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Habib Ammari

Publications and source records attributed to Habib Ammari.

At least 163 records · Page 9Linked to original sources
arXiv2015

Surface Plasmon Resonance of Nanoparticles and Applications in Imaging

In this paper we provide a mathematical framework for localized plasmon resonance of nanoparticles. Using layer potential techniques associated with the full Maxwell equations, we derive small-volume expansions for the electromagnetic fields, which are uniformly valid with respect to the nanoparticle's bulk electron relaxation rate. Then, we discuss the scattering and absorption enhancements by plasmon resonant nanoparticles. We study both the cases of a single and multiple nanoparticles. We present numerical simulations of the localized surface plasmonic resonances associated to multiple particles in terms of their separation distance.

math-ph↗
arXiv2015

Mathematical analysis of plasmonic nanoparticles: the scalar case

Localized surface plasmons are charge density oscillations confined to metallic nanoparticles. Excitation of localized surface plasmons by an electromagnetic field at an incident wavelength where resonance occurs results in a strong light scattering and an enhancement of the local electromagnetic fields. This paper is devoted to the mathematical modeling of plasmonic nanoparticles. Its aim is threefold: (i) to mathematically define the notion of plasmonic resonance and to analyze the shift and broadening of the plasmon resonance with changes in size and shape of the nanoparticles; (ii) to study the scattering and absorption enhancements by plasmon resonant nanoparticles and express them in terms of the polarization tensor of the nanoparticle. Optimal bounds on the enhancement factors are also derived; (iii) to show, by analyzing the imaginary part of the Green function, that one can achieve super-resolution and super-focusing using plasmonic nanoparticles. For simplicity, the Helmholtz equation is used to model electromagnetic wave propagation.

math.AP↗
arXiv2015

Well-posedness of a Pulsed Electric Field Model in Biological Media and its Finite Element Approximation

This work aims at providing a mathematical and numerical framework for the analysis on the effects of pulsed electric fields on biological media. Biological tissues and cell suspensions are described as having a heteregeneous permittivity and a heteregeneous conductivity. Well-posedness of the model problem and the regularity of its solution are established. A fully discrete finite element scheme is proposed for the numerical approximation of the potential distribution as a function of time and space simultaneously for an arbitrary shaped pulse, and it is demonstrated to enjoy the optimal convergence order in both space and time. The proposed numerical scheme has potential applications in the fields of medicine, food sciences, and biotechnology.

math.AP↗
arXiv2014

Super-resolution in high contrast media

A mathematical theory is developed to explain the super-resolution and super-focusing in high contrast media. The approach is based on the resonance expansion of the Green function associated with the medium. It is shown that the super-resolution is due to sub-wavelength resonant modes excited in the medium which can propagate into the far-field.

math.AP↗
arXiv2014

Multi-frequency acousto-electromagnetic tomography

This paper focuses on the acousto-electromagnetic tomography, a recently introduced hybrid imaging technique. In a previous work, the reconstruction of the electric permittivity of the medium from internal data was achieved under the Born approximation assumption. In this work, we tackle the general problem by a Landweber iteration algorithm. The convergence of such scheme is guaranteed with the use of a multiple frequency approach, that ensures uniqueness and stability for the corresponding linearized inverse problem. Numerical simulations are presented.

math.AP↗
arXiv2014

Super-resolution in Imaging High Contrast Targets from the Perspective of Scattering Coefficients

In this paper we consider the inverse scattering problem for high-contrast targets. We mathematically analyze the experimentally-observed phenomenon of super-resolution in imaging the target shape. This is the first time that a mathematical theory of super-resolution has been established in the context of imaging high contrast inclusions. We illustrate our main findings with a variety of numerical examples. Our analysis is based on the novel concept of scattering coefficients. These findings may help in developing resonant structures for resolution enhancement.

math-ph↗
arXiv2014

Magnetic resonance-based reconstruction method of conductivity and permittivity distributions at the Larmor frequency

Magnetic resonance electrical property tomography is a recent medical imaging modality for visualizing the electrical tissue properties of the human body using radio-frequency magnetic fields. It uses the fact that in magnetic resonance imaging systems the eddy currents induced by the radio-frequency magnetic fields reflect the conductivity ($σ$) and permittivity ($ε$) distributions inside the tissues through Maxwell's equations. The corresponding inverse problem consists of reconstructing the admittivity distribution ($γ=σ+iωε$) at the Larmor frequency ($ω/2π=$128 MHz for a 3 tesla MRI machine) from the positive circularly polarized component of the magnetic field ${\bf H}=(H_x,H_y,H_z)$. Previous methods are usually based on an assumption of local homogeneity ($\nablaγ\approx 0$) which simplifies the governing equation. However, previous methods that include the assumption of homogeneity are prone to artifacts in the region where $γ$ varies. Hence, recent work has sought a reconstruction method that does not assume local-homogeneity. This paper presents a new magnetic resonance electrical property tomography reconstruction method which does not require any local homogeneity assumption on $γ$. We find that $γ$ is a solution of a semi-elliptic partial differential equation with its coefficients depending only on the measured data $H^+$, which enable us to compute a blurred version of $γ$. To improve the resolution of the reconstructed image, we developed a new optimization algorithm that minimizes the mismatch between the data and the model data as a highly nonlinear function of $γ$. Numerical simulations are presented to illustrate the potential of the proposed reconstruction method.

math.AP↗
arXiv2014

Stability Analysis for Magnetic Resonance Elastography

We consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues. The measurements are made on the entirety of a smooth domain. Since tissues can be modeled as quasi-incompressible fluids, we examine the Stokes system and consider only the recovery of shear modulus distributions. Our main result is to establish