Abstract

Filtered external noise has been an important tool in characterizing the spatial-frequency sensitivity of perceptual templates. Typically, low-pass- and/or high-pass-filtered external noise is added to the signal stimulus. Thresholds, the signal energy necessary to maintain given criterion performance levels, are measured as functions of the spatial-frequency passband of the external noise. An observer model is postulated to segregate the impact of the external noise and the internal noise. The spatial-frequency sensitivity of the perceptual template is determined by the relative impact exerted by external noise in each frequency band. The perceptual template model (PTM) is a general observer model that provides an excellent account of human performance in white external noise [Vision Res. 38, 1183 (1998); J. Opt. Soc. Am. A 16, 764 (1999)]. We further develop the PTM for filtered external noise and apply it to derive the spatial-frequency sensitivity of perceptual templates.

© 2001 Optical Society of America

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  75. This approximation becomes exact if (i) Ts(fx, fy)=Ts(f ) and Ss(fx, fy)=Ss(f ), that is, the template and the signal are radially symmetric in Fourier space; or (ii) Ts(fx, fy)= kSs(fx, fy),∀fx,fy; that is, the template is perfectly matched to the signal stimulus; or (iii) either the template or the signal stimulus (or both) are uniform for every radius where the template and the signal overlap in Fourier space. In the current application, condition (iii) holds because the stimulus is a pair of points in Fourier space, and condition (ii) is approximately true because humans tend to use nearly optimal templates in simple stimulus situations.60 If one or more of these conditions holds approximately, then the equations should provide reasonable approximations.
  76. This approximation becomes exact if (i) Ts(fx, fy)=Ts(f ) and F(fx, fy)=F(f ); that is, the template and the experimenter-applied filter are radially symmetric in Fourier space; or (ii) Ts(fx, fy)=kF(fx, fy),∀fx,fy; that is, the template is matched to the noise filter; or (iii) either the template or the noise filter (or both) are uniform for every radius where the template and the noise filter overlap in Fourier space. In the current application, F(fx, fy)= F(f )=constant, the expected spectrum of the Gaussian noise is uniform, and condition (iii) is met. In most applications it will be possible to construct filters such that condition (iii) holds.
  77. In the current development, cross products in the form (β2c2+Next2)γare eliminated in order to yield analytical solutions. The effects of the cross terms have been evaluated in two of our previous publications. In one study,59PTMs with full cross-product forms were fit to the data by methods of iterative solution. The results were equivalent in pattern to those from fits of PTMs without cross products, and the cross-product terms were small. In the other study,72the analytical PTMs without cross products were compared with full stochastic PTMs. The analytical form was found to be a good approximation of the stochastic model.
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  89. Implemented in Matlab 5.3, the procedure for a given model consisted of the following. (1) For a given set of the model parameters, using Eq. (21) to compute log(cτtheory)from the model for each external-noise condition at three different performance levels. (2) Computing the squared difference between the log threshold prediction from the model and the observed sqdiff=[log(cτ theory)-log(cτ)]2for each threshold. The log approximately equates the standard error over large ranges in contrast thresholds, corresponding to weighted least squares, an equivalent to the maximum likelihood solution for continuous data. In the current data set, this assumption is true. (3) Computing L: summation of sqdiff from all the thresholds across all the external-noise conditions. (4) Using a gradient-descent method to adjust the model parameters to find the minimum L. (5) After obtaining the minimum L, computing the r2statistic to evaluate the goodness of the model fit: (25)r2=1.0-∑[log(cτtheory)-log(cτ)]2∑{log(cτ)-mean[log(cτ)]}2,where Σ and mean( )run over all the thresholds for a particular observer. An Ftest for nested models was used to statistically compare the four models. An Fis defined: (26)F(df1, df2)=[(rfull2-rreduced2)/df1]/[(1-rfull2)/df2],where df1=kfull-kreduced,and df2=N-kfull.The kvariables are the number of parameters in each model, and Nis the number of predicted data points.
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  91. Full sets of maximum-likelihood fits were performed on the psychometric functions for all four models: uPTM, cPTM, uLAM, and cLAM. For an observer who is correct in Kijtrials among a total of Nijtrials in the jthsignal contrast and ithfilter condition, the likelihood of a model that predicts a fraction of Pijcorrect in each condition is defined as (27)likelihood=∏i=1I∏j=1JNij!Kij!(Nij-Kij)! PijKij(1-Pij)Nij-Kij,where Pijis defined by Eq. (22). Asymptotically, χ2(df )statistics could be used to compare the proper set of models:(28)χ2(df )=2.0×loglikelihoodfulllikelihoodreduced,where dfis the difference of the number of parameters between the full and the reduced models.
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2000 (4)

B. A. Dosher, Z.-L. Lu, “Noise exclusion in spatial attention,” Psychol. Sci. 11, 139–146 (2000).
[CrossRef]

B. A. Dosher, Z.-L. Lu, “Mechanisms of perceptual attention in precuing of location,” Vision Res. 40, 1269–1292 (2000).
[CrossRef] [PubMed]

B. A. Dosher, Z.-L. Lu, “Perceptual templates in spatial attention,” Invest. Ophthalmol. Visual Sci. 41, S750 (2000).

Z.-L. Lu, B. A. Dosher, “Spatial attention: different mechanisms for central and peripheral temporal precues?” J. Exp. Psychol. 26, 1534–1548 (2000).

1999 (7)

Z.-L. Lu, B. A. Dosher, “Attention fine-tunes perceptual templates in spatial cuing,” Bull. Psychonom. Soc. 40, 52 (1999).

