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Research Article  |   July 2008
Some observations on contrast detection in noise
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Journal of Vision July 2008, Vol.8, 4. doi:10.1167/8.9.4
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      Robbe L. T. Goris, Peter Zaenen, Johan Wagemans; Some observations on contrast detection in noise. Journal of Vision 2008;8(9):4. doi: 10.1167/8.9.4.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

The standard psychophysical model of our early visual system consists of a linear filter stage, followed by a nonlinearity and an internal noise source. If a rectification mechanism is introduced at the output of the linear filter stage, as has been suggested on some occasions, this model actually predicts that human performance in a classical contrast detection task might benefit from the addition of weak levels of noise. Here, this prediction was tested and confirmed in two contrast detection tasks. In Experiment 1, observers had to discriminate a low-contrast Gabor pattern from a blank. In Experiment 2, observers had to discriminate two low-contrast Gabor patterns identical on all dimensions, except for orientation (−45° vs. −45°). In both experiments, weak-to-modest levels of 2-D, white noise were added to the stimuli. Detection thresholds vary nonmonotonically with noise power, i.e., some noise levels improve contrast detection performance. Both simple uncertainty reduction and an energy discrimination strategy can be excluded as possible explanations for this effect. We present a quantitative model consistent with the effects and discuss the implications.

Introduction
Computational models describing the psychophysical mechanisms mediating pattern vision typically consist of a linear filter stage, followed by a static nonlinearity and a late, internal noise source (i.e., prior to the decisional stage, but after the nonlinearity; Nachmias, 1989). Further, detection threshold studies, i.e., experiments that involve barely visible, low-contrast stimuli, have provided much evidence consistent with a model of our visual system in which visual processing is accomplished in many, relatively independent spatially localized spatial-frequency and orientation-selective filters (Blakemore & Campbell, 1969; Campbell, Carpenter, & Levinson, 1969; Campbell & Robson, 1968; DeValois & DeValois, 1988; Graham & Nachmias, 1971; Henning, Hertz, & Hinton, 1981; Stromeyer & Julesz, 1972; Wilson, McFarlane, & Phillips, 1983). The psychophysical receptive field profiles of these filters, while slightly asymmetric on double logarithmic coordinates (Henning, 1988; Henning et al., 1981), are nonetheless typically approximated by Gabor functions (Figure 1). Although not necessarily an “optimal” stimulus (Watson, Barlow, & Robson, 1983; but see Henning, Derrington, & Madden, 1983), Gabors are popular stimuli to explore contrast detection because they are band-limited in spatial frequency and localized in space (Daugman, 1985). 
Figure 1
 
These stimuli ought to be viewed from a distance of 60 cm. (a) A Gabor function at maximal contrast. This stimulus had a spatial frequency of 7 cycles/deg and resembles the stimulus used in Experiment 1. (b) The same stimulus at 5% signal contrast. When presented for about 25 ms, this would be around detection threshold for the majority of observers. Weak, 2-D white noise has been added to this display. When viewed from a distance of 60 cm, the noise power matches the lowest level of noise used in our experiments. (c) The same stimulus at 5% signal contrast, embedded in the “optimal” level of noise used in our experiments. (d) The same stimulus at 5% signal contrast, embedded in the highest level of noise used in our experiments.
Figure 1
 
These stimuli ought to be viewed from a distance of 60 cm. (a) A Gabor function at maximal contrast. This stimulus had a spatial frequency of 7 cycles/deg and resembles the stimulus used in Experiment 1. (b) The same stimulus at 5% signal contrast. When presented for about 25 ms, this would be around detection threshold for the majority of observers. Weak, 2-D white noise has been added to this display. When viewed from a distance of 60 cm, the noise power matches the lowest level of noise used in our experiments. (c) The same stimulus at 5% signal contrast, embedded in the “optimal” level of noise used in our experiments. (d) The same stimulus at 5% signal contrast, embedded in the highest level of noise used in our experiments.
To gain insight in the visual system's processing at suprathreshold contrasts, sinusoidal contrast discrimination has been studied extensively (e.g., Bird, Henning, & Wichmann, 2002; Foley, 1994; Foley & Chen, 1997; Foley & Legge, 1981; Gorea & Sagi, 2001; Henning & Wichmann, 2007; Kontsevich, Chen, & Tyler, 2002; Legge, 1981; Legge & Foley, 1980; Legge, Kersten, & Burgess, 1987; Nachmias & Sansbury, 1974; Wichmann, 1999; Yang & Makous, 1995). The main finding of these studies is the pedestal or dipper effect: A sinusoidal grating is much more detectable when added to a low-contrast masking or pedestal grating of identical spatial frequency, orientation, phase, and temporal duration. Once the contrast of the pedestal grating exceeds a certain value, discrimination thresholds rise in a Weber's law-like fashion. To explain this dipper-shaped threshold-vs.-contrast function, contrast perception models typically include a nonlinear post-filter stage (e.g., Foley, 1994; Foley & Chen, 1997; Legge & Foley, 1980; Wichmann, 1999; Yang & Makous, 1995). The response expansion that is believed to occur at low contrasts (and thus weak filter responses) is sometimes thought to underlie the initial threshold decrease although this view has recently been questioned (Henning & Wichmann, 2007). Because human performance is not perfect (i.e., both in contrast detection and discrimination experiments, human thresholds differ from ideal observer thresholds), psychophysical models must either assume or postulate limited (neural) efficiency and/or an internal noise source. There has been some debate regarding the question whether the crucial, performance-limiting noise source is to be found in the early or rather in the later processing stages (e.g., Henning, Bird, & Wichmann, 2002; Nachmias, 1989; Pelli, 1991). Furthermore, this might differ for contrast processing in detection and discrimination circumstances (Henning et al., 2002). It is thus not surprising that models describing the psychophysical mechanisms underlying detection in noise have considered several possible locations for the main, internal noise source (Lu & Dosher, 2008). Nevertheless, the standard model of the psychophysical mechanisms mediating pattern vision that proposes a linear filter stage, a static nonlinearity, and a late internal noise source is fairly consistent with most contrast discrimination data. 
There is a clear analogue for this model at the neural level. The neural substrate for contrast processing is likely to be found in primary visual cortex (e.g., DeValois & DeValois, 1988). The classical view on neurons in these early stages of visual processing is well described by models that include a linear filter as their first stage (e.g., Carandini et al., 2005). In a subsequent stage, contrast normalization via nonspecific suppression takes place, leading to contrast–response functions that can be described by Naka–Rushton functions (e.g., Albrecht, Geisler, & Crane, 2003; Carandini, Heeger, & Senn, 2002; Freeman, Durand, Kiper, & Carandini, 2002; Geisler & Albrecht, 1995, 1997; Heeger, 1992a). It has been speculated that this type of image analysis serves, among other goals, to produce efficient coding (Bethge, 2006; Schwartz & Simoncelli, 2001). One crucial element of these models has not been mentioned yet. Because cells cannot have negative firing rates and linear filters do give negative responses, a mechanism removing negative responses must be introduced before the normalization stage (e.g., Carandini et al., 2005; Heeger, 1992b). Different suggestions have been made, among which half-, full- and over-rectification, and half- and full-squaring (in the latter case, (rectified) on- and off-cells may be sources of both halves of the response). All achieve the removal of negative responses. Similar rectification mechanisms have successfully been introduced in psychophysical contrast perception models (e.g., Foley, 1994; Legge & Foley, 1980). One consequence of the rectification of a linear filter's output is that it leads to a monotonically rising filter response as a function of noise power. In other words, if a rectification mechanism is implemented, externally added noise not only affects internal response variability but also raises the mean response to a stimulus, at least at low-contrast levels. Because weak filter responses are followed by response expansion in pattern vision models, these models actually predict that contrast detection performance should improve slightly if weak levels of white noise are added to the stimuli. Due to the introduced stimulus variability and the random phase relation between signal and noise, this improvement is likely to be relatively mild compared to the classical pedestal effect. At higher noise levels, thresholds are expected to rise linearly (see Figure 2). 
Figure 2
 
