Flicker perception was investigated using two-alternative forced-choice detection and discrimination tasks with four different types of external noise: (1) broadband noise, (2) 5-Hz notched-noise—broadband noise with a 5-Hz band centered on the signal frequency removed, (3) 10-Hz notched-noise, and (4) no external noise. The signal was a burst of 10-Hz sinusoidal flicker presented in one of two observation intervals. In discrimination experiments, a pedestal—sinusoidal flicker with the same frequency, duration, and phase as the signal—was added to both observation intervals. With no noise, observers' performance first improved with increasing pedestal modulation, before deteriorating in accordance with Weber's Law, producing the typical “dipper” shaped plot of signal versus pedestal modulation. Noise affects performance, but the dipper effect persisted in each type of noise. The results exclude three models: the ideal-observer in which the pedestal improves performance by specifying the signal exactly; off-frequency-looking models in which the dipper depends on detection by channels tuned to temporal frequencies different from that of the signal; and strict energy detectors. Our data are consistent with signal processing by a single mechanism with an expansive non-linearity for near-threshold signal modulations (with an exponent of six) and a compressive “Weberian” non-linearity for high modulations.

- The effect is the result of a specific nonlinear transducer function (e.g., Foley & Legge, 1981; Legge & Foley, 1981; Nachmias & Sansbury, 1974), such that the early part of the function is accelerating and the later part decelerating. The accelerating portion generates the dipper, because the difference in output between signal-plus-pedestal and the pedestal alone is larger than the difference in output between the signal alone and no signal, while the decelerating portion produces Weber's Law by compression. In some versions, the deceleration is produced by a divisive gain control (e.g., Boynton & Foley, 1999; Foley, 1994).
- The effect is due to a specific nonlinear transducer function combined with a signal-dependent internal noise (e.g., Green, 1967; Kontsevich, Chen, & Tyler, 2002), such that the accelerating nonlinearity produces the dipper at low pedestal levels, while the noise produces Weber's Law at high levels.

- Perhaps the most radical proposal is that the effect, in spatial vision at least, is due not to the characteristics of a single mechanism but to the pooled characteristics of many mechanisms with non-linear transducer functions that are insufficient in themselves to produce substantial dippers. The dipper is assumed to be produced by the recruitment of mechanisms that are mistuned away from the signal and pedestal as the pedestal contrast first increases (Goris, Wichmann, & Henning, 2009; Henning & Wichmann, 2007). We refer to these models as the “off-frequency-looking” model.

^{2}, which was sufficient to guarantee rod saturation.

*alone*should be detected with performance levels of 60% and 90%, respectively. In the transition region, the performance results from a mixture of detection-like trials, in which flicker with the temporal and spatial characteristics of the signal is seen in only one observation interval, and discrimination-like trials in which that flicker is seen in both intervals and the interval containing the more pronounced flicker (or flicker more like that of the signal) is chosen as having contained the signal. This region in Figure 4 is very small and corresponds, in effect, to the width of the psychometric function relating the percentage of correct responses to the depth of signal modulation in the absence of a pedestal.

*M*is the pedestal modulation, Δ

*M*is the added signal modulation,

*m*is the slope and

*c*the intercept. All three observers produce results of the form of Figure 4 in the condition with no external noise. In all cases, the intercepts,

*c,*are close to zero. The largest 95% confidence interval for the intercept, −0.030 to 0.029, was for the 90% performance contour for observer HES; all the remaining confidence intervals were within 0.01 of zero. This result is important, because it implies that in the regions in which performance can be described by Equation 1 it is governed, as in many discrimination tasks, by something like Weber's law; and it also means that the Weber fraction, Δ

*M*/

*M,*can be extracted from the slopes of the linear fits. Rearranging Equation 1 with

*c*= 0 gives:

Observer | % | Weber Fraction | % | “Intercept” |
---|---|---|---|---|

GBH | 60 | 0.091 | 60 | −.00155 |

75 | 0.139 | 75 | .00101 | |

90 | 0.176 | 90 | .00696 | |

HES | 60 | 0.089 | 60 | .00007 |

75 | 0.208 | 75 | −.00077 | |

90 | 0.348 | 90 | −.00089 | |

AS | 60 | 0.127 | 60 | −.00414 |

75 | 0.183 | 75 | −.00161 | |

90 | 0.218 | 90 | .00604 | |

Average | 60 | 0.102 | 60 | −.00187 |

75 | 0.177 | 75 | −.00046 | |

90 | 0.247 | 90 | .00403 |

*M*with Δ

*M*+

*M*at low pedestal modulations is equivalent to a slope of −1 in the logarithmic TvC plots, whereas the linear growth of Δ

*M*with Δ

*M*+

*M*with zero intercept at high pedestal modulations is equivalent to a slope of +1 in the logarithmic TvC plots. Neither graphical representation explains the data; any model that fits the underlying psychometric functions must have the characteristics of the data in both types of figure.

