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.

*D*is the same regardless of the stimulus from which it came.

*what*stimuli are can strongly affect stereoacuity. Specifically, stereoacuity depends on how similar two stimuli are in spatial frequency and orientation (Farell, 2003a, 2006). These are effects of

*inter-stimulus*similarity. They are not limitations imposed by the range or precision of absolute disparity signals of individual stimuli. Rather, they are limitations on the relative disparity signal that can be recovered between two stimuli.

^{1}However, the particular plaids used in Farell's (2006) study did not exploit this property. There, the plaid's components were orthogonally oriented and constrained in their disparities. The result was a confounding of the plaid's disparity direction and its orientation content, just as in the case of individual gratings. Thus, the finding of that study—that perceived depth depends jointly on relative component orientation and relative component disparity between the two stimuli—is consistent with two hypotheses. It is consistent with the hypothesis that a depth match is seen when the two stimuli have

*components*with both similar orientations and disparity magnitudes. And it is consistent with the hypothesis that a depth match is seen when the two

*patterns*have similar disparity magnitudes and directions, independent of the disparities of their components. Thus, either relative component orientation or relative stimulus disparity direction, or both, might be effective mediators of relative stereoscopic depth.

*σ*. The peaks of the center and surround Gaussian envelopes were separated by a distance of 2° visual angle. Spatial frequencies of the sinusoidal carriers were 2 cycles/degree (c/d) in most experiments; 1 c/d (and in a few cases, 3 c/d) patterns were examined in several experiments and produced data similar to those reported here.

^{2}, which was also the patterns' mean luminance. Observers used a chin rest and viewed the displays with natural pupils in a moderately lit room. Stimuli generation and presentation were controlled by a Matlab (Mathworks, Inc.) program incorporating elements of Psychophysical Toolbox software (Brainard, 1997; Pelli, 1997).

_{+}’ gratings) or the component with zero disparity (‘R

_{0}’ gratings). We hypothesized that the disparity of the R

_{+}grating yielding a depth match with the plaid would vary directly with the disparity of the plaid's component parallel to the grating (that is, +5°, +15°, or +30° phase disparity). The depth match between the R

_{0}grating and the plaid should occur at a grating disparity of zero regardless of the plaid's disparity. This is because the plaid's component parallel to the R

_{0}grating had a disparity of zero. For both R

_{+}and R

_{0}conditions, the depth match was hypothesized to be independent of plaid angle. The reason is that variations in plaid angle change the pattern disparity (both in magnitude and direction) but do not change the component disparities or the parallel orientations of test grating and plaid component. To preview, the results agreed with expectations under the parallel-components hypothesis; subsequent experiments, however, led us to reevaluate, and finally to reject, this hypothesis.

_{+}conditions

_{+}conditions; the disparity of the other component was zero. The mean PSEs for these conditions appear in Figure 2. Here the disparities of the grating that yielded depth matches with the plaid are plotted as a function of plaid angle. Data for the three phase disparities of the plaid's non-zero component (5°, 15°, or 30°) appear separately. To match the perceived depth of the plaid, the test grating required a greater disparity as the component disparity increased, and for each of the three component disparities the perceived depth was approximately the same regardless of the plaid angle. Data for individual observers also appear in Figure 2.

*SD*), respectively. Thus, the central grating appeared at a farther distance than a plaid with the same component disparity. The size of this offset (6.7° ± 6.1° [mean ±

*SD*]) shows no obvious relation to the size of the plaid's component disparity or to its pattern disparity. Therefore, it is best regarded as a bias induced by the center-surround arrangement of the stimuli rather than a consequence of processing the carrier disparities.

_{0}conditions

_{0}conditions the test grating had the same orientation as the plaid's zero-disparity component. PSEs for these conditions, shown in Figure 3, differed markedly from those for the R

_{+}conditions. Observers perceived depth matches between plaids and R

_{0}test gratings when the phase disparity of the grating was approximately 0°. Varying the disparity magnitude of the plaid's positive-disparity component had small effects on PSEs. This contrasts with the robust effect of the plaid's component disparity magnitude in R

_{+}conditions. Note that R

_{0}and R

_{+}test gratings do not differ in the disparity magnitudes that give them the same horizontal disparity as the reference plaids. But R

_{0}and R

_{+}test gratings required quite different disparity magnitudes in order to match the depth of the same set of reference plaids ( Figure 2 vs. Figure 3). So, relative horizontal disparity does not predict depth matches between gratings and plaids.

