Abstract

Existing observer models developed for studies with the external noise paradigm are strictly applicable only to target detection or identification/discrimination of orthogonal target(s). We elaborated the perceptual template model (PTM) to account for contrast thresholds in identifying nonorthogonal targets. Full contrast psychometric functions were measured in an orientation identification task with four orientation differences across a wide range of external noise levels. We showed that observer performance can be modeled by the elaborated PTM with two templates that correspond to the two stimulus categories. Sampling efficiencies of the human observers were also estimated. The elaborated PTM provides a theoretical framework for characterizing joint feature and contrast sensitivity of human observers.

© 2009 Optical Society of America

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2008

Z. L. Lu and B. A. Dosher, “Characterizing observers using noise and observer models: assessing internal representations with external noise,” Psychol. Rev. 115, 44-82 (2008).
[CrossRef] [PubMed]

2006

J. A. Solomon and M. J. Morgan, “Stochastic re-calibration: contextual effects on perceived tilt,” Proc. R. Soc., London, Ser. B 273, 2681-2686 (2006).
[CrossRef]

2004

B. A. Dosher, S.-H. Liu, N. Blair, and Z.-L. Lu, “The spatial window of the perceptual template and endogenous attention,” Vision Res. 44, 1257-1271 (2004).
[CrossRef] [PubMed]

2003

X. Li, Z.-L. Lu, P. Xu, J. Jin, and Y. Zhou, “Generating high gray-level resolution monochrome displays with conventional computer graphics cards and color monitors,” J. Neurosci. Methods 130, 9-18 (2003).
[CrossRef] [PubMed]

2002

L. L. Kontsevich, C. C. Chen, and C. W. Tyler, “Separating the effects of response nonlinearity and internal noise psychophysically,” Vision Res. 42, 1771-1784 (2002).
[CrossRef] [PubMed]

2001

F. A. Wichmann and N. J. Hill, “The psychometric function: I. Fitting, sampling, and goodness of fit,” Percept. Psychophys. 63, 1293-1313 (2001).
[CrossRef]

A. Gorea and D. Sagi, “Disentangling signal from noise in visual contrast discrimination,” Nat. Neurosci. 4, 1146-1150 (2001).
[CrossRef] [PubMed]

Z.-L. Lu and B. A. Dosher, “Characterizing the spatial-frequency sensitivity of perceptual templates,” J. Opt. Soc. Am. A 18, 2041-2053 (2001).
[CrossRef]

2000

L. Itti, C. Koch, and J. Braun, “Revisiting spatial vision: toward a unifying model,” J. Opt. Soc. Am. A 17, 1899-1917 (2000).
[CrossRef]

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

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

P. Vazquez, M. Cano, and C. Acuna, “Discrimination of line orientation in humans and monkeys,” J. Neurophysiol. 83, 2639-2648 (2000).
[PubMed]

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

1999

1998

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

L. Kiorpes and J. A. Movshon, “Peripheral and central factors limiting the development of contrast sensitivity in Macaque monkeys,” Vision Res. 38, 61-70 (1998).
[CrossRef] [PubMed]

M. D'Zmura and K. Knoblauch, “Spectral bandwidths for the detection of color,” Vision Res. 38, 3117-3128 (1998).
[CrossRef]

1997

1995

B. S. Tjan, W. L. Braje, G. E. Legge, and D. Kersten, “Human efficiency for recognizing 3-D objects in luminance noise,” Vision Res. 35, 3053-3069 (1995).
[CrossRef] [PubMed]

1994

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

J. M. Foley, “Human luminance pattern-vision mechanisms: masking experiments require a new model,” J. Opt. Soc. Am. A 11, 1710-1719 (1994).
[CrossRef]

1993

P. Makela, D. Whitaker, and J. Rovamo, “Modelling of orientation discrimination across the visual field,” Vision Res. 33, 723-730 (1993).
[CrossRef] [PubMed]

S. J. Waugh, D. M. Levi, and T. Carney, “Orientation, masking, and vernier acuity for line targets,” Vision Res. 33, 1619-1638 (1993).
[CrossRef] [PubMed]