D. G. Pelli, “Close encounters—an artist shows that size affects shape,” Science 285, 844–846 (1999).
[CrossRef] [PubMed]

Z.-L. Lu, G. Sperling, “Second-order reversed phi,” Percept. Psychophys. 61, 1075–1088 (1999).
[CrossRef] [PubMed]

Z.-L. Lu, B. A. Dosher, “Characterizing human perceptual inefficiencies with equivalent internal noise,” J. Opt. Soc. Am. A 16, 764–778 (1999).
[CrossRef]

B. A. Dosher, Z.-L. Lu, “Mechanisms of perceptual learning,” Vision Res. 39, 3197–3221 (1999).
[CrossRef]

B. L. Beard, A. J. Ahumada, “Detection in fixed and random noise in foveal and parafoveal vision explained by template learning,” J. Opt. Soc. Am. A 16, 755–763 (1999).
[CrossRef]

J. A. Solomon, M. J. Morgan, “Reverse correlation reveals psychophysical receptive fields,” Invest. Ophthalmol. Visual Sci. 40, S572 (1999).

1998 (2)

Z.-L. Lu, B. A. Dosher, “External noise distinguishes mechanisms of attention,” Vision Res. 38, 1183–1198 (1998).
[CrossRef] [PubMed]

Y. Yeshurun, M. Carrasco, “Attention improves or impairs visual performance by enhancing spatial resolution,” Nature 396, 72–75 (1998).
[CrossRef] [PubMed]

1997 (1)

1995 (1)

1994 (1)

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

1992 (1)

D. J. Heeger, “Normalization of cell responses in cat striate cortex,” Visual Neurosci. 9, 181–197 (1992).
[CrossRef]

1991 (3)

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

D. H. Parish, G. Sperling, “Object spatial frequencies, retinal spatial frequencies, noise, and the efficiency of letter discrimination,” Vision Res. 31, 1399–1415 (1991).
[CrossRef] [PubMed]

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

1990 (1)

L. T. Maloney, “Confidence intervals for the parameters of psychometric functions,” Percept. Psychophys. 47, 127–134 (1990).
[CrossRef] [PubMed]

1988 (3)

1987 (1)

1985 (5)

1983 (2)

M. C. Morrone, D. C. Burr, J. Ross, “Added noise restores recognizability of coarse quantized images,” Nature 305, 226–228 (1983).
[CrossRef] [PubMed]

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]

1982 (2)

J. P. Thomas, J. Gille, R. Barker, “Simultaneous detection and identification: theory and data,” J. Opt. Soc. Am. 72, 1642–1651 (1982).
[CrossRef] [PubMed]

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

1981 (5)

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]

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vision Res. 21, 1115–1122 (1981).
[CrossRef] [PubMed]

L. Olzak, J. P. Thomas, “Gratings: why frequency discrimination is sometimes better than detection,” J. Opt. Soc. Am. 71, 64–70 (1981).
[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]

J. M. Foley, G. E. Legge, “Contrast detection and near-threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
[CrossRef] [PubMed]

1980 (2)

G. E. Legge, J. M. Foley, “Contrast masking in human vision,” J. Opt. Soc. Am. 70, 1458–1471 (1980).
[CrossRef] [PubMed]

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

1979 (3)

H. R. Wilson, J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–32 (1979).
[CrossRef] [PubMed]

J. P. Thomas, J. Gille, “Bandwidths of orientation channels in human vision,” J. Opt. Soc. Am. 69, 652–660 (1979).
[CrossRef] [PubMed]

A. P. Ginsburg, D. W. Evans, “Predicting visual illusions from filtered images based on biological data,” J. Opt. Soc. Am. 69, 1443 (1979) (abstract).

1978 (1)

S. P. Mckee, G. Westheimer, “Improvement in vernier acuity with practice,” Percept. Psychophys. 24, 258–262 (1978).
[CrossRef] [PubMed]

1977 (2)

N. V. S. Graham, “Visual detection of aperiodic spatial stimuli by probability summation among narrowband channels,” Vision Res. 17, 637–652 (1977).
[CrossRef] [PubMed]

J. P. Thomas, “Model of the function of receptive fields in human vision,” Psychol. Rev. 77, 121–134 (1977).
[CrossRef]

1976 (2)

P. H. Schiller, B. L. Finlay, S. F. Volman, “Quantitative studies of single cell properties in monkey striate cortex. III. Spatial-frequency,” J. Neurophysiol. 39, 1334–1351 (1976).
[PubMed]

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

1975 (3)

P. E. King-Smith, J. J. Kulikowski, “The detection of gratings by independent activation of line detectors,” J. Physiol. 247, 237–271 (1975).
[PubMed]

J. Nachmias, A. Weber, “Discrimination of simple and complex gratings,” Vision Res. 15, 217–223 (1975).
[CrossRef] [PubMed]

D. J. Tolhurst, R. S. Dealey, “The detection and identification of lines and edges,” Vision Res. 15, 1367–1372 (1975).
[CrossRef] [PubMed]

1974 (2)

D. H. Hubel, T. N. Wiesel, “Uniformity of monkey striate cortex: a parallel relationship between field size, scatter, and magnification factor,” J. Comp. Neurol. 158, 295–306 (1974).
[CrossRef] [PubMed]

J. Nachmias, R. V. Sansbury, “Grating contrast: Discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
[CrossRef] [PubMed]

1973 (1)

L. D. Harmon, B. Julesz, “Masking in visual recognition: effects of two-dimensional filtered noise,” Science 180, 1194–1197 (1973).
[CrossRef] [PubMed]

1972 (1)

C. F. Stromeyer, B. Julesz, “Spatial-frequency masking in vision: critical bands and spread of masking,” J. Opt. Soc. Am. 64, 1221–1232 (1972).
[CrossRef]

1971 (4)

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

C. B. Blakemore, J. Nachmias, “The orientation specificity of two visual after-effects,” J. Physiol. 213, 157–174 (1971).
[PubMed]

B. E. Carter, G. B. Henning, “The detection of gratings in narrow-band visual noise,” J. Physiol. 219, 355–365 (1971).
[PubMed]

A. P. Ginsburg, “Psychological correlates of a model of the human visual system,” IEEE Trans. Aerosp. Electron. Syst. 71-C-AES, 283–290 (1971).