*in general*. However, off-frequency looking across spatial-frequency channels cannot be excluded by the results of our experiment.

*M*(in our case, the pedestal modulation) and the difference in magnitude, Δ

*M*(in our case, the added signal modulation) that is needed to make the combined modulation

*M*+ Δ

*M*just noticeably different from

*M*. In its simplest form, the Weber relation is Δ

*M*=

*wM,*where the proportionality constant,

*w,*is called the Weber fraction.

*R,*where Δ

*R*= Δ

*M*/

*M,*Equation 3 follows by integration:

_{e}(

*M*′) = −

*C*(where

*M*′ is the value of

*M*for which

*R*(

*M*) = 0), Equation 3 becomes:

*M*′ = 0.1, with

*M*on a linear scale in the upper panel and on a logarithmic scale in the lower panel. Now Δ

*R,*the difference in

*R*corresponding to a just noticeable stimulus difference Δ

*M*=

*wM,*is always log

_{e}(1 +

*w*), independent of

*M,*as shown in the derivation of Equation 5:

*R*(

*M*) = log

_{e}(

*M*/

*M*′) by the addition of Gaussian noise having a standard deviation,

*σ,*independent of

*R*(i.e., we assume the internal noise after the transducer is constant), then equal differences in

*R*(and correspondingly equal fractional increases in

*M*) will be detected with equal reliability whatever the starting value of

*R*.

*w*is set by the noise standard deviation

*σ,*which has the same units as

*R*and can be thought of as the equivalent root-mean-squared (r.m.s.) variation in the stimulus modulation from observation-interval to observation-interval, expressed as a fraction of the mean modulation

*M*. The difference in

*R*between two intervals with modulations

*M*and (1 +

*w*)

*M*is distributed with standard deviation √2

*σ*around its mean of log

_{e}(1 +

*w*), which is approximately

*w*when

*w*is small. Referring this to the cumulative Gaussian distribution,

*σ*is equal to the Weber fraction

*w*for a criterion of 76% correct 2AFC performance.

*R*(

*M*) as defined above decreases smoothly toward zero as the modulation

*M*decreases to

*M*′. But when

*M*is less than

*M*′,

*R*becomes negative, and it becomes increasingly negative without limit as

*M*approaches zero (as indicated by the black curve of Figure 6). Fechner (1860) dealt with this unwelcome feature of the log transform by suggesting that the negative values of

*R*correspond to ‘unconscious sensations’ that are all introspectively equivalent to one another, since none are consciously registered. As Fechner's contemporaries were quick to point out (e.g., Müller, 1878), a simple and natural alternative proposal is that the sensory response

*R*simply remains zero for all

*M*<

*M*′. With this assumption, Fechner's log transform is truncated, replacing the negative values by zero (i.e., the lower-most blue line in Figure 6). The threshold modulation for eliciting a nonzero response,

*M*′, divides the response-modulation function into two regions. Below

*M*′ the response is zero, above

*M*′ it is positive and logarithmically compressed (though, approximately linear just above threshold where M is not much greater than

*M*′):

*M*′ in Equation 7 provides a Fechnerian basis for the dipper. All subthreshold modulations

*M*£

*M*′ yield the same (zero) response, so pedestal and signal modulations that by themselves produce zero response can combine to produce a modulation that is discriminable from the (zero) response generated by the pedestal alone.

