Effects of binocular disparity on binocular luminance combination

Abstract

This study aimed to examine the effects of binocular disparity on binocular combination of brightness information coming from luminance increments and decrements. The point of subjective equality was determined by asking the observers to judge which stimulus appeared brighter—a bar stimulus with variable disparity or another stimulus with zero disparity. For the bar stimulus, the interocular luminance ratio was varied to trace an equal brightness curve. Binocular disparity had no effect on luminance increments presented on a gray or black background. In contrast, when luminance decrements were presented on a gray background, non-zero disparities elevated points of subjective equality for stimuli with interocular luminance differences. This means that the binocular brightness combination of the two monocular signals shifted from winner-take-all summation toward linear averaging. It has been argued that this effect may be caused by non-zero binocular disparities attenuating interocular suppression, which is deemed to operate normally with zero disparity.

Introduction

Researchers have examined how visual signals for brightness delivered to the two eyes are combined to yield brightness perception in binocular viewing, hereafter referred to as binocular brightness. In their review of previous studies, Blake and Fox (1973) argued that, although viewing with both eyes open is not dramatically different from viewing with only one eye, sensitive experiments can reveal differences between binocular and monocular brightness. This trend dates back to the work of Fechner (1860), who noted that the binocular brightness of the light spot decreases as the luminance in one eye increases from zero. This phenomenon is referred to as Fechner's paradox. Levelt (1965) and Engel (1970) conducted controlled experiments on binocular brightness combinations. They found that, when both eyes were sufficiently illuminated, binocular brightness appeared to be perceptually matched among the conditions in which the linear averages of the two monocular luminance values were equivalent. In contrast, when the luminance in one eye was much lower than that in the other eye, the eye with higher luminance determined the binocular brightness (winner-take-all), which is consistent with Fechner's paradox. 

The way of binocular brightness combination is known to vary with the luminance polarity and the luminance difference between a stimulus and its background. According to Anstis and Ho (1998), when a luminance decrement (a dark spot) was presented on a white background, the binocular brightness combination took the form of winner-take-all. That is, a greater decrement in one eye always determined the binocular brightness without binocular averaging. Moreover, Baker, Wallis, Georgeson, and Meese (2012) adopted a gray background, instead of black or white, to test the binocular brightness of light and dark spots. Their results showed that the equal brightness functions for luminance increments deviated from the prediction of linear averaging, being close to winner-take-all. The functions for luminance decrements have also exhibited compromised linearity for small decrements. Ding and Levi (2017) reported similar results for a wider range of target and background luminance. In Ding and Levi's (2017) model, interocular suppression is assumed to be weak when the total luminance of the stimuli is low and the luminance values are balanced between the two eyes, resulting in linear averaging. If the total luminance is at a medium level, the binocular brightness combination is expected to show compromised linearity for both luminance increments and decrements. This model was recently expanded to explain depth perception (Ding & Levi, 2021). 

Interestingly, the binocular brightness of Ganzfeld stimuli followed linear averaging over the entire range of stimulus luminance, with no indication of Fechner's paradox (Bourassa & Rule, 1994). This could be due to a weaker interocular suppression by Ganzfeld stimuli than by spatially localized stimuli. In contrast, binocular combinations have been reported to shift from winner-take-all to linear averaging when binocularly matched features are added to stimuli in dichoptic color-saturation mixtures (Kingdom & Libenson, 2015) and when the luminance contrasts of surroundings are the same for both eyes in dichoptic contrast mixtures (Wang, Ding, Levi, & Cooper, 2022). This evidence suggests that suppression of the eye with the lower contrast by the eye with the higher contrast results in winner-take-all behavior. Binocularly matched features would keep the interocular suppression balanced between the two eyes, resulting in a shift of the binocular combination toward linear averaging. This notion is supported by studies showing that binocularly matched features reduce the effects of dichoptic masking (Jennings & Kingdom, 2019; Kingdom & Wang, 2015). 

The aforementioned studies focused on situations in which binocular stimuli were presented at the corresponding retinal positions. Recently, Maehara and Murakami (2020) reported that suprathreshold grating stimuli with non-zero binocular disparity appeared to have a higher luminance contrast than those with zero binocular disparity. As simultaneous stimulations at corresponding retinal points are known to elicit strong interocular suppression (Blake, 1989; for an opposing view, see a dynamical model proposed by Said & Heeger, 2013), the authors attributed the perceptual enhancement of luminance contrast to weaker interocular suppression in stimuli with binocular disparities. In contrast, binocular summation is assumed to be functional even for stimuli with binocular disparity, because the same mechanisms may underlie binocular summation and stereopsis (Kingdom, Yared, Hibbard, & May, 2020; Lema & Blake, 1977; May & Zhaoping, 2022; Rose, Blake, & Halpern, 1988). 

