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Article  |   August 2015
Testing the role of luminance edges in White's illusion with contour adaptation
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Journal of Vision August 2015, Vol.15, 14. doi:10.1167/15.11.14
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      Torsten Betz, Robert Shapley, Felix A. Wichmann, Marianne Maertens; Testing the role of luminance edges in White's illusion with contour adaptation. Journal of Vision 2015;15(11):14. doi: 10.1167/15.11.14.

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

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Abstract

White's illusion is the perceptual effect that two equiluminant gray patches superimposed on a black-and-white square-wave grating appear different in lightness: A test patch placed on a dark stripe of the grating looks lighter than one placed on a light stripe. Although the effect does not depend on the aspect ratio of the test patches, and thus on the amount of border that is shared with either the dark or the light stripe, the context of each patch must, in a yet to be specified way, influence their lightness. We employed a contour adaptation paradigm (Anstis, 2013) to test the contribution of each of the test patches' edges to the perceived lightness of the test patches. We found that adapting to the edges that are oriented parallel to the grating slightly increased the lightness illusion, whereas adapting to the orthogonal edges abolished, or for some observers even reversed, the lightness illusion. We implemented a temporal adaptation mechanism in three spatial filtering models of lightness perception, and show that the models cannot account for the observed adaptation effects. We conclude that White's illusion is largely determined by edge contrast across the edge orthogonal to the grating, whereas the parallel edge has little or no influence. We suggest mechanisms that could explain this asymmetry.

Introduction
In 1979 White presented a stimulus that caused a problem for contrast-based accounts of lightness perception (White, 1979). He showed that when gray patches were superimposed on a black-and-white square-wave grating, those patches that were placed on the black bars of the grating looked lighter than the patches that were placed on the white bars (Figure 1). A lightness computation based on luminance ratios would predict the opposite effect, because test patches placed on black bars share more border with white bars, and hence should appear darker than patches placed on white bars. Since the original publication, numerous attempts have been made to explain White's illusion and to integrate it with existing theories of lightness perception (Anderson, 1997; Anstis, 2005; Blakeslee & McCourt, 1999; Gilchrist et al., 1999; Howe, 2005; Kingdom & Moulden, 1991; Ripamonti & Gerbino, 2001; Robinson, Hammon, & de Sa, 2007; Salmela & Laurinen, 2009; Spehar, Gilchrist, & Arend, 1995; Taya, Ehrenstein, & Cavonius, 1995; White, 1981). However, to this day there seems to be no consensus about how White's illusion can be explained. 
Figure 1
 
White's illusion. The test patch on the black bar shares more border with the white flanking bars than with the black coaxial bar, so based on edge contrast alone, it should appear darker than the test patch on the white bar. The opposite is the case.
Figure 1
 
White's illusion. The test patch on the black bar shares more border with the white flanking bars than with the black coaxial bar, so based on edge contrast alone, it should appear darker than the test patch on the white bar. The opposite is the case.
The most prominent low-level mechanistic explanation of the illusion is given by spatial filtering models (Blakeslee & McCourt, 1999; Dakin & Bex, 2003; Otazu, Vanrell, & Alejandro Párraga, 2008; Robinson et al., 2007). We have recently shown (Betz, Shapley, Wichmann, & Maertens, in press) that these models are unlikely to include the correct mechanisms to explain White's illusion, because they cannot account for the effect of narrowband luminance noise on the perceived lightness of the test patches (Salmela & Laurinen, 2009). Instead, our analyses suggested that luminance edges at the test patch boundaries are crucial for the computation of their lightness. However, this raises the question of how exactly the luminance contrast across the edges determines perceived lightness in White's stimulus. As we have pointed out above, a simple account that weights the edge contrast by the length of the border is insufficient because it makes the incorrect prediction that the test patch on the dark bar should appear darker than the test patch on the light bar. Therefore, if edge contrast is indeed critical for perceived patch lightness, and if the length of the border is not critical for perceived patch lightness, then an alternative hypothesis would be that it is the orientation relationship between the edges and the grating that is a critical factor. 
Here, we test the contribution of luminance edges to perceived lightness in White's illusion experimentally by employing the method of contour adaptation. Anstis (2013) showed that adapting to a high-contrast flickering contour for a few seconds strongly reduced the perceived contrast of a luminance edge that was subsequently presented at the adapted position. Luminance edges that are presented with a low contrast can disappear completely when presented subsequently to an adapting contour. We used this technique to adapt the edges of the test patches in White's illusion and to observe the effect of contour adaptation on the perceived lightness of the test patch. If the luminance contrast across edges is indeed a major determinant of a surface's lightness, then adaptation should have a measurable effect on test patch lightness in White's illusion. In particular, incremental edges (i.e., higher intensity inside than outside) should have a brightening effect and decremental edges (i.e., lower intensity inside than outside) should have a darkening effect on the surface. When the influence of the edge is weakened by means of contour adaptation then its role in determining the lightness of a surface should be weakened as well. Specifically, adapting at the location of incremental edges should reduce the brightening effect and hence make a patch appear darker, whereas adapting at the location of decremental edges should reduce the darkening effect and make a patch appear brighter. Thus, adaptation to the edges parallel to the grating should make the patch on the dark stripe lighter and the patch on the light stripe darker, increasing White's illusion. Adapting to the edges orthogonal to the grating should make the patch on the dark stripe darker and the patch on the light stripe lighter, decreasing White's illusion. Movies 1 and 2 illustrate adaptation to the orthogonal and to the parallel edges, respectively. Informal observations generally confirmed the edge-based predictions above. These demos suggest a role for both types of luminance edges in White's illusion. However, adapting to the edges orthogonal to the grating seems to have a larger effect on perceived lightness. To substantiate and quantify the informal observations, we tested the effect of contour adaptation on White's effect in a psychophysical experiment. 
Movie 1.
 
Contour adaptation of the test patch edges orthogonal to the grating. Most observers see the test patches merge with the grating bar on which they are placed.
Movie 1.
 
Contour adaptation of the test patch edges orthogonal to the grating. Most observers see the test patches merge with the grating bar on which they are placed.
Movie 2.
 
Contour adaptation of the test patch edges parallel to the grating. Most observers see the test patches bleed out into the neighboring bars.
Movie 2.
 
