Abstract

In the study of the spatial characteristics of the visual channels, the power spectrum model of visual masking is one of the most widely used. When the task is to detect a signal masked by visual noise, this classical model assumes that the signal and the noise are previously processed by a bank of linear channels and that the power of the signal at threshold is proportional to the power of the noise passing through the visual channel that mediates detection. The model also assumes that this visual channel will have the highest ratio of signal power to noise power at its output. According to this, there are masking conditions where the highest signal-to-noise ratio (SNR) occurs in a channel centered in a spatial frequency different from the spatial frequency of the signal (off-frequency looking). Under these conditions the channel mediating detection could vary with the type of noise used in the masking experiment and this could affect the estimation of the shape and the bandwidth of the visual channels. It is generally believed that notched noise, white noise and double bandpass noise prevent off-frequency looking, and high-pass, low-pass and bandpass noises can promote it independently of the channel’s shape. In this study, by means of a procedure that finds the channel that maximizes the SNR at its output, we performed numerical simulations using the power spectrum model to study the characteristics of masking caused by six types of one-dimensional noise (white, high-pass, low-pass, bandpass, notched, and double bandpass) for two types of channel’s shape (symmetric and asymmetric). Our simulations confirm that (1) high-pass, low-pass, and bandpass noises do not prevent the off-frequency looking, (2) white noise satisfactorily prevents the off-frequency looking independently of the shape and bandwidth of the visual channel, and interestingly we proved for the first time that (3) notched and double bandpass noises prevent off-frequency looking only when the noise cutoffs around the spatial frequency of the signal match the shape of the visual channel (symmetric or asymmetric) involved in the detection. In order to test the explanatory power of the model with empirical data, we performed six visual masking experiments. We show that this model, with only two free parameters, fits the empirical masking data with high precision. Finally, we provide equations of the power spectrum model for six masking noises used in the simulations and in the experiments.

© 2013 Optical Society of America

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    [CrossRef]
  30. G. B. Henning, B. G. Hertz, and J. L. Hinton, “Effects of different hypothetical detection mechanisms on the shape of spatial-frequency filters inferred from masking,” J. Opt. Soc. Am. 71, 574–581 (1981).
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  31. G. B. Henning and F. A. Wichmann, “Some observations on the pedestal effect,” J. Vis. 7(1):3, 1–15 (2007).
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  32. B. Leshowitz and F. L. Wightman, “On-frequency masking with continuous sinusoids,” J. Acoust. Soc. Am. 49, 1180–1190 (1971).
  33. R. D. Patterson, “Auditory filter shapes derived with noise stimuli,” J. Acoust. Soc. Am. 59, 640–654 (1976).
    [CrossRef]
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    [CrossRef]
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  41. M. A. García-Pérez and V. Sierra-Vázquez, “Deriving channel gains from large-area sine-wave contrast sensitivity data,” Spatial Vis. 9, 235–260 (1995).
  42. A. Schofield and M. A. Georgeson, “Sensitivity to contrast modulation: the spatial frequency dependence of second-order vision,” Vis. Res. 43, 243–259 (2003).
    [CrossRef]
  43. H. R. Wilson, D. K. McFarlane, and G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vis. Res. 23, 873–882 (1983).
    [CrossRef]
  44. J. Rovamo, R. Fransilla, and R. Näsänen, “Contrast sensitivity as a function of spatial frequency, viewing distance and eccentricity with and without spatial noise,” Vis. Res. 32, 631–637 (1992).
    [CrossRef]
  45. A. Schofield and M. A. Georgeson, “Sensitivity to modulations of luminance and contrast in visual white noise: separate mechanisms with similar behaviour,” Vis. Res. 39, 2697–2716 (1999).
    [CrossRef]
  46. P. E. King-Smith, S. S. Grigsby, A. J. Vingrys, S. C. Benes, and A. Supowit, “Efficient and unbiased modifications of the QUEST threshold method: theory, simulations, experimental evaluation and practical implementation,” Vis. Res. 34, 885–912 (1994).
    [CrossRef]
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2013 (1)

Z. M. Westrick, C. A. Henry, and M. S. Landy, “Inconsistent channel bandwidth estimates suggest winner-take-all nonlinearity in second-order vision,” Vis. Res. 81, 58–68 (2013).
[CrossRef]

2010 (1)

I. Oruç and J. J. S. Barton, “Critical frequencies in the perception of letters, faces, and novel shapes: evidence for limited scale invariance for faces,” J. Vis. 10(12):20, 1–12 (2010).

2009 (1)

I. Oruc and M. S. Landy, “Scale dependence and channel switching in letter identification,” J. Vis. 9(9):4, 1–19 (2009).

2007 (1)

G. B. Henning and F. A. Wichmann, “Some observations on the pedestal effect,” J. Vis. 7(1):3, 1–15 (2007).
[CrossRef]

2006 (3)

I. Serrano-Pedraza and V. Sierra-Vazquez, “The effect of white-noise mask level on sinewave contrast detection thresholds and the critical-band-masking model,” Spanish J. Psychol. 9, 249–262 (2006).

I. Serrano-Pedraza and V. Sierra-Vázquez, “The effect of white-noise mask level on sine-wave contrast threshold and the critical band masking model,” Spanish J. Psychol. 9, 249–262 (2006).

I. Oruc, M. S. Landy, and D. G. Pelli, “Noise masking reveals channels for second-order letters,” Vis. Res. 46, 1493–1506 (2006).
[CrossRef]

2004 (3)

C. P. Talgar, D. G. Pelli, and M. Carrasco, “Covert attention enhances letter identification without affecting channel tuning,” J. Vis. 4(1):3, 22–31 (2004).