The present study aimed to examine the effects of binocular disparity on the binocular brightness combination of luminance increments and decrements. We measured the point of subjective equality (PSE) in brightness between a standard stimulus with zero disparity and a comparison stimulus with a certain interocular excursion ratio and a zero, crossed, or uncrossed binocular disparity. If binocular disparity caused weaker interocular suppression, then the binocular brightness combination for a stimulus with non-zero as opposed to zero disparity would be shifted away from winner-take-all toward averaging. There were three conditions in the luminance excursion (i.e., a generic term to indicate a difference, either incremental or decremental, in the luminance values between the stimulus and background) of the comparison stimulus: a decrement on a gray background, an increment on a gray background, and an increment on a black background. We used a short vertical bar-shaped stimulus as the comparison stimulus so that it did not stimulate the corresponding retinal positions in the two eyes when presented with non-zero binocular disparity. To avoid a masking effect on the comparison stimulus, the standard stimulus was a pair of relatively large rectangles presented within a region outside the location of the comparison stimulus. Binocular disparities within the fusion limit were selected because preliminary observers had difficulty judging the brightness in double vision when the luminance excursions differed between the eyes. 

Methods

Observers

Four adults naïve to the purpose of the experiment and two authors (observers 5 and 6) participated in Experiment 1. Four different naïve adults and the two authors participated in Experiment 2. All observers had normal or corrected-to-normal visual acuity with normal stereopsis and provided written informed consent. The present study followed protocols approved by the institutional ethics committee and was in accordance with the tenets of the Declaration of Helsinki. 

Apparatus

In a dimly lit room, a ViSaGe MKII Stimulus Generator (Cambridge Research Systems, Rochester, UK) with a 14-bit gray-level resolution was used to present the stimuli on a CRT video monitor (RDF223H; Mitsubishi, Tokyo, Japan). The display resolution was 800 × 600 pixels, with a refresh rate of 100 Hz. The observers viewed the screen using a mirror stereoscope (Chuo Precision Industrial, Tokyo, Japan). The presentation area subtended a 13° × 13° arc in each eye. The viewing distance was 57 cm. 

Stimuli

There were three excursion conditions: a decrement on a gray background, an increment on a gray background, and an increment on a black background. The gray and black backgrounds had luminance values of 35.1 cd/m² and 0.2 cd/m², respectively. A square frame subtending 9.9° × 9.9° was presented to each eye to aid binocular fusion (Figure 1). A fixation dot was presented at the center of the presentation area with two flanking dots positioned 1.5° to the left and right (Figure 1a). Nonius dots were also presented 1.5° above and below the fixation dot to the left and right eyes, respectively. The luminance of these dots was 21.0 cd/m² higher than the background luminance. 

Figure 1.

 

Stimulus examples. (a) A fixation dot and Nonius dots. (b) The standard stimulus for the luminance decrement on a gray background. (c) The standard stimulus for the luminance increment on a gray background. (d) The comparison stimulus for the luminance decrement on a gray background. (e) The comparison stimulus for the luminance increment on a black background.

The standard stimulus was a pair of rectangles each subtending 9.9° × 1.0° (Figures 1b and 1c). The stimuli were presented 3.0° above and below the fixation dot. In Experiment 1, their luminance was 14 cd/m² lower in the luminance decrement condition (Figure 1b) and 14 cd/m² higher in the luminance increment conditions (Figure 1c) than the background luminance. In Experiment 2, the luminance of the standard stimulus was 21 cd/m² lower than the background luminance. 

Figure 2.

 

Mean brightness PSEs in three binocular disparity conditions and three interocular excursion ratios. The ordinate indicates the excursion (i.e., the absolute value of increment/decrement from the background luminance) required to establish the brightness match to the brightness of the standard stimulus that was fixed across all conditions. The larger excursions (base PSEs) are plotted in the dichoptic conditions (1:0.5 and 0.5:1). The letters L and R in the panels indicate the eye (left and right) that received the designated excursion. The error bars indicate the 95% CIs of the mean PSEs. (a) Results for the luminance decrement on a gray background. (b) Results for the luminance increment on a gray background. (c) Results for the luminance increment on a black background.

As a comparison stimulus, a vertical bar subtending 0.2° × 1.1° was presented at the center (Figures 1d and 1e). In Experiment 1, the excursion levels, either decremental or incremental, of the comparison stimuli were chosen as 8.8, 10, 11.4, 13.1, 15, 17.2, and 19.7 cd/m² except that, for one of the observers, the excursion range had to be shifted to 10 to 22.5 cd/m² for the luminance increment on a gray background after the first session. In Experiment 1, three interocular excursion ratios were chosen for the comparison stimulus. In the equal-ratio (L:R = 1:1) condition, the images of the left and right eyes had the same excursion. In the dichoptic conditions (L:R = 1:0.5 or 0.5:1), the excursion in one eye was half the “base” excursion value (i.e., one of the above-described excursion levels). In Experiment 2, the excursion levels of the comparison stimuli were 13.2, 15.1, 17.2, 19.7, 22.5, 25.8, and 29.6 cd/m², and five interocular excursion ratios (1:1, 1:0.75, 1:0.5, 0.75:1, and 0.5:1) were selected for the comparison stimuli. 

The binocular disparity of the comparison stimuli had three levels: zero (0°), crossed (−0.2°), and uncrossed (0.2°) disparities. In the zero-disparity condition, monocular images were displayed at the center. The images were horizontally shifted by 0.1° in opposite directions for the left and right eyes to create non-zero disparities. 

