April 2015
Volume 15, Issue 5
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Article  |   April 2015
Dichoptic color saturation mixture: Binocular luminance contrast promotes perceptual averaging
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Journal of Vision April 2015, Vol.15, 2. doi:10.1167/15.5.2
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      Frederick A. A. Kingdom, Lauren Libenson; Dichoptic color saturation mixture: Binocular luminance contrast promotes perceptual averaging. Journal of Vision 2015;15(5):2. doi: 10.1167/15.5.2.

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

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Abstract

We demonstrate a new type of interaction between suprathreshold color (chromatic) and luminance contrast in the context of binocular vision. When two isoluminant colored disks of identical hue but different saturations are presented to different eyes, the apparent saturation of the resulting “dichoptic” mix is close to that of the more saturated patch if presented binocularly. This result is commensurate with previous findings using luminance contrast and is close to the scenario termed “winner-take-all.” However, when binocularly matched luminance contrast is added to the dichoptic saturation mixture, the apparent saturation of the mixture shifts away from winner-take-all towards the average of the two dichoptic saturations. The likely cause of this effect is that the matched luminance contrasts reduce the interocular suppression between the unmatched color saturations. We suggest that the presence of binocularly matched luminance contrast promotes the interpretation that the dichoptic color saturations, even though unmatched, nevertheless originate from the same object. We term this idea the “object commonality” hypothesis.

Introduction
Many studies have considered how we perceive objects that are unmatched in the two eyes in dimensions other than position, for example in contrast or hue. The earliest studies of such “dichoptic” mixtures dealt with the appearance of stimuli with different luminances, (e.g., Engel, 1970; Fechner, 1860; Levelt, 1965), an issue that continues to engage researchers today (e.g., Anstis & Ho, 1998; Baker, Wallis, Georgeson, & Meese, 2012). The study of chromatic (color) dichoptic mixture has invariably dealt with dichoptic differences in hue (reviewed by Hovis, 1989). However the chromatic analog of dichoptic luminance mixture, namely dichoptic saturation mixture, which deals with the appearance of dichoptic colors that are identical in hue but different in saturation, has to our knowledge never been studied. In this communication we ask two questions. The first: Are the rules governing the appearance of dichoptic saturation mixtures different to those governing the appearance of dichoptic luminance mixtures? The second: What effect does binocularly matched luminance contrast have on the appearance of dichoptic saturation mixtures? 
To motivate our experiments, consider first what is known about dichoptic luminance mixture. If the stimuli are disks on a grey background, as in the experiments described here with color, the brightness (perceived luminance) of the dichoptic mixture follows closely that of the higher of the two luminances if they are increments, or the lower of the two luminances if they are decrements (many of the relevant studies are discussed in Baker et al., 2012). This result is a form of “winner-take-all,” and may be counterposed to the theoretical alternative scenario of “averaging,” in which the brightness/darkness of the dichoptic mixture is closer to the average of the two luminances. The winner-take-all scenario found with dichoptic mixtures of not just luminance but also luminance contrast (e.g., when the stimuli are periodic stimuli such as gratings whose apparent contrasts are the dependent variable) is assumed to result from interocular suppression, that is, mutual inhibition between the eyes (Baker et al., 2012; Legge, 1979; Legge & Rubin, 1981; Meese, Georgeson, & Baker, 2006). The question arises as to whether this behavior is also found in the chromatic domain. There is some evidence that binocular summation at threshold is greater for chromatic than luminance stimuli (Simmons, 2005; Simmons & Kingdom, 1998), so it is possible that at suprathreshold contrast levels there is also more summation, i.e., more averaging between dichoptic chromatic contrasts than with dichoptic luminance contrasts. 
A number of studies have shown that the presence of matched features in the two eyes reduces the interocular competition that causes phenomena such as binocular rivalry and dichoptic masking (Blake & Boothroyd, 1985; Buckthought & Wilson, 2007; Meese & Hess, 2005; O'Shea, 1987). For example Blake and Boothroyd (1985) showed that binocular rivalry from mismatched grating orientations was reduced if a third grating was introduced to one eye, such that one eye saw a plaid and the other a grating. Blake and Boothroyd argued that the grating component that was now matched in orientation in the two eyes served to reduce the between-eye suppression that caused rivalry. Meese and Hess (2005) found that a thin black ring in one eye surrounding a Gabor in the other eye strongly masked the Gabor, but that when the ring was present in both eyes, i.e., was matched, the masking was reduced. In Buckthought and Wilson's (2007) study, binocular rivalry from orthogonal oblique orientations was reduced by the presence of matched vertical gratings. Interestingly, the reduction in rivalry was contingent upon the spatial-frequency similarity of the matched and unmatched gratings. If the matched and unmatched features therefore need to be similar along dimensions other than the one defining the mismatch in order for the matched features to reduce the interocular suppression between the unmatched features, one might suppose that matched luminance contrasts would have little impact on the appearance of unmatched color saturations. However, Mullen, Kim, and Gheiratmand (2014) have found that a luminance grating in one eye strongly masks an orthogonally oriented color grating in the other eye, suggesting that color and luminance contrast strongly interact in dichoptic vision. We might therefore expect matched luminance contrasts to reduce the interocular suppression between unmatched color saturations, causing them to average. 
In summary the aim of this communication is to determine how we perceive dichoptic mixtures of color saturations and to examine the influence on the mixture's appearance of binocularly matched luminance contrast. A brief report of this study has been given elsewhere (Libenson & Kingdom, 2014). 
Before proceeding, a note about terminology: Although our stimuli are uniform disks on a fixed grey background, one can think of them as varying in luminance or luminance contrast when achromatic, or varying in saturation or color contrast when chromatic. Since all our measurements are in terms of contrast, we will often use the terms luminance contrast and color contrast in reference to our stimuli; but because saturation enjoys common usage in the color vision literature, we will also use the terms luminance and saturation. 
Methods
Subjects
Two subjects participated. DW was a female undergraduate volunteer who was naïve to the purposes of the experiment. FK is the senior author. Both subjects had normal or corrected-to-normal vision and normal color vision as tested by the Ishihara Color Plates. Data collected from the second author LL was not included as it was very inconsistent, and we surmise that this was probably due to her markedly reduced stereo-vision. 
Stimuli–generation and display
The stimuli were generated by a VISAGE graphics card (Cambridge Research Systems) and displayed on a Sony Trinitron F500 flat-screen monitor. The R (red), G (green) and B (blue) gun outputs of the monitor were gamma-corrected after calibration with an Optical photometer (Cambridge Research Systems). The spectral emission functions of the R, G and B phosphors were measured using a PR 640 spectral radiometer (Photo Research), with the monitor screen filled with red, green, or blue at maximum luminance. The CIE coordinates of the monitor phosphors were R: x = 0.624, y = 0.341; G: x = 0.293, y = 0.609; B: x = 0.148, y = 0.075. The members of each dichoptic pair were presented either side of the monitor screen and fused via a custom-built, 8-mirror Wheatstone stereoscope, with an aperture of 10° × 10° and a viewing distance along the light path of 55 cm. 
Stimuli colors and contrasts
The four color (red, cyan, violet and lime) and two achromatic (black and white) stimuli lay along the six axes of the DKL color space (Derrington, Krauskopf, & Lennie, 1984) whose isoluminant plane is shown in Figure 1a. The isoluminant plane of the DKL consists of two cardinal axes defined by combinations of long-wavelength-sensitive (L), middle-wavelength-sensitive (M), and short-wavelength-sensitive (S) cone contrasts. The three cone contrasts are defined as: Lc = ΔL/Lb, Mc = ΔM/Mb and Sc = ΔS/Sb (Cole, Hine, & McIlhagga, 1993; Norlander & Koenderink, 1983; Stromeyer, Cole, & Kronauer, 1985). The denominator in each cone-contrast term refers to the cone excitation of the background, which was a mid-grey color with CIE chromaticity x = 0.282 and y = 0.311, and luminance 40 cd/m2. The numerator in each cone contrast term represents the difference in cone excitation between the disk and background. The LMS cone excitations assigned to disk and background were converted to RGB phosphor intensities using the cone spectral sensitivity functions provided by Smith and Pokorny (1975) and the measured RGB spectral functions of the monitor. 
Figure 1.
 
