**We investigated suprathreshold binocular combination, measuring both the perceived phase and perceived contrast of a cyclopean sine wave. We used a paradigm adapted from Ding and Sperling (2006, 2007) to measure the perceived phase by indicating the apparent location (phase) of the dark trough in the horizontal cyclopean sine wave relative to a black horizontal reference line, and we used the same stimuli to measure perceived contrast by matching the binocular combined contrast to a standard contrast presented to one eye. We found that under normal viewing conditions (high contrast and long stimulus duration), perceived contrast is constant, independent of the interocular contrast ratio and the interocular phase difference, while the perceived phase shifts smoothly from one eye to the other eye depending on the contrast ratios. However, at low contrasts and short stimulus durations, binocular combination is more linear and contrast summation is phase-dependent. To account for phase-dependent contrast summation, we incorporated a fusion remapping mechanism into our model, using disparity energy to shift the monocular phases towards the cyclopean phase in order to align the two eyes' images through motor/sensory fusion. The Ding-Sperling model with motor/sensory fusion mechanism gives a reasonable account of the phase dependence of binocular contrast combination and can account for either the perceived phase or the perceived contrast of a cyclopean sine wave separately; however it requires different model parameters for the two. However, when fit to both phase and contrast data simultaneously, the Ding-Sperling model fails. Incorporating interocular gain enhancement into the model results in a significant improvement in fitting both phase and contrast data simultaneously, successfully accounting for both linear summation at low contrast energy and strong nonlinearity at high contrast energy.**

*I*and

_{L}*I*are the signal inputs to a narrow-band and orientation-selective spatial frequency channel for each eye, and ℰ

_{R}*and ℰ*

_{L}*are the total weighted contrast energy of two eyes' images across all orientations and all spatial frequency channels. However, this one-layer gain-control model violates contrast constraints in binocular combination; at high contrast, the model predicts that the binocular combined contrast would be much smaller than monocular contrast because of strong mutual interocular inhibition in binocular viewing but no inhibition from the other eye in monocular viewing when the other eye is closed. To address this violation, they introduced a second layer of interocular gain control that mutually inhibits the gain control in the first layer, i.e., This successfully predicts that the perceived contrast is the same whether one eye is closed or both eyes remain open under normal viewing conditions (at high contrast for long stimulus durations). To test their model, they used an adaptive procedure to measure the perceived phase of a horizontal cyclopean sine wave by indicating the apparent location (phase) of the dark trough of the cyclopean sine wave relative to a black horizontal reference line. They performed six experiments to test the predictions of this two-layer gain-control model: (a) At high contrast, the eye with higher contrast contributes more than predicted by linear summation; (b) at low contrast, the binocular combination behaves like linear summation, and when contrast increases, the behavior of binocular combination becomes more and more nonlinear; (c) the eye with noise will dominate in the combination because, with noise contrast, it has more total contrast. All these predictions were confirmed by their experiments.*

_{R}^{1}model by explicitly including interocular enhancement—multiplying the other eye's contrast in one eye's gain operation.

*I*=

_{L}*I*

_{0}+

*m*cos(2

_{L}*πf*+

_{s}y*Ϭ*) and

_{L}*I*=

_{R}*I*

_{0}+

*m*cos(2

_{R}*πf*+

_{s}y*Ϭ*), were used as stimuli.

_{R}*I*

_{0}is the luminance of the background and the mean luminance of the sine-wave gratings;

*f*is the spatial frequency, identical in both eyes;

_{s}*m*and

_{L}*m*are the modulation contrasts of the left- and right-eye sine-wave gratings, respectively;

_{R}*Ϭ*and

_{L}*Ϭ*are the corresponding phases. The stimuli were windowed in a rectangular window both spatially (3° × 3°) and temporally (1 s or 117 ms). There were exactly two cycles visible in each eye's sine wave.

