1. Introduction

Stereoscopic three-dimensional (3D) display is a device used to present two slightly different images to each of an observer’s two eyes to produce three-dimensional sense of scenes. At present, it has been widely used in TV entertainment, video games, vision research, remote device operation, medical imaging, vocational training, virtual reality (VR), augmented reality (AR) and other fields. When two different images are presented to the two eyes and the brain can fuse the two images into a single perception, binocular fusion is said to occur. In contrast, when the images are different and they appear to alternate, it is called binocular rivalry [1]. In stereoscopic three-dimensional (S3D) displays, if there is binocular rivalry, there are many reasons, such as luminance mismatch, chromaticity mismatch, and structure mismatch [2]. These mismatches can be induced at several stages in the process of 3D contents, including content generation, coding and transmission, and rendering on 3D displays [3]. Long-term viewing of mismatched image information will result in visual discomfort or visual fatigue, and in severe cases, it is prone to dizziness and vomiting. Over the last decades, visual discomfort has always been described as the number one health issue restricting the further development of stereoscopic display technology [4,5].

Human perception becomes difficult and uncomfortable in the event of binocular fusion when the level of mismatches exceeds a certain threshold [6,7]. Specifically, when the color difference presented for the left and right eyes exceeds the binocular color fusion limit, color rivalry is said to occur [2]. The first experimental result on the phenomenon of binocular color rivalry was observed in the early 18th century, using different colors of silk to observe through holes [8]. When binocular color fusion occurs, people can perceive a fixed color called “binocular color mixture” by colorists [9,10]. It is essential to establish the “appropriate” conditions for binocular color fusion. This is an essential requirement of 3D content creation and 3D system design.

Some researchers have studied the effect of hues or wavelength differences on binocular color fusion using stimuli of various sizes, intensities, and saturation. Ikeda et al. studied the fusion limits of spectral colors and white light [2]. Their experiments were to determine the differences in wavelengths that caused color rivalry. The color fusion limit is quantified as a wavelength function of the spectral colors and the stimulation wavelengths range from 500 to 660 nanometers. Their results indicated that the color fusion limit is between 10 and 50 nanometers, depending on the wavelength region studied. In 2011, Jung et al. carried out experiments to measure the color fusion limits of eight chroma points selected from CIE 1976 chromaticity diagram at 10 cd/m² luminance level [7]. The percentage of color fusion was expressed by Euclidean distance along straight lines in $u^{\prime}v^{\prime}$ chromaticity diagram ($\Delta E_{{u}^{\prime} v^{\prime}}$), and they quantified the color fusion limit by using ellipses in the chromaticity diagram. On Jung's experimental model, Chen et al. measured the color fusion limits for different disparities [11]. The experimental results show that when the disparity increases from -120 to +120 arc minutes, the color fusion limit increases accordingly. In the MacLeod–Boynton color space, Malkoc and Kingdom defined the binocular color rivalry threshold (BCRT) as angular color difference, they obtained that mean BCRT was 48.9° of color angle when the stimuli had a mean luminance of 25.7 cd/m² and stimulus contrast was fixed at 80% [1]. Qin et al. examined the effects of luminance and size of stimuli upon the binocular fusion limit [12,13], and reported that color fusion occurs more difficult with central visual field than peripheral visual field [14].

It is clear that color fusion occurs under conditions that apparently minimize chromatic differences between the corresponding retinal areas. Hering noted that stable color fusion was more challenging to obtain as hue difference between the two binocular stimuli increased [15]. According to Hering, opponent colors would have the most significant tendency for rivalry, because these stimuli have the most considerable difference in hue. Further, Ikeda and Nakashima hypothesized that color fusion still occurred when opponent chromatic responses from each eye was less than a threshold value [16]. Their model worked well in predicting rivalry between wavelengths in the mid-spectral region. However, very little data exists which quantify the relationship between opponent chromatic saturation and binocular fusion of opponent colors for non-spectral colors.

