Geometric Constraints of Visual Space.

Perspective space has been introduced as a computational model of visual space. The model is based on geometric features of visual space. The model has proven to describe a range of phenomena related to the visual perception of distance and size. Until now, the model lacks a mathematical description that holds for complete 3D space. Starting from a previously derived equation for perceived distance in the viewing direction, the suitability of various functions is analyzed. Functions must fulfill the requirement that straight lines, oriented in whatever direction in physical space, transfer to straight lines in visual space. A second requirement is that parallel lines oriented in depth in physical space, converge to a finite vanishing point in visual space. A rational function for perceived distance, compatible with the perspective-space model of visual space, satisfies the requirements. The function is unique. Analysis of alternative functions shows there is little tolerance for deviations. Conservation of the straightness of lines constrains visual space to having a single geometry. Visual space is described by an analytical function having one free parameter, that is, the distance of the vanishing point.

Multiple Photographs of a Perspective Scene Reveal the Principles of Picture Perception.

A picture is a powerful and convenient medium for inducing the illusion that one perceives a three-dimensional scene. The relative invariance of picture perception across viewing positions has aroused the interest of painters, photographers, and visual scientists. This study explores variables that may underlie the invariance. It presents a computational analysis of distances and directions in sets of two photographs of perspective scenes taken from different camera positions. Focal lengths of the lens and picture sizes are chosen such that the sizes of one of the familiar objects are equally large in both photographs. The selected object is perceived at the same distance in both photographs, independent of viewing distance, showing that pictorial distance is fully determined by angular size of the object. Pictorial distance is independent of camera position, focal length of the lens, and picture size. Distances and directions of pictorial objects are computed as a function of viewing distance, and compared with distances and directions of the physical objects as a function of camera position. The computations show that ratios between pictorial distances, directions, and angular sizes of objects in a photograph are constant, as a function of viewing distance. The constant ratios are proposed as the reason for invariance of picture perception over a range of viewing distances. Reanalysis of distance judgments obtained from the literature shows that perspective space, previously proposed as the model for visual space, is also a good model for pictorial space. The geometry of pictorial space contradicts some conceptions about picture perception.

Perspective Space as a Model for Distance and Size Perception.

In the literature, perspective space has been introduced as a model of visual space. Perspective space is grounded on the perspective nature of visual space during both binocular and monocular vision. A single parameter, that is, the distance of the vanishing point, transforms the geometry of physical space into that of perspective space. The perspective-space model predicts perceived angles, distances, and sizes. The model is compared with other models for distance and size perception. Perspective space predicts that perceived distance and size as a function of physical distance are described by hyperbolic functions. Alternatively, power functions have been widely used to describe perceived distance and size. Comparison of power and hyperbolic functions shows that both functions are equivalent within the range of distances that have been judged in experiments. Two models describing perceived distance on the ground plane appear to be equivalent with the perspective-space model too. The conclusion is that perspective space unifies a number of models of distance and size perception.

Equidistant Intervals in Perspective Photographs and Paintings.

Human vision is extremely sensitive to equidistance of spatial intervals in the frontal plane. Thresholds for spatial equidistance have been extensively measured in bisecting tasks. Despite the vast number of studies, the informational basis for equidistance perception is unknown. There are three possible sources of information for spatial equidistance in pictures, namely, distances in the picture plane, in physical space, and visual space. For each source, equidistant intervals were computed for perspective photographs of walls and canals. Intervals appear equidistant if equidistance is defined in visual space. Equidistance was further investigated in paintings of perspective scenes. In appraisals of the perspective skill of painters, emphasis has been on accurate use of vanishing points. The current study investigated the skill of painters to depict equidistant intervals. Depicted rows of equidistant columns, tiles, tapestries, or trees were analyzed in 30 paintings and engravings. Computational analysis shows that from the middle ages until now, artists either represented equidistance in physical space or in a visual space of very limited depth. Among the painters and engravers who depict equidistance in a highly nonveridical visual space are renowned experts of linear perspective.

Intracranial Recordings of Occipital Cortex Responses to Illusory Visual Events.