_{R}*m*= max{

*m*,

_{L}*m*}, varied from 6% to 96%, interocular contrast ratio,

_{R}*δ*=

*m*/

_{R}*m*, varied from ¼ to four, the spatial frequency was 0.68, 1.36, or 2.72 cpd (cycles per degree), and the phase difference,

_{L}*Ϭ*= |

*Ϭ*–

_{R}*Ϭ*|, was fixed at 90°. Figure 2 shows the 45 test points of right (RE) versus left eye (LE) contrast at which the perceived phase was measured. Points in one solid curve have the same base contrast

_{L}*m*and the points along a dashed line have the same interocular contrast ratio δ that is labeled near the line. For each base contrast, when

*δ*≤ 1 (vertical solid lines),

*m*=

_{L}*m*and

*m*=

_{R}*δm*; when

*δ*> 1 (horizontal solid lines),

*m*=

_{L}*m*/

*δ*and

*m*=

_{R}*m*. When

*δ*increases from zero to ∞, the contrast of the LE's grating remains constant at base contrast

*m*while the RE's contrast increases from zero to

*m*(points from bottom to top along a vertical line), and then the RE's contrast remains constant at base contrast

*m*while the LE's contrast decreases from

*m*to zero (points from right to left in a horizontal line).

*m*and

*δ*, there are two displays: (a) the phase of the LE's grating is lower-shifted (

*Ϭ*= −

_{L}*Ϭ*/2) and the phase of the RE's is higher-shifted (

*Ϭ*=

_{R}*Ϭ*/2) (Figure 3A); (b) the phase of the LE's is higher-shifted (

*Ϭ*=

_{L}*Ϭ*/2) and the phase of the RE's is lower-shifted (

*Ϭ*= −

_{R}*Ϭ*/2) (Figure 3B). For the two displays, two staircases were randomly interleaved to measure the perceived phases,

*Ϭ̂*

_{1}or

*Ϭ̂*

_{2}, in the cyclopean sine waves (LE + RE) concurrently. When only the LE is presented with a grating (i.e.,

*δ*= RE/LE = 0), the perceived phase is the same as in LE's, i.e.,

*Ϭ̂*

_{1}=

*Ϭ*= −

_{L}*Ϭ*/2 for the display shown in Figure 3A or

*Ϭ̂*

_{2}=

*Ϭ*=

_{L}*Ϭ*/2 for the display shown in Figure 3B. On the other hand, when a grating is presented only to the RE (i.e.,

*δ*= ∞), the perceived phase is the same as in RE's, i.e.,

*Ϭ̂*

_{1}=

*Ϭ*=

_{R}*Ϭ*/2 (Figure 3A) or

*Ϭ̂*

_{2}=

*Ϭ*= −Ϭ/2 (Figure 3B). To cancel any possible vertical position bias, the perceived phase is averaged as

_{R}*Ϭ̂*= (

*Ϭ̂*

_{1}−

*Ϭ̂*

_{2})/2, which varies from the LE's phase (−

*Ϭ*/2) to the RE's phase (

*Ϭ*/2) when

*δ*increases from zero to ∞.

^{2}; the luminance with all pixels set to the maximum value was 46.0 cd/m

^{2}. The background level

*I*

_{0}surrounding the sine-wave gratings was set to 26.2 cd/m

^{2}, and this was also the average luminance of the sine waves themselves. Displays were viewed in a mirror stereoscope and positioned optically at 68 (0.68 cpd of sine waves), 136 (1.36 cpd of sine waves), or 272 cm (2.72 cpd of sine waves) from the observer.

*Ϭ̂*= (

*Ϭ̂*

_{1}−

*Ϭ̂*

_{2})/2, was calculated as the dependent variable of the experiment. Typically, for each run, there were 18 concurrent staircases interleaved to measure the perceived phase for nine interocular contrast ratios. Observers JP and MD each ran a total of 3 (Spatial Frequency) × 6 (Base Contrast) × 9 (Contrast Ratio) × 2 (Displays) × 50 (Repeats) = 16,200 trials. Observers CF, CG, KT, and JS each ran a total of 1 (Spatial Frequency) × 3, 4, or 5 (Base Contrast) × 9 (Contrast Ratio) × 2 (Displays) × 50 (Repeats) ≈ 3,600 trials.