1992

1991

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

V. M. Richards, L. M. Heller, and D. M. Green, “The detection of a tone added to a narrow band of noise: the energy model revisited,” Q. J. Exp. Psychol. 43, 481-501 (1991).
[CrossRef]

1990

S. F. Bowne, “Contrast discrimination cannot explain spatial frequency, orientation or temporal frequency discrimination,” Vision Res. 30, 449-461 (1990).
[CrossRef] [PubMed]

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

R. Vogels and G. A. Orban, “How well do response changes of striate neurons signal differences in orientation: a study in the discriminating monkey,” J. Neurosci. 10, 3543-3558 (1990).
[PubMed]

1989

B. G. Smith and J. P. Thomas, “Why are some spatial discriminations independent of contrast?” J. Opt. Soc. Am. A 6, 713-724 (1989).
[CrossRef] [PubMed]

L. E. Humes and W. Jesteadt, “Models of the additivity of masking,” J. Acoust. Soc. Am. 85, 1285-1294 (1989).
[CrossRef] [PubMed]

W. S. Geisler, “Sequential ideal-observer analysis of visual discriminations,” Psychol. Rev. 96, 267-314 (1989).
[CrossRef] [PubMed]

1988

W. M. Hartmann and J. Pumplin, “Noise power fluctuations and the masking of sine signals,” J. Acoust. Soc. Am. 83, 2277-2289 (1988).
[CrossRef] [PubMed]

A. E. Burgess and B. Colborne, “Visual signal detection. IV. Observer inconsistency,” J. Opt. Soc. Am. A 5, 617-627 (1988).
[CrossRef] [PubMed]

1987

G. E. Legge, D. Kersten, and A. E. Burgess, “Contrast discrimination in noise,” J. Opt. Soc. Am. A 4, 391-404 (1987).
[CrossRef] [PubMed]

A. B. Watson, “Estimation of local spatial scale,” J. Opt. Soc. Am. A 4, 1579-1582 (1987).
[CrossRef] [PubMed]

A. J. Ahumada, “Putting the visual system noise back in the picture,” J. Opt. Soc. Am. A 4, 2372-2378 (1987).
[CrossRef] [PubMed]

A. Bradley, B. C. Skottun, I. Ohzawa, G. Sclar, and R. D. Freeman, “Visual orientation and spatial frequency discrimination: a comparison of single neurons and behavior,” J. Neurophysiol. 57, 755-772 (1987).
[PubMed]

B. C. Skottun, A. Bradley, G. Sclar, I. Ohzawa, and R. D. Freeman, “The effects of contrast on visual orientation and spatial frequency discrimination: a comparison of single cells and behavior,” J. Neurophysiol. 57, 773-786 (1987).
[PubMed]

1985

1984

1982

R. L. De Valois, E. William Yund, and N. Hepler, “The orientation and direction selectivity of cells in macaque visual cortex,” Vision Res. 22, 531-544 (1982).
[CrossRef] [PubMed]

1981

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

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This logic could be extended to consider more than two templates with an appropriate decision rule for identification tasks with more than two stimuli.

Dosher and Lu (2000) showed that the stochastic PTM exhibits all the key characteristics derived for the simplified (analytic) PTM. In general, the analytic PTM is a close approximation to the stochastic PTM and provides a good approach to model testing: The (analytic) PTM fits all the data we have collected very well. In the special case when γ = 1.0, the (analytic) PTM is identical to the stochastic PTM. In the two extreme regions of the external noise manipulation, i. e., when internal additive noise dominates or when external noise dominates, the (analytic) PTM model approaches the stochastic model asymptotically.

In the ePTM development, the external noise in the stimulus had a Gaussian distribution, corresponding to white external noise. After nonlinear transduction, the distribution of the external noise might deviate from the Gaussian distribution. Spatial and temporal summation in the perceptual system should reduce this deviation. When combined with additive and multiplicative noises, both of which are Gaussian distributed, we assume that the sum of the noises is approximately Gaussian. However, we restrict ourselves to performance levels below 90% so as to avoid the tails of the distribution. The Gaussian assumption is not central to the development of the PTM outlined above, but it does simplify the application to signal detection estimation: the Gaussian noise distribution allows us to use the Gaussian form of signal detection calculations.

