Canonical Coordinates for Retino-Cortical Magnification

Luc Florack

doi:10.3390/axioms3010070

Abstract

A geometric model for a biologically-inspired visual front-end is proposed, based on an isotropic, scale-invariant two-form field. The model incorporates a foveal property typical of biological visual systems, with an approximately linear decrease of resolution as a function of eccentricity, and by a physical size constant that measures the radius of the geometric foveola, the central region characterized by maximal resolving power. It admits a description in singularity-free canonical coordinates generalizing the familiar log-polar coordinates and reducing to these in the asymptotic case of negligibly-sized geometric foveola or, equivalently, at peripheral locations in the visual field. It has predictive power to the extent that quantitative geometric relationships pertaining to retino-cortical magnification along the primary visual pathway, such as receptive field size distribution and spatial arrangement in retina and striate cortex, can be deduced in a principled manner. The biological plausibility of the model is demonstrated by comparison with known facts of human vision.

Highlights

The visual system of humans interfaces the optical world via a sensorium that is characterized by receptive fields, i.e., light sensitive cells, of various sizes and aperture profiles
Scale space theory provides a foundation for a rigorous taxonomy and functional interpretation of visual receptive fields as non-infinitesimal differential operators
Retino-cortical magnification can be conveniently described in terms of canonical coordinates(recall that the primary visual cortex on one side of the brain represents a hemifield, the bounds on φ):

Summary

Introduction

The visual system of humans (and other mammalian species) interfaces the optical world via a sensorium (or visual front-end, the internal embodiment of the visual field) that is characterized by receptive fields, i.e., light sensitive cells, of various sizes and aperture profiles. No previously proposed theory seems to provide a principled account of the spatial organization of the entire retina, including the fovea centralis (the central, avascular zone of the retina with maximal acuity), where the log-polar model breaks down, due to its physically void singularity. This shortcoming is reflected in the design of log-polar mapping algorithms and space-variant cameras, in which one typically employs some heuristics to handle the transition between periphery and central retina.

Modeling the Sensorium

Modeling Retino-Cortical Magnification

Canonical Coordinates

Conclusion and Discussion