*and ℰ*

_{L}*are the total weighted contrast energy presented to the two eyes which should be summed over space and time and also over spatial frequency channels and all orientations. In this study, because the stimuli are in a narrow spatial frequency band and have a fixed size and duration at one spatial frequency, the total contrast energy in one channel can be written as where*

_{R}*g*is a gain-control threshold at which the contrast gain control becomes apparent.

_{c}*μ*) in the RE, and Equation 3 was rewritten as follows:

*I*which was repeatedly used in constructing the Ding-Sperling model (Figure 4B) and DSKL (Ding-Sperling-Klein-Levi) model (Figure 4E). The Ding-Sperling model consists of left and right eye channels, each containing two gain control mechanisms: one based on total contrast energy (TCE) in the gain-control layer (blue) which is nonselective for orientation and spatial frequency and the other which is selective along those dimensions in the signal layer (black). The two TCE components exert reciprocal inhibition on one another in the gain-control layers (blue) in proportion to their respective TCE outputs, and the outputs of those TCE components exert gain control on the other eye's selective gain control in the signal layer (black). The outputs are summed linearly to determine the binocular signal.

*μ*) in the RE, the model output is given by When the left eye's contrast increases, the right eye's gain decreases because of increasing gain control from the left eye, while the left eye's gain increases through decreasing gain control from the right eye (and vice-versa). At high contrast, i.e.,

*m*>>

_{L}*g*and

_{c}*m*>>

_{R}*g*, the Ding-Sperling model (Equation 6) can be simplified to the contrast-weighted summation model (Equation 5). In order to fit both the phase and contrast data simultaneously, we also fit several variations of the Ding-Sperling model.

_{c}*α*, which is assumed to be one in both layers for Ding-Sperling model. We first modified the model by adding an asymmetry between the two layers, the gain control and the gain-control of gain control having different gain-control efficiency, i.e., where

*α*is the relative gain-control efficiency in the nonselective layer (blue) when the gain-control efficiency (black) in the selective layer is assumed to be one.

*g*.

_{e}*β*is the relative gain-control efficiency in the gain control to the gain enhancement. In the full model (Figure 4E), there are three layers for each eyes: (a) the selective signal layer (black) that receives both gain control (black filled circle) and gain enhancement (red open circle) from the other eye and outputs the signal to the binocular summation site; (b) the nonselective gain-control layer (blue) that first extracts and sums image contrast energy across frequency channels and orientations (TCE) and then exerts gain control to the other eye's three layers separately with different gain-control efficiencies (1,

*α*, and

*β*); (c) the gain-enhancement layer that extracts image contrast energy (TCE*) and exerts gain enhancement only to the other eye's signal layer. Figure 4F illustrates the left eye's part of the full model to show how to calculate the left eye's output (the first summand in Equation 10). Before output, the left eye's signal receives gain control from the right eye's gain-control layer that itself receives gain control from the left eye and also the left eye's signal receives gain enhancement from the right eye's gain-enhancement layer that receives gain control from the left eye. The right eye's part in the full model is symmetric to the left eye's in the normal vision.

*ℰ*= 0,

_{L}*ℰ*= 0,

_{R}*Ϭ̂*) of a cyclopean grating varies as a function of the interocular contrast ratio (

*δ*). The (physical) phase difference in two eyes was fixed at 90° (

*Ϭ*= −45° and Ϭ

_{L}*= 45° indicated by arrows on the sides of Figure 6), the base contrast was 96% (*), 48% (×), 24% (○), 12% (▿), or 6% (□), and the spatial frequency was 0.68 (top), 1.36 (middle), or 2.72 cpd (bottom). When the interocular contrast ratio*

