A variational formulation of the sub-Riemannian model of the primary visual cortex


In this talk we present a new mathematical model for the human primary visual cortex V1 and its applications to image processing, based on the well-known sub-Riemannian Citti-Petitot-Sarti model. In this model the primary visual cortex is represented as the bundle $PT\mathbb R^2$ of directions of the plane, where each point corresponds to a neuron with both spatial location and local orientation preferences. This structure is then endowed with a sub-Riemannian metric, mimicking the long and short range connections between neurons.
The main novelty of ou approach is the definition of the lift of 2D images to this space, and more generally of image restoration tasks (e.g. denoising or inpainting), through a dissipation of the energy by the cortical structure. More precisely, we define a variational procedure that lifts a visual stimulus (i.e. an image) to the cortical state that both minimizes its sub-Riemannian $H^1$ norm (minimum effort of the cortex hypothesis) and at the same time re-projects back to the same image (consistency hypothesis). In sharp contrast with previous definitions of lifts (e.g. via Gabor wavelets), this method does not make use of an a-priori fixed lift procedure. This allows us to integrate into this procedure the sub-Riemannian structure of the cortex, which in turn, makes the lift sensitive to the curvature of objects composing the image, without resorting to any post-processing diffusion over the cortical surface.
We will focus, in particular, on the characterization of this lift as a wavelet-type transform lift where the wavelet is a highly anisotropic distribution on $\mathbb R^2$, which allows for a rigorous derivation of the so-called extra-classical receptive fields.
This is joint work with G. Peyré, J.-M. Mirebeau and A. Sarti.

Voss, Norway