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In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group [Formula omitted] on the space [Formula omitted] of ternary forms of even degree 2d. The construction relies on two key ingredients: on the one hand, the Slice Lemma allows us to reduce the problem to determining the invariants for the action on a subspace of the finite subgroup [Formula omitted] of signed permutations. On the other hand, our construction relies in a fundamental way on specific bases of harmonic polynomials. These bases provide maps with prescribed [Formula omitted]-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the [Formula omitted]-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the [Formula omitted]-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed [Formula omitted]-invariants to determine the [Formula omitted]-orbit locus and provide an algorithm for the inverse problem of finding an element in [Formula omitted] with prescribed values for its invariants. These computational issues are relevant in brain imaging., Author(s): Paul Görlach [sup.1] [sup.2] , Evelyne Hubert [sup.1] , Théo Papadopoulo [sup.1] Author Affiliations: (1) INRIA Méditerranée, , Sophia Antipolis, France (2) grid.419532.8, Max Planck Institute for Mathematics in [...]

The forward electroencephalography (EEG) problem involves finding a potential &nable; from the Poisson equation &nable; · (σ &nable;V) = f, in which f represents electrical sources in the brain, and a the conductivity of the head tissues. In the piecewise constant conductivity head model, this can be accomplished by the boundary element method (BEM) using a suitable integral formulation. Most previous work uses the same integral formulation, corresponding to a double-layer potential. In this paper we present a conceptual framework based on a well-known theorem (Theorem 1) that characterizes harmonic functions defined on the complement of a bounded smooth surface. This theorem says that such harmonic functions are completely defined by their values and those of their normal derivatives on this surface. It allows us to cast the previous BEM approaches in a unified setting and to develop two new approaches corresponding to different ways of exploiting the same theorem. Specifically, we first present a dual approach which involves a single-layer potential. Then, we propose a symmetric formulation, which combines single- and double-layer potentials, and which is new to the field of LEG, although it has been applied to other problems in electromagnetism. The three methods have been evaluated numerically using a spherical geometry with known analytical solution, and the symmetric formulation achieves a significantly higher accuracy than the alternative methods. Additionally, we present results with realistically shaped meshes. Beside providing a better understanding of the foundations of BEM methods, our approach appears to lead also to more efficient algorithms. [ABSTRACT FROM AUTHOR]

An important issue in electroencephalography (EEG) experiments is to measure accurately the three dimensional (3D) positions of electrodes. We propose a system where these positions are automatically estimated from several images using computer vision techniques. Yet, only a set of undifferentiated points are recovered this way and remains the problem of labeling them, i.e. of finding which electrode corresponds to each point. This paper proposes a fast and robust solution to this latter problem based on combinatorial optimization. We design a specific energy that we minimize with a modified version of the Loopy Belief Propagation algorithm. Experiments on real data show that, with our method, a manual labeling of two or three electrodes only is sufficient to get the complete labeling of a 64 electrodes cap in less than 10 seconds. However, it is shown to be robust to missing electrodes in the reconstructed data.