_{R}*δ*increased, the perceived phase of the cyclopean sine waves shifted systematically from the left eye's phase (−45°) to the right eye's phase (45°). The results are consistent with previous studies (Ding & Sperling, 2006, 2007). The solid curves are the best fits from the DSKL model (Model 3c). The black dashed curve is the prediction from algebraic (linear) summation of two eyes' sine waves with attenuation in the right eye for ocular imbalanced contrast perception (the asymptote of Models 2 and 3a–c at zero contrast energy—see Models section). All data points except equal-physical-contrast (

*δ*= 1) points are shifted away from this linear-summation line, biased to the eye presented with stronger contrast (to the LE when

*δ*< 1 and to the RE when

*δ*> 1). This bias toward the eye with stronger contrast (beyond linear summation) demonstrates interocular contrast gain control. At the lowest spatial frequency (0.68 cpd), all curves (solid and dashed) intercept almost at the same point, with almost no perceived phase shifted at equal physical contrast (

*δ*= 1, dashed vertical line), indicating almost no eye-bias or balanced vision when the two eyes are presented with identical contrast. However, at the highest spatial frequency (2.72 cpd), both observers showed some eye bias; JP biased toward the LE and MD biased toward the RE.

*δ*for gratings of 0.68, 1.36, and 2.72 cpd when stimulus duration was 1 s and interocular phase difference was either 0° (blue circles) or 90° out of phase (red stars). The contrast of the two eyes' sine waves was normalized by the standard contrast. The contours are almost symmetrical across the

*δ*= 1 line (dashed 45° black line) at 0.68 and 1.36 cpd, and they are very similar when the interocular phase difference varies from 0° to 90°. The DSKL model (Model 3c) with an added motor/sensory fusion mechanism with the same model parameters used for fitting the phase data (Experiment 1) accurately predicts this phase-independence of binocular contrast combination at high contrast levels (solid curves). The horizontal and vertical dashed lines are predictions from the winner-take-all model; the stimulus in the eye with stronger contrast wins the competition to give the contrast percept of the binocular-combined gratings while completely ignoring the other eye's stimulus. Interestingly, this winner-take-all model also provides a reasonable fit to the data, reflecting binocular perceived contrast being nearly constant at all interocular contrast ratios.

*N*be the number of model parameters and

_{p}*N*be the number of observed data points. We have the number of degrees of freedom

_{data}*ν*=

*N*−

_{data}*N*, and the reduced chi-square is given by

_{p}*χ*

^{2}/

*ν*. If Model A is nested within Model B, the

*F*test that tests whether Model B significantly improves data fitting is given by which compares the variance between Models A and B with the variance inside Model B and has an

*F*distribution with [

*ν*(

*a*) –

*ν*(

*b*),

*ν*(

*b*)] degrees of freedom. When the

*F*-value is large enough, Model A can be rejected at a small false-rejection probability

*p*(

*F*).

*F*test with the

*F*-value given in the row of the second model. With three steps of modification (Models 3a–c) of the Ding-Sperling model (Model 2), Models 3a and 3c achieved significant improvement in data fitting; the previous model could be rejected with a very small (<0.001) probability of false rejection. However, without gain control of the gain enhancement (Model 3c), the gain enhancement itself in Model 3b failed to further improve the data fitting in three observers.

*μ*) was assumed to be one; Model 1 has one parameter (

*γ*) and Model 2 has two parameters (

*γ*and

*g*

_{c}). The top-left panels in Figures 11A and B demonstrate that fitting Model 1 only to the phase data (blue curve) provides a reasonable fit, but its prediction of the contrast contour (blue curve in top-right) is far removed from the data. The predicted contrast contour shows a strong Fechner's paradox, i.e., inhibition of one eye's monocular contrast perception by a small input in the other eye; however, the observed data shows a winner-take-all phenomenon, with no apparent inhibition from the other eye's image (which has smaller contrast). When fitting Model 1 only to the contrast data (red curve in top-right of Figures 11A and B), the fit again seems reasonable, but with different model parameters from fitting the phase only. However, the predicted phase (red curve in top-left) switches from one eye to the other much more rapidly than shown by the data. Although the same model can fit either phase or contrast data separately, the best fitting model parameters are not consistent, large