Procedures

The observers pressed a key to start each trial when the Nonius and fixation dots appeared to be aligned along the vertical meridian. After a 400-ms blank, the standard stimulus was presented for 200 ms. After another 600-ms blank, the comparison stimulus was presented for 200 ms. The fixation dot disappeared 100 ms before the onset of the comparison stimulus but the other dots remained. The duration of each stimulus presentation was too short to evoke vergence in response to a disparate stimulus. The observers judged which stimulus was darker in the luminance-decrement condition or brighter in the luminance-increment condition. 

In Experiment 1, each experimental block had 189 trials, consisting of three repetitive trials for each combination of three binocular disparities (zero, crossed, and uncrossed), three interocular excursion ratios (1:1, 1:0.5, and 0.5:1), and seven excursion levels (see above). The order of the trials was randomized within a block. The observers rested after completing a block and adapted to the background luminance for 30 seconds before starting each block. During this adaptation period, the standard stimulus of the impending block was presented for 200 ms every 3 seconds. The observers completed four blocks in a session devoted to one of three excursion conditions (a decrement on a gray background, an increment on a gray background, and an increment on a black background). Three sessions were conducted for each excursion condition; thus, 6804 (189 × 4 × 3 × 3) trials were conducted for each observer. The order of the sessions was counterbalanced among the observers. 

In Experiment 2, each experimental block had 210 trials consisting of two repetitive trials for each combination of the three binocular disparities (zero, crossed, and uncrossed), five interocular excursion ratios (1:1, 1:0.75, 1:0.5, 0.75:1, and 0.5:1), and seven excursion levels (see above). The observers completed four blocks per session. Four sessions were conducted for a luminance decrement on a gray background; thus, 3360 (210 × 4 × 4) trials were conducted for each observer. All other procedures were the same as those in Experiment 1. 

We determined the brightness PSEs and 95% confidence intervals (CIs) of the comparison stimuli that should be perceptual matches to the standard stimuli by fitting the log-Quick function to the data using the Palamedes toolbox (Kingdom & Prins, 2016). There were 252 trials for each fit (36 trials × 7 excursion levels) in Experiment 1 and 224 trials (32 × 7 excursion levels) in Experiment 2. PSE was defined as the excursion corresponding to a 50% probability of judging the comparison stimulus as having greater excursion than the standard stimulus. 

Results of Experiment 1

Comparisons of PSEs

Figure 2 shows the mean PSEs for the three binocular disparities (zero, crossed, and uncrossed) and the three interocular excursion ratios (L:R = 1:1, 1:0.5, and 0.5:1) plotted separately for the three excursion conditions. When the excursions were unequal between the eyes, a larger excursion, hereafter referred to as the base PSE, was plotted. When the luminance excursions were equal (1:1) for both eyes, the PSE was expected to be approximately 14 cd/m², which was the luminance excursion of the standard stimulus. The actual results were roughly consistent with the predictions. As shown in Figure 2, the PSE for the equal ratio (1:1) was approximately 12 to 13 cd/m². Although these measurements were slightly lower than the standard excursion, this was possibly because the comparison stimulus was presented near the fovea, whereas the standard stimulus was presented within the parafovea (3.0° of eccentricity). 

If the binocular brightness combination was winner-take-all, then the lengths of the bars would be the same regardless of the interocular excursion ratio. However, Figure 2 suggests the contrary because the bars are generally longer for unequal ratios (1:0.5 and 0.5:1) than for the equal ratio (1:1). In contrast, if the binocular brightness combination was a linear average, then the length of the bars would be 4/3 times longer for the unequal ratios than for the equal ratio. This prediction was more or less applicable to the luminance increment on a black background (Figure 2c). On the other hand, for the luminance decrement on a gray background (Figure 2a), the bars in the unequal ratios (1:0.5 and 0.5:1) did not support this linear-average prediction. The data in the luminance increment on a gray background were somewhat intermediate (Figure 2b). Similar trends were observed in the individual data, as shown in Figure 3 for two representative observers. 

Figure 3.

 

Brightness PSEs for two representative observers plotted separately in the left and right panels. The error bars indicate the 95% CIs of the estimated PSEs. (a) Results for the luminance decrement on a gray background. (b) Results for the luminance increment on a gray background. (c) Results for the luminance increment on a black background.

We subjected the base PSE values to a two-way repeated-measures analysis of variance (ANOVA) with the interocular excursion ratio (1:1, 1:0.5, or 0.5:1) and binocular disparity (zero, crossed, or uncrossed) as factors. In addition to the two-way ANOVAs, we conducted Dunnett's multiple-comparison tests for each interocular excursion ratio. In this multiple comparison, the base PSE at zero binocular disparity was considered as the control condition because our main concern was the effect of binocular disparity on the binocular luminance combination. 

