Vertical disparity pooling and the induced effect

Attentional selection in judgments of stereo depth

Stereoscopic depth is most useful when it comes from relative rather than absolute disparities. However, the depth perceived from relative disparities can vary with stimulus parameters that have no connection with depth or are irrelevant to the task. We investigated observers' ability to judge the stereo depth of task-relevant stimuli while ignoring irrelevant stimuli. The calculation of depth from disparity differs for 1-D and 2-D stimuli and we investigated the role this difference plays in observers' ability to selectively process relevant information. We show that the presence of irrelevant disparities affects perceived depth differently depending on stimulus dimensionality. Observers could not ignore disparities of irrelevant stimuli when they judged the relative depth between a 1-D stimulus (a grating) and a 2-D stimulus (a plaid). Yet these irrelevant disparities did not affect judgments of the relative depth between 2-D stimuli. Two processes contributing to stereo depth were identified, only one of which computes depth from a horizontal disparity metric and permits attentional selection. The other uses all stimuli, relevant and irrelevant, to calculate an effective disparity direction for comparing disparity magnitudes. These processes produce inseparable effects in most data sets. Using multiple disparity directions and comparing 1-D and 2-D stimuli can distinguish them.

Perceived depth in non-transitive stereo displays

The separation between the eyes shapes the distribution of binocular disparities and gives a special role to horizontal disparities. However, for one-dimensional stimuli, disparity direction, like motion direction, is linked to stimulus orientation. This makes the perceived depth of one-dimensional stimuli orientation dependent and generally non-veridical. It also allows perceived depth to violate transitivity. Three stimuli, A, B, and C, can be arranged such that A > B (stimulus A is seen as farther than stimulus B when they are presented together) and B > C, yet A ⩽ C. This study examines how the visual system handles the depth of A, B, and C when they are presented together, forming a pairwise inconsistent stereo display. Observers' depth judgments of displays containing a grating and two plaids resolved transitivity violations among the component stimulus pairs. However, these judgments were inconsistent with judgments of the same stimuli within depth-consistent displays containing no transitivity violations. To understand the contribution of individual disparity signals, observers were instructed in subsequent experiments to judge the depth of a subset of display stimuli. This attentional instruction was ineffective; relevant and irrelevant stimuli contributed equally to depth judgments. Thus, the perceived depth separating a pair of stimuli depended on the disparities of the other stimuli presented concurrently. This context dependence of stereo depth can be approximated by an obligatory pooling and comparison of the disparities of one- and two-dimensional stimuli along an axis defined locally by the stimuli.

Latitude and longitude vertical disparities

The literature on vertical disparity is complicated by the fact that several different definitions of the term "vertical disparity" are in common use, often without a clear statement about which is intended or a widespread appreciation of the properties of the different definitions. Here, we examine two definitions of retinal vertical disparity: elevation-latitude and elevation-longitude disparities. Near the fixation point, these definitions become equivalent, but in general, they have quite different dependences on object distance and binocular eye posture, which have not previously been spelt out. We present analytical approximations for each type of vertical disparity, valid for more general conditions than previous derivations in the literature: we do not restrict ourselves to objects near the fixation point or near the plane of regard, and we allow for non-zero torsion, cyclovergence, and vertical misalignments of the eyes. We use these expressions to derive estimates of the latitude and longitude vertical disparities expected at each point in the visual field, averaged over all natural viewing. Finally, we present analytical expressions showing how binocular eye position-gaze direction, convergence, torsion, cyclovergence, and vertical misalignment-can be derived from the vertical disparity field and its derivatives at the fovea.

The third dimension in the primary visual cortex

Anatomical superposition of the cortical projections from the overlapping visual fields of the two eyes does not make it obvious how the disposition of objects in the third dimension is encoded. Hubel and Wiesel's demonstration that units in the primary visual cortex of the mammal respond preferentially to elongated contours of specific orientation encouraged the inquiry into whether binocular disparity might not similarly be represented as an attribute interdigitated within the orderly progression of position. When this was found to indeed be the case, this entrained a brisk research activity into the disparity of receptive fields of single units in the primary visual cortex and the influence on their response of the three-dimensional locations of outside world stimuli. That cells' preferred orientations covered the whole gamut whereas space perception required only horizontal disparity was an apparent paradox that needed resolution. A connection with an observer's stereoscopic performance was made by the discovery that cells in the primate primary visual cortex display good tuning to the disparity in random-dot stereograms. But a wide gap still remains between the properties of these cortical units and human stereo thresholds in simple target configurations, let alone depth judgments in which perceptual and cognitive factors enter. When the neural circuits in the primary visual cortex that are involved in processing depth are eventually traced in detail they will also need to have properties that allow for the plasticity in learning and experience.

