We provide a mathematical analysis and a numerical framework for Lorentz force electrical conductivity imaging. Ultrasonic vibration of a tissue in the presence of a static magnetic field induces an electrical current by the Lorentz force. This current can be detected by electrodes placed around the tissue; it is proportional to the velocity of the ultrasonic pulse, but depends nonlinearly on the conductivity distribution. The imaging problem is to reconstruct the conductivity distribution from measurements of the induced current. To solve this nonlinear inverse problem, we first make use of a virtual potential to relate explicitly the current measurements to the conductivity distribution and the velocity of the ultrasonic pulse. Then, by applying a Wiener filter to the measured data, we reduce the problem to imaging the conductivity from an internal electric current density. We first introduce an optimal control method for solving such a problem. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscosity-type regularization to a transport equation satisfied by the current density field. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. We also test both schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance.© H. Ammari et al. / J. Math. Pures Appl. 103 (2015) 1390-1409 1391 de l'onde ultrasonore et dépend non linéairement de la conductivité électrique. Le problème est de reconstruire cette conductivité à partir des mesures de courant. En premier lieu, on utilise une fonction test (potentiel virtuel) pour quantifier le lien entre le signal et la conductivité. Ensuite, à l'aide d'une déconvolution et d'un filtrage, il est possible de ramener le problème à la reconstruction d'une carte de conductivité à partir de la donnée d'un courant électrique interne sur l'ensemble du domaine. On donne d'abord une méthode d'optimisation pour résoudre ce problème. Une seconde méthode de reconstruction directe, utilisant une méthode de viscosité et la résolution d'une équation de transport à coefficients discontinus, est ensuite proposée. On démontre que la résolution de ce problème donne une reconstruction exacte de la conductivité lorsque le paramètre de régularisation tend vers zéro. On illustre les deux méthodes numériquement et on compare leur performances (résolution et stabilité en présence de bruit de mesure).

The aim of this paper is to develop an efficient reconstruction algorithm for ultrasound-modulated diffuse optical tomography. In diffuse optical imaging, the resolution is in general low. By mechanically perturbing the medium, we show that it is possible to achieve a significant resolution enhancement. When a spherical acoustic wave is propagating inside the medium, the optical parameter of the medium is perturbed. Using cross-correlations of the boundary measurements of the intensity of the light propagating in the perturbed medium and in the unperturbed one, we provide an iterative algorithm for reconstructing the optical absorption coefficient. Using a spherical Radon transform inversion, we first establish an equation that the optical absorption satisfies. This equation together with the diffusion model constitutes a nonlinear system. Then, solving iteratively such a nonlinear coupled system, we obtain the true absorption parameter. We prove the convergence of the algorithm and present numerical results to illustrate its resolution and stability performances.

The aim of this paper is to develop a mathematical framework for acousto-electromagnetic tomography and to introduce an efficient reconstruction algorithm. In electromagnetic wave imaging, the resolution is limited by the Rayleigh criterion, that is, half the operating wavelength. By mechanically perturbing the medium, we show that it is possible to achieve a significant resolution enhancement. We provide a new inversion formula for the permittivity distribution from cross-correlations between the electromagnetic boundary measurements in the perturbed medium and in the unperturbed one. We present numerical results to illustrate the resolution and the stability performances of the proposed reconstruction algorithm.

<p style='text-indent:20px;'>Localization and reconstruction of small defects in acoustic or electromagnetic waveguides is of crucial interest in nondestructive evaluation of structures. The aim of this work is to present a new multi-frequency inversion method to reconstruct small defects in a 2D waveguide. Given one-side multi-frequency wave field measurements of propagating modes, we use a Born approximation to provide a <inline-formula><tex-math id="M1">\begin{document}$ \text{L}^2 $\end{document}</tex-math></inline-formula>-stable reconstruction of three types of defects: a local perturbation inside the waveguide, a bending of the waveguide, and a localized defect in the geometry of the waveguide. This method is based on a mode-by-mode spacial Fourier inversion from the available partial data in the Fourier domain. Indeed, in the available data, some high and low spatial frequency information on the defect are missing. We overcome this issue using both a compact support hypothesis and a minimal smoothness hypothesis on the defects. We also provide a suitable numerical method for efficient reconstruction of such defects and we discuss its applications and limits.</p>

The aim of this paper is to tackle the nonlinear optical reconstruction problem. Given a set of acousto-optic measurements, we develop a mathematical framework for the reconstruction problem in the case where the optical absorption distribution is supposed to be a perturbation of a piecewise constant function. Analyzing the acousto-optic measurements, we prove that the optical absorption coefficient satisfies, in the sense of distributions, a new equation. For doing so, we introduce a weak Helmholtz decomposition and interpret in a weak sense the cross-correlation measurements using the spherical Radon transform. We next show how to find an initial guess for the unknown coefficient. Finally, in order to construct the true coefficient we provide a Landweber type iteration and prove that the resulting sequence converges to the solution of the system constituted by the optical diffusion equation and the new equation mentioned above. Our results in this paper generalize the acousto-optic process proposed in [5] for piecewise smooth optical absorption distributions.