The geometry of the space of symmetric positive-definite matrices and its applications in smoothing positive-definite tensor fields

English

Séminaire Modèles et Algorithmes Déterministes: EDP-MOISE

15/12/2011 - 11:00 Mr Maher Moakher (Ecole Nationale d'Ingénieurs de Tunis) Salle 1 - Tour IRMA

We start by presenting the differential geometry of the space of symmetric positive-definite matrices. We give explicit forms of the metric tensor, Christoffel symbols, differential operators and geodesics on this Riemannian manifold. We then use the harmonic map and minimal immersion theories to construct three flows that drive a noisy field of symmetric positive-definite data into a smooth one. The harmonic map flow is equivalent to the heat flow or isotropic linear diffusion which smooths data everywhere. A modification of the harmonic flow leads to a Perona-Malik like flow which is a selective smoother that preserves edges. The minimal immersion flow gives rise to a nonlinear system of coupled diffusion equations with anisotropic diffusivity.