(a) Isoluminant plane of DKL color space. L = long-wavelength-sensitive, M = middle-wavelength-sensitive, S = short-wavelength-sensitive cones. (b) The four color (red, cyan, violet, lime) and two achromatic (black, white) conditions used in the experiments. (c) Stimulus construction and procedure using violet as the example. LE = left eye; RE = right eye. See text for further details. Note that the colors will unlikely be accurate due to limitations in photographic reproduction.
Figure 1.
 
(a) Isoluminant plane of DKL color space. L = long-wavelength-sensitive, M = middle-wavelength-sensitive, S = short-wavelength-sensitive cones. (b) The four color (red, cyan, violet, lime) and two achromatic (black, white) conditions used in the experiments. (c) Stimulus construction and procedure using violet as the example. LE = left eye; RE = right eye. See text for further details. Note that the colors will unlikely be accurate due to limitations in photographic reproduction.
The term “cardinal” implies that the colors uniquely stimulate the three post-receptoral mechanisms. The relative cone contrast inputs to these mechanisms have been estimated to be as follows: kLc + Mc for the luminance (LUM) mechanism, producing black and white; LcMc for the mechanism that differences L and M cone-contrasts, producing red and cyan; Sc − (Lc + Mc)/2 for the mechanism that differences S from the sum of L and M cone-contrasts, producing violet and lime (Cole et al., 1993; Stromeyer et al., 1985). The parameter k determines the relative weightings of the L and M cone-contrast inputs to the luminance mechanism, and this value varies between observers, and was established for each subject (see below). In order to isolate the three cardinal mechanisms, the stimuli must be constructed such that the L-M stimulus does not activate either the LUM or the S mechanism, the S stimulus does not activate either the LUM or L-M mechanism, and the LUM stimulus does not activate either the S or L-M mechanism. Kingdom, Rangwala, and Hammamji (2005) used the following combinations of Lc, Mc, and Sc to achieve this:    The measures of contrast were calculated as follows: for L-M, the difference between Lc and Mc; for S, simply Sc; and for “LUM,” the contrast assigned to each cone, e.g., Lc.  
The contrasts of the fixed-reference disks (Figure 1c) were set to either 10 or 20 times their individual detection thresholds, as given in Table 1. The lower value of ×10 for the cyan and lime disks was found to be necessary to prevent the colors in the adjustable disk exceeding the monitor look-up-table when a decrement in luminance contrast was added. For each color, ten values of test contrast were chosen for DW and five for FK that spanned the range linearly from zero to the contrast of the reference. 
Table 1.
 
