The monocular and binocular ideal observers performed very similarly to each other. For example,
Figure 11 compares the two visibility-based ideal observers for the short bar targets. This was the condition in which the monocular and binocular ideal observers had the biggest difference, and this difference was quite small. To understand why the monocular and binocular performance was so similar, consider what happens at various Δ
Z. For small Δ
Z, the expected visibilities of the two targets are nearly the same, and so the monocular visibility-based observer performs at near chance. The binocular observer has little advantage here, since averaging the visibilities of the two eyes reduces the variability of the visibilities only slightly, and the variability remains much greater than Δ
Z (recall error bars in
Figure 2b). For example, if the visibilities of the two eyes were independent, then averaging the visibilities of the two eyes would reduce the standard deviation of visibility only by a factor 1 /
Display Formula but, in fact, the visibilities of a target in the two eyes are not independent and so averaging the visibilities reduces the standard deviation of visibility by less than that factor. As Δ
Z increases, the performance of both the monocular and binocular visibility-based ideal observers increase. However, the binocular observers gain less from the second view than they did in the case that Δ
Z was small since, as Δ
Z increases, the disparity differences between the near target and the occluders of the near target become smaller, because the near target on average is closer to its occluders. It follows that the visibilities of the left and right views of the near target become more similar, and hence the second view of the near target becomes more correlated with the first. A different argument holds for the far target but the result is the same. As Δ
Z increases, the visibility of the far target approaches zero in both left and right views, so again the binocular visibility-based ideal observer benefits little from the second view.