  The affine Gaussian derivative model can in several respects be regarded as a
canonical model for receptive fields over a spatial image domain: (i) it can be
derived by necessity from scale-space axioms that reflect structural properties
of the world, (ii) it constitutes an excellent model for the receptive fields
of simple cells in the primary visual cortex and (iii) it is covariant under
affine image deformations, which enables more accurate modelling of image
measurements under the local image deformations caused by the perspective
mapping, compared to the more commonly used Gaussian derivative model based on
derivatives of the rotationally symmetric Gaussian kernel.
  This paper presents a theory for discretizing the affine Gaussian scale-space
concept underlying the affine Gaussian derivative model, so that scale-space
properties hold also for the discrete implementation.
  Two ways of discretizing spatial smoothing with affine Gaussian kernels are
presented: (i) by solving semi-discretized affine diffusion equation, which has
derived by necessity from the requirements of a semi-group structure over scale
as parameterized by a family of spatial covariance matrices and obeying
non-creation of new structures from any finer to any coarser scale in terms of
non-enhancement of local extrema and (ii) approximating these semi-discrete
affine receptive fields by parameterized families of 3x3-kernels as obtained
from an additional discretization along the scale direction. The latter
discrete approach can be optionally complemented by spatial subsampling at
coarser scales, leading to the notion of affine hybrid pyramids.
  Using these theoretical results, we outline hybrid architectures for discrete
approximations of affine covariant receptive field families, to be used as a
first processing layer for affine covariant and affine invariant visual
operations at higher levels.
