Functions confined in space and spatial frequency include Gabor-like functions as well as intrinsically two-dimensional functions resembling "non-Cartesian" gratings

Abstract

Established theoretical reasons recommend functions that are limited in space and spatial frequency as models for receptive fields of visual neurons. On the basis of an interpretation of spread as variance and via analogy to quantum mechanics, the Gabor functions are often stated to be maximally limited in space and spatial frequency (Daugman, 1985). Some Gabor functions indeed resemble receptive fields in V1, but this view provides little insight into why V1 receptive fields have only a small number of lobes, nor into receptive field shape beyond V1. Here we consider an alternative interpretation of “limited in space and spatial frequency.” We consider a function to be “confined” in space (or spatial frequency) if it is unchanged by windowing in space (or spatial frequency). While no function can be simultaneously confined in both space and spatial frequency, there is a rigorous sense in which the 2-dimensional Hermite functions achieve simultaneous confinement as nearly as possible. The 2-dimensional Hermite functions are a complete basis set and form a natural hierarchy. The first levels of this hierarchy contain functions that resemble Gabor functions with a small number of lobes, and thus resemble V1 receptive fields. Further down the hierarchy are intrinsically 2-dimensional functions, some of which resemble the non-Cartesian gratings, to which some V4 neurons respond preferentially (Gallant, 1996). In addition to their many interesting mathematical properties, the two-dimensional Hermite functions allow for efficient (“sparse”) local synthesis of images, including natural scenes, faces, and letters. While we make no claim that this view suffices to account for receptive field structure, we suggest that it provides a framework for a principled study of receptive fields, and that it is useful to think of receptive fields (along the V1-to-V2-to-V4 pathway) as not only expanding, but also increasing in their combined space-bandwidth aperture.