*γ*values (12.5 for JP; 3.9 for MD) for contrast fitting and small

*γ*values (1.3 for JP and 0.8 for MD) for phase fitting. When fitting Model 1 to the two data sets simultaneously (the black curves), with a trade-off

*γ*value (2.6 ± 0.5 for JP and 1.7 ± 0.3 for MD), the fits to both data sets are poor. When fitting Model 2 to either phase or contrast data separately, similar to fitting Model 1, the best fitting model parameters are not consistent although each separate fit appears reasonable. However, when fitting Model 2 to both phase and contrast data, the goodness of fit was improved in comparison to Model 1.

*g*(0.22 ± 0.02 for JP; and 0.14 ± 0.03 for MD), the monocular contrast perception in one eye would not be suppressed by the other eye's small input if its contrast is smaller than

_{c}*g*, resulting in a better fit to the data. However, when

_{c}*g*is not zero (as supposed in Model 1) or small enough, the predicted binocular-combined contrast when both eyes are presented with identical images (contrast ratio = one) would not be the same as the contrast in monocular view; the prediction falls far from the observed data, as shown in the bottom-left panels of Figures 11A and 11B. To solve this problem, we introduced interocular gain enhancement (Model 3c, dashed black curves in Figure 11). By selecting suitable gain-control and gain-enhancement thresholds,

_{c}*g*and

_{c}*g*, and suitable

_{e}*α*,

*β*, and

*γ*values, interocular gain control and gain enhancement maintain a reasonable balance in binocular vision, achieving constant contrast perception (apparent winner-take-all phenomenon) and smoothly shifting phase perception (with a reasonable exponent parameter) when interocular contrast ratio varies. To better understand how Model 3c works this way, we simulated the model using model parameters fitted to both phase and contrast data.

*γ*value is reasonable for binocular phase combination) when the contrast ratio RE/LE increases. Note that this is quite different from winner-take-all models (e.g., the Legge model with an infinite exponent), which fail to predict the smooth phase shift in binocular phase combination. However, at low base contrast, Model 3c predicts binocular linear summation. Figure 12B shows the LE's apparent contrast predicted from Model 3c when the base contrast decreases from 96% to 3%. When the base contrast is above 12%, the normalized LE apparent contrast curves are almost overlaid. When the base contrast decreases to 6% and 3%, the LE's apparent contrast curves shift toward the LE's input curve (dashed blue curve) and is almost identical to the input curve at 3% base contrast, because the system becomes more linear at lower base contrast.

*f*) channel, there are seven model parameters for binocular summation but four of them are shared across different frequency channels. For the disparity energy calculation in the motor/sensory fusion mechanism, there are two model parameters. Therefore, for observers CG, CF, JS, and KT who were tested only at 0.68 cpd, there are nine parameters in total. For observers JP and MD who were tested at three spatial frequencies, there were 13 parameters in total; parameters for disparity energy were fixed (

_{s}*g*= 0.038 averaged from observers CG and CF and

_{f}*γ*= 1).

_{f}*δ*=

*m*/

_{R}*m*be interocular contrast ratio, from the Appendix, the perceived phase and contrast from the Legge model (Figure 14A) are given by respectively. When

_{L}*γ*= 1, the Legge model is identical to the linear summation model; when

*γ*> 1, the model predicts that the eye with higher contrast would have more weight in binocular combination than would be predicted by the linear summation; when

*γ*= ∞, the model is identical to the winner-take-all model, predicting the perceived phase switching from one eye to the other at

*δ*= 1 and the perceived contrast is the higher contrast of the two eyes.

