Reconstruction and stability in acousto-optic imaging for absorption maps with bounded variation

The aim of this paper is to propose for the first time a reconstruction scheme and a stability result for recovering from acoustic-optic data absorption distributions with bounded variation. The paper extends earlier results on smooth absorption distributions. It opens a door for a mathematical and numerical framework for imaging, from internal data, parameter distributions with high contrast in biological tissues

Radiative transfer and diffusion limits for wave field correlations in locally shifted random media

The aim of this paper is to develop a mathematical framework for opto-elastography. In opto-elastography, a mechanical perturbation of the medium produces a decorrelation of optical speckle patterns due to the displacements of optical scatterers. To model this, we consider two optically random media, with the second medium obtained by shifting the first medium in some local region. We derive the radiative transfer equation for the cross-correlation of the wave fields in the media. Then we derive its diffusion approximation. In both the radiative transfer and the diffusion regimes, we relate the correlation of speckle patterns to the solutions of the radiative transfer and the diffusion equations. We present numerical simulations based on our model which are in agreement with recent experimental measurements

A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography

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

A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography

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. © 2014 Elsevier Masson SAS

Mathematical modelling of hybrid biomedical imaging by mechanical perturbations

Dans cette thèse, nous introduisons et développons une approche mathématiques originale des techniques d'imagerie biomédicales dites "hybrides". L'idée et d'appliquer une méthode d'imagerie mal posée, tout en perturbant le milieu à imager par des déplacements mécaniques. Ces déplacements provenant d'une équation de type onde élastique perturbent les mesures effectuées. En utilisant ces mesures perturbées, et profitant du caractère local des perturbations mécaniques, il est possible d'augmenter considérablement la résolution de la méthode de base. Le problème direct est donc un couplage d'une EDP décrivant la propagation utilisée pour la méthode de base et d'une seconde décrivant les champs de déplacement mécaniques. Dans toutes cette thèse, on fait l'hypothèse d'un milieu mécaniquement homogène afin d'assurer le contrôle et la géométrie des ondes perturbatrices utilisées. A partir des mesures perturbées, une étape d'interprétation permet de construire une donnée interne au domaine considéré. Cette étape nécessite en général l'inversion d'opérateurs géométriques intégraux de type Radon, afin d'utiliser le caractère localisant des perturbations utilisées. A partir de cette donnée interne, il est possible d'initier une procédure de reconstruction du paramètre physique recherché. Dans le chapitre 1, il est question d'un couplage entre micro-ondes et perturbations sphériques. Dans les chapitres 2, 3 et 4, nous étudions l'imagerie optique diffuse toujours couplée avec des perturbations sphériques. Enfin dans le chapitre cinq, nous donnons une méthode originale de reconstruction de la conductivité électrique par un couplage entre champs magnétique et perturbations acoustiques focalisées.This thesis aims at developing an original mathematical approach for modeling hybrid biomedical imaging modalities. The core idea is to run an ill-posed imaging method while perturbing the medium using mechanical displacements. These displacements described by an elastic wave equation perturb the collected measurements. Using these perturbed measurements and taking advantage of the perturbation localizing e↵ect, it is possible to significantly overcome the resolution of the basic method. The direct problem here is a coupling between a PDE describing the propagation used for the basic method and a second one describing the mechanical displacements fields. In the whole thesis, we only consider mechanically homogeneous medium in order to assure the control and the geometry of the perturbing wavefronts. From these perturbed measurements, an interpretation step leads to an internal data map inside the considered medium. This step usually requires inversion of geometric integral operators such as Radon transform. This allows to use the geometrical localizing behavior of the perturbations. From this internal data, one can start a recovering procedure for the unknown physical parameter. This recovering step involves a new non physical PDE, non linearly coupled with the main modality equation. In the first chapter, we study a coupling between micro-waves and spherical perturbations. In chapter 2, 3 and 4, we propose a model for di↵use optical imaging coupled with spherical perturbations. In chapter 5, we introduce a new method for imaging the electric conductivity by a coupling between magnetic field and focused acoustic perturbation

Orthogonal modes of a fully anisotropic and heterogeneous elastic medium

The aim of this short note is to give a synthetic presentation of the mathematical elements that are used to solve the elastic wave system of equations in a bounded anisotropic elastic body, in a general framework. In particular, the proof of existence of a basis of orthogonal modes is given. We explain how these modes can by used to efficiently approach dynamic problems in time or harmonic regimes

Regularization for seismic sources inversion from interferometric data

International audienceIn this talk, we deal with the inverse wave source problem from interferometric measurements. Interferometric data are made of correlations of the wave field recorded at different receiver positions and frequencies. In some configurations, these correlations are stable with respect to phase shift due to some wave-speed uncertainties inside the medium. Hence, trying to recover sources from this data is of large interest

Modélisation de l'imagerie biomédicale hybride par perturbations mécaniques

This thesis aims at developing an original mathematical approach for modeling hybrid biomedical imaging modalities. The core idea is to run an ill-posed imaging method while perturbing the medium using mechanical displacements. These displacements described by an elastic wave equation perturb the collected measurements. Using these perturbed measurements and taking advantage of the perturbation localizing e↵ect, it is possible to significantly overcome the resolution of the basic method. The direct problem here is a coupling between a PDE describing the propagation used for the basic method and a second one describing the mechanical displacements fields. In the whole thesis, we only consider mechanically homogeneous medium in order to assure the control and the geometry of the perturbing wavefronts. From these perturbed measurements, an interpretation step leads to an internal data map inside the considered medium. This step usually requires inversion of geometric integral operators such as Radon transform. This allows to use the geometrical localizing behavior of the perturbations. From this internal data, one can start a recovering procedure for the unknown physical parameter. This recovering step involves a new non physical PDE, non linearly coupled with the main modality equation. In the first chapter, we study a coupling between micro-waves and spherical perturbations. In chapter 2, 3 and 4, we propose a model for di↵use optical imaging coupled with spherical perturbations. In chapter 5, we introduce a new method for imaging the electric conductivity by a coupling between magnetic field and focused acoustic perturbationsDans cette thèse, nous introduisons et développons une approche mathématiques originale des techniques d'imagerie biomédicales dites "hybrides". L'idée et d'appliquer une méthode d'imagerie mal posée, tout en perturbant le milieu à imager par des déplacements mécaniques. Ces déplacements provenant d'une équation de type onde élastique perturbent les mesures effectuées. En utilisant ces mesures perturbées, et profitant du caractère local des perturbations mécaniques, il est possible d'augmenter considérablement la résolution de la méthode de base. Le problème direct est donc un couplage d'une EDP décrivant la propagation utilisée pour la méthode de base et d'une seconde décrivant les champs de déplacement mécaniques. Dans toutes cette thèse, on fait l'hypothèse d'un milieu mécaniquement homogène afin d'assurer le contrôle et la géométrie des ondes perturbatrices utilisées. A partir des mesures perturbées, une étape d'interprétation permet de construire une donnée interne au domaine considéré. Cette étape nécessite en général l'inversion d'opérateurs géométriques intégraux de type Radon, afin d'utiliser le caractère localisant des perturbations utilisées. A partir de cette donnée interne, il est possible d'initier une procédure de reconstruction du paramètre physique recherché. Dans le chapitre 1, il est question d'un couplage entre micro-ondes et perturbations sphériques. Dans les chapitres 2, 3 et 4, nous étudions l'imagerie optique diffuse toujours couplée avec des perturbations sphériques. Enfin dans le chapitre cinq, nous donnons une méthode originale de reconstruction de la conductivité électrique par un couplage entre champs magnétique et perturbations acoustiques focalisées