Contour adaptation of the test patch edges parallel to the grating. Most observers see the test patches bleed out into the neighboring bars.
In the second part of this work, we analyze the implications of our findings for spatial filtering models of lightness perception. We focus our efforts on these types of models because they are currently the only computational models that allow the prediction of the lightness of an image region based on the luminance of an input image. In their original form, the models are not equipped with mechanisms that are sensitive to temporal adaptation, and hence did not allow to make predictions about the response to contour adaptation. We therefore implemented an adaptation mechanism in the most widely cited model (oriented difference of gaussians [ODOG]; Blakeslee & McCourt, 1999), and in two more recent spatial filtering models that employ local normalization mechanisms and might thus be better suited to account for contour adaptation (Otazu et al., 2008; Robinson et al., 2007). We show that regardless of the choice of adaptation parameters, the models cannot reproduce the contour adaptation effects that were observed psychophysically. This analysis adds further evidence to our recent claim (Betz et al., in press) that pure spatial filtering accounts are unlikely to be a correct explanation of lightness phenomena. 
Methods
Participants
Ten naive observers (four female) participated in the experiment. Observers' mean age was 29 years (min. 22 years, max. 37 years). With the exception of one observer, they had normal or corrected-to-normal vision. Observer 6 had 20/25 vision. Observers were financially compensated for their time. 
Stimuli and apparatus
The test stimuli consisted of a version of White's illusion in which a square test patch was embedded. The grating contained six bars, three light and three dark bars, with a total size of 7.32° × 7.32°. The test patch was 1.22° × 1.22° wide. The dark bars had a luminance of 39.6 cd/m2 and the light bars a luminance of 48.4 cd/m2 corresponding to a Michelson contrast of 0.1. The contrast was deliberately chosen to be low, because the effect of contour adaptation is most pronounced for low-contrast edges (Anstis, 2013). The background luminance was 44 cd/m2. The comparison square was also 1.22° × 1.22° in size. Its initial luminance was randomly set in each trial to a value between 26 and 44 cd/m2. It was presented on top of a random checkered background that consisted of 13 × 13 checks of size 0.58° × 0.58° with gray values sampled uniformly from values between 35 and 53 cd/m2
The test grating was centered horizontally on the screen and placed above a central fixation circle, so that its lower boundary was directly adjacent to the fixation circle. The comparison background was placed directly below, and in each trial the comparison patch was placed at the same distance from the fixation circle as the test patch (Figure 2). 
Figure 2
 
Illustration of the screen during matching. The observer adjusted the comparison square in the lower half of the screen to match the lightness of the test patch in the grating. The gray background was actually larger, and has been cropped for this illustration.
Figure 2
 
Illustration of the screen during matching. The observer adjusted the comparison square in the lower half of the screen to match the lightness of the test patch in the grating. The gray background was actually larger, and has been cropped for this illustration.
The adapting stimuli consisted of two parallel bars, each 0.13° (4 px) wide and 1.22° tall, separated by 1.22°. Adapting stimuli were centered on the location of the edge, such that they had 2-px overlap with the test patch and 2-px overlap with the background stripe of the grating. 
Stimuli were presented on a linearized 21-in. Siemens SMM21106LS monitor (400 × 300 mm, 1024 × 768 px, 130 Hz) controlled by a DataPixx (VPixx Technologies Inc. Saint-Bruno, QC, Canada) and custom presentation software developed in our lab and published at https://github.com/TUBvision/hrl. Observers were seated 70 cm from the screen, and their position was fixed with a chin-rest. Responses were recorded with a ResponsePixx button-box (VPixx Technologies, Inc.). 
Procedure
Our goal was to measure the effect of edge adaptation on the perceived lightness of the test patches in White's illusion. Our task and presentation parameters were similar to those employed by Anstis (2013). Adaptors and test stimuli were presented in a loop as long as required by the observers to complete their lightness setting. A trial started with the flickering adaptors that were shown for 5 s. The adaptors were contrast-reversing with a frequency of 5 Hz and were presented at 100% contrast (luminance changed between 0.24 and 88 cd/m2; background luminance was 44 cd/m2). After the adaptation period, the test stimulus was shown for 1 s, and then the adaptation cycle started again. 
Observers' task was to adjust the lightness of the comparison patch so as to match the perceived lightness of the target patch that was embedded in the square wave grating of the White stimulus. Observers indicated when they were satisfied with their setting by a button press and continued to the next trial. 
Experimental design
The independent variable was the type of adaptor used, which could be orthogonal to the grating, parallel to the grating, or none. The latter condition was included as a baseline in order to measure the strength of White's illusion for our experimental stimuli and presentation parameters. We included two more controls to test the importance of the exact alignment between adaptors and luminance edges. In these controls, the adaptors were shifted by half a bar width orthogonal to their orientation and in a random direction (see Movies 3 and 4). Figure 3 illustrates all types of adaptors used. 
Figure 3
 
Illustration of the different types of adaptors used in the experiment. From top left to bottom center: no adaptor, orthogonal to the grating, parallel to the grating, orthogonal and shifted, parallel and shifted. The bottom right stimulus illustrates the simultaneous-contrast control condition, which did not have an adaptor. Note that in the experiment, presentation of grating and adaptors was separated in time, and adaptors flickered (i.e., they switched luminance between black and white). Also note that the grating was phase shifted by 180° in half of the trials, making the test patch at that location an increment instead of a decrement. The background of the simultaneous contrast condition was dark in half of the trials.
Figure 3
 
Illustration of the different types of adaptors used in the experiment. From top left to bottom center: no adaptor, orthogonal to the grating, parallel to the grating, orthogonal and shifted, parallel and shifted. The bottom right stimulus illustrates the simultaneous-contrast control condition, which did not have an adaptor. Note that in the experiment, presentation of grating and adaptors was separated in time, and adaptors flickered (i.e., they switched luminance between black and white). Also note that the grating was phase shifted by 180° in half of the trials, making the test patch at that location an increment instead of a decrement. The background of the simultaneous contrast condition was dark in half of the trials.
Movie 3.
 
Contour adaptation with shifted orthogonal adaptors. Most observers see little effect of these adaptors, and perceive White′s illusion similar to the unadapted condition.
Movie 3.
 
Contour adaptation with shifted orthogonal adaptors. Most observers see little effect of these adaptors, and perceive White′s illusion similar to the unadapted condition.
Movie 4.
 
Contour adaptation with shifted parallel adaptors. Most observers see little effect of these adaptors, and perceive White′s illusion similar to the unadapted condition.
Movie 4.
 
