All these features involve comparing some functions of the incoming intensity values at nearby spatio-temporal locations and this points very strongly to the notion of derivatives. Visual perception, computational or biological, depends upon the extraction from the raw flow of images incoming on the retina of a number of image features such as edges, corners, textures or directions of motion, at a variety of spatio-temporal scales. We also propose some preliminary ideas for putting our theory to the test by actual measurements of brain activity. It is surprising that the geometry that emerges from our investigations is not the usual Euclidean geometry but the much less familiar hyperbolic, non-Euclidean, geometry. Assuming that their representations are invariant to these transformations, we predict the appearance of specific patterns of activity which we call hyperbolic planforms. Geometric transformations of the retinal image, caused say by eye movements, affect these derivatives. It is based on the simple assumption that the spatial variations of the image intensity, its derivatives, are extracted and represented in some visual brain areas by populations of neurons that excite and inhibit each other according to the values of these derivatives. We propose a theory to account for an invariance that pertains to such image features as edges and textures. This must be the effect of the neuronal organization of the visual areas of our brains that succeed in maintaining in our consciouness a representation that seems to be protected from brutal variations. Our visual perception of the world is remarkably stable despite the fact that we move our gaze and body. If these patterns could be observed through brain imaging techniques they would reveal the built-in or acquired invariance of the neural organization to the action of the corresponding subgroups. These patterns can be described by what we call hyperbolic or H-planforms that are reminiscent of Euclidean planar waves and of the planforms that were used in previous work to account for some visual hallucinations. By studying the bifurcations of the solutions of the structure tensor equations, the analog of the classical Wilson and Cowan equations, under the assumption of invariance with respect to the action of these subgroups, we predict the appearance of characteristic patterns. This brings in the beautiful structure of its group of isometries and certain of its subgroups which have a direct interpretation in terms of the organization of the neural populations that are assumed to encode the structure tensor. We then extend the classical ring model to this case and show that its natural framework is the non-Euclidean hyperbolic geometry.
We study the specific problem of visual edges and textures perception and suggest that these features may be represented at the population level in the visual cortex as a specific second-order tensor, the structure tensor, perhaps within a hypercolumn. This allows us to make predictions about the kinds of patterns that should be observed in the activity of real brains through, e.g., optical imaging, and opens the door to the design of experiments to test these hypotheses. We propose to use bifurcation theory and pattern formation as theoretical probes for various hypotheses about the neural organization of the brain.
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