Orientation selectivity properties for the affine Gaussian derivative and the affine Gabor models for visual receptive fields

This paper presents a theoretical analysis of the orientation selectivity of simple and complex cells that can be well modelled by the generalized Gaussian derivative model for visual receptive fields, with the purely spatial component of the receptive fields determined by oriented affine Gaussian derivatives for different orders of spatial differentiation. A detailed mathematical analysis is presented for the three different cases of either: (i) purely spatial receptive fields, (ii) space-time separable spatio-temporal receptive fields and (iii) velocity-adapted spatio-temporal receptive fields. Closed-form theoretical expressions for the orientation selectivity curves for idealized models of simple and complex cells are derived for all these main cases, and it is shown that the orientation selectivity of the receptive fields becomes more narrow, as a scale parameter ratio $\kappa$, defined as the ratio between the scale parameters in the directions perpendicular to vs. parallel with the preferred orientation of the receptive field, increases. It is also shown that the orientation selectivity becomes more narrow with increasing order of spatial differentiation in the underlying affine Gaussian derivative operators over the spatial domain. For comparison, we also present a corresponding theoretical orientation selectivity analysis for purely spatial receptive fields according to an affine Gabor model. The results from that analysis are consistent with the results obtained from the affine Gaussian derivative model,in the respect that the orientation selectivity becomes more narrow when making the receptive fields wider in the direction perpendicular to the preferred orientation of the receptive field.