Contour adaptation with shifted parallel adaptors. Most observers see little effect of these adaptors, and perceive White′s illusion similar to the unadapted condition.
We used two grating orientations (horizontal and vertical) so that the absolute adaptor orientation was not confounded with the adaptor type (i.e., parallel and orthogonal to the grating). Test patches appeared in one of four positions: on the second (near) or third bar (far) relative to fixation, and immediately to the left or to the right of the screen center. We had noticed in pilot experiments, and participants confirmed this observation, that the near position was easier to match. The far condition, on the other hand, had the advantage that the test patch was located more centrally within the stimulus, which is a more typical configuration for White's illusion. Therefore, we included both conditions. Matches did not differ between the near and far condition, however, and thus we collapsed the data from the two in the analysis. The first (i.e., topmost or leftmost) bar of the grating could either be light or dark, so that every test patch position could become an increment or a decrement with respect to the bar on which it was placed. 
In order to prevent observers from using cognitive strategies that departed from pure lightness perception for their matching (e.g., use the background luminance as a reference for their judgments if they assumed that the test patch is always mean gray), we measured lightness matches for three different patch luminances (42.2, 44, and 45.8 cd/m2). Finally, we included a control condition without adaptors in which the test patch was placed on a homogeneous background (either 39.6 or 48.4 cd/m2), so that we could compare our effect sizes to simultaneous lightness contrast. Overall, the experiment consisted of 288 trials (2 grating orientations × 5 adaptor types × 4 test patch positions × 2 grating phases × 3 test patch luminances + 2 × 4 × 2 × 3 simultaneous contrast controls). 
Results
The critical features of the results are easiest to understand by looking first at data from individual observers (Figures 4 and 5). Averages across observers are shown in Figure 6, and individual data from all observers are available as supplemental material (Supplementary Figures S1 through S10). 
Figure 4
 
Lightness matching results for one observer. The small icons indicate the stimulus condition. The x-axis denotes the three different levels of test patch luminance and the y-axis the luminance of the matching patch. Light gray data points are for test patches on a light gray background (either a light gray bar in White's illusion, or a light gray square in simultaneous contrast), dark gray points are for dark backgrounds. Small circles are individual trials, the squares indicate means across trials. Outliers are marked as red plus signs, and where excluded for the computation of the means. The black horizontal line indicates the mean luminance of the display, the solid gray lines the luminances of the light and dark grating bars, and the dotted lines the different test patch luminances. The strength of the illusion corresponds to the difference between the light and dark squares at a given test patch luminance. Note that only in the condition where the edges orthogonal to the grating have been adapted (middle panel, top row) does the illusion disappear.
Figure 4
 
Lightness matching results for one observer. The small icons indicate the stimulus condition. The x-axis denotes the three different levels of test patch luminance and the y-axis the luminance of the matching patch. Light gray data points are for test patches on a light gray background (either a light gray bar in White's illusion, or a light gray square in simultaneous contrast), dark gray points are for dark backgrounds. Small circles are individual trials, the squares indicate means across trials. Outliers are marked as red plus signs, and where excluded for the computation of the means. The black horizontal line indicates the mean luminance of the display, the solid gray lines the luminances of the light and dark grating bars, and the dotted lines the different test patch luminances. The strength of the illusion corresponds to the difference between the light and dark squares at a given test patch luminance. Note that only in the condition where the edges orthogonal to the grating have been adapted (middle panel, top row) does the illusion disappear.
Figure 5
 
Lightness matching results for a different observer (fourth from the right in Figure 6). Same conventions as in Figure 4. Note that for this observer, White's illusion is not only absent, but reversed in the orthogonal adaptor condition (middle panel, top row). Test patches on dark stripes appear dark in this condition, and test patches on light stripes appear light.
Figure 5
 
Lightness matching results for a different observer (fourth from the right in Figure 6). Same conventions as in Figure 4. Note that for this observer, White's illusion is not only absent, but reversed in the orthogonal adaptor condition (middle panel, top row). Test patches on dark stripes appear dark in this condition, and test patches on light stripes appear light.
Figure 6
 
Illusion strength for the six observers showing White's illusion in the no-adaptor condition. Colored circles represent mean values for individual observers; error bars indicate bootstrapped 95% confidence intervals for the means. Large red circles are means across observers. Observers are ordered by the illusion strength they show for White's illusion without adaptors. The observers plotted in the two previous figures correspond to the third and fourth datapoints from the right in this figure.
Figure 6
 
Illusion strength for the six observers showing White's illusion in the no-adaptor condition. Colored circles represent mean values for individual observers; error bars indicate bootstrapped 95% confidence intervals for the means. Large red circles are means across observers. Observers are ordered by the illusion strength they show for White's illusion without adaptors. The observers plotted in the two previous figures correspond to the third and fourth datapoints from the right in this figure.
In the single observer data (Figure 4), light gray symbols indicate the matches that were made to a test patch placed on a light bar, and dark gray symbols indicate the matches that were made to a test patch placed on a dark bar of the grating. Since in White's illusion test patches on light bars appear darker and test patches on dark bars appear lighter, in this representation White's illusion shows itself as light gray symbols having lower values than dark gray symbols. This is what was observed in the no-adaptor condition (Figure 4, upper left panel) for all three test patch luminances. Illusion strength can be expressed as the difference in match luminance between the test patch on a dark and on a light bar. The average magnitude of White's effect across patch luminances for this observer was 3.3 cd/m2
The lower right panel in Figure 4 shows the simultaneous lightness contrast effect, which amounted to 3.9 cd/m2, and was hence of similar magnitude as White's effect. Note that, somewhat unconventionally, in our depiction a simultaneous lightness contrast effect has the same sign as White's illusion. This is because we label the conditions with respect to the luminance of the bar on which the test patch is placed, and not with respect to the luminance of the flanking bar. We chose this type of labeling, because in our version of the stimulus, the test patches were squares and thus shared an equal amount of border with the carrier and the flanking bars. We found using the carrier bar as the reference most intuitive. 
The effect of contour adaptation can be seen by comparing the data for the orthogonal adaptor with those for the parallel one. After adapting to the edges orthogonal to the grating, White's illusion for this observer was reduced to zero (Figure 4, upper middle panel). Similar matches were made for test patches on light bars and on dark bars. On the other hand, adapting to the edges parallel to the grating had a smaller effect on the perceived lightness of the test patches. For this observer, the magnitude of White's effect was 5.0 cd/m2 after adaptation with parallel adaptors. Similarly, adaptors that were shifted with respect to the grating edges had no effect on White's illusion. The data for shifted adaptors look similar to the data in the no-adaptor condition (Figure 4, lower left and middle panel). 
Figure 5 shows the results for a different observer. The results are comparable to the first observer except for the orthogonal adaptation condition (upper middle panel). For this observer adapting to the edges orthogonal to the grating did not only reduce the lightness difference between the test patches in White's illusion but instead led to a reversal. The observer reported that adaptation made the test patches invisible and hence they merged with the bar on which they were placed. Thus, a test patch on a dark bar appeared as dark as the bar itself and vice versa for a test patch on a light bar (Figure 5). Such a reversal of the effect occurred for four out of 10 observers. 
To summarize the results across observers, we expressed illusion strength as the difference in match luminance between the test patch on a dark and on a light bar. We further averaged across the three test patch luminances. We found that six out of 10 observers showed a significant White's effect in the condition without adaptors (Figure 6); the other four did not (Figure 7). For the six observers that did perceive White's effect in the no-adaptor condition, it is evident from Figure 6 that the strength of White's illusion was reduced, and for some observers even reversed, after adaptation to the edges orthogonal to the grating. Adapting to the edges parallel to the grating slightly increased the illusion, while the shifted adaptors had little or no effect. 
Figure 7
 