Vertical-size disparities are temporally integrated for slant perception

We investigated temporal properties of vertical-size and horizontal-size disparity processing for slant perception. Subjects indicated perceived slants for a stereoscopic stimulus in which the two magnitudes of vertical-size or horizontal-size disparities were oscillated stepwise with various frequencies (from 0.2 to 10 Hz). For the stimulus with vertical-size disparity oscillation, two slants corresponding to the two magnitudes of disparity were perceived for low-frequency conditions, whereas only a static mean slant of the two slants was perceived for high frequencies (5 and 10 Hz). For the stimulus with horizontal-size disparity oscillation, two slants were perceived for all the temporal frequency conditions. These results indicate that temporal properties of vertical- and horizontal-size disparity processing are clearly different and vertical-size disparities are temporally integrated over a period of around 500 ms for slant perception.

Vertical-disparity gradients are processed independently in different depth planes

We examined the effects of vertical-disparity gradients on apparent depth curvature of textured surfaces. In Experiment 1, vertical disparities induced expected curvatures when the surface had a horizontal disparity of < +/-40.34'. A central row of elements, lacking vertical disparities, ceased to have the same apparent curvature as the surface when the horizontal disparity between row and surface exceeded +/-5'. In Experiment 2, vertical disparities were not pooled between superimposed surfaces separated by horizontal disparities > +/-10'. Thus, vertical-disparity gradients are not pooled over depth for curvature perception. Our results suggest that vertical disparities are used to determine distances to surfaces directly, rather than to estimate vergence.

Evidence for implication of primate area V1 in neural 3-D spatial localization processing

We investigated the neural mechanisms underlying visual localization in 3-D space in area V1 of behaving monkeys. Three different sources of information, retinal disparity, viewing distance and gaze direction, that participate in these neural mechanisms are being reviewed. The way they interact with each other is studied by combining retinal and extraretinal signals. Interactions between retinal disparity and viewing distance have been shown in foveal V1; we have observed a strong modulation of the spontaneous activity and of the visual response of most V1 cells that was highly correlated with the vergence angle. As a consequence of these gain effects, neural horizontal disparity coding is favoured or refined for particular distances of fixation. Changing the gaze direction in the fronto-parallel plane also produces strong gains in the visual response of half of the cells in foveal V1. Cells tested for horizontal disparity and orientation selectivities show gain effects that occur coherently for the same spatial coordinates of the eyes. Shifts in preferred disparity also occurred in several neurons. Cells tested in calcarine V1 at retinal eccentricities larger than 10 degrees , show that horizontal disparity is encoded at least up to 20 degrees around both the horizontal and vertical meridians. At these large retinal eccentricities we found that vertical disparity is also encoded with tuning profiles similar to those of horizontal disparity coding. Combinations of horizontal and vertical disparity signals show that most cells encode both properties. In fact the expression of horizontal disparity coding depends on the vertical disparity signals that produce strong gain effects and frequent changes in peak selectivities. We conclude that the vertical disparity signal and the eye position signal serve to disambiguate the horizontal disparity signal to provide information on 3-D spatial coordinates in terms of distance, gaze direction and retinal eccentricity. We suggest that the relative weight among these different signals is the determining factor involved in the neural processing that gives information on 3-D spatial localization.

