Section: New Results

Source recovery problems

Participants : Laurent Baratchart, Kateryna Bashtova, Sylvain Chevillard, Juliette Leblond, Dmitry Ponomarev.

This section is concerned with inverse problems for 3-D Poisson-Laplace equations. Though the geometrical settings differ in the 2 sections below, the characterization of silent sources (that give rise to a vanishing potential at measurement points) is one of the common problems to both which has been recently achieved in the magnetization setup, see [14] .

Application to EEG

This work is conducted in collaboration with Maureen Clerc and Théo Papadopoulo from the Athena Project-Team, and with Jean-Paul Marmorat (Centre de mathématiques appliquées - CMA, École des Mines de Paris).

In 3-D, functional or clinical active regions in the cortex are often modeled by point-wise sources that have to be localized from measurements on the scalp of a potential satisfying a Laplace equation (EEG, electroencephalography). In the work [3] it was shown how to proceed via best rational approximation on a sequence of 2-D disks cut along the inner sphere, for the case where there are at most 2 sources. Last year, a milestone was reached in the research on the behavior of poles in best rational approximants of fixed degree to functions with branch points [6] , to the effect that the technique carries over to finitely many sources (see Section  4.2 ).

In this connection, a dedicated software “FindSources3D” is being developed, in collaboration with the team Athena and the CMA. We took on this year algorithmic developments, prompted by recent and promising contacts with the firm BESA (see Section  5.6 ), namely automatic detection of the number of sources (which is left to the user at the moment) and simultaneous processing of data from several time instants. It appears that in the rational approximation step, multiple poles possess a nice behavior with respect to branched singularities. This is due to the very physical assumptions on the model (for EEG data, one should consider triple poles). Though numerically observed in [8] , there is no mathematical justification so far why multiple poles generate such strong accumulation of the poles of the approximants. This intriguing property, however, is definitely helping source recovery. It is used in order to automatically estimate the “most plausible” number of sources (numerically: up to 2, at the moment).

In connection with the work [14] related to inverse magnetization issues (see Section  6.1.2 ), the characterization of silent sources for EEG has been carried out [42] . These are sums of (distributional) derivatives of Sobolev functions vanishing on the boundary.

In a near future, magnetic data from MEG (magneto-encephalography) will become available along with EEG data; indeed, it is now possible to use simultaneously corresponding measurement devices, in order to measure both electrical and magnetic fields. This should enhance the accuracy of our source recovery algorithms.

Let us mention that discretization issues in geophysics can also be approached by such techniques. Namely, in geodesy or for GPS computations, one is led to seek a discrete approximation of the gravitational potential on the Earth's surface, from partial data collected there. This is the topic of a beginning collaboration with physicist colleagues (IGN, LAREG, geodesy). Related geometrical issues (finding out the geoid, level surface of the gravitational potential) are worthy of consideration as well.

Magnetization issues

This work is carried out in the framework of the “équipe associée Inria” IMPINGE, comprising Eduardo Andrade Lima and Benjamin Weiss from the Earth Sciences department at MIT (Boston, USA) and Douglas Hardin and Edward Saff from the Mathematics department at Vanderbilt University (Nashville, USA),

Localizing magnetic sources from measurements of the magnetic field away from the support of the magnetization is the fundamental issue under investigation by IMPINGE The goal is to determine magnetic properties of rock samples (e.g. meteorites or stalactites) from fine field measurements close to the sample that can nowadays be obtained using SQUIDs (supraconducting coil devices). Currently, rock samples are cut into thin slabs and the magnetization distribution is considered to lie in a plane, which makes for a somewhat less indeterminate framework than EEG as regards inverse problems because “less” magnetizations can produce the same field (for the slab has no inner volume).

The magnetization operator is the Riesz potential of the divergence of the magnetization, see (1 ). Last year, the problem of recovering a thin plate magnetization distribution from measurements of the field in a plane above the sample led us to an analysis of the kernel of this operator, which we characterized in various functional and distributional spaces [14] . Using a generalization of the Hodge decomposition, we were able to describe all magnetizations equivalent to a given one. Here, equivalent means that the magnetizations generate the same field from above and from below if, say, the slab is horizontal. When magnetizations have bounded support, which is the case for rock samples, we proved that magnetizations equivalent from above are also equivalent from below, but this is no longer true for unbounded supports. In fact, even for unidirectional magnetizations, uniqueness of a magnetization generating a given field depends on the boundedness of the support, as we proved that any magnetization is equivalent from above to a unidirectional one (with infinite support in general). This helps explaining why methods in the Fourier domain (which essentially loose track of the support information) do encounter problems. It also shows that information on the support must be used in a crucial way to solve the problem.

This year, we produced a fast inversion scheme for magnetic field maps of unidirectional planar geological magnetization with discrete support located on a regular grid, based on discrete Fourier transform [18] . Figures 5 , 6 , 7 and 8 show an example of reconstruction. As the just mentioned article shows, the Fourier approach is computationally attractive but undergoes aliasing phenomena that tend to offset its efficiency. In particular, estimating the total moment of the magnetization sample seems to require data extrapolation techniques which are to take place in the space domain. This is why we have started to study regularization schemes based on truncation of the support in connection with singular values analysis of the discretized problem.

In a joint effort by all members of IMPINGE, we set up a heuristics to recover dipolar magnetizations, using a discrete least square criterion. At the moment, it is solved by a singular value decomposition procedure of the magnetization-to-field operator, along with a regularization technique based on truncation of the support. Preliminary experiments on synthetic data give quite accurate results to recover the net moment of a sample, see the preliminary document http://www-sop.inria.fr/apics/IMPINGE/Documents/NotesSyntheticExample.pdf . We also ran the procedure on real data (measurements of the field generated by Lunar spherules) for which the net moment can be estimated by other methods. The net moment thus recovered matches well the expected moment.

This shows that the technique we use to reduce the support, which is based on thresholding contributions of dipoles to the observations, is capable of eliminating some nearly silent dipole distributions which flaw the singular value analysis. In order to better understand the geometric nature of such distributions, and thus affirm theoretical bases to the above mentioned heuristics, we raised the question of determining an eigenbasis for the positive self adjoint operator mapping a $L^2$ magnetization on a rectangle to the field it generates on a rectangle parallel to the initial one. Once ordered according to decreasing eigenvalues, such a basis should retain “as much information as possible” granted the order of truncation.

This is not such an easy problem and currently, in the framework of the PhD thesis of D. Ponomarev, we investigate a simplified two-dimensional analog, defined via convolution of a function on a segment with the Poisson kernel of the upper half-plane and then restriction to a parallel segment in that half-plane. Surprisingly perhaps, this issue was apparently not considered in spite of its natural character and the fact that it makes contact with classical spectral theory. Specifically, it amounts to spectral representation of certain compressed Toeplitz operators with exponential-of-modulus symbols. Beyond the bibliographical research needed to understand the status of this question, only preliminary results have been attained so far.