Illusion strength for the four observers not showing White's illusion in the no-adaptor condition. Same conventions as in Figure 6.
Figure 7
 
Illusion strength for the four observers not showing White's illusion in the no-adaptor condition. Same conventions as in Figure 6.
To determine confidence intervals for the illusion strengths we drew 10,000 bootstrap samples from the data. We resampled with replacement from the eight data points that were measured in each condition (test patch luminance × carrier bar luminance) and then computed illusion strength as described above for each of the samples. Data points that lay more than twice the interquartile range away from the median of a condition (marked as outliers in Figure 4) were excluded from this analysis, as they were likely due to response lapses. For five out of the six observers who showed White's illusion, the 95% confidence intervals show no overlap between the no-adaptor and the orthogonal-adaptor conditions, indicating a statistically significant difference. 
There was considerable variability across observers. As mentioned above, four observers did not report perceiving White's illusion (Figure 7), and they also did not show an effect of simultaneous contrast. Some of these observers, on inquiry after the experiment, explained that they had tried to compensate their matches for putatively illusory lightness effects, although we had instructed them explicitly to match lightness as they see it and not as they infer it. According to their reports, one can assume that cognitive factors had some influence on these observers' judgments. The effect of the adaptors, however, was preserved even in the four observers who reported no White's illusion, as orthogonal adaptors reduced illusion strength, parallel adaptors slightly enhanced it, and shifted adaptors had no effect. We address the issue of cognitive effects on perceptual judgments in more detail below (see Difficulties with matching tasks section). 
Computational modeling
The effects of contour adaptation on White's illusion provide a challenge for spatial filtering models with contrast normalization (e.g., Blakeslee & McCourt, 1999; Otazu et al., 2008; Robinson et al., 2007). These models explain White's illusion as the result of cortical filtering of the input stimulus with a set of filters that span a large range of spatial frequencies and orientations. The outputs of the individual filters are weighted depending on the energy in each filter response and then recombined to create a lightness image. The resulting output image has lower values at locations that observers tend to perceive as dark, and higher values at locations that observers tend to perceive as bright, although the luminances at these locations are identical in the input image. Spatial filtering models were not designed to treat luminance borders explicitly, but the orientation and spatial frequency selectivity of their component filters make the models responsive to contrast edges. We have recently analyzed these models (Betz et al., under revision), and concluded that they are unlikely to be a correct explanation of lightness perception. However, the adaptation effects reported here might be the result of mechanisms that interfere with visual processing at the same early level of processing at which spatial filtering models presumably operate. We therefore wanted to test whether or to what extent the models could capture the experimentally observed adaptation effect. 
The models in their original form did not contain a mechanism that would implement a temporal adaptation effect. We therefore augmented the ODOG (Blakeslee & McCourt, 1999), FLODOG (Robinson et al., 2007) and BIWAM (Otazu et al., 2008) models by a respective adaptation mechanism. To give the models a fair chance we tested them with favorable, even potentially unrealistic, choices of adaptation parameters. For the implementation we had to decide between two candidate mechanisms of adaptation. Retinal adaptation would be modeled by changing the input to the model. Cortical adaptation would require a modification of the model implementation; in particular, it would require the possibility of attenuating the individual filter responses depending on a filter's response to the adapting stimulus. Anstis (2013) has argued that the contour adaptation mechanism is unlikely to be of retinal origin, because the mean luminance of the adaptors over time is equal to the background, and the adaptors do not create an afterimage. Also, subjective contours do not adapt, which has been interpreted to suggest that the relevant adaptation happens in primary visual cortex (Anstis, 2013). 
We therefore implemented a variant of cortical adaptation as a reduction in the gain of the linear filters, in close analogy to previous work (Goris, Putzeys, Wagemans, & Wichmann, 2013). First, we compute the response of each filter to the adaptor. We then use these adaptor responses to determine the magnitude of attenuation for the filter responses to the stimulus. The filter responses at each location of the image are attenuated in proportion to the adaptor responses at that location. The adaptors are changing luminance (back and forth between black and white on a mean gray background) over time. However, for simplicity, we compute the response to the adaptor by computing the response of the filter to a dark adaptor on a gray background and by taking the absolute value as the maximal response of the filter to the flickering adaptor over time. This approach allows us to test whether it is possible to predict the observed contour adaptation effects qualitatively with the ODOG, FLODOG, or BIWAM model. 
In order to model this type of adaptation we need to consider two factors, the effect of maximum adaptation and the amount of response reduction to any stimulus between no adaptation and maximal adaptation. The maximum adaptation effect is the magnitude of response reduction when the filter has previously been adapted with an optimal stimulus. We express the maximum adaptation effect as a fraction α of the response of an unadapted filter. It has been shown that firing rates in cat simple cells decrease on average to about 20% of their pre-adaptation value after prolonged adaptation (Sanchez-Vives, Nowak, & McCormick, 2000). The (physiologically unrealistic) limit for adaptation would be a filter response of zero at the adapted locations. Second, we need to determine how the response of a filter changes with varying amounts of adaptation due to stimuli that are less than optimal. This second factor is more complicated to implement, because none of the models contains any nonlinearities that would cause the filter responses to saturate. Therefore, the level of stimulation for complete adaptation of the filter is undefined, and so is the function that relates the magnitude of adaptation to the level of stimulation. We modeled the adaptation level as an inverse cumulative Gaussian function of the adaptor response. Free parameters are the mean and the standard deviation of the Gaussian, μ and σ. μ defines the amount of response to the adapting stimulus that is required to reach 50% of the maximum adaptation effect. σ determines how quickly adaptation changes for different values of the adaptor response. Higher values of σ imply that adaptation drops quickly for suboptimal stimulation, whereas lower values imply that even a weak response to the adapting stimulus would still lead to substantial adaptation. Thus, the adapted response of each filter is given as  where Φ is the cumulative normal distribution function, r0 is the unadapted response, and ra is the response to the adapting stimulus. Examples of this function for different parameter values are shown in Figure 8. These adapted filter responses are then added and normalized in exactly the same way as the unadapted responses would be in the standard versions of the models.  
Figure 8
 