_{0}conditions and 5°, 15°, or 30° in R

_{+}conditions. Figures 2 and 3 show this, too, to be the case. However, both R

_{0}and R

_{+}conditions also show minor effects of plaid angle; as discussed below, either component disparities or pattern disparities might be the source of these influences. These effects, evident in Figures 2 and 3, contributed to statistically significant interactions between reference grating type, R

_{+}or R

_{0}, with component disparity (

*F*(2, 16) = 107.7), plaid angle (

*F*(4, 32) = 8.1), and both component disparity and plaid angle (

*F*(8, 64) = 2.8), as determined by ANOVA, with all

*p*s < 0.01, except the last (

*p*= 0.011). The main effect of test grating type, R

_{+}vs. R

_{0}, (

*F*(1, 8) = 76.7), was also significant (

*p*< 0.01).

_{+}data across plaid angles and observers, this function was quite linear with a near-unity slope (0.96) at the 150 ms duration. At the 1 s duration, however, the slope was reduced to 0.79 ( Figure 4). A proportionately similar reduction occurred for the R

_{0}conditions. One possibility is that contrast envelopes, having zero disparity, influenced the perceived depth of the carrier at the long duration. The annulus has lengthy edges; perhaps at long durations eye movements take the edges to retinal areas that can readily extract the envelope's zero disparity. This is admittedly without the support of precedents—while McKee, Verghese, Ma-Wyatt, & Petrov (2007) found a time-varying effect of envelope disparity on the wallpaper illusion, this effect dissipated, rather than increased, over time—yet it's the only idea we've come up with.

_{0}and the R

_{+}conditions.

*D*

_{ c}(15° or 30°); the plaid angle,

*α*, varied between 60° and 105° (we found that observers had difficulty fusing plaids with a plaid angle of 120° under some disparity conditions). Component disparity and plaid angle jointly control the pattern disparity magnitude:

*p*< 0.01, Bonferroni

*t*-tests).

*D*

_{ p},

*ϕ*], projects onto the grating's perpendicular disparity axis. This projection gives the disparity the grating would have if it had been a component of the plaid:

*D*

_{ g}is the grating's depth-matching disparity and

*θ*is its orientation (0° ≤

*θ*< 180° and 0° ≤ |

*ϕ*| ≤ 180°).

^{2}This says, in effect, that a stimulus disparity direction, not the horizontal meridian, defines a reference axis along which the projected disparity magnitudes of the plaid and the grating are compared when judging their relative depths. This disparity direction can be taken to be that either of the plaid or of the grating (Chai & Farell, 2008).

^{3}

*r*

^{2}> 0.954 for 5 of the 6 correlations, and

*r*

^{2}= 0.866 for the sixth, for which the disparity direction was horizontal and the observer was S4; all

*p*s < 0.01 by

*F*-test). The best-fitting linear function has a slope of 0.76 for one observer and 0.72 for the other. The data show the previously observed bias to see the center as farther than expected from its disparity, especially at large disparities, that accounts for the less-than-unity slope of these functions.

_{+}and R

_{0}conditions. It is equal to the disparity of the plaid's component that is parallel to the grating in the R

_{+}condition and is equal to zero in the R

_{0}condition. In Experiment 2, the plaid and the grating had the same disparity direction. Thus, the projection prediction is that their disparity magnitudes along this common direction should be equal at the depth match. The data of both experiments are in good agreement with these predictions.

*b*) that a depth match that is predicted (and observed, see lower left points in Figure 7) between a plaid with positive horizontal disparity and a grating with negative horizontal disparity. Such a prediction arises only for non-horizontal plaid disparities. Specifically, the prediction holds when the grating is oriented between the plaid's disparity vector and the horizontal. Thus, among the plaid and grating pairs that are seen at the same depth are those that, when viewed individually relative to an isotropic zero-disparity reference stimulus, appear not only at different depths, but also at different depth polarities: The plaid appears behind the reference stimulus and the grating appears in front of it (Farell, Chai, & Fernandez, 2009). One can find a kin of this phenomenon in 'depth-reversed' plaids, where the superposition of one grating with zero-disparity, for example, and another with positive (‘far’) disparity, produces a negative-disparity (‘near’) plaid (Farell, 1998).