Contrast detection thresholds and reference contrasts for the four color and two achromatic disks used in the experiments, for both subjects DW and FK.
Table 1.
 
Contrast detection thresholds and reference contrasts for the four color and two achromatic disks used in the experiments, for both subjects DW and FK.
Stimuli–disks
Example stimuli are shown in Figures 1c and 2. The diameter of each disk was 1.5° at the viewing distance of 55 cm. The disks were positioned above and below the fixation point inside a black circular ring 1 pixel wide and 7.0° in diameter. Each pair of disks was separated by 3° along a virtual line connecting their centers. The orientation of the virtual line was randomized on each trial within the range −30° to +30° from vertical in order to minimize the buildup of after-images during trials. 
Figure 2.
 
Example stimuli as the observer would see them. Top: achromatic conditions; bottom: color saturation conditions using lime as the example color. The task on each trial was to choose the disk, above or below fixation, according to the following criteria: black, the darker disk; white, the brighter disk; cyan, the more saturated disk. The cyan color will unlikely be accurate due to limitations in photographic reproduction.
Figure 2.
 
Example stimuli as the observer would see them. Top: achromatic conditions; bottom: color saturation conditions using lime as the example color. The task on each trial was to choose the disk, above or below fixation, according to the following criteria: black, the darker disk; white, the brighter disk; cyan, the more saturated disk. The cyan color will unlikely be accurate due to limitations in photographic reproduction.
Stimuli–added luminance contrast conditions
For the four color conditions (red, cyan, violet, lime), the perceived saturation of the dichoptic mixture was measured both at isoluminance and with added, binocularly matched luminance contrast. By the latter we mean that the luminance contrast was added equally to all four disks in the stimulus (adjustable, test, left-eye-reference, and right-eye-reference). We used two types of luminance contrast, an increment and a decrement. The decrement was used with all four color conditions, whereas the increment was used with just the cyan and lime conditions. We set the decrement to a contrast of −0.25. We set the increment to the same multiples of detection threshold as the decrement for each subject, resulting in increment contrasts of 0.19 for DW and 0.29 for FK. In retrospect it would have been more balanced to set the decrement contrast to the same multiples of its own detection threshold for each subject rather than use the same contrast of −0.25. However, in terms of the results and the conclusions of the study, the contrasts of the added luminance contrast conditions are not critical. 
Procedure–measurement of isoluminance
Because of inter-subject variation in the relative weightings of the L and M cones that feed the luminance mechanism, it was necessary to ensure that the colors combining L and M cone modulations were isoluminant. We used the criterion of minimum perceived motion. A 0.025 contrast, 0.5 cpd L-M (red-cyan) sinusoidal grating was set to drift at about 1.0 Hz. Subjects pressed a key to add or subtract luminance (L + M) contrast to the grating until the perceived motion was at a minimum. Each subject made between 20 and 30 settings. The average amount of luminance contrast added (or subtracted) was used to calculate the parameter k in Equation 1a, which is the ratio of Lc to Mc in the putative luminance mechanism. The values of k were 0.43 for DW and 1.05 for FK. Although S cones only contribute to the luminance mechanisms under extreme conditions (Eskew, McLellan, & Giulianini, 1999; Ripamonti, Woo, Crowther, & Stockman, 2009), there is always the possibility of calibration error with S stimuli, so for each observer we also measured the isoluminant point for a drifting 0.25 contrast S (violet-lime) grating with the same spatiotemporal parameters as that used for the L-M stimulus. The ratio of L+M to S contrast needed to make the S stimuli isoluminant was 0.068 for DW and 0.1 for FK. 
Procedure–contrast thresholds
To equate the contrasts of the four color and two achromatic disks we first measured their individual contrast detection thresholds. On each trial a binocular disk target was presented either above or below fixation according to the same temporal and position rules as in the main experiment (see below). The subject indicated by a key press whether the target was above or below fixation. We used a three-up-one-down staircase procedure (Kingdom & Prins, 2010) with a step-fraction of 1.5 for the first eight trials and 1.3 for the remaining trials. After eight reversals the session was terminated and the threshold contrast was calculated as the geometric mean contrast over the eight reversals. Table 1 presents contrast thresholds for the six disks and on the basis of these thresholds the calculated contrasts of the reference disks. 
Procedure–saturation matching
The procedure involved a matching task. For the color conditions subjects adjusted on each trial the saturation of the adjustable disk in one eye (the fixed-test saturation was in the other eye) until the perceived saturation of the dichoptic mixture matched that of the binocular reference disk on the other side of fixation (see Figures 1c and 2). During a trial both the dichoptic and binocular disks were slowly pulsed on and off in synchrony, with each cycle consisting of both disks being presented in a cosine temporal envelope of 1s followed by an inter-stimulus-interval of 1 s. The orientation of the disk pair was randomized on each cycle within the −30° to +30° range. The task for the subject was to indicate, using a key press at the end of each cycle, the disk (top or bottom) that appeared most saturated. The cycling was continuous, i.e., irrespective of whether a button press was recorded. After each button press, the saturation of the adjustable disk was increased or decreased by a factor of 1.1 for the first six cycles of stimulus presentation and 1.05 thereafter, towards the point-of-subjective-equality. When the observer was satisfied with the match, he/she pressed another key to terminate the trial and the match was recorded. A key press initiated the next trial with a different value of test saturation. Five to six matches were made for each condition, and it is the means and standard errors of the settings that are shown in the graphs below. For the achromatic black and white disks, the procedure was identical except that the judgement was to equate the brightnesses or darknesses of the disks. 
Results
Achromatic mixtures
We first show the results for the achromatic black and white disks. Each graph in Figure 3 plots the luminance contrast of the adjustable disk in one eye as a function of the fixed-test luminance contrast in the other eye that was needed to match the reference disk on the other side of fixation. 
Figure 3.
 