Illustration of the stimuli (left) and model components (right) discussed in the text. Stimuli with a narrow-band spatial-frequency spectrum are often assumed to be processed in a single spatially localized, spatial-frequency and orientation-selective filter. Ideally, an observer uses a filter that is an exact template of the signal (iii). Human deviations from ideal observer performance are typically interpreted as stemming from internal noise sources (i and vi) on the one hand and limited efficiency or sampling (ii) on the other hand (this refers to using only a sample of the available image information, for instance only the image pixels corresponding to the white pixels in ii). Early noise has the same dimensionality as the signal (i), while late noise is introduced prior to the decision, where the signal representation is 1-D (vi). If the signal representation at the decisional stage is directly based on the (rectified) filter response, a linear threshold-vs.-noise-power function is to be expected. To explain the pedestal effect seen in contrast discrimination, models of pattern vision must include a nonlinear post-filter stage, for instance a Naka–Rushton transformation (v). If the output of the human psychophysical filters mediating detection of a Gabor stimulus in noise is subject to rectification (iv), nonlinear transduction (v) and dominant late noise (vi) prior to the decisional stage, a dipper-shaped threshold-vs.-noise-power function is to be expected. The rectifying and Naka–Rushton nonlinearities are presented as different stages in this figure to disentangle their effects on the internal signal representation. They could, however, be collapsed into a single positively accelerated nonlinearity. Further, as mentioned in the text, the rectification need not be full-wave.
Figure 2
 
Illustration of the stimuli (left) and model components (right) discussed in the text. Stimuli with a narrow-band spatial-frequency spectrum are often assumed to be processed in a single spatially localized, spatial-frequency and orientation-selective filter. Ideally, an observer uses a filter that is an exact template of the signal (iii). Human deviations from ideal observer performance are typically interpreted as stemming from internal noise sources (i and vi) on the one hand and limited efficiency or sampling (ii) on the other hand (this refers to using only a sample of the available image information, for instance only the image pixels corresponding to the white pixels in ii). Early noise has the same dimensionality as the signal (i), while late noise is introduced prior to the decision, where the signal representation is 1-D (vi). If the signal representation at the decisional stage is directly based on the (rectified) filter response, a linear threshold-vs.-noise-power function is to be expected. To explain the pedestal effect seen in contrast discrimination, models of pattern vision must include a nonlinear post-filter stage, for instance a Naka–Rushton transformation (v). If the output of the human psychophysical filters mediating detection of a Gabor stimulus in noise is subject to rectification (iv), nonlinear transduction (v) and dominant late noise (vi) prior to the decisional stage, a dipper-shaped threshold-vs.-noise-power function is to be expected. The rectifying and Naka–Rushton nonlinearities are presented as different stages in this figure to disentangle their effects on the internal signal representation. They could, however, be collapsed into a single positively accelerated nonlinearity. Further, as mentioned in the text, the rectification need not be full-wave.
The combination of rectification and response expansion has typically not been included in contrast perception models describing detection in noise, so that a linear threshold rise is predicted at all noise levels (e.g., Dosher & Lu, 1998; Lu & Dosher, 2008; Pelli, 1985; Pelli & Farell, 1999), as illustrated in Figure 2. Exploration of detection in weak noise can thus be seen as a way to test whether both rectification and response expansion are necessary components of the standard early vision model of detection in noise. 
Detection in noise has been studied on many occasions, and all but one study (Blackwell, 1998) reported monotonically rising detection thresholds as a function of noise power (e.g., Carter & Henning, 1971; Dosher & Lu, 1998; Pelli, 1985; Pelli & Farell, 1999). This inconsistency might simply be a consequence of the different noise levels considered in these studies: Blackwell (1998) used lower noise levels than the other studies mentioned above. She reported small facilitation effects in a classical detection-in-noise task, i.e., the discrimination of a low-contrast signal from a uniform field of mean luminance. The performance improvement was interpreted as resulting from uncertainty reduction, i.e., certain noise levels reduce temporal uncertainty about the stimulus. 
In the experiments reported here, we measured detection thresholds in the presence of weak levels of 2-D, white noise while temporal and spatial uncertainty regarding stimulus presentation was minimized by means of a high-contrast visual marker and a constant temporal task profile. Feedback was used to induce good performance. Stimulus presentation time was very short to obtain relatively high-detection thresholds (around 5% Michelson contrast) and avoid artefact effects due to the limited luminance sampling of our CRT. Finally, to rule out the possibility that the performance improvement found in a classical detection task (i.e., Experiment 1) was due to a strategy based on global energy discrimination instead of truly enhanced signal visibility, we also measured contrast thresholds in an additional experiment. In this experiment, subjects had to discriminate two Gabors identical on all dimensions except that their orientation differed by 90°. It has been reported that orientation discrimination results in similar contrast threshold estimates as classical signal detection (e.g., Solomon & Pelli, 1994). In both experiments, a decrease of contrast thresholds at certain weak noise levels was observed. 
Methods
Equipment
The experiments were run using a Power Macintosh G4 computer, with the software packages MATLAB (Mathworks, Natick, MA) and PSYCHTOOLBOX (Brainard, 1997; Pelli, 1997). Luminance was measured with a KONICA MINOLTA CS-100 Spot Chroma Meter. Gamma correction used an 8-bit lookup table and ensured that the monitor was linear over the entire luminance range used in the experiments. The stimuli were presented on a SONY Trinitron GDM-FW900 monitor, with a spatial resolution of 1920 × 1440 pixels and a temporal resolution of 75 Hz. The experiment was run in a darkened room and the screen's mean gray background luminance was set to 42 cd/m2. Viewing distance was 120 cm, leading to a pixel-size of .009° of visual angle. 
Observers
Three observers participated in Experiment 1 (R.V., B.B., and R.G.), and three observers participated in Experiment 2 (I.P., H.H., and E.G.). All were well practiced with the task and stimuli before data collection began and had normal or optically corrected-to-normal vision. All observers, except for R.G., were naive to the purpose of the experiment. 
Stimuli
Signal stimuli
The Gabor stimuli consisted of orientated sinusoidal gratings with a spatial extent of 2.35° of visual angle and a spatial frequency of 7 cycles/deg, which were then multiplied by a two-dimensional spatial Gaussian envelope with sigma equal to 0.27°. In Experiment 1, the Gabor stimulus was the target or signal, orientated horizontally (0°). The nontarget stimulus was a gray field of the same mean luminance as the signal. In Experiment 2, the target stimulus was arbitrarily defined as a Gabor pattern orientated +45° (to the right); an otherwise identical Gabor pattern orientated −45° (to the left) was defined as the nontarget stimulus. These angles were chosen to avoid effects of pixel contamination (due to pixel contamination, the effective contrast of vertically orientated gratings may be lower than the contrast of horizontally orientated gratings). Further, contrast sensitivity does not differ for these oblique orientations. Examples of the Gabor pattern used in Experiment 1 are shown in Figure 1
Background noise stimuli
For each stimulus presentation, a fresh noise sample, of which each pixel luminance value was sampled from a Gaussian distribution centered at mean luminance, was generated. Ten levels of white noise were used, ranging between 0 and 38 × 10−7 deg2 noise power spectral density. Noise power spectral density is defined as the variance in luminance (relative to the space-average luminance) multiplied with the pixel area, expressed in visual degrees squared. It represents the average power at the different frequencies present in the noise. The maximal amount of clipping (i.e., pixels set to the minimal or maximal luminance values because of the limited dynamical range of the monitor and video card) at the highest noise level was below 2%. Estimates of the effective images ensured that the nonlinear monitor operations (i.e., power saving functions, gamma correction, luminance rounding and the gamma function) did not distort the spectral properties of the noise stimuli. 
Procedure
In both Experiments 1 and 2, a temporal two-alternative-forced-choice (2AFC) task was used. Stimulus presentation time was approximately 25 ms (presentation time was doubled for subject E.G. and halved for subjects B.B. and R.G. to allow them to reach similar thresholds as the other subjects). Each stimulus presentation, the stimulus was surrounded by a red square (2.35°) at maximal luminance. This square appeared and disappeared with the stimulus onset and offset. All trials started with the 266-ms presentation of a gray field. The first stimulus presentation was followed by an interstimulus interval (ISI) of 466 ms and then followed by the second stimulus presentation. The ISI was thus about 20 times longer than the stimulus duration and more than 3 times as long as the temporal impulse response (Graham, 1989). Response time was limited to 1,000 ms and was indicated by a gray field of mean luminance, presented after the second stimulus presentation. After the response screen, a screen with a green square (2.35°) appeared for about 750 ms. During this presentation, auditory feedback indicated which interval contained the signal (low tone of 1,000 Hz: first interval; high tone of 2,000 Hz: second interval). Trials in which responses fell outside the one second answer interval (about 2% of the trials) were repeated. On these occasions, the interval containing the signal was selected randomly again, and the noise was also refreshed again. After a few training blocks, participants were very familiar with this steady temporal task profile. 
To obtain psychometric functions, seven Michelson contrast levels were tested at each noise level. The Michelson contrast of an image is defined as the maximal luminance minus the minimal luminance divided by their sum and thus ranges between 0 (no spatial luminance variation) and 1 (black and white present in the image). In Experiment 1, an interval contained either a uniform field of mean luminance or a nonzero contrast Gabor stimulus. In Experiment 2, all stimuli had the same contrast within a trial but, as mentioned above, the stimuli were orientated either +45° or −45°. 
Within each block of 50 trials, the contrast and noise levels were randomized. During an entire block, a red square of 3.5° of visual angle, surrounding the stimuli, at half of the maximal contrast was presented all the time. Subject I.P., E.G., R.V., B.B., and R.G. completed 7,000 trials in total; subject H.H. completed 4,000 trials in total. 
Results
The results are summarized in Figure 3. The 75% correct Michelson contrast detection thresholds are plotted as a function of noise power spectral density. The upper row shows the results for two observers (H.H. and R.V.), and the lower row shows the results averaged across observers in Experiment 1 (left) and 2 (right). Thresholds vary nonmonotonically with the external noise level: Before increasing, thresholds reach a minimum at a particular noise level. Stated differently, some noise levels improve human contrast detection. This effect is present for all observers, in both experiments. 
Figure 3
 
