Stereo vision is an area in which we are increasingly able to construct detailed numerical models of the computations carried out by cerebral cortex. Piecewise-frontoparallel cross-correlation is one such model, closely based on the known physiology and able to explain important aspects of human stereo depth perception. Here, we show that it predicts important differences in the ability to detect disparity gratings with square-wave vs. sine-wave profiles. In particular, the model can detect square-wave gratings up to much higher disparity amplitudes than sine-wave gratings. We test this prediction in human subjects and find that it is not borne out. Rather there seems to be little or no difference between the detectability of square- and sine-wave disparity gratings for human subjects. We conclude that the model needs further refinement in order to capture this aspect of human stereo vision.

^{2}and black pixels were 0.26 cd/m

^{2}.

*A*, the disparity noise dots had disparity +

*A*or −

*A*with equal probability. On sine-wave trials, they had a disparity in the range [−

*A*, +

*A*]. In the grating stimuli, all dots at a given vertical position had the same disparity, but in the noise stimuli, disparity was picked without reference to vertical position, so dots in the same row would have different disparities.

*a*= 0.583,

*s*

_{1}= 0.443 arcmin, and

*s*

_{2}= 2.04 arcmin (Filippini & Banks, 2009; Geisler & Davila, 1985). The images were then scaled to make the distance between rows and columns 0.6 arcmin. This was done to make sure the resolution of the images was no higher than the spacing between cones at the fovea (Filippini & Banks, 2009; Geisler & Davila, 1985; Rossi & Roorda, 2009).

*L*

_{ w }and

*R*

_{ w }are the contents of the windows in the left and right images multiplied by the window function and cov is the covariance. The window functions used to obtain the main results presented here were Gaussians centered on the current window position (that is, (Δ

*x*/2,

*y*) in one eye and (−Δ

*x*/2,

*y*) in the other) and cut off two standard deviations from the center in each direction. The output from the cross-correlator was a two-dimensional image of correlation as a function of the horizontal disparity, Δ

*x*, between the windows as well as the vertical position of the windows,

*y*(see Figure 3). The disparities used were in the range from −25 to 25 arcmin with a step of 0.6 arcmin (1 pixel in the scaled images). The step in the

*y*-position was also 1 pixel in the scaled images.

*x*, for each vertical window position,

*y*, and recording the difference in horizontal position between the two windows as an estimate of the horizontal disparity at that vertical position:

*x*

_{est}(

*y*) was then calculated as

*μ*is the mean and

*σ*is the standard deviation of Δ

*x*

_{est}. Two examples of what the autocorrelograms looked like are given in Figure 4. Finally, both a sine wave and a triangular wave, which are the autocorrelation functions of a sine wave and a square wave, respectively, with the same frequency used in the stimulus were fit to the autocorrelogram and the

*r*

^{2}value of the best fit was recorded. For each pair of a wave and a noise pattern, making up a single trial, the image pair that got the highest

*r*

^{2}value was guessed to contain the grating (Figure 5).

*CO*is the correlator output,

*T*

_{ n }is one of the templates,

*μ*

_{ CO }and

*μ*

_{ Tn }are the means over all disparities Δ

*x*and all

*y*-positions of the correlator output and template

*T*

_{ n }, respectively. All sums were performed over disparity and

*y*-position. The interval for which the difference between the correlation to the grating template and the correlation to the noise template was the highest was guessed to contain the grating.

*y*-position and the detector for 0.6 arcmin can only be the most strongly activated one when the entire window or very close to the entire window is seeing 0.3 arcmin. This can only happen for the square waves, and it is only for the lowest frequency that it happens for a large enough range of

*y*-values to allow detection.

*y*-positions to identify a grating; high correlation in small regions close to the peaks of the sine waves (see Figure 13) may be enough since the relevant template has the same pattern. The drop in performance for the lowest amplitude happens only to a lesser degree for the template matching rule than for the autocorrelation-based rule. This is because the template matching rule uses the outputs from all the correlation detectors and not just the one that has the strongest response at each

*y*-position. However, critically, both decision rules show the same key features highlighted at the end of the previous section. In particular, as disparity amplitude increases, performance remains high for the square-wave gratings and declines for the sine wave. The alternative template matching approach mentioned in the Methods section also showed this behavior (results not shown). Thus, this key behavior is not dependent on any particular decision rule. As explained in the previous section, we attribute it to the properties of the initial disparity encoding performed by correlation detectors tuned to uniform disparity.

*π*its peak-to-trough range, plus successive lower amplitude sine waves. As the grating period decreases to the limit of detectable frequencies, a point is reached where the fundamental frequency is still above threshold, but the third harmonic is already below threshold. Sine- and square-wave gratings thus become indistinguishable. Most of our data fall within this domain, since for most subjects the highest frequency tested was just at the threshold of discriminability, whereas the lowest frequency tested was more than one-third of this value. This means that even at the lowest frequency tested, the third harmonic distinguishing the square-wave from the sine-wave grating would be nearly undetectable if presented alone. Thus if the linear theory is correct, if we plot performance as a function of the amplitude of the fundamental, instead of the whole-waveform amplitude used so far in this paper, performance should become the same for square-wave and sine-wave gratings.

*π*. To assess whether this manipulation brings performance for the two waveforms closer together, we used the curves fitted to each set of data. For each frequency, we computed the integral of the absolute difference between the curves for the sine waves and the square waves, first for the original data and then for the adjusted data. If this integral was smaller for the adjusted data, this indicated that the shift to fundamental amplitude had brought the results closer together. This is indicated with a + symbol at the bottom left of the panels in Figure 10; a − symbol indicates that the shift to fundamental amplitude brought the fits further apart. Bootstrap resampling was used to estimate the significance of any change. The asterisks in Figure 10 indicate

*p*< 0.05 (two-tailed test), while NS indicate that the adjustment had no significant effect either way.

*y*-positions that the different curves passed through was computed at each

*x*-position and the mean of this standard deviation over all

*x*-positions was used as a measure of how closely the curves were superimposed. Bootstrap resampling was used to estimate the significance of any difference between the two ways of plotting the data. The curves were found to be significantly more superimposed (

*p*< 0.05) when plotting against disparity gradient for three out of four subjects (PFA, OO, and ISP). For the fourth subject, no significant difference either way was found. Thus, our data are consistent with the idea that performance at high amplitudes is limited by the highest perceivable disparity gradient.

*y*-positions that the different curves passed through was computed at each

*x*-position and the mean of this standard deviation over all

*x*-positions was used as a measure of how closely the curves were superimposed. Bootstrap resampling was used to estimate the significance of any difference between the two ways of plotting the data. No significant difference was found for the results with either of the decision rules. Thus, for the model results, the curves do not superimpose any better when the data are plotted against disparity gradient. Rather, the performance of the model depends separately on frequency and amplitude, and not simply on disparity gradient (amplitude × frequency). This is not surprising given that the model has no mechanisms that specifically detect disparity gradient. The observed dependence of frequency and amplitude may be because the correlation output from the first stage of the model has the highest correlation in the regions close to the flat parts of the sine wave (see Figure 13). Thus, performance may be limited by the size of the regions that are flat enough to generate high correlation, rather than by the maximum disparity gradient in the stimulus.

*closer*to sine wave. This deficiency, therefore, also cannot explain the discrepancy between model and human results.