D. G. Pelli, D. M. Levi, and S. T. L. Chung, “Using visual noise to characterize amblyopic letter identification,” J. Vis. 4(10):6, 904–920 (2004).

C. V. Hutchinson and T. Ledgeway, “Spatial frequency selective masking of first-order and second-order motion in the absence of off-frequency looking,” Vis. Res. 44, 1499–1510 (2004).
[CrossRef]

2003 (1)

A. Schofield and M. A. Georgeson, “Sensitivity to contrast modulation: the spatial frequency dependence of second-order vision,” Vis. Res. 43, 243–259 (2003).
[CrossRef]

2002 (1)

N. J. Majaj, D. G. Pelli, P. Kurshan, and M. Palomares, “The role of spatial frequency channels in letter identification,” Vis. Res. 42, 1165–1184 (2002).
[CrossRef]

2000 (1)

1999 (2)

A. Schofield and M. A. Georgeson, “Sensitivity to modulations of luminance and contrast in visual white noise: separate mechanisms with similar behaviour,” Vis. Res. 39, 2697–2716 (1999).
[CrossRef]

K. T. Mullen and M. A. Losada, “The spatial tuning of color and luminance peripheral vision measured with notch filtered noise masking,” Vis. Res. 39, 721–731 (1999).
[CrossRef]

1998 (1)

K. T. Blackwell, “The effect of white and filtered noise on contrast detection thresholds,” Vis. Res. 38, 267–280 (1998).
[CrossRef]

1995 (2)

M. A. García-Pérez and V. Sierra-Vázquez, “Deriving channel gains from large-area sine-wave contrast sensitivity data,” Spatial Vis. 9, 235–260 (1995).

M. A. Losada and 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]

1994 (2)

P. E. King-Smith, S. S. Grigsby, A. J. Vingrys, S. C. Benes, and A. Supowit, “Efficient and unbiased modifications of the QUEST threshold method: theory, simulations, experimental evaluation and practical implementation,” Vis. Res. 34, 885–912 (1994).
[CrossRef]

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

1992 (1)

J. Rovamo, R. Fransilla, and R. Näsänen, “Contrast sensitivity as a function of spatial frequency, viewing distance and eccentricity with and without spatial noise,” Vis. Res. 32, 631–637 (1992).
[CrossRef]

1991 (1)

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

1988 (1)

M. C. Morrone and D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. Lond. B 235, 221–245 (1988).

1983 (1)

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

1981 (1)

1980 (1)

R. D. Patterson and I. Nimmo-Smith, “Off-frequency listening and auditory-filter asymmetry,” J. Acoust. Soc. Am. 67, 229–245 (1980).
[CrossRef]

1977 (1)

R. D. Patterson and G. B. Henning, “Stimulus variability and auditory filter shape,” J. Acoust. Soc. Am. 62, 649–664 (1977).
[CrossRef]

1976 (1)

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

1975 (2)

D. H. Kelly, “Spatial frequency selectivity in the retina,” Vis. Res. 15, 665–672 (1975).
[CrossRef]

C. F. Stromeyer and S. Klein, “Evidence against narrow-band spatial frequency channels in human vision: the detectability of frequency modulated gratings,” Vis. Res. 15, 899–910 (1975).
[CrossRef]

1974 (1)

R. D. Patterson, “Auditory filter shape,” J. Acoust. Soc. Am. 55, 802–809 (1974).
[CrossRef]

1972 (1)

1971 (3)

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

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

B. Leshowitz and F. L. Wightman, “On-frequency masking with continuous sinusoids,” J. Acoust. Soc. Am. 49, 1180–1190 (1971).

1969 (1)

C. Blakemore and F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. 203, 237–260 (1969).

1968 (1)

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

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

1940 (1)

H. Fletcher, “Auditory patterns,” Rev. Mod. Phys. 12, 47–65 (1940).
[CrossRef]

Atkinson, J.

O. Braddick, F. W. Campbell, and J. Atkinson, “Channels in vision: basic aspects,” in Handbook of Sensory Physiology, R. Held, H. W. Leibowitz, and H. L. Teuber, eds. (Springer Verlag, 1978), pp. 3–38.

Barton, J. J. S.

I. Oruç and J. J. S. Barton, “Critical frequencies in the perception of letters, faces, and novel shapes: evidence for limited scale invariance for faces,” J. Vis. 10(12):20, 1–12 (2010).

Benes, S. C.

P. E. King-Smith, S. S. Grigsby, A. J. Vingrys, S. C. Benes, and A. Supowit, “Efficient and unbiased modifications of the QUEST threshold method: theory, simulations, experimental evaluation and practical implementation,” Vis. Res. 34, 885–912 (1994).
[CrossRef]

Blackwell, K. T.

K. T. Blackwell, “The effect of white and filtered noise on contrast detection thresholds,” Vis. Res. 38, 267–280 (1998).
[CrossRef]

Blakemore, C.

C. Blakemore and F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. 203, 237–260 (1969).

Braddick, O.

O. Braddick, F. W. Campbell, and J. Atkinson, “Channels in vision: basic aspects,” in Handbook of Sensory Physiology, R. Held, H. W. Leibowitz, and H. L. Teuber, eds. (Springer Verlag, 1978), pp. 3–38.

Burr, D. C.

M. C. Morrone and D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. Lond. B 235, 221–245 (1988).

Campbell, F. W.

C. Blakemore and F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. 203, 237–260 (1969).

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

O. Braddick, F. W. Campbell, and J. Atkinson, “Channels in vision: basic aspects,” in Handbook of Sensory Physiology, R. Held, H. W. Leibowitz, and H. L. Teuber, eds. (Springer Verlag, 1978), pp. 3–38.

Carrasco, M.

