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The affine Gaussian derivative model can in several respects be regarded as acanonical model for receptive fields over a spatial image domain: (i) it can bederived by necessity from scale-space axioms that reflect structural propertiesof the world, (ii) it constitutes an excellent model for the receptive fieldsof simple cells in the primary visual cortex and (iii) it is covariant underaffine image deformations, which enables more accurate modelling of imagemeasurements under the local image deformations caused by the perspectivemapping, compared to the more commonly used Gaussian derivative model based onderivatives of the rotationally symmetric Gaussian kernel.  This paper presents a theory for discretizing the affine Gaussian scale-spaceconcept underlying the affine Gaussian derivative model, so that scale-spaceproperties hold also for the discrete implementation.  Two ways of discretizing spatial smoothing with affine Gaussian kernels arepresented: (i) by solving semi-discretized affine diffusion equation, which hasderived by necessity from the requirements of a semi-group structure over scaleas parameterized by a family of spatial covariance matrices and obeyingnon-creation of new structures from any finer to any coarser scale in terms ofnon-enhancement of local extrema and (ii) approximating these semi-discreteaffine receptive fields by parameterized families of 3x3-kernels as obtainedfrom an additional discretization along the scale direction. The latterdiscrete approach can be optionally complemented by spatial subsampling atcoarser scales, leading to the notion of affine hybrid pyramids.  Using these theoretical results, we outline hybrid architectures for discreteapproximations of affine covariant receptive field families, to be used as afirst processing layer for affine covariant and affine invariant visualoperations at higher levels.

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