The 75% correct contrast detection thresholds as a function of noise spectral density on double logarithmic coordinates. The most leftward point denotes the noiseless detection threshold. Error bars, where visible, show estimates of ±1 SD, calculated across observers for panels c and d. The left column shows results for Experiment 1, the right column results for Experiment 2. (a) Results for observer R.V. (b) Results for observer H.H. (c) Results for the signal detection task, averaged across two observers (R.G. and B.B.; the data of R.V. have not been used for panel c because the noise levels used for this observer were slightly different). (d) Results for the orientation discrimination task, averaged across three observers (I.P., E.G., and H.H.).
Figure 3
 
The 75% correct contrast detection thresholds as a function of noise spectral density on double logarithmic coordinates. The most leftward point denotes the noiseless detection threshold. Error bars, where visible, show estimates of ±1 SD, calculated across observers for panels c and d. The left column shows results for Experiment 1, the right column results for Experiment 2. (a) Results for observer R.V. (b) Results for observer H.H. (c) Results for the signal detection task, averaged across two observers (R.G. and B.B.; the data of R.V. have not been used for panel c because the noise levels used for this observer were slightly different). (d) Results for the orientation discrimination task, averaged across three observers (I.P., E.G., and H.H.).
As can be seen in Figure 4, the depth of the dip at 75% correct varies somewhat between observers, similar to the classical pedestal effect. To estimate the average maximal threshold reduction as a function of noise, thresholds for each observer were first normalized by their noiseless detection threshold. This way, the noiseless detection threshold equals one and all other thresholds are expressed as products of this detection threshold. These normalized thresholds were then averaged over observers. Via the same bootstrap procedure used to estimate confidence intervals for each participant, the average maximal 75% correct threshold reduction was estimated to be of a factor 1.35, with the 95% confidence interval ranging from 1.25 to 1.46. At 60% correct, the threshold reduction factor was estimated to be 1.63 (95% CI: 1.40ı.89), and at 90% correct, this factor equaled 1.14 (95% CI: 1.06ı.23). The strength of the noise benefit thus depends on the performance level considered, as is the case for the classical pedestal effect seen in contrast discrimination. However, these values are considerably smaller than the improvement factors usually observed in classical pedestal experiments. Estimates from a contrast discrimination study, making use of the same signal and procedure as reported here, suggest an average improvement factor of 3.44 at 75% correct (Goris, Wagemans, & Wichmann, in press), which is in line with other estimates (e.g., Bird et al., 2002; Henning & Wichmann, 2007). Nevertheless, the same mechanism that underlies the dipper effect in contrast discrimination might give rise to the noise benefit observed here. The pedestal effect depends on the phase relation between signal and pedestal—the dipper effect disappears with signal and pedestal 90° out-of-phase; a phase difference of 180° leads to a “bumper effect” (e.g., Yang & Makous, 1995). Therefore, with random phase relation between signal and noise, only a slight benefit is expected. 
Figure 4
 