For the luminance decrement on a gray background (Figure 2a), the main effect of interocular excursion ratio was significant, F(2, 10) = 40.6, p < 0.001, η_p² = 0.890. Between the two bilaterally symmetrical conditions for unequal ratios (1:0.5 and 0.5:1), the base PSEs were expected to be comparable. Thus, the significant main effect presumably reflects the result that the base PSE was smaller for the equal (1:1) ratio (12.3 cd/m²) than for the unequal ratios (14.6 and 13.7 cd/m² for the 1:0.5 and 0.5:1 ratios, respectively). The two-way ANOVA also revealed a significant main effect of binocular disparity, F(2, 10) = 11.2, p = 0.003, η_p² = 0.691, and a significant interaction, F(4, 20) = 3.45, p = 0.027, η_p² = 0.409. According to Dunnett's test to clarify the effect of binocular disparity, the crossed disparity yielded a significant elevation in the base PSEs in the dichoptic conditions (1:0.5 and 0.5:1) but not in the equal ratio (1:1) condition (p > 0.05). The 95% CIs of the PSE elevations ranged from 0.230 to 1.80 cd/m² for the 1:0.5 ratio (p = 0.014) and from 0.320 to 1.81 cd/m² for the 0.5:1 ratio (p = 0.008). However, the uncrossed disparity showed no significant effect (p > 0.05). 

For the luminance increment on a gray background (Figure 2b), the two-way ANOVA yielded a significant main effect of interocular excursion ratio, F(2, 10) = 88.7, p < 0.001, η_p² = 0.947, indicating that the base PSE was smaller for the equal (1:1) ratio (12.9 cd/m²) than for the unequal ratios (16.6 and 15.7 cd/m² for the 1:0.5 and 0.5:1 ratios, respectively). The main effect of binocular disparity was not statistically significant (p > 0.05). The interaction between the two factors was significant, F(4, 20) = 4.81, p = 0.007, η_p² = 0.490. However, in Dunnett's multiple-comparison tests, neither crossed nor uncrossed disparities produced significant changes in the base PSEs from the PSE under the control condition (zero disparity) in any of the interocular excursion ratios (p > 0.05). The significant interaction may reflect some ratio-dependent differences between the crossed and uncrossed disparities, the origin of which could not be identified. 

For the luminance increment on a black background (Figure 2c), the main effect of interocular excursion ratio was significant, F(2, 10) = 53.1, p < 0.001, η_p² = 0.914. In the same way as the conditions described above, the base PSE was smaller for the equal (1:1) ratio (11.7 cd/m²) than for the unequal ratios (15.7 and 14.8 cd/m² for the 1:0.5 and 0.5:1 ratios, respectively). The main effects of disparity and the interaction were not significant (p > 0.05). Dunnett's multiple-comparison test showed no significant effect of binocular disparity (p > 0.05). 

Because the standard and comparison stimuli had different shapes and appeared in a fixed temporal order, the observers were always able to distinguish those stimuli; thus, response bias could have caused an overall shift of PSEs in individual data. However, this type of constant error is supposed to influence the measurements equally among different conditions and therefore is orthogonal to our experimental purpose. In addition, the six observers’ PSEs were distributed within a limited range of excursion; for example, the standard deviation of PSE was only 1.21 cd/m² for the decremental stimuli with zero disparity. The mean PSEs at the equal ratio (1:1) were also comparable with the standard excursion, as described above. Taken together, it is unlikely that response bias artificially created the observed differences across conditions in the present experiment. 

Model fitting

Kingdom and Libenson (2015) used a binocular processing model to describe the relationship between binocular color-saturation perception and dichoptic stimuli with an identical hue but different saturation values. We used their model to visualize the effects of binocular disparity on binocular brightness combination. The model is based on a model of contrast normalization proposed by Legge and Foley (1980) and its extensions for binocular processing (Ding & Sperling, 2006; Maehara & Goryo, 2005; Meese, Georgeson, & Baker, 2006). Although Ding and Levi (2017) recently proposed a binocular processing model that can explain various aspects of binocular perception, including binocular brightness combinations, its computations are much more complex than the model used by Kingdom and Libenson (2015). Therefore, for simplicity, we adopted their model as a descriptive model for the present data, which can be expressed by the following equation:  

\begin{eqnarray*}B = \frac{{C_L^p}}{{z + C_L^q + wC_R^q}} + \frac{{C_R^p}}{{z + C_R^q + wC_L^q}} \end{eqnarray*}

 

The right side of the equation consists of two terms that represent two monocular responses. C_L and C_R are the luminance excursions in the left and right eyes, respectively. These monocular inputs are first raised to the power of p for the numerator and q for the denominator. The raised signals from the two eyes and a constant (z) are summed in denominators. Before this summation, the raised signal from the other eye is multiplied by a weighting parameter (w) that controls how much one eye is suppressed by the other eye. The numerator expression in each monocular response is then divided by the denominator expression. Following the division, two monocular responses are summed to yield binocular brightness (B). 

The weighting parameter, w, is important for the analysis of binocular luminance combinations. If the weighting parameter is approximately 1, then the raised signals from the two eyes are simply summed in the denominator expressions. Because this process works as interocular suppression, the eye receiving the higher excursion predominates the binocular brightness when the luminance is unbalanced between the two eyes. On the other hand, if the weighting parameter is near 0, then each monocular response is virtually free from the divisive suppression from the other eye. Therefore, binocular brightness is determined based on the linear summation of the two monocular responses. See Kingdom and Libenson (2015) for more details about the weighting parameters. 