*σ*insignificant in data fitting. In other words, the gain control in the normalization model could be considered as a binocular contrast gain control, which accounts for the contrast transfer function in monocular and binocular vision. Therefore, its behavior in fitting our data is identical to the Legge model; its gain-control path plays no role in binocular combination. However, if different weights are assumed for monocular and interocular gain controls in the normalization model, the model would contain both monocular and interocular mechanisms. From this modified normalization model, the perceived contrast is given by where

*w*is the relative weight for interocular gain control when the weight for monocular gain control is assumed to be one. The perceived phase is given by Equation A18 with apparent contrast ratio calculated by Equation A49.

*m*is presented in one eye only, the perceived contrast predicted from the Ding-Sperling model or any other model in this study always equals to

*m*, i.e.,

*m̂*=

*m*when one eye is closed. Without adding monocular mechanisms, it is impossible to account for contrast discrimination data from our models.

*Z*, which might be determined from contrast transfer function or with the power in the control path

*q*depending on the power in the signal path

*γ*. We reduced the model by letting

*q*=

*γ*− 1, and we have To include interocular mechanisms in the model, different weights should be assumed for monocular and interocular gain controls. We have The perceived phase is given by Equation A18 with apparent contrast ratio calculated by The two-stage model (Figure 14D) was proposed by the same group as the Meese-Hess model to extend it to account for the finding of less nonlinearity at lower contrast, using gain control with an exponent near-to-one in the first stage before binocular combination and to account for high nonlinearity at higher contrast using the gain control in a second stage after binocular combination. However, like the Meese-Hess model, the gain control in the first stage could be considered as a binocular contrast gain control after the binocular combination as shown in Figure 14D′. The parameters in the control path have no effect on the prediction of perceived phase, which is also given by Equation 12 deduced from the Legge model. When fitting both phase and contrast data, the model has to be reduced to a two-parameter model with only the first stage; the parameters in the second stage couldn't be determined from our data. From this reduced model, the perceived contrast is given by Again, to include interocular mechanism, different weights are assumed for monocular and interocular gain controls. We have The perceived phase is given by Equation A18 with apparent contrast ratio calculated by Without the asymmetry assumption of monocular and interocular gain controls, all these models deduced from contrast combination have no effective interocular mechanisms. The contrast gain control in the normalization, Meese-Hess, or two-stage model is actually a binocular gain control because both eyes gains increase or decrease in exactly the same way no matter which eye's contrast increases or decreases. The perceived phase is only determined by the signal path of the model; all four models give the same prediction of the perceived phase given by Equation 12. On the other side, in the Ding-Sperling model, increasing the left eye's contrast would decrease the right eye's gain but, at the same time, it would increase the left eye's gain through suppressing the gain control from the right eye, and vice versa.

*μ*= 1 for all our five models and

*γ** =

*γ*for Models 3b and 3c.

*N*be the number of model parameters and

_{p}*L*hais the maximized value of the likelihood function for the estimated model; AIC is defined as

_{Max}*AIC*= 2

*N*– 2 ln

_{p}*L*. Assuming that the errors are normally distributed and independent, after ignoring the constant term, AIC is given by

_{Max}*AIC*=

*χ*

^{2}+ 2

*N*.

_{p}*g*= 0, giving the same prediction at all contrast levels, it would give a better fit only at high contrast than the Legge model. When fitting a range of contrast levels, the data was more linear at lower contrast levels and became more nonlinear as contrast increased. Model 1 failed to pick up the linear feature at low contrast, making it worse in data fitting than the Legge model. For the purpose of fitting our data, the normalization model (Figure 14B) is essentially identical to the Legge model, and both models have the same chi-squared statistics.

