A fundamental task of the visual system is to infer depth by using binocular disparity. To encode binocular disparity, the visual cortex performs two distinct computations: one detects matched patterns in paired images (matching computation); the other constructs the cross-correlation between the images (correlation computation). How the two computations are used in stereoscopic perception is unclear. We dissociated their contributions in near/far discrimination by varying the magnitude of the disparity across separate sessions. For small disparity (0.03°), subjects performed at chance level to a binocularly opposite-contrast (anti-correlated) random-dot stereogram (RDS) but improved their performance with the proportion of contrast-matched (correlated) dots. For large disparity (0.48°), the direction of perceived depth reversed with an anti-correlated RDS relative to that for a correlated one. Neither reversed nor normal depth was perceived when anti-correlation was applied to half of the dots. We explain the decision process as a weighted average of the two computations, with the relative weight of the correlation computation increasing with the disparity magnitude. We conclude that matching computation dominates fine depth perception, while both computations contribute to coarser depth perception. Thus, stereoscopic depth perception recruits different computations depending on the disparity magnitude.

^{2}. Because of their short decay time, we only used red phosphors to minimize interocular crosstalk (<3% of the background).

^{2}) and dark (0.01 cd/m

^{2}) dots. The dot size was 0.14 × 0.14° with anti-aliasing. The dot density (the fraction of the non-background area to total area) was 24%. One dot occluded another dot where dots overlapped. The probability of a contrast-matched dot being occluded by a contrast-reversed dot was equal to the probability of a reversed dot being occluded by a matched dot.

*P*

_{d}) that describes the proportion of correct choices as follows:

*x*is the match level (percentage of contrast-matched dots),

*γ*is the

*y*-intercept (i.e.,

*P*

_{d}(0)),

*α*is the value of

*x*at the psychophysical threshold such that

*P*

_{d}(

*α*) = 1 − (1 −

*γ*) × 0.368, and

*β*is proportional to the slope at

*P*

_{d}(

*α*). We searched for a set of parameters that maximized the likelihood of observing the data by assuming a binomial distribution. For each subject, 15 free parameters (3 parameters × 5 psychometric functions tested at different disparity magnitudes) were required to describe the entire data set.

*F*), defined as

*x*

_{c}is the value of

*x*satisfying

*P*

_{d}(

*x*) = 0.5. Substituting Equation 1 gives

*F*measures the contribution of the odd symmetric component, centered at 50% match and 50% correct. In cases where

*P*

_{d}(

*x*) did not cross 0.5,

*x*

_{c}was set to zero. The denominator in Equation 2 is equal to the total area of the deviation from

*P*

_{d}(

*x*) = 0.5 (Figure 3A, inset). The numerator is equal to twice the area of only the downward deviation from

*P*

_{d}(

*x*) = 0.5. The fractional area is zero for the matching computation prediction and unity for the correlation computation prediction (Figure 1C).

*P*

_{w}from the weighted average of the two computations (see 1 for derivation) such that

*a*is the response amplitude of the encoding detectors,

*w*is the relative weight of the correlation computation over the matching computation, and

*f*

_{1}(

*x*) and

*f*

_{2}(

*x*) describe the dependency of the detectors' responses on the match level

*x*for the correlation computation and for the matching computation, respectively.

*f*

_{1}(

*x*) linearly transforms the match level

*x*into a value ranging from −1 to 1 (Equation A2).

*f*

_{2}(

*x*), which has two parameters (

*u*and

*l*), is a sigmoidal function that transforms

*x*into a value ranging from 0 to 1 (Equation A3). We fitted this function to the data obtained in the five experiments using different disparity magnitudes while solving for

*a, u,*and

*l*over all five experiments and

*w*for each experiment, resulting in a total of eight free parameters (

*a, u, l,*and five weights,

*w*

_{1},

*w*

_{2},

*w*

_{3},

*w*

_{4}, and

*w*

_{5}) for each subject. As in the fit of the descriptive psychometric function (Equation 1), we used the maximum likelihood estimation and assumed a binominal distribution for the observed number of correct choices. All data analyses were done with MATLAB (Mathworks, Natick, MA).

*SD*), 0.06 ± 0.037,

*n*= 4), but much larger at the largest disparity (0.48°; mean ±

*SD*, 0.42 ± 0.074,

*n*= 4). The fractional area increased with the disparity magnitude (Figure 3B; regression slope of 0.32 for data pooled across all subjects,

*p*= 2.8 × 10

^{−6}, H0: linear-regression slope against the common log of the disparity magnitude is 0). In two subjects (TO and MT), the fractional area increased gradually, while in the other two (TD and MH) it abruptly increased at 0.2°. As the disparity magnitude increased, the psychometric function changed from following the prediction of the matching computation to partially following the prediction of the correlation computation.

