Shape equivalence under perspective and projective transformations

Abstract

When a planar shape is viewed obliquely, it is deformed by a perspective deformation. If the visual system were to pick up geometrical invariants from such projections, these would necessarily be invariant under the wider class of projective transformations. Towhat extent can the visual system tell the difference between perspective and nonperspective but still projective deformations of shapes? To investigate this, observers were asked to indicate which of two test patterns most resembled a standard pattern. The test patterns were related to the standard pattern by a perspective or projective transformation, or they were completely unrelated, Performance was slightly better in a matching task with perspective and unrelated test patterns (92.6%) than in a projective-random matching task (88.8%). In a direct comparison, participants had a small preference (58.5%) for the perspectively related patterns over the projectively related ones. Preferences were based on the values of the transformation parameters (slant and shear). Hence, perspective and projective transformations yielded perceptual differences, but they were not treated in a categorically different manner by the human visual system. When do two shapes look alike? Obviously, when you take a shape and translate, reflect, or rotate it, it will still be perceived as the same shape (except perhaps for some highly familiar shapes, such as the outline of the United States when presented in a strange orientation; see Rock, 1973). Similarly, an object seen at different distances, although producing retinal images of different sizes, is perceived as having the same shape. Moreover, an object and its cast shadow or its projected image are usually considered shape equivalent.It is widely believed that shape constancy is based on the visual system's sensitivity to the geometric congruence under an increasingly larger set ofoperations in the above cases: from Euclidean, to similarity, to projective transformations. This kind ofinnate geometry can take the form of unconscious or automatic recovery of shape constancy based on implicit knowledge of the laws of geometry (e.g., Helmholtz, 1868Helmholtz, /1968Rock, 1983), or it can take the form of sensitivity to the invariants under each group of transformations (e.g., Cutting, 1986;Gibson, 1950).How large the set of operations can be for shapes before and after transformation to be considered perceptually equivalent is not clear yet. This may seem strange beThe work presented in this paper was supported by an Esprit Basic Research Action ("VIVA," or "Viewpoint-Invariant Visual Acquisition"), Grant F. (August 1995). The quality of the presentation has been improved thanks to helpful suggestions by James Cutting, Michael Kubovy, and two anonymous reviewers. Correspondence should be addressed to 1. Wagemans, Department of Psychology, University of Leuven, Tiensestraat 102, B-3000 Leuven, Belgium (e-mail: johan.wagemans@kuleuven.ac.be).cause psychological experiments could test specific predictions derived from a strong ma...

Minimal information to determine affine shape equivalence.

“…Even so, it is evident that an invariants-based approach can be useful, and its geometric foundations can lead to predictions that are testable and relevant. Thus, the idea of minimal information is based on a hierarchy of transformation geometries (which has found application in other contexts; e.g., Tittle, Todd, Perotti, & Norman, 1995), and this framework allows one to derive predictions about shape equivalence under different sets of transformations (e.g., perspective versus projective; Wagemans, Lamote, & Van Gool, 1997;also Foster, 1975).…”

A Mental Space Similarity Group Model of Shape Constancy

“…In our present experimental procedure, a standard form was presented together with a rotatedÂreflected image of the standard form, and the subject was required to respond to the sameÂdifferent object by an appropriate key-press. In contrast to this type of discrimination procedure was the three-pattern configuration procedure used by Wagemans et al (1997). They used two patterns below the standard pattern, and these two patterns were images of the standard form by perspective or projective transformations or of a new pseudo-random pattern.…”

“…However, Wagemans et al (1997) prefer to attribute their results to quantitative parametric differences rather than the fundamental qualitative difference between perspective and projective transformations. Another conclusion of their study shows that although compactness varies with parameter values, the observer's decision cannot be determined by a simple preference for similar compactness in shapes.…”

“…An investigation (Wagemans, Lamote, 6 Van Gool, 1997) was carried out to tackle the issue of perspective and projective invariance on shape equivalence. The results suggest that the visual system does not deal with perspectivities in a categorically different way from projectivities.…”

Can the Comparisons of Feature Locations Explain the Difficulty in Discriminating Mirror-Reflected Pairs of Geometrical Figures from Disoriented Identical Pairs?

“…Wagemans, Lamote and van Gool [22] compared the perceived resemblance of "projectively equivalent" pairs of figures with that of more specific "perspectively equivalent" pairs of figures. The results showed that people judged both types of pairs as identical, although there was a preference toward equivalent pairs after perspective transformation compared to equivalent pairs after projective transformation.…”