*R*is contaminated by additive Gaussian internal noise of fixed variance, Equation 7 predicts performance in our experiments fairly well. Figure 7 shows the data for observer HES (replotted from the center panel of Figure 1) and the solid lines show the performance contours predicted by the model, and fitted with

*M*′ and

*σ*as free parameters, estimated iteratively by using MATLAB's

*fminsearch*function (based on the Nelder-Mead algorithm) to minimize the mean squared error of prediction in log

_{e}(

*M*). On each iteration, Equation 7 was used to evaluate the mean response,

*R*

_{ p}for each experimental pedestal modulation,

*M*

_{ p}(assuming the trial value for

*M*′); the mean signal-plus-pedestal response required for criterion discrimination performance was then obtained as

*R*

_{ crit}=

*R*

_{ p}+ √2

*σz*

_{ crit}, where

*z*

_{ crit}is the standard normal deviate corresponding to the criterion percent correct, respectively 0.253, 0.674 and 1.282 for 60%, 75% and 90% correct responses. Equation 7 was inverted to determine the total modulation of signal and pedestal

*M*

_{ crit}needed for the response

*R*

_{ crit}, and then the required signal modulation

*M*

_{ s}was obtained as

*M*

_{ s}=

*M*

_{ crit}−

*M*

_{ p}.

*M*′ = 0.0075, and

*σ*= 0.118.

*M*=

*wM*implies discriminative capacity that improves without limit as background magnitude is decreased, contrary to observation. For many discrimination tasks, where no dipper is observed, a modified form of Weber's Law applies: the detectable stimulus increment has a progressive, linear relation to the combination of background stimulus magnitude

*M*and a constant

*M*′ to which it is added:

*M*′ is no longer the stimulus associated with zero response. In early discussions of intensity discrimination (Delboeuf, 1873),

*M*′ was regarded as the equivalent intensity of an effective background stimulus or ‘intrinsic light,’ always present and added to any external stimulus.

*M*+

*M*′) for

*M*in Fechner's logarithmic formula, yielding:

*dR*/

*dM*to increase with various degrees of smoothness in the near-threshold range:

*n*adjusts the “hardness” of the threshold while

*M*′ no longer necessarily corresponds to intrinsic light. Equations 7, 9 and 10 are asymptotically equivalent. The family of curves plotted with blue lines in Figure 6 show

*R*as a function of

*M*using Equation 10, for different values of the parameter

*n*.

*R*; (ii) fixed internal noise added to

*R*; and (iii) a decision mechanism. The shape of the predicted TvC function is strongly determined by the form of the response function provided the noise that limits the observers' behavior does not precede the nonlinearity (Lasley & Cohn, 1981; Peterson & Birdsall, 1953) and the dipper is typically modeled, as it is here, by assuming a response nonlinearity that is accelerative in the region of

*M*′. In Equation 10, just as in Equation 7,

*M*′ is in that sense the “threshold” modulation, even though in Equation 10, a stimulus less than

*M*′ can elicit a response, and may be detectable without a pedestal if

*w*< 1.

*n*in Equation 10. Softening the assumed threshold nonlinearity in Equation 10 rounds off and slightly elevates the bottom of the dipper, and also increases the predicted separation of the performance contours when the pedestal is sub-threshold or absent. With no pedestal, and small

*M,*the contour spacing in a logarithmic plot is reduced when

*n*is high, since the more accelerated the response function, the less is the change in stimulus modulation needed for a criterion change in response. But for pedestal modulations

*M*≫

*M*′, where Weber's Law applies (at least asymptotically) for any

*n,*the signal modulation must increase the natural log of the total modulation by √2

*σz*

_{ crit}, making the contour spacing wider and independent of

*n*.

*M*′,

*σ*and

*n*) were varied iteratively for a best (least-squares) fit. The best fitting values of

*n*were strikingly high (8, 7, and 5 for HES, AS and GBH, respectively), implying a very abrupt “threshold” nonlinearity. The large values of

*n*that were required to fit the data illustrate the common failure of energy detectors (

*n*= 2) to fit data of the sort we obtained (Wichmann, 1999). A value of

*n*= 2 generates predictions that are obviously inaccurate (0.14 r.m.s. error in log

_{10}modulation) in two respects: the dipper is clearly too shallow, and the spread between high and low criteria when no pedestal is present is too wide.

