This paper gives an in-depth theoretical analysis of the direction and speed selectivity properties of idealized models of the spatio-temporal receptive fields of simple cells and complex cells, based on the generalized Gaussian derivative model for visual receptive fields. According to this theory, the receptive fields are modelled as velocity-adapted affine Gaussian derivatives for different image velocities and different degrees of elongation.  By probing such idealized receptive field models of visual neurons to moving sine waves with different angular frequencies and image velocities, we characterize the computational models to a structurally similar probing method as is used for characterizing the direction and speed selective properties of biological neurons. It is shown that the direction selective properties become sharper with increasing order of spatial differentiation and increasing degree of elongation in the spatial components of the visual receptive fields. It is also shown that the speed selectivity properties are sharper for increasing order of spatial differentiation, while they are for the inclination angle $\theta = 0$ independent of the degree of elongation.

By comparison to results of neurophysiological measurements of direction and speed selectivity for biological neurons in the primary visual cortex, we find that our theoretical results are consistent with (i) velocity-tuned visual neurons that are sensitive to particular motion directions and speeds, and (ii) different visual neurons having broader vs. sharper direction and speed selective properties.  Our theoretical results in combination with results from neurophysiological characterizations of motion-sensitive visual neurons are also consistent with a previously formulated hypothesis that the simple cells in the primary visual cortex ought to be covariant under local Galilean transformations, so as to enable processing of visual stimuli with different motion directions and speeds.