Vertical disparities influence the perception of 3D depth, but little is known about the neuronal mechanisms underlying this. One possibility is that these perceptual effects are mediated by an explicit encoding of two-dimensional disparity. Recently, J. C. A. Read and B. G. Cumming (2006) pointed out that current psychophysical and physiological evidence is consistent with a much more economical one-dimensional encoding. Almost all relevant information about vertical disparity could in theory be extracted from the activity of purely horizontal-disparity sensors. Read and Cumming demonstrated that such a 1D system would experience Ogle's induced effect, a famous illusion produced by vertical disparity. Here, we test whether the brain employs this 1D encoding, using a version of the induced effect stimulus that simulates the viewing geometry at infinity and thus removes the cues which are otherwise available to the 1D model. This condition was compared to the standard induced effect stimulus, presented on a frontoparallel screen at finite viewing distance. We show that the induced effects experienced under the two conditions are indistinguishable. This rules out the 1D model proposed by Read and Cumming and shows that vertical disparity, including sign, must be explicitly encoded across the visual field.

*V*depends on the receptive field size

*σ*.

*H*and

*V*, respectively, then a sensor tuned to a horizontal disparity

*H*

_{pref}will report an effective interocular correlation of

*C*

_{eff}=

*C*exp(−0.25[(

*H*−

*H*

_{pref})

^{2}+

*V*

^{2}] /

*σ*

^{2}) (Read & Cumming, 2006). Thus, the response of a single neuron in this model confounds horizontal disparity, vertical disparity, and interocular correlation, but the three can be distinguished by their different effects on the population as a whole. Horizontal disparity can be read off from the preferred disparity of the maximally responding sensors in the population, while the magnitude of vertical disparity can be deduced from the effective interocular correlation sensed by these maximally responding sensors. Vertical disparity can be distinguished from a genuine reduction in stimulus interocular correlation because vertical disparity affects predominantly the smallest receptive fields, whereas reductions in interocular correlation affect all scales equally. However, only the magnitude of vertical disparity, ∣

*V*∣, can be deduced from the activity in this 1D population, not the vertical disparity

*V*itself. Read and Cumming (2006) showed that this suffices to explain illusions such as the induced effect. In fact, they pointed out that under most circumstances the fully signed vertical disparity,

*V,*can be deduced from a knowledge of how the magnitude ∣

*V*∣ varies across the retina. In normal viewing, the pattern of vertical disparity across the retina is highly constrained by viewing geometry. Figures 2A and 2B show vertical disparity fields for two example eye positions. In each case, vertical disparity is zero along the horizontal retinal meridian and along a vertical line whose position depends on the gaze azimuth. The convergence angle controls the rate at which vertical disparity magnitude increases away from the “cross” formed by these two lines. Vertical disparity is positive in the 1st and 3rd, and negative in the 2nd and 4th, quadrants of this cross. Figures 2C and 2D show the effective interocular correlation sensed by the population tuned to the stimulus horizontal disparity, and Figures 2E and 2F show the vertical disparity magnitude reconstructed from this effective correlation. Although this only gives the magnitude, not the sign of vertical disparity, the sign can be deduced from the position relative to the “cross” of zero vertical disparity, as indicated by the symbols. Thus, the sign of vertical disparity anywhere in the retina can usually be deduced from the overall pattern of vertical disparity magnitude.

*on the retina*. This is convenient, given that cells in early visual cortex encode visual information in retinotopic coordinates. In this system, the directions “horizontal” and “vertical” on the retina are defined when the eyes are in

*primary position,*i.e., looking straight ahead to infinity (Figure 3A). With the eyes in primary position, the two retinal images of an object, such as the black dot at the corner of the square in Figure 3A, differ only in their horizontal coordinate in this coordinate system. Thus, whatever an object's position in space, it can have only horizontal disparity on the retina (blue vector in Figure 3B). When the eyes move away from primary position, this is no longer the case. An example is shown in Figures 3C and 3D, where the eyes are converging at 40°. Now, the images of the black dot differ both in their horizontal and vertical coordinates (blue vector in Figure 3D). In other words, the object has a vertical disparity on the retina. However, most physiologists have used “vertical disparity” to refer to vertical displacements on the computer screen used to display the stimuli. This produces a

*non-epipolar disparity,*i.e., one which could not be produced by any real object, given the current position of the eyes, but which can be produced experimentally. So for example we might arrange matters such that the left eye views the black dot at the bottom-right corner of the screen in Figure 3C, but the right eye views the dot color-coded green. Since the two dots are directly above one another on the screen, they have a purely vertical disparity on the screen. But as the green vector in Figure 3D shows, they project to the same vertical position on the retina. Thus, experimentally adding in vertical disparity

*on the screen*has produced a vertical disparity

*on the retina*of zero.

*on-screen*vertical disparity would remove this vertical disparity on the retina, enhancing the cells' response. Thus, the fact that Durand et al. (2002) reported preferred vertical disparities up to 0.6° (their Figure 3A) does not enable us to rule out the possibility that all cells were tuned to zero vertical disparity on the retina. To convert a cell's preferred on-screen vertical disparity into its preferred retinal disparity requires a knowledge of its precise location in the visual field, but this is not usually provided. Thus, existing physiological studies tell us little about the distribution of preferred disparities in visual cortex, especially in the periphery where the distinction between screen and retinal vertical disparity is most crucial.