*x*

_{c}(the match level at chance performance). In contrast with these predictions,

*x*

_{c}increased as the task became easier (Figure 4A; regression slope of 20.1,

*p*= 1.2 × 10

^{−3}), although the increase was not monotonic in two subjects (TD and MH). The high performance at 50% match decreased toward chance as the task became easier (Figure 4B; regression slope of −13.7,

*p*= 0.0051). We found only one tendency that agreed with the expected results if task difficulty was the cause; the performance at 0% match deviated away from chance as the task became easier (Figure 4C; regression slope of −12.6,

*p*= 0.016). Nevertheless, in general, task difficulty does not explain the observed shift of the psychometric functions.

*p*= 0.18; subject TO,

*p*= 2.5 × 10

^{−5}; subject MH,

*p*= 4.6 × 10

^{−5}; subject MT,

*p*= 4.6 × 10

^{−5}; subject ST,

*p*= 3.2 × 10

^{−9}; subject SH,

*p*= 8.7 × 10

^{−7}; binomial test), indicating that the five subjects perceived reversed depth for contrast-reversed RDSs. The performance of the remaining subject (TD) with contrast-reversed RDSs was also numerically lower than 50%. The probability of observing such bias across six subjects by chance is 0.016 (1/2

^{6}). Therefore, reversed depth in contrast-reversed RDSs was unlikely to be caused by vergence eye movements, suggesting that correlation computation contributes to stereopsis without the execution of vergence eye movements.

*SD*, 96 ± 2.6%,

*n*= 4). The weight of the correlation computation relative to that of the matching computation increased with the disparity magnitude (Figure 7D; regression slope of 0.22,

*p*= 1.0 × 10

^{−4}) in a manner similar to the fractional area increase (

*r*= 0.88,

*p*= 3.2 × 10

^{−7}). We could replicate the above results even when using Gabor function disparity tunings with various preferred disparities (normalized log likelihood, 95 ± 3.1%; regression slope of relative weight, 0.13,

*p*= 6.3 × 10

^{−4}; correlation of relative weight between the two fits,

*r*= 0.85,

*p*= 2.2 × 10

^{−6}). The incorporation of Gabor functions makes the analysis more physiologically relevant. Overall, most changes in the psychometric function were explained by changing the relative weight of the two computations.

*x*

_{c}, the point of intersection with chance performance (Figure 8A), while varying the parameters

*u*and

*l*(Equation A3) did not shift

*P*

_{w}(0), the

*y*-intercept (Figures 8B and 8C).

*s*(−1 for “near” disparity and +1 for “far” disparity) with different dependencies on the % binocular match level

*x*. The disparity magnitude is not given as an input. However, one parameter, the weight

*w,*can be assigned different values depending on the disparity magnitude.

*R*

_{ ij }) of these detectors are defined as follows:

*i*and

*j*are binary values identifying the subsystem and the detector, respectively. Throughout this section,

*i*= 1 denotes the subsystem of the correlation computation and

*i*= 2 denotes the subsystem of the matching computation,

*j*= 1 denotes the near detector and

*j*= 2 denotes the far detector. The coefficient

*c*

_{ j }is defined as (

*c*

_{1},

*c*

_{2}) = (−1, +1). The correspondence between the coefficient

*c*

_{ j }and disparity sign

*s*determines the sign of the first term. Parameter

*a*is the response amplitude. The function

*f*

_{ i }(

*x*) maps the match level

*x*to a value between −1 and 1 (

*i*= 1, the correlation computation) or a value between 0 and 1 (

*i*= 2, the matching computation). Parameter

*b*is the response baseline.

*ε*

_{ ij }represents noise and is the only random variable; all other parameters are deterministic. We assume that the response amplitude,

*a,*and baseline,

*b,*are the same between the two subsystems.

*f*

_{1}(

*x*) and that of the matching subsystem

*f*

_{2}(

*x*) are

*f*

_{1}(

*x*) is a linear map, and

*f*

_{2}(

*x*) is a sigmoidal function that consists of four parts: zero, expansive non-linearity, compressive non-linearity, and one. Expansive and compressive non-linearities are odd symmetric with each other. The parameters

*u*and

*l*determine the upper and lower limits of the dynamic range for

*f*

_{2}(

*x*), respectively.

*u*was set to be larger than

*l*.

*ε*

_{ ij }is defined as Gaussian noise centered at zero. The noise variance is assumed to be proportional to the mean responses of the detectors with a scaling factor

*ϕ*(Dean, 1981). Thus,

*E*and

*V*indicate the expectation and variance of the random variables, respectively. We assume that

*ϕ*is the same between the correlation and matching subsystems.

