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The Perspective Structure of Visual Space

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ABSTRACT

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.

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Transformations from physical to perspective space. Panels show stimuli (blue dots and lines) in a plane of physical space and their equivalents in perspective space (red dots and lines). Gray dots indicate the positions of the eyes. Panel (a) shows a set of seven dots (blue) arranged along a straight line in physical space. The dots are lying on equidistant lines (blue) that vanish at infinity in the straight-ahead viewing direction. In perspective space the lines (red) converge to a finite vanishing point. Dots have identical egocentric directions in physical and perspective space (dashed blue lines). Panel (b) shows dots (blue) arranged along two parallel lines in physical space and their equivalents (red) in perspective space. Panel (c) shows dots (blue) arranged along two orthogonal lines in physical space and their non-orthogonal equivalents (red) in perspective space. For reasons of clarity the underlying directional and perspective lines are not drawn in panels (b) and (c).
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fig1-2041669515613672: Transformations from physical to perspective space. Panels show stimuli (blue dots and lines) in a plane of physical space and their equivalents in perspective space (red dots and lines). Gray dots indicate the positions of the eyes. Panel (a) shows a set of seven dots (blue) arranged along a straight line in physical space. The dots are lying on equidistant lines (blue) that vanish at infinity in the straight-ahead viewing direction. In perspective space the lines (red) converge to a finite vanishing point. Dots have identical egocentric directions in physical and perspective space (dashed blue lines). Panel (b) shows dots (blue) arranged along two parallel lines in physical space and their equivalents (red) in perspective space. Panel (c) shows dots (blue) arranged along two orthogonal lines in physical space and their non-orthogonal equivalents (red) in perspective space. For reasons of clarity the underlying directional and perspective lines are not drawn in panels (b) and (c).

Mentions: To explore the geometry of perspective spaces, computations were made on points and lines in physical space. Figure 1a shows the basic idea behind the computations. The Cartesian grid represents a plane in physical space that includes the centers of both eyes. The blue lines represent parallel and equidistant lines in the straight-ahead viewing direction in physical space. Lines converging to a vanishing point at finite distance in perspective space, which are associated with lines in parallel to the viewing direction in physical space, are dubbed perspective lines from now on. The red lines in Figure 1(a) are perspective lines associated with viewing in the straight-ahead direction. Each position in physical space is specified by an egocentric direction and distance. Using a polar coordinate system is appropriate for vision because directions and distances result from different processes. Information about directions follows from retinal and eye position signals whereas distances require interpretation of nonpositional properties of the retinal stimulus (Erkelens, 2012). In the computations we assume that directions of objects in perspective space are identical to their egocentric directions in physical space. The position of a stimulus in perspective space is computed from its position in physical space by finding the intersections between directional and perspective lines. In this way, the distances of stimuli are given by the structure of perspective space. The distance of the vanishing point determines the distance of stimuli in perspective space. The distance of the vanishing point represents the weighted sum of all depth cues. If depth cues such as disparity, size, blur, and ocular vergence together would indicate veridical depth, then the vanishing point would be located at infinity. Depth is not veridical in perspective space. The distance of the vanishing point indicates the underestimation of depth. Figure 1(a) shows that a straight line-piece in physical space (the row of blue lines) transfers to a straight line-piece in perspective space (the row of red dots and lines). Conserved straightness implies that collinearity is preserved in perspective space. Figure 1(b) shows that two parallel line-pieces in physical space are generally not parallel in perspective space. Fronto-parallel line-pieces are the exception. Implication of the directional differences is that parallelism is not preserved in perspective space. Both properties, that is, violated parallelism and preserved collinearity, are also properties of visual space. Evidence comes from experiments in which subjects were asked to set bars in parallel (Cuijpers et al., 2000) and collinear (Cuijpers et al., 2002). Figure 1(c) shows that angles between orthogonal lines in physical space are non-orthogonal in perspective space. Similar differences between physical and perspective angles were measured in a recent study in which subjects judged angles between bars oriented in depth (Erkelens, 2015b). Agreement between computations and experimental results regarding preserved collinearity, violated parallelism, and large angular deviations shows that perspective space is an attractive candidate model of visual space as far as directions and angles are concerned.Figure 1.