*sign*of the pattern inverts depending on whether the image is expanded or compressed. The 1D population of Figure 1B cannot encode this sign. Yet, the classic induced effect is still experienced in this apparatus, with the direction of perceived slant depending on the sign of magnification (Backus et al., 1999). This at last is evidence that the visual system can measure the sign of vertical disparity, yet even this does not absolutely prove the existence of a 2D distribution of disparity detectors across the visual field. Since the stimuli were presented for several seconds, it remains possible that a signed measurement of vertical disparity is made only at the fovea, and a map of stimulus vertical disparity is built up by fixating different regions of the visual field. It is suggestive in this context that several authors report that vertical disparity illusions depend on long presentations, building up gradually over time (Allison, Howard, Rogers, & Bridge, 1998; Kaneko & Howard, 1997; Ogle, 1938; Westheimer, 1984). The analogous possibility for horizontal disparity was considered when stereopsis was first discovered: “It may be supposed that … [horizontal disparity] is appreciated by successively directing the point of convergence of the optic axes successively to a sufficient number of its points to enable us to judge accurately of its form” (Wheatstone, 1838). Wheatstone (1838) and countless others have presented compelling evidence that horizontal disparity is not measured solely via vergence, for example the fact that we can perceive multiple horizontal disparities even when a stimulus is presented too briefly to allow eye movements. Only one study to date has used short presentations in this apparatus (Banks et al., 2001). The results suggest that the sign of retinal vertical disparity is detected, but the study was not designed to address this, and this conclusion was not explicitly drawn by the authors.

*X*′

_{L},

*Y*′

_{L}) are the coordinates of a point on the virtual screen perpendicular to the left eye, then this point projects to coordinates (

*X*

_{L},

*Y*

_{L}) on the physical screen ( Figure 5), where

*Z*is the viewing distance (165 cm),

*θ*is half the vergence angle, and the origin of both coordinate systems is the fixation point. The analogous equations for the right eye are

*θ*depends on the interocular distance

*I*of the observer:

*θ*= arctan(

*I*/ (2

*Z*)). We used a value of 1.1° for all observers, corresponding to

*I*= 6.5 cm. Measured distances for our observers ranged from 6.2 to 6.5 cm, only slightly larger than the error on the measurement. For an observer whose interocular distance is in fact 5.0 cm (the bottom end of the adult distribution; Dodgson, 2004), running the experiment with

*I*= 6.5 cm introduces a maximum error of about a pixel at the edge of the image where the correction is largest. This is similar to the alignment error between the projectors.

*σ,*of the cumulative Gaussian was taken as the threshold. Confidence intervals on the fitted threshold were obtained by boot-strap resampling (Wichmann & Hill, 2001). Briefly, we simulated each experiment by using the fitted psychometric function to the original data (see Figure 8) as the model for the observer and using the same number of samples per point as in the original data. A new psychometric function was fitted to each set of simulated data and the value of the threshold (

*σ*′) was recorded. Vertical and horizontal black lines crossing the data points in Figure 9 represent the central 95% range of the distribution of 2000 simulated thresholds

*σ*′.

*σ*) estimated from the fitted psychometric functions for both viewing geometries are similar. These results confirm that the geometric effect produces a strong percept of a slanted surface, in both viewing conditions (“standard” vs. “infinite distance”).

*epipolar,*i.e., consistent with the binocular geometry. Thus, there is only one cell in this population which can be reporting the correct binocular correspondence at this point in the visual field; activity in the others must reflect false matches. The correspondence problem would be made easier to solve if neurons tuned to epipolar disparities were somehow boosted. This would only be possible if neurons were available tuned to a range of vertical disparities (or if neurons dynamically adjusted their disparity tuning so as to ensure they were tuned to the current epipolar geometry). Using epipolar geometry to aid stereo correspondence is routine in multiple-camera machine vision (Hartley & Zisserman, 2000), but it must be said there is as yet no evidence for it in human stereopsis.

*X*′ and

*Y*′, respectively, on the virtual screen. The red dot shows where the corresponding image has to be drawn on the physical screen (black line). It has coordinates (

*X, Y*) on the physical screen. We now derive Equation 1 relating (

*X*

_{L},

*Y*

_{L}) to (

*X*′

_{L},

*Y*′

_{L}) for the left eye. To do this, it will be useful to introduce a head-centered coordinates system (

*X*

_{H},

*Y*

_{H},

*Z*

_{H}) as indicated in Figure A1.

*X*

_{H}and

*Y*

_{H}are parallel to the coordinate axes on the physical screen; the

*Y*

_{H}therefore comes “out of the paper” towards the reader. The virtual white dot therefore has head-centered coordinates

*X*

_{H}=

*X*′

_{L}cos

*θ, Y*

_{H}=

*Y*′

_{L}, and

*Z*

_{H}=

*Z*+

*X*′sin

*θ,*where

*Z*is the distance to the screen and

*θ*is half the vergence angle. The nodal point of the left eye has coordinates

*X*

_{H}=

*Z*tan

*θ, Y*

_{H}=

*Z*

_{H}= 0. A line passing through the nodal point and the virtual image has vector equation

*Z*

_{H}=

*Z,*at

*λ*=

*Z*/ (

*Z*+

*X*′sin

*θ*). Substituting this value of

*λ*into Equation A1 gives us the coordinates of the red dot on the physical screen:

*X*

_{H}= −

*Z*tan

*θ*. A similar derivation then yields Equation 2.