Illustration of the effect of the three parameters on the function relating the response to the adapting stimulus to the level of adaptation. The x-axis shows the response to the adapting stimulus; the y-axis indicates the response strength resulting from this adaptation level. Each panel illustrates the effect of changing one of the three parameters while keeping the other two fixed. Black lines indicate smaller parameter values.
Figure 8
 
Illustration of the effect of the three parameters on the function relating the response to the adapting stimulus to the level of adaptation. The x-axis shows the response to the adapting stimulus; the y-axis indicates the response strength resulting from this adaptation level. Each panel illustrates the effect of changing one of the three parameters while keeping the other two fixed. Black lines indicate smaller parameter values.
To evaluate whether the models are in principle capable of reproducing the observed effects of contour adaptation, we explored the parameter space of α, μ, and σ. We first looked at combinations of low, medium, and high values for the three parameters. For α, the most extreme parameter setting is zero, which would correspond to complete response reduction of the filter (Figure 8, left panel, black line). A medium adaptation effect would require a more realistic parameter setting of about 0.2, and a mild adaptation effect could be modeled with a parameter setting of 0.5. A meaningful range of parameter settings for μ and σ is less obvious. An important indicator for the upper limit is the largest possible response of a filter. Since all our stimuli are in the range [0, 1], the maximum response of a filter would be achieved by a stimulus that is 1 where the filter is positive, and 0 where the filter is negative. This response is simply the sum of positive values in the filters integration field. For filters in the ODOG and FLODOG models, this value is ≈0.32. The exact value depends on the specific parameters used in our model implementation. Since μ defines the value where the cumulative Gaussian function has a value of 0.5, setting μ to a value of 0.32 would mean that even filters that are optimally responsive to the adaptors will reach only 50% of the maximum adaptation level. On the other hand, setting μ to 0 would mean that even filters that do not at all respond to the adapting stimulus will reach 50% of the maximum adaptation level. Both options are too extreme, because an optimal stimulus should cause full adaptation, and a stimulus that causes no response in the filter should not cause any adaptation. A meaningful range is somewhere in between these two values. We chose values of μ ∈ {0.08, 0.01, 0.001} for the present exploration. A finer sampling of the parameter space was required for the BIWAM model because for that model, the effect of changing the μ parameter was not monotonic. For BIWAM we tested μ values between 0.1 and 0.0005. Having determined values for μ somewhat constrained the plausible range for σ. In order to ensure that there is (almost) no adaptation in filters that have no response to the adaptor, σ should be at most half as large as μ. We tested model responses for values of σ ∈ {0.04, 0.005, 0.0001}, skipping those combinations where Image not available For the BIWAM, we tested eight linearly spaced values between 0.005 and 0.0001.  
Our results show that the effects of adaptation in ODOG and FLODOG are similar. In both models, it was not possible to reproduce the effect of contour adaptation observed psychophysically (Figure 9). Most of the tested parameter settings caused only mild adaptation effects, and failed to reproduce the strong and specific effect of orthogonal adaptors on perceived patch lightness. Only with extreme parameter settings the reduction or even reversal of the illusion in the orthogonal adaptor condition could be reproduced. However, these settings caused unspecific adaptation effects in all conditions, including the parallel and the shifted adaptor conditions, which is in contradiction to what was observed experimentally. 
Figure 9
 
Effects of adaptation in the two models. Top panel: ODOG model. Bottom panel: FLODOG model. Different symbols, colors, and sizes encode different values of the adaptation parameters α, μ, and σ, respectively. Red circles replot means across observers from Figure 6, normalized to an illusion strength of 1 in the unadapted condition.
Figure 9
 
Effects of adaptation in the two models. Top panel: ODOG model. Bottom panel: FLODOG model. Different symbols, colors, and sizes encode different values of the adaptation parameters α, μ, and σ, respectively. Red circles replot means across observers from Figure 6, normalized to an illusion strength of 1 in the unadapted condition.
To accomplish this strong adaptation we had to use small values for μ, which implies that even filters that responded only weakly to the adaptor were strongly inhibited. This allowed the adaptation to affect also large filters in the models. Such large filters are important for determining surface lightness far away from the edges, but they are only weakly stimulated by the small adaptors. These filters lack location specificity and hence the location specificity of the adaptors was lost. Results for the BIWAM model are difficult to visualize due to the larger parameter space sampled, so results are not shown here. However, none of the 3,200 parameter combinations tested could reduce the illusion strength in the orthogonal adaptor condition close to or even below zero, without also affecting illusion strength in at least one of the other adaptor conditions in a manner inconsistent with the data. We conclude that it is not possible to explain the effect of contour adaptation on perceived lightness in White's illusion purely within the framework of spatial filtering models with contrast normalization. Our results leave open the possibility that the relevant adaptation takes place prior to the processing modeled by ODOG, FLODOG, or BIWAM. In that case, the correct way to simulate the effects of adaptation would be to modify the input to the models, rather than adapting the filter responses. However, without an explicit theory about the processing steps that occur prior to the filtering stage of the model, it is unclear what type of input modification should exactly be performed. We would argue that if the preprocessing is complex enough to explain the effect of contour adaptation on perceived lightness, for example involving filling-in across weakened borders, it might be that the spatial filtering computations are not needed as an explanation of lightness perception at all. At the very least, it would appear questionable that the models are usually tested with unmodified images if the assumption is that their actual input is already preprocessed. 
Discussion
We have shown that perceiving the luminance step between the test patch and the grating bar on which it is placed is a critical condition for perceiving White's illusion. If the contrast along this edge is reduced or even nulled due to contour adaptation, White's illusion is no longer perceived, and for some observers, the illusion even reverses. For this effect to appear, it is essential that contour adaptation takes place exactly at the location of the luminance edge. Shifting the adapting lines by half the bar width of the grating eliminated the effect of adaptation and rendered the results indistinguishable from those obtained without adaptors. These results are consistent with previously reported effects of contour adaptation on perceived lightness (Anstis & Greenlee, 2014), and point to a critical role of luminance borders for surface lightness. Our experiments extend that work by showing that not all luminance borders that enclose a surface are treated equally in the computation of lightness in White's illusion. 
Potential causes of the effect
We found that temporal adaptation selectively interfered with lightness perception in White's illusion when we adapted the edges of the test patches that were orthogonal to the bars of the carrier grating. This finding is mirrored by results obtained when masking White's stimulus using narrowband noise of different orientations (Salmela & Laurinen, 2009). In that study, noise oriented parallel to the grating increased the strength of White's illusion, whereas noise oriented orthogonal to the grating reduced or even reversed the illusion. Both sets of data show that White's illusion is predominantly modulated by the luminance contrast across the test patch edges that are oriented orthogonal to the grating. However, the question how the visual system distinguishes between both types of edges, and in which ways they are subsequently treated differently, remains unanswered. 
In the following we propose iso-orientation surround suppression (see Gheorghiu, Kingdom, & Petkov, 2014, for a recent review) as a low-level candidate mechanism that could potentially account for the observed asymmetry. Iso-orientation surround suppression is a cortical mechanism by which neurons are inhibited by activity of other neurons in their local surround. The inhibition is orientation selective (i.e., cells with a certain orientation preference are most strongly inhibited by other cells with the same preference; Blakemore & Tobin, 1972; Cavanaugh, Bair, & Movshon, 2002; Henry, Joshi, Xing, Shapley, & Hawken, 2013; Nothdurft, Gallant, & Van Essen, 1999). The above-mentioned results suggest that lightness perception depends on the activity of edge-sensitive mechanisms such as oriented odd-phase filters. Weakening the response of filters with an orientation preference parallel to the grating while at the same time leaving the response of filters with an orientation preference orthogonal to the grating intact, would lead exactly to the asymmetry that was observed in White's illusion (see Figure 10). 
Figure 10
 
