Joint covariance property under geometric image transformations for spatio-temporal receptive fields according to the generalized Gaussian derivative model for visual receptive fields

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 joint covariance property under compositions of spatial scaling transformations, spatial affine transformations, Galilean transformations and temporal scaling transformations, which makes it possible to characterize how different types of image transformations interact with each other. Specifically, the derived relations show how the receptive field parameters need to be transformed, in order to match the output from spatio-temporal receptive fields with the underlying spatio-temporal image transformations.