Abstract
We present a Riemannian geometry theory to examine the systematically warped geometry of perceived visual space attributable to the size–distance relationship of retinal images associated with the optics of the human eye. Starting with the notion of a vector field of retinal image features over cortical hypercolumns endowed with a metric compatible with that size–distance relationship, we use Riemannian geometry to construct a place-encoded theory of spatial representation within the human visual system. The theory draws on the concepts of geodesic spray fields, covariant derivatives, geodesics, Christoffel symbols, curvature tensors, vector bundles and fibre bundles to produce a neurally-feasible geometric theory of visuospatial memory. The characteristics of perceived 3D visual space are examined by means of a series of simulations around the egocentre. Perceptions of size and shape are elucidated by the geometry as are the removal of occlusions and the generation of 3D images of objects. Predictions of the theory are compared with experimental observations in the literature. We hold that the variety of reported geometries is accounted for by cognitive perturbations of the invariant physically-determined geometry derived here. When combined with previous description of the Riemannian geometry of human movement this work promises to account for the non-linear dynamical invertible visual-proprioceptive maps and selection of task-compatible movement synergies required for the planning and execution of visuomotor tasks.
Highlights
From the time of Euclid (300 BC) onwards builders and surveyors and the like have found the three-dimensional (3D) world in which they function to be adequately described by the theorems of Euclidean geometry
While the retinal image itself changes from one viewpoint to another, the geometry of the 3D perceived visual space derived from stereoscopic vision with estimates of Euclidean depth based on triangulation remains the same regardless of the scene and of the place of the head in the environment
Given an internal reference metric that can be moved to any point in (G, g), Equation (36) provides an implementation of a perceptual tape measure able to measure the actual size of perceived objects and distances between points in the outside world taking the warped geometry of the perceived visual space into account
Summary
From the time of Euclid (300 BC) onwards builders and surveyors and the like have found the three-dimensional (3D) world in which they function to be adequately described by the theorems of Euclidean geometry. Rather than rejecting the existence of a geometrically invariant perceived visual space we suggest that the various measured geometries are accounted for by top-down cognitive mechanisms perturbing the underlying invariant geometry derivable mathematically from the size–distance relationship between the size of the image on the retina and the Euclidean distance between the nodal point of the eye and the object in the environment. This relationship is attributable to the anatomy of the human eye functioning as an optical system. Together with similar descriptions in our previous paper [52], can assist in making the power and the elegance of this remarkable geometry accessible to an interdisciplinary readership
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