**Binocular combination for first-order (luminance-defined) stimuli has been widely studied, but we know rather little about this binocular process for spatial modulations of contrast (second-order stimuli). We used phase-matching and amplitude-matching tasks to assess binocular combination of second-order phase and modulation depth simultaneously. With fixed modulation in one eye, we found that binocularly perceived phase was shifted, and perceived amplitude increased almost linearly as modulation depth in the other eye increased. At larger disparities, the phase shift was larger and the amplitude change was smaller. The degree of interocular correlation of the carriers had no influence. These results can be explained by an initial extraction of the contrast envelopes before binocular combination (consistent with the lack of dependence on carrier correlation) followed by a weighted linear summation of second-order modulations in which the weights (gains) for each eye are driven by the first-order carrier contrasts as previously found for first-order binocular combination. Perceived modulation depth fell markedly with increasing phase disparity unlike previous findings that perceived first-order contrast was almost independent of phase disparity. We present a simple revision to a widely used interocular gain-control theory that unifies first- and second-order binocular summation with a single principle— contrast-weighted summation—and we further elaborate the model for first-order combination. Conclusion: Second-order combination is controlled by first-order contrast.**

*for first-order combination only*. The details of this analysis and model fitting are in Appendix 2.

^{2}in each eye. These organic light-emitting diode microdisplays are linear in luminance response (Black, Thompson, Maehara, & Hess, 2011) and exhibit pixel independence in image presentation (Cooper, Jiang, Vildavski, Farrell, & Norcia, 2013); thus we would not expect any nonlinear distortions of our stimuli due to the display equipment (Klein, Hu, & Carney, 1996).

*θ/2*(

*θ*= 0°, 45°, 90°) were presented dichoptically to the left side of the central fixation point in the two eyes. This pair of CM test gratings was viewed through the goggles and combined perceptually to create a single cyclopean grating. A monocular horizontal sine-wave CM grating (the “probe”) was presented to the right side of fixation in the left or the right eye (Figure 1). Observers adjusted the phase and modulation depth of the monocular probe grating to match the binocular percept of the test stimulus.

*θ/2*in the nondominant eye and −

*θ/2*in the dominant eye or vice versa. Perceived phase at each interocular modulation ratio (

*δ*) was quantified as half the difference between the matched phases in these two configurations.

*θ*.

*L*

_{0}is the background luminance, and

*g*

_{1}(

*x*,

*y*) and

*g*

_{2}(

*x*,

*y*) are the 2-D binary, white-noise carriers in the two eyes with correlated carriers

*g*

_{1}=

*g*

_{2}, with anticorrelated carriers

*g*

_{1}= −

*g*

_{2}, and with uncorrelated carriers

*g*

_{1}≠

*g*

_{2}.

*C*= 0.2 is the contrast of carrier,

_{g}*f*= 0.29 c/° is the spatial frequency of the sine-wave envelope,

*M*

_{0}= 0.8 is the fixed modulation depth in the nondominant eye, and

*δ*(the

*modulation ratio*) is the interocular ratio of modulation depths (

*δ*= 0, 0.2, 0.4, 0.6, 0.8, 1.0). The two dichoptic gratings in the test had an equal and opposite phase shift of

*θ*/2 (relative to the center of the screen), where

*θ*= 0°, 45°, 90°.

*δ*= 0), the test images presented to the left side of fixation were CM gratings with modulation depth of 0.8 in the nondominant eye and 0 in the dominant eye (i.e., carrier noise without modulation). This configuration was different from that of the stimuli presented to the right of fixation, i.e., a monocular probe grating in one eye and uniform mean luminance (no carrier noise) in the other eye.

*θ′*for an interocular phase difference

*θ*and modulation ratio

*was predicted to be and the matching probe modulation depth*

**δ***M′*was

*F*(5, 10) = 124.82,

*p*< 0.001, and phase difference,

*F*(2, 4) = 151.14,

*p*< 0.001, with a significant interaction,

*F*(10, 20) = 28.29,

*p*< 0.001. Such phase dependency of second-order combination is different from previous reports on first-order combination with the same paradigm (Huang et al., 2011; Huang et al., 2010), in which the perceived contrast of first-order gratings did not depend on the interocular phase difference. However, all our data—for both phase and amplitude matching—are in good agreement with contrast-weighted linear summation (solid curves in Figure 2a, b).

