Center for Interdisciplinary Mathematics
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Previous talk

Time:Mar. 3, 2015, 4:00pm-5:00pm.
Place:106 McAllister Building.
Speaker: Ugo Boscain, CMAP, École Polytechnique CNRS

In this talk I will present a model of geometry of vision due to Petitot, Citti Sarti and our research group. One of the main features of this model is that the primary visual cortex V1 lifts an image from R^2 to the bundle of directions of the plane. Neurons are grouped into orientation columns, each of them corresponding to a point of this bundle.

In this model a corrupted image is reconstructed by minimizing the energy necessary for the activation of the orientation columns corresponding to regions in which the image is corrupted. The minimization process intrinsically defines an hypoelliptic heat equation on the bundle of directions of the plane.

The numerical integration of this equation is difficult and require techniques of non-commutative Fourier analysis.

The purpose of this research is to validate the biological model and to obtain an algorithm of image inpainting going beyond the state of the art.

[1] U. Boscain, J. Duplaix, J.P. Gauthier, F. Rossi, “Anthropomorphic image reconstruction via hypoelliptic diffusion”. SIAM J. CONTROL OPTIM.Vol. 50, No. 3, pp. 1309–1336, 2012.

[2] U. Boscain, R. Chertovskih, J.P. Gauthier, A. Remizov. Hypoelliptic diffusion and human vision: a semi-discrete new twist. SIAM Journal on Imaging Sciences 2014, Vol. 7, No. 2, pp. 669-695.