A Riemannian Geometry Theory of Three-Dimensional Binocular Visual Perception

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.

Osteopathic Palpation of the Heart

“…The same pathology involving cardiac tissue will impose morpho-functional modifications, with impaired ability to move [ 28 ]. Probably, to understand the cardiac movement dictated by the disposition of its fibers, one could trace the concepts of Riemannian differential geometry, where the geometries of curves with arbitrary dimensions are taken into consideration, knowing the measurement variables [ 29 ].…”

A Riemannian Geometry Theory of Synergy Selection for Visually-Guided Movement

“…In this paper we combine our previous separate applications of Riemannian geometry to action [5] and to vision [6] to develop a Riemannian geometry theory of computational processes required in the planning and execution of minimum-effort visually-guided movement synergies to achieve specified visual goals. In so doing we construct a somatosensoryhippocampal-visual map of the body and describe its instantiation in visuospatial memory.…”

“…For those wishing to venture further we include a tutorial appendix on the major concepts of this remarkable geometric tool. We also direct the reader to our two previous papers [5,6] where the separate applications of Riemannian geometry to synergy formation and to perceived visual space are given in full detail.…”

“…Section 3: Background. We summarize our previous two papers [5,6] concerning application of Riemannian geometry to analysis of action and vision. This includes a more detailed approach to the Riemannian geometry used in the theory.…”

“…Section 4: Here we take the previous place-encoded theory of visuospatial memory [6] in which visual images of the environment as seen from different places in the environment are stored in corresponding partitions of visuospatial memory and extend it to a posture-andplace-encoded theory in which visual images of one's own body seen in different postures are added to the visuospatial partitioning. The result is a geometric (fibre-bundle) structure of partitioned visuospatial memory that stores these place-and-posture-encoded visual images to provide a 3D representation of the environment and of the body in that environment as seen from any place and in any posture.…”