Type de poste : PhD Thesis or Post-doc.
Lieu de travail : Sophia-Antipolis
Thème de recherche : Systèmes numériques
Projet : APICS
Inverse potential problems for MEG/EEG.
The inverse MEG/EEG (magnetoencephalography/electro-encephalography) problem that we consider consists in recovering the location of finitely many dipolar sources, located inside the brain, from electric and/or magnetic measurements on the scalp.
The APICS team has developed an approach to EEG for spherical models of the head which is based on function theory and approximation theory. However, processing magnetic data which may now be obtained from MEG equipments is necessary in order to handle all the information available and get more accuracy on the localization. While Maxwell's equations still model the behavior of the magnetic quantities, which are still solutions to Laplace's equation, a pre-processing of the data is needed in this case, namely, they must first be deconvolved.
From these data, a preliminary "cortical mapping" step has to be performed, that consists in transmitting the available (pointwise) data from the scalp onto the brain's boundary. To this extent, one can use either boundary elements methods or best approximation techniques by harmonic fields (bounded extremal problems in 3-D Hardy classes), for which only preliminary algorithms exist. Many theoretical and computational developments have to be carried out, in particular when the measurements are restricted to a single hemisphere of the outer boundary.
The proposed internship is mainly concerned with the above items. Also, an interesting extension is to consider more general geometries, like ellipsoidal ones. This raises the issue of solving 3D bounded extremal problems in non-spherical domains. For further references, as well as more details on the source localization techniques that are used, see the articles by Baratchart, Leblond, and Yattselev at the url: /ralyx.inria.fr/2006/Raweb/apics
In a first step, the applicant will be asked to study the magnetic model of MEG and to set up appropriate deconvolution methods in order to convert the available data into Dirichlet and/or Neumann ones for Laplace's equation. Next, emphasis will be put on bounded extremal problems on harmonic gradients. The algorithm we have in mind consist by and large in solving generalized Toeplitz and Hankel spectral equations in appropriate bases (e.g. spherical harmonics).
Modelling (Maxwell's equations), function theory, harmonic analysis, elliptic PDE's, numerical analysis; linux, matlab.
The internship will take place at INRIA Sophia-Antipolis Contacts: Juliette.Leblond@sophia.inria.fr, Laurent.Baratchart@sophia.inria.fr