In this paper, we conducted a quantitative measurement experiment for detecting the chromatic saturation limit of binocular opponent-color fusion. The main independent variable used to determine the binocular fusion limit is the chromatic saturation of the opponent colors, given by the distance from the center of the color space along a particular color direction on an equal lightness plane. CIELAB is a device-independent and color-opponent space based on nonlinearly compressed coordinates (e.g. CIE XYZ color space) and physiological characteristics. It can use digital methods to describe human visual perception and ensure accurate color calculation. CIELAB is the most widely used color system in the fields of science and engineering because of its good color constancy, which can minimize the influence of illumination in color experiments [17]. The color in CIELAB is expressed as three unitless coordinate values: ${L^\ast },\,{a^\ast }$ and ${b^\ast }$. The vertical axis ${L^\ast }$ is the lightness, and the other two axes are the color-opponent dimensions forming the chroma and hue plane. The relationships between chroma ($C$), hue ($H$) and values ${a^\ast },\,{b^\ast }$ are shown as Eq. (1) and (2):

The color stimuli of left and right eyes are sampled from the equal lightness plane and have the most enormous difference in hue but the same chroma. The chromatic fusion limit (CFL) is measured by increasing the chroma value to detect whether the opposite colors are fused.

In the second part, we describe the experimental method. A pair of circular blocks present the relatively opposite colors with same lightness in the CIELAB color space. Five subjects were invited to make a forced-choice whether they can fuse a total of 32 opponent color pairs. Due to the binocular color fusion may be affected by color inconsistency between eyes, we swapped the colors of the left and right stimuli. 320 trials were tested for each subject, and 1600 test results were collected. In the third part, experimental results are presented, and we use ellipses to quantify the chromatic fusion limits for opposing colors. Finally, we summarize this article in the fourth part.

2. Method

2.1 Apparatus and viewing conditions

The color stimulus presentation in the experiment was provided by a Samsung 3D display (S23A950D). The screen size is 23 inches, and the resolution is 1920 (horizontal) ${\times} $ 1080 (vertical) pixels. It is connected to the graphics card (NVIDIA GeForce GTX 1080) of a personal computer (Intel Core 2 Duo CPU, 3 GHz processing speed, 4 GB RAM, Microsoft Windows 7), and the digital number is 8 bits. The display has 2D/3D switching functions, and the three-dimensional images can be seen by wearing active shutter glasses. A Photo Research PR-715 spectroradiometer was used to measure the luminance and chromaticity at the center point of the display through the 3D glasses. The CIEXYZ of the black point of the collection were 0.247/0.241/0.467, the CIE1931 chromaticity (x, y) of the white point was (0.274, 0.280). As shown in Fig. 1(a), the color gamut of the 3D display is within the triangular area encapsulated by the reddest chromaticity point (0.617, 0.333), the greenest chromaticity point (0.330, 0.616), and the bluest chromaticity point (0.152, 0.067). Figure 1(b) shows the tone reproduction curve (TRC) of the 3D displays.

 

Fig. 1. (a) Chromaticity coordinate distribution of primary colors and the triangle vertices indicate chromaticity coordinates of each channel at maximum brightness; (b) Tone reproduction curve (TRC) of the 3D displays.

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The experimental process was performed in a closed dark room. Participants were asked to sit 58 cm from the 3D displays, where lighting conditions were held constant for all participants at all sessions. Figure 2 shows the apparatus and observation environment used in our experiment.

 

Fig. 2. The equipment (a) and observation environment (b) used for the research in the experiment.

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2.2 Subject

Five college students aged between 22 and 25 were selected to participate in the experiment by passing vision and color vision tests, showing normal visual acuity, normal color perception and normal stereo vision. The experiment conforms to the standards stipulated in the Declaration of Helsinki [18]. They read and signed the informed consent form and could withdraw from the test at any stage.