*E*from the amplitudes of the 200 Fourier components of the stimulus, it can be obtained directly as the sum of the squares of the time-varying excursion in relative luminance:

*E*is proportional to the square of the signal amplitude, which is half the square of the modulation depth

*M*in Equations 7, 9 and 10. Those equations can therefore be restated in terms of

*E*/

*E*′ instead of

*M*/

*M*′, with

*E*=

*M*

^{2}/2 and the exponent

*n*replaced by

*n*/2, so that the best fitting exponent of

*n*= 6 becomes

*n*= 3 thus:

*E*£

*E*′) to a power-law (in this case a cubic) transform. But precise squaring of the deviations before integration is not critical to the predictions of energy-detection schemes, so long as the model prevents cancellation of positive and negative deviations. The energy detector is in this sense representative of a family of ‘

*rectified transient*’ detectors. When only external noise has to be considered, all detectors that base decisions on a monotonic function of energy perform equivalently (Lasley & Cohn, 1981; Peterson & Birdsall, 1953) and are effectively energy detectors. But if significant noise is added after the non-linearity the exponent in Equation 12 becomes critical. As noted above, linearity with energy (an exponent of 2 in Equation 10, or 1 in Equation 12) does not yield visually acceptable fits; linearity with modulation (halving the exponent) is even worse, predicting (in the absence of external noise) no dipper at all; but an energy detector with cubic response growth (Equation 12) gives a good account of our results without external noise. We consider next whether the energy-cubed model can predict performance with external noise as well.

*E*on any trial, expressed as a multiple of the expected energy of each noise component, is a sample from the chi-square distribution with the degrees of freedom equal to the number of independent noise components (e.g., 200 for the no-notch noise). When a signal or pedestal is present, the flicker energy is a sample from the non-central chi-square distribution, where the non-centrality parameter is the energy due to the sum of pedestal and signal. For each simulated presentation, the stimulus energy was generated by a random draw from the appropriate distribution, and the resulting response was obtained from Equation 12. Independent Gaussian internal noise of standard deviation

*σ*was then added to the responses for each of the two presentations in a simulated 2AFC trial, and the decision was counted as correct if the response to the signal presentation was greater than to the no-signal presentation. We adopted the values for

*M*′ and

*σ*that best fit the no-noise data for each subject, and a threshold hardness exponent

*n*= 3 in accordance with Equation 12. Simulations were run on a range of test modulations spanning the full range of the psychometric function, with 10000 simulated trials per test modulation per pedestal, and the test modulations required for criterion performance were estimated by interpolation. For observers GBH, HES and AS, the best-fitting values of

*M*′ were 0.0165, 0.0081 and 0.0226 respectively, and the best-fitting values of

*σ*were 0.1761, 0.2314 and 0.1961.

*N*is the exponent of the rising low-frequency part of the modulation sensitivity function. The high frequency cutoff is also steeper for large

*N,*so increasing

*N*makes the passband narrower, preserving full transmission at 10 Hz. The bandwidth-narrowing exponent

*N*was determined iteratively, with a complete simulation run for each iteration.

*N*between 1 and 2 gave a good account of the data ( Figure 3 shows the model predictions for

*N*= 1.4, with a root mean square prediction error of 0.116).

*N*yields good estimates of the overall amount of masking for the notch noises as well as for the broadband noise. The rightward shift of the dipper in the external noise conditions is also predicted (perhaps over-predicted) by the model, because threshold is set by total noise at the output, and the contribution of external noise to this total is greater for weak pedestals, where the gradient of the function relating energy to output ( Equation 12) is steep. The required passband of the early filter is quite broad, ranging from about 3 to 25 Hz at half-height. This is quite comparable with the width of the temporal modulation sensitivity function, although the peak and width of that function vary considerably with the conditions of observation (Kelly, 1977; Robson, 1966). The filter bandwidth is, however, narrower than the bandwidth at the retinal output, which exceeds the psychophysical detection bandwidth (Lee et al., 2007). Evidently, most if not all of the visible noise is effective in reducing sensitivity to the test signal, as if the observer's decision is based on the total visibility-weighted flicker energy integrated over frequency.

*e*

^{ M}. But internal noise may also be introduced in the form of random fluctuations in signals representing luminance—noise present in the input to the stages responsible for rectification and compressive nonlinearity. Although Figure 3 shows that such noise need not be invoked to provide an approximate account of the detection thresholds, it is expected a priori and indeed provides an important functional justification for threshold nonlinearity, as the nonlinearity would be helpful in rejecting small inputs that are likely to be due to internal noise at the input to the nonlinear stage (Morgan, Chubb, & Solomon, 2008; Simoncelli & Adelson, 1996).

*Mémoires couronnés et autres memoires publies par l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique*(vol. 23). Bruxelles: Hayez.