Illustration of the effect of oriented surround suppression on the filters responding to orthogonal and parallel edges of a test patch in White's stimulus. Filters that respond to the test patch's edges parallel to the grating are inhibited by the response to the edges between the grating bars. The orthogonal edges are unaffected by surround suppression because there are no iso-oriented edges in their surround. This leads to the asymmetry between the two types of edges that is required for explaining White's illusion.
Figure 10
 
Illustration of the effect of oriented surround suppression on the filters responding to orthogonal and parallel edges of a test patch in White's stimulus. Filters that respond to the test patch's edges parallel to the grating are inhibited by the response to the edges between the grating bars. The orthogonal edges are unaffected by surround suppression because there are no iso-oriented edges in their surround. This leads to the asymmetry between the two types of edges that is required for explaining White's illusion.
In particular, filters that are responsive to the grating would inhibit filters that are responsive to test patch edges parallel to the grating. Filters that are responsive to the test patch edge orthogonal to the grating would not experience such inhibition, and would consequently have a stronger response. If surface lightness is determined by a mechanism that integrates the information from all edges of a surface, the test patch will appear more similar to the neighboring bars because of the reduced filter response. Hence a test patch on a dark bar will look light and a test patch on a light bar will look dark. This is what is observed in White's illusion. Such a mechanism should also lead to differences in the perceived contrast at the two types of edges of the test patch. This could be tested in future work. 
We have suggested that White's illusion results from a differential weighting of the contrasts at edges orthogonal and parallel to the grating: A full contrast effect across the orthogonal edges outweighs a reduced contrast effect across the parallel edges. This interpretation is challenged by previous reports that tried to quantify the relative contribution of the two types of edges (Anstis, 2005; Clifford & Spehar, 2003). In these studies, the authors used a colored inducing grating, and found that the perceived color of the gray test patches was influenced by two factors: contrast with the color of the carrier bar and assimilation to the color of the flanking bars. However, both studies found assimilation effects specifically at spatial frequencies higher than the 0.4 cpd of the grating that was employed in our study. With gratings of low spatial frequencies, only contrast effects were observed. It is possible that the assimilation effects at higher spatial frequencies result from chromatic aberration, and are thus an optical effect rather than the result of neural processing. It has previously been shown that White's illusion increases with increasing spatial frequency of the grating (Blakeslee & McCourt, 2004). The increase was most prominent for grating frequencies above 2 cpd. It is thus possible that multiple mechanisms contribute to the perceptual effect, one being the edge-based contrast modulated by iso-orientation surround suppression, as proposed here, and another that depends on assimilation due to spatial smoothing at high spatial frequencies. Whether this spatial smoothing is of optical or neural origin is an open question. 
An experimental approach to test the question whether the edges parallel to the grating contribute at all to the perceived lightness of the test patches, would be to quantitatively compare the effect sizes of White's illusion and simultaneous lightness contrast within observers. Our model sketch predicts White's illusion to be at most as strong as simultaneous contrast (assuming that the inhibition of the parallel edges reduces their effect to zero). Due to the difficulties discussed in the Difficulties with matching tasks section, it is difficult to draw conclusions from the present contrast data. The literature is divided with respect to the question of which of the two effects is stronger: Anderson (1997) and White (1981) reported that White's illusion is stronger, and Blakeslee and McCourt (1999) reported the opposite. Settling this question in future research would be helpful to determine which individual factors determine the perceptual outcome known as White's illusion. 
The filling-in problem
The above description raises the question how surface lightness is computed from edge contrast. An edge-based account of lightness perception is clearly incomplete as long as it does not address this so-called filling-in problem. The connection between contour adaptation and filling in has been discussed before (Anstis, 2013; Anstis & Greenlee, 2014), but at present there exists no satisfactory mechanistic model of the process. 
Some authors conceive of filling-in as the spread of information from one neuron to the next in retinotopically organized visual areas, and the percept of a surface depends on the firing of a corresponding patch of neurons (Grossberg & Mingolla, 1985; Paradiso & Nakayama, 1991; Ramachandran, 1992; Ramachandran & Gregory, 1991; Rossi & Paradiso, 1996). Others have rejected this isomorphic notion of a surface representation either on philosophical grounds (O'Regan, 1992), or because of psychophysical (Blakeslee & McCourt, 2008; Robinson & de Sa, 2013) and electrophysiological evidence (von der Heydt, Friedman, & Zhou, 2003; Zurawel, Ayzenshtat, Zweig, Shapley, & Slovin, 2014) that is inconsistent with the idea. Our results do not settle the discrepancy, but they underline the importance of solving the filling-in problem in order to understand lightness perception. One casual observation that we found noteworthy is that, for some observers, the orthogonal adaptors rendered the test patch indistinguishable from the bar on which it was placed. In other words, the test patch was perceptually filled in with the lightness of its background bar. This is interesting, because the parallel edges of the test patch to the neighboring bars were not adapted in this condition. These edges were perfectly visible, having the same contrast as in the no-adaptor condition in which they were always perceived, and the contrast across these edges could have signaled that the lightness of the test patch was different from that of the bar. To us this suggests that perceived surface lightness need not be fully consistent with the edge contrast signaled across all edges of a surface. 
Difficulties with matching tasks
In the following we will discuss some general difficulties with matching tasks as we think that they were not specific to our setup or our stimuli. We would like to encourage an open discussion of these issues because that would make the comparison and evaluation of results from different experiments more transparent, in particular when they are inconsistent (cf. Bindman & Chubb, 2004). 
It is known that instructions can have an effect on lightness matching (Arend & Reeves, 1986; Bäuml, 1999; Reeves, Amano, & Foster, 2008; Troost & de Weert, 1991). We observed that, although observers were explicitly instructed to set the match luminance according to the perceived lightness of the test, some of them still adopted a different strategy and tried to act like a photometer (i.e., they tried to infer the veridical luminance of the test). This tendency might be attributable to observers' strong desire to please the experimenter (cf. Durgin et al., 2009). 
If observers believe that they are confronted with an optical illusion, and if they further believe that they found a way of knowing the “correct” answer, then their responses are likely to be biased by their cognitive strategy (see Runeson, 1977, for an illustrative example of such strategic behavior). For example, when the test patches have the same luminance as the background of the screen (which is a common experimental setup), observers could adjust their matches relative to the background luminance instead of matching test patch lightness. To counteract such tendencies, we adopted a perturbation approach and measured lightness matches for three different test patch luminances. Observers did not know in advance how many values of test patch luminance were tested. If they had used a strategy such as setting the match luminance always to the same value to counteract a presumed illusion, this would have been evident in constant match values across the different test luminances. Thus, although this perturbation does not necessarily prevent observers from strategic matching, it would at least allow us to identify such flawed matches. 
In our data, match luminance generally increased with increasing test patch luminance, indicating that observers were indeed matching individual stimuli. However, three of the four observers that did not perceive White's illusion and that did not show a simultaneous contrast effect (Figure 7), also did not show this increase in matched luminance with increasing test luminance in the simultaneous contrast condition. This is a further indication that these observers followed some strategy, but it does not allow us to understand what exactly they were doing. The downside of this kind of control condition is that it considerably increases the number of experimental trials without contributing much to answering the main research question. Clearly, even with such safeguards in place observers might still engage in other strategies. 
Another observation is the large variability across and within observers. Variability within observers might not simply be variance in setting the responses. With simple stimuli of the kind used here, the perceived lightness of different image parts can change over time. Readers can try that for themselves. It is very much possible to see test patches on dark and light backgrounds as identical, especially if one has the background knowledge that they are in fact identical. Thus, response differences across trials may reflect the fact that observers did indeed perceive the stimulus differently in some trials. In that case, the mean across trials may not be a good indicator of subjective experience. The variability across observers in lightness illusions seems to indicate that there really are large individual differences in how pronounced certain lightness phenomena are (e.g., Ripamonti et al., 2004; Robinson et al., 2007). For example, Ekroll and Faul (2013) have demonstrated that observers perceived an additional transparent layer for similar stimuli when they were of low contrast. So in stimulus conditions with relatively low contrast, some observers may have perceived an additional transparent layer that could make the matching task ill-defined. This makes it difficult to argue which of two lightness illusions is stronger, and it might also explain the inconsistent results with respect to the relative strengths of White's illusion and simultaneous contrast. 
Finally, as can be seen in the data from individual observers (see supplemental material), observers sometimes selected match lightnesses that were outside the luminance range spanned by the grating. This is surprising, as it is difficult to imagine how a test patch that has a luminance somewhere in between the two luminances of the grating could appear much brighter than the bright bar or darker than the dark bar. These data points cannot be easily attributed to response lapses, because they were not accompanied by particularly short or long response times and they were different from the random start value. Thus, observers seem to have deliberately chosen these extreme luminances to match the test patch luminance. A possible explanation for this finding might be that the luminance range in the match background was larger than the luminance range in the grating. If observers performed a contrast normalization between the test display and the match display (see Singh & Anderson, 2006; Zeiner & Maertens, 2014), this could explain why they used a higher luminance range for their matches than the range that was spanned by the grating. This is just one further example that lightness matching is not such an easy and straightforward task as it might appear on the surface. 
Conclusion
Our experiments show that luminance edges play a central role in White's illusion. The illusion seems to be predominantly caused by the luminance edge between the test patch and its background bar, while the edge contrast to neighboring bars is largely ignored. The effect of contour adaptation on White's illusion could not be replicated by spatial filtering models, which adds further evidence against the adequacy of such models as a mechanistic explanation of White's illusion in particular, and lightness perception in general. Our results highlight the importance of further investigating the question of how surface lightness is computed from edge contrast. 
Acknowledgments
This work was supported by the German Research Foundation (GRK 1589/1 Sensory Computation in Neural Systems, and DFG MA5127/1-1 to Marianne Maertens). Felix Wichmann was funded, in part, by the German Federal Ministry of Education and Research (BMBF) through the Bernstein Computational Neuroscience Program Tübingen (FKZ: 01GQ1002). We would like to thank Alan Robinson for the code of his Matlab implementations of the ODOG and FLODOG models. 
Commercial relationships: none. 
Corresponding author: Torsten Betz. 
Email: torsten.betz@bccn-berlin.de. 
Address: Technische Universität Berlin, Sekretariat MAR 5-3, Marchstr. 23, 10587, Berlin, Germany. 
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Figure 1
 