1970 (2)

U. Greis, R. Rohler, “Untersuchung der subjektiven Detailerkennbarkeit mit Hilfe der Ortsfrequenzfilterung,” Opt. Acta 17, 515–526 (1970).
[CrossRef]

H. Pollehn, H. Roehrig, “Effect of noise on the MTF of the visual channel,” J. Opt. Soc. Am. 60, 842–848 (1970).
[CrossRef] [PubMed]

1969 (2)

A. Pantle, R. Sekuler, “Size detecting mechanisms in human vision,” Science 162, 1146–1148 (1969).
[CrossRef]

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

1968 (2)

A. S. Gilinsky, “Orientation-specific effects of patterns of adapting light on visual acuity,” J. Opt. Soc. Am. 58, 13–18 (1968).
[CrossRef] [PubMed]

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

1966 (2)

1964 (1)

1963 (1)

1962 (2)

D. H. Hubel, T. N. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s striate cortex,” J. Physiol. 160, 106–154 (1962).
[PubMed]

J. A. Swets, D. M. Green, W. P. Tanner, “On the width of critical bands,” J. Acoust. Soc. Am. 34, 108–113 (1962).
[CrossRef]

1960 (1)

G. Sperling, “The information available in brief visual presentations,” Psychol. Monogr. 11, 1–74 (1960).
[CrossRef]

1957 (1)

H. B. Barlow, “Incremental thresholds at low intensities considered as signal/noise discrimination,” J. Physiol. 136, 469–488 (1957).
[PubMed]

1940 (1)

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

1939 (1)

C. H. Graham, R. H. Brown, F. A. Mote, “The relation of size stimulus and intensity in the human eye. I. Intensity threshold for white light,” J. Exp. Psychol. 24, 554–573 (1939).

Ahumada, A. J.

Albrecht, D. G.

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

Barker, R.

Barlow, H. B.

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]

H. B. Barlow, “Incremental thresholds at low intensities considered as signal/noise discrimination,” J. Physiol. 136, 469–488 (1957).
[PubMed]

Beard, B. L.

B. L. Beard, A. J. Ahumada, “Detection in fixed and random noise in foveal and parafoveal vision explained by template learning,” J. Opt. Soc. Am. A 16, 755–763 (1999).
[CrossRef]

B. L. Beard, A. J. Ahumada, “Technique to extract relevant image features for visual tasks,” in Human Vision and Electronic Imaging III, B. E. Rogowitz, T. N. Pappas, eds., Proc. SPIE3299, 79–85 (1998).
[CrossRef]

Bergen, J. R.

H. R. Wilson, J. R. Bergen, “A four mechanism model for threshold spatial vision,” Vision Res. 19, 19–32 (1979).
[CrossRef] [PubMed]

Blackwell, H. R.

Blakemore, C. B.

C. B. Blakemore, J. Nachmias, “The orientation specificity of two visual after-effects,” J. Physiol. 213, 157–174 (1971).
[PubMed]

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

Brown, R. H.

C. H. Graham, R. H. Brown, F. A. Mote, “The relation of size stimulus and intensity in the human eye. I. Intensity threshold for white light,” J. Exp. Psychol. 24, 554–573 (1939).

Burgess, A. E.

Burr, D. C.

M. C. Morrone, D. C. Burr, J. Ross, “Added noise restores recognizability of coarse quantized images,” Nature 305, 226–228 (1983).
[CrossRef] [PubMed]

Campbell, F. W.

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

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

Carrasco, M.

Y. Yeshurun, M. Carrasco, “Attention improves or impairs visual performance by enhancing spatial resolution,” Nature 396, 72–75 (1998).
[CrossRef] [PubMed]

Carter, B. E.

B. E. Carter, G. B. Henning, “The detection of gratings in narrow-band visual noise,” J. Physiol. 219, 355–365 (1971).
[PubMed]

Colborne, B.

Creelman, C. D.

N. A. MacMillan, C. D. Creelman, Detection Theory: A User’s Guide (Cambridge U. Press, New York, 1991).

de Valois, R. L.

Dealey, R. S.

D. J. Tolhurst, R. S. Dealey, “The detection and identification of lines and edges,” Vision Res. 15, 1367–1372 (1975).
[CrossRef] [PubMed]

DeValois, R. L.

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

Dosher, B.

G. Sperling, B. Dosher, “Strategy and optimization in human information processing,” in Handbook of Perception and Performance, K. Boff, L. Kaufmon, J. Thomas, eds. (Wiley, New York, 1986), Vol. 1, Chap. 2, pp. 1–65.

Dosher, B. A.