Maximal facilitation ratio at the 75% correct detection threshold for each observer. The green symbols show the average maximal facilitation ratio, error bars indicate the 95% confidence interval. To estimate these quantities, 100,000 threshold estimates for each condition were generated by means of a bootstrap procedure, as described in Wichmann and Hill (2001b).
Figure 4
 
Maximal facilitation ratio at the 75% correct detection threshold for each observer. The green symbols show the average maximal facilitation ratio, error bars indicate the 95% confidence interval. To estimate these quantities, 100,000 threshold estimates for each condition were generated by means of a bootstrap procedure, as described in Wichmann and Hill (2001b).
The noise facilitation effect appears to be somewhat stronger in Experiment 2 than in Experiment 1. This may be due to interobserver variability, which is relatively high (see Figure 4). On the other hand, this difference may also be due to modified strategies or stimulus representations, caused by the changed stimulus conditions in Experiment 2 (e.g., the overall energy between blank stimuli and orientated Gabor stimuli differs). 
Given that our experiments were designed to minimize uncertainty about space and time of the stimulus presentation by means of visual markers and a steady temporal task profile, a straightforward uncertainty reduction effect of some noise levels seems not to be a likely explanation for these results—uncertainty models of pattern vision predict no noise benefit in contrast detection (Lu & Dosher, 2008; Pelli, 1985). Furthermore, the similarity between the data of Experiments 1 and 2 excludes the possibility that a global energy-discrimination strategy underlies these results. Although this explanation is not inconsistent with the results of Experiment 1, the results of Experiment 2 rather suggest that stimulus identity was retained. Altogether, these results seem to be consistent with the standard psychophysical model of pattern vision, wherein a linear filter stage is followed by a rectification mechanism, expansion of weak filter responses due to a static nonlinearity and a late internal noise source. We provide quantitative examination of this suggestion in the next section. 
Model: Equations, fitting, evaluation
Equations
We elaborate the standard divisive gain-control model as formalized by Wichmann (1999) for sinusoidal contrast discrimination to describe sinusoidal contrast detection in noise. In order to do so, the gain-control model is extended with two parameters typically used by linear detection in noise models, i.e., early noise and sampling. Further, the filter stage—which need not be specified explicitly for sinusoidal contrast discrimination—is chosen to consist of (optimal) template matching (e.g., Lu & Dosher, 2008). Observer's responses are modelled within the framework of signal detection theory (SDT; Green & Swets, 1966). Three hypothetical stages are specified in SDT models: First, a stimulus theory describes how a transduction mechanism maps physical stimuli to internal states; second, a probabilistic theory of internal states describes the probability distribution of the internal states that results from repeated presentation of the same stimulus; and finally, a deterministic response theory describes a decision rule that maps internal states to a response. 
Stimulus theory
As in some linear detection in noise models, the linear part of the transduction mechanism consists of early signal-independent or additive 2-D noise, σadd2, image sampling or calculation efficiency, k, and template matching (Lu & Dosher, 2008). The parameter k expresses the proportion of available information used by the observer and ranges between 0 and 1. Cross-correlating the noisy, sampled input image, Isampled, with an optimal signal template, Tsignal, transforms the 2-D input stimuli to 1-D responses, Rk, as given by Equation 1. 
Rk=Tsignal(x,y)Isampled(x,y)dxdy.
(1)
Subsequently, this filter response is rectified. The effects of sampling and the rectification on the mean internal representation, ∣Rk∣, were estimated via simulations with the noise and signal contrast levels used in our experiments as input. The scale of these responses depends on the image size used and therefore these responses were normalized by the filter response to a full contrast signal so that filter responses to a noiseless, unsampled signal became identical to the Michelson contrast of that signal. As explained above and illustrated in Figure 2, the aforementioned model components give rise to a linear relationship between image contrast and internal contrast representation. To describe the nonlinear mapping of stimulus contrast to internal contrast representation, R(C), the second part of the transduction mechanism consisted of the three parameter Naka–Rushton function (free parameters α, β, and p), which is illustrated in Figure 2v and given by Equation 2. 
R(C)=αCpβp+Cp.
(2)
The transduction mechanism is thus fully determined by specifying the sampling (k) and the parameters of the Naka–Rushton equation (α, β, and p). Equation 3 expresses this transduction, t, as a function of the signal contrast (C) and the effective total noise spectral density (Ntotal) given a certain sampling value k. Because the early, internal noise is additive, Ntotal is the sum of the external noise level σext2 and the early noise σadd2. 
t(C,Ntotal|k)=α|Rk(C,Ntotal)|pβp+|Rk(C,Ntotal)|p .
(3)
It is important to note that the rectified filter responses, ∣Rk∣, used in the expansive, i.e., the nominator, and the compressive, i.e., the denominator, parts of the Naka–Rushton function were the same. Although some evidence points to the existence of a broadly tuned contrast gain-control pool (e.g., Foley, 1994; Holmes & Meese, 2004), we opted to use only within-channel suppression in this model to avoid an increase of the number of free parameters. 
Theory of internal states
The variability of internal states is determined by internal noise on the one hand and the use of stochastic stimuli on the other hand. Three free parameters were used to describe the late 1-D internal noise, assumed to be Gaussian and having both a signal-independent (free parameter σlate2) and signal-dependent source (free parameters γ and ξ, with ξ a proportional constant and γ an exponent). 
The external noise used in our experiments will inevitably contribute to the total variability of the internal representation. Because the Naka–Rushton function is—in general—nonlinear, the effect of its parameter values on the variability of the internal representation has to be estimated by means of simulations. To obtain estimates of the variance of the noise representations, we ran simulations with the noise levels used in our experiments as input. Equal variance of the signal and noise representations was assumed as a first approximation. We varied k, p, and β and fitted descriptive functions to the simulated variances. These descriptive functions allowed us to formalize the full model behavior. 
Response theory
As it is standard, it was assumed that the observer's response (“interval 1” or “interval 2”) is determined by the stimulus interval that led to the highest internal state. The model equations (see Equations 46) were arranged to express percent correct, p(C, Ntotal), as a function of the signal contrast (C) and the effective total noise spectral density (Ntotal) in a 2-AFC task.  
p ( C , N t o t a l ) = 0 1 2 π g ( C , N t o t a l ) e ( z f ( C , N t o t a l ) ) 2 2 g ( C , N t o t a l ) d z ,
(4)
where z is a dummy variable and f(C, Ntotal) and g(C, Ntotal) are given by  
f ( C , N t o t a l ) = α ( ( | R k ( N t o t a l C ) | ) p β p + ( | R k ( N t o t a l C ) | ) p ( | R k ( N t o t a l ) | ) p β p + ( | R k ( N t o t a l ) | ) p ) ,
(5)
and  
g ( C , N t o t a l ) = 2 σ l a t e 2 + 2 ( V A R ( α ( | R k ( N t o t a l ) | ) p β p + ( | R k ( N t o t a l ) | ) p ) ) + ζ 2 ( ( α ( | R k ( N t o t a l C ) | ) p β p + ( | R k ( N t o t a l C ) | ) p ) 2 γ + ( α ( | R k ( N t o t a l ) | ) p β p + ( | R k ( N t o t a l ) | ) p ) 2 γ ) .
(6)
In sum, by combining simulations and analytical descriptions, we obtained parametric descriptions of performance in a 2-AFC task as a function of eight free parameters (σadd2, k, α, β, p, σlate2, γ, and ξ) and external noise level N and signal contrast C
Fitting
One of the eight parameters can be arbitrarily set to any value. To follow the usual convention, σlate2 was taken to be 1, resulting in a seven free parameter model. An additional free parameter λ (“lapse rate”) was introduced in the fitting of the model to avoid biased parameter estimates (Wichmann, 1999; Wichmann & Hill, 2001a). Priors were introduced for each parameter to constrain estimates to realistic values. To find the surface p(C, Ntotal) that maximizes the likelihood that the data were generated from a process with success probability given by p(C, Ntotal), the log-likelihood of the surface p(C, Ntotal) given the parameters (σadd2, k, α, β, p, γ, ξ, and λ) was maximized using purpose-written software in MATLAB (fminsearch, which makes use of the Nelder–Mead simplex search method). The log-likelihood of the surface p(C, Ntotal) given parameter vector θ, containing {σadd2, k, α, β, p, γ, ξ, σlate2, and λ} with σlate2 = 1 is given by Equation 7: 
l(θ)=j=1Zi=1Kjlog(njiyjinji)+yjinjilog(p(Cji,Nj;θ))+(1yji)njilog(1p(Cji,Nj;θ)) ,
(7)
with nji the number of trials (block size) measured at noise level Nj and signal contrast Cji and yji the proportion of correct responses in that condition. Because the problem is nonconvex due to λ, a multistart procedure with pseudorandomly selected initial values was used to find the probable global minimum for each participant. 
Evaluation
To evaluate model fits, we considered the overall distance between model prediction and data and the presence of systematic errors in the residuals. Quality of the overall fit can be assessed by judging total deviance (see Equation 8), i.e., the log-likelihood ratio of the saturated model and the best fitting model (the saturated model is the model with no residual error between model predictions and data). What deviance does not assess, however, are systematic trends in the deviance residuals (see Equation 9), i.e., the agreement between individual data points and the corresponding model prediction. For binomial data, deviance is expressed by Equation 8.  
D = 2 j = 1 Z i = 1 K j { n j i y j i log ( y j i p ( C j i , N j ) ) + n j i ( 1 y j i ) log ( 1 y j i 1 p ( C j i , N j ) ) } ,
(8)
This statistic indicates how well a model describes data. Asymptotically, it can be shown to be χ2-distributed, with degrees-of-freedom equal to the number of data blocks minus the number of free parameters if the model is correct and the observer's behavior were perfectly stationary during the whole experiment (such an observer would thus generate truly binomially distributed data). Often, due to a variety of reasons, this is not the case. Responses of nonstationary observers are more variable than binomially distributed data and thus lead to higher deviances (overdispersion). Wichmann (1999) has shown that, due to the typically relatively small number of measurements, the asymptotically derived distributions often fail to approximate the real distribution of D for psychophysical data sets. The real distribution of D can be estimated easily by means of Monte Carlo simulations. As suggested by Wichmann (1999), we estimated the distribution of D for each model fit by means of 10,000 simulated data sets for an observer whose correct responses in our experiment are binomially distributed as specified by the model fit. From these simulations, we derived critical values for each reported fit. 
Each deviance residual di is defined as the square root of the deviance value calculated for data point i in isolation, signed according to the direction of the arithmetic residual yip(Cji, Ni). For binomial data, this is expressed by Equation 9.  
d j i = sgn ( y j i p ( C j i , N j ) ) 2 [ n j i y j i log ( y j i p ( C j i , N j ) ) + n j i ( 1 y j i ) log ( 1 y j i 1 p ( C j i , N j ) ) ] .
(9)
Note that D =
j = 1 Z i = 1 K j
dji2, as for RMSE. Systematic trends in deviance residuals indicate a systematic misfit of the model. 
Model: Results
For each observer, the eight free parameter gain-control model was fitted to all data. A detailed example of such a fit to the data of observer I.P. and R.V. is shown in Figure 5. Both the psychometric functions derived from the model fit (full red lines) and Weibull functions fitted to the data (dotted red lines) are plotted. The gain-control model has roughly 1 free parameter per noise level (i.e., eight free parameters for 10 noise levels) as compared to the four free parameters per noise level used by the Weibull function (Wichmann & Hill, 2001a). Nevertheless, the fits to the data are relatively similar. The more parsimonious gain-control model thus seems to give a reasonable approximation of these data. The whole model fit is shown in Figure 6 for all observers. Performance, indicated by color, is plotted as a function of noise spectral density and signal contrast. The thresholds derived from the model fits are dipper-shaped as a function of noise spectral density. Furthermore, the threshold reduction depends on performance level considered—being stronger at lower performance levels—thus capturing the same trend seen in the raw data. Table 1 lists the parameter estimates and normalized total deviance for all observers. 
Figure 5
 
