In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group O 3 on the space R[x, y, z] 2d of ternary forms of even degree 2d. The construction relies on two key ingredients: On 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 B 3 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 B 3 -equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the B 3 -invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the O 3 -invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed B 3 -invariants to determine the O 3 -orbit locus and provide an algorithm for the inverse problem of finding an element in R[x, y, z] 2d with prescribed values for its invariants. These computational issues are relevant in brain imaging.the action of O 3 on R[x, y, z] 4 is determined as a subset of a minimal generating set of polynomial invariants of the action of O 3 on the elasticity tensor in [OKA17]. There, the problem is mapped to the joint action of SL 2 (C) on binary forms of different degrees and resolved by Gordan's algorithm [GY10, Oli17] so that the invariants are given as transvectants.A generating set of rational invariants separates general orbits [PV94, Ros56] -this remains true for any group, even for non-reductive groups. Rational invariants can thus prove to be sufficient, and sometimes more relevant, in applications [HL12, HL13, HL16] and in connection with other mathematical disciplines [HK07b,Hub12]. A practical and very general algorithm to compute a generating set of these first appeared in [HK07a]; see also [DK15]. The case of the action of O 3 on R[x, y, z] 4 , a 15-dimensional space, is nonetheless not easily tractable by this algorithm. In the case of R[x, y, z] 4 , the 12 generating invariants we construct in this article are seen as being uniquely determined by their restrictions to a slice Λ 4 , which is here a 12dimensional subspace. The knowledge of these restrictions is proved to be sufficient to evaluate the invariants at any point in the space R[x, y, z] 4 . The underlying slice method is a technique used to show rationality of invariant fields [CTS07]. We demonstrate here its power for the computational aspects of Invariant Theory.

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This paper concerns the modelisation and simulation of the forward problem of Electrical Impedance Tomography (EIT), i.e. the computation of an electrical potential due to an applied boundary current. The underlying goal is the estimation of the conductivities of the head tissues, in particular the skull, in order to improve the head models used in Electroencephalography (EEG). In the quasistatic approximation, the problem can be modeled by a Poisson equation with a non-vanishing Neumann boundary condition. We introduce a symmetric boundary integral formulation, which is discretized using mixed finite elements, and show its application to an EIT experiment in view of estimating skull conductivity. Résumé. Nous nous intéressonsà la modélisation et la simulation du problème direct de la Tomographie par ImpédanceÉlectrique (EIT), i.e. le calcul du potentielélectrique dans un volume résultant de l'application d'un courant sur la surface. Le but sous-jacent est l'estimation des conductivités des tissus de la tête, en particulier celle du crâne, afin d'améliorer les modèles de tête utilisés en Electroencéphalographie (EEG). Par une approximation quasistatique, on modélise le problème par uné equation de Poisson, avec condition de Neumann non-nulle sur le bord. Nous introduisons une formulation symétrique paréquations intégrales, ainsi que sa discrétisation paréléments finis mixtes. Nous montrons l'application de cette méthodeà une expérience d'EIT, en vue d'estimer la conductivité du crâne.

Abstract. An important issue in electroencephalographiy (EEG) experiments is to measure accurately the three dimensional (3D) positions of the 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.