2.3 Stimuli

In the CIELAB color space, all stimulus colors are selected on the plane with lightness ${L^\ast }$ of 30 (luminance of 15 ± 0.5 cd/m²). As shown in Fig. 3(a), ${L^\ast }{a^\ast }{b^\ast }$(30, 0, 0) is selected as the center of sampling points, and the selection of the opposite color pair is based on the values of ${a^\ast }$ and ${b^\ast }$ channels. There are four groups of color sample points on four opposite color directions, including color pairs: Red-Green (R-G), Yellow-Blue (Y-B), RedYellow-GreenBlue (RY-GB) and RedBlue-GreenYellow (RB-GY). Taking the direction of the opposite color pair R-G as an example, as shown in Fig. 3 (b), the chromatic value ${a^\ast }$ starts from 8 and increases with a step size of 2, and a total of 8 sample point pairs are obtained.

 

Fig. 3. Selection of stimulus points: (a) the equal lightness plane (L^*=30), (b) the sampled points of opposite color pairs.

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The stimulus colors of the experiment are generated and controlled by specially written software in C++. Researchers generally agree that that color fusion is present more often with a dark surround than with an equal-brightness white surround. Therefore, the stimulus consists of a pair of circular patches on a black background, as shown in Fig. 4. The circular patch has a viewing angle of 2 degrees in diameter. The dialogue box in the upper left corner is used to interact with the observer. Chen et al. suggested that the color fusion limit changed under different disparities [11], in order to avoid the influence of visual depth perception, the disparity of the stimulus was set to zero. A frame with the gray color serves as a zero-disparity reference. In addition, the gray frame can always provide a corresponding reference color in the field of view. A total of 64 color points (4 color directions × 16 sample points) were selected. The flicker of shutter glasses can easily lead to interocular crosstalk when data are collected at the most minor disparities. Therefore, rough stimuli colors with specific CIE-1931 coordinates were generated by the look-up tables (LUT) method [19], then the Photo Research PR-715 spectroradiometer was used to measure the luminance and chromaticity of the RGB color presented on display through the 3D glasses. The color difference between the desired value and the measured value is controlled within 1$\Delta E_{ab}^\ast $ by fine-tuning the RGB value. The RGB value of each color point is stored before starting the experiment.

 

Fig. 4. Example of a stimulus used in the experiment. The size of the circular patch was 2° in diameter, and the left and right colors of stimuli are opposite colors, which were generated by the software.

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2.4 Procedure

The software controls the beginning and end of the experiment. Many factors affect the experiment. To maximize the accuracy of the experimental data and exclude the influence of color inconsistency between eyes, the experiment was divided into two sessions. In session 1, we selected the stimulus colors for the left eye as R, Y, RY and GY. In session 2, we selected the stimulus colors for the left eye as G, B, GB and RB. Firstly, for each session, left and right circular blocks are filled randomly with a pair of opposite colors on a black background to avoid effects of memory. Each pair of stimulus presentation time is 15s, so that the subject has enough time to perceive whether the colors are fused.

According to the observer's description during the experiment, there are four stages in the process of binocular fusion into binocular rivalry, as shown in Fig. 5: stage 1, binocular colors merge into a single color, this color appears to be gray; stage 2, between binocular fusion and binocular rivalry, color fusion appears glossy; stage 3, mixed colors appear in the binocular color rivalry, and the mixed colors can sense the colors of the left and right eyes at the same time (The mixed color does not produce periodic alternation); and stage 4, binocular color rivalry, the color of the left and right eyes alternates periodically. If the color difference between the two eyes starts to increase from zero, a point is reached where the binocular image will appear slight gloss or glitter. At this point, the difference between the two eyes is noticeable [20]. In our experiment, binocular fusion should be the single color sense. Since the color perceived by the gloss is single, it belongs to binocular fusion. If the perceived color is not single, it is binocular rivalry. If mixed color appears, the perceived color is not a single color and it belongs to binocular rivalry. We try to measure chromatic fusion limits for opposite colors. Therefore, the observer needs to make a forced-choice for either fusion or rivalry.

 

Fig. 5. Four stages from binocular fusion to binocular rivalry.

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The observer has two buttons to choose: “Y” and “N”. The “Y” button indicates that the observer can fuse the opposite color pair, and the perceived color is a single color. The “N” button indicates that the brain cannot merge the colors seen by the left and right eyes, and the perceived color is not a single color. After traversing all opposite color pairs, all keyboard information will be automatically recorded to a document. Each opposing color pair will be randomly repeated ten times (5 times for session 1, 5 times for session 2). There were 320 trials (4 groups of opposite directions × 8 pairs of opposite colors × 10 repeats) for an observer.