White's illusion. The test patch on the black bar shares more border with the white flanking bars than with the black coaxial bar, so based on edge contrast alone, it should appear darker than the test patch on the white bar. The opposite is the case.
Figure 1
 
White's illusion. The test patch on the black bar shares more border with the white flanking bars than with the black coaxial bar, so based on edge contrast alone, it should appear darker than the test patch on the white bar. The opposite is the case.
Movie 1.
 
Contour adaptation of the test patch edges orthogonal to the grating. Most observers see the test patches merge with the grating bar on which they are placed.
Movie 1.
 
Contour adaptation of the test patch edges orthogonal to the grating. Most observers see the test patches merge with the grating bar on which they are placed.
Movie 2.
 
Contour adaptation of the test patch edges parallel to the grating. Most observers see the test patches bleed out into the neighboring bars.
Movie 2.
 
Contour adaptation of the test patch edges parallel to the grating. Most observers see the test patches bleed out into the neighboring bars.
Figure 2
 
Illustration of the screen during matching. The observer adjusted the comparison square in the lower half of the screen to match the lightness of the test patch in the grating. The gray background was actually larger, and has been cropped for this illustration.
Figure 2
 
Illustration of the screen during matching. The observer adjusted the comparison square in the lower half of the screen to match the lightness of the test patch in the grating. The gray background was actually larger, and has been cropped for this illustration.
Figure 3
 
Illustration of the different types of adaptors used in the experiment. From top left to bottom center: no adaptor, orthogonal to the grating, parallel to the grating, orthogonal and shifted, parallel and shifted. The bottom right stimulus illustrates the simultaneous-contrast control condition, which did not have an adaptor. Note that in the experiment, presentation of grating and adaptors was separated in time, and adaptors flickered (i.e., they switched luminance between black and white). Also note that the grating was phase shifted by 180° in half of the trials, making the test patch at that location an increment instead of a decrement. The background of the simultaneous contrast condition was dark in half of the trials.
Figure 3
 
Illustration of the different types of adaptors used in the experiment. From top left to bottom center: no adaptor, orthogonal to the grating, parallel to the grating, orthogonal and shifted, parallel and shifted. The bottom right stimulus illustrates the simultaneous-contrast control condition, which did not have an adaptor. Note that in the experiment, presentation of grating and adaptors was separated in time, and adaptors flickered (i.e., they switched luminance between black and white). Also note that the grating was phase shifted by 180° in half of the trials, making the test patch at that location an increment instead of a decrement. The background of the simultaneous contrast condition was dark in half of the trials.
Movie 3.
 
