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 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.