3. Results and discussion

3.1 Calculation of chromatic fusion limit

The results were averaged for each observer. We calculated the low fusion probability for all observers on different opposite color directions. The fusion probability $({p{\%}} )$ is calculated as follows [11]:

If the fusion probability of binocular colors is 50%, the chromatism $\Delta E_{ab}^{\ast }$ corresponding to the opposite color pair is selected as the chromatic fusion limit. The linear interpolation formula may be used to calculate the chromatic fusion limits for different opposite directions as in the literature [7]. However, when the fusion probability $(p\%)$ never exceeds 50%, linear interpolation will not work. Therefore, we use a psychometric function to fit the probability [21]:

 

Fig. 6. Fusion probability fitting curve in session 1 (Observer HL): (a) R-G direction, R/G vs. LE/RE (b) RY-GB direction, RY/GB vs. LE/RE, (c) Y-B direction, Y/B vs. LE/RE, (d) RB-GY direction, GY/RB vs. LE/RE. See Dataset 1 for underlying values [22].

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From Fig. 6, it can be observed that the chromatic fusion limit is about 42 $\Delta E_{ab}^\ast $ down to above 25 $\Delta E_{ab}^\ast $. In this interval, the slope of fusion curves in the Y-B and RB-GY directions decreases, while the slope of the fusion curve increases in the R-G and RB-GY directions. We calculated the chromatism $\Delta E_{ab}^\ast $ as chromatic fusion limit when the fusion probability $(p\%)$ was 50% and observed that the fusion rate in the RY-GB and Y-B directions decreased rapidly. When the fusion probability $(p\%)$ is 50%, chromatic fusion limits are 30.57 $\Delta E_{ab}^\ast $ on the Y-B direction, 27.67 $\Delta E_{ab}^\ast $ on the RY-GB direction, 41.67 $\Delta E_{ab}^\ast $ on the R-G direction, and 39.59 $\Delta E_{ab}^\ast $ on the RB-GY direction, respectively. The chromatic fusion limit on the R-G direction is more extensive than on other directions, and the chromatic fusion limit on the RY-GB direction is the smallest.

The fusion curve of the subject HL in session 2 is shown in Fig. 7. The results have some changes, the chromatic fusion limit is between 25 $\Delta E_{ab}^\ast $ and 45 $\Delta E_{ab}^\ast $. The slope of the fusion curve in different directions is not much different from session 1. When the fusion rate $(p\%)$ is 50%, chromatic fusion limits are 28.44 $\Delta E_{ab}^\ast $ on the Y-B direction, 33.89 $\Delta E_{ab}^\ast $ on the RY-GB direction, 41.33 $\Delta E_{ab}^\ast $ on the R-G direction, and 44.64 $\Delta E_{ab}^\ast $ on the RB-GY direction, respectively. These results can also be found in Table 1.

 

Fig. 7. Fusion probability fitting curve in session 2 (Observer HL): (a) G-R direction, G/R vs. LE/RE, (b) GB-RY direction, GB/RY vs. LE/RE, (c) B-Y direction, B/Y vs. LE/RE, (d) RB-GY direction, RB/GY vs. LE/RE.

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Table 1. The chromatic aberration $E_{ab}^{\ast}$ of the subjects in different opposite color directions when p% is 50%.

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Table 1 lists all of observers’ chromatic fusion limits, where e represents the difference between session 1 and session 2. It is observed that all observers produced the minimum value in Y-B or RY-GB directions and the maximum value in R-G or RB-GY directions. Obviously, the average fusion limit in RB-GY direction is bigger than that of other directions. In comparison, the fusion limit in the direction of RY-GB is smaller than that of other directions.

