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Habib Ammari

Publications and source records attributed to Habib Ammari.

At least 181 records · Page 10Linked to original sources

Admittivity imaging from multi-frequency micro-electrical impedance tomography

The aim of this paper is to propose an optimal control optimization algorithm for reconstructing admittivity distributions (i.e., both conductivity and permittivity) from multi-frequency micro-electrical impedance tomography. A convergent and stable optimization scheme is shown to be obtainable from multi-frequency data. The results of this paper have potential applicability in cancer imaging, cell culturing and differentiation, food sciences, and biotechnology.

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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.

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The Concept of Heterogeneous Scattering Coefficients and Its Application in Inverse Medium Scattering

This work investigates the scattering coefficients for inverse medium scattering problems. It shows some fundamental properties of the coefficients such as symmetry and tensorial properties. The relationship between the scattering coefficients and the far-field pattern is also derived. Furthermore, the sensitivity of the scattering coefficients with respect to changes in the permittivity and permeability distributions is investigated. In the linearized case, explicit formulas for reconstructing permittivity and permeability distributions from the scattering coefficients is proposed. They relate the exponentially ill-posed character of the inverse medium scattering problem at a fixed frequency to the exponential decay of the scattering coefficients. Moreover, they show the stability of the reconstruction from multifrequency measurements. This provides a new direction for solving inverse medium scattering problems.

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Optimal Shape Design by Partial Spectral Data

In this paper, we are concerned with a shape design problem, in which our target is to design, up to rigid transformations and scaling, the shape of an object given either its polarization tensor at multiple contrasts or the partial eigenvalues of its Neumann-Poincaré operator, which are known as the Fredholm eigenvalues. We begin by proposing to recover the eigenvalues of the Neumann-Poincaré operator from the polarization tensor by means of the holomorphic functional calculus. Then we develop a regularized Gauss-Newton optimization method for the shape reconstruction process. We present numerical results to demonstrate the effectiveness of the proposed methods and to illustrate important properties of the Fredholm eigenvalues and their associated eigenfunctions. Our results are expected to have important applications in the design of plasmon resonances in nanoparticles as well as in the multifrequency or pulsed imaging of small anomalies.

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Wavelet methods for shape perception in electro-sensing

This paper aims at presenting a new approach to the electro-sensing problem using wavelets. It provides an efficient algorithm for recognizing the shape of a target from micro-electrical impedance measurements. Stability and resolution capabilities of the proposed algorithm are quantified in numerical simulations.

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Spectroscopic imaging of a dilute cell suspension

A rigorous homogenization theory is derived to describe the effective admittivity of cell suspensions. A new formula is reported for dilute cases that gives the frequency-dependent effective admittivity with respect to the membrane polarization. Different microstructures are shown to be distinguishable via spectroscopic measurements of the overall admittivity using the spectral properties of the membrane polarization. The Debye relaxation times associated with the membrane polarization tensor are shown to be able to give the microscopic structure of the medium. A natural measure of the admittivity anisotropy is introduced and its dependence on the frequency of applied current is derived. A Maxwell-Wagner-Fricke formula is given for concentric circular cells, and the results can be extended to the random cases. A randomly deformed periodic medium is also considered and a new formula is derived for the overall admittivity of a dilute suspension of randomly deformed cells.

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Shape identification and classification in echolocation

The paper aims at proposing the first shape identification and classification algorithm in echolocation. The approach is based on first extracting geometric features from the reflected waves and then matching them with precomputed ones associated with a dictionary of targets. The construction of such frequency-dependent shape descriptors is based on some important properties of the scattering coefficients and new invariants. The stability and resolution of the proposed identification algorithm with respect to measurement noise and the limited-view aspect are analytically and numerically quantified.

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Reconstruction of Inhomogeneous Conductivities via the Concept of Generalized Polarization Tensors

This paper extends the concept of generalized polarization tensors (GPTs), which was previously defined for inclusions with homogeneous conductivities, to inhomogeneous conductivity inclusions. We begin by giving two slightly different but equivalent definitions of the GPTs for inhomogeneous inclusions. We then show that, as in the homogeneous case, the GPTs are the basic building blocks for the far-field expansion of the voltage in the presence of the conductivity inclusion. Relating the GPTs to the Neumann-to-Dirichlet (NtD) map, it follows that the full knowledge of the GPTs allows unique determination of the conductivity distribution. Furthermore, we show important properties of the the GPTs, such as symmetry and positivity, and derive bounds satisfied by their harmonic sums. We also compute the sensitivity of the GPTs with respect to changes in the conductivity distribution and propose an algorithm for reconstructing conductivity distributions from their GPTs. This provides a new strategy for solving the highly nonlinear and ill-posed inverse conductivity problem. We demonstrate the viability of the proposed algorithm by preforming a sensitivity analysis and giving some numerical examples.

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Backpropagation Imaging in Nonlinear Harmonic Holography in the Presence of Measurement and Medium Noises

In this paper, the detection of a small reflector in a randomly heterogenous medium using second-harmonic generation is investigated. The medium is illuminated by a time-harmonic plane wave at frequency omega. It is assumed that the reflector has a non-zero second-order nonlinear susceptibility, and thus emits a wave at frequency two omega in addition to the fundamental frequency linear scattering. It is shown how the fundamental frequency signal and the second-harmonic signal propagate in the medium. A statistical study of the images obtained by migrating the boundary data is performed. It is proved that the second-harmonic image is more stable with respect to medium noise than the one obtained with the fundamental signal. Moreover, the signal-to-noise ratio for the second-harmonic image does not depend neither on the second-order susceptibility tensor nor on the volume of the particle.