*F*(5, 5) = 52.06,

*p*< 0.001, and this effect was not different for the three dichoptic carrier types—no significant interaction between modulation ratio and carrier type,

*F*(10, 10) = 0.31,

*p*= 0.96. The PvRs for all three dichoptic carriers (Figures 2a, 3a, b) are in good agreement with the prediction of linear summation. Similarly, the MvRs for anticorrelated and uncorrelated carriers were almost identical to those for correlated carriers and again closely matched the prediction of contrast-weighted linear summation. This was supported by the ANOVA, which showed that perceived amplitude of the cyclopean image depended significantly on the interocular phase difference,

*F*(2, 2) = 58.44,

*p*= 0.017, and this was true for all three dichoptic carriers because the interaction of interocular phase difference and carrier type was not significant,

*F*(4, 4) = 0.30,

*p*= 0.86.

*W*,

_{L}*W*for binocular combination, is influenced directly by

_{R}*first-order*phase disparity (

*θ*). Thus generalizing from Equations A1 and A2 in Appendix 1, where

*g*(

*θ*)

*= max*(1

*– h + h.cosθ*, 0), and

*h*is a constant (

*0 ≤ h ≤ 1*) that controls the degree of disparity dependence. Note from this definition that, to prevent negative suppression,

*g ≥*0. Increasing

*h*increases the extent to which interocular suppression depends on phase disparity: The strength factor

*g(θ*) for interocular suppression falls from 1 to 1

*– h*as

*θ*increases from 0 to 90°. The exponent

*n*corresponds to the Ding and Sperling (2006) model's

*γ*term (see Appendix 1), and shapes the way the weights vary with overall contrast level.

*h*> 0, the strength factor

*g*for interocular suppression decreases when disparity is present. As a result, for first-order gratings, the weight for each eye

*increases*with phase disparity (blue curves in Figure 4c) because suppression from the other eye decreases. This serves to eliminate (Figure 4a) the disparity dependence of contrast matching that must otherwise be expected.

*,*

**a**_{L}*for each eye are defined by where*

**a**_{R}*,*

**r**_{L}*are the unweighted monocular response amplitudes, and the binocular response amplitude*

**r**_{R}*is given from the vector sum of the two weighted responses*

**r**_{B}*,*

**a**_{L}*, taking phase difference into account (Equation A7). For simplicity and parsimony, we assume here that*

**a**_{R}*=*

**r**_{L}*,*

**C**_{L}*=*

**r**_{R}*, but in general, the monocular contrast response is likely to be more complex. If the system uses the combined response*

**C**_{R}*to evaluate contrast, two gratings will appear to match in contrast when their*

**r**_{B}*values are the same.*

**r**_{B}*fminsearch*in

*Matlab*) to minimize the value of chi-square summed over all the group-mean data (contrast and phase matches; cf. Ding et al., 2013b), we found that best-fitting values

*h*= 0.69,

*s*= 0.04, and

*n*= 1.53 gave a very good quantitative account of the first-order results of Huang et al. (2010) for both contrast matching (Figure 4a) and phase matching (Figure 4b). When we switched off phase disparity dependence in the gain control (set

*h*= 0) or refitted the model with

*h*= 0, the predicted contrast matches decreased markedly at 90° disparity (not shown) and were a poor fit to the data, but phase matches were almost unchanged. Thus contrast matches were sensitive to the influence of first-order phase disparity on gain control (Equations 5 and 6), but phase matches were not.

*h*= 0 in our model; hence

*g*= 1), only made matters worse: The predictions collapsed onto linear summation, making contrast matching vary sharply with the cosine of disparity (Equation A7) at all contrast levels (Figure 5c), quite unlike the data. Something else is needed.

*If binocular summation will reduce response amplitude, then don't do it*. This idea is easily implemented by supposing that contrast judgments are based on a modified response

**r**_{B}**′**:

*is too low, it is replaced by the larger of the two unweighted monocular responses. This substitution, using the*

**r**_{B}*MAX*operator, can be seen as a form of WTA rivalry, and it occurs mostly at large disparities at which

*falls dramatically. With no free parameters, it strikingly improved contrast-matching predictions for large disparities as shown in Figure 5d (goodness of fit,*

**r**_{B}*r*

^{2}= 0.805; RMS error = 0.62 dB), compared with the poor fit of Figure 5b and c. Thus, two different forms of suppression are likely to play a role in binocular combination: (a) the graded, disparity-sensitive interocular suppression that sets the balance (the weights, Equations 5 and 6) for left- and right-eye inputs to the summation process and (b) a more profound, perhaps later stage, WTA suppression that effectively compares the two monocular inputs with the binocularly combined response and picks the largest of the three while vetoing the other two. Selection of the monocular response at large disparities serves to prevent the binocular cancellation of antiphase signals from feeding through to perception. (See Appendix 2 for further analysis and discussion.)

