ANTOINE LAURAIN
PHD IN MATHEMATICS
INVERSE PROBLEMS

Keywords: inverse problems, electrical impedance tomography, fluorescence optical tomography, magnetic impedance tomography

Electrical impedance tomography

Electrical Impedance Tomography (EIT) is a non-destructive imaging technique which has various applications. Its purpose is to reconstruct the electric conductivity of hidden objects inside a medium with the help of boundary field measurements. Our approach consists in minimizing the L_2-distance between the potentials pertinent to a certain given number of applied currents and corresponding measurements and assuming that the conductivity is piecewise constant. In this case, the objects are assumed to have sharp interfaces. Such an assumption allows us to pursue a shape and topology optimization approach. A levet set method is used to model the interface inclusion/background: the levelset function is updated using the shape gradient. The interface is initialized using the notion of topological derivative which is adapted to the framework of EIT.

> Domain expression of the shape derivative and application to electrical impedance tomography
> Second-order topological expansion for electrical impedance tomography
> Electrical Impedance Tomography: From Topology to Shape

Fluorescence optical tomography

Fluorescence tomography is a non-invasive imaging modality that reconstructs fluorophore distributions inside a small animal from boundary measurements of the fluorescence light. The associated inverse problem is stabilized by a priori properties or information. In this paper, cases are considered where the fluorescent inclusions are well separated from the background and have a spatially constant concentration. Under these a priori assumptions, the identification process may be formulated as a shape optimization problem, where the interface between the fluorescent inclusion and the background constitutes the unknown shape. In this paper, we focus on the computation of the so-called topological derivative for fluorescence tomography which could be used as a stand-alone tool for the reconstruction of the fluorophore distributions or as the initialization in a level-set-based method for determining the shape of the inclusions.

> Topological sensitivity analysis in fluorescence optical tomography

Magnetic impedance tomography

In this paper the shape derivative of an objective depending on the solution of an eddy current approximation of Maxwell's equations is obtained. Using a Lagrangian approach in the spirit of Delfour and Zol\'esio, the computation of the shape derivative of the solution of the state equation is bypassed. This theoretical result is applied to magnetic impedance tomography, which is an imaging modality aiming at the contactless mapping (identification) of the unknown electrical conductivities inside an object given measurements recorded by receiver coils.

> Shape sensitivities for an inverse problem in magnetic induction tomography based on the eddy current model
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