Results with the achromatic stimuli. Each graph plots the adjustable luminance contrast in one eye as a function of the test luminance contrast in the other eye, required to make the resulting dichoptic mix the same brightness/darkness as the reference disk. Points with vertical error bars are when the adjustable contrast is in the left eye (LE) and test contrast in the right eye (RE). Points with horizontal error bars are for RE adjustable and LE test contrasts. Filled circles are for luminance decrements; open circles, luminance increments. Error bars are standard errors of the mean of each set of matches. The oblique dotted line represents the prediction if the dichoptic mixture is perceived as the average of the two luminance contrasts, whereas the horizontal and vertical dashed lines represent the prediction if the mixture is perceived as that of the larger excursion, or “winner-take-all.” Top graphs for observer DW; bottom graphs for FK.
Figure 3.
 
Results with the achromatic stimuli. Each graph plots the adjustable luminance contrast in one eye as a function of the test luminance contrast in the other eye, required to make the resulting dichoptic mix the same brightness/darkness as the reference disk. Points with vertical error bars are when the adjustable contrast is in the left eye (LE) and test contrast in the right eye (RE). Points with horizontal error bars are for RE adjustable and LE test contrasts. Filled circles are for luminance decrements; open circles, luminance increments. Error bars are standard errors of the mean of each set of matches. The oblique dotted line represents the prediction if the dichoptic mixture is perceived as the average of the two luminance contrasts, whereas the horizontal and vertical dashed lines represent the prediction if the mixture is perceived as that of the larger excursion, or “winner-take-all.” Top graphs for observer DW; bottom graphs for FK.
The data are “iso-darkness” and “iso-brightness” curves, that is, combinations of the adjustable and test disks that produce a fixed-criterion level of darkness or brightness. The matched and test contrasts are expressed as a proportion of the reference contrast: Note that the test contrast is always less than or equal to the reference contrast, and thus less than or equal to unity in the figure. Thus, adjustable disk contrasts that are greater than unity imply that the match was set higher than the reference contrast, whereas values less than unity imply that the match was set lower than the reference contrast. 
The dashed and dotted lines in Figure 3 are the predictions for winner-take-all and averaging. Consider first winner-take-all. According to this scenario, providing the test contrast is lower than the reference contrast, observers will always set the adjustable (adj) contrast to that of the reference, as under these conditions the test contrast is suppressed by the higher contrast in the other eye and contributes nothing to the mixture—hence, the horizontal and vertical dashed lines in the figure. Note that in this scenario the reference (ref) disks in the two eyes suppress each other equally. Winner-take-all means that max (adj, test) = ref, and because test < = ref, adj = ref. The prediction for averaging is very different. Now, both test and adjustable luminance contrasts contribute proportionately to the mixture. Thus if the test contrast is zero, the adjustable contrast needs to be twice that of the reference in order to make the average of test and adjustable equal to that of the reference. Thus for averaging (test + adj)/2 = (ref + ref)/2, i.e., adj = 2ref − test. In other words as the test increases from zero, the adjustable contrast must decrease proportionately to keep the average constant. This is the oblique dotted line in the figure. 
The data in Figure 3 fall closely to winner-take-all, especially for the decrements. These results are very similar to those reported by Baker et al. (2012) who used a similar methodology. As with Baker et al., there is a hint in the data of the contrast equivalent of “Fechner's paradox,” especially with the increments: As the test contrast is increased from zero, the adjustable contrast first increases slightly before decreasing as the test contrast approaches that of the reference. We will return to a discussion of this feature of the data in the Model section. 
Color saturation mixtures
Figures 4 and 5 show iso-saturation curves for DW and FK's color data respectively. The isoluminant data are the filled circles, whereas the data for the added luminance contrast conditions are the filled squares for added decrements, and filled triangles for added increments. It is worth reiterating that the luminance contrast was a fixed amount added equally to all four disks (adjustable, test, left-eye reference, and right-eye reference) and was therefore ostensibly irrelevant to the task, which was always to match the color saturations of the disks. The data points have been color coded to correspond both to the colors of the isoluminant disks (violet, red, cyan, lime), as well as the disks with added decremental (dark-violet, dark-red, dark-cyan, dark-lime) or incremental (light-cyan, light-lime) luminance contrast. 
Figure 4.
 
Results for observer DW. Circles are for isoluminant disks (iso.); squares for disks with added luminance contrast. Darker squares are for added decrement luminance contrast (dec.); lighter triangles for added increment luminance contrast (inc.).
Figure 4.
 