Illusory depth perception of oblique lines produced by overlaid vertical disparity

Our visual system matches images from both eyes to establish a single view and stereo depth even when they contain a certain amount of vertical disparity. This paper demonstrates a new stereo effect showing an aspect of vertical disparity processing. When oblique lines without disparity are overlaid with sparse random dots with vertical disparity, the lines look closer or farther in depth. The characteristics of this stereo illusion were experimentally investigated. The results showed that the sign of the perceived depth of the oblique lines depended on the combination of the line orientation and the vertical disparity sign, and that the amount of perceived depth became larger as the line orientation became more horizontal. The depth illusion robustly existed even under conditions that ruled out eye movements (i.e., vertical vergence and cyclovergence) by local-parallel or brief presentations of the stereo figures. This phenomenon suggests that the visual system locally measures vertical disparity and is not simply tolerating a small amount of vertical disparity. Stereo capture of vertical disparity and horizontal matching after vertical image shifts were proposed as possible explanations for the depth illusion.

Understanding the cortical specialization for horizontal disparity

Because the eyes are displaced horizontally, binocular vision is inherently anisotropic. Recent experimental work has uncovered evidence of this anisotropy in primary visual cortex (V1): neurons respond over a wider range of horizontal than vertical disparity, regardless of their orientation tuning. This probably reflects the horizontally elongated distribution of two-dimensional disparity experienced by the visual system, but it conflicts with all existing models of disparity selectivity, in which the relative response range to vertical and horizontal disparities is determined by the preferred orientation. Potentially, this discrepancy could require us to abandon the widely held view that processing in V1 neurons is initially linear. Here, we show that these new experimental data can be reconciled with an initial linear stage; we present two physiologically plausible ways of extending existing models to achieve this. First, we allow neurons to receive input from multiple binocular subunits with different position disparities (previous models have assumed all subunits have identical position and phase disparity). Then we incorporate a form of divisive normalization, which has successfully explained many response properties of V1 neurons but has not previously been incorporated into a model of disparity selectivity. We show that either of these mechanisms decouples disparity tuning from orientation tuning and discuss how the models could be tested experimentally. This represents the first explanation of how the cortical specialization for horizontal disparity may be achieved.

Neurons in parafoveal areas V1 and V2 encode vertical and horizontal disparities

Stereoscopic vision mainly relies on binocular horizontal disparity (HD), and its cortical encoding is well established in the foveal representation of the visual field. The role of vertical disparity (VD) is more controversial. Thus far, in the monkey, very few studies have investigated the HD sensitivity beyond 5 degrees of retinal eccentricity and no evidence of a real encoding of VD exists in the parafoveal representation of areas V1 and V2. Using dynamic random dot stereograms, we have tested both HD and VD selectivities in the parafoveal representation of V1 (calcarine V1) and V2 (eccentricities > 10 degrees ) in a behaving monkey. HD and VD selectivities have been characterized using fitting with Gabor function. A large proportion of the tested cells were both HD and VD selective (47%) and, to a lesser extent, HD selective only (8%) or VD selective only (23%). We found a real encoding of VD, with the same diversity in the tuning profiles as described for HD, that cannot be assimilated to a simple perturbation of the HD matching process. Moreover, the VD encoding had a finer scale than the HD one, which is coherent with the smaller range of naturally occurring VD. For the HD encoding, both the percentage of selective cells and the tuning parameters were close to those reported in foveal V1. These results show that, at parafoveal eccentricities in V1 and V2, disparity detectors are tuned to both horizontal and vertical dimensions of the positional disparity existing between matched features in both retinas.

Is vertical disparity used to determine azimuth?

The azimuth of a stimulus relative to the head can be determined from an extra-retinal, eye-position signal plus an estimate of the retinal eccentricity of the image. Alternatively, azimuth could be determined from retinal-image information alone. Specifically, stimulus azimuth could be estimated from two derivatives of vertical disparity: vertical size ratio (which varies with azimuth), and the horizontal gradient of vertical size ratio (a measure of distance). Here we examine the determinants of perceived azimuth in viewing conditions that, theoretically, should favor the use of vertical disparity. We find no evidence that vertical disparity is used. Perceived azimuth was determined completely by felt eye position and the retinal eccentricity of the image.