For a Gaussian random variable R with mean 0 and standard deviation NextσTN, the standard deviation of sign(R)abs(R)γ1 is Nextγ1σTNγ1Fγ1(γ1)...F(γ1)=1.00, 1.07, 1.14, 1.20, 1.26, 1.32, 1.37, 1.42, 1.47, 1.52, and 1.57, for = 1.0, 1.2, ..., 3.0.

JS could not perform better than 85% correct in the highest external noise condition in some of the orientation difference conditions. We excluded his data in that condition in all the analysis.

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Figures (6)

Fig. 1
Fig. 1

Examples of a simple Gabor orientation identification task in the (a) contrast and (b) feature domain.

Fig. 2
Fig. 2

Full psychometric functions in all the experimental conditions. Smooth curves represent the best fitting Weibull functions.

Fig. 3
Fig. 3

TvC funcitons at 65%, 75%, and 85% performance levels in the four orientation difference conditions. Smooth curves represent the best-fitting ePTM. Error bars denote one standard deviation.

Fig. 4
Fig. 4

Schematic representations of the original PTM and the ePTM. In the ePTM, two detectors, one better matched to the signal stimulus in a given trial (with gain β B ) and the less-well-matched to the signal stimulus (with gain β W ), are used to model identification of two nonorthogonal targets.

Fig. 5
Fig. 5

Schematic representation of the perceptual template based on normalized β W .

Fig. 6
Fig. 6

TvC functions at 65%, 75%, and 85% performance levels in the four orientation difference conditions, plotted as squared contrast thresholds versus variance of external noise for the three human observers (first three rows) and the ideal observer (last row). The lines represent the results of the linear regression analysis.

Tables (4)

Tables Icon

Table 1 Parameters of the Best-Fitting Weibull Functions

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Table 2 Parameters of the Best-Fitting ePTM

Tables Icon

Table 3 Slopes and Intercepts of the Squared Threshold Contrast versus External Noise Variance Functions

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Table 4 Sampling Efficiencies of the Human Observers

Equations (36)

