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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.

No MeSH data available.


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Transformation of distances from physical to perspective space. Panels show stimuli (blue dots and lines) in physical space and their equivalents in perspective space (red dots and lines). Gray dots indicate the positions of the eyes. Panel (a) shows two sets of blue dots in a plane in physical space arranged along straight, frontal line-pieces at different distances from the viewer (big dots at short distance and small dots at long distance). Their perspective equivalents are shown in red. Line segments are numbered from 1 to 6. Ratios between interdot distances in perspective and physical space are shown at the bottom. The horizontal lines through the data indicate that ratios are constant for dots placed along a frontal line. Panel (b) shows similar sets of dots and line segments arranged along other orientations. Ratios between distances in perspective and physical space are shown at the bottom. Lines are best exponential fits to the data. Panel (c) shows the Pappus condition in physical (blue) and perspective (red) space.
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fig2-2041669515613672: Transformation of distances from physical to perspective space. Panels show stimuli (blue dots and lines) in physical space and their equivalents in perspective space (red dots and lines). Gray dots indicate the positions of the eyes. Panel (a) shows two sets of blue dots in a plane in physical space arranged along straight, frontal line-pieces at different distances from the viewer (big dots at short distance and small dots at long distance). Their perspective equivalents are shown in red. Line segments are numbered from 1 to 6. Ratios between interdot distances in perspective and physical space are shown at the bottom. The horizontal lines through the data indicate that ratios are constant for dots placed along a frontal line. Panel (b) shows similar sets of dots and line segments arranged along other orientations. Ratios between distances in perspective and physical space are shown at the bottom. Lines are best exponential fits to the data. Panel (c) shows the Pappus condition in physical (blue) and perspective (red) space.

Mentions: Figure 2 shows relations between distances and distances in physical and perspective space. Due to the finite vanishing point in perspective space, distances are shorter than their equivalents in physical space. Distances in perspective space decrease relative to distances in physical space with increasing egocentric distance. The graph at the bottom of Figure 2(a) illustrates the relationship by showing the highest ratio between distances in perspective and physical space at the shortest egocentric distance. Figure 2(b) shows the relationships for sizes of differently oriented line-pieces. The graph at the bottom indicates that ratios between perspective and physical sizes decrease approximately exponentially as a function of distance in the depth direction. Figure 2(a) and (b) shows that sizes are not invariant under translation and rotation in perspective space. The size ratios are consistent with experimental evidence that perceived sizes are underestimated (Gilinsky, 1951) and compression of size in depth relative to frontal size increases with distance (Li, Phillips, & Durgin, 2011). Perspective space as a model of visual space provides geometrical explanations for these experimental findings.Figure 2.


The Perspective Structure of Visual Space
Transformation of distances from physical to perspective space. Panels show stimuli (blue dots and lines) in physical space and their equivalents in perspective space (red dots and lines). Gray dots indicate the positions of the eyes. Panel (a) shows two sets of blue dots in a plane in physical space arranged along straight, frontal line-pieces at different distances from the viewer (big dots at short distance and small dots at long distance). Their perspective equivalents are shown in red. Line segments are numbered from 1 to 6. Ratios between interdot distances in perspective and physical space are shown at the bottom. The horizontal lines through the data indicate that ratios are constant for dots placed along a frontal line. Panel (b) shows similar sets of dots and line segments arranged along other orientations. Ratios between distances in perspective and physical space are shown at the bottom. Lines are best exponential fits to the data. Panel (c) shows the Pappus condition in physical (blue) and perspective (red) space.
© Copyright Policy - creative-commons
Related In: Results  -  Collection

License 1 - License 2 - License 3
Show All Figures
getmorefigures.php?uid=PMC5016827&req=5

fig2-2041669515613672: Transformation of distances from physical to perspective space. Panels show stimuli (blue dots and lines) in physical space and their equivalents in perspective space (red dots and lines). Gray dots indicate the positions of the eyes. Panel (a) shows two sets of blue dots in a plane in physical space arranged along straight, frontal line-pieces at different distances from the viewer (big dots at short distance and small dots at long distance). Their perspective equivalents are shown in red. Line segments are numbered from 1 to 6. Ratios between interdot distances in perspective and physical space are shown at the bottom. The horizontal lines through the data indicate that ratios are constant for dots placed along a frontal line. Panel (b) shows similar sets of dots and line segments arranged along other orientations. Ratios between distances in perspective and physical space are shown at the bottom. Lines are best exponential fits to the data. Panel (c) shows the Pappus condition in physical (blue) and perspective (red) space.
Mentions: Figure 2 shows relations between distances and distances in physical and perspective space. Due to the finite vanishing point in perspective space, distances are shorter than their equivalents in physical space. Distances in perspective space decrease relative to distances in physical space with increasing egocentric distance. The graph at the bottom of Figure 2(a) illustrates the relationship by showing the highest ratio between distances in perspective and physical space at the shortest egocentric distance. Figure 2(b) shows the relationships for sizes of differently oriented line-pieces. The graph at the bottom indicates that ratios between perspective and physical sizes decrease approximately exponentially as a function of distance in the depth direction. Figure 2(a) and (b) shows that sizes are not invariant under translation and rotation in perspective space. The size ratios are consistent with experimental evidence that perceived sizes are underestimated (Gilinsky, 1951) and compression of size in depth relative to frontal size increases with distance (Li, Phillips, & Durgin, 2011). Perspective space as a model of visual space provides geometrical explanations for these experimental findings.Figure 2.

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