Effects of horizontal and vertical additive disparity noise on stereoscopic corrugation detection

Stereoscopic corrugation detection in the presence of horizontal- and vertical- additive disparity noise was examined using a signal detection paradigm. Random-dot stereograms either represented a 3-D square-wave surface with various amounts of Gaussian-distributed additive disparity noise or had the same disparity values randomly redistributed. Stereoscopic detection of 2 arcmin peak amplitude corrugations was found to tolerate significantly greater amplitudes of vertical-disparity noise than horizontal-disparity noise--irrespective of whether the corrugations were horizontally or vertically oriented. However, this directional difference in tolerance to disparity noise was found to reverse when the corrugation and noise amplitudes were increased (so as to produce equivalent signal-to-noise ratios). These results suggest that horizontal- and vertical-disparity noise pose different problems for dot-matching and post-matching surface reconstruction as corrugation and noise amplitudes increase.

Adaptation to disparity but not to perceived depth

The purpose of the present study was to investigate whether adaptation can occur to disparity per se. The adapting stimuli were large random-dot patterns of which the two half-images were transformed such that the depth effects induced by the vertical transformations were nulled by horizontal transformations. Thus, the adapting stimuli were perceptually the same, whereas the disparity fields differed from each other. The adapting stimuli were presented for five minutes. During that period, the percept of a fronto-parallel surface did not change. After the adapting period, subjects perceived a thin untransformed strip as either slanted or curved depending on the adapting transformation. The thin strips provided negligible information about the vertical disparity field. In a forced-choice task we measured the amount of horizontal transformation that was required to null the acquired adaptation. We found that the amounts of horizontal transformation required to perceive the test strip fronto-parallel were significantly different from zero. We conclude that the visual system can adapt to disparity signals in the absence of a perceptual drive.

Strength of depth effects induced by three types of vertical disparity

The goal of the present study is to compare the strengths of depth effects induced by different types of vertical disparity. We use a nulling task, in which the depth effects induced by vertical disparity are nulled by horizontal disparity. The advantage of this method is that it prevents cue conflicts from arising between disparity and other depth cues. The ratios between horizontal and vertical disparity that evoke the percept of a fronto-parallel stimulus vary per type of vertical disparity. The ratios determined for vertical scale and vertical quadratic mix (vertical scale with a horizontal gradient) vary strongly across subjects. The ratios for vertical shear are constant, since all subjects needed the same amount of horizontal and vertical shear to perceive a fronto-parallel plane. In these experiments, one conflict cannot be avoided, namely the conflict between vertical disparity and oculomotor signals. This conflict may cause differential weighting of vertical disparity and oculomotor signals, which could explain the individual differences. The different ratios for different types of vertical disparity suggest that weighting is specific for each type of vertical disparity and the associated oculomotor signal.

How vertical disparities assist judgements of distance

The ratio of the vertical sizes of corresponding features in the two eyes' retinal images depends both on the associated object's distance and on its horizontal direction relative to the head (eccentricity). It is known that manipulations of vertical size ratio can affect perceived distance, size, depth and shape. We examined how observers use the vertical size ratio to determine the viewing distance. Do they use the horizontal gradient of vertical size ratio, or do they combine the vertical size ratio itself with the eccentricity at which it is found? Distance scaling (as measured by having subjects set an ellipsoid's size and shape to match a tennis ball) was no better when the judged object was 30 degrees to the right of the head (where vertical size ratios vary considerably with distance) than when it was located straight ahead. Distance scaling improved when vertical disparities were presented within larger visual fields, irrespective of where this was relative to the head. Our results support the proposal that subjects use the horizontal gradient of vertical size ratio to estimate the distance of an object that they are looking at.

Temporal aspects of slant and inclination perception

Linear transformations (shear or scale transformations) of either horizontal or vertical disparity give rise to the percept of slant or inclination. It has been proposed that the percept of slant induced by vertical size disparity, known as Ogle's induced-size effect, and the analogous induced-shear effect, compensate for scale and shear distortions arising from aniseikonia, eccentric viewing, and cyclodisparity. We hypothesised that these linear transformations of vertical disparity are processed more slowly than equivalent transformations of horizontal disparity (horizontal shear and size disparity). We studied the temporal properties of the stereoscopic slant and inclination percepts that arose when subjects viewed stereograms with various combinations of horizontal and vertical size or shear disparities. We found no evidence to support our hypothesis. There were no clear differences in the build-up of percepts of slant or inclination induced by step changes in horizontal size or shear disparity and those induced by step changes in vertical size or shear disparity. Perceived slant and inclination decreased in a similar manner with increasing temporal frequency for modulations of transformations of both horizontal and vertical disparity. Considerable individual differences were found and several subjects experienced slant reversal, particularly with oscillating stimuli. An interesting finding was that perceived slant induced by modulations of dilation disparity was in the direction of the vertical component. This suggests the vertical size disparity mechanism has a higher temporal bandwidth than the horizontal size disparity mechanism. However, conflicting perspective information may play a dominant role in determining the temporal properties of perceived slant and inclination.

