When a surface covered with a regular texture is viewed in perspective, the projected texture provides a number of cues to 3D surface orientation. For oriented textures, one cue is perspective convergence: symmetry lines that are parallel along the surface project to lines that vary systematically in orientation. We investigated the contribution of perspective convergence to perception of 3D slant and tested whether slant from convergence depends on oriented spectral components. Subjects judged the sign of slant about a vertical axis of rotation. Textures were composed of filled circles in three spatial arrangements: a hex grid with symmetry lines at 0 and ±60 deg relative to the tilt direction (aligned condition), a hex grid with symmetry lines at 90 and ±30 deg (perpendicular condition), and random arrangements with similar average spacing (isotropic condition). The two hex grid textures differed in the amount of spectral energy present in the tilt direction (horizontal) but were otherwise closely matched. Slant discrimination thresholds for monocular stimuli were higher for isotropic textures than for either of the two hex grid textures and were higher for the perpendicular texture than for the aligned texture. In a second experiment, we measured the weight given to texture relative to binocular slant information for cue conflict stimuli (±5 deg). Weights were found to agree with individual subjects' monocular thresholds, in accordance with optimal estimation theory. We conclude that the visual system uses perspective convergence to perceive slant and that effective use of convergence requires the presence of spectral components aligned with the tilt direction.

*texture gradient*to refer to such variations. There is now a considerable body of work studying the role of texture in human vision, for perceiving both the 3D shape of curved surfaces (e.g., Cumming, Johnston, & Parker, 1993; Li & Zaidi, 2000, 2001, 2003; Todd & Akerstrom, 1987; Todd & Oomes, 2002; Todd, Oomes, Koenderink, & Kappers, 2004) and the 3D orientation of planar surfaces (e.g., Andersen, Braunstein, & Saidpour, 1998; Beck, 1960; Buckley, Frisby, & Blake, 1996; Knill, 1998a, 1998b; Passmore & Johnston, 1995; Rosas, Wichmann, & Wagemans, 2004; Saunders, 2003; Todd, Thaler, & Dijkstra, 2005). Texture has also been studied as a 3D cue in the area of computational and computer vision (Aloimonos & Swain, 1988; Blostein & Ahuja, 1989; Gårding, 1993; Ikeuchi, 1984; Knill, 1998c; Sakai & Finkel, 1995, 1997; Stevens, 1981b; Super & Bovik, 1995; Turner, Gerstein, & Bajcsy, 1991; Witkin, 1981).

*perspective convergence*. In the example shown in Figure 1, dominant orientations result from the symmetric arrangement of elements, which are organized into regular rows and diagonals. These symmetry lines are parallel along the surface, but due to perspective, their projections vary in orientation, converging toward a point on the horizon associated with the surface. As illustrated in Figure 2, there is a simple mathematical relation between the direction and amount of convergence in an image and the 3D slant and tilt of the surface

^{1}(see also Saunders & Knill, 2001). In these examples, the textures are highly regular, but textures can also have dominant orientations in a more statistical sense. To provide a perspective convergence cue, it suffices that a texture has some anisotropy that is homogeneous across the surface, such as that of wood grain.

*aligned*texture, dots were centered at points of a hexagon grid, with a center-to-center spacing of 1 cm (∼1.4 deg). The

*perpendicular*texture was identical to the aligned texture, except for it being rotated 90 deg within its plane. For the

*isotropic*textures, dot positions were determined by starting with a random sample drawn from a uniform distribution and then applying an iterative procedure to increase the uniformity of spacing to make uniformity of spacing similar to that of the two hex grid textures. This procedure consisted of simulating mutual repulsion between elements and iterating until the configuration stabilized. Opposite edges of the square region were treated as being connected (i.e., elements distributed on a torus) so that the resulting configuration would tile. The magnitude of simulated repulsion decreased as a Gaussian function of distance, with Gaussian width equal to half of the distance between neighboring points in a hex arrangement with the same area and density. Ten different texture samples were generated by this procedure, each with 256 points within a 14.9 × 14.9 cm square region (∼21 × 21 deg). On each trial in the isotropic condition, one sample from this set was randomly chosen and tiled to extend throughout the surface. The isotropic textures were also randomly rotated and translated on each trial to add further variability. Random translations were also added to the hex grid textures.

