Axiomatic theory of receptive fields
Receptive field profiles registered by cell recordings have shown that mammalian vision has developed receptive fields tuned to different sizes and orientations in the image domain as well as to different image velocities in space-time. Corresponding cell recordings in the auditory system has shown that mammals have developed receptive fields tuned to different frequencies as well as temporal transients. This article describes normative theories that have been developed to explain these properties of sensory receptive fields based on structural properties of the environment. Beyond theoretical explanation of biological phenomena, these theories can also be used for computational modelling of biological receptive fields and for building algorithms for artificial perception based on sensory data.
Idealized models of visual receptive fields similar to those found in the retina, the lateral geniculate nucleus and the primary visual cortex of higher mammals can be derived in an axiomatic way from structural requirements on the first stages of visual processing that reflect symmetry properties of the surrounding world in combination with additional assumptions to ensure internally consistent image representations at multiple spatial and temporal scales. Specifically, idealized functional models for linear spatio-temporal receptive fields can be derived in a principled manner to constitute a combination of Gaussian derivatives over the spatial domain and either non-causal Gaussian derivatives or truly time-causal temporal scale-space kernels over the temporal domain:
where
Correspondingly, and with similar notation idealized functional models for spatial receptive fields can be expressed of the form
This model specifically generalizes the receptive field model in terms of Gaussian derivatives
from directional derivatives of rotationally Gaussian kernels to directional derivatives of affine Gaussian kernels .
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