*Fechner paradox*that is not seen in the first-order data. The (predicted) paradox is that when one eye's contrast is fixed and the other eye's contrast is increased from zero, then the combined binocular amplitude goes down before going up. The paradoxical reduction is predicted when interocular suppression outweighs the increase in signal strength given by adding contrast to the second eye. Although the Fechner paradox is reliably found in dichoptic brightness matching against a dark background (Anstis & Ho, 1998; D. H. Baker, Wallis, Georgeson, & Meese, 2012b; Engel, 1970; Levelt, 1965), it seems clear from several recent detailed studies that it does not normally occur for luminance contrast against a mid-gray background—neither for increments, decrements, step edges, nor gratings (D. H. Baker et al., 2012b; Ding et al., 2013b). Instead, perceived (matched) contrast is usually close to the higher of the two eyes' contrasts: WTA (although in amblyopic vision, a substantial, asymmetric Fechner paradox has been observed; Ding et al., 2013a). The deviation between model and data (Figure 4a) is not large, but it is systematic and deserves attention.

*noise*, a WTA process (the

*MAX*operator, Equation 8) can give a good account not only of perceived contrast (Figures 5d, A2a, and A4), but also of perceived phase for dichoptic first-order gratings (Figure A2b). Without noise, the predicted phase switched far too abruptly with increasing contrast ratio, just as Ding et al. (2013b) had supposed (not shown). This analysis strengthens the case for a fairly direct role of monocular signals in binocular vision. In short:

*When the summed binocular response is lower than both its monocular inputs, use the larger monocular response instead*.

*same*rule (contrast weighting) is being applied to different circumstances; relative contrast varies, or it doesn't.

*not*depend on CM phase disparity because the weights are driven by (first-order) carrier contrast, and this is unaffected by the absolute or relative phase of modulation. For CM, the required lack of phase dependence is achieved by setting

*h*= 0 in the weights for CM stimuli. This is a logical requirement that represents the change in type of stimulus rather than a change in the model itself. We also find from model fitting that for CM the constant

*s*has to be at or close to 0. These two values (

*h*= 0,

*s*= 0) lead to the simple linear averaging behavior for dichoptic CM seen in Figures 2, 3, and 6.

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*M*,

_{L}*M*with carrier contrasts

_{R}*C*,

_{L}*C*. We define the carrier-dependent weights

_{R}*W*,

_{L}*W*assigned to the left and right eyes as where

_{R}*s*is typically a small-valued constant that prevents division by 0 and ensures that the weight for one eye is 1 when the other eye's contrast is 0. Here, because

*C*,

_{L}*C*are never both 0 at the same time, we can simplify and let

_{R}*s*= 0. With this simplification, the model has no free parameters and so makes direct predictions about the observer's matching of CM phase and amplitude (shown in Figures 2 and 3).

*W*=

_{L}*W*= 0.5, implying simple averaging. But for the monocular probe (assumed here to be in the left eye),

_{R}*C*= 0, so

_{R}*W*= 1, and

_{L}*W*= 0, implying no binocular combination of any kind.

_{R}*R*,

_{L}*R*) to modulation at spatial frequency

_{R}*f*, with phases

*ϕ*,

_{L}*ϕ*, as a function of position (

_{R}*y*) are given by

*R*(

_{B}*y*) =

*R*(

_{L}*y*) +

*R*(

_{R}*y*). Thus Equations A3 and A4 sum linearly; each can be treated as a vector with amplitudes

*W*and phases

_{L}M_{L}, W_{R}M_{R}*ϕ*=

_{L}*θ*/2,

*ϕ*= −

_{R}*θ*/2. The resultant phase

*θ′*for interocular phase difference

*θ*(=

*ϕ*−

_{L}*ϕ*) and modulation ratio

_{R}*(=*

**δ***M*) is

_{R}/M_{L}*M*in the left eye,

_{0}*a*=

_{L}*W*

_{L}M_{0},

*a*=

_{R}*δ*.