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*Vision Research**I*and

_{L}*I*be the stimuli presented to the left and right eyes, respectively, given by and The perceived sine wave is supposed to be the arithmetic summation of the two eyes' sine waves, i.e.,

_{R}*Î*=

*I*+

_{L}*I*. Figure A1 shows vector presentations of sine waves and their arithmetic summation in the complex plane. We have where

_{R}*Î*

_{0}= 2

*I*

_{0}, and

*Ϭ*(black) to

_{L}*Ϭ*(black) to

_{R}*g*is the contrast threshold at which the motor/sensory fusion becomes apparent and

_{f}*γ*is the exponent value for the gain control in the motor/sensory fusion mechanism. At very low contrast, when

_{f}*D*<<

*Ϭ*and

_{L}*Ϭ*, and the perceived contrast and phase are still given by Equations A4 and A5, respectively. At very high contrast, when

_{R}*D*>>

*Ϭ̂*, and the perceived contrast is given by

*m*+

_{L}*m*but the perceived phase is still given by Equation A5. Generally, after motor/sensory fusion, the perceived contrast is given by and the perceived phase is given by Let

_{R}*Ϭ*=

_{R}*Ϭ*/2,

*Ϭ*= −

_{L}*Ϭ*/2, and

*m*=

_{R}*δm*, where

_{L}*Ϭ*=

*Ϭ*–

_{R}*Ϭ*is the phase difference between two eyes and

_{L}*δ*=

*m*/

_{R}*m*is interocular contrast ratio, then the perceived contrast and phase before motor/sensory fusion are given by After motor/sensory fusion, the perceived contrast is given by Equation A9 and the perceived phase is given by When only the left eye is presented with the sine wave, i.e.,

_{L}*m*= 0 or

_{R}*δ*= 0 , the perceived phase is the same as the input from the left eye, i.e.,

*Ϭ̂*′ =

*Ϭ̂*=

*Ϭ*= −

_{L}*Ϭ*/2. When only right eye is presented with the sine wave, i.e.,

*m*= 0 or

_{L}*δ*= ∞, the perceived phase is the same as the input from the right eye, i.e.,

*Ϭ̂*′ =

*Ϭ̂*=

*Ϭ*=

_{R}*Ϭ*/2. When the two eyes are presented with the sine waves with identical contrast, i.e.,

*m*=

_{L}*m*or

_{R}*δ*= 1, the perceived phase is zero, i.e.,

*Ϭ̂*′ =

*Ϭ̂*=

*0*. When interocular contrast ratio varies from zero to ∞, the perceived phase varies from the phase of the left eye sine wave, −

*Ϭ*/2, to the phase of the right eye sine wave,

*Ϭ*/2. As shown in Figure 17B, before and after motor/sensory fusion the perceived phases are very close to each other, i.e.,

*Ϭ̂*′ ≈

*Ϭ̂*, for all interocular contrast ratios at all contrast levels.

*Ϭ*=

_{R}*Ϭ*/2,

*Ϭ*= −

_{L}*Ϭ*/2 (the case for Experiment 1), the perceived phase of a cyclopean sine wave is given by With the fraction of the disparity remapping demand given by After remapping of two eyes' corresponding points through motor/sensory fusion, the perceived contrast and phase are given by

*m*= max{

*m*,

_{L}*m*}, interocular contrast ratio

_{R}*δ*=

*m*/

_{R}*m*

_{L}_{,}and two eyes phase difference

*Ϭ*=

*Ϭ*–

_{R}*Ϭ*. At each set of

_{L}*m*,

*δ*,

*Ϭ*, we measured the perceived phase

*m*,

*δ*, and Ϭ, i.e.,

*m*, interocular contrast ratio

_{st}*δ*=

*m*/

_{R}*m*, and interocular phase difference

_{L}*Ϭ*, we measured the base contrast

*m̄*″ = max{

*m*,

_{L}*m*} with standard error of

_{R}*δ*when the perceived contrast

*m̄*′ matches the perception

*m*=

_{L}*m*and

_{st}*m*= 0, we have the perception of the standard contrast

_{R}*m̂*=

_{st}*m*. On the other side, at a combination of base contrast

_{st}*m̂*, contrast ratio

*δ*, and phase difference

*Ϭ*the perceived contrast

*m̂*′ could be written as of function of

*m̂*,

*δ*and

*Ϭ*from a model. When it matches the perception of the standard contrast

*δ*and a fixed phase difference

*Ϭ*, the base contrast at which the binocular-combined contrast matches monocular standard contrast could be calculated from a model, i.e., To minimize the weighted sum of squared errors given by where