Performance as a function of signal contrast for observers R.V. (squares, upper row) and I.P. (circles, lower row) at three different noise levels: No noise (left), “optimal” noise (middle), and the highest noise level used in the experiment (right). Gray circles indicate measured performance, the full red lines the fit of the gain-control model discussed in the paper. Dotted red lines indicate the best fitting Weibull function.
Figure 5
 
Performance as a function of signal contrast for observers R.V. (squares, upper row) and I.P. (circles, lower row) at three different noise levels: No noise (left), “optimal” noise (middle), and the highest noise level used in the experiment (right). Gray circles indicate measured performance, the full red lines the fit of the gain-control model discussed in the paper. Dotted red lines indicate the best fitting Weibull function.
Figure 6
 
Model fits for all observers. Performance, indicated by color, is plotted as a function of noise spectral density and signal contrast on double logarithmic coordinates.
Figure 6
 
Model fits for all observers. Performance, indicated by color, is plotted as a function of noise spectral density and signal contrast on double logarithmic coordinates.
Table 1
 
The parameter estimates and deviance of the model described in the text for all observers.
Table 1
 
The parameter estimates and deviance of the model described in the text for all observers.
R.V. B.B. R.G. I.P. E.G. H.H.
σ add 2 0.020 0.034 0.026 0.025 0.023 0.012
k 0.007 0.013 0.013 0.012 0.010 0.008
α 3.6e39 4.2e39 8.1e39 3.6e39 1.2e39 4.8e33
β 0.054 0.054 0.054 0.056 0.056 0.14
p 19.9 22.6 22.1 21.46 20.57 13.72
σ late 2 1 1 1 1 1 1
γ 0.49 0.70 0.45 0.46 1.01 0.07
ξ 6.6e19 4.5e19 7.3e19 5.4e19 2.1e19 1.2e17
λ 0.025 0.022 0.035 0.024 0.053 0.013
D 0.74 2.16* 1.95* 1.50 2.40* 0.74
 

Note: Bold symbols and numbers indicate frozen parameter values.

