Abstract:In the present paper, we develop a novel Bayesian approach to the problem of estimating neural currents in the brain from a fixed distribution of magnetic field (called topography), measured by magnetoencephalography. Differently from recent studies that describe inversion techniques, such as spatio-temporal regularization/filtering, in which neural dynamics always plays a role, we face here a purely static inverse problem. Neural currents are modelled as an unknown number of current dipoles, whose state space is described in terms of a variable-dimension model. Within the resulting Bayesian framework, we set up a sequential Monte Carlo sampler to explore the posterior distribution. An adaptation technique is employed in order to effectively balance the computational cost and the quality of the sample approximation. Then, both the number and the parameters of the unknown current dipoles are simultaneously estimated. The performance of the method is assessed by means of synthetic data, generated by source configurations containing up to four dipoles. Eventually, we describe the results obtained by analyzing data from a real experiment, involving somatosensory evoked fields, and compare them to those provided by three other methods.

and m.i.d.a. (methods for image and data analysis) group

In this paper we explain the linear sampling method and its performances in various scattering conditions by means of an analysis of the far-field equation based on the principle of energy conservation. Specifically, we consider the conservation of energy along the flow strips of the Poynting vector associated with the scattered field whose far-field pattern is one of the two terms in the far-field equation. The behavior of these flow lines is numerically investigated and theoretically described. Appropriate assumptions on the flow lines, based on the numerical results, allow characterizing a set of approximate solutions of the far-field equation which can be used to visualize the boundary of the scatterer in the framework of the linear sampling method. In particular, under the same assumptions, we can show that Tikhonov regularized solutions belong to this set of approximate solutions for appropriate choices of the regularization parameter.

The linear sampling method is a qualitative procedure for the visualization of both impenetrable and inhomogeneous scatterers, which requires the regularized solution of a linear illposed integral equation of the first kind. An open issue in this technique is the one of determining the optimal scatterer profile from the visualization maps in an automatic manner. In the present paper this problem is addressed in two steps. First, linear sampling is optimized by using a new regularization algorithm for the solution of the integral equation, which provides more accurate maps for different levels of the noise affecting the data. Then an edge detection technique based on active contours is applied to the optimized maps. Our computation exploits a recently introduced implementation of the linear sampling method, which enhances both the accuracy and the numerical effectiveness of the approach. Introduction.The linear sampling method [11,14] is a qualitative procedure for the solution of acoustical and electromagnetic inverse scattering problems in the resonance region. It is based on what we will call the general theorem [3,4], which is concerned with a linear ill-posed integral equation of the first kind, named the far-field equation, whose integral kernel is the far-field pattern of the scattered field and whose data is a known analytical function. According to the general theorem, for each point in the physical space an approximate solution of the far-field equation exists, such that the L 2 -norm of this solution blows up to infinity when the point approaches the boundary of the scatterer from inside, while it stays arbitrarily large outside. This behavior inspires a visualization algorithm [14] based on the following steps for each point of a computational grid containing the object:1. (approximately) solve a discretized version of the far-field equation; 2. plot the indicator function, which is an appropriate monotonic function of the Euclidean norm of this approximate solution: then the scatterer profile will be highlighted by all points in the physical space, whereby the indicator function is significantly large or small with respect to the surrounding points. In step 1 of the method, it is crucial to account for the ill-posedness of the linear inverse problem of solving the far-field equation. This is done by applying a regularization procedure like the Tikhonov method, where the integral equation is replaced by a convex minimum problem, and the optimal trade-off between the stability of the solution and its reliability in reproducing the data function is realized through a judicious choice of a regularization parameter. In most traditional sampling imple-

In this paper we present a hybrid approach to numerically solve two-dimensional electromagnetic inverse scattering problems, whereby the unknown scatterer is hosted by a possibly inhomogeneous background. The approach is 'hybrid' in that it merges a qualitative and a quantitative method to optimize the way of exploiting the a priori information on the background within the inversion procedure, thus improving the quality of the reconstruction and reducing the data amount necessary for a satisfactory result. In the qualitative step, this a priori knowledge is utilized to implement the linear sampling method in its near-field formulation for an inhomogeneous background, in order to identify the region where the scatterer is located. On the other hand, the same a priori information is also encoded in the quantitative step by extending and applying the contrast source inversion method to what we call the 'inhomogeneous Lippmann-Schwinger equation': the latter is a generalization of the classical Lippmann-Schwinger equation to the case of an inhomogeneous background, and in our paper is deduced from the differential formulation of the direct scattering problem to provide the reconstruction algorithm with an appropriate theoretical basis. Then, the point values of the refractive index are computed only in the region identified by the linear sampling method at the previous step. The effectiveness of this hybrid approach is supported by numerical simulations presented at the end of the paper.

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We describe a very fast and automatic formulation of the linear sampling method for three-dimensional electromagnetic inverse scattering problems. This formulation is an extension of a no-sampling implementation recently proposed for twodimensional configurations. In this 3D framework, regularization occurs independently not only of the sampling point but even of the polarization of the fundamental solution used as known term. Furthermore, a very effective automatic procedure for the selection of the optimal surface describing the scatterer is introduced.

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