Orientation selectivity properties for integrated affine quasi quadrature models of complex cells

This paper presents an analysis of the orientation selectivity properties of idealized models of complex cells in terms of affine quasi quadrature measures, which combine the responses of idealized models of simple cells in terms of affine Gaussian derivatives by (i) pointwise squaring, (ii) summation of responses for different orders of spatial derivation and (iii) spatial integration. Specifically, this paper explores the consequences of assuming that the family of spatial receptive fields should be covariant under spatial affine transformations, thereby implying that the receptive fields ought to span a variability over the degree of elongation. We investigate the theoretical properties of three main ways of defining idealized models of complex cells and compare the predictions from these models to neurophysiologically obtained receptive field histograms over the resultant of biological orientation selectivity curves. It is shown that the extended modelling mechanisms lead to more uniform behaviour and a wider span over the values of the resultant that are covered, compared to an earlier presented idealized model of complex cells without spatial integration. More generally, we propose to, based on the presented results: (i) include an explicit variability over the degree of elongation of the receptive fields in functional models of complex cells, and that (ii) the suggested methodology with comparisons to biological orientation selectivity curves and orientation selectivity histograms could be used as a new tool to evaluate other computational models of complex cells in relation to biological measurements.