**The effects of signal and noise on contrast discrimination are difficult to separate because of a singularity in the signal-detection-theory model of two-alternative forced-choice contrast discrimination (Katkov, Tsodyks, & Sagi, 2006). In this article, we show that it is possible to eliminate the singularity by combining that model with a binocular combination model to fit monocular, dichoptic, and binocular contrast discrimination. We performed three experiments using identical stimuli to measure the perceived phase, perceived contrast, and contrast discrimination of a cyclopean sine wave. In the absence of a fixation point, we found a binocular advantage in contrast discrimination both at low contrasts (<4%), consistent with previous studies, and at high contrasts (≥34%), which has not been previously reported. However, control experiments showed no binocular advantage at high contrasts in the presence of a fixation point or for observers without accommodation. We evaluated two putative contrast-discrimination mechanisms: a nonlinear contrast transducer and multiplicative noise (MN). A binocular combination model (the DSKL model; Ding, Klein, & Levi, 2013b) was first fitted to both the perceived-phase and the perceived-contrast data sets, then combined with either the nonlinear contrast transducer or the MN mechanism to fit the contrast-discrimination data. We found that the best model combined the DSKL model with early MN. Model simulations showed that, after going through interocular suppression, the uncorrelated noise in the two eyes became anticorrelated, resulting in less binocular noise and therefore a binocular advantage in the discrimination task. Combining a nonlinear contrast transducer or MN with a binocular combination model (DSKL) provides a powerful method for evaluating the two putative contrast-discrimination mechanisms.**

*d*′) depends on the signal-to-noise ratio, remaining constant when both signal and noise increase or decrease. However, with only discrimination measurements, it is difficult or even impossible to separate the effects of the signal and the noise on discrimination performance. For example, enhanced performance could be equally well accounted for by an increase in signal or a decrease in noise, resulting in a singularity in the signal-detection-theory (SDT) model of two-alternative forced-choice (2AFC) contrast-discrimination data (Katkov, Tsodyks, & Sagi, 2006). To avoid the singularity, a prior assumption is needed for fitting a model (Katkov et al., 2006). With an assumption of constant noise, the discrimination data can be accounted for by a nonlinear contrast transducer (NCT), such as monocular or binocular contrast-gain control. With an assumption of linear contrast transformation, the same data could also be accounted for by multiplicative noise (MN), in which the variance is dependent on stimulus. Despite a great deal of effort (e.g., Georgeson & Meese, 2006; Klein, 2006; Kontsevich, Chen, & Tyler, 2002), no previous study has succeeded in distinguishing between NCT and MN in contrast discrimination because of the singularity (Katkov et al., 2006).

*x*causes

*d*′ to double. This could be due to one of two possibilities: Before manipulation

*x*, the mean response difference was 4 units and the noise standard deviation was 4 units, so that

*d*′ = 4/4 = 1; after manipulation

*x*,

*d*′ could become 2 either because the mean response difference doubled (

*d*′ = 8/4 = 2) or because internal noise halved (

*d*′ = 4/2 = 2). Now suppose we carry out another manipulation

*y*, and we find that

*d*′ goes from 1 to 4; again, there are two possibilities: 16/4 = 4 and 4/1 = 4. With only the monocular measurement, every time we apply a manipulation and see a

*d*′ change, we cannot know whether it is because the numerator goes up or the denominator goes down. But now we measure the output from both eyes, and we apply the two manipulations separately but simultaneously to both eyes. Suppose we know how the outputs from the two eyes are combined, and that they are simply summed. Now we apply manipulation

*x*to one eye and manipulation

*y*to the other. Under the hypothesis that our manipulations change mean response, the final

*d*′ should be . Under the hypothesis that the manipulations reduce noise standard deviation, it should be . So the two numbers are different, and we can tell them apart. More generally, we can apply a continuum for our manipulation, and if we plot all possible paired values between the two eyes, the resulting surface for binocular

*d*′ will have a form that is different under the two hypotheses of changing mean response and changing noise standard deviation. Theoretically, through combining the SDT model and a binocular combination model the singularity could be removed and a unique solution obtained to the question of how the signal and noise affect contrast discrimination. For this purpose, we need a robust binocular combination model that works in multiple binocular tasks.

*I*=

_{L}*I*

_{0}

*(1 +*

_{L}*m*cos(2

_{L}*πf*+

_{s}y*θ*)) and

_{L}*I*=

_{R}*I*

_{0}

*(1 +*

_{R}*m*cos(2

_{R}*πf*+

_{s}y*θ*)) were used as stimuli.