An orientation anisotropy in the effects of scaling vertical disparities

Gårding et al. (Vis Res 1995;35:703-722) proposed a two-stage theory of stereopsis. The first uses horizontal disparities for relief computations after they have been subjected to a process called disparity correction that utilises vertical disparities. The second stage, termed disparity normalisation, is concerned with computing metric representations from the output of stage one. It uses vertical disparities to a much lesser extent, if at all, for small field stimuli. We report two psychophysical experiments that tested whether human vision implements this two-stage theory. They tested the prediction that scaling vertical disparities to simulate different viewing distances to the fixation point should affect the perceived amplitudes of vertically but not horizontally oriented ridges. The first used elliptical half-cylinders and the 'apparently circular cylinder' judgement task of Johnston (Vis Res 1991;31:1351-1360). The second experiment used parabolic ridges and the amplitude judgement task of Buckley and Frisby (Vis Res 1993;33:919-934). Both studies broadly confirmed the anisotropy prediction by finding that large scalings of vertical disparities simulating near distances had a strong effect on the perceived amplitudes of the vertically oriented stimuli but little effect on the horizontal ones. When distances > 25 cm were simulated there were no significant differential effects and various methodological reasons are offered for this departure from expectations.

Robust and optimal use of information in stereo vision

Differences between the left and right eye's views of the world carry information about three-dimensional scene structure and about the position of the eyes in the head. The contemporary Bayesian approach to perception implies that human performance in using this source of eye-position information can be analysed most usefully by comparison with the performance of a statistically optimal observer. Here we argue that the comparison observer should also be statistically robust, and we find that this requirement leads to qualitatively new behaviours. For example, when presented with a class of stereoscopic stimuli containing inconsistent information about eccentricity of gaze, estimates of this gaze parameter recorded from one robust ideal observer bifurcate at a critical value of stimulus inconsistency. We report an experiment in which human observers also show this phenomenon and we use the experimentally determined critical value to estimate the vertical acuity of the visual system. The Bayesian analysis also provides a highly reliable and biologically plausible algorithm that can recover eye positions even before the classic stereo-correspondence problem is solved, that is, before deciding which features in the left and right images are to be matched.

Temporal aspects of stereoscopic slant estimation: an evaluation and extension of Howard and Kaneko's theory

We investigated temporal aspects of stereoscopically perceived slant produced by the following transformations: horizontal scale, horizontal shear, vertical scale, vertical shear, divergence and rotation, between the half-images of a stereogram. Six subjects viewed large field stimuli (70 degrees diameter) both in the presence and in the absence of a visual reference. The presentation duration was: 0.1, 0.4, 1.6, 6.4 or 25.6 s. Without reference we found the following: rotation and divergence evoked considerable perceived slant in a number of subjects. This finding violates the recently published results of Howard and Kaneko. Slant evoked by vertical scale and shear was similar to slant evoked by horizontal scale and shear but was generally less. With reference we found the following: vertical scale and vertical shear did not evoke slant. Slant due to rotation and divergence was similar to slant due to horizontal scale and shear but was generally less. According to the theory of Howard and Kaneko, perceived slant depends on the difference between horizontal and vertical scale and shear disparities. We made their theory more explicit by translating their proposals into linear mathematical expressions that contain weighting factors that allow for both slant evoked by rotation or divergence, subject-dependent underestimation of slant and other related phenomena reported in the literature. Our data for all stimulus durations and for all subjects is explained by this 'unequal-weighting' extension of Howard and Kaneko's theory.