UNLABELLED: Ambiguous visual stimuli elicit different perceptual interpretations over time, creating the illusion that a constant stimulus is changing. We investigate whether such spontaneous changes in visual perception involve occipital brain regions specialized for processing visual information, despite the absence of concomitant changes in stimulation. Spontaneous perceptual changes observed while viewing a binocular rivalry stimulus or an ambiguous structure-from-motion stimulus were compared with stimulus-induced perceptual changes that occurred in response to an actual stimulus change. Intracranial recordings from human occipital cortex revealed that spontaneous and stimulus-induced perceptual changes were both associated with an early transient increase in high-frequency power that was more spatially confined than a later transient decrease in low-frequency power. We suggest that the observed high-frequency and low-frequency modulations relate to initiation and maintenance of a percept, respectively. Our results are compatible with the idea that spontaneous changes in perception originate from competitive interactions within visual neural networks. SIGNIFICANCE STATEMENT: Ambiguous visual stimuli elicit different perceptual interpretations over time, creating the illusion that a constant stimulus is changing. The literature on the neural correlates of conscious visual perception remains inconclusive regarding the extent to which such spontaneous changes in perception involve sensory brain regions. In an attempt to bridge the gap between existing animal and human studies, we recorded from intracranial electrodes placed on the human occipital lobe. We compared two different kinds of ambiguous stimuli, binocular rivalry and the phenomenon of ambiguous structure-from-motion, enabling generalization of our findings across different stimuli. Our results indicate that spontaneous and stimulus-induced changes in perception (i.e., "illusory" and "real" changes in the stimulus, respectively) may involve sensory regions to a similar extent.

The extent of visual space inferred from perspective angles.

Retinal images are perspective projections of the visual environment. Perspective projections do not explain why we perceive perspective in 3-D space. Analysis of underlying spatial transformations shows that visual space is a perspective transformation of physical space if parallel lines in physical space vanish at finite distance in visual space. Perspective angles, i.e., the angle perceived between parallel lines in physical space, were estimated for rails of a straight railway track. Perspective angles were also estimated from pictures taken from the same point of view. Perspective angles between rails ranged from 27% to 83% of their angular size in the retinal image. Perspective angles prescribe the distance of vanishing points of visual space. All computed distances were shorter than 6 m. The shallow depth of a hypothetical space inferred from perspective angles does not match the depth of visual space, as it is perceived. Incongruity between the perceived shape of a railway line on the one hand and the experienced ratio between width and length of the line on the other hand is huge, but apparently so unobtrusive that it has remained unnoticed. The incompatibility between perspective angles and perceived distances casts doubt on evidence for a curved visual space that has been presented in the literature and was obtained from combining judgments of distances and angles with physical positions.

Perception of Perspective Angles.

We perceive perspective angles, that is, angles that have an orientation in depth, differently from what they are in physical space. Extreme examples are angles between rails of a railway line or between lane dividers of a long and straight road. In this study, subjects judged perspective angles between bars lying on the floor of the laboratory. Perspective angles were also estimated from pictures taken from the same point of view. Converging and diverging angles were judged to test three models of visual space. Four subjects evaluated the perspective angles by matching them to nonperspective angles, that is, angles between the legs of a compass oriented in the frontal plane. All subjects judged both converging and diverging angles larger than the physical angle and smaller than the angles in the proximal stimuli. A model of shallow visual space describes the results. According to the model, lines parallel to visual lines, vanishing at infinity in physical space, converge to visual lines in visual space. The perceived shape of perspective angles is incompatible with the perceived length and width of the bars. The results have significance for models of visual perception and practical implications for driving and flying in poor visibility conditions.

The Perspective Structure of Visual Space.

