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Representation of local geometry in the visual system. (English) Zbl 0617.92024

It is shown that a convolution with certain reasonable receptive field (RF) profiles yields the exact partial derivatives of the retinal illuminance blurred to a specified degree. It is known that local geometry depends on the differential structure of the illuminance at given points and thus the partial derivatives of the illuminance are important. Therefore the authors consider the partial derivatives with respect to the spatial parameters throughout the multiresolution representation.
The relation is: the derivatives of the blurred illuminances are equal to the convolution of the original image with certain RF profiles that may aptly be called ”fuzzy derivatives”. Two-dimensional RF profiles are obtained through multiplication and addition in the same way as the partial derivatives.
Next, specific visual processors are considered using differential geometry and implementing the resulting expressions from analysis through the simple artifice of substituting RF’s for the partial derivatives. By replacing the illuminance with its third order jet extension, position dependent geometries are obtained. It is shown how such a representation can function as the substrate for ”point processors” computing geometrical features such as edge curvature.
A clear dichotomy between local and multilocal visual routines is obtained. The terms of the truncated Taylor series representing the jets are partial derivatives whose corresponding RF profiles closely mimic the well known units in the primary visual cortex. This description provides a novel means to understand and classify these units. A discussion ends this biomathematical paper, where the local geometry in terms of the Taylor series is studied.
Reviewer: T.Postelnicu

MSC:

91E30 Psychophysics and psychophysiology; perception
53B99 Local differential geometry
92Cxx Physiological, cellular and medical topics
91E99 Mathematical psychology
53A99 Classical differential geometry
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