Modelling thin tissue compartiments using the immersed FEM (continuous Galerkin)

International audienceThis presentation describes a trilinear immersed finite element method for solving the electroencephalography forward problem, which is a three-dimensional elliptic interface problem in the head geometry. The method uses hexahedral Cartesian meshes (i.e. 3D images which can be explored using standard visualization tools for MR images) independent of the interfaces between head tissues, thus avoiding the sometimes difficult task of generating geometry fitting meshes (which is exacerbated for child brains which contains close interfaces requiring a very high number of elements to obtain numerically good mesh representations). Brain interfaces are provided as level sets representations, which are also 3D images. Such levelset representations can directly be used in head segmentation tools but can be also easily obtained from meshes. The finite element space is locally modified to better approximating the continuity properties of the solution (continuous potential and normal currents despite a discontinuity of the conductivity). Numerical results show that this method achieves the same accuracy as the standard linear finite element method with geometry fitting meshes without the hassle of creating meshes for the complex domain that is the head

Rational invariants of even ternary forms under the orthogonal group

In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group $\mathrm{O}_3$ on the space $\mathbb{R}[x,y,z]_{2d}$ 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 dermining the invariants for the action on a subspace of the finite subgroup $\mathrm{B}_3$ 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 $\mathrm{B}_3$-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the $\mathrm{B}_3$-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the $\mathrm{O}_3$-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed $\mathrm{B}_3$-invariants to determine the $\mathrm{O}_3$-orbit locus and provide an algorithm for the inverse problem of finding an element in $\mathbb{R}[x,y,z]_{2d}$ with prescribed values for its invariants. These are the computational issues relevant in brain imaging.Comment: v3 Changes: Reworked presentation of Neuroimaging application, refinement of Definition 3.1. To appear in "Foundations of Computational Mathematics

A Closed-Form Solution of Rotation Invariant Spherical Harmonic Features in Diffusion MRI

International audienceRotation invariant features are an indispensable tool for characterizing diffusion Magnetic Resonance Imaging (MRI) and in particular for brain tissue microstructure estimation. In this work, we propose a new mathematical framework for efficiently calculating a complete set of such invariants from any spherical function. Specifically, our method is based on the spherical harmonics series expansion of a given function of any order and can be applied directly to the resulting coefficients by performing a simple integral operation analytically. This enable us to derive a general closed-form equation for the invariants. We test our invariants on the diffusion MRI fiber orientation distribution function obtained from the diffusion signal both in-vivo and in synthetic data. Results show how it is possible to use these invariants for characterizing the white matter using a small but complete set of features

A Theory of the Motion Fields of Curves

This paper is a study of the motion field generated by moving 3D curves which are observed by a camera. We first discuss the relationship between optical flow and motion field and show that the assumptions made in the computation of the optical flow are a bit difficult to defend. We then go ahead to study the motion field of a general curve. We first study the general case of a curve moving nonrigidly and introduce the notion of isometric motion. In order to do this, we introduce the notion of spatio-temporal surface and study its differential properties up to the second order. We show that, contrarily to what is commonly believed, the full motion field of the curve (i.e the component tangent to the curve) cannot be recovered from this surface. We also give the equations that characterize the spatio-temporal surface completely up to a rigid transformation. Those equations are the expressions of the first and second fundamental forms and the Gauss and Codazzi-Mainardi equations. We then..

Rigid Motion and Structure from Curves Using Scale Space

This article presents an extension of works by Olivier Faugeras and Th'eo Papadopoulo on moving 3D curves. They have developed a theory for the motion fields of 3D curves seen in a monocular sequence of 2D images and described methods for computing the kinematic screw (\Omega ; V) of the curve and its time derivative ( \Omega ; V) in the case of rigid 3D motion. Here the results of implementing one of these methods are reported and compared to a previous implementation of another of the methods. Furthermore aspects of a scale space extension of the solution are discussed. 1 Introduction Faugeras and Papadopoulo [3] have presented a theory for the motion fields of 3D curves seen in a sequence of 2D images and related it to the well-known concept of optic flow. They introduced the concept of spatio-temporal surface and presented equations relating the kinematic screw (\Omega ; V) of the curve and its time derivative ( \Omega ; V) to derivatives defined on this surface, in the ..

Inverse source estimation problem in EEG

International audienceBeing given pointwise measurements of the electric potential taken by electrodes on part of the scalp, the EEG (electroencephalography) inverse problem consists in estimating current sources within the brain that account for this activity. A model for the behaviour of the potential rests on Maxwell equation in the quasi-static case, under the form of a Poisson-Laplace equation. We describe our approach for solving the inverse problem in spherical geometry, for piecewise constant electric conductivity values, and pointwise dipolar source terms

Inverse source estimation problem in EEG

International audienceBeing given pointwise measurements of the electric potential taken by electrodes on part of the scalp, the EEG (electroencephalography) inverse problem consists in estimating current sources within the brain that account for this activity. A model for the behaviour of the potential rests on Maxwell equation in the quasi-static case, under the form of a Poisson-Laplace equation. We describe our approach for solving the inverse problem in spherical geometry, for piecewise constant electric conductivity values, and pointwise dipolar source terms

Automatic labeling of EEG electrodes using combinatorial optimization

Abstract — 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. I