C. P. Talgar, D. G. Pelli, and M. Carrasco, “Covert attention enhances letter identification without affecting channel tuning,” J. Vis. 4(1):3, 22–31 (2004).

Chung, S. T. L.

D. G. Pelli, D. M. Levi, and S. T. L. Chung, “Using visual noise to characterize amblyopic letter identification,” J. Vis. 4(10):6, 904–920 (2004).

Davenport, W. B.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (Wiley-IEEE, 1958).

Fletcher, H.

H. Fletcher, “Auditory patterns,” Rev. Mod. Phys. 12, 47–65 (1940).
[CrossRef]

Fransilla, R.

J. Rovamo, R. Fransilla, and R. Näsänen, “Contrast sensitivity as a function of spatial frequency, viewing distance and eccentricity with and without spatial noise,” Vis. Res. 32, 631–637 (1992).
[CrossRef]

García-Pérez, M. A.

M. A. García-Pérez and V. Sierra-Vázquez, “Deriving channel gains from large-area sine-wave contrast sensitivity data,” Spatial Vis. 9, 235–260 (1995).

Georgeson, M. A.

A. Schofield and M. A. Georgeson, “Sensitivity to contrast modulation: the spatial frequency dependence of second-order vision,” Vis. Res. 43, 243–259 (2003).
[CrossRef]

A. Schofield and M. A. Georgeson, “Sensitivity to modulations of luminance and contrast in visual white noise: separate mechanisms with similar behaviour,” Vis. Res. 39, 2697–2716 (1999).
[CrossRef]

Graham, N.

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

Green, D. M.

D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, 1966). Reprinted with corrections, 1974.

Grigsby, S. S.

P. E. King-Smith, S. S. Grigsby, A. J. Vingrys, S. C. Benes, and A. Supowit, “Efficient and unbiased modifications of the QUEST threshold method: theory, simulations, experimental evaluation and practical implementation,” Vis. Res. 34, 885–912 (1994).
[CrossRef]

Henning, G. B.

G. B. Henning and F. A. Wichmann, “Some observations on the pedestal effect,” J. Vis. 7(1):3, 1–15 (2007).
[CrossRef]

G. B. Henning, B. G. Hertz, and J. L. Hinton, “Effects of different hypothetical detection mechanisms on the shape of spatial-frequency filters inferred from masking,” J. Opt. Soc. Am. 71, 574–581 (1981).
[CrossRef]

R. D. Patterson and G. B. Henning, “Stimulus variability and auditory filter shape,” J. Acoust. Soc. Am. 62, 649–664 (1977).
[CrossRef]

Henry, C. A.

Z. M. Westrick, C. A. Henry, and M. S. Landy, “Inconsistent channel bandwidth estimates suggest winner-take-all nonlinearity in second-order vision,” Vis. Res. 81, 58–68 (2013).
[CrossRef]

Hertz, B. G.

Hinton, J. L.

Hutchinson, C. V.

C. V. Hutchinson and T. Ledgeway, “Spatial frequency selective masking of first-order and second-order motion in the absence of off-frequency looking,” Vis. Res. 44, 1499–1510 (2004).
[CrossRef]

Julesz, B.

Kelly, D. H.

D. H. Kelly, “Spatial frequency selectivity in the retina,” Vis. Res. 15, 665–672 (1975).
[CrossRef]

King-Smith, P. E.

P. E. King-Smith, S. S. Grigsby, A. J. Vingrys, S. C. Benes, and A. Supowit, “Efficient and unbiased modifications of the QUEST threshold method: theory, simulations, experimental evaluation and practical implementation,” Vis. Res. 34, 885–912 (1994).
[CrossRef]

Klein, S.

C. F. Stromeyer and S. Klein, “Evidence against narrow-band spatial frequency channels in human vision: the detectability of frequency modulated gratings,” Vis. Res. 15, 899–910 (1975).
[CrossRef]

Kurshan, P.

N. J. Majaj, D. G. Pelli, P. Kurshan, and M. Palomares, “The role of spatial frequency channels in letter identification,” Vis. Res. 42, 1165–1184 (2002).
[CrossRef]

Landy, M. S.

Z. M. Westrick, C. A. Henry, and M. S. Landy, “Inconsistent channel bandwidth estimates suggest winner-take-all nonlinearity in second-order vision,” Vis. Res. 81, 58–68 (2013).
[CrossRef]

I. Oruc and M. S. Landy, “Scale dependence and channel switching in letter identification,” J. Vis. 9(9):4, 1–19 (2009).

I. Oruc, M. S. Landy, and D. G. Pelli, “Noise masking reveals channels for second-order letters,” Vis. Res. 46, 1493–1506 (2006).
[CrossRef]

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

Ledgeway, T.

C. V. Hutchinson and T. Ledgeway, “Spatial frequency selective masking of first-order and second-order motion in the absence of off-frequency looking,” Vis. Res. 44, 1499–1510 (2004).
[CrossRef]

Leshowitz, B.

B. Leshowitz and F. L. Wightman, “On-frequency masking with continuous sinusoids,” J. Acoust. Soc. Am. 49, 1180–1190 (1971).

Levi, D. M.

D. G. Pelli, D. M. Levi, and S. T. L. Chung, “Using visual noise to characterize amblyopic letter identification,” J. Vis. 4(10):6, 904–920 (2004).

Losada, M. A.

K. T. Mullen and M. A. Losada, “The spatial tuning of color and luminance peripheral vision measured with notch filtered noise masking,” Vis. Res. 39, 721–731 (1999).
[CrossRef]

M. A. Losada and 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]

Majaj, N. J.

N. J. Majaj, D. G. Pelli, P. Kurshan, and M. Palomares, “The role of spatial frequency channels in letter identification,” Vis. Res. 42, 1165–1184 (2002).
[CrossRef]

McFarlane, D. K.