Luneburg's model has been the reference for experimental studies of visual space for almost seventy years. His claim for a curved visual space has been a source of inspiration for visual scientists as well as philosophers. The conclusion of many experimental studies has been that Luneburg's model does not describe visual space in various tasks and conditions. Remarkably, no alternative model has been suggested. The current study explores perspective transformations of Euclidean space as a model for visual space. Computations show that the geometry of perspective spaces is considerably different from that of Euclidean space. Collinearity but not parallelism is preserved in perspective space and angles are not invariant under translation and rotation. Similar relationships have shown to be properties of visual space. Alley experiments performed early in the nineteenth century have been instrumental in hypothesizing curved visual spaces. Alleys were computed in perspective space and compared with reconstructed alleys of Blumenfeld. Parallel alleys were accurately described by perspective geometry. Accurate distance alleys were derived from parallel alleys by adjusting the interstimulus distances according to the size-distance invariance hypothesis. Agreement between computed and experimental alleys and accommodation of experimental results that rejected Luneburg's model show that perspective space is an appropriate model for how we perceive orientations and angles. The model is also appropriate for perceived distance ratios between stimuli but fails to predict perceived distances.

Computation and measurement of slant specified by linear perspective.

One of the striking features of vision is that we can experience depth in two-dimensional images. Since the Renaissance, artists have used linear perspective to create sensations of depth and slant. What is not known is how the brain measures linear perspective information from the retinal image. Here, an experimental technique and geometric computations were used to isolate slant related to linear perspective from slant induced by other cues. Grid stimuli, designed to induce strong impressions of slant, were sufficiently simple to allow accurate predictions on the basis of numeric computations. Measurement of slant about the vertical axis as functions of slant depicted on the screen and slant of the screen relative to the observer showed that linear perspective explained 95% of the slant judgments. Precision and accuracy of the judgments suggest a neural substrate that is able to make highly accurate comparisons between orientations of lines imaged at different retinal locations. The neural basis of slant from the linear perspective has not yet been clarified. Long-range connections in V1, however, and cells in V2, V4, lateral occipital cortex, and caudal intraparietal sulcus have features that suggest an involvement in slant perception.

Virtual slant explains perceived slant, distortion, and motion in pictorial scenes.

Virtual slants of obliquely viewed figures consisting of skewed grids, computed as a function of depicted slant and slant of the physical surface, were compared with perceived slants in both monocular and binocular viewing conditions. Computations were based on an assumption of parallelism. Perceived slant was well-correlated with virtual slant, even though slants were generally underestimated. The qualitative relationship between virtual slants and physical surface slants was variable, and liable to produce different degrees and directions of apparent rotation as well as of apparent compensation, depending on the depicted slant of the stimulus. Contributions of screen-related cues were small and mainly limited to small slants. Remarkably, binocular disparity did not have any effect on perceived slant. The results imply that many past findings of both transformation and (apparent) compensation in pictorial viewing may in principle be straightforwardly explained as a function of the virtual stimulus.

A dual visual-local feedback model of the vergence eye movement system.

Pure vergence movements are the eye movements that we make when we change our binocular fixation between targets differing in distance but not in direction relative to the head. Pure vergence is slow and controlled by visual feedback. Saccades are the rapid eye movements that we make between targets differing in direction. Saccades are extremely fast and controlled by a local, non-visual feedback loop. Usually, we change our fixation between targets that differ in both distance and direction. Then, vergence eye movements are combined with saccades. A number of models have been proposed to explain the dynamics of saccade-related vergence movements. The models have in common that visual input is ignored for the duration of the responses. This type of control is realistic for saccades but not for vergence. Here, I present computations performed to investigate if a model using dual visual and local feedback can replace the current models. Simulations and stability analysis lead to a model that computes an estimate of target vergence instead of retinal disparity and uses this signal as the main drive. Further analysis shows that the model describes the dynamics of pure vergence responses over the full physiological range, saccade-related vergence movements, and vergence adaptation. The structure of the model leads to new hypotheses about the control of vergence.

Stereo-vision: head-centric coding of retinal signals.

Stereo-vision is generally considered to provide information about depth in a visual scene derived from disparities in the positions of an image on the two eyes; a new study has found evidence that retinal-image coding relative to the head is also important.

The effect of changing size on vergence is mediated by changing disparity.

