We measured the just-noticeable difference (JND) in orientation variance between two textures ( Figure 1) as we varied the baseline (pedestal) variance present in both textures. JND's first fell as pedestal variance increased and then rose, producing a ‘dipper’ function similar to those previously reported for contrast, blur, and orientation-contrast discriminations. A dipper function (both facilitation and masking) is predicted on purely statistical grounds by a noisy variance-discrimination mechanism. However, for two out of three observers, the dipper function was significantly better fit when the mechanism was made incapable of discriminating between small sample variances. We speculate that a threshold nonlinearity like this prevents the visual system from including its intrinsic noise in texture representations and suggest that similar thresholds prevent the visibility of other artifacts that sensory coding would otherwise introduce, such as blur.

*σ*from its true value. A possible resolution of this paradox is that when we see a texture as uniform, we are not seeing the orientation of every element in the texture, but rather the output of a specialized mechanism that computes orientation variance. If stimulation of this mechanism were subject to a threshold nonlinearity, then the perceived uniformity of a uniform texture could be explained.

*luminance*contrasts is normally taken as evidence for a threshold nonlinearity (e.g., Foley & Legge, 1981; Legge & Foley, 1980). However, in the case of variance discrimination, facilitation is expected simply on the basis of intrinsic noise (Laming, 1986; Paakkonen & Morgan, 1993).

^{1}The full derivation is given in the 1, but the informal argument runs thus. Suppose, in a 2AFC experiment, the observer compares two sample variances, each of which reflects the visual system's internal noise as well as the stimulus variance. The function mapping stimulus variance to sample variance will thus have two distinct parts; a flat part, in which the stimulus variance is negligible compared to the internal noise, and a steadily increasing part, in which the internal noise is negligible. Because of the flat part, any criterion increase in sample variance will require a larger increase in stimulus variance when sample variance is low.

^{2}. The viewing distance was approximately 57 cm so that the pixel size was approximately 0.018 deg of visual angle. The texture elements were Gabor wavelets of maximum contrast. Specifically, the Weber contrast

*g*varied as a function of position

*x, y*with respect to the center of the wavelet as follows:

*λ*(the wavelength of the windowed grating) is 0.1198 deg,

*σ*(the space constant of the window) is

*λ*/2, and

*θ*gives the angle normal to grating orientation; that is,

*λ,*slightly perturbed by displacing each elements randomly in

*x*and

*y*by an amount drawn from a uniform pdf with width 1.5

*λ*. Thus, the whole array subtended approximately 3.6 deg of visual angle. The jitter was resampled between each of the two stimulus presentations on every trial.

*θ*were drawn from Gaussian probability density functions. For one of the two textures, the density had random mean and “pedestal” variance

*σ*

_{p}

^{2}. The density for the other texture had a different random mean and greater variance (

*σ*

_{p}+ Δ

*σ*)

^{2}. The mean orientation was randomized between presentations, to prevent the use of any one orientation-tuned channel by the observer, and spatial position of the elements was jittered between presentations.

*σ*at which the observer was 82% correct. There was no feedback to indicate whether the response was correct or not. The pedestal variance was randomly selected on each trial from a set of preset values. A block of trials terminated when each of these preset values had been presented 50 times. Thus, when

*σ*

_{p}∈ {0°, 1°, 2°, 4°, 8°, 16°}, as was the case for observers MM (6 blocks) and JAS (9 blocks) and IM (4 blocks), each block contained 300 trials. The four naive observers experienced only one block, with interleaved pedestal levels {0°, 1°, 2°, 4°} only.

σ _{int} | n + 1 | c | ln L | |
---|---|---|---|---|

MM no thresh | 2.87 | 9 | – | −647.10 |

MM + thresh | 2.23 | 8 | 3.16 | −642.11** |

IM no thresh | 2.49 | 5 | – | −504.83 |

IM + thresh | 0.98 | 4 | 2.86 | −495.91** |

JAS no thresh | 4.80 | 10 | – | −1010.6 |

JAS + thresh | 4.82 | 9 | 1.04 | −1010.6 (NS) |

*L*

_{C}and

*L*

_{U}be the likelihoods of the best-fitting constrained and unconstrained models. As is well-known (e.g., Hoel, Port, & Stone, 1971), under the null hypothesis that the constrained model captures the true state of the world,

*L*

_{U}). The chi-square values were significant (

*p*< .010) for observers MM and IM, but not for observer JAS. To give a more intuitive impression of the success of the two models, Figure 3 plots the relative likelihoods in comparison to two extreme baselines. The ‘coin flipping’ model has the simulated observer choose between the two intervals with equal probability, independently of the stimulus level or pedestal. This is as poor as a fit could be. The ‘Weibull fits’ model shows the best fit of a set of 2-parameter Weibull psychometric functions to the each of the pedestal conditions separately. This model has 2

*n*free parameters, where

*n*is the number of pedestals, in comparison to the 2 and 3 parameters of the models described in Table 1, and it is as good as a fit could be given the noise in the observer's data. It is satisfying to see that the models are much closer to the Weibull fits than to ‘coin flipping’. The two versions of the intrinsic noise model, with and without an additional threshold, are seen to be very close.

*σ*

_{p}+ Δ

*σ*)

^{2}and the second alternative can be described as having the smaller variance

*σ*

_{p}

^{2}. The model observer collects a sample of size

*n*+ 1 from the first interval, a sample of the same size from the second interval, and responds correctly when the variance of the former sample exceeds that of the latter. These two sample variances can be denoted by the independent random variables

*S*and

*N,*respectively. The expected response accuracy is given by the formula

*F*

_{ N}(

*x*) is the cumulative distribution function (CDF) of

*N,*and

*f*

_{ S}(

*x*) is the probability density function (PDF) of

*S*. The lower limit of integration is zero because neither

*S*nor

*N*can ever be negative.

*σ*

_{int}

^{2}. In that case,

*S*will be [(

*σ*

_{p}+ Δ

*σ*)

^{2}+

*σ*

_{int}

^{2}]/(

*n*+ 1) times a chi-square random variable (call it

*U*), having

*n*degrees of freedom; and

*N*will be (

*σ*

_{p}

^{2}+

*σ*

_{int}

^{2})/(

*n*+ 1) times an independent chi-square random variable (call it

*V*), also having

*n*degrees of freedom; and probability correct is given by the formula

*F*is the

*F*-distribution, with degrees of freedom

*n*and

*n*.

*f*

_{ X}(

*x;n*) and

*F*

_{ X}(

*x;n*) are the PDF and CDF for a chi-square random variable

*X,*with

*n*degrees of freedom; then

*f*

_{ X}(

*x*/

*a*;

*n*)/

*a*and

*F*

_{ X}(

*x*/

*a*;

*n*) will be the PDF and CDF for

*aX,*as long as

*a*> 0. Therefore, the CDF for

*N*can be written as

*S*are

*c,*below which all sample variances are indistinguishable from zero. Either the sample variance from the first interval (the one with the larger variance) could be bigger than that from the second interval, or neither sample variance could exceed the threshold and the observer makes a lucky guess. Therefore, the expected response accuracy for 2AFC would be

*F*

_{X}(

*x*), is such that

*x*) is the CDF of

*S*

_{X}

^{2}, the sample variance of

*X,*and

*x*) is the CDF of

*S*

^{2}, a same-sized sample of

*X*/

*y*=

*x*/var

*S,*and substitute into Equation A1:

*σ*

_{p}

^{2}≫

*σ*

_{int}

^{2},

*c*). In that case,

*σ*/

*σ*

_{p}. This is Weber's Law.

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