nonadditivity of masking by narrow-band noises,” Vision Res. 31, 1053–1065 (1991).
[CrossRef] [PubMed]

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

Other (8)

D. G. Pelli, “Effects of visual noise,” Ph.D. dissertation (University of Cambridge, Cambridge, UK, 1981).

A pixel with luminance l has contrast (l-l0)/l0, where l0 is the expected or mean luminance. Each pixel’s power is its squared contrast. Note: Because target and mask are presented on alternate frames, the maximum power any pixel can have is 0.25.

J. Majaj, D. G. Pelli, P. Kurshan, M. Palomares, New York University, Department of Psychology, 6 Washington Place, New York, N.Y. 10003-6634 (personal communication, December1998).

When the axes in Fig. 7(a) are rotated so that the dashed line (which intersects the regression line where it intersects the mean threshold elevation) becomes the abscissa, the slope of the data points remains significant (p<0.04). For Figs. 7(b) and 7(c) the p values are approximately 0.5 and 0.4, respectively.

J. Yao, “Predicting human performance by model observers,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1995).

S. Wolfram, The Mathematica Book, 3rd ed. (Wolfram Media/Cambridge U. Press, Cambridge, UK, 1996).

D. G. Pelli, “The quantum efficiency of vision,” in Vision: Coding and Efficiency, C. B. Blakemore, ed. (Cambridge U. Press, Cambridge, UK, 1990).

In another section of their paper, Losada and Mullen report absolute thresholds (where N=0, not shown in Fig. 3). These data were fitted simultaneously with the data in Fig. 3. If these data are excluded, the addition of parameters pi still does not improve the fit to subject MAL’s data, but it does reduce the RMS error of the fit to subject MJS’s data by 1%.

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Figures (7)

Fig. 1
Fig. 1

Off-frequency looking. This diagram illustrates power gain for two channels. Both channels, particularly the one tuned to lower spatial frequencies (on the left), are sensitive to a sinusoidal grating whose frequency is represented by the arrow. The two channels have different amounts of internal noise, shown in gray. The spectrum of a low-pass stimulus noise is shown in black. When the spectral density of this noise, N, is small, the channel on the left has a greater signal-to-noise ratio. When N is large, the channel on the right has the greater signal-to-noise ratio.

Fig. 2
Fig. 2

Threshold curves determined by two channels. Each thin black curve describes the performance of an observer restricted to using just one of the two channels in Fig. 1. The thick gray curves describe the best possible performance for an observer free to select either of the two channels in Fig. 1. The dashed curves describe the performance of an ideal observer with access to both channels in Fig. 1. Threshold curves in (a) have been transformed into threshold-elevation curves in (b). Detection strategies based on the output of a single channel produce threshold-elevation curves with unit slope. Detection strategies based on the output of multiple channels can produce shallower threshold-elevation curves.

Fig. 3
Fig. 3

Linear fit to the data of Losada and Mullen.5 Thresholds are for luminance gratings in low-pass (filled squares, solid curves) and high-pass (open squares, dashed curves) noise. For each observer, curves satisfy the equation P=aiN+b, where P is threshold power (i.e., contrast squared), N is power spectral density, and low-pass and high-pass noise have different ais. The fit is good. On log–log coordinates these curves have an asymptotic slope of 1, but Losada and Mullen argue that these thresholds rise with less than unit slope. For comparison, I fitted the data with the alternative function: P=aiNpi+b, allowing the low-pass and high-pass noise to have different pis. This alternative fit was no better.22

Fig. 4
Fig. 4

Power thresholds c2 for grating detection in the presence of ideal high-pass, ideal low-pass, and white noise of various power spectral densities N. Each individual plot shows log c2-versus-log N for a particular combination of target frequency and mask spectrum. No-noise (absolute) thresholds are indicated on the vertical axes of the white-noise plots. Curves show the models’ fits to the data (open circles). In most plots the two curves are indistinguishable, indicating similar predictions from the fixed- and best-channels models. The two models’ fits differ when the noise was low pass and the cutoff frequency was lower than the target frequency. In these cases the best-channels model always predicts a shallower log c2-versus-log-N curve.

Fig. 5
Fig. 5

Sensitivities (1/c02), estimated by both the arbitrary-channels model (black dots) and the ad hoc description (gray dots). The dashed curve shows Valeton and Watson’s17 parabolic fit to their similarly obtained data. They used stimuli identical to mine, but they did not measure thresholds below 0.5 c/deg. The solid curve is parabolic above 2.28 c/deg and flat below. Thus it faithfully describes my data while retaining the spirit of Valeton and Watson’s fit.

Fig. 6
Fig. 6

Channel bandwidths, as described by the fixed- (solid line) and best- (dashed line) channels models when fitted to the data of experiment 1. Both models fit best when octave bandwidths decrease at a similar rate with center frequency. The fixed-channels model requires channel bandwidths to be larger than does the best-channels model.

Fig. 7
Fig. 7

Off-frequency looking. Each plot shows threshold elevation for a 1-c/deg target as a function of the spectral density of low-pass noise with a 0.5-c/deg cutoff for three observers. Dashed lines have unit slope. Regression lines (solid) are shown. Each has less than unit slope, indicating a nonlinear relationship between spectral density and threshold power.

Tables (1)

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Table 1 Model Parameters and Values

Equations (22)

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W(x, y)=exp-π x2+y2λ2.
H2(f )=NfLf<fH0otherwise,
R2(N, fL, fH, f0, fi)=G2(f0, fi)Ni(fi)+NfLfHG2(f, fi)d f.
c2(N, fL, fH, f0)=1sR2(N, fL, fH, f0, f0),
c2(N, fL, fH, f0)=1sR2(N, fL, fH, f0, fi),
G2(f, fi)=2-4[log(f )-log(fi)]2/log[2w(fi)]2,
c02(f0)c2(0, fL, fH, f0)=Ni(f0)s,
Ni(fi)=sc02(fi)
w(fi)=m log(fi)+w0.
log[1/c02(f )]
=1.82f<2.281.82-1.85(log f-log 2.28)2f2.28 .
 AN{c, [NA+N0fHGA2(f )d f ]1/2},
BN{βc, [NB+N0fHGB2(f )d f ]1/2},
cov(A, B)=N0fHGB2(f )GA2(f )df.
(d)2=[AB]AAABBABB-1AB,
(d)2=[cβc]var(B)/q-cov(A, B)/q-cov(A, B)/qvar(A)/q cβc,
q=var(A)var(B)-cov2(A, B).
(d)2=c2 var(B)-2β cov(A, B)+β2 var(A)q.
c2var(A)var(B)-cov2(A, B)var(B)-2β cov(A, B)+β2 var(A).
c2(N, fL, fH, f0)-c02(f0)=k(fL, fH, f0)N.
c2(f0, fL, fH, N)
=max[c02(f0), k(f0, fL, fH)Np(f0, fL, fH)].

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