We fit this descriptive model to the data using a procedure similar to that of Kingdom and Libenson (2015). Each PSE was normalized to the PSE at an equal ratio (1:1). For example, if the luminance excursions in the left and right eyes are 15 and 7.5 cd/m², respectively, at the PSE for an unequal ratio (1:0.5) and if the PSE for the equal ratio is 12 cd/m², then their normalized values are 1.25 and 0.625, respectively. The normalized luminance excursions in the two eyes, L and R, are at the PSE when the binocular brightness (B) equals 1. An equal brightness curve, which corresponds to the family of PSEs to the standard stimulus predicted by the model, can be plotted on the plane in which the x- and y-axes indicate normalized luminance in the left and right eyes, respectively. The model prediction was constrained to pass through the PSE for an equal ratio because its normalized value was one by definition and a constant z was fixed to be near zero. Therefore, each fit aimed to find the curve that ran closest to the two data points in the dichoptic conditions. The model contained four parameters, but p, q, and z were fixed at 4.25, 3.0, and 0.001, respectively. These parameter values are the same as the constants used by Kingdom and Libenson (2015). We used the MATLAB (MathWorks, Natick, MA) “fminbnd” function to find the w value that minimized the root mean squared error (RMSE) between the PSE data and the predicted PSEs in the radial direction. The w-value range was set to be 0 to 1.25. 

Table 1 presents the mean w values, their 95% CIs, and mean RMSEs. The mean RMSEs were reasonably small compared with \(\sqrt 2 \) of the radial distance to the standard value, suggesting that the model fit the data well. The smooth curves in the left column of Figure 4 correspond to the model fits with interobserver mean w values. In this figure, the normalized luminance was reconverted back to cd/m² before plotting. The best fit to the data from the two representative observers is shown in the middle and right columns (see Figure A1 in the Appendix for all individual data in the condition with the decrement on a gray background). The equal brightness curves had greater curvatures, especially at the center, for the luminance decrement on a gray background (top row), compared to the conditions of the luminance increments (middle and bottom rows). The curves show a quasilinear function for the luminance increment on a black background (bottom row). The curves for the luminance increment on a gray background were in between (middle row). 

Table 1.

 

Mean w Values, 95% CI Limits, and Mean RMSEs for Experiment 1.

 

Figure 4.

 

Equal brightness curves for the decrement on a gray background (top panels), the increment on a gray background (middle panels), and the increment on a black background (bottom panels). Smooth curves in the left panels correspond to the model curves with mean w values for the zero-disparity (blue), crossed-disparity (red), and uncrossed-disparity (yellow) conditions. The blue dotted lines represent the winner-take-all prediction (vertical and horizontal lines) and the linear-averaging prediction (diagonal line) in the zero-disparity condition. The central and right panels show the fit to the individual data of two representative observers. Symbols in the figure represent the PSEs. The error bars indicate the 95% CIs of the estimated PSEs. Most error bars are smaller than the symbols.

As noted previously, ANOVAs and Dunnett's multiple comparison tests of PSEs indicate that crossed disparity had a significant effect on how the visual system integrates luminance decrements presented to the left and right eyes. This significant effect can be observed as a change in the equal brightness curves for the luminance decrement on a gray background (top rows of Figure 4). The curves appear to be less bent for the crossed disparity (red) than for the zero disparity (blue). The mean w values reflect this trend (Table 1a). This was smaller for the crossed disparity (0.631) than for the zero disparity (0.780). Although we subjected w values to Dunnett's test, the difference between zero and crossed disparity reached a statistical significance level of p = 0.064, which is rather marginal. However, this may be because three (p, q, and z) of the four model parameters were fixed at the constants used by Kingdom and Libenson (2015) and thus are not guaranteed to be optimal for our observers. The uncrossed disparity also yielded a smaller exponent parameter (0.675) than zero disparity, resulting in slightly less bent curves (Figure 4, yellow); however, this trend was not significant in Dunnett's test. For the luminance increments on the gray and black backgrounds (Figure 4, middle and bottom rows), the curves almost overlap. The mean w values were comparable across the three binocular disparities under these conditions (Tables 1b and 1c), indicating that the binocular disparity had little effect when luminance increments were presented. 

Results of Experiment 2: Luminance decrement with a larger excursion

In Experiment 1, the crossed disparity produced a significant change in the binocular brightness combination of luminance decrement on a gray background, although there was no significant effect of disparity for luminance increment on a gray and black background. The effect of disparity may depend on the pattern of binocular combination; that is, disparity had no effect on luminance increments because the binocular combination was already close to linear averaging at zero disparity. If this was the case, then the effect of disparity might become larger in a situation in which binocular combination at zero disparity is apparently winner-take-all. According to Baker et al. (2012) and Ding and Levi (2017), the binocular combination of decrements is intermediate between winner-take-all and linear averaging at small excursions but gradually gets closer to winner-take-all with increasing excursion. Therefore, in the next experiment, we used stimuli with a larger decremental excursion (1.5 times) than those in Experiment 1. The levels of interocular excursion ratios were also increased from three to five for more precise estimations of equal brightness curves. 