In order to understand whether fusion limit is statistically significant for different color opposite directions, we carried on an analysis of variance (ANOVA). We did one-way ANOVA for session 1 and session 2, respectively. The first step is to test whether the data meets the basic assumptions of ANOVA, that is, to test the normality and homogeneity of the variances. The Anderson-Darling test was performed on the data of 5 groups (4 opposite color directions). The return value $h = 0$ for each group indicated that the null hypothesis was rejected by the test results at the default significance level of 5%, and the fusion results of the 5 groups all followed the normal distribution. The homogeneity test of variance was carried out for the 4 opposite color directions, and the test results were shown in Table 2, with a higher p value ($p$ = 0.98827), indicating that the null hypothesis was accepted at the significance level of 0.05. The binocular fusion limits of the 5 groups were in line with the normal distribution with the same variance, satisfying the primary hypothesis of one-way ANOVA.

Table 2. Summary of homogeneous test of variance

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In Table 3, ANOVA shows inter-group variation (columns) and intra-group variation (errors). $SS$ is the sum of squares. $DF$ is the degree of freedom. The total degree of freedom is the number of observations minus 1. The degree of freedom between groups is the number of groups minus 1. Degrees of freedom within groups is equal to the total degrees of freedom minus the degrees of freedom between groups. $MS$ is the mean square error or $SS/{\; }DF$ for each source of variation. $F{\; }$statistic is the ratio of mean square error (328.197/64.513). P value is the probability that the test statistic is greater than the calculated test statistic value, i.e., $P(F$ > 5.09). In Session1 and Session 2, p values were 0.0116 and 0.0027, respectively, both lower than 0.05, indicating significant differences between column mean values. Variance analysis results have shown that binocular fusion limits in different opposite color directions have a significant difference, but this does not mean that any two opposite color direction significant differences in the binocular fusion limit. Therefore, a pair comparison test, namely multiple comparisons, is needed to determine the opposite color direction of the binocular fusion limit with differences. Finding out the significantly different color directions can help us determine the position and direction of semi major axis and semi minor axis of ellipse fitting.

Table 3. Result of ANOVA

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In Fig. 8, the circle represents the mean of each group, and the line segment represents the confidence level of the mean of each group (the confidence level defaults to 95%). According to the position relationship of each line segment, the difference between the two groups was judged. The two lines are projected on the X-axis. If the projection positions overlapped, there was no significant difference between the two groups (the significance level defaults to 0.05). If a certain two lines segments do not intersect, there is no overlap, and then the difference between the two groups is significant. If there is an overlap between two lines segments, the difference between the two groups is not significant.

 

Fig. 8. Multiple comparison: (a) session 1. (b) session 2.

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In Fig. 8(a), the lines of opposite color directions GY-RB and Y-B are projected onto the X-axis, and the two lines overlap, so there is no significant difference in hue between GY-RB and Y-B. In Fig. 8 (b), the lines of opposite color direction GY-RB and Y-B are projected onto the X-axis without overlapping, so they are significant differences. As it can be seen from Fig. 8, there are substantial differences between GY-RB and RY-GB in session 1, and significant differences between GY-RB and the other color directions in session 2. Table 4 shows the multiple comparison results of session 1 and session 2, respectively. The first two columns are the series number of the comparison group. The fourth column is the mean difference between the two groups. The third and fifth columns are the lower and upper limit of the confidence interval. The 95% confidence interval of mean difference does not include 0, which indicates that the difference between the two groups is significant under the significance level of 0.05. Obviously, in session 1 and session 2, the sequence number is 4, that is GY-RB direction, which is significantly different from other directions.

Table 4. Multiple comparison results

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3.2 Quantification of chromatic fusion limit

We obtained chromatic fusion limits where binocular fusion probability is 50%. The fusion limits of sample point pairs can be displayed in a two-dimensional plane. Different subjects have different fusion limits at different opposite color directions. Regarding the expression of the color fusion limit, Ikeda et al. [23] and Qin et al. [24] studied spectral colors and used $\mathrm{\Delta }\lambda $ to represent the fusion limit of wave length $\lambda $. For non-spectral colors, Ikeda et al. [25] suggested that although the color fusion limit of each observer varies from person to person, a circle can be approximately fitted to represent the fusion limit of white light. However, Jung et al. suggested that the shape of color fusion limit could be more accurately represented by fitting ellipse [7]. The ellipse equation is defined as:

 

Fig. 9. Fusion ellipse boundaries for all subjects: (a) ellipse of fusion limits for session 1, and (b) ellipse of fusion limits for session 2. See Dataset 1 for underlying values [22].

