Date(s) : 07/03/2013 iCal
14 h 00 min - 15 h 00 min
Harmonic analysis on manifolds: recent applications in neuroimaging data By Julien Lefèvre, LSIS.
This talk aims at introducing some mathematical tools of harmonic analysis on non-euclidian domains\, typically Riemannian manifolds\, and to show illustrative applications for surfaces in R^3. In parallel I will refer to a discrete version of this framework by considering graph laplacian.\nFirst\, I will present classical results on eigenfunctions of Laplace-Beltrami operator\, sometimes called Manifold harmonics in computer graphics community. These harmonics must be seen as Fourier modes for an arbitrary domain that can be used for analysis/synthesis of signals living on the manifold. Besides\, these modes reveal important properties related to the underlying geometry which are related to a famous question raised by Kac « can one hear the shape of a drum ? » I will show also how to compute practically eigenfunctions and eigenvectors with Finite Elements Method in the case of triangulated meshes as well as unstructured point clouds.\nIn a second part I will focus on possible applications of harmonic analysis for 3D surfaces such as geometric filtering\, registration and clustering. Several topics in neuroanatomy will be addressed such as brain variability or cortical development. At last I will present some open issues that have emerged from observation of data and that can be formulated in terms of a mathematical conjecture.