Results for observer DW. Circles are for isoluminant disks (iso.); squares for disks with added luminance contrast. Darker squares are for added decrement luminance contrast (dec.); lighter triangles for added increment luminance contrast (inc.).
Figure 5.
 
Results for observer FK. Symbols and labels as in Figure 2.
Figure 5.
 
Results for observer FK. Symbols and labels as in Figure 2.
For both subjects, the isoluminant data are similar to the achromatic data in Figure 3. With regard to the added binocular luminance contrast data, we hypothesized that the data would shift towards averaging. Although the shift varies depending on color and observer, it is evident in all conditions. 
Luminance or luminance contrast?
The question arises as to whether the effect of adding binocularly matched luminance contrast to the color disks is contingent on the luminance and color contrasts being spatially contiguous. Another way to pose this question is whether the effect of adding luminance contrast is due to the luminance that is added/subtracted, or the luminance contrast that is added/subtracted. To test between the two alternatives, we used violet as the test color and the addition of a luminance decrement. The luminance decrement was either spatially contiguous with the 1.25° violet disk, as before, or extended outwards to a diameter of 6.5° such that it replaced the grey background around the violet. With the extended luminance decrement, the color disk became isoluminant again, this time with respect to a dark-grey rather than mid-grey background. We made measurements for five levels of decrement contrast, for both the extended (6.5°) and non-extended (1.25°) conditions with naïve observer DW. The results are shown in Figure 6. With the contiguous background the matched saturation increases with luminance contrast in keeping with the previous results. However, with the extended luminance contrast there is little or no effect of luminance contrast on matched saturation. This result suggests that spatial contiguity of the luminance and color contrasts is necessary for the shift in perceived saturation towards averaging to occur. 
Figure 6.
 
Effect of increasing the spatial extent of the added decrement luminance contrast for the violet condition. Contiguous = same size as violet disk; extended = larger than violet disk. Left: adjustable contrast in left eye; right: adjustable contrast in right eye.
Figure 6.
 
Effect of increasing the spatial extent of the added decrement luminance contrast for the violet condition. Contiguous = same size as violet disk; extended = larger than violet disk. Left: adjustable contrast in left eye; right: adjustable contrast in right eye.
Model
There are many approaches to modelling binocular interactions. Some incorporate a dynamic component in order to model the perceptual alternations that characterize binocular rivalry (e.g., Wilson, Blake, & Lee, 2001). Of those that deal with static percepts, as here, there are a number of approaches (e.g., Baker et al., 2012; Ding & Sperling, 2006, 2007; Kim, Gheiratmand, & Mullen, 2013; Maehara & Goryo, 2005; Meese et al., 2006; Meese & Baker, 2009; Zhou, Georgeson, & Hess, 2014). The majority of static models, however, are based on a model of contrast transduction first proposed by Legge and Foley (1980), namely:  where R is the internal response to contrast C, while z, p and q are constants. In Equation 2, the exponents p and q are typically assumed to be positive, ranging between 2 and 5 with p > q. The equation is designed to model contrast transduction as an accelerating function near threshold followed by a deceleration at high contrasts, with p determining the acceleration, q the deceleration, and z the contrast of the transition between the two.  
The static binocular interaction models based on Equation 2 incorporate a term in the denominator to model inhibition from the other eye. Thus if we use the terms CL and CR for the contrasts in the left and right eyes, the expressions for each eye (L and R) are:  and  If we add the two expressions together to obtain the model summed binocular response, we obtain  Equation 5 is termed the “late summation” model by Meese and Hess (2004), but it is generic in the sense that the majority of static dichoptic contrast models are variants of it; for example some models omit the exponent q and/or impose additional nonlinearities on the output binocular response term (many of these models are described in Meese et al., 2006).  
Our model is a modification of Equation 5. It incorporates a weighting function w on the interocular suppression component that is not a constant but is dependent on the amount of matched luminance contrast in the stimulus (see also Zhou, Georgeson, & Hess, 2014). If we prefix the contrast input from the “other” eye with w in Equations 2 and 3, then add the two equations together, the model binocular response becomes  To model our data using Equation 6, we use CL and CR for the contrasts of the colors in units relative to that of the fixed-reference contrast, and employ iterative search to find that value of the adjustable contrast that produces the same model response to the dichoptic and reference disks. Thus if we consider the case in which the adjustable contrast is in the left eye and the test in the right eye, we solve the following equality:  where Ladj is the model response to the adjustable contrast in the left eye, Rtest the model response to the test contrast in the right eye, and Lref and Rref the model responses to the left- and right-eye reference contrasts. A corresponding equation is used when the adjustable contrast is in the right eye and test contrast in the left eye.  
Inspection of Figures 4 and 5 reveals that there is variability both within and between the data across subjects and conditions, particularly in the degree to which the added luminance contrast promotes averaging. These variations could be modelled by selecting a unique set of the four parameters p, q, z and w for each subject and condition. Unfortunately our dichoptic saturation data is insufficient to determine accurately the values of p, q, and z. To do this, one needs to measure the transducer function for color saturation, either directly using a scaling method, or as most studies have done in the achromatic domain, indirectly using contrast discrimination measures gathered across a range of contrasts (e.g., Baker, Meese, & Georgeson, 2007; Meese et al., 2006). Our approach is therefore necessarily more modest, and we have only attempted to demonstrate how in principle a conventional dichoptic contrast model that incorporates w is able to account for the broad pattern of findings. 
Figure 7 shows the results of the modelling. The left-hand graph shows model predictions for a reference contrast of 0.23 (DW's lime condition), with p fixed at 4.25, q at 3.0 and z at 0.001, with w ranging from 0.0 to 1.25. The choice of p and q was slightly higher than previous estimates for dichoptic luminance contrast, which we found enabled w to exert a more significant influence on the amount of averaging. Note that when w = 0.0, it implies that the interocular suppression is “switched off.” The graph shows a number of interesting features. First, when w > 1, the part of the curve close to a test contrast of zero falls slightly below the winner-take-all prediction for the left-eye data, or to the left of it for the right-eye data, and only crosses the winner-take-all line mid-way along the line as test contrast approaches its maximum value. To our knowledge this model outcome has not been reported before in the literature on dichoptic contrast mixture, but it is important because much of the isoluminant as well as black and white data show such a trend (see Figures 3, 4, and 5). Second, it can be seen that the effect of reducing w from its maximum of 1.25 to 0.0 is to systematically shift all the curves away from winner-take-all towards averaging, such that when w = 0.0, i.e., no interocular suppression, the data are close to perfect averaging. 
Figure 7.
 
