  This paper presents a theory for how geometric image transformations can be
handled by a first layer of linear receptive fields, in terms of true
covariance properties, which, in turn, enable geometric invariance properties
at higher levels in the visual hierarchy. Specifically, we develop this theory
for a generalized Gaussian derivative model for visual receptive fields, which
is derived in an axiomatic manner from first principles, that reflect symmetry
properties of the environment, complemented by structural assumptions to
guarantee internally consistent treatment of image structures over multiple
spatio-temporal scales.
  It is shown how the studied generalized Gaussian derivative model for visual
receptive fields obeys true covariance properties under spatial scaling
transformations, spatial affine transformations, Galilean transformations and
temporal scaling transformations, implying that a vision system, based on image
and video measurements in terms of the receptive fields according to this
model, can to first order of approximation handle the image and video
deformations between multiple views of objects delimited by smooth surfaces, as
well as between multiple views of spatio-temporal events, under varying
relative motions between the objects and events in the world and the observer.
  We conclude by describing implications of the presented theory for biological
vision, regarding connections between the variabilities of the shapes of
biological visual receptive fields and the variabilities of spatial and
spatio-temporal image structures under natural image transformations.