B. A. Dosher, Z.-L. Lu, “Noise exclusion in spatial attention,” Psychol. Sci. 11, 139–146 (2000).
[CrossRef]

B. A. Dosher, Z.-L. Lu, “Mechanisms of perceptual attention in precuing of location,” Vision Res. 40, 1269–1292 (2000).
[CrossRef] [PubMed]

B. A. Dosher, Z.-L. Lu, “Perceptual templates in spatial attention,” Invest. Ophthalmol. Visual Sci. 41, S750 (2000).

Z.-L. Lu, B. A. Dosher, “Spatial attention: different mechanisms for central and peripheral temporal precues?” J. Exp. Psychol. 26, 1534–1548 (2000).

Z.-L. Lu, B. A. Dosher, “Attention fine-tunes perceptual templates in spatial cuing,” Bull. Psychonom. Soc. 40, 52 (1999).

Z.-L. Lu, B. A. Dosher, “Characterizing human perceptual inefficiencies with equivalent internal noise,” J. Opt. Soc. Am. A 16, 764–778 (1999).
[CrossRef]

B. A. Dosher, Z.-L. Lu, “Mechanisms of perceptual learning,” Vision Res. 39, 3197–3221 (1999).
[CrossRef]

Z.-L. Lu, B. A. Dosher, “External noise distinguishes mechanisms of attention,” Vision Res. 38, 1183–1198 (1998).
[CrossRef] [PubMed]

B. A. Dosher, Z.-L. Lu, “Perceptual learning reflects external noise filtering and internal noise reduction through channel reweighting,” in Proc. Natl. Acad. Sci. U.S.A. 95, 13 988–13 993.

Eckstein, M. P.

Enroth-Cugell, C.

C. Enroth-Cugell, J. G. Robson, “The contrast sensitivity of retinal ganglion cells of the cat,” J. Physiol. 258, 517–552 (1966).

Evans, D. W.

A. P. Ginsburg, D. W. Evans, “Predicting visual illusions from filtered images based on biological data,” J. Opt. Soc. Am. 69, 1443 (1979) (abstract).

Finlay, B. L.

P. H. Schiller, B. L. Finlay, S. F. Volman, “Quantitative studies of single cell properties in monkey striate cortex. III. Spatial-frequency,” J. Neurophysiol. 39, 1334–1351 (1976).
[PubMed]

Fletcher, H.

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

Foley, J. M.

J. M. Foley, G. E. Legge, “Contrast detection and near-threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
[CrossRef] [PubMed]

G. E. Legge, J. M. Foley, “Contrast masking in human vision,” J. Opt. Soc. Am. 70, 1458–1471 (1980).
[CrossRef] [PubMed]

Gilinsky, A. S.

Gille, J.

Ginsburg, A. P.

A. P. Ginsburg, D. W. Evans, “Predicting visual illusions from filtered images based on biological data,” J. Opt. Soc. Am. 69, 1443 (1979) (abstract).

A. P. Ginsburg, “Psychological correlates of a model of the human visual system,” IEEE Trans. Aerosp. Electron. Syst. 71-C-AES, 283–290 (1971).

A. P. Ginsburg, “Visual information processing based on spatial filters constrained by biological data,” Ph.D dissertation (University of Cambridge, Cambridge, UK, 1978), Library of Congress 79-600156.

Graham, C. H.

C. H. Graham, R. H. Brown, F. A. Mote, “The relation of size stimulus and intensity in the human eye. I. Intensity threshold for white light,” J. Exp. Psychol. 24, 554–573 (1939).

Graham, N.

N. Graham, “Spatial frequency channels in human vision: detecting edges without edge detectors,” in Visual Coding and Adaptability, C. S. Harris, ed. (Erlbaum, Hillsdale, N. J., 1980), pp. 215–262.

Graham, N. V. S.

N. V. S. Graham, “Visual detection of aperiodic spatial stimuli by probability summation among narrowband channels,” Vision Res. 17, 637–652 (1977).
[CrossRef] [PubMed]

N. V. S. Graham, Visual Pattern Analyzers (Oxford U. Press, New York, 1989).

Green, D. M.

J. A. Swets, D. M. Green, W. P. Tanner, “On the width of critical bands,” J. Acoust. Soc. Am. 34, 108–113 (1962).
[CrossRef]

Greis, U.

U. Greis, R. Rohler, “Untersuchung der subjektiven Detailerkennbarkeit mit Hilfe der Ortsfrequenzfilterung,” Opt. Acta 17, 515–526 (1970).
[CrossRef]

Harmon, L. D.

L. D. Harmon, B. Julesz, “Masking in visual recognition: effects of two-dimensional filtered noise,” Science 180, 1194–1197 (1973).
[CrossRef] [PubMed]

Hays, W. L.

W. L. Hays, Statistics, 3rd ed. (CBS College Publishing, New York, 1981).

Heeger, D. J.

D. J. Heeger, “Normalization of cell responses in cat striate cortex,” Visual Neurosci. 9, 181–197 (1992).
[CrossRef]

Henning, G. B.

Hertz, B. G.

Hill, N. J.

For an excellent discussion on fitting psychometric functions, see F. A. Wichmann, N. J. Hill, “The psychometric function I: fitting, sampling and goodness-of-fit,” Percept. Psychophys. accepted for publication.

F. A. Wichmann, N. J. Hill, “The psychometric function II: bootstrap based confidence intervals and sampling” Percept. Psychophys. (to be published).

Hinton, J. L.

Hubel, D. H.

D. H. Hubel, T. N. Wiesel, “Uniformity of monkey striate cortex: a parallel relationship between field size, scatter, and magnification factor,” J. Comp. Neurol. 158, 295–306 (1974).
[CrossRef] [PubMed]

D. H. Hubel, T. N. Wiesel, “Receptive fields, binocular interaction and functional architecture in the cat’s striate cortex,” J. Physiol. 160, 106–154 (1962).
[PubMed]

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]

Julesz, B.