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

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Moore, B. C. J.

R. D. Patterson and B. C. J. Moore, “Auditory filters and excitation patterns as representations of frequency resolution,” in Frequency Selectivity in Hearing, B. C. J. Moore, ed. (Academic, 1986), pp. 123–177.

B. C. J. Moore, An Introduction to the Psychology of Hearing (Academic, 1997).

B. C. J. Moore, An Introduction to the Psychology of Hearing (Academic, 1997).

Morrone, M. C.

M. C. Morrone and D. C. Burr, “Feature detection in human vision: a phase-dependent energy model,” Proc. R. Soc. Lond. B 235, 221–245 (1988).

Mullen, K. T.

K. T. Mullen and M. A. Losada, “The spatial tuning of color and luminance peripheral vision measured with notch filtered noise masking,” Vis. Res. 39, 721–731 (1999).
[CrossRef]

M. A. Losada and 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]

Nachmias, J.

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

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

Näsänen, R.

J. Rovamo, R. Fransilla, and R. Näsänen, “Contrast sensitivity as a function of spatial frequency, viewing distance and eccentricity with and without spatial noise,” Vis. Res. 32, 631–637 (1992).
[CrossRef]

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

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R. D. Patterson and I. Nimmo-Smith, “Off-frequency listening and auditory-filter asymmetry,” J. Acoust. Soc. Am. 67, 229–245 (1980).
[CrossRef]

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I. Oruc and M. S. Landy, “Scale dependence and channel switching in letter identification,” J. Vis. 9(9):4, 1–19 (2009).

I. Oruc, M. S. Landy, and D. G. Pelli, “Noise masking reveals channels for second-order letters,” Vis. Res. 46, 1493–1506 (2006).
[CrossRef]

Oruç, I.

I. Oruç and J. J. S. Barton, “Critical frequencies in the perception of letters, faces, and novel shapes: evidence for limited scale invariance for faces,” J. Vis. 10(12):20, 1–12 (2010).

Palomares, M.

N. J. Majaj, D. G. Pelli, P. Kurshan, and M. Palomares, “The role of spatial frequency channels in letter identification,” Vis. Res. 42, 1165–1184 (2002).
[CrossRef]

Patterson, R. D.

R. D. Patterson and I. Nimmo-Smith, “Off-frequency listening and auditory-filter asymmetry,” J. Acoust. Soc. Am. 67, 229–245 (1980).
[CrossRef]

R. D. Patterson and G. B. Henning, “Stimulus variability and auditory filter shape,” J. Acoust. Soc. Am. 62, 649–664 (1977).
[CrossRef]

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

R. D. Patterson, “Auditory filter shape,” J. Acoust. Soc. Am. 55, 802–809 (1974).
[CrossRef]

R. D. Patterson and B. C. J. Moore, “Auditory filters and excitation patterns as representations of frequency resolution,” in Frequency Selectivity in Hearing, B. C. J. Moore, ed. (Academic, 1986), pp. 123–177.

Pelli, D. G.

I. Oruc, M. S. Landy, and D. G. Pelli, “Noise masking reveals channels for second-order letters,” Vis. Res. 46, 1493–1506 (2006).
[CrossRef]

C. P. Talgar, D. G. Pelli, and M. Carrasco, “Covert attention enhances letter identification without affecting channel tuning,” J. Vis. 4(1):3, 22–31 (2004).

D. G. Pelli, D. M. Levi, and S. T. L. Chung, “Using visual noise to characterize amblyopic letter identification,” J. Vis. 4(10):6, 904–920 (2004).

N. J. Majaj, D. G. Pelli, P. Kurshan, and M. Palomares, “The role of spatial frequency channels in letter identification,” Vis. Res. 42, 1165–1184 (2002).
[CrossRef]

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[CrossRef]

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

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

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W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (Wiley-IEEE, 1958).

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J. Rovamo, R. Fransilla, and R. Näsänen, “Contrast sensitivity as a function of spatial frequency, viewing distance and eccentricity with and without spatial noise,” Vis. Res. 32, 631–637 (1992).
[CrossRef]

Sachs, M. B.

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A. Schofield and M. A. Georgeson, “Sensitivity to contrast modulation: the spatial frequency dependence of second-order vision,” Vis. Res. 43, 243–259 (2003).
[CrossRef]

A. Schofield and M. A. Georgeson, “Sensitivity to modulations of luminance and contrast in visual white noise: separate mechanisms with similar behaviour,” Vis. Res. 39, 2697–2716 (1999).
[CrossRef]

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I. Serrano-Pedraza and V. Sierra-Vazquez, “The effect of white-noise mask level on sinewave contrast detection thresholds and the critical-band-masking model,” Spanish J. Psychol. 9, 249–262 (2006).

I. Serrano-Pedraza and V. Sierra-Vázquez, “The effect of white-noise mask level on sine-wave contrast threshold and the critical band masking model,” Spanish J. Psychol. 9, 249–262 (2006).

I. Serrano-Pedraza, “Procesos visuales de demodulación espacial,” Doctoral dissertation (Universidad Complutense, 2005) (unpublished). Available at http://www.ucm.es/BUCM/tesis/psi/ucm-t28909.pdf .

Sierra-Vazquez, V.

I. Serrano-Pedraza and V. Sierra-Vazquez, “The effect of white-noise mask level on sinewave contrast detection thresholds and the critical-band-masking model,” Spanish J. Psychol. 9, 249–262 (2006).

Sierra-Vázquez, V.

I. Serrano-Pedraza and V. Sierra-Vázquez, “The effect of white-noise mask level on sine-wave contrast threshold and the critical band masking model,” Spanish J. Psychol. 9, 249–262 (2006).