A computational model of depth perception based on headcentric disparity

It is now well established that depth is coded by local horizontal disparity and global vertical disparity. We present a computational model which explains how depth is extracted from these two types of disparities. The model uses the two (one for each eye) headcentric directions of binocular targets, derived from retinal signals and oculomotor signals. Headcentric disparity is defined as the difference between headcentric directions of corresponding features in the left and right eye's images. Using Helmholtz's coordinate systems we decompose headcentric disparity into azimuthal and elevational disparity. Elevational disparities of real objects are zero if the signals which contribute to headcentric disparity do not contain any errors. Azimuthal headcentric disparity is a 1D quantity from which an exact equation relating distance and disparity can be derived. The equation is valid for all headcentric directions and for all binocular fixation positions. Such an equation does not exist if disparity is expressed in retinal coordinates. Possible types of errors in oculomotor signals (six) produce global elevational disparity fields which are characterised by different gradients in the azimuthal and elevational directions. Computations show that the elevational disparity fields uniquely characterise both the type and size of the errors in oculomotor signals. Our model uses a measure of the global elevational disparity field together with local azimuthal disparity to accurately derive headcentric distance throughout the visual field. The model explains existing data on whole-field disparity transformations as well as hitherto unexplained aspects of stereoscopic depth perception.

Types of size disparity and the perception of surface slant

We examined (i) perceived slant of a textured surface about a vertical axis as a function of disparity magnitude for horizontal-size disparity, vertical-size disparity, and overall-size disparity; and (ii) interactions between patterns with various types and magnitudes of size disparity and superimposed or adjacent zero-disparity stimuli. Horizontal-size disparity produced slant which increased with increasing disparity, was enhanced by superimposed zero-disparity stimuli, and induced contrasting slant in superimposed or adjacent zero-disparity stimuli. Vertical-size disparity produced opposite slant (induced effect) which was reduced to near zero by a superimposed zero-disparity pattern and both patterns appeared as one surface. Adjacent vertical-size-disparity and zero-disparity patterns appeared as separate surfaces with a wide curved boundary. Overall-size disparity produced slant which was enhanced by a superimposed zero-disparity pattern and less so by a zero-disparity line, and induced more slant in a zero-disparity line than in a zero-disparity pattern. The results are discussed in terms of depth underestimation of isolated surfaces, depth enhancement, depth contrast, and the processing of deformation disparity.

Spatial limitation of vertical-size disparity processing

We investigated the upper limit of horizontal spatial modulation of vertical-size disparity in a textured surface for the perception of depth. In Experiment 1 subjects matched the appearance of a surface with modulated horizontal-size disparity to that of a surface with modulated vertical-size disparity. In Experiment 2 we determined the threshold amplitude of modulation of vertical-size disparity required for the perception of depth as a function of the spatial frequency of disparity modulation. The results indicate that sensations of depth are not elicited by modulations of vertical-size disparity of any amplitude at spatial frequencies higher than about 0.04 c/deg. We conclude that vertical disparities are averaged within about 20 deg-wide areas and suggest that this global measurement is used to scale local horizontal disparities for the perception of surface slant.

Spatial properties of shear disparity processing

We investigated whether vertical-shear disparity was extracted from the whole visual field or from a more local area and how global estimates of vertical disparity are derived. We also investigated the role of cyclovergence in processing shear disparity. Random-dot stereoscopic displays in various configurations were presented with horizontal-shear disparity, vertical-shear disparity or same-sign horizontal- and vertical-shear (rotation) disparity. Vertical-shear disparity introduced into only the right half of a 60 deg-wide display produced perceived inclination of the whole display when the center of shear was on the fovea, but did not produce inclination, either of the whole display or of a local area when the centre of shear was in an eccentric retinal position. A display containing dots with vertical-shear disparity mixed with dots with zero-disparity produced one inclined surface. Horizontal-shear disparity always produced inclination confined to the local area of disparity. Rotation disparity produced no inclination when introduced into the whole display, but when introduced with zero-disparity dots. It produced an inclined plane distinct from the plane defined by the zero-disparity dots. These results could be attributed to cyclovergence, which we therefore eliminated in our last experiment. We conclude that the perception of surface inclination is based on the difference between local horizontal-shear disparity and global vertical-shear disparity averaged over the whole visual field.