Contour adaptation with shifted orthogonal adaptors. Most observers see little effect of these adaptors, and perceive White′s illusion similar to the unadapted condition.
Movie 3.
 
Contour adaptation with shifted orthogonal adaptors. Most observers see little effect of these adaptors, and perceive White′s illusion similar to the unadapted condition.
Movie 4.
 
Contour adaptation with shifted parallel adaptors. Most observers see little effect of these adaptors, and perceive White′s illusion similar to the unadapted condition.
Movie 4.
 
Contour adaptation with shifted parallel adaptors. Most observers see little effect of these adaptors, and perceive White′s illusion similar to the unadapted condition.
Figure 4
 
Lightness matching results for one observer. The small icons indicate the stimulus condition. The x-axis denotes the three different levels of test patch luminance and the y-axis the luminance of the matching patch. Light gray data points are for test patches on a light gray background (either a light gray bar in White's illusion, or a light gray square in simultaneous contrast), dark gray points are for dark backgrounds. Small circles are individual trials, the squares indicate means across trials. Outliers are marked as red plus signs, and where excluded for the computation of the means. The black horizontal line indicates the mean luminance of the display, the solid gray lines the luminances of the light and dark grating bars, and the dotted lines the different test patch luminances. The strength of the illusion corresponds to the difference between the light and dark squares at a given test patch luminance. Note that only in the condition where the edges orthogonal to the grating have been adapted (middle panel, top row) does the illusion disappear.
Figure 4
 
Lightness matching results for one observer. The small icons indicate the stimulus condition. The x-axis denotes the three different levels of test patch luminance and the y-axis the luminance of the matching patch. Light gray data points are for test patches on a light gray background (either a light gray bar in White's illusion, or a light gray square in simultaneous contrast), dark gray points are for dark backgrounds. Small circles are individual trials, the squares indicate means across trials. Outliers are marked as red plus signs, and where excluded for the computation of the means. The black horizontal line indicates the mean luminance of the display, the solid gray lines the luminances of the light and dark grating bars, and the dotted lines the different test patch luminances. The strength of the illusion corresponds to the difference between the light and dark squares at a given test patch luminance. Note that only in the condition where the edges orthogonal to the grating have been adapted (middle panel, top row) does the illusion disappear.
Figure 5
 
Lightness matching results for a different observer (fourth from the right in Figure 6). Same conventions as in Figure 4. Note that for this observer, White's illusion is not only absent, but reversed in the orthogonal adaptor condition (middle panel, top row). Test patches on dark stripes appear dark in this condition, and test patches on light stripes appear light.
Figure 5
 
Lightness matching results for a different observer (fourth from the right in Figure 6). Same conventions as in Figure 4. Note that for this observer, White's illusion is not only absent, but reversed in the orthogonal adaptor condition (middle panel, top row). Test patches on dark stripes appear dark in this condition, and test patches on light stripes appear light.
Figure 6
 
Illusion strength for the six observers showing White's illusion in the no-adaptor condition. Colored circles represent mean values for individual observers; error bars indicate bootstrapped 95% confidence intervals for the means. Large red circles are means across observers. Observers are ordered by the illusion strength they show for White's illusion without adaptors. The observers plotted in the two previous figures correspond to the third and fourth datapoints from the right in this figure.
Figure 6
 
Illusion strength for the six observers showing White's illusion in the no-adaptor condition. Colored circles represent mean values for individual observers; error bars indicate bootstrapped 95% confidence intervals for the means. Large red circles are means across observers. Observers are ordered by the illusion strength they show for White's illusion without adaptors. The observers plotted in the two previous figures correspond to the third and fourth datapoints from the right in this figure.
Figure 7
 
Illusion strength for the four observers not showing White's illusion in the no-adaptor condition. Same conventions as in Figure 6.
Figure 7
 
Illusion strength for the four observers not showing White's illusion in the no-adaptor condition. Same conventions as in Figure 6.
Figure 8
 
Illustration of the effect of the three parameters on the function relating the response to the adapting stimulus to the level of adaptation. The x-axis shows the response to the adapting stimulus; the y-axis indicates the response strength resulting from this adaptation level. Each panel illustrates the effect of changing one of the three parameters while keeping the other two fixed. Black lines indicate smaller parameter values.
Figure 8
 
Illustration of the effect of the three parameters on the function relating the response to the adapting stimulus to the level of adaptation. The x-axis shows the response to the adapting stimulus; the y-axis indicates the response strength resulting from this adaptation level. Each panel illustrates the effect of changing one of the three parameters while keeping the other two fixed. Black lines indicate smaller parameter values.
Figure 9
 
Effects of adaptation in the two models. Top panel: ODOG model. Bottom panel: FLODOG model. Different symbols, colors, and sizes encode different values of the adaptation parameters α, μ, and σ, respectively. Red circles replot means across observers from Figure 6, normalized to an illusion strength of 1 in the unadapted condition.
Figure 9
 
Effects of adaptation in the two models. Top panel: ODOG model. Bottom panel: FLODOG model. Different symbols, colors, and sizes encode different values of the adaptation parameters α, μ, and σ, respectively. Red circles replot means across observers from Figure 6, normalized to an illusion strength of 1 in the unadapted condition.
Figure 10
 
Illustration of the effect of oriented surround suppression on the filters responding to orthogonal and parallel edges of a test patch in White's stimulus. Filters that respond to the test patch's edges parallel to the grating are inhibited by the response to the edges between the grating bars. The orthogonal edges are unaffected by surround suppression because there are no iso-oriented edges in their surround. This leads to the asymmetry between the two types of edges that is required for explaining White's illusion.
Figure 10
 
Illustration of the effect of oriented surround suppression on the filters responding to orthogonal and parallel edges of a test patch in White's stimulus. Filters that respond to the test patch's edges parallel to the grating are inhibited by the response to the edges between the grating bars. The orthogonal edges are unaffected by surround suppression because there are no iso-oriented edges in their surround. This leads to the asymmetry between the two types of edges that is required for explaining White's illusion.
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