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D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, 1966). Reprinted with corrections, 1974.

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C. P. Talgar, D. G. Pelli, and M. Carrasco, “Covert attention enhances letter identification without affecting channel tuning,” J. Vis. 4(1):3, 22–31 (2004).

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P. E. King-Smith, S. S. Grigsby, A. J. Vingrys, S. C. Benes, and A. Supowit, “Efficient and unbiased modifications of the QUEST threshold method: theory, simulations, experimental evaluation and practical implementation,” Vis. Res. 34, 885–912 (1994).
[CrossRef]

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Z. M. Westrick, C. A. Henry, and M. S. Landy, “Inconsistent channel bandwidth estimates suggest winner-take-all nonlinearity in second-order vision,” Vis. Res. 81, 58–68 (2013).
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H. R. Wilson, D. K. McFarlane, and G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vis. Res. 23, 873–882 (1983).
[CrossRef]

Comput. J. (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

J. Acoust. Soc. Am. (5)

R. D. Patterson and G. B. Henning, “Stimulus variability and auditory filter shape,” J. Acoust. Soc. Am. 62, 649–664 (1977).
[CrossRef]

B. Leshowitz and F. L. Wightman, “On-frequency masking with continuous sinusoids,” J. Acoust. Soc. Am. 49, 1180–1190 (1971).

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

R. D. Patterson and I. Nimmo-Smith, “Off-frequency listening and auditory-filter asymmetry,” J. Acoust. Soc. Am. 67, 229–245 (1980).
[CrossRef]

R. D. Patterson, “Auditory filter shape,” J. Acoust. Soc. Am. 55, 802–809 (1974).
[CrossRef]

J. Opt. Soc. Am. (3)

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

J. Physiol. (2)

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

C. Blakemore and F. W. Campbell, “On the existence of neurones in the human visual system selectively sensitive to the orientation and size of retinal images,” J. Physiol. 203, 237–260 (1969).

J. Vis. (5)

I. Oruc and M. S. Landy, “Scale dependence and channel switching in letter identification,” J. Vis. 9(9):4, 1–19 (2009).

I. Oruç and J. J. S. Barton, “Critical frequencies in the perception of letters, faces, and novel shapes: evidence for limited scale invariance for faces,” J. Vis. 10(12):20, 1–12 (2010).

D. G. Pelli, D. M. Levi, and S. T. L. Chung, “Using visual noise to characterize amblyopic letter identification,” J. Vis. 4(10):6, 904–920 (2004).

C. P. Talgar, D. G. Pelli, and M. Carrasco, “Covert attention enhances letter identification without affecting channel tuning,” J. Vis. 4(1):3, 22–31 (2004).

G. B. Henning and F. A. Wichmann, “Some observations on the pedestal effect,” J. Vis. 7(1):3, 1–15 (2007).
[CrossRef]

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

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[CrossRef]

Spanish J. Psychol. (2)

I. Serrano-Pedraza and V. Sierra-Vázquez, “The effect of white-noise mask level on sine-wave contrast threshold and the critical band masking model,” Spanish J. Psychol. 9, 249–262 (2006).

I. Serrano-Pedraza and V. Sierra-Vazquez, “The effect of white-noise mask level on sinewave contrast detection thresholds and the critical-band-masking model,” Spanish J. Psychol. 9, 249–262 (2006).

Spatial Vis. (1)

M. A. García-Pérez and V. Sierra-Vázquez, “Deriving channel gains from large-area sine-wave contrast sensitivity data,” Spatial Vis. 9, 235–260 (1995).

Vis. Res. (15)

A. Schofield and M. A. Georgeson, “Sensitivity to contrast modulation: the spatial frequency dependence of second-order vision,” Vis. Res. 43, 243–259 (2003).
[CrossRef]

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

J. Rovamo, R. Fransilla, and R. Näsänen, “Contrast sensitivity as a function of spatial frequency, viewing distance and eccentricity with and without spatial noise,” Vis. Res. 32, 631–637 (1992).
[CrossRef]

A. Schofield and M. A. Georgeson, “Sensitivity to modulations of luminance and contrast in visual white noise: separate mechanisms with similar behaviour,” Vis. Res. 39, 2697–2716 (1999).
[CrossRef]

P. E. King-Smith, S. S. Grigsby, A. J. Vingrys, S. C. Benes, and A. Supowit, “Efficient and unbiased modifications of the QUEST threshold method: theory, simulations, experimental evaluation and practical implementation,” Vis. Res. 34, 885–912 (1994).
[CrossRef]

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

K. T. Mullen and M. A. Losada, “The spatial tuning of color and luminance peripheral vision measured with notch filtered noise masking,” Vis. Res. 39, 721–731 (1999).
[CrossRef]

K. T. Blackwell, “The effect of white and filtered noise on contrast detection thresholds,” Vis. Res. 38, 267–280 (1998).
[CrossRef]

D. H. Kelly, “Spatial frequency selectivity in the retina,” Vis. Res. 15, 665–672 (1975).
[CrossRef]

C. V. Hutchinson and T. Ledgeway, “Spatial frequency selective masking of first-order and second-order motion in the absence of off-frequency looking,” Vis. Res. 44, 1499–1510 (2004).
[CrossRef]

Z. M. Westrick, C. A. Henry, and M. S. Landy, “Inconsistent channel bandwidth estimates suggest winner-take-all nonlinearity in second-order vision,” Vis. Res. 81, 58–68 (2013).
[CrossRef]

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

C. F. Stromeyer and S. Klein, “Evidence against narrow-band spatial frequency channels in human vision: the detectability of frequency modulated gratings,” Vis. Res. 15, 899–910 (1975).
[CrossRef]

N. J. Majaj, D. G. Pelli, P. Kurshan, and M. Palomares, “The role of spatial frequency channels in letter identification,” Vis. Res. 42, 1165–1184 (2002).
[CrossRef]

I. Oruc, M. S. Landy, and D. G. Pelli, “Noise masking reveals channels for second-order letters,” Vis. Res. 46, 1493–1506 (2006).
[CrossRef]

Other (8)

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (Wiley-IEEE, 1958).