_{R}*I*

_{0}

*and*

_{L}*I*

_{0}

*are the luminance of the background and the mean luminance of the sine-wave gratings in the two eyes (= 26.2 cd/m*

_{R}^{2});

*f*is the spatial frequency (= 0.68 c/°), identical in both eyes;

_{s}*m*and

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

_{R}*θ*and

_{L}*θ*are the corresponding phases. The stimuli were windowed in a circular window spatially with a blurred edge (3° in diameter) and a square window temporally (117 ms). The observation distance was 68 cm.

_{R}**Figure 1**

**Figure 1**

**Figure 2**

**Figure 2**

*m*

_{1}and

*m*

_{2}be the contrast of a pair of stimuli,

*R*

_{1}and

*R*

_{2}their internal responses, and

*σ*

_{1}and

*σ*

_{2}the corresponding standard deviations. Using the cumulative Gaussian distribution the percentage of correct discriminations of the two contrasts is given by

*R*

_{1}= 0 and

*σ*

_{1}= 1—Equation 1 still has two unknown parameters,

*R*

_{2}and

*σ*

_{2}. With only one measurement

*P*, the solution of Equation 1 is highly ambiguous; multiple solutions exist. Even with multiple pairs of contrasts, the singularity still exists when fitting Equation 1 to the data, making the two mechanisms of NCT and MN inseparable (Katkov et al., 2006). To eliminate the ambiguity in the solution of Equation 1, one more measurement is needed. If we have a robust binocular combination model whose parameters are already known from different binocular tasks, the binocular responses can be deduced from monocular responses, therefore adding one more measurement without additional unknown parameters. Let

*R*and

*σ*

^{2}be the monocular mean response and variance to stimulus

*m*; its binocular mean response and variance are given by

*R̂*(

*R*) and

*σ̂*

^{2}(

*R*,

*σ*

^{2}), where

*R̂*and

*σ̂*

^{2}can be deduced from the binocular combination model. For a pair of stimuli

*m*

_{1}and

*m*

_{2}, we have another measurement, percentage of correct binocular discriminations:

- The signal goes through both accelerating (Acc) and compressive (Comp) nonlinearities, and the noise is assumed to be a constant Gaussian additive noise (AN), which is independent of the input. We refer this as the NCT mechanism (Figure 3A).
- The signal first goes through an Acc nonlinearity, and both MN (dependent on the input) and AN (independent of the input) are added to the signal. We refer this as the MN mechanism (Figure 3B).

**Figure 3**

**Figure 3**

*m*is given by with constant variance

*σ*

^{2}of Gaussian noise. Equation 3 is identical to the binocular contrast-gain control (CGC), the second stage of the two-stage model (Meese et al., 2006); it is accelerating when

*m*<

*Z*and compressive when

*m*>

*Z*. For the MN mechanism, the internal response is given by with contrast-dependent variance of MN and constant variance

*σ*

^{2}of AN. When

*m*<

*Z*, Equation 4 is an Acc contrast transducer, accounting for the “dipper” in the threshold-versus-contrast (TVC) function. When

*m*>

*Z*, it becomes a linear contrast transducer, and the compression is accounted for by MN.

*α*, and

*β*); and (c) the gain-enhancement layer that extracts image contrast energy (ℰ*) and exerts gain enhancement only to the other eye's signal layer. As shown in Figure 3C, before output the LE's signal (black) receives gain control from the RE's gain-control layer (blue), which itself receives gain control from the LE, and also the LE's signal (black) receives gain enhancement from the RE's gain-enhancement layer (red), which receives gain control from the LE. The model output is given by

*m*and

_{L}*m*are presented to the two eyes, the image contrast energy for gain control is given by and the image contrast energy for gain enhancement is given by where

_{R}*g*is the contrast gain-control threshold and

_{c}*g*is the contrast gain-enhancement threshold.

_{e}**Figure 4**

**Figure 4**

*p*,

*q*,

*Z*, and

*σ*), of which

*Z*reflects the contrast-detection threshold; it would be an Acc contrast transducer when contrast <

*Z*and a Comp contrast transducer when contrast >

*Z*.

**Figure 5**

**Figure 5**

*Z*, Equation 4 is an Acc contrast transducer, accounting for the dipper in the TVC function. When contrast >

*Z*, Equation 4 becomes a linear contrast transducer and the compression is accounted for by MN (Equation 5).