*j*= 1) and far (

*j*= 2) detectors. The output of the second stage

*S*

_{ i }(

*x, s*) is

*i*= 2) and correlation (

*i*= 1) subsystems. This is the decision variable

*D*(

*x, s*):

*w*controls the relative contribution of the correlation and matching subsystems to the decision variable. Again, the noise distribution of the decision variable becomes Gaussian noise centered at zero. The noise variance is a weighted summation of the variances of the second-stage outputs from the matching and correlation subsystems. The weights are squared when the variance of a summed distribution is calculated. Thus, the distribution of the decision variable

*D*(

*x, s*) becomes

*D*(

*x, s*) through a step function:

*T*= +1 denotes a far choice, and

*T*= −1 indicates a near choice. The output is correct if choice

*T*and the stimulus disparity sign

*s*have the same sign.

*ϕ*) and baseline (

*b*) parameters are redundant (Equation A11), we set

*ϕ*= 1. Then, the amplitude (

*a*) and baseline (

*b*) parameters are also redundant because they determine the Gaussian center and width, respectively; increasing (or decreasing) the Gaussian center and decreasing (or increasing) the Gaussian width have the same effects on the psychometric function. Therefore, we fix

*b*= 1. Thus, the variance of the decision variable

*D*(

*x, s*) can be rewritten as

*D*(

*x, s*), we can calculate the probability of a correct choice

*P*

_{w}(

*x*). We redefine the expectation of the decision variable for a “far” disparity stimulus

*E*[

*D*(

*x,*+1)] as

*μ*(

*x*). Hence, the expectation for a “near” disparity

*E*[

*D*(

*x,*−1)] is rewritten as −

*μ*(

*x*). The variance

*V*[

*D*(

*x, s*)] is replaced with

*σ*

^{2}. Following these changes, the probability of a correct choice can be expressed as

*N*(

*z*∣

*μ*(

*x*),

*σ*

^{2}) denotes a normal distribution with mean

*μ*(

*x*) and variance

*σ*

^{2}.

*P*indicates probability. If we transform

*z*into

*t*such that

*P*

_{w}(

*x*) can be rewritten as

*s*= 1,

*ϕ*= 1, and

*b*= 1 into Equations A10 and A11 gives the final solution of the psychometric function:

*a*), relative weight (

*w*), and two non-linearity parameters

*u*and

*l*in

*f*

_{2}(

*x*) (see Equation A3). In the fitting procedure,

*a, u,*and

*l*were kept the same in the five experiments using different disparity magnitudes. The weight parameter

*w*was varied, however. A total of eight free parameters were used to explain the entire data set for each subject (

*a, u, l*and

*w*

_{1},

*w*

_{2},…,

*w*

_{5}). The parameter

*a*was constrained to be positive. The weight parameters (

*w*

_{1},

*w*

_{2},…,

*w*

_{5}) were constrained between 0 and 1. The parameters

*u*and

*l*were constrained between 0 and 100 with

*u*>

*l*.

*f*

_{2}(

*x*) as a non-linear function, we began by defining

*f*

_{2}(

*x*) as a linear function that transforms the match level

*x*into a value ranging from 0 to 1. This definition makes

*f*

_{2}(

*x*) more comparable to

*f*

_{1}(

*x*). However, this version could not explain the psychometric data, especially for fine near/far discrimination (open circles in Figure 2). The linear

*f*

_{2}(

*x*) yields psychometric functions steeply rising at around 0% match level, although the percentage of correct choices by the subjects gradually increased. The percentages were almost flat around chance performance between 0% and 25% match levels for subjects TO and MH. Thus,

*f*

_{2}(

*x*) was extended to a non-linear function. In contrast,

*f*

_{1}(

*x*) was kept linear because this minimal version could explain the data well and is consistent with the disparity energy model.

*k*th detector of near (

*j*= 1) and far (

*j*= 2) population had the following tuning function,

*G*

_{ k }

^{ j }, for disparity (

*d*):

*o*

_{ k }

^{ j }and

*ξ*indicate the center position and size of the Gaussian envelope, respectively (

*ξ*= 0.2). The parameters

*q*and

*θ*

_{ j }indicate the frequency and phase of the cosine carrier, respectively (

*q*= 0.25,

*θ*

_{1}= −0.5

*π,*and

*θ*

_{2}= 0.5

*π*). We defined the center position

*o*

_{ k }

^{1}(

*k*= 1, 2,…, 10) as −0.1, −2/30, −1/30, 0, 1/30, 2/30, 0.1, 0.2, 0.45, and 0.7 so that the distribution of preferred disparities is matched to the MT data (DeAngelis & Uka, 2003).

*o*

_{ k }

^{2}= −

*o*

_{ k }

^{1}for

*k*= 1, 2,…, 10. In the fitting procedure, original

*a*in Equation 4 was replaced with

*a*′(

*d*) =

*G*

_{ k }

^{2}(

*d*) −

*G*

_{ k }

^{1}(

*d*)} so that the decision variable is based on subtraction between the pooled response of the near population from that of far.

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