  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 for
spatio-temporal receptive fields in terms of spatio-temporal derivative
operators applied to spatio-temporally smoothed image data under compositions
of spatial scaling transformations, spatial affine transformations, Galilean
transformations and temporal scaling transformations. Specifically, 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.
  For this purpose, we also fundamentally extend the notion of scale-normalized
derivatives to affine-normalized derivatives, that are computed based on
spatial smoothing with affine Gaussian kernels, and analyze the covariance
properties of the resulting affine-normalized derivatives for the affine group
as well as for important subgroups thereof.
  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.