D. M. Green and J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, 1966). Reprinted with corrections, 1974.

O. Braddick, F. W. Campbell, and J. Atkinson, “Channels in vision: basic aspects,” in Handbook of Sensory Physiology, R. Held, H. W. Leibowitz, and H. L. Teuber, eds. (Springer Verlag, 1978), pp. 3–38.

R. D. Patterson and B. C. J. Moore, “Auditory filters and excitation patterns as representations of frequency resolution,” in Frequency Selectivity in Hearing, B. C. J. Moore, ed. (Academic, 1986), pp. 123–177.

B. C. J. Moore, An Introduction to the Psychology of Hearing (Academic, 1997).

B. C. J. Moore, An Introduction to the Psychology of Hearing (Academic, 1997).

I. Serrano-Pedraza, “Procesos visuales de demodulación espacial,” Doctoral dissertation (Universidad Complutense, 2005) (unpublished). Available at http://www.ucm.es/BUCM/tesis/psi/ucm-t28909.pdf .

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

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Figures (8)

Fig. 1.
Fig. 1.

Components of the power spectrum model of visual masking. (a) CSF ( A = 337.323 and a = 0.4455 ). (b) Symmetric and asymmetric MTFs. The peak spatial frequency is 4 c / deg and the bandwidth is 1.25 octaves. Black line, symmetric channel ( σ = 0.1148 ); gray line, asymmetric channel ( α = 0.3679 ). (c) Relationship between channel bandwidth (octaves) and its peak spatial frequency. Continuous line, exponential decreasing function ( B = 4.498 and b = 0.6822 ); dashed line, constant function ( B = 0 and b = 1.25 ). The parameters for the panels (a) and (c) are taken from Serrano–Pedraza and Sierra–Vázquez (2006, subject IS).

Fig. 2.
Fig. 2.

Computational procedure used to obtain the peak frequency of the best channel (the channel that maximizes the SNR) using a fixed signal and a range of different masking bandpass noises of varying center frequency and constant bandwidth. (a) Power of the signal (grating of 4 c / deg ) at the output of each channel (using the asymmetrical shape for the MTF of the channels). (b) Power of each bandpass noise at the output of each channel. The different bandpass noises are represented with different gray levels. (c) Ratio between the functions in (a) and (b). Each circle represents the peak frequency of the channel that maximizes the SNR. Note that when the center frequency of the bandpass noise does not match the spatial frequency of the signal there is off-frequency looking. The off-frequency looking is more extreme when the central frequency of the bandpass noise is closer to the signal’s spatial frequency.

Fig. 3.
Fig. 3.

Simulation results for asymmetrical channels. Circles show results for the best-channel detection model; squares show results for the fixed-channel detection model. (a) Upper panel: estimated thresholds (in log units) for a grating of 4 c / deg masked by a white noise as a function of the power spectral density. Lower panel: estimated peak spatial frequency of the channel mediating detection as a function of the power spectral density. (b) Upper panel: estimated thresholds masked by a low-pass noise as a function of the cutoff frequency. Lower panel: estimated peak frequency of the channel mediating detection as a function of the cutoff frequency. (c) Upper panel: estimated thresholds masked by a high-pass noise as a function of the cutoff frequency. Lower panel: estimated peak frequency of the channel mediating detection as a function of the cutoff frequency. (d) Upper panel: estimated thresholds masked by asymmetrical notched noise as a function of the spectral notch size. Lower panel: estimated peak frequency of the channel mediating detection as a function of the spectral notch size. (e) Upper panel: estimated thresholds masked by bandpass noise as a function of the center frequency of the noise. Lower panel: estimated peak frequency of the channel mediating detection as a function of the center frequency of the noise. (f) Upper panel: estimated thresholds masked by asymmetrical double bandpass noise as a function of the spectral gap. Lower panel: estimated peak frequency of the channel mediating detection as a function of the spectral gap. The horizontal gray line in the upper panels shows the threshold value for this grating in the absence of masking. The horizontal black line in the lower panels shows the peak spatial frequency of the channel centered on the signal frequency. R 2 is the coefficient of determination between thresholds estimated by the two detection models. RMSE is the root mean squared error between peak channel frequencies estimated by the two detection models.

Fig. 4.
Fig. 4.

Simulation results for symmetrical channels. Circles show results for best-channel detection model; squares show results for fixed-channel detection model. (a) Upper panel: estimated contrast detection thresholds (in log units) for a grating of 4 c / deg masked by a white noise as a function of the power spectral density. Lower panel: estimated peak frequency of the channel mediating detection as a function of the power spectral density. (b) Upper panel: estimated thresholds masked by a low-pass noise as a function of the cutoff frequency. Lower panel: estimated peak frequency of the channel mediating detection as a function of the cutoff frequency. (c) Upper panel: estimated thresholds masked by a high-pass noise as a function of the cutoff frequency. Lower panel: estimated peak frequency of the channel mediating detection as a function of the cutoff frequency. (d) Upper panel: estimated thresholds masked by symmetrical notched noise as a function of the spectral notch size. Lower panel: estimated peak frequency of the channel mediating detection as a function of the spectral notch size. (e) Upper panel: estimated thresholds masked by bandpass noise as a function of the center frequency of the noise. Lower panel: estimated peak frequency of the channel mediating detection as a function of the center frequency of the noise. (f) Upper panel: estimated thresholds masked by symmetrical double bandpass noise as a function of the spectral gap. Lower panel: estimated peak frequency of the channel mediating detection as a function of the spectral gap. The horizontal gray line in the upper panels shows the threshold value for this grating in the absence of masking. The horizontal black line in the lower panels shows the frequency of the channel centered on the signal frequency. R 2 and RMSE the same as Fig. 3.