In this study, we investigated the effect of changing size on vergence. Erkelens and Regan (1986) proposed that this cue to motion in depth affects vergence in a similar way as it affects perception. The measured effect on vergence was small and we wondered why the vergence system would use changing size as an additional cue to changing disparity. To elucidate the effect of changing size on vergence, we used an annulus carrying both changing size and changing disparity signals to motion in depth. The cues were either congruent or signaled a different depth. The results showed that vergence was affected by changing size, however in an opposite way than that perception was affected. These results were incongruent with those reported by Erkelens and Regan (1986). We therefore additionally measured the effects on vergence of the individual parameters associated with changing size, i.e., stimulus area, retinal eccentricity, and luminance. Stimulus (retinal) eccentricity was inversely related to vergence gain. Luminance, on the other hand, had a smaller but positive relation to vergence gain. Thus, changing size affected the disparity signal two-fold: it changed the retinal location of the disparity signal and it changed the strength of the disparity signal (luminance change). These effects of changing size on disparity can explain both our results (change in retinal location of the disparity signal) and those of Erkelens and Regan (1986; change in luminance). We thus conclude that changing size did not in itself contribute to vergence, rather its effect on vergence was mediated by disparity.

The role of temporally coarse form processing during binocular rivalry.

Presenting the eyes with spatially mismatched images causes a phenomenon known as binocular rivalry-a fluctuation of awareness whereby each eye's image alternately determines perception. Binocular rivalry is used to study interocular conflict resolution and the formation of conscious awareness from retinal images. Although the spatial determinants of rivalry have been well-characterized, the temporal determinants are still largely unstudied. We confirm a previous observation that conflicting images do not need to be presented continuously or simultaneously to elicit binocular rivalry. This process has a temporal limit of about 350 ms, which is an order of magnitude larger than the visual system's temporal resolution. We characterize this temporal limit of binocular rivalry by showing that it is independent of low-level information such as interocular timing differences, contrast-reversals, stimulus energy, and eye-of-origin information. This suggests the temporal factors maintaining rivalry relate more to higher-level form information, than to low-level visual information. Systematically comparing the role of form and motion-the processing of which may be assigned to ventral and dorsal visual pathways, respectively-reveals that this temporal limit is determined by form conflict rather than motion conflict. Together, our findings demonstrate that binocular conflict resolution depends on temporally coarse form-based processing, possibly originating in the ventral visual pathway.

Removal of monocular interactions equates rivalry behavior for monocular, binocular, and stimulus rivalries.

When the two eyes are presented with conflicting stimuli, perception starts to fluctuate over time (i.e., binocular rivalry). A similar fluctuation occurs when two patterns are presented to a single eye (i.e., monocular rivalry), or when they are swapped rapidly and repeatedly between the eyes (i.e., stimulus rivalry). Although all these cases lead to rivalry, in quantitative terms these modes of rivalry are generally found to differ significantly. We studied these different modes of rivalry with identical intermittently shown stimuli while varying the temporal layout of stimulation. We show that the quantitative differences between the modes of rivalry are caused by the presence of monocular interactions between the rivaling patterns; the introduction of a blank period just before a stimulus swap changed the number of rivalry reports to the extent that monocular and stimulus rivalries were inducible over ranges of spatial frequency content and contrast values that were nearly identical to binocular rivalry. Moreover when monocular interactions did not occur the perceptual dynamics of monocular, binocular, and stimulus rivalries were statistically indistinguishable. This range of identical behavior exhibited a monocular (approximately 50 ms) and a binocular (approximately 350 ms) limit. We argue that a common binocular, or pattern-based, mechanism determines the temporal constraints for these modes of rivalry.

Monocular symmetry in binocular vision.