*W*

_{R}M_{0}, and from Equations A1 and A2, both weights

*W*,

_{L}*W*are 0.5. Inserting these into Equation A7, we get

_{R}*not*the same as simply assuming that the monocular envelopes are averaged. The difference lies in the probe response. With averaging, the response to the probe would be

*M′*/2 (average of

*M′*and 0) instead of

*M′*. All predicted amplitude matches would double and completely fail to fit the data. The key assumption in our model is that the weights are driven by luminance contrast, not by modulation depth, and with this contrast weighting, the same combination rule gives different behavior for the binocular test and the monocular probe. For the test grating, when the carrier contrast is present and equal in both eyes, we get binocular averaging, but for the probe, when one eye has no carrier, we get WTA (weights of 1 and 0). The surprising consequence is that a monocular (

*δ*= 0) test amplitude of 0.8 is matched by a monocular probe amplitude of 0.4. These two rules (averaging and WTA) emerge from the same weighting scheme, which can be seen as a mechanism for achieving “ocularity invariance”—perception of form and contrast remain the same with one eye and two eyes (D. H. Baker, Meese, & Georgeson, 2007; Ding & Sperling, 2006).

*ρ*is the gain-control efficiency of the signal sine-wave grating, and

*γ*is the exponent of the nonlinear transducer. Equivalence with Equations A1 and A2 is exact when

*s*= 1/

*ρ*and

*γ*= 1. Note that the meaning of

*C*(or

_{L}*C*) remains unchanged; it is the luminance contrast of the image, determined by the carrier for second-order and the grating itself for first-order.

_{R}*C*=

_{L}*C*, the Ding and Sperling Equations A11 and A12 would always give equal weights, and

_{R}*W*=

_{L}*W*= 0.5, if

_{R}*ρ*≫ 1, again implying simple averaging. Note that when

*C*=

_{L}*C*this equality of weights holds true for any value of the nonlinear exponent (

_{R}*γ*). And similarly for a monocular (left eye) probe

*C*= 0, so

_{R}*W*= 1 and

_{L}*W*= 0, again implying no binocular combination for the probe. This also is true for any

_{R}*γ*and implies that the present experiments on CM are insensitive to the value of

*γ*. If the ratio

*C*:

_{L}*C*is systematically varied, then the ocular weights vary—favoring the eye that has the higher contrast—and the value of

_{R}*γ*can be estimated from phase-matching data. This has been done for first-order signals but not yet for second-order. For first-order summation, the geometric mean estimate of

*γ*was close to 1 (1.16 ± 0.18, pooled over two previous studies with a total of seven observers; Ding & Sperling, 2007; Huang et al., 2010). Our formulation (Equations A1 and A2) is equivalent to assuming

*γ*= 1 for CM.

*C*,

_{L}*C*needed to compute the weights (Equations A1 and A2) are the mean pixel contrasts or Michelson contrasts (which are the same for binary noise). This is simple, and it makes the contrast values and the weights independent of modulation depth, and so the weights are equal at all modulation ratios and equal to 0.5 when

_{R}*s*= 0. But is mean contrast the most relevant measure? The sharp-eyed reader may notice that in Figures 2b and 3 the observed amplitude matches at low modulation ratios (0, 0.2) lie slightly but consistently

*above*the model's asymptote of 0.4 that is predicted by equal weighting. It turns out that this discrepancy disappears, and the model fit improves, if RMS contrast is used in place of mean contrast. For binary noise (as used here) Schofield and Georgeson (1999) showed (their equation 8) that

*C*=

_{RMS}*C*is the carrier pixel contrast and

*m*the contrast modulation depth. Thus RMS contrast increases with modulation. In our experiments, at a modulation ratio of zero, the modulated image in one eye has a higher RMS contrast than its unmodulated partner. The result of using RMS contrast to compute the weights is shown in Figure 6. Model predictions, still with no free parameters, were compared with all the data points from Experiments 1 and 2. The fit was excellent: For the amplitude matches

*r*

^{2}= 0.960 (RMS error = 0.335 dB), compared with

*r*

^{2}= 0.943 (RMS error = 0.461 dB) when mean contrast was used for the weights (as in Figures 2 and 3). The improvement clearly lay at the low modulation ratios, and Figure 6c reveals why: The weight is a little higher for the eye with more modulation. This pushes the binocular sum a bit higher than it would be if the weights were equal, but the difference in weights shrinks as the other eye's modulation increases toward equality. We conclude that, although the improvement in model fit is small, it looks convincing and has a clear rationale: Ocular weights for CM are driven by RMS contrast not mean or Michelson contrast. (Note: This analysis has no implications for first-order combination because, for luminance gratings, Michelson contrast and RMS contrast are directly proportional to each other and therefore functionally interchangeable.)

*first-order*contrast-matching across all phase disparities seen in Figure 5d. Then we show that adding independent noise to each of the three signals before the MAX operator can yield good predictions for perception of binocular phase while, at the same time, improving the predictions for contrast-matching by reducing the Fechner paradox.