Stereoscopic depth perception and vertical disparity: neural mechanisms

The additivity assumption relates to the various stereo-disparity components in the vertical and horizontal meridians, each of which is assumed to be independent of the other, with the total disparity in each dimension being the linear sum of the separate components. Information about the position of the eyes provided by the corollary discharge leads to compensatory changes in the lateral geniculate nuclei whereby the angle of gaze disparity component at retinal level is offset by equal and opposite changes at geniculate level. These geniculate changes concern only eye position. Changes in the retinal images such as those produced by lenses (i.e. induced effect) are passed on to the cortex without modification at the geniculate level. Discrimination of the local depth disparity component can be achieved by subtracting the local vertical eccentricity component from the total horizontal disparity.

Relative size disparities and the perception of surface slant

Perceived slant produced by size disparities in random-dot displays was measured by tactile matching. For a 60 deg surface, slant produced by vertical-size disparity (the induced effect) was opposite to that produced by horizontal-size disparity. Overall-size disparity produced a little slant. With small displays, effects of horizontal and vertical disparities were reduced but not those of overall disparity. A zero-disparity surround increased effects of horizontal and overall disparities but reduced the induced effect. A mixture of horizontally disparate and zero-disparity dots produced two slanted surfaces. Vertically disparate and zero-disparity dots produced one slanted surface. Abutting opposite horizontal disparities produced surfaces with a sharp boundary. Abutting vertical disparities produced surfaces with a gradual boundary. Perceived slant depends on the difference between horizontal-size disparity detected locally and mean vertical-size disparity over a relatively large area.

Pooling of vertical disparities by the human visual system

Two experiments are described in which the effects of scaling vertical disparities on the perceived amplitudes of dome-shaped surfaces depicted with horizontal disparities were examined. The Mayhew and Longuet-Higgins's theory and the regional-disparity-correction theory of Garding et al predict that scaling should generate a change in perceived depth appropriate to the viewing distance simulated by the scaled vertical disparities. Significant depth changes were observed, by means of a nulling task in which the vertical-disparity-scaling effect was cancelled by the observer choosing a pattern of horizontal disparities that made the dome-shaped surface appear flat. The sizes of the scaling effects were less than those predicted by either theory, suggesting that other cues to fixation distance such as oculomotor information played an appreciable role. In conditions in which 50% of the texture elements were given one value of vertical-disparity scaling and the remaining 50% were left unscaled, the size of the scaling effect on perceived depth could be accounted for by equally weighted pooling of the vertical-disparity information unless the two scalings were very dissimilar, in which case the lower scaling factor tended to dominate. These findings are discussed in terms of a Hough parameter estimation model of the vertical-disparity-pooling process.

Interaction of stereo and texture cues in the perception of three-dimensional steps

A computational method for calibrating stereo using shape-from-texture is described together with five experiments that tested whether the human visual system implements the method. The experiments all tested the prediction that the perceived size of a step between two planar and slanted real surfaces should be affected by texture slant cues projected on to them that are inconsistent with the disparity cues. The predicted effect was observed but the results could be accounted for by a new phenomenon revealed in control conditions: the perceived size of a step between two slanted planes is in part determined by the size of the slants even when texture and stereo cues are held consistent. We conclude that the hypothesis that human stereo is calibrated by texture is not confirmed.

Stereopsis, vertical disparity and relief transformations

The pattern of retinal binocular disparities acquired by a fixating visual system depends on both the depth structure of the scene and the viewing geometry. This paper treats the problem of interpreting the disparity pattern in terms of scene structure without relying on estimates of fixation position from eye movement control and proprioception mechanisms. We propose a sequential decomposition of this interpretation process into disparity correction, which is used to compute three-dimensional structure up to a relief transformation, and disparity normalization, which is used to resolve the relief ambiguity to obtain metric structure. We point out that the disparity normalization stage can often be omitted, since relief transformations preserve important properties such as depth ordering and coplanarity. Based on this framework we analyse three previously proposed computational models of disparity processing; the Mayhew and Longuet-Higgins model, the deformation model and the polar angle disparity model. We show how these models are related, and argue that none of them can account satisfactorily for available psychophysical data. We therefore propose an alternative model, regional disparity correction. Using this model we derive predictions for a number of experiments based on vertical disparity manipulations, and compare them to available experimental data. The paper is concluded with a summary and a discussion of the possible architectures and mechanisms underling stereopsis in the human visual system.

