Discrete approximations of the affine Gaussian derivative model for visual receptive fields

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 nonenhancement of local extrema and (ii) approximating these semi-discrete affine receptive fields by parameterized families of 3× 3-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. For the first approach, we show how the solutions can be computed from 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 achieve higher-order discrete approximation of the angular dependency of the discrete affine Gaussian receptive fields. For the second approach, we analyse how the step length in the scale direction can be determined, given the requirements of a positive discretization. We do also show how discrete directional derivative approximations can be efficiently implemented to approximate affine Gaussian derivatives. 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.