Figure 5 illustrates the individual results for the six observers. As expected from the above-mentioned literature, the equal brightness curves were more bent than those in Experiment 1, indicating that binocular brightness was determined in a winner-take-all manner. However, observer 10 showed a somewhat different pattern (see the bottom-right panel in Figure 5). Even when the excursion was larger for the left eye than for the right eye, binocular combination suggested a strong dominance of the right eye. Due to these asymmetrical results between the two eyes, the model failed to fit the data for observer 10. In addition, observer 10 exhibited very large 95% CIs (error bars in Figure 5) compared with those for the other observers. Notably, despite the principle that the range should ideally be spanning 0% to 100% to derive a reliable sigmoidal psychometric function by best-fitting, inspection of the raw data confirmed that in only five of the 15 conditions did the response rates in the method of constant stimuli span <25% to >75%. Due to these indications of atypical binocular vision and poor brightness discriminability, we excluded observer 10 from the following analysis. 

Figure 5.

 

Equal brightness curves for luminance decrement on a gray background in Experiment 2. Each panel shows the fit to the individual data. Symbols represent the PSEs. Error bars indicate the 95% CIs of the estimated PSEs. Most error bars are smaller than the symbols.

Comparisons of PSEs

We subjected the base PSE values to a two-way repeated-measures ANOVA with the interocular excursion ratio (1:1, 1:0.75, 1:0.5, 0.75:1, or 0.5:1) and binocular disparity (zero, crossed, or uncrossed) as factors. The base PSEs significantly differed across interocular excursion ratios, F(4, 16) = 7.09, p < 0.001, η_p² = 0.639. This significant main effect indicates that the base PSE was smaller for the equal (1:1) ratio (19.1 cd/m²) than for the dichoptic conditions (20.4, 20.3, 20.2, and 20.2 cd/m² for the 1:0.75, 1:0.5, 0.75:1, and 0.5:1 ratios, respectively). Although the main effect of binocular disparity was not significant (p > 0.05), the two factors significantly interacted, F(8, 32) = 3.59, p < 0.001, η_p² = 0.473. To clarify the interaction, we conducted Dunnett's multiple comparison test for each interocular ratio. The base PSEs were significantly higher for the crossed disparity than for zero disparity at the interocular excursion ratios of 0.5:1 (20.7 cd/m² vs. 19.6 cd/m²) and 0.75:1 (20.6 cd/m² vs. 19.7 cd/m²). The 95% CIs of the PSE elevations ranged from 0.410 to 1.75 cd/m² for the 0.5:1 ratio (p = 0.005) and from 0.758 to 1.56 cd/m² for the 0.75:1 ratio (p = 0.033). These elevations in the PSEs are consistent with the results of Experiment 1, although there was no significant difference at the interocular excursion ratios of 1:0.5 and 1:0.75 (p > 0.05). The uncrossed disparity produced no significant elevation in PSEs (p > 0.05), but the base PSEs were significantly lower for the uncrossed disparity (18.8 cd/m²) than for zero disparity (19.6 cd/m²) at the equal ratio (1:1). The 95% CIs of the PSE reduction ranged from 0.152 to 1.37 cd/m² (p = 0.018). 

The crossed disparity caused some elevation of PSEs in the dichoptic conditions but not in the equal-ratio condition, resulting in the equal brightness curve being less bent for the crossed disparity than for zero disparity. In contrast, the uncrossed disparity caused the reduction of PSEs at the equal ratio but not in the dichoptic conditions. This result is also consistent with the shape change in the equal brightness curve in a manner similar to the above-described case of the crossed disparity. The aforementioned shape changes from winner-take-all to slightly less curvatures at non-zero disparities can be seen in Figure 5, especially in the curves of observers 5 and 9. 

Model fitting

We fit the model to the PSE data to visualize the changes in equal brightness curves using the same procedure as that in Experiment 1. Table 2 presents the mean w values, their 95% CIs, and mean RMSEs. The mean RMSEs were comparable to those in Experiment 1 and reasonably small. The smooth curves in Figure 5 are the fits to the individual data. Figure 6 shows the model curves with the interobserver mean w values. Consistent with the results of previous studies (Baker et al., 2012; Ding & Levi, 2017), the curvatures were greater in Experiment 2 (Figure 6) than in Experiment 1 (Figure 4). Dunnett's test confirmed that the crossed and uncrossed disparities produced smaller w values (0.916 and 0.904, respectively) than those in the zero-disparity condition (1.15). The 95% CIs of the difference ranged from 0.0841 to 0.381 for the crossed disparity (p = 0.006) and from 0.0957 to 0.393 for the uncrossed disparity (p = 0.004). Because of the reduction in w values, the curvatures were less bent for the crossed and uncrossed disparities than for zero disparity (Figures 5 and 6). These results indicate that, in the situation of luminance decrement with a larger excursion, binocular disparities weakened interocular suppression, and consequently the binocular combination was less of a winner-take-all scenario. 

Table 2.

 

Mean w Values, 95% CI Limits, and Mean RMSEs for Decrement on a Gray Background in Experiment 2.