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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.

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Passive Array Correlation-Based Imaging in a Random Waveguide

We consider reflector imaging in a weakly random waveguide. We address the situation in which the source is farther from the reflector to be imaged than the energy equipartition distance, but the receiver array is closer to the reflector to be imaged than the energy equipartition distance. As a consequence, the reflector is illuminated by a partially coherent field and the signals recorded by the receiver array are noisy. This paper shows that migration of the recorded signals cannot give a good image, but an appropriate migration of the cross correlations of the recorded signals can give a very good image. The resolution and stability analysis of this original functional shows that the reflector can be localized with an accuracy of the order of the wavelength even when the receiver array has small aperture, and that broadband sources are necessary to ensure statistical stability, whatever the aperture of the array.

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Target Detection and Characterization from Electromagnetic Induction Data

The goal of this paper is to contribute to the field of nondestructive testing by eddy currents. We provide a mathematical analysis and a numerical framework for simulating the imaging of arbitrarily shaped small volume conductive inclusions from electromagnetic induction data. We derive, with proof, a small-volume expansion of the eddy current data measured away from the conductive inclusion. The formula involves two polarization tensors: one associated with the magnetic contrast and the second with the conductivity of the inclusion. Based on this new formula, we design a location search algorithm. We include in this paper a discussion on data sampling, noise reduction, and on probability of detection. We provide numerical examples that support our findings.

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Modeling active electrolocation in weakly electric fish

In this paper, we provide a mathematical model for the electrolocation in weakly electric fishes. We first investigate the forward complex conductivity problem and derive the approximate boundary conditions on the skin of the fish. Then we provide a dipole approximation for small targets away from the fish. Based on this approximation, we obtain a non-iterative location search algorithm using multi-frequency measurements. We present numerical experiments to illustrate the performance and the stability of the proposed multi-frequency location search algorithm. Finally, in the case of disk- and ellipse-shaped targets, we provide a method to reconstruct separately the conductivity, the permittivity, and the size of the targets from multi-frequency measurements.

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Target Identification Using Dictionary Matching of Generalized Polarization Tensors

The aim of this paper is to provide a fast and efficient procedure for (real-time) target identification in imaging based on matching on a dictionary of precomputed generalized polarization tensors (GPTs). The approach is based on some important properties of the GPTs and new invariants. A new shape representation is given and numerically tested in the presence of measurement noise. The stability and resolution of the proposed identification algorithm is numerically quantified.

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Shape recognition and classification in electro-sensing

This paper aims at advancing the field of electro-sensing. It exhibits the physical mechanism underlying shape perception for weakly electric fish. These fish orient themselves at night in complete darkness by employing their active electrolocation system. They generate a stable, high-frequency, weak electric field and perceive the transdermal potential modulations caused by a nearby target with different admittivity than the surrounding water. In this paper, we explain how weakly electric fish might identify and classify a target, knowing by advance that the latter belongs to a certain collection of shapes. Our model of the weakly electric fish relies on differential imaging, i.e., by forming an image from the perturbations of the field due to targets, and physics-based classification. The electric fish would first locate the target using a specific location search algorithm. Then it could extract, from the perturbations of the electric field, generalized (or high-order) polarization tensors of the target. Computing, from the extracted features, invariants under rigid motions and scaling yields shape descriptors. The weakly electric fish might classify a target by comparing its invariants with those of a set of learned shapes. On the other hand, when measurements are taken at multiple frequencies, the fish might exploit the shifts and use the spectral content of the generalized polarization tensors to dramatically improve the stability with respect to measurement noise of the classification procedure in electro-sensing. Surprisingly, it turns out that the first-order polarization tensor at multiple frequencies could be enough for the purpose of classification. A procedure to eliminate the background field in the case where the permittivity of the surrounding medium can be neglected, and hence improve further the stability of the classification process, is also discussed.

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Anomalous localized resonance using a folded geometry in three dimensions

If a body of dielectric material is coated by a plasmonic structure of negative dielectric material with nonzero loss parameter, then cloaking by anomalous localized resonance (CALR) may occur as the loss parameter tends to zero. It was proved in other papers by authors that if the coated structure is circular (2D) and dielectric constant of the shell is a negative constant (with loss parameter), then CALR occurs, and if the coated structure is spherical (3D), then CALR does not occur. The aim of this paper is to show that the CALR takes place if the spherical coated structure has a specially designed anisotropic dielectric tensor. The anisotropic dielectric tensor is designed by unfolding a folded geometry.

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Strong convergence of the solutions of the linear elasticity and uniformity of asymptotic expansions in the presence of small inclusions

We consider the Lamé system of linear elasticity when the inclusion has the extreme elastic constants. We show that the solutions to the Lamé system converge in appropriate $H^1$-norms when the shear modulus tends to infinity (the other modulus, the compressional modulus is fixed), and when the bulk modulus and the shear modulus tend to zero. Using this result, we show that the asymptotic expansion of the displacement vector in the presence of small inclusion is uniform with respect to Lamé parameters.

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