 

*Indicates that D is outside the (Bonferroni corrected) 95% confidence interval of a stationary observer.

The normalized total deviance averaged across observers equals 1.58, with all observers ranging between 0.74 and 2.4. This indicates that the quality of fit is fairly good for some observers, but poorer for others. Overall, the gain-control model explains much of the variance in these data. Nevertheless, for three of six observers, total deviance does not belong to the 95% confidence interval of the deviance distribution expected if observers' behavior were stationary. These confidence intervals were Bonferroni corrected, thus ensuring that the overall probability of making a Type I error, i.e., falsely rejecting the null-hypothesis, equals .05. Figure 7a shows the model deviance residual distribution across observers, based on 420 blocks of 100 trials each. In Figure 7b, model deviance residuals are plotted as a function of noise spectral density with different color labels for different observers and different symbols for Experiments 1 and 2 (squares and circles, respectively). The thick red line describes the mean deviance residual as a function of noise spectral density; the dotted lines indicate the 99% confidence interval. As can be seen in these figures, the distribution of deviance residuals is centered on zero and approximately Gaussian in shape. Because the deviance residuals suggest that the noiseless detection threshold is, perhaps, a bit underestimated and the thresholds in some “optimal” noise levels a bit overestimated, the model may underestimate the noise facilitation effect slightly. Overall, however, the lack in quality of fit for some observers seems more due to overdispersion (i.e., nonstationary behavior in some conditions, which cannot be fixed by any other model) than to a systematic mismatch. 
Figure 7
 
(a) The distribution of model deviance residuals across all observers. (b) Model deviance residuals as a function of noise spectral density plotted on semi-logarithmic coordinates. Dotted red lines indicate the 99% confidence interval.
Figure 7
 
(a) The distribution of model deviance residuals across all observers. (b) Model deviance residuals as a function of noise spectral density plotted on semi-logarithmic coordinates. Dotted red lines indicate the 99% confidence interval.
The parameter estimates listed in Table 1 are relatively similar across observers. For all participants, the level of early, internal noise (σadd2) is estimated to be approximately equal to the weakest external noise level that leads to a threshold rise. This is not inconsistent with the notion of “equivalent input noise” used in some linear detection in noise models (Lu & Dosher, 2008; Nagaraja, 1964). Sampling (k) is estimated to be approximately 1%, which is in line with some other reported estimates (e.g., Legge et al., 1987). Figure 8 illustrates the effect of sampling on a signal, a noise and a signal plus noise image. As sampling decreases from 100% to 10% (i.e., one log unit), the response of an optimal filter to a signal image decreases by a log unit, while the response to a noise image only decreases by the square root of a log unit (i.e., approximately a factor of 3.16). The response to a signal plus noise image decreases with half a log unit (i.e., approximately a factor of 5) for the noise and contrast levels used in this example. The sampling parameter thus mainly serves to scale the (ratio of) responses to signal, noise, and signal plus noise images. Inefficiencies in the visual system need not be conceptualized as sampling. Alternatively, this parameter could be thought of as reflecting the use of a suboptimal filter, for instance a spatial-frequency-tuned channel that has an effective bandwidth that is broader than the narrowband Gabor signal to be detected. As can be seen in Equations 1 and 2, α is simply a rescaling parameter that determines the response range. More interestingly, β reflects the semi-saturation contrast of the contrast response function (Heeger, 1992a, 1992b). For all observers but one (H.H.), β is estimated to be in the vicinity of the 75% correct (noiseless) detection threshold. The estimates of the response exponent p may seem fairly high compared to fits of the gain-control model to contrast discrimination data (e.g., Foley, 1994; Wichmann, 1999). This is a consequence of the use of external (and early internal) noise: Because early noise linearizes nonlinear systems (see discussion), the exponent of the accelerating part of the nonlinearity needs to be high enough to fit the experimentally observed noise benefit. 
Figure 8
 
Illustration of the effect of image sampling. Upper row: A maximal contrast stimulus sampled at 100%, 50%, 25%, and 10%. Second row: A noise stimulus sampled at 100%, 50%, 25%, and 10%. Third row: A signal plus noise stimulus sampled at 100%, 50%, 25%, and 10%. Lower row: The (normalized) response of an optimal template to the signal (S), noise (N), and signal plus noise (SN) stimulus. The filter response to the noiseless signal decreases with 1 log unit if sampling decreases with 1 log unit. The response to a noise stimulus decreases with the square root of a log unit, while the response to a signal plus noise stimulus decreases with half a log unit (for these particular noise and contrast levels).
Figure 8
 
