This paper presents a theory for discretizing the affine Gaussian scale-space concept 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 as derived by necessity from the requirement of a semi-group structure over a continuum of scale parameters as parameterized by a family of spatial covariance matrices and obeying non-creation of new structures from any finer to any coarser scale as formalized by the requirement of non-enhancement of local extrema and (ii) a set of parameterized 3x3-kernels as derived from an additional discretization of the above theory along the scale direction and with the parameters of the kernels having a direct interpretation in terms of the covariance matrix of the composed discrete smoothing operation.

We show how convolutions with the first family of kernels can be implemented in terms of a closed form expression for the Fourier transform and analyse how a remaining degree of freedom in the theory can be explored to ensure a positive discretization and optionally also achieve higher-order discrete approximation of the angular dependency of the shapes of the affine Gaussian kernels.

We do also show how discrete directional derivative approximations can be efficiently implemented to approximate affine Gaussian derivatives as constituting a canonical model for receptive fields over a purely spatial image domain and with close relations to receptive fields in biological vision.