From disparity to depth: How to make a grating and a plaid appear in the same depth plane

Abstract

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.

The horizontal disparity direction vs. the stimulus disparity direction in the perception of the depth of two-dimensional patterns

“…The plaids’ contrast-envelope parameters were identical to those used for the central test grating and surrounding reference plaid by Chai and Farell (2009), whose results we want to compare with data for plaid pairs. In addition to allowing a comparison of the two studies, using the center-surround configuration avoids certain spatial alignment effects that might otherwise contribute unwanted variability to the data.…”

“…The disparities of the contrast envelopes were fixed at zero, as in the experiments of Chai and Farell (2009). The only non-zero disparities were interocular phase shifts of the carriers.…”

“…The contours nearest these patterns were the edges of the monitors’ screens, which were located approximately 3.8° vertically and 5.8° horizontally from the outer visible limit of the annulus. As described elsewhere (Chai & Farell, 2009), occluders rendered the vertical contours of the monitor visible only monocularly. This, too, was to prevent stimuli other than the annular plaid from functioning as a reference for observers’ judgments of the depth of the central plaid.…”

“…If D 1 and D 2 are constrained to be horizontal, the two disparity magnitudes will, other things being equal, have to be the same in order to produce a depth match. However, in the general case of arbitrary disparity directions (so that D 1 and D 2 must be denoted by vector rather than scalar values), equal disparity magnitudes, even equal horizontal and vertical disparity magnitudes, do not necessarily give rise to equal perceived depths (Chai & Farell, 2009; Farell, Chai, & Fernandez, 2009). …”

“…Specifically, D 2 is proportional to the sine of the angular difference between the orientation of the grating and the disparity direction of the plaid. Horizontal disparity is not a factor; in general, gratings and plaids have different horizontal disparities when they have equal perceptual depths (Chai & Farell, 2009). Their horizontal disparities can even differ in sign (Farell et al, 2009).…”