*σ*

_{1}was reliably larger than another threshold

*σ*

_{2}, we assumed that the likelihood function for each condition represented the true probability of getting the observed results (as a function of possible threshold values) and computed from these the probability of the null hypothesis,

*σ*

_{1}≤

*σ*

_{2}.

*p*< .01).

*s*

_{tex}) differed from the slant specified by stereo information (

*s*

_{st}) by ±5 deg. These cue conflicts were generated by a double-projection method (e.g., Hillis et al., 2004; Knill & Saunders, 2003, after Banks & Backus, 1998). Surface texture was first projected from a slant

*s*

_{tex}onto an image plane relative to a cyclopean eye. Then, the result was back projected onto a surface with slant

*s*

_{st}, resulting in a slightly distorted pattern of texture along the stereo-specified surface. The final binocular images were the left and right eyes' accurate perspective views of this surface.

*predicted*relative cue weights, given subjects' discrimination thresholds, under the assumption of optimal integration. The details of these analyses are given in 2.

*p*< .01). Subject R.W.S. also gave nonzero weight to texture information provided by isotropic textures, whereas subjects J.A.S. and D.A.M. relied entirely on stereo information in this condition (J.A.S.:

*p*= .14,

*ns*; D.A.M.:

*p*= .27,

*ns*). For all subjects except M.J.R., there was a significant difference between the weight given to aligned grid texture and the weight given to either the isotropic texture or the perpendicular grid texture (

*p*< .01). The difference between the weights given to perpendicular textures and isotropic textures was not reliable for these subjects (J.A.S.:

*p*= .12,

*ns*; R.W.S.:

*p*= .47,

*ns*; D.A.M.:

*p*= .17,

*ns*). Given the low texture weights, this lack of difference could likely be a floor effect.

*p*< .01). Subject D.A.M. showed intermediate results: a clear improvement in thresholds with the addition of stereo, with residual differences across the texture types in the direction observed previously. Thus, the observed thresholds for the binocular stimuli were generally consistent with the observed cue weights.

*minimized expected entropy*method. For a given trial, the probe slants and responses from previous trials in the same condition, {

*s*

_{ k},

*r*

_{ k}}, were used to estimate a posterior probability distribution

*P*(

*μ,σ*|

*s*

_{1},

*r*

_{1},

*s*

_{2},

*r*

_{2},…

*s*

_{ n},

*r*

_{ n}), where

*μ*is the PSE and

*σ*is the difference between the PSE and the 75% point. The next probe

*s*

_{ n + 1}was chosen to minimize the expected entropy, −

*p*log(

*p*), of the posttrial posterior function,

*P*(

*μ,σ*|

*s*

_{1},

*r*

_{1},

*s*

_{2},

*r*

_{2},…

*s*

_{ n + 1},

*r*

_{ n + 1}). The entropy cost function rewards probes that would be expected to result in a more peaked and concentrated posterior distribution over the space of possible combinations of

*μ*and

*σ,*consistent with the goal of estimating

*μ*and

*σ*with minimal bounds of uncertainty. There are only two possibilities for the next response, 0 or 1, and for each of these possibilities, one can compute what the new postresponse likelihood distribution would be, as well as its entropy. The expected value of entropy is simply a weighted average of the two possible results, where weights are proportional to their probabilities,

*P*(

*r*

_{ n + 1}= 0|

*s*

_{ n + 1}) and

*P*(

*r*

_{ n + 1}= 1|

*s*

_{ n + 1}). If

*μ*and

*s*were known, these probabilities would be directly determined by the model psychometric function. Thus, to estimate