*c*in Figure 8). We used reference plaids both with and without horizontal components and having both oblique and horizontal disparity directions. We collected data from three observers (those who contributed to Figure 9). The PSEs for all three observers were hard to evaluate because of the shallowness of the slopes of the psychometric functions when the test grating was horizontal. Their slopes were, on average, less than one-seventh the value of those for non-horizontal test gratings. Thus, varying the disparity of a horizontal test grating produced no more than a weak change in the relative depth of the grating and the plaid (and this might be only because the grating wasn't coded as strictly horizontal). Yet, according to Equation 2, the change should have been the same as for test gratings with other orientations. Thus, while the disparity of a horizontal 1-D component contributes to the perceived depth of the pattern of which it is a part ( Figure 9), the disparity of a lone horizontal 1-D stimuli appears not to enter into the calculation of stereo depth. In light of Figure 8c, one could use this finding to argue that for 1-D stimuli the perpendicular disparity direction is the functional disparity direction.

^{4}This does not mean that absolute orientation and disparity direction do not affect perceived depth, only that they do not make a major contribution to depth matches, where perceived depth separation is zero. There is also no evidence, aside from the horizontal-grating exception, that the disparity projection calculation applies only over a limited range of absolute disparity directions. Therefore, disparity encoding must vary with the magnitude and polarity of both vertical and horizontal disparity components—the information available, up to a global rotation, in a 2-space vector code for both 1-D and 2-D stimuli (for a recent discussion of this issue, see Serrano-Pedraza & Read, 2009).

^{2}value of 0.834.

*F*(1, 42) = 30.43,

*p*< 0.001). As noted earlier, there was no analogous effect of duration on PSEs. Vergence eye movements are a possible source of the difference. We did not record eye movements and so cannot test for interactions between vergence changes and stimulus orientation. We do have some preliminary evidence that eye movements in themselves do not affect the outcome of Experiment 1 (where, however, the gratings were non-vertical). In an unreported variation of that experiment, we instructed observers either to change fixation from the center to the surround approximately midway through 1-second trials or to maintain fixation on the center. They reported the instructions easy to comply with. We found no significant difference between the two conditions in slopes or in PSEs, which closely resembled the PSEs of Figures 2 and 3.

*a priori*knowledge of the vertical disparity) would have to control it.

^{1}A pattern with a 2-D spatial structure can have an ambiguous disparity. It might be periodic along one direction and therefore have multiple correspondence magnitudes, or along multiple dimensions and have multiple correspondence directions and magnitudes (as is the case with plaids). For our purposes, it is in their nearest-neighbor matches that the disparities of 2-D patterns differ importantly from those of 1-D patterns. Nearest-neighbor matches are independent of the monocular spatial structure of 2-D patterns, but they are in the direction normal to the orientation of 1-D patterns. Hence, the nearest-neighbor matches of 2-D patterns, which we use to define the disparity vector of these patterns, can be given an arbitrary direction. To measure the disparity vector of a stimulus, we consider the superposition of the two retinas and measure the magnitude and the right-eye-to-left-eye direction of the offset of nearest corresponding points.

^{2}If orientations are relabeled as −180° ≤

*θ*< 0°, then the angular difference in Equation 2 should be reckoned as (

*ϕ*−

*θ*).

^{3}Equivalently, Equation 2 predicts a depth match when the grating has the same disparity magnitude as the plaid in the direction of the plaid's disparity. Equivalently again, it predicts a depth match when the plaid's disparity vector is consistent with the grating's disparity constraint line, or when

**A**•

**B**= |

**B**|

^{2}, where

**A**is the plaid's disparity vector and

**B**is the grating's perpendicular disparity vector.

^{4}This makes it unlikely that disparity signals from second-order features have a systematic effect on depth matches in our experiments. We checked this by calculating the disparities of second-order features at the sum and difference of the spatial frequencies of the components of our plaids. Their low spatial-frequency content makes these second-order features the most likely to have an influence (Delicato & Qian, 2005). We found no significant relationship between these disparities and those of the test gratings in our PSE data.