L. D. Harmon, B. Julesz, “Masking in visual recognition: effects of two-dimensional filtered noise,” Science 180, 1194–1197 (1973).
[CrossRef] [PubMed]

C. F. Stromeyer, B. Julesz, “Spatial-frequency masking in vision: critical bands and spread of masking,” J. Opt. Soc. Am. 64, 1221–1232 (1972).
[CrossRef]

King-Smith, P. E.

P. E. King-Smith, J. J. Kulikowski, “The detection of gratings by independent activation of line detectors,” J. Physiol. 247, 237–271 (1975).
[PubMed]

Kulikowski, J. J.

P. E. King-Smith, J. J. Kulikowski, “The detection of gratings by independent activation of line detectors,” J. Physiol. 247, 237–271 (1975).
[PubMed]

Landy, M. S.

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

Legge, G. E.

G. E. Legge, D. G. Pelli, G. S. Rubin, M. M. Schleske, “Psychophysics of reading: I. Normal vision,” Vision Res. 25, 239–252 (1985).
[CrossRef]

J. M. Foley, G. E. Legge, “Contrast detection and near-threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
[CrossRef] [PubMed]

G. E. Legge, J. M. Foley, “Contrast masking in human vision,” J. Opt. Soc. Am. 70, 1458–1471 (1980).
[CrossRef] [PubMed]

Losada, M. A.

Lu, Z.-L.

B. A. Dosher, Z.-L. Lu, “Mechanisms of perceptual attention in precuing of location,” Vision Res. 40, 1269–1292 (2000).
[CrossRef] [PubMed]

B. A. Dosher, Z.-L. Lu, “Noise exclusion in spatial attention,” Psychol. Sci. 11, 139–146 (2000).
[CrossRef]

Z.-L. Lu, B. A. Dosher, “Spatial attention: different mechanisms for central and peripheral temporal precues?” J. Exp. Psychol. 26, 1534–1548 (2000).

B. A. Dosher, Z.-L. Lu, “Perceptual templates in spatial attention,” Invest. Ophthalmol. Visual Sci. 41, S750 (2000).

Z.-L. Lu, B. A. Dosher, “Attention fine-tunes perceptual templates in spatial cuing,” Bull. Psychonom. Soc. 40, 52 (1999).

Z.-L. Lu, G. Sperling, “Second-order reversed phi,” Percept. Psychophys. 61, 1075–1088 (1999).
[CrossRef] [PubMed]

Z.-L. Lu, B. A. Dosher, “Characterizing human perceptual inefficiencies with equivalent internal noise,” J. Opt. Soc. Am. A 16, 764–778 (1999).
[CrossRef]

B. A. Dosher, Z.-L. Lu, “Mechanisms of perceptual learning,” Vision Res. 39, 3197–3221 (1999).
[CrossRef]

Z.-L. Lu, B. A. Dosher, “External noise distinguishes mechanisms of attention,” Vision Res. 38, 1183–1198 (1998).
[CrossRef] [PubMed]

B. A. Dosher, Z.-L. Lu, “Perceptual learning reflects external noise filtering and internal noise reduction through channel reweighting,” in Proc. Natl. Acad. Sci. U.S.A. 95, 13 988–13 993.

MacMillan, N. A.

N. A. MacMillan, C. D. Creelman, Detection Theory: A User’s Guide (Cambridge U. Press, New York, 1991).

Maloney, L. T.

L. T. Maloney, “Confidence intervals for the parameters of psychometric functions,” Percept. Psychophys. 47, 127–134 (1990).
[CrossRef] [PubMed]

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]

Mckee, S. P.

S. P. Mckee, G. Westheimer, “Improvement in vernier acuity with practice,” Percept. Psychophys. 24, 258–262 (1978).
[CrossRef] [PubMed]

Morgan, M. J.

J. A. Solomon, M. J. Morgan, “Reverse correlation reveals psychophysical receptive fields,” Invest. Ophthalmol. Visual Sci. 40, S572 (1999).

Morrone, M. C.

M. C. Morrone, D. C. Burr, J. Ross, “Added noise restores recognizability of coarse quantized images,” Nature 305, 226–228 (1983).
[CrossRef] [PubMed]

Mote, F. A.

C. H. Graham, R. H. Brown, F. A. Mote, “The relation of size stimulus and intensity in the human eye. I. Intensity threshold for white light,” J. Exp. Psychol. 24, 554–573 (1939).

Muller, K. T.

Nachmias, J.

J. Nachmias, A. Weber, “Discrimination of simple and complex gratings,” Vision Res. 15, 217–223 (1975).
[CrossRef] [PubMed]

J. Nachmias, R. V. Sansbury, “Grating contrast: Discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
[CrossRef] [PubMed]

C. B. Blakemore, J. Nachmias, “The orientation specificity of two visual after-effects,” J. Physiol. 213, 157–174 (1971).
[PubMed]

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

J. Nachmias, “Signal detection theory and its application to problems in vision,” in Handbook of Sensory Physiology, D. Jameson, L. M. Hurvich, eds. (Springer–Verlag, Berlin, 1972), Vol. 7/4, Chap. 8.

Nagaraja, N. S.

Olzak, L.

Pantle, A.

A. Pantle, R. Sekuler, “Size detecting mechanisms in human vision,” Science 162, 1146–1148 (1969).
[CrossRef]

Parish, D. H.