Human vision is highly sensitive to bilateral symmetry in 2-D images. It is, however, not clear yet whether this visual sensitivity relates to symmetry of 3-D objects or whether it relates to symmetry of the 2-D image itself. We used a stereoscopically presented stimulus and a 3-D bisection task that enable us to dissociate object symmetry from image symmetry. The bisection stimulus consisted of three parallel lines, of which two lines were located in one depth plane and the third line in another. Bisection judgments were different for horizontal and vertical lines, which can be explained by taking into account the distinct viewpoints of the left and right eyes for either of the visible sides of the 3-D object. Image symmetry from a monocular vantage point predicts 3-D bisection better than object symmetry. We conclude that observers use either of the two monocular 2-D images separately but not a single cyclopean view-nicely dovetailing with what they do when they assess both 3-D visual direction and 3-D shape-to assess 3-D symmetry.

Dichoptic masking and binocular rivalry share common perceptual dynamics.

Two of the strongest tools to manipulate visual awareness of potentially salient stimuli are binocular rivalry and dichoptic masking. Binocular rivalry is induced by presenting incompatible images to the two eyes over prolonged periods of time, leading to an alternating perception of the two images. Dichoptic masking is induced when two images are presented once in rapid succession, leading to the perception of just one of the images. Although these phenomena share some key characteristics, most notably the ability to erase from awareness potentially very salient stimuli, their relationship is poorly understood. We investigated the perceptual dynamics during long-lasting dynamic stimulation leading to binocular rivalry or dichoptic masking. We show that the perceptual dynamics during dichoptic masking conditions meet the classifiers used to classify a process as binocular rivalry; that is, (1) Levelt's 2nd proposition is obeyed; (2) perceptual dominance durations follow a gamma distribution; and (3) dominance durations are sequentially independent. We suggest that binocular rivalry and dichoptic masking may be mediated by the same inhibitory mechanisms.

A single motion system suffices for global-motion perception.

Global-motion perception is the perception of coherent motion in a noisy motion stimulus. Thresholds for coherent motion perception were measured for different combinations of signal and noise speeds. Previous research [Edwards, M., Badcock, D. R., & Smith, A. T. (1998). Independent speed-tuned global-motion systems. Vision Research, 38 (11), 1573-1580; Khuu, S. K., & Badcock, D. R. (2002). Global speed processing: evidence for local averaging within, but not across two speed ranges. Vision Research, 42 (28), 3031-3042.] showed that thresholds were elevated when signal and noise speeds were similar, but not when they were different. The regions of increased threshold values for low and high signal speeds showed little overlap. On the basis of this evidence two independent speed-tuned systems were proposed: one for slow and one for fast-motion. However, in those studies only two signal speeds were used. We expanded the results by measuring threshold-curves for four different signal speeds. Considerable overlap of the threshold-curves was found between conditions. These results speak against a bipartite global-motion system. Model simulations indicate that present and previous experimental results can be produced by a single motion system providing that the mechanisms within it are speed-tuned.

A single system explains human speed perception.

Motion is fully described by a direction and a speed. The processing of direction information by the visual system has been extensively studied; much less is known, however, about the processing of speed. Although it is generally accepted that the direction of motion is processed by a single motion system, no such consensus exists for speed. Psychophysical data from humans suggest two separate systems processing luminance-based fast and slow speeds, whereas neurophysiological recordings in monkeys generally show continuous speed representation, hinting at a single system. Although the neurophysiological findings hint at a single system, they remain inconclusive as only a limited amount of cells can be measured per study and, possibly, the putative different motion systems are anatomically separate. In three psychophysical motion adaptation experiments, we show that predictions on the basis of the two-motion system hypothesis are not met. Instead, concurrent modeling showed that both here-presented and previous data are consistent with a single system subserving human speed perception. These findings have important implications for computational models of motion processing and the low-level organization of the process.

A quantitative measure for the uniformity of fixation density: The Voronoi method.

In order to characterize the uniformity of fixation density, we propose aquantitative measure based on Voronoi diagrams, in which cells are defined around fixation locations. We examined how normalized cell size distributions are related to homogeneous and inhomogeneous fixation densities. Two possible measures for use with the Voronoi method are discussed. Both show good correlation with subjective visual evaluations of the uniformity of fixation densities. Not only are these measures objective and quantitative, they also have a simple intuitive meaning: They may be thought of as reflecting the clustering of fixations.