Fig. 5.
Fig. 5.

Effect of the shape of the notched noise on threshold estimation for different channel shapes. Circles show results for the best-channel model; squares show results for the fixed-channel model. Left panels show the MTF of the channel and an example of the notched noise used in the simulations, the arrow indicates the spatial frequency of the signal. Center panels, estimated contrast detection thresholds (in log units) for a grating of 4 c / deg masked by notched noise as a function of notch size; right panels, peak channel frequency as a function of notch size. (a) Results for asymmetrical channels of constant bandwidth of 1.25 octaves. Upper panels, results for asymmetrical notched noise; left panel, sample of one asymmetrical channel with peak frequency of 4 c / deg (black line) and one sample of the mask with a spectral gap of 1 octave (shaded area). Lower panels, results for symmetrical notched noise; left panel, sample of one asymmetrical channel with peak frequency of 4 c / deg and one sample of the mask with a spectral gap of 2 c / deg . (b) Results for symmetrical channels of constant bandwidth of 1.25 octaves. Upper panels, results for symmetrical notched noise; left panel, sample of one symmetrical channel centered on 4 c / deg and one sample of the mask with a spectral gap of 2 c / deg . Lower panels, results for asymmetrical notched noise; left panel, sample of one symmetrical channel centered on 4 c / deg and one sample of the mask with a spectral gap of 1 octave. The horizontal gray line in the center panels shows the threshold value for this grating in the absence of masking. The horizontal black line in the right panels shows the frequency of the channel centered on the signal frequency. R 2 is the coefficient of determination between thresholds. RMSE is the root mean squared error between peak channel frequencies.

Fig. 6.
Fig. 6.

Effect of the shape of the double bandpass noise on threshold estimation depending of the channel shape. Circles show results for best-channel model; squares show results for fixed-channel model. Left panels show the MTF of the channel and an example of the notched noise used in the simulations, the arrow indicates the spatial frequency of the signal. Center panels, estimated contrast detection thresholds (in log units) for a grating of 4 c / deg masked by double bandpass noise as a function of the spectral gap; right panels, peak channel frequency as a function of the spectral gap. (a) Results for asymmetrical channels of constant bandwidth of 1.25 octaves. Upper panels, results for asymmetrical spectral gap; left panel, sample of one asymmetrical channel with peak frequency of 4 c / deg (black line) and one sample of the mask with a spectral gap of 1 octave (shaded area). Lower panels, results for symmetrical spectral gap; left panel, sample of one asymmetrical channel with peak frequency of 4 c / deg and one sample of the mask with a spectral gap of 2 c / deg . (b) Results for symmetrical channels of constant bandwidth of 1.25 octaves. Upper panels, results for symmetrical spectral gap; left panel, sample of one symmetrical channel centered on 4 c / deg and one sample of the mask with a spectral gap of 2 c / deg . Lower panels, results for asymmetrical spectral gap; left panel, sample of one symmetrical channel centered on 4 c / deg and one sample of the mask with a spectral gap of 1 octave. The horizontal gray line in the center panels shows the threshold value for this grating in the absence of masking. The horizontal black line in the right panels shows the frequency of the channel centered on the signal frequency. R 2 is the coefficient of determination between thresholds. RMSE is the root mean squared error between peak channel frequencies.

Fig. 7.
Fig. 7.

Effect of white noise level and bandwidth of channels on the CSF. (a) Simulation results with constant bandwidth for six spatial frequencies and five power spectral levels. Upper panel, bandwidth in octaves as a function of the peak channel frequency; lower panel, estimated contrast detection thresholds (in log units) of the test as a function of the spatial frequency parameterized by the noise level of the white noise. Circles, results for the best-channel detection model; squares, results for the fixed-channel detection model. The thick black line represents the inverse CSF (detection thresholds in absence of masking). Note that thresholds obtained with the lowest power density level are closer to this curve. (b) Simulation results with decreasing bandwidth for six spatial frequencies and five power spectral levels. Upper panel, bandwidth in octaves as a function of the peak channel frequency; lower panel, log-thresholds of the test as a function of the spatial frequency parameterized by the noise level of the white noise. The thick black line represents the inverse function of the CSF.

Fig. 8.
Fig. 8.

Results from visual masking experiments for two subjects (ISP and GB). Each column shows the contrast detection thresholds in log units (circles; in the low-pass and high-pass column, the squares show thresholds for high-pass noise) for a Gabor patch of 1 c / deg masked by a particular type of noise (all-pass, low-pass and high-pass, notched, bandpass, and double bandpass noise). In each panel there is a sketch of the amplitude spectrum of the noise used in the experiment. (a), (b) Both rows show the contrast thresholds ( mean ± S.D. ) of the subject ISP as a function of the power spectral density (all-pass), cutoff frequency (low- and high-pass), notch size, center spatial frequency (bandpass) and spectral gap (double bandpass). The empirical thresholds in the two rows are the same. (c), (d) These rows show the contrast thresholds ( mean ± S.D. ) of the subject GB. (a), (c) In these two rows, the black line represents the fitted power spectrum masking model assuming asymmetric channel MTF [see Eq. (3)], constant bandwidth, and best-channel detection model. The left-most panel from the left shows the parameters estimated: b (bandwidth in octaves) and s (sensitivity). (b), (d) In these two rows, the black line is the fitted power spectrum masking model assuming asymmetric channel MTF, constant bandwidth, and fixed-channel detection model. The horizontal gray line in all panels shows the threshold for the Gabor patch of 1 c / deg without masking. The value of R 2 shown in the left panel of each row is the coefficient of determination between all masking thresholds from the five panels and the model predictions.

