The influence of natural image transformations on receptive field responses is crucial for modelling visual operations in computer vision and biological vision. In this regard, covariance properties with respect to geometric image transformations in the earliest layers of the visual hierarchy are essential for expressing robust image operations, and for formulating invariant visual operations at higher levels.

This paper defines and proves a set of joint covariance properties under compositions of spatial scaling transformations, spatial affine transformations, Galilean transformations and temporal scaling transformations, which make it possible to characterize how different types of image transformations interact with each other and the associated spatio-temporal receptive field responses. In this regard, we also extend the notion of scale-normalized derivatives to affine-normalized derivatives, to be able to obtain true affine-covariant properties of spatial derivatives, that are computed based on spatial smoothing with affine Gaussian kernels.

The derived relations show how the parameters of the receptive fields need to be transformed, in order to match the output from spatio-temporal receptive fields under composed spatio-temporal image transformations. As a side effect, the presented proof for the joint covariance property over the integrated combination of the different geometric image transformations also provides specific proofs for the individual transformation properties, which have not previously been fully reported in the literature.

We conclude with a geometric analysis, showing how the derived joint covariance properties make it possible to relate or match spatio-temporal receptive field responses, when observing, possibly moving, local surface patches from different views, under locally linearized perspective or projective transformations, as well as when observing different instances of spatio-temporal events that may occur either faster or slower between different views of similar spatio-temporal events. We do furthermore describe how the parameters in the studied composed spatio-temporal image transformation models directly relate to geometric entities in the image formation process and the 3-D scene structure.

In relation to these geometric interpretations, the derived explicit transformation properties for receptive field responses, defined in terms of spatio-temporal derivatives of the underlying covariant spatio-temporal smoothing kernels, do specifically show how to both interpret and relate spatio-temporal receptive field responses, when viewing dynamic scenes under different composed geometric viewing conditions.

Specifically, we propose that this theoretical analysis should have direct relevance, when interpreting the functional properties of biological receptive fields, both computationally and with regard to how the simple cells in the primary visual cortex, whose functional properties we here model with an idealized axiomatically derived spatio-temporal receptive field model. From the viewpoint of the here presented theory, in combination with previous biological modelling results that demonstrate a very good qualitative agreement between idealized receptive field models according to this theory and neurophysiological recordings of actual biological receptive fields in the primary visual cortex of higher mammals, the shapes of these joint spatio-temporal receptive fields can, from this viewpoint, be regarded as very well adapted to the structural properties of the environment.