Illustration of the effect of image sampling. Upper row: A maximal contrast stimulus sampled at 100%, 50%, 25%, and 10%. Second row: A noise stimulus sampled at 100%, 50%, 25%, and 10%. Third row: A signal plus noise stimulus sampled at 100%, 50%, 25%, and 10%. Lower row: The (normalized) response of an optimal template to the signal (S), noise (N), and signal plus noise (SN) stimulus. The filter response to the noiseless signal decreases with 1 log unit if sampling decreases with 1 log unit. The response to a noise stimulus decreases with the square root of a log unit, while the response to a signal plus noise stimulus decreases with half a log unit (for these particular noise and contrast levels).
Inspection of γ reveals that for three of six observers, the exponent of the level-dependent noise source is estimated to be approximately 0.5, which corresponds to a “neural” noise scheme with noise variance proportional to the mean firing rate (e.g., Geisler & Albrecht, 1997). For one observer, γ is close to one, which corresponds to the standard deviation being proportional to the mean response. With high α-estimates, the proportional constant of the level-dependent noise source, ξ, needs to have high values too to have any influence on the slope of the psychometric functions. Lapse rate (λ) estimates are around 2%, which is not unusual (Wichmann & Hill, 2001a). 
In sum, the overall quality of fit provided by the contrast gain-control model is reasonable. While some observers clearly show nonstationary behavior, there is only a minor indication of systematic errors in the model fit. If any, the depth of the dip is slightly underestimated. Inspection of the parameter estimates reveals that they are not inconsistent with other applications of the contrast gain-control model in spatial vision. 
Discussion
Much evidence has accumulated that suggests that our early visual system consists of spatial-frequency and orientation-selective channels. In contrast detection circumstances, many data suggest that these channels operate independently and approximately linear, i.e., superposition holds. At higher contrasts, effects of a nonlinear post-filter stage have been reported frequently: Contrast discrimination thresholds initially drop as a function of pedestal contrast, before they rise in a way that is roughly consistent with Weber's law. The standard early vision model explains both types of data as stemming from a linear filter stage, followed by a static nonlinearity and an internal noise source. It is important to mention that some studies, which made use of slightly more complex stimuli than simple sinusoidal gratings, have reported data that are hard to reconcile with independent spatial-frequency channels (e.g., Derrington & Henning, 1989; Henning, Hertz, & Broadbent, 1975). Furthermore, recent work has questioned to what degree the standard model explains the well-known pedestal effect: This effect might be caused by the pooling of information from several channels rather than stemming from the characteristics of a single spatial-frequency-tuned channel (Henning & Wichmann, 2007). Nevertheless, the standard early vision model describes classical contrast discrimination data to a satisfying degree (Wichmann, 1999). As in the neural analogue of this model, a rectification mechanism at the output of the linear filter stage has been adopted on some occasions as well as here. This rectification mechanism causes external noise to increase the mean filter response to a low-contrast stimulus. Because weak filter responses are expanded subsequently, weak amounts of externally added noise could improve contrast detection. This prediction was tested and confirmed in the experiments reported here. Furthermore, both a simple uncertainty reduction mechanism and a global energy discrimination strategy seem to be unlikely explanations. It was also demonstrated that an eight-parameter operationalization of the contrast gain-control model fitted the data to a reasonable degree, making use of plausible parameter estimates. 
At higher noise levels, thresholds rise linearly, as has been reported on many occasions. This linear rise, despite a presumably nonlinear underlying contrast response function, can be understood as a consequence of Birdsall's theorem (Green & Swets, 1966): In a multistage system, sensitive to order, effects of nonlinear transformations occurring after the strongest independent noise injection cannot be measured if this noise source is the crucial performance-limiting factor. Thus, once the external noise becomes the crucial performance-limiting factor, it effectively linearizes the system. A second implication of this theorem is that a dominant late internal noise source is also a necessary condition for facilitation as a function of noise to occur. Thus, at least in the conditions where facilitation is present, the crucial, performance-limiting internal noise source in these experiments is located in the later processing stages, i.e., after the response expansion. Were it located earlier, e.g., in the filter stage, thresholds would increase linearly at all external noise levels. 
Finally, it is interesting to note that improved signal transmission as a function of externally added noise is a well-known phenomenon in physics, labeled stochastic resonance (e.g., Wiesenfeld & Moss, 1995). Stochastic resonance is the signature of nonlinear information processing and has been reported in many man-made and biological systems, including the human tactile (Collins, Imhoff, & Grigg, 1996) and auditory system (Zeng, Fu, & Morse, 2000). Stochastic resonance is never observed in linear systems, i.e., systems in which superposition holds. 
Conclusions
Addition of weak levels of 2-D, white noise to simple Gabor stimuli, which have typically been used in contrast detection tasks, improves human detection and discrimination performance. This effect is neither caused by reduction of uncertainty about spatial and temporal stimulus occurrence nor is it based on global energy discrimination. We interpret these results as consistent with contrast gain-control models, thus stemming from a rectified linear filter response, followed by response expansion in a post-filter stage and a dominant late internal noise source at low external noise levels. These results imply that detection in noise is better treated as signal discrimination instead of signal detection at the level of a single channel. 
Acknowledgments
R.G. analyzed the data and contributed most to the writing of the paper. P.Z. designed the experiments and the model and contributed to the writing of the paper. J.W. supervised the project and contributed to the writing of the paper. R.G. is Research Assistant of the Fund for Scientific Research-Flanders (FWO-Vlaanderen) under the supervision of J.W. Costs for equipment and salary (P.Z.) were paid from grants by the Research Fund of the University of Leuven (IDO/98/002) and by the FWO-Vlaanderen (G.0095.03) awarded to J.W. 
We would like to thank colleagues who provided helpful comments, especially F. Wichmann, B. Henning, M. Lages, H. Op de Beeck, and two anonymous reviewers. 
Commercial relationships: none. 
Corresponding author: Johan Wagemans. 
Email: Johan.Wagemans@psy.kuleuven.be. 
Address: Laboratory for Experimental Psychology, University of Leuven, Tiensestraat 102, 3000 Leuven, Belgium. 
Footnote
Footnotes
 *These authors contributed equally to this work and should be considered joint first authors.
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Figure 1
 
These stimuli ought to be viewed from a distance of 60 cm. (a) A Gabor function at maximal contrast. This stimulus had a spatial frequency of 7 cycles/deg and resembles the stimulus used in Experiment 1. (b) The same stimulus at 5% signal contrast. When presented for about 25 ms, this would be around detection threshold for the majority of observers. Weak, 2-D white noise has been added to this display. When viewed from a distance of 60 cm, the noise power matches the lowest level of noise used in our experiments. (c) The same stimulus at 5% signal contrast, embedded in the “optimal” level of noise used in our experiments. (d) The same stimulus at 5% signal contrast, embedded in the highest level of noise used in our experiments.
Figure 1
 
These stimuli ought to be viewed from a distance of 60 cm. (a) A Gabor function at maximal contrast. This stimulus had a spatial frequency of 7 cycles/deg and resembles the stimulus used in Experiment 1. (b) The same stimulus at 5% signal contrast. When presented for about 25 ms, this would be around detection threshold for the majority of observers. Weak, 2-D white noise has been added to this display. When viewed from a distance of 60 cm, the noise power matches the lowest level of noise used in our experiments. (c) The same stimulus at 5% signal contrast, embedded in the “optimal” level of noise used in our experiments. (d) The same stimulus at 5% signal contrast, embedded in the highest level of noise used in our experiments.
Figure 2
 
Illustration of the stimuli (left) and model components (right) discussed in the text. Stimuli with a narrow-band spatial-frequency spectrum are often assumed to be processed in a single spatially localized, spatial-frequency and orientation-selective filter. Ideally, an observer uses a filter that is an exact template of the signal (iii). Human deviations from ideal observer performance are typically interpreted as stemming from internal noise sources (i and vi) on the one hand and limited efficiency or sampling (ii) on the other hand (this refers to using only a sample of the available image information, for instance only the image pixels corresponding to the white pixels in ii). Early noise has the same dimensionality as the signal (i), while late noise is introduced prior to the decision, where the signal representation is 1-D (vi). If the signal representation at the decisional stage is directly based on the (rectified) filter response, a linear threshold-vs.-noise-power function is to be expected. To explain the pedestal effect seen in contrast discrimination, models of pattern vision must include a nonlinear post-filter stage, for instance a Naka–Rushton transformation (v). If the output of the human psychophysical filters mediating detection of a Gabor stimulus in noise is subject to rectification (iv), nonlinear transduction (v) and dominant late noise (vi) prior to the decisional stage, a dipper-shaped threshold-vs.-noise-power function is to be expected. The rectifying and Naka–Rushton nonlinearities are presented as different stages in this figure to disentangle their effects on the internal signal representation. They could, however, be collapsed into a single positively accelerated nonlinearity. Further, as mentioned in the text, the rectification need not be full-wave.
Figure 2
 