*r*varies with disparity for two reasons: (a) the vector sum varies with phase difference (Equation A7), and (b) the component amplitudes in the vector sum also vary with phase difference because the ocular weights vary with disparity (when

_{B}*h*> 0; Equations 5 and 6). Red curves in Figure A1 illustrate how the resultant amplitude

*r*varies with phase disparity, falling steeply to zero as disparity increases from ±120° to ±180°. On the other hand, the monocular responses do not depend on disparity and plot as circles (blue) in these polar diagrams.

_{B}*r*as the test grating does.) Importantly, the experimental contrast matches (green squares; mean of 3 Ss; D. H. Baker et al., 2012a) lie close to the predicted ones (gray symbols) at all disparities and all contrast levels. This figure makes it clear how, on this model, contrast-matching up to ±120° disparity is explained by the weighted binocular sum,

_{B}′*r*(red curve), and at larger disparities, it is the monocular response (blue curve) that matters. More subtly, the relationship between monocular and binocular response curves varies with contrast level because the binocular weights vary with contrast (Equations 5 and 6). The monocular response is relatively weaker at low contrast, and this accounts for the interaction between disparity and contrast level seen most clearly in Figure 5a and d, a stronger effect of phase disparity at low contrasts.

_{B}*r*

^{2}= 0.996), and the Fechner paradox was reduced, but the phase matches were poor (

*r*

^{2}= 0.64, RMS error = 10.1° of phase; not shown). As expected, they were not graded with contrast ratio (unlike Figure 4b), but were constant, determined by the higher contrast input until the contrast ratio was close to 1.0. However, if (as seems likely) the three inputs to the MAX operator are noisy over time or from trial to trial, then all three inputs should contribute somewhat to any given condition, and perceived phase might show a smoother transition as contrast ratio increased from 0 to 1 (like the data in Figure 4b). This idea was tested with the same first-order model as before (

*WTA 1*) except that the effects of adding independent Gaussian noise to the signals before the MAX operator were also included. Rather than running a Monte Carlo simulation (too slow), we wrote a

*Matlab*function that computed the statistics of the MAX operator (mean,

*SD*, and probability distribution of the output) given the means and standard deviations of the

*N*input signals (

*N*= 3). This useful function is available as Supplementary Material to this paper.

*,*

**r**_{L}*,*

**r**_{B}*with a common standard deviation defined as*

**r**_{R}*σ*=

*noise*.max(

*,*

**r**_{L}*,*

**r**_{B}*), where*

**r**_{R}*noise*is a single new parameter of the model. Thus three small inputs were less noisy than three large inputs, but one large input would make all three inputs more noisy. Contrast matches were, as before, obtained when the mean output for a test grating matched the mean output for the comparison grating. Phase matches were given by the probability-weighted vector sum of the three signal phases (

*ϕ*,

_{L}*ϕ*,

_{B}*ϕ*). Thus, where

_{R}*p*is the probability that in a given test condition the

_{j}*j*th signal (

*j*= L, B, R) yielded the

*max*response. The probability

*p*is also returned by the

_{j}*Matlab*function described above, and we refer to it as the

*contribution*made by the

*j*th signal to perception of a given test condition. Obviously, the three contributions always sum to 1.

*s*,

*h*,

*n*, and

*noise*. The fit to the contrast data (Figure A2a) was excellent (

*r*

^{2}= 0.995; RMS error = 0.31 dB). But now the fit to the phase data (Figure A2b) was also very good (

*r*

^{2}= 0.963; RMS error = 3.2° of phase). Phase-matching behavior of the model was sensitive to the level of noise. For example, doubling or halving the

*noise*parameter degraded the quality of phase predictions much more than the contrast matches.

*C*is fixed (0.32) while

_{L}*C*rises from 0 to 0.32. Not surprisingly the

_{R}*R*contribution rises with

*C*. But a surprising implication is that the binocular (B) contribution is greatest (0.5) for a monocular input and falls to about one third when the input is fully binocular (i.e., left and right eye contrasts are equal), irrespective of disparity. Using the same parameters, Figure A3 gives more detail on these contributions across all phase disparities.

_{R}_{B}falling, but when this happens, the MAX operator automatically draws a greater contribution from

*r*(Figures A2c and A3), and the paradox is averted. Thus, not only at large disparities but also at intermediate contrast ratios, monocular responses substitute for reduced binocular ones and render perceived contrast more nearly constant than it would otherwise be.

_{L}