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Table 5. Statistics of the estimated parameter values of each subject and the Goodness-of-Fit of the ellipse.

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In the standard uniform chromaticity diagram, the chromatic fusion limit is not modeled as a circle of equal size. The chromaticity fusion limits are measured experimentally, and a group of ellipses similar to MacAdam ellipses are obtained. MacAdam ellipses are used to determine color differences. They constitute the areas in the color diagram in which the comparative colors surrounding a reference color are perceived to be an equal color distance apart [26]. Because the observation points of binocular chromaticity fusion limits are different from the observers, the points on the ellipse are not continuous. Therefore, those ellipses converted into the chromaticity diagram are not ellipses in strict sense (their shapes are very similar to ellipses).

The ellipse's long axis is determined on the RB-GY direction, and the short axis is on the RY-GB direction. Some observers have experienced varying degrees of lateralization in session 2. The observer KC has a certain positive deflection from the positive and negative values, and WL has a certain reverse deflection in session 2. Previously, the dominant eye of each observer was recorded. The dominant eyes of KC and WL were the left and right eyes, respectively. In session 2, the two observers showed a certain deflection. We guessed that it was affected by the dominant eye.

In the experiments of Phuangsuwan and Ikeda, they found that adapting color and adapted color are not opposite, and there are different angles of deviation [27,28]. Their experiment is not a stereo vision experiment, but it gives us some hints. Because the color space selected in the experiment is CIELAB color space, the opposite color is determined based on the maximum hue difference. According to the description of Phuangsuwan and Ikeda, the ellipse curve fitted by us also has some deflection, which also shows that CIELAB color space is not a uniform color space.

Two color scientists, Shevell and Martin, introduced the topic of color opponency from the perspective of color appearance (psychophysics) and neuronal response measurement (physiology) [29]. Overall, unique hues are essential for understanding the rivalry of perceived color, but any cell-preferential stimuli may not represent them. On the contrary, for example, unique hues may come from the average population activity of many cone-shaped hand cells rather than individual cells’ specific responses. The opposition between perception and physiology does not necessarily converge at the level of an individual's highly tuned nerve cells. Conversely, the convergence may lie in analyzing the minimum response, or it may be a relative response between cells that form a whole. It was found that the cortical receiving area had excitatory input from long-wavelength sensitive cone cells (L +) and inhibitory input from medium-wave sensitive cone cells (M). However, in the primary visual cortex, most of the receiving areas are elongated rather than circular. This elongation makes them selective for the positioning of edges in the visual world [30,31]. According to literature reports, in the retina, cones sensitive to red and green account for about 90% of the total cones, while blue cones only account for 10% of the total cones and are constantly surrounded by longer wavelength cones [32,33]. Our experimental results show that the color fusion limit varies with the opposite color direction, which is independent of the distribution of cells.

We represented the averaged ellipse fitting curves of session 1 and session 2 together in the two-dimensional plane. The dotted line curves in Fig. 10 (a) are respectably the ellipse fitting curves of session 1 and session 2, and the straight-line curve is the average ellipse curves of session 1 and session 2. The average ellipse curves are transformed into CIELUV space, as shown in Fig. 10 (b), the curve is the binocular color fusion boundary, the scatter points indicate the direction of color selection, and the dotted lines are the major axis and minor axis in CIELAB space. In CIELUV space, the length of the semi-major axis is 0.074 $\Delta E_{ab}^{\ast }$ and the length of semi-minor axis is 0.059 $\Delta E_{ab}^{\ast }$. Compared with June's ellipse fitting [7], with the maximum semi-major axis 0.156 $\Delta E_{ab}^{\ast }$ and the minimum semi-minor axis 0.0415 $\Delta E_{ab}^{\ast }$, and Chen's ellipse fitting [11] at the disparity ${0^\circ }$ with the semi-major axis 0.092 $\Delta E_{ab}^{\ast }$ and the semi-minor axis 0.036 $\Delta E_{ab}^{\ast }$, our limits of color fusion are within reasonable bounds.