Illustration of the stimuli (left) and model components (right) discussed in the text. Stimuli with a narrow-band spatial-frequency spectrum are often assumed to be processed in a single spatially localized, spatial-frequency and orientation-selective filter. Ideally, an observer uses a filter that is an exact template of the signal (iii). Human deviations from ideal observer performance are typically interpreted as stemming from internal noise sources (i and vi) on the one hand and limited efficiency or sampling (ii) on the other hand (this refers to using only a sample of the available image information, for instance only the image pixels corresponding to the white pixels in ii). Early noise has the same dimensionality as the signal (i), while late noise is introduced prior to the decision, where the signal representation is 1-D (vi). If the signal representation at the decisional stage is directly based on the (rectified) filter response, a linear threshold-vs.-noise-power function is to be expected. To explain the pedestal effect seen in contrast discrimination, models of pattern vision must include a nonlinear post-filter stage, for instance a Naka–Rushton transformation (v). If the output of the human psychophysical filters mediating detection of a Gabor stimulus in noise is subject to rectification (iv), nonlinear transduction (v) and dominant late noise (vi) prior to the decisional stage, a dipper-shaped threshold-vs.-noise-power function is to be expected. The rectifying and Naka–Rushton nonlinearities are presented as different stages in this figure to disentangle their effects on the internal signal representation. They could, however, be collapsed into a single positively accelerated nonlinearity. Further, as mentioned in the text, the rectification need not be full-wave.
Figure 3
 
The 75% correct contrast detection thresholds as a function of noise spectral density on double logarithmic coordinates. The most leftward point denotes the noiseless detection threshold. Error bars, where visible, show estimates of ±1 SD, calculated across observers for panels c and d. The left column shows results for Experiment 1, the right column results for Experiment 2. (a) Results for observer R.V. (b) Results for observer H.H. (c) Results for the signal detection task, averaged across two observers (R.G. and B.B.; the data of R.V. have not been used for panel c because the noise levels used for this observer were slightly different). (d) Results for the orientation discrimination task, averaged across three observers (I.P., E.G., and H.H.).
Figure 3
 
The 75% correct contrast detection thresholds as a function of noise spectral density on double logarithmic coordinates. The most leftward point denotes the noiseless detection threshold. Error bars, where visible, show estimates of ±1 SD, calculated across observers for panels c and d. The left column shows results for Experiment 1, the right column results for Experiment 2. (a) Results for observer R.V. (b) Results for observer H.H. (c) Results for the signal detection task, averaged across two observers (R.G. and B.B.; the data of R.V. have not been used for panel c because the noise levels used for this observer were slightly different). (d) Results for the orientation discrimination task, averaged across three observers (I.P., E.G., and H.H.).
Figure 4
 
Maximal facilitation ratio at the 75% correct detection threshold for each observer. The green symbols show the average maximal facilitation ratio, error bars indicate the 95% confidence interval. To estimate these quantities, 100,000 threshold estimates for each condition were generated by means of a bootstrap procedure, as described in Wichmann and Hill (2001b).
Figure 4
 
Maximal facilitation ratio at the 75% correct detection threshold for each observer. The green symbols show the average maximal facilitation ratio, error bars indicate the 95% confidence interval. To estimate these quantities, 100,000 threshold estimates for each condition were generated by means of a bootstrap procedure, as described in Wichmann and Hill (2001b).
Figure 5
 
Performance as a function of signal contrast for observers R.V. (squares, upper row) and I.P. (circles, lower row) at three different noise levels: No noise (left), “optimal” noise (middle), and the highest noise level used in the experiment (right). Gray circles indicate measured performance, the full red lines the fit of the gain-control model discussed in the paper. Dotted red lines indicate the best fitting Weibull function.
Figure 5
 
Performance as a function of signal contrast for observers R.V. (squares, upper row) and I.P. (circles, lower row) at three different noise levels: No noise (left), “optimal” noise (middle), and the highest noise level used in the experiment (right). Gray circles indicate measured performance, the full red lines the fit of the gain-control model discussed in the paper. Dotted red lines indicate the best fitting Weibull function.
Figure 6
 
Model fits for all observers. Performance, indicated by color, is plotted as a function of noise spectral density and signal contrast on double logarithmic coordinates.
Figure 6
 
Model fits for all observers. Performance, indicated by color, is plotted as a function of noise spectral density and signal contrast on double logarithmic coordinates.
Figure 7
 
(a) The distribution of model deviance residuals across all observers. (b) Model deviance residuals as a function of noise spectral density plotted on semi-logarithmic coordinates. Dotted red lines indicate the 99% confidence interval.
Figure 7
 
(a) The distribution of model deviance residuals across all observers. (b) Model deviance residuals as a function of noise spectral density plotted on semi-logarithmic coordinates. Dotted red lines indicate the 99% confidence interval.
Figure 8
 
Illustration of the effect of image sampling. Upper row: A maximal contrast stimulus sampled at 100%, 50%, 25%, and 10%. Second row: A noise stimulus sampled at 100%, 50%, 25%, and 10%. Third row: A signal plus noise stimulus sampled at 100%, 50%, 25%, and 10%. Lower row: The (normalized) response of an optimal template to the signal (S), noise (N), and signal plus noise (SN) stimulus. The filter response to the noiseless signal decreases with 1 log unit if sampling decreases with 1 log unit. The response to a noise stimulus decreases with the square root of a log unit, while the response to a signal plus noise stimulus decreases with half a log unit (for these particular noise and contrast levels).
Figure 8
 
Illustration of the effect of image sampling. Upper row: A maximal contrast stimulus sampled at 100%, 50%, 25%, and 10%. Second row: A noise stimulus sampled at 100%, 50%, 25%, and 10%. Third row: A signal plus noise stimulus sampled at 100%, 50%, 25%, and 10%. Lower row: The (normalized) response of an optimal template to the signal (S), noise (N), and signal plus noise (SN) stimulus. The filter response to the noiseless signal decreases with 1 log unit if sampling decreases with 1 log unit. The response to a noise stimulus decreases with the square root of a log unit, while the response to a signal plus noise stimulus decreases with half a log unit (for these particular noise and contrast levels).
Table 1
 
The parameter estimates and deviance of the model described in the text for all observers.
Table 1
 
The parameter estimates and deviance of the model described in the text for all observers.
R.V. B.B. R.G. I.P. E.G. H.H.
σ add 2 0.020 0.034 0.026 0.025 0.023 0.012
k 0.007 0.013 0.013 0.012 0.010 0.008
α 3.6e39 4.2e39 8.1e39 3.6e39 1.2e39 4.8e33
β 0.054 0.054 0.054 0.056 0.056 0.14
p 19.9 22.6 22.1 21.46 20.57 13.72
σ late 2 1 1 1 1 1 1
γ 0.49 0.70 0.45 0.46 1.01 0.07
ξ 6.6e19 4.5e19 7.3e19 5.4e19 2.1e19 1.2e17
λ 0.025 0.022 0.035 0.024 0.053 0.013
D 0.74 2.16* 1.95* 1.50 2.40* 0.74
 

Note: Bold symbols and numbers indicate frozen parameter values.

 

*Indicates that D is outside the (Bonferroni corrected) 95% confidence interval of a stationary observer.

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