Left: the effect of changing the weighting parameter w on the interocular suppression in the model predictions. Right: application of the model to DW's lime data. The values of w have been selected to give the left-eye (LE) data an approximate fit based on visual inspection.
Figure 7.
 
Left: the effect of changing the weighting parameter w on the interocular suppression in the model predictions. Right: application of the model to DW's lime data. The values of w have been selected to give the left-eye (LE) data an approximate fit based on visual inspection.
On the right of the figure, using the same values of p, q, and z, the model has been applied to DW's lime data, with w set to 1.2 for the isoluminant and 0.4 for the added luminance contrast condition. These values of w were chosen to fit DW's left-eye data based on visual inspection. The fit is good, but the set of parameters that fit well for her left-eye data fare less well for her right-eye data, suggesting that another combination of parameters would be optimal for that condition. Using different values of w for the left and right eyes is an obvious candidate for improving the fits, but again we must reiterate that without knowing accurately the values of the exponents p and q for each eye, it is speculation as to whether the between-eye differences are really due to differences in w rather than in the other parameters. 
Baker et al. (2012) modelled their dichoptic luminance mixture data using a descriptive model based on an equation first proposed by the physicist Schrödinger. The equation had a single free parameter: the exponent on the equation's luminance values. The model predictions were formulated within the same contrast space as in Figure 7. Although Baker et al. showed that reducing the value of the exponent shifted the middle part of the iso-brightness curves away from winner-take-all towards averaging, the curves always folded-back to winner-take-all at the axes at [1,0] for the RE adjustable and [0.1] for the LE adjustable contrasts. This fold-back is not seen in any of our color-plus-luminance contrast conditions, and therefore a change in the value of the exponent in Baker et al.'s model is not appropriate for modeling this part of our data. 
The fold-back is, however, seen in our isoluminant and achromatic data, most noticeably for the white disks (Figure 3). As noted earlier, the fold-back is the contrast equivalent of “Fechner's paradox.” It is captured by our model: see the curve in the left panel of Figure 7 with w = 0. Why does it happen? Meese et al. (2006) explain, “Suppose that the left eye is presented with a high-contrast stimulus (say 50%). As the contrast of a similar stimulus in the right increases, it will contribute to both suppression of and summation with the contrast in the left eye. For low to intermediate contrasts in the right eye, the level of suppression turns out to be greater than the contribution to summation and the overall response decreases. Hence, if the contrast in the right eye is reset to zero, the overall response goes back up again and perceived contrast increases, explaining the paradox” (p. 1234). 
Discussion
The isoluminant data for both subjects were close to winner-take-all, and thus we can conclude that the rules for dichoptic saturation mixtures are similar to those for dichoptic luminance mixtures. However, the addition of binocularly matched luminance contrast significantly shifted the saturation mixtures towards averaging. The shift towards averaging was contingent on the added luminance contrast being spatially contiguous with the color patches. A modelling exercise revealed that a simple and plausible explanation for the effects of the binocularly added luminance contrast was that it reduced the gain on the interocular suppression between the color contrasts. 
Are there other plausible explanations for these results? Adding luminance contrast to color contrast has the effect of desaturating the colors (e.g., Bimler, Paramei, & Izmailov, 2009), so could desaturation produce the shifts towards averaging we observed? We argue no. Remember that the luminance contrasts were added to both dichoptic and reference disks, so any overall desaturation would apply equally to both, and so would be “taken into account.” One can however think of the effect of the added luminance contrast as having a specifically dichoptic desaturating effect. As the isoluminant data testifies (i.e., without the added luminance contrast), a saturated color in only one eye appears approximately as saturated as when the color is in both eyes; this is winner-take-all. However, with the added luminance contrast, a saturated color in only one eye will tend to average with the zero level of saturation in the other, desaturating the perceived color of the mixture. For those who can free-fuse, two demonstrations of “dichoptic desaturation” can be observed in Figures 8 and 9. In the interests of advancing the generality of the findings here, Figure 8 presents the stimulus arrangement in a different form from that used in the experiments described above, a form somewhat commensurate with the method employed by Meese and Hess (2005) in their study of dichoptic masking with luminance Gabor patches described in the Introduction. In Figure 8, violet disks are surrounded by a thick black ring, unlike in our experiments in which spatially contiguous luminance contrast was added to the colors. As one might expect, the effects, however, are similar: Surrounding an isoluminant color with a black annulus adds to that color an increment in luminance contrast. Free-fusion of Figure 7 reveals that without the black annulus, a violet in just one eye appears almost as saturated as when in both eyes. However, with the black annulus the violet-in-one-eye mixture appears desaturated. Figure 9 provides another example of dichoptic desaturation, this time using an image of a natural scene. Free-fusion of the top two image pairs results in perceived saturations that are similar. In other words, it makes little difference to the perceived saturation of the mix if the left half is a plain grey rather than a copy of the right half. On the other hand if the left half has luminance contrast instead of being uniformly grey, as in the bottom figure, the fused stimulus appears desaturated. The matched luminance contrasts in the bottom figure promote perceptual averaging of the disparate color saturations. 
Figure 8.
 
