Relationships between the degrees of freedom in the affine Gaussian derivative model for visual receptive fields and 2-D affine image transformations, with application to covariance properties of simple cells in the primary visual cortex

When observing the surface patterns of objects delimited by smooth surfaces, the projections of the surface patterns to the image domain will be subject to substantial variabilities, as induced by variabilities in the geometric viewing conditions, and as generated by either monocular or binocular imaging conditions, or by relative motions between the object and the observer over time. To first order of approximation, the image deformations of such projected surface patterns can be modelled as local linearizations in terms of local 2-D spatial affine transformations. This paper presents a theoretical analysis of relationships between the degrees of freedom in 2-D spatial affine image transformations and the degrees of freedom in the affine Gaussian derivative model for visual receptive fields. For this purpose, we first describe a canonical decomposition of 2-D affine transformations on a product form, closely related to a singular value decomposition, while in closed form, and which reveals the degrees of freedom in terms of (i) uniform scaling transformations, (ii) an overall amount of global rotation, (iii) a complementary non-uniform scaling transformation and (iv) a relative normalization to a preferred symmetry orientation in the image domain. Then, we show how these degrees of freedom relate to the degrees of freedom in the affine Gaussian derivative model. Finally, we use these theoretical results to consider whether we could regard the biological receptive fields in the primary visual cortex of higher mammals as being able to span the degrees of freedom of 2-D spatial affine transformations, based on interpretations of existing neurophysiological experimental results.