 

Figure 6.

 

The model fits with mean w values for the zero-disparity (blue), crossed-disparity (red), and uncrossed-disparity (yellow) conditions in Experiment 2. The blue dotted lines represent the winner-take-all prediction (vertical and horizontal lines) and the linear-averaging prediction (diagonal line) in the zero-disparity condition.

Discussion

The present study examined how binocular disparity changed binocular brightness combination patterns. The results of this study were consistent with those of previous studies in that luminance decrements in the left and right eyes were combined in a winner-take-all fashion, whereas the combination of luminance increments was close to linear averaging when the luminance values in the two eyes were not very different (Anstis & Ho, 1998; Baker et al., 2012; Ding & Levi, 2017). Our main finding was that the binocular brightness combination slightly deviated from the winner-take-all prediction when a luminance decrement was accompanied by a crossed disparity (Experiment 1) or by a crossed or uncrossed disparity (Experiment 2). In contrast, for luminance increments, binocular disparity had no effect on the binocular brightness combination. 

It was expected that the presence of binocular disparity would cause binocular brightness combinations to approach linear averaging if stimuli with non-zero binocular disparity weakened interocular suppression. However, the effect of binocular disparity was significant only for a luminance decrement and not for increments. Previous studies have reported that the monocular luminance values of decrements are combined in a nonlinear fashion (that is, winner-take-all), whereas those of increments are combined in a linear fashion, except for the range corresponding to Fechner's paradox (Anstis & Ho, 1998; Baker et al., 2012; Ding & Levi, 2017; Engel, 1970; Levelt, 1965). The reduction in interocular suppression due to non-zero disparities may be effective only when the binocular brightness combination for stimuli with zero disparity is strongly nonlinear in the first place. According to Baker et al. (2012), the effect of luminance excursion is asymmetrical between increments and decrements. In their results, the equal brightness curves for decrements became slightly shifted toward linear averaging as the luminance excursion decreased, whereas the curves for increments on a gray background did not change. The effects of excursion level on decrements could also be due to the reduction in interocular suppression because interocular suppression is generally considered to weaken as the luminance contrast of stimuli decreases (Maehara, Huang, & Hess, 2009; Schor & Heckmann, 1989). That is, the results of Baker et al. (2012) are arguably consistent with the present results in that the reduction of interocular suppression is a viable idea in both cases to explain the reduction of strong nonlinearity in the binocular combination of luminance decrements. 

Moreover, the luminance contrasts of the stimuli might have caused the difference between the results of luminance decrements and increments. In Experiment 1, we equated Weber contrasts between the decrement and increment on a gray background. Consequently, the Michelson contrast, (L_max − L_min)/(L_max + L_min), of the decremental standard (25%) was higher than that of the incremental standard (17%). Because interocular suppression is assumed to be greater for stimuli with higher contrasts, the effects of binocular disparity could increase for decrements more than for increments due to the difference in the Michelson contrast. Nonetheless, according to Ding and Levi (2017), who examined binocular combinations within a wider range of luminance excursions, incremental stimuli show little change in the equal brightness curve among different excursions when background luminance is sufficiently high (16.2 cd/m²). In light of their finding, we expect that the present results of the incremental conditions would not change dramatically even if the luminance contrast was increased. Future investigations with a wider range of excursions with disparity manipulations will resolve this interesting issue. 

Lu and Sperling (2012) summarized psychophysical studies concerning the asymmetry between luminance increments and decrements by proposing that black (decremental) stimuli are represented with larger magnitudes in the visual system than white (incremental) stimuli when they have the same physical intensity in virtually all asymmetric cases. The asymmetry between luminance increments and decrements may reflect the characteristics of natural images. Based on natural image statistics, researchers have suggested that retinal images contain a greater number of dark features than bright features (Cooper & Norcia, 2015; Ratliff, Borghuis, Kao, Sterling, & Balasubramanian, 2010). Notably, one study suggested that non-zero binocular disparities appear more frequently in dark features than in bright features in natural images (Cooper & Norcia, 2015). Therefore, it is possible that our visual system represents dark features more dominantly than bright features during binocular disparity processing. This asymmetry may have caused a difference in the effects of binocular disparity between luminance increments and decrements. Cooper and Norcia (2015) reported that dark features were more prominent at lower spatial frequencies. In our study, the observers had to judge the brightness of the surfaces but not the luminance contrast of the edges. Judgment of the former depends on information from components with relatively low spatial frequencies. This may be another reason why the effect of binocular disparity was significant only for the luminance decrement. 

The effect of binocular disparity was significant for crossed disparity but not for uncrossed disparity in Experiment 1, although both types of disparity produced significant effects in Experiment 2. Various studies have demonstrated separate mechanisms that selectively process crossed and uncrossed disparities (see for a review, see Mustillo 1985). The results of Experiment 1 presumably reflected the differential characteristics of these mechanisms. Manning, Finlay, Neill, and Frost (1987) found that the duration threshold for detection was substantially longer for stimuli with uncrossed disparity than for those with crossed disparity. Patterson et al. (1995) also showed that a longer duration was required to perceive depth from an uncrossed disparity than from a crossed disparity. These studies suggest that the mechanism for crossed disparity is more sensitive than that for uncrossed disparity. Although our stimuli were suprathreshold and 100% detectable, they are only briefly presented. The stimuli with crossed disparity might have elicited a larger neural response than those with uncrossed disparity in the present experiment, causing a difference in the effectiveness of the binocular brightness combination in Experiment 1. 