*P*(

*r*

_{ n + 1}|

*s*

_{ n + 1}), we marginalized over

*μ*and

*σ,*using the posterior distribution computed from previous response history as an estimate of

*P*(

*μ,σ*):

*P*(

*r*

_{ n + 1}|

*s*

_{ n + 1},

*μ,σ*), rather than a more standard cumulative Gaussian, to simplify computation. The function was scaled to range from 0.025 to 0.975 rather than from 0 to 1, to reduce the effect of lapses of attention and guessing on the probe selection. Informal testing of the procedure revealed it to be highly efficient and robust. The space of possible bias and threshold values was discretely sampled to carry out the marginalization, with

*μ*sampled linearly from −30 and 30 deg and with

*σ*sampled quadratically from 0.25 to 36.

*S*

_{per}is a weighted average of the slants estimated from stereo information,

*S*

_{ster}, and from texture information,

*S*

_{tex}:

*S*

_{per}= 0 corresponds to the situation where stereo and texture information null each other, such that the perceived orientation from the combined slant information appears frontal. As illustrated in Figure 8, the combination of

*S*

_{tex}and

*S*

_{st}that produces a frontal percept is related to the relative magnitude of stereo and texture weights.

*w*

_{tex}and

*w*

_{st}sum to 1. This corresponds to ignoring the potential contributions from slant cues other than stereo or texture, such as accommodation. However, any such slant cues would be consistent with frontal plane—the orientation of the display monitors—and, therefore, would not be expected to affect what combination of texture and stereo cues null each other (Backus, Banks, van Ee, & Crowell, 1999).

*S*

_{tex}=

*S*

_{st}+ Δ

*S,*where Δ

*S*on each trial was ±5 deg. Perceived slant can therefore be expressed as:

*S*

_{st}in the same manner as in Experiment 1. As there could be systematic bias in an observer's internal standard for frontal, we did not interpret PSEs as a direct measure of where

*S*

_{per}= 0. Rather, we combined PSEs for positive and negative conflict stimuli to compute texture weights:

*S*

_{st}

^{+}and

*S*

_{st}

^{−}are the stereo slants that null perceived slant for Δ

*S*> 0 and Δ

*S*< 0, respectively (see, e.g., Backus & Matza-Brown, 2003).

*S*

_{tex}and

*S*

_{st}is assumed to be Gaussian, then the optimal Bayesian estimate of slant can be described as a weighted sum, as in Equation B1, with weights being inversely proportional to the variances of the noise (see Knill & Saunders, 2003):

*T*

_{tex}and

*T*

_{st}, can be taken as representing the standard deviations of the single-cue measurement noise,

*σ*

_{tex}and

*σ*

_{st}:

*T*

_{tex}, for the various texture types. However, we did not measure slant discrimination for stimuli that provide only stereo information, as was done in Knill and Saunders (2003) and Hillis et al. (2004). Instead, we inferred a stereo discrimination threshold

*T*

_{st}for each subject, based on the combined-cue discrimination thresholds observed in Experiment 2. If stereo and texture information are optimally integrated in a linear model (i.e., weights satisfy Equation B4) and if performance is limited primarily by sensory noise, then the thresholds for combined-cue stimuli can be predicted based on the thresholds for single-cue stimuli:

*T*

_{st + tex}for each of the three texture types. We inferred the stereo-only threshold for a subject,

*T*

_{st}, to be that which minimizes the sum squared error between the observed thresholds

*T*

_{st + tex}and the thresholds predicted by Equation B6 using the single-cue thresholds from Experiment 1 as

*T*

_{tex}. We then used this inferred stereo threshold,

*T*

_{st}, together with the texture-only thresholds,

*T*

_{tex}, for each texture condition to compute predicted texture cue weights,

*w*

_{tex}, from Equation B5.