“Dichoptic desaturation” by binocular luminance contrast revealed in a disk-annulus arrangement. If one free-fuses the four stereo-pairs, a pronounced desaturation will likely only be observed in the bottom pair.
Figure 8.
 
“Dichoptic desaturation” by binocular luminance contrast revealed in a disk-annulus arrangement. If one free-fuses the four stereo-pairs, a pronounced desaturation will likely only be observed in the bottom pair.
Figure 9.
 
“Dichoptic desaturation” in an image of a natural scene. When the stereo-pairs are free-fused, readers may observe that the colors in the top and middle pair remain equally saturated, but the colors in the bottom pair appear desaturated.
Figure 9.
 
“Dichoptic desaturation” in an image of a natural scene. When the stereo-pairs are free-fused, readers may observe that the colors in the top and middle pair remain equally saturated, but the colors in the bottom pair appear desaturated.
It appears therefore that binocularly matched luminance contrast reduces interocular suppression between unmatched colors. A related conclusion was reached by Blake and Boothroyd (1985) in their study of orientation-based binocular rivalry, and by Meese and Hess (2005) in their study of dichoptic contrast masking (see also Buckthought & Wilson, 2007). As Blake and Boothroyd (1985) put it, “We conclude that the presence of matching features in the two eyes' views stabilizes the binocular percept. Evidently, the binocular visual system first seeks to establish correspondence between image features contained in the two monocular views. Failure to establish such correspondence leads to binocular suppression, an effective means for eliminating diplopia and/or confusion. When correspondence is established, however, binocular fusion takes precedence over suppression. Only those monocular features with no interocular counterpart participate in the rivalry process” (p. 123). 
If our hypothesis is correct that binocularly matched luminance contrast reduces interocular suppression between unmatched colors, the question is why? Objects that lie in different depth planes along the line of sight will often partially overlap in depth, and it is presumably important for vision that the overlapping parts do not “blend” perceptually as this blending would introduce noise into the process of segmenting the three-dimensional image into objects and layers. One of the functions of interocular suppression might be to suppress the weaker of the two overlapping signals in order to minimize such noise. Thus, the presence of binocularly matched luminance contrast promotes the interpretation that dimensions that are unmatched but contiguous, for example in color, are part of the same object. And if part of the same object, it would be safe to assume that the between-eye differences between them likely are caused internally rather than externally, and that it is therefore best to combine their signals rather than suppress the weaker of the two. We term this the “object commonality” hypothesis. 
The results from the present study have revealed a new type of interaction between suprathreshold color and luminance contrast in the context of binocular vision. We mentioned earlier that Mullen et al. (2014) found strong dichoptic masking of colored target gratings by orthogonally oriented luminance gratings. It seems likely that the dichoptic masking of color contrast by luminance contrast revealed by Mullen et al., and the promotion of dichoptic saturation averaging by matched luminance contrast in the present study are mediated by similar mechanisms. A future challenge will be to develop a model of binocular interaction that is able to accommodate both sets of findings. 
Acknowledgments
This study was supported by Canadian Institute of Health Research grant #MOP 123349 given to F. K. Special thanks to Mark Georgeson and Daniel Baker for many helpful discussions concerning the significance of the findings. 
Commercial relationships: none 
Corresponding author: Frederick Kingdom. 
Email: frederick.kingdom@mcgill.ca. 
Address: McGill Vision Research, Montreal, Quebec, Canada. 
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Figure 1.
 
(a) Isoluminant plane of DKL color space. L = long-wavelength-sensitive, M = middle-wavelength-sensitive, S = short-wavelength-sensitive cones. (b) The four color (red, cyan, violet, lime) and two achromatic (black, white) conditions used in the experiments. (c) Stimulus construction and procedure using violet as the example. LE = left eye; RE = right eye. See text for further details. Note that the colors will unlikely be accurate due to limitations in photographic reproduction.
Figure 1.
 