 

Fig. 10. (a) The dotted line curves are the average fitting ellipses of session 1 and session 2, the grey line is the fitting ellipse of session 1, and the orange line is the fitting ellipse of session 2, (b) corresponding fusion boundary in CIELUV color space.

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In a classic study of sensory physiology, Derrington, Krauskopf, and Lennie (DKL) studied how the S, M, and L pyramids promote adversary responses in monkey lateral geniculate nucleus (LGN) [34]. DKL's strategy is to exchange points on the opposite sides of the iso-luminescent circle and measure the spike rate of a single LGN cell. The results show that the so-called red-green cells do not accept the functional input of S-cones but only accept the contraposition signal input of L-cones and M-cones. We transformed the experimental results into CIELUV space and got almond shaped drag ellipses similar to DKL color space. Their conclusion shows that the cone-optic cells in LGN do not line up along the Red–Green or blue-yellow Hering color optic axes. Hering's unique hues are not represented in the visual pathway at early processing stages.

We swapped the left and right stimuli and measured the chromatic fusion limits in two sessions. The results show that the semi-major axis of the ellipse in session 1 is 27.13$E$, and the semi-minor axis is 17.12 $\Delta E_{ab}^{\ast }$. In session 2, the semi-major axis is 27.97 $\Delta E_{ab}^{\ast }$ and the semi-minor axis is 16.83 $\Delta E_{ab}^{\ast }$. The average semi-major axis is 27.55 $\Delta E_{ab}^{\ast }$ and the semi-minor axis is 16.98 $\Delta E_{ab}^{\ast}$. Though the individual fitting results are somewhat different in session 1 and session 2, the fitting ellipses are approximately equal. We used a black background and a circular object filled with the sampled colors to measure the binocular chromatic fusion limit. The measured results should be greater than that on an equal luminant gray background because a bright surround could induce blackness in the stimulus fields, which increased the color's saturation and made the dichoptic mixtures less stable. And a bright surround might produce stronger complementary afterimages which were responsible for driving rivalry [35], or the high luminance contrasts of stimuli on dark backgrounds promote perceptual averaging of dichoptic colors [36]. It is worth noting that the dominant eye may influence the results of KC and WL. The experiment does not exclude whether the dominant eye affect the measurement of the color fusion boundary. We do not rule out the speculation that the binocular color fusion is affected by the dominant eye, or the binocular color rivalry is not only visual perception, this process involves the participation of brain cognition.

4. Conclusion

In this paper, the chromatic fusion limit of opposite colors in CIELAB color space with constant brightness is measured, and the results are analyzed and discussed. To better quantify the chromatic fusion limit, we obtained accurate information by selecting a lot of color sample point pairs. The number of observations became very large for a subject, and visual discomfort occurred during the experiment. Many factors affect each person's color fusion limits. We divided the investigation into two sessions to test whether color inconsistency between eyes affects the binocular color fusion. Results show that the human eye's fusion tolerance of the opposite colors on the directions of RB-GY and R-G is greater than that of the opposite colors on the directions of Y-B and RY-GB. We quantified the chromatic fusion limit for contrasting colors by ellipses in the CIELAB chromaticity diagram. The average semi-major axis of the ellipses is 27.55$\mathrm{\Delta }E_{ab}^\ast $, and the average semi-minor axis is 16.98$\mathrm{\Delta }E_{ab}^\ast $.

We suggested that the color fusion limit changes with the change of the opposite color direction. The fusion limit is independent of the distribution of cells and has nothing to do with the color inconsistency between eyes. The dominant eye may have some effects on binocular color fusion, but binocular color rivalry mainly involves the participation of brain cognition. Our measurement could be applied for some applications, such as the creation of stereo contents, design of stereoscopic display devices, and comfortable visual assessment of stereoscopic displays.