*Ϭ̂*′ and

*m̄*were predicted from a model given by Equations A30 and A32, respectively,

*Ϭ̂*″ and

*m̄*″ and

*f*, we have one set of model parameters,

_{s}*g*and

_{c}*g*, for computation of contrast energy. For three spatial frequencies, we fit the model to all data for all three spatial frequencies from both experiments by minimizing the weighted sum of squared errors given by

_{e}*m̂*′ =

*m̂*. Because we always have

*Ϭ̂*′ ≈

*Ϭ̂*, for simplicity, we only show how to fit Model 1 without considering motor/sensory fusion. From Equations A17 and A18, we have for prediction of the perceived phase, which is independent of base contrast

*m*in Model 1. For contrast matching experiment, at base contrast

*m̄*, we have

*m*=

_{L}*m̄*and

*m*=

_{R}*δm̄*when

*δ*≤ 1, and

*m*=

_{L}*m̄*/

*δ*and

*m*=

_{R}*δ*when

*δ*= > 1. From Equations A16 and A20 with

*Ϭ*= 0, the perceived contrast is given by From Equation A31 and

*m̂*′ =

*m̂*, the base contrast at which the binocular-combined contrast matches the monocular standard contrast is given by Applying Equations A35 and A37 in a chi-squared estimator (Equation A33 or A34), we could fit both phase and contrast data into Model 1 without considering motor/sensory fusion. Generally, it would be difficult to write down the explicit formula for Equation A33 or A34, and we had to use Matlab program to find a digital solution of

_{st}*m̄*at a combination of

*m*,

_{st}*δ*, and

*Ϭ*.

^{γ}that only operates on contrast and has no effect on phase, the two eye inputs become Therefore, the apparent interocular contrast ratio is given by When

*Ϭ*–

_{R}*Ϭ*=

_{L}*Ϭ*, from Equations A18 and A39, the perceived phase is given by The calculation after the binocular combination has no effect on the perceived phase. When the two inputs have identical phase, i.e.,

*Ϭ*=

_{L}*Ϭ*, after the combination and the calculation of (·)

_{R}^{1/γ}, the perceived contrast is given by When the standard contrast

*m*is only presented in one eye, say,

_{st}*m*=

_{L}*m*and

_{st}*m*= 0, we have its perception

_{R}*m̂*=

_{st}*m*Like the five models in this study, the Legge model doesn't include any monocular mechanism, but unlike these models, it doesn't include any interocular mechanism either. Similarly, as was the case in deducing Equation A37, the base contrast

_{st}*m̄*, at which the binocular-combined contrast matches the monocular standard contrast, is given by Applying Equations A40 and A42 in a chi-squared estimator (Equation A33 with

*Ϭ*= 0 in the second term), we fitted the Legge model to both phase and contrast data at 48% contrast and 0.68 cpd of spatial frequency (blue curve in Figure 15).

*Ϭ*=

_{L}*Ϭ*, the perceived contrast is given by, The monocular contrast perception of

_{R}*m*is given by Unlike the five models in this article and the Legge model, the normalization model includes monocular contrast gain control. When the perceived contrast at

_{st}*m*=

_{L}*m̄*and

*m*=

_{R}*δm̄*(

*δ*≤ 1) is matched to

*m̂*, we have Obviously, we have

_{st}*δ*)

^{γ}*m̄*when

^{γ}*δ*≤ 1. Similarly, we have

*δ*)

^{−γ}*m̄*when

^{γ}*δ*> 1. Therefore, we also have Equation A42 in the normalization model to predict the base contrast

*m̄*when binocularly-combined contrast matches the monocular standard contrast. The model-fitting curve is exactly the same as the one from the Legge model (blue in Figure 15).

*w*is the relative weight for interocular gain control when the weight for monocular gain control is assumed to be one. Therefore, the perceived contrast is given by And the apparent interocular contrast ratio is given by Using Equation A49, the perceived phase is given by Equation A18.