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L ( x , y ) = L 0 ( 1.0 + c sin ( 2 π f ( x cos θ + y sin θ ) ) ) e [ ( x 2 + y 2 ) 2 σ 2 ] ,
P ( c ) = ξ + ( 1 ξ λ ) ( 1 e ( c τ ) η ) ,
likelihood = N i ! K i ! ( N i K i ) ! P i K i ( 1 P i ) N i K i ,
χ 2 ( d f ) = 2 log ( max likelihood full max likelihood reduced ) ,
S ( x , y , t ) = c S 0 ( x , y , t ) + N ext g ( x , y , t ) ,
Y B 1 = T B ( x , y , t ) S ( x , y , t ) d x d y d t = c T B ( x , y , t ) S 0 ( x , y , t ) d x d y d t + N ext T B ( x , y , t ) g ( x , y , t ) d x d y d t ,
Y W 1 = T W ( x , y , t ) S ( x , y , t ) d x d y d t = c T W ( x , y , t ) S 0 ( x , y , t ) d x d y d t + N ext T W ( x , y , t ) g ( x , y , t ) d x d y d t .
Y B 1 = M B c + N ext σ T N g ̃ 1 ( 0 , 1 ) ,
Y W 1 = M W c + N ext σ T N g ̃ 2 ( 0 , 1 ) ,
Y B 2 = ( M B c ) γ 1 + N ext γ 1 σ T N γ 1 F γ 1 ( γ 1 ) g ̃ 1 ( 0 , 1 ) ,
Y W 2 = ( M W c ) γ 1 + N ext γ 1 σ T N γ 1 F γ 1 ( γ 1 ) g ̃ 2 ( 0 , 1 ) ,
β B = M B σ T N F ( γ 1 ) = T B ( x , y , t ) S 0 ( x , y , t ) d x d y d t σ T N F ( γ 1 ) ,
β W = M W σ T N F ( γ 1 ) = T W ( x , y , t ) S 0 ( x , y , t ) d x d y d t σ T N F ( γ 1 ) ,
Y B 2 = ( β B c ) γ 1 + N ext γ 1 g ̃ 1 ( 0 , 1 ) ,
Y W 2 = ( β W c ) γ 2 + N ext γ 2 g ̃ 2 ( 0 , 1 ) .
σ m B 2 = N m 2 [ N ext 2 γ 2 + ( β B c ) 2 γ 2 ] .
σ m W 2 = N m 2 [ N ext 2 γ 2 + ( β W c ) 2 γ 2 ] .
Y B 3 = ( β B c ) γ 1 + N ext γ 1 g ̃ 1 ( 0 , 1 ) + N a g ̃ 3 ( 0 , 1 ) + σ m B g ̃ 5 ( 0 , 1 ) ,
Y W 3 = ( β W c ) γ 1 + N ext γ 1 g ̃ 2 ( 0 , 1 ) + N a g ̃ 4 ( 0 , 1 ) + σ m W g ̃ 6 ( 0 , 1 ) ,
D = Y B 3 Y W 3 = [ ( β B c ) γ 1 ( β W c ) γ 1 ] + N ext γ 1 [ g ̃ 1 ( 0 , 1 ) g ̃ 2 ( 0 , 1 ) ] + N a [ g ̃ 3 ( 0 , 1 ) g ̃ 4 ( 0 , 1 ) ] + [ σ m B g ̃ 5 ( 0 , 1 ) σ m W g ̃ 6 ( 0 , 1 ) ] .
σ total 2 = 2 1 β W β B N ext 2 γ 1 + N m 2 [ 2 N ext 2 γ 2 + ( β B c ) 2 γ 2 + ( β W c ) 2 γ 2 ] + 2 N a 2 .
d = mean ( Y B 3 ) mean ( Y W 3 ) σ total 2 2 = ( β B c ) γ 1 ( β W c ) γ 1 1 β W β B N ext 2 γ 1 + N m 2 [ N ext 2 γ 2 + ( β B c ) 2 γ 2 + ( β W c ) 2 γ 2 2 ] + N a 2 .
c τ = { [ ( 1 β W β B + N m 2 ) N ext 2 γ + N a 2 ] ( β B 2 γ β W 2 γ ) d 2 N m 2 ( β B 2 γ + β W 2 γ ) 2 } 1 2 γ .
c τ 2 c τ 1 = [ ( β M 2 γ β U 2 γ ) d 1 2 N m 2 ( β M 2 γ + β U 2 γ ) 2 ( β M 2 γ β U 2 γ ) d 2 2 N m 2 ( β M 2 γ + β U 2 γ ) 2 ] 1 2 γ .
c τ = [ d 2 β B 2 ( N ext 2 + N a 2 ) ] 1 2 .
c τ 2 = d 2 β B 2 ( N ext 2 + N a 2 ) .
β B = υ β I B .
c τ 2 = d 2 υ β I B 2 ( N ext 2 + N a 2 ) = 1 υ k ( N ext 2 + N a 2 ) ,
υ = 1 a k = d 2 a β I B 2 .
β B = υ β I B .
R S S = [ log ( c τ predicted ) log ( c τ measured ) ] 2 ,
r 2 = 1.0 [ log ( c τ predicted ) log ( c τ measured ) ] 2 [ log ( c τ measured ) mean ( log ( c τ measured ) ) ] 2 ,
F ( d f 1 , d f 2 ) = ( r full 2 r reduced 2 ) d f 1 ( 1 r full 2 ) d f 2 ,
c Ideal 2 ( P c | task ) = μ Ideal ( P c | task ) N ext 2 .
c Human 2 ( P c | task , obs ) = μ Human ( P c | task , obs ) N ext 2 + b .
υ ( P c | task , obs ) = μ Ideal ( P c | task ) μ Human ( P c | task , obs ) .

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