Tables (3)

Tables Icon

Table 1. Parameter Values of the Power Spectrum Model of Visual Masking

Tables Icon

Table 2. Noise Types and Parameter Values (See Appendix A)

Tables Icon

Table 3. Comparison Between Power Density Levelsa

Equations (21)

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m T 2 ( u 0 ) = m 0 2 ( ξ k ) + 4 s 0 + ρ ( u ) | H ( u ; ξ k ) | 2 d u | H ( u 0 ; ξ k ) | 2 ,
CS ( u ) = A u exp [ a u ] .
| H ( u ; ξ i ) | = { exp [ ln 2 ( | u | / ξ i ) 2 α i 2 ] u 0 0 u = 0 ,
B oct ( ξ i ) = 2 2 ln 2 α i .
| H ( u ; ξ i ) | = exp [ 2 π 2 σ i 2 ( u ξ i ) 2 ] + exp [ 2 π 2 σ i 2 ( u + ξ i ) 2 ] ,
B oct ( ξ i ) = log 2 [ 2 π σ i ξ i + ln 2 2 π σ i ξ i ln 2 ] .
B oct ( ξ i ) = b + B exp ( ξ i ) ,
α i = ln 2 2 2 × B oct ( ξ i ) ,
σ i = [ 2 B oct ( ξ i ) + 1 ] [ 2 B oct ( ξ i ) 1 ] × ln 2 π 2 × 1 ξ i ,
m T 2 ( u 0 ) = m 0 2 ( u 0 ) + 4 s 0 + ρ ( u ) | H ( u ; u 0 ) | 2 d u
r ( u 0 , ξ i , m T ) = P s ( u 0 , ξ i , m s 0 ) P N ( ξ i ) = m T 2 ( u 0 ) 2 | H ( u 0 ; ξ i ) | 2 s m 0 2 ( ξ i ) 2 + 2 0 + ρ ( u ) | H ( u ; ξ i ) | 2 d u = m T 2 ( u 0 ) | H ( u 0 ; ξ i ) | 2 s m 0 2 ( ξ i ) + 4 0 + ρ ( u ) | H ( u ; ξ i ) | 2 d u .
r ( u 0 , ξ k , m T ) = max i { r ( u 0 , ξ i , m T ) } ,
m T ( u 0 ) ξ i [ | H ( u 0 ; ξ i ) | 2 s m 0 2 ( ξ i ) + 4 0 + ρ ( u ) | H ( u ; ξ i ) | 2 d u ] | ξ i = ξ k = 0
u lo u hi | H ( u ; ξ i ) | 2 d u = u lo u hi exp [ ln 2 ( u / ξ i ) α i 2 ] d u = π α i ξ i 2 exp [ α i 2 4 ] × { erf [ ln ( u hi / ξ i ) ( α i 2 / 2 ) α i ] erf [ ln ( u lo / ξ i ) ( α i 2 / 2 ) α i ] } ,
erf ( x ) = 2 π 0 x exp [ t 2 ] d t
u lo u hi | H ( u ; ξ i ) | 2 d u = u lo u hi { exp [ 4 π 2 σ i 2 ( u ξ i ) 2 ] + exp [ 4 π 2 σ i 2 ( u + ξ i ) 2 ] + 2 exp [ 4 π 2 σ i 2 ( u 2 + ξ i 2 ) ] } d u = 1 2 π σ { exp [ 4 π 2 σ i 2 ξ i 2 ] × [ erf ( 2 π σ i u hi ) erf ( 2 π σ i u lo ) ] + 1 2 [ erf ( 2 π σ i ( u hi + ξ i ) ) erf ( 2 π σ i ( u lo + ξ i ) ) ] + 1 2 [ erf ( 2 π σ i ( u hi ξ i ) ) erf ( 2 π σ i ( u lo ξ i ) ) ] } ,
m T 2 ( u 0 , u hi , N 0 ) = m 0 2 ( ξ k ) + 4 N 0 s 0 u hi | H ( u ; ξ k ) | 2 d u | H ( u 0 ; ξ k ) | 2 .
m T 2 ( u 0 , u l o , N 0 ) = m 0 2 ( ξ k ) + 4 N 0 s u lo u hi | H ( u ; ξ k ) | 2 d u | H ( u 0 ; ξ k ) | 2 ,
m T 2 ( u 0 , u lo , u hi , N 0 ) = m 0 2 ( ξ k ) + 4 N 0 s [ 0 u lo | H ( u ; ξ k ) | 2 d u + u hi 32 | H ( u ; ξ k ) | 2 d u ] | H ( u 0 ; ξ k ) | 2 ,
m T 2 ( u 0 , B u , u C , N 0 ) = m 0 2 ( ξ k ) + 4 N 0 s u C B u 2 u C + B u 2 | H ( u ; ξ k ) | 2 d u | H ( u 0 ; ξ k ) | 2 ,
m T 2 ( u 0 , B u , u C lo , u C hi , N 0 ) = m 0 2 ( ξ k ) + 4 N 0 s [ u C lo B u 2 u C lo + B u 2 | H ( u ; ξ k ) | 2 d u + u C hi B u 2 u C hi + B u 2 | H ( u ; ξ k ) | 2 d u ] | H ( u 0 ; ξ k ) | 2 ,

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