In the present experiments, the largest excursion difference between the two eyes was found in the excursion ratios of 1:0.5 and 0.5:1. If we could measure PSEs in even greater unbalanced ratios, we could make a more precise evaluation of the nonlinearity of equal-brightness data. We tested the interocular excursion ratios of L:R = 1:0.25 and 0.25:1 in a preliminary experiment. When non-zero disparities were added, however, observers were not able to fuse the stimuli as in cyclopean vision but instead saw only two dichoptic images, one brighter and the other darker, making it difficult to perform the brightness task as in the main experiment. Therefore, our PSE measurements had to be limited to range between 1:0.5 and 0.5:1 cases. One possible extension of the stimuli is to add a high-contrast rectangular contour to support binocular fusion of our comparison stimulus, whereas the luminance inside the contour varies for each eye. This will enable us to examine the effects of binocular disparity under conditions of greater luminance differences between the eyes. However, in such cases, the luminance of the contour can affect the brightness perception of the interior. In addition, a process of perceptual filling-in may be triggered by the spatial profiles of luminance around the contour. Therefore, elucidating the effects of contours on the binocular brightness combination requires future studies. 

Maehara and Murakami (2020) reported that perceived contrast was approximately 1.1 times higher at 1° binocular disparity than at zero disparity. If this perceptual enhancement had occurred, the position of the equal brightness curves would shrink in the lower-left direction when the stimuli had a non-zero binocular disparity. However, this effect was not observed in Experiment 1 (Figure 4) and was smaller in Experiment 2 (Figure 6) than that reported by Maehara and Murakami (2020). The absence or reduction of perceptual enhancement may be owing to differences in the range of binocular disparities. We presented a short, thin bar as a stimulus for the binocular brightness combination and set its binocular disparity within the fusion limit. This is because brightness judgment in double vision is difficult under dichoptic conditions. In contrast, the Gabor patches used in the study by Maehara and Murakami (2020) presented disparities over the fusion limit, but within the stereopsis limit. Another possible explanation for the above discrepancy is that binocular luster is involved in the perceptual enhancement. Previous studies have suggested that its occurrence depends on stimulus size (for a review, see Wendt & Faul, 2022). The size of the comparison stimulus in the present study was arguably smaller than the lower limit of binocular luster, whereas the Gabor patches used in the study by Maehara and Murakami (2020) were sufficiently large. Nonetheless, the perceptual enhancement observed in their study was also vigorous when the left-eye and right-eye images were vertically disparate with virtually no spatial overlap, which is an unfavorable situation for luster. 

Conclusions

We found that, although the binocular brightness combination of the luminance decrement tended to be nonlinear and more like winner-take-all, the combination became less of a winner-take-all scenario when presented with non-zero disparity. This shift in binocular combination is possibly due to weaker interocular suppression caused by non-zero binocular disparity. This is presumably because simultaneous stimulations at corresponding retinal points are required for strong suppression. Another possible mechanism is that stereopsis, which is processed when disparity is signaled, inhibits interocular suppression to avoid eye rivalry and to promote binocular fusion, resulting in the maximal use of precise position information originating from two monocular signals. The mechanism of attenuation of the interocular suppression we found still remains unclear, and we argue that one of the possible extensions would be to introduce vertical disparity to ascertain whether the attenuation requires the process for stereopsis specifically or just any misalignment from the corresponding retinal points. 

Previous studies have suggested that interocular suppression is reduced in Ganzfeld stimuli (Bourassa & Rule, 1994) and balanced between the two eyes in binocularly matched features (Jennings & Kingdom, 2019; Kingdom & Libenson, 2015; Kingdom & Wang, 2015; Wang et al., 2022). These techniques could be effective tools for training binocular vision in stereoblind observers or patients with amblyopia. Here, we suggest that binocular disparity manipulation potentially offers another technique to make both eyes functional. However, the effects of binocular disparity were small and limited to a luminance decrement in the present experiment. For such training purposes, future studies are required to determine the optimal stimulus conditions to elicit the largest attenuation effect on interocular suppression in the domain of brightness. 

Acknowledgments

IM was supported by a JSPS KAKENHI Grant-in-Aid for Scientific Research (21H04426). 

Commercial relationships: none. 

Corresponding author: Goro Maehara. 

Email: [e-mail]. 

Address: Department of Human Science, Kanagawa University, Kanagawa, Japan. 

Appendix

Figure A1.

 

Individual results for the decrement on a gray background in Experiment 1. Smooth curves in the left panels correspond to the model fit with mean w values for the zero-disparity (blue), crossed-disparity (red), and uncrossed-disparity (yellow) conditions. Symbols in the figure represent the PSEs. Error bars indicate the 95% CIs of the estimated PSEs. Most error bars are smaller than symbols.