Size constancy, depth constancy and vertical disparities: a further quantitative interpretation

The size and depth constancies considered here operate only at near distances (< about 2 m) in a static stimulus situation with vergence as the only cue to distance. The innervation of the extraocular muscles, as evidenced by the corollary discharge, provides information about the vergence of the eyes and hence about the egocentric distance both for symmetrical and asymmetrical vergences. Size and depth constancies are regarded as the first and second stages of a linked two-stage process. In the lateral geniculate nuclei compensatory adjustments are separately applied to each retinal image as they are received from the two eyes. The modified ocular images, with their associated vertical and horizontal disparities, now provide synaptic inputs to binocularly activated cells in the visual cortex. Then, by a process akin to the induced effect, cortical cells with geniculate afferents with vertical disparities will have their outputs expressed in terms of horizontal disparities. The horizontal disparity outputs of these cortical cells are then further multiplied by the outputs from cortical cells with geniculate afferents with horizontal disparities. It is this second multiplicative process that brings about the quadratic relationship between horizontal retinal disparity and egocentric distance. The proposed mechanisms involve the known ability of the visual system to detect and respond to vertical as well as horizontal disparities and provide a definite role for the induced effect in the perceptual process. The above neural model is based on fairly simple equations that give a remarkably adequate description of the operation of the two constancies.

Vertical disparities, differential perspective and binocular stereopsis

To calculate the depth difference between a pair of points on a three-dimensional surface from binocular disparities, it is necessary to know the absolute distance to the surface. Traditionally, it has been assumed that this information is derived from non-visual sources such as the vergence angle of the eyes. It has been shown that the horizontal gradient of vertical disparity between the images in the two eyes also contains information about the fixation distance. Recent results, however indicated that manipulations of the vertical disparity gradient have no effect on either the perceived shape or the perceived depth of surfaces defined by horizontal disparities. Following the reasoning of Longuet-Higgins and Tyler, we suggest that vertical disparities are best understood as a consequence of perspective viewing from two different vantage points and the results we report here show that the human visual system is able to exploit vertical disparities and use them to scale the perceived depth and size of stereoscopic surfaces, if the field of view is sufficiently large.

The recovery of structure from motion: no evidence for a special link with the convergent disparity mechanism

Convergent and divergent stereo mechanisms were compared in their ability to recover structure from motion. Contrary to a recent result reported by Richards and Lieberman, no difference in their performance was found; both mechanisms appeared equally capable of supporting the perception of good structure from motion. Possible reasons for the disparate results are discussed.

Monocular aniseikonia: a motion parallax analogue of the disparity-induced effect

Mayhew and Longuet-Higgins have recently outlined a computational model of binocular depth perception in which the small vertical disparities between the two eyes' views of a three-dimensional scene are used to determine the 'viewing parameters' of fixation distance (d) and the angle of asymmetric convergence of the eyes (g). The d/g hypothesis, as it has been called, correctly predicts that a fronto-parallel surface, viewed with a vertically magnifying lens over one eye, should appear to be rotated in depth about a vertical axis. We report here a comparable illusion for surfaces specified by monocular motion parallax information, which can be explained more simply by considering the differential invariants of the optic flow field. In addition, our observations suggest that the disparity-induced effect is not a 'whole field' phenomenon nor one limited to small magnification differences between the eyes.

Reinstatement of binocular depth perception by amphetamine and visual experience after visual cortex ablation

In adult cats with bilateral visual cortex ablation the complete deficit in binocular depth perception, as measured on a visual cliff, was reversed by 4 doses of amphetamine. The amphetamine-induced recovery endured after the amphetamine treatment was discontinued. This enduring recovery of function was not obtained if the animals were housed in the dark during drug intoxication. Therefore, both amphetamine intoxication and visual experience are simultaneously required for recovery of binocular depth perception after visual cortex ablation.