(a) Isoluminant plane of DKL color space. L = long-wavelength-sensitive, M = middle-wavelength-sensitive, S = short-wavelength-sensitive cones. (b) The four color (red, cyan, violet, lime) and two achromatic (black, white) conditions used in the experiments. (c) Stimulus construction and procedure using violet as the example. LE = left eye; RE = right eye. See text for further details. Note that the colors will unlikely be accurate due to limitations in photographic reproduction.
Figure 2.
 
Example stimuli as the observer would see them. Top: achromatic conditions; bottom: color saturation conditions using lime as the example color. The task on each trial was to choose the disk, above or below fixation, according to the following criteria: black, the darker disk; white, the brighter disk; cyan, the more saturated disk. The cyan color will unlikely be accurate due to limitations in photographic reproduction.
Figure 2.
 
Example stimuli as the observer would see them. Top: achromatic conditions; bottom: color saturation conditions using lime as the example color. The task on each trial was to choose the disk, above or below fixation, according to the following criteria: black, the darker disk; white, the brighter disk; cyan, the more saturated disk. The cyan color will unlikely be accurate due to limitations in photographic reproduction.
Figure 3.
 
Results with the achromatic stimuli. Each graph plots the adjustable luminance contrast in one eye as a function of the test luminance contrast in the other eye, required to make the resulting dichoptic mix the same brightness/darkness as the reference disk. Points with vertical error bars are when the adjustable contrast is in the left eye (LE) and test contrast in the right eye (RE). Points with horizontal error bars are for RE adjustable and LE test contrasts. Filled circles are for luminance decrements; open circles, luminance increments. Error bars are standard errors of the mean of each set of matches. The oblique dotted line represents the prediction if the dichoptic mixture is perceived as the average of the two luminance contrasts, whereas the horizontal and vertical dashed lines represent the prediction if the mixture is perceived as that of the larger excursion, or “winner-take-all.” Top graphs for observer DW; bottom graphs for FK.
Figure 3.
 
Results with the achromatic stimuli. Each graph plots the adjustable luminance contrast in one eye as a function of the test luminance contrast in the other eye, required to make the resulting dichoptic mix the same brightness/darkness as the reference disk. Points with vertical error bars are when the adjustable contrast is in the left eye (LE) and test contrast in the right eye (RE). Points with horizontal error bars are for RE adjustable and LE test contrasts. Filled circles are for luminance decrements; open circles, luminance increments. Error bars are standard errors of the mean of each set of matches. The oblique dotted line represents the prediction if the dichoptic mixture is perceived as the average of the two luminance contrasts, whereas the horizontal and vertical dashed lines represent the prediction if the mixture is perceived as that of the larger excursion, or “winner-take-all.” Top graphs for observer DW; bottom graphs for FK.
Figure 4.
 
Results for observer DW. Circles are for isoluminant disks (iso.); squares for disks with added luminance contrast. Darker squares are for added decrement luminance contrast (dec.); lighter triangles for added increment luminance contrast (inc.).
Figure 4.
 
Results for observer DW. Circles are for isoluminant disks (iso.); squares for disks with added luminance contrast. Darker squares are for added decrement luminance contrast (dec.); lighter triangles for added increment luminance contrast (inc.).
Figure 5.
 
Results for observer FK. Symbols and labels as in Figure 2.
Figure 5.
 
Results for observer FK. Symbols and labels as in Figure 2.
Figure 6.
 
Effect of increasing the spatial extent of the added decrement luminance contrast for the violet condition. Contiguous = same size as violet disk; extended = larger than violet disk. Left: adjustable contrast in left eye; right: adjustable contrast in right eye.
Figure 6.
 
Effect of increasing the spatial extent of the added decrement luminance contrast for the violet condition. Contiguous = same size as violet disk; extended = larger than violet disk. Left: adjustable contrast in left eye; right: adjustable contrast in right eye.
Figure 7.
 
Left: the effect of changing the weighting parameter w on the interocular suppression in the model predictions. Right: application of the model to DW's lime data. The values of w have been selected to give the left-eye (LE) data an approximate fit based on visual inspection.
Figure 7.
 
Left: the effect of changing the weighting parameter w on the interocular suppression in the model predictions. Right: application of the model to DW's lime data. The values of w have been selected to give the left-eye (LE) data an approximate fit based on visual inspection.
Figure 8.
 
“Dichoptic desaturation” by binocular luminance contrast revealed in a disk-annulus arrangement. If one free-fuses the four stereo-pairs, a pronounced desaturation will likely only be observed in the bottom pair.
Figure 8.
 
“Dichoptic desaturation” by binocular luminance contrast revealed in a disk-annulus arrangement. If one free-fuses the four stereo-pairs, a pronounced desaturation will likely only be observed in the bottom pair.
Figure 9.
 
“Dichoptic desaturation” in an image of a natural scene. When the stereo-pairs are free-fused, readers may observe that the colors in the top and middle pair remain equally saturated, but the colors in the bottom pair appear desaturated.
Figure 9.
 
“Dichoptic desaturation” in an image of a natural scene. When the stereo-pairs are free-fused, readers may observe that the colors in the top and middle pair remain equally saturated, but the colors in the bottom pair appear desaturated.
Table 1.
 
Contrast detection thresholds and reference contrasts for the four color and two achromatic disks used in the experiments, for both subjects DW and FK.
Table 1.
 
Contrast detection thresholds and reference contrasts for the four color and two achromatic disks used in the experiments, for both subjects DW and FK.
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