Differentiating between Affine and Perspective-Based Models for the Geometry of Visual Space Based on Judgments of the Interior Angles of Squares

Abstract

This paper attempts to differentiate between two models of visual space. One model suggests that visual space is a simple affine transformation of physical space. The other proposes that it is a transformation of physical space via the laws of perspective. The present paper reports two experiments in which participants are asked to judge the size of the interior angles of squares at five different distances from the participant. The perspective-based model predicts that the angles within each square on the side nearest to the participant should seem smaller than those on the far side. The simple affine model under our conditions predicts that the perceived size of the angles of each square should remain 90 • . Results of both experiments were most consistent with the perspective-based model. The angles of each square on the near side were estimated to be significantly smaller than the angles on the far side for all five squares in both experiments. In addition, the sum of the estimated size of the four angles of each square declined with increasing distance from the participant to the square and was less than 360 • for all but the nearest square.

New Approaches to 3D Vision

“…But Gibson's proposal has been given a variety of interpretations by the contemporary literature. Wagner and colleagues [ 237 ] suggest that ‘Gibson's … doctrine of realism implies that visual space should be strictly Euclidean … ’ Warren [ 238 ] advances an ‘affine’ interpretation: ‘humans do not in fact recover Euclidean structure—rather, they reliably perceive qualitative shape (hills, dales, courses and ridges), which is specified by the second-order differential structure of images.’ Finally, Tsao & Tsao [ 239 ] argue for a ‘topological’ approach.…”

“…Perspective space: Train tracks appear to converge as they recede in distance, suggesting that instead of an affine transform (which preserves parallel lines), visual space is a perspective projection of Euclidean space (with visual space converging to a vanishing point) [329][330][331][332][333]. Wagner et al [237] find that human judgements correspond better to a perspective projection than an affine transform. But what's so surprising about the perspective space account is just how shallow visual space appears to be.…”

Geometric Constraints of Visual Space

“…The two rules are treated as axioms for the mathematical theory of perspective space. The relationship between distances in the viewing direction described by equation ( 1 ) has been derived from just these two axioms in other studies ( Erkelens, 2017 ; Wagner et al, 2018 ).…”

A Riemannian Geometry Theory of Three-Dimensional Binocular Visual Perception

“…Since at least the eighteenth century, philosophers, artists and scientists have theorized on the nature of perceived visual space and various geometries have been proposed [4,5]. Beyond the simple demonstration above there has long been a wealth of formal experimental evidence to demonstrate that what we perceive is a warped transformation of physical space [2,3,6,7,8,9,10,11,12,13,14].…”

“…From the continued work of this group [21,22,23,24,25,26,27,28,29,30,31] along with that of other researchers, it is now apparent that experimental measures of the geometry of perceived visual space are not just task-dependent. They vary according to the many contextual factors that affect the spatial judgements that provide those measures [4,5,32]. Along with the nature of the task these can include what is contained in the visual stimuli, the availability of external reference frames, the setting (indoors vs. outdoors), cue conditions, judgement methods, instructions, observer variables such as age, and the presence of illusions.…”