D. H. Parish, G. Sperling, “Object spatial frequencies, retinal spatial frequencies, noise, and the efficiency of letter discrimination,” Vision Res. 31, 1399–1415 (1991).
[CrossRef] [PubMed]

Patterson, R. D.

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

Pavel, M.

Pelli, D. G.

D. G. Pelli, “Close encounters—an artist shows that size affects shape,” Science 285, 844–846 (1999).
[CrossRef] [PubMed]

J. A. Solomon, D. G. Pelli, “The visual channel that mediates 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]

G. E. Legge, D. G. Pelli, G. S. Rubin, M. M. Schleske, “Psychophysics of reading: I. Normal vision,” Vision Res. 25, 239–252 (1985).
[CrossRef]

D. G. Pelli, “Uncertainty explains many aspects of visual contrast detection and discrimination,” J. Opt. Soc. Am. A 2, 1508–1532 (1985).
[CrossRef] [PubMed]

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

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

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]

Pollehn, H.

Riedl, T.

Riedl, T. R.

Robson, J. G.

Roehrig, H.

Rohler, R.

U. Greis, R. Rohler, “Untersuchung der subjektiven Detailerkennbarkeit mit Hilfe der Ortsfrequenzfilterung,” Opt. Acta 17, 515–526 (1970).
[CrossRef]

Ross, J.

M. C. Morrone, D. C. Burr, J. Ross, “Added noise restores recognizability of coarse quantized images,” Nature 305, 226–228 (1983).
[CrossRef] [PubMed]

Rubin, G. S.

G. E. Legge, D. G. Pelli, G. S. Rubin, M. M. Schleske, “Psychophysics of reading: I. Normal vision,” Vision Res. 25, 239–252 (1985).
[CrossRef]

Sachs, M. B.

Sansbury, R. V.

J. Nachmias, R. V. Sansbury, “Grating contrast: Discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
[CrossRef] [PubMed]

Schiller, P. H.

P. H. Schiller, B. L. Finlay, S. F. Volman, “Quantitative studies of single cell properties in monkey striate cortex. III. Spatial-frequency,” J. Neurophysiol. 39, 1334–1351 (1976).
[PubMed]

Schleske, M. M.

G. E. Legge, D. G. Pelli, G. S. Rubin, M. M. Schleske, “Psychophysics of reading: I. Normal vision,” Vision Res. 25, 239–252 (1985).
[CrossRef]

Schlosberg, H.

R. S. Woodworth, H. Schlosberg, Experimental Psychology, 2nd ed. (Holt, Rinehart & Winston, New York, 1954).

Sekuler, R.

A. Pantle, R. Sekuler, “Size detecting mechanisms in human vision,” Science 162, 1146–1148 (1969).
[CrossRef]

Solomon, J. A.

J. A. Solomon, M. J. Morgan, “Reverse correlation reveals psychophysical receptive fields,” Invest. Ophthalmol. Visual Sci. 40, S572 (1999).

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

Sperling, G.

Z.-L. Lu, G. Sperling, “Second-order reversed phi,” Percept. Psychophys. 61, 1075–1088 (1999).
[CrossRef] [PubMed]

D. H. Parish, G. Sperling, “Object spatial frequencies, retinal spatial frequencies, noise, and the efficiency of letter discrimination,” Vision Res. 31, 1399–1415 (1991).
[CrossRef] [PubMed]

T. R. Riedl, G. Sperling, “Spatial-frequency bands in complex visual stimuli: American Sign Language,” J. Opt. Soc. Am. A 5, 606–616 (1988).
[CrossRef] [PubMed]

M. Pavel, G. Sperling, T. Riedl, A. Vanderbeek, “Limits of visual communication: the effect of signal-to-noise ratio on the intelligibility of American Sign Language,” J. Opt. Soc. Am. A 4, 2355–2365 (1987).
[CrossRef] [PubMed]

G. Sperling, “The information available in brief visual presentations,” Psychol. Monogr. 11, 1–74 (1960).
[CrossRef]

G. Sperling, B. Dosher, “Strategy and optimization in human information processing,” in Handbook of Perception and Performance, K. Boff, L. Kaufmon, J. Thomas, eds. (Wiley, New York, 1986), Vol. 1, Chap. 2, pp. 1–65.

Stromeyer, C. F.

C. F. Stromeyer, B. Julesz, “Spatial-frequency masking in vision: critical bands and spread of masking,” J. Opt. Soc. Am. 64, 1221–1232 (1972).
[CrossRef]

Swets, J. A.

J. A. Swets, D. M. Green, W. P. Tanner, “On the width of critical bands,” J. Acoust. Soc. Am. 34, 108–113 (1962).
[CrossRef]

Tanner, W. P.

J. A. Swets, D. M. Green, W. P. Tanner, “On the width of critical bands,” J. Acoust. Soc. Am. 34, 108–113 (1962).
[CrossRef]

Thomas, J. P.

Thorell, L. G.

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

Tolhurst, D. J.

D. J. Tolhurst, R. S. Dealey, “The detection and identification of lines and edges,” Vision Res. 15, 1367–1372 (1975).
[CrossRef] [PubMed]

Vanderbeek, A.

Volman, S. F.