The Perspective Structure of Visual Space
Transformations from physical to perspective space. Panels show stimuli (blue dots and lines) in a plane of physical space and their equivalents in perspective space (red dots and lines). Gray dots indicate the positions of the eyes. Panel (a) shows a set of seven dots (blue) arranged along a straight line in physical space. The dots are lying on equidistant lines (blue) that vanish at infinity in the straight-ahead viewing direction. In perspective space the lines (red) converge to a finite vanishing point. Dots have identical egocentric directions in physical and perspective space (dashed blue lines). Panel (b) shows dots (blue) arranged along two parallel lines in physical space and their equivalents (red) in perspective space. Panel (c) shows dots (blue) arranged along two orthogonal lines in physical space and their non-orthogonal equivalents (red) in perspective space. For reasons of clarity the underlying directional and perspective lines are not drawn in panels (b) and (c).
© Copyright Policy - creative-commons
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC5016827&req=5

fig1-2041669515613672: Transformations from physical to perspective space. Panels show stimuli (blue dots and lines) in a plane of physical space and their equivalents in perspective space (red dots and lines). Gray dots indicate the positions of the eyes. Panel (a) shows a set of seven dots (blue) arranged along a straight line in physical space. The dots are lying on equidistant lines (blue) that vanish at infinity in the straight-ahead viewing direction. In perspective space the lines (red) converge to a finite vanishing point. Dots have identical egocentric directions in physical and perspective space (dashed blue lines). Panel (b) shows dots (blue) arranged along two parallel lines in physical space and their equivalents (red) in perspective space. Panel (c) shows dots (blue) arranged along two orthogonal lines in physical space and their non-orthogonal equivalents (red) in perspective space. For reasons of clarity the underlying directional and perspective lines are not drawn in panels (b) and (c).
Mentions: To explore the geometry of perspective spaces, computations were made on points and lines in physical space. Figure 1a shows the basic idea behind the computations. The Cartesian grid represents a plane in physical space that includes the centers of both eyes. The blue lines represent parallel and equidistant lines in the straight-ahead viewing direction in physical space. Lines converging to a vanishing point at finite distance in perspective space, which are associated with lines in parallel to the viewing direction in physical space, are dubbed perspective lines from now on. The red lines in Figure 1(a) are perspective lines associated with viewing in the straight-ahead direction. Each position in physical space is specified by an egocentric direction and distance. Using a polar coordinate system is appropriate for vision because directions and distances result from different processes. Information about directions follows from retinal and eye position signals whereas distances require interpretation of nonpositional properties of the retinal stimulus (Erkelens, 2012). In the computations we assume that directions of objects in perspective space are identical to their egocentric directions in physical space. The position of a stimulus in perspective space is computed from its position in physical space by finding the intersections between directional and perspective lines. In this way, the distances of stimuli are given by the structure of perspective space. The distance of the vanishing point determines the distance of stimuli in perspective space. The distance of the vanishing point represents the weighted sum of all depth cues. If depth cues such as disparity, size, blur, and ocular vergence together would indicate veridical depth, then the vanishing point would be located at infinity. Depth is not veridical in perspective space. The distance of the vanishing point indicates the underestimation of depth. Figure 1(a) shows that a straight line-piece in physical space (the row of blue lines) transfers to a straight line-piece in perspective space (the row of red dots and lines). Conserved straightness implies that collinearity is preserved in perspective space. Figure 1(b) shows that two parallel line-pieces in physical space are generally not parallel in perspective space. Fronto-parallel line-pieces are the exception. Implication of the directional differences is that parallelism is not preserved in perspective space. Both properties, that is, violated parallelism and preserved collinearity, are also properties of visual space. Evidence comes from experiments in which subjects were asked to set bars in parallel (Cuijpers et al., 2000) and collinear (Cuijpers et al., 2002). Figure 1(c) shows that angles between orthogonal lines in physical space are non-orthogonal in perspective space. Similar differences between physical and perspective angles were measured in a recent study in which subjects judged angles between bars oriented in depth (Erkelens, 2015b). Agreement between computations and experimental results regarding preserved collinearity, violated parallelism, and large angular deviations shows that perspective space is an attractive candidate model of visual space as far as directions and angles are concerned.Figure 1.

View Article: PubMed Central - PubMed

ABSTRACT

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.

No MeSH data available.


Related in: MedlinePlus