**Table 1**

**Figure 6**

**Figure 6**

**Table 2**

**Figure 7**

**Figure 7**

**Table 3**

**Figure 8**

**Figure 8**

*m*≈

*Z*(the shallowest point) with little effect on the other thresholds. At low contrast (when

*m*<

*Z*), MN is too small and the AN dominates performance, while at high contrast (when

*m*>

*Z*), the Acc (Equation 4) becomes a linear contrast transducer. Similar to the combined model with late NCT (Figure 8A), the combined model with late MN overestimates binocular discrimination thresholds at high contrast (Figure 8B), although it captures the main features of the TVC functions ( = 14.26). Moving MN before the binocular combination site but after interocular interactions (middle MN, Figure 8C) improves the prediction ( = 12.11), and further moving MN before interocular interactions (early MN, Figure 8D) makes the prediction nearly perfect ( = 4.82).

**Figure 9**

**Figure 9**

**Table 4**

**Figure 10**

**Figure 10**

**Table 5**

*m*with an early Gaussian MN—i.e., LE =

*m*+ 𝒩

*(0,*

_{L}*σ*) and RE =

_{m}*m*+ 𝒩

*(0,*

_{R}*σ*), where

_{m}*σ*=

_{m}*km*(

^{a}*k*= 0.08,

*a*= 0.70, from Table 1). The model was simulated without considering the NCT and AN after the binocular combination site. For the two-stage model (parameters from Table 4), we simulated only the first stage.

**Figure 11**

**Figure 11**

**Figure 12**

**Figure 12**

**Figure 13**

**Figure 13**

*p*values, and confidence intervals. Consistent with a previous study (Summers & Meese, 2009), the FP had no significant masking effect on monocular contrast discrimination (Figure 14A). Averaged across observers, the masking effect on monocular contrast discrimination was only a factor of 1.10 (95% CI [0.94, 1.29],

*p*= 0.115). However, under binocular viewing, contrast discrimination was significantly improved when the FP was absent. This phenomenon has not been reported previously and cannot be simply explained by the FP's masking effect. Averaged across observers, the improvement in binocular contrast discrimination with no FP was a factor of 1.45 (95% CI [1.19, 1.91],

*p*= 0.025).

**Figure 14**

**Figure 14**

*p*= 0.21). However, when the FP was absent, binocular contrast discrimination was significantly better than monocular contrast discrimination, a key phenomenon observed in this study which has not been reported previously. Averaged across observers, when the FP was absent the binocular advantage was a factor of 1.39 (95% CI [1.09, 1.60],

*p*= 0.001).

**Figure 15**

**Figure 15**

*Spatial Vision*, 20 (5), 397–413.

*Vision Research*, 48 (15), 1625–1640.

*PLoS One*, 7 (4), e34696.

*Journal of the Optical Society of America*, 50 (7), 738–738.

*The Journal of Physiology*, 145 (3), 579–594.

*Ophthalmic and Physiological Optics*, 8 (2), 153–164.

*Ophthalmic and Physiological Optics*, 35 (5), 476–499.

*Optics Express*, 16 (19), 14731–14745.

*Physica*, 10 (7), 553–564.

*Journal of Vision*, 13 (9): 550, doi:10.1167/13.9.550. [Abstract]

*Proceedings of the National Academy of Sciences, USA*, 103 (4), 1141–1146.

*Computational vision in neural and machine systems*(pp. 257–305). Cambridge, UK: Cambridge University Press.

*Vision Research*, 46 (25), 4294–4303.

*Signal detection theory and psychophysics*. New York: Wiley.

*Optometry & Vision Science*, 66 (2), 123–129.

*PLoS One*, 5 (12), e15075.

*Vision Research*, 46 (1), 259–266.

*Vision Research*, 46 (25), 4279–4293.

*Vision Research*, 42 (14), 1771–1784.

*Vision Research*, 24 (4), 373–383.

*Vision Research*, 24 (4), 385–394.

*Optical Review*, 12 (2), 76–82.

*Ophthalmic and Physiological Optics*, 29 (3), 256–263.

*Computational models of visual processing*(pp. 147–152). Cambridge, MA: MIT Press.

*Proceedings of the IRE*, 30 (6), 293–300.

*Journal of the Optical Society of America*, 38 (2), 196–208.

*Optometry & Vision Science*, 82 (4), 279–285.

*Vision Research*, 49 (14), 1894–1900.

*Sensation and perception*(4th ed.). Sunderland, MA: Sinauer Associates.