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Implemented in Matlab 5.3, the procedure for a given model consisted of the following. (1) For a given set of the model parameters, using Eq. (21) to compute log(cτtheory)from the model for each external-noise condition at three different performance levels. (2) Computing the squared difference between the log threshold prediction from the model and the observed sqdiff=[log(cτ theory)-log(cτ)]2for each threshold. The log approximately equates the standard error over large ranges in contrast thresholds, corresponding to weighted least squares, an equivalent to the maximum likelihood solution for continuous data. In the current data set, this assumption is true. (3) Computing L: summation of sqdiff from all the thresholds across all the external-noise conditions. (4) Using a gradient-descent method to adjust the model parameters to find the minimum L. (5) After obtaining the minimum L, computing the r2statistic to evaluate the goodness of the model fit: (25)r2=1.0-∑[log(cτtheory)-log(cτ)]2∑{log(cτ)-mean[log(cτ)]}2,where Σ and mean( )run over all the thresholds for a particular observer. An Ftest for nested models was used to statistically compare the four models. An Fis defined: (26)F(df1, df2)=[(rfull2-rreduced2)/df1]/[(1-rfull2)/df2],where df1=kfull-kreduced,and df2=N-kfull.The kvariables are the number of parameters in each model, and Nis the number of predicted data points.

Full sets of maximum-likelihood fits were performed on the psychometric functions for all four models: uPTM, cPTM, uLAM, and cLAM. For an observer who is correct in Kijtrials among a total of Nijtrials in the jthsignal contrast and ithfilter condition, the likelihood of a model that predicts a fraction of Pijcorrect in each condition is defined as (27)likelihood=∏i=1I∏j=1JNij!Kij!(Nij-Kij)! PijKij(1-Pij)Nij-Kij,where Pijis defined by Eq. (22). Asymptotically, χ2(df )statistics could be used to compare the proper set of models:(28)χ2(df )=2.0×loglikelihoodfulllikelihoodreduced,where dfis the difference of the number of parameters between the full and the reduced models.

F. A. Wichmann, N. J. Hill, “The psychometric function II: bootstrap based confidence intervals and sampling” Percept. Psychophys. (to be published).

For an excellent discussion on fitting psychometric functions, see F. A. Wichmann, N. J. Hill, “The psychometric function I: fitting, sampling and goodness-of-fit,” Percept. Psychophys. accepted for publication.

A resampling method (Refs. 84, 85) was used to compute the standard deviation of each threshold. We assumed that the number of correct responses at each signal contrast level on every psychometric function has a binomial distribution with a single-even probability p, which is the measured percent correct. We then generated a theoretically resampled psychometric function for a given condition by independently replacing the number of correct responses at each signal stimulus contrast on the psychometric function with a sample from the corresponding binomial distribution. Repeating this process 2000 times, we generated 2000 theoretically resampled psychometric functions in every external-noise condition. We estimated the standard deviation for each threshold by fitting Weibull to these theoretically resampled psychometric functions and computing the standard deviation of the 2000 resampled thresholds for each external-noise condition.

This approximation becomes exact if (i) Ts(fx, fy)=Ts(f ) and Ss(fx, fy)=Ss(f ), that is, the template and the signal are radially symmetric in Fourier space; or (ii) Ts(fx, fy)= kSs(fx, fy),∀fx,fy; that is, the template is perfectly matched to the signal stimulus; or (iii) either the template or the signal stimulus (or both) are uniform for every radius where the template and the signal overlap in Fourier space. In the current application, condition (iii) holds because the stimulus is a pair of points in Fourier space, and condition (ii) is approximately true because humans tend to use nearly optimal templates in simple stimulus situations.60 If one or more of these conditions holds approximately, then the equations should provide reasonable approximations.

This approximation becomes exact if (i) Ts(fx, fy)=Ts(f ) and F(fx, fy)=F(f ); that is, the template and the experimenter-applied filter are radially symmetric in Fourier space; or (ii) Ts(fx, fy)=kF(fx, fy),∀fx,fy; that is, the template is matched to the noise filter; or (iii) either the template or the noise filter (or both) are uniform for every radius where the template and the noise filter overlap in Fourier space. In the current application, F(fx, fy)= F(f )=constant, the expected spectrum of the Gaussian noise is uniform, and condition (iii) is met. In most applications it will be possible to construct filters such that condition (iii) holds.

In the current development, cross products in the form (β2c2+Next2)γare eliminated in order to yield analytical solutions. The effects of the cross terms have been evaluated in two of our previous publications. In one study,59PTMs with full cross-product forms were fit to the data by methods of iterative solution. The results were equivalent in pattern to those from fits of PTMs without cross products, and the cross-product terms were small. In the other study,72the analytical PTMs without cross products were compared with full stochastic PTMs. The analytical form was found to be a good approximation of the stochastic model.

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Figures (4)

Fig. 1
Fig. 1

a, Noisy PTM. There are five major components: (1) a perceptual template, (2) nonlinear transducer functions, (3) a multiplicative internal-noise source, (4) an additive internal-noise source, and (5) a decision process. A good example of a perceptual template is a spatial-frequency filter F(f ), with a center frequency and a bandwidth such that a range of frequencies adjacent to the center frequency pass through with smaller gains. The nonlinear transducer function takes the form of an expansive power function. Limitations of human observers are modeled as equivalent internal noise. Multiplicative noise is an independent noise source whose amplitude is proportional to the (average) amplitude of the output from the perceptual template. Additive internal noise is another noise source whose amplitude does not vary with signal strength. Both multiplicative and additive noise is added to the output from template matching, and the noisy signal is the input to a task-appropriate decision process. b, threshold versus contrast functions from a PTM model. The signal contrast necessary to achieve different criteria (d=1.0, 1.4, 2.0) is a function of the contrast of external noise.

