Binocular disparities have a straightforward geometric relation to object depth, but the computation that humans use to turn disparity signals into depth percepts is neither straightforward nor well understood. One seemingly solid result, which came out of Wheatstone’s work in the 1830’s, is that the sign and magnitude of horizontal disparity predict the perceived depth of an object: ‘Positive’ horizontal disparities yield the perception of ‘far’ depth, ‘negative’ horizontal disparities yield the perception of ‘near’ depth, and variations in the magnitude of horizontal disparity monotonically increase or decrease the perceived extent of depth. Here we show that this classic link between horizontal disparity and the perception of ‘near’ versus ‘far’ breaks down when the stimuli are one-dimensional. For these stimuli, horizontal is not a privileged disparity direction. Instead of relying on horizontal disparities to determine their depth relative to that of two-dimensional stimuli, the visual system uses a disparity calculation that is non-veridical yet well suited to deal with the joint coding of disparity and orientation.

Even though binocular disparity is a very well-studied cue to depth, the function relating disparity and perceived depth has been characterized only for the case of horizontal disparities. We sought to determine the general relationship between disparity and depth for a particular set of stimuli. The horizontal disparity direction is a special case, albeit an especially important one. Non-horizontal disparities arise from a number of sources under natural viewing condition. Moreover, they are implicit in patterns that are one-dimensional, such as gratings, lines, and edges, and in one-dimensional components of two-dimensional patterns, where a stereo matching direction is not well-defined. What function describes perceived depth in these cases? To find out, we measured the phase disparities that produced depth matches between a reference stimulus and a test stimulus. The reference stimulus was two-dimensional, a plaid; the test stimulus was one-dimensional, a grating. We find that horizontal disparity is no more important than other disparity directions in determining depth matches between these two stimuli. As a result, a grating and a plaid appear equal in depth when their horizontal disparities are, in general, unequal. Depth matches are well predicted by a simple disparity vector calculation; they survive changes in component parameters that conserve these vector quantities. The disparity vector rule also describes how the disparities of 1-D components might contribute to the perceived depth of 2-D stimuli.

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Horizontal binocular disparity has long been the conventional predictor of stereo depth. Surprisingly, an alternative predictor fairs just as well. This alternative predicts the relative depth of two stimuli from the relation between their disparity vectors, without regard to horizontal disparities. These predictions can differ; horizontal disparities accurately predict the perceived depth of a grating and a plaid only when the grating is vertical, while the vector calculation accurately predicts it at all except near-horizontal grating orientations. For spatially two-dimensional stimulus pairs, such as plaids, dots, and textures, the predictions cannot be distinguished when the stimuli have the same disparity direction or when the disparity direction of one of the stimuli is horizontal or has a magnitude of zero. These are the conditions that have prevailed in earlier studies. We tested whether the perceived depth of two-dimensional stimuli depends on relative horizontal disparity magnitudes or on relative disparity magnitudes along a disparity axis. On both measures tested—depth matches and depth-interval matches—the perceived depth of plaids varied with their horizontal disparities and not with disparity direction differences as observed for grating-plaid pairs. Differences in disparity directions as great as 120° did not affect depth judgments. This result, though opposite the grating-plaid data, is consistent with them and provides a view into the construction of orientation-invariant disparity representations.

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