Fig. 2
Fig. 2

a, Two-dimensional low-pass spatial-frequency filters with seven different passbands. b, two-dimensional high-pass spatial-frequency filters with seven different passbands. c, examples of low-pass-filtered noise. From left to right, the examples resulted from filtering a white Gaussian noise image through the seven filters in a. d, examples of high-pass-filtered noise. From left to right, the examples result from filtering white Gaussian noise images through the seven filters in b. e, a Gabor template in real space. f, the spatial-frequency power spectrum of the Gabor template in e. g, the gain (amplification factor applied to each spatial frequency) of the template in e as a function of spatial-frequency (distance from the origin in f). h, the amount of noise energy passed through the perceptual template in e, f, and g as a function of the passband of the low-pass filters (increasing with cutoff frequency) in a and the passband of the high-pass filters (decreasing with cutoff frequency) in b. i, the predicted contrast thresholds as functions of the passband of low-pass filters (increasing with cutoff frequency) in a and the passband of the high-pass filters (decreasing with cutoff frequency) in b at three performance levels for a perceptual template model with its template described in e, f, and g.

Fig. 3
Fig. 3

A trial in the two-interval forced-choice Gabor detection task. A trial consists of two intervals. The first interval starts with a 250-ms fixation cross. In the next three stimulus frames, each lasting 8.3 ms, one external-noise frame, one signal/blank frame, and another noise frame, appear in the center of the display. The second interval starts 500 ms after the end of the first. Only one of the two intervals contains a signal Gabor. The method of constant stimuli was used to measure psychometric functions at seven different contrast levels.

Fig. 4
Fig. 4

First row, contrast threshold as function of passband of the low-pass and high-pass filters for three observers at 70% correct performance level. The curves were from fits to the PTM. Second row, the best-fitting spatial-frequency sensitivity of the perceptual templates (estimated from fitting the TVF functions). The arrows on the x axis indicate the center frequency of the Gabor stimulus. Third row, the best-fitting spatial-frequency sensitivity of the perceptual templates (estimated from fitting the full psychometric functions). The arrows on the x axis indicate the center frequency of the Gabor stimulus.

Tables (1)

Tables Icon

Table 1 Parameters of the Perceptual Template Model That Best Fits the TVFs a

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

T(.)=T(fx, fy, t)=Ts(fx, fy)Tt(t),
Ts2(fx, fy)dfxdfy=1.0.
S(fx, fy, t)=cSs(fx, fy)St(t),
Next(fx, fy, t)=Next,s(fx, fy)Next,t(t)=σextG(fx, fy)F(fx, fy)Next,t(t),
S1=T(fx, fy, t)S(fx, fy, t)dfxdfydt
=cTs(fx, fy)Ss(fx, fy)dfxdfyTt(t)St(t)dt.
S12πcTs(f )Ss(f )fdfTt(t)St(t)dt.
N1=T(fx, fy, t)Next(fx, fy, t)dfxdfydt
=Ts(fx, fy)Next,s(fx, fy)dfxdfyTt(t)Next,t(t)dt
=σextTs(fx, fy)G(fx, fy)F(fx, fy)dfxdfy×Tt(t)Next,t(t)dt.
σN12=σext2Ts2(fx, fy)F2(fx, fy)dfxdfy×Tt(t)Next,t(t)dt2.
σN122πσext2Ts2(f )F2(f )fdf Tt(t)Next,t(t)dt2.
α=2πTt(t)St(t)dt,
2πTt(t)Next,t(t)dt2=1.0.
S1=αcTs(f )Ss(f )fdf,
σN12=σext2Ts2(f )F2(f )fdf.
Filowpass(f )=j=1iAi(f ).
Fihighpass(f )=j=N-i-1NAj(f ).
T(j)=Ts(f )Aj(f )fdffj2-fj-12.
S(j)=Ss(f )Aj(f )fdffj2-fj-12.
S1=acj=1N[T(j)S(j)(fj2-fj-12)]
σlowpass2(i)=σext2j=1iTs2(f )Aj2(f )fdf=σext2j=1i[T2(j)(fj2-fj-12)2].
σhighpass2(i)=σext2j=N-i-1NTs2(f )Aj2(f )fdf=σext2j=N-i-1N[T2(j)(fj2-fj-12)2].
S2=S1γ,
Varext=σpassband2γ(i).
Vartotal=Varext+Varmul+Varadd
=σpassband2γ(i)+Nmul2[S22+σpassband2γ(i)]+Nadd2
=(1+Nmul2)σpassband2γ(i)+Nmul2S12γ+Nadd2.
d=S2(Vartotal)1/2=S1γ[(1+Nmul2)σpassband2γ(i)+Nmul2S12γ+Nadd2]1/2,
cr(i)
=1αT(j)S(j)(fj2-fj-12)×(1+Nmul2)T2(j)(fj2-fj-12)2γ+Nadd21/d2-Nmul21/2γ.
PercentCorrect=-+g(x-d)G(x)M-1dx,
l(x, y)=l01.0+cpeaksin(2πfx)exp-x2+y22σ2,
cpeaksin(2πfx)exp-x2+y22σ2.
Pc=max-(max-0.5)×2-(c/ρ)η
r2=1.0-[log(cτtheory)-log(cτ)]2{log(cτ)-mean[log(cτ)]}2,
F(df1, df2)=[(rfull2-rreduced2)/df1]/[(1-rfull2)/df2],
likelihood=i=1Ij=1JNij!Kij!(Nij-Kij)! PijKij(1-Pij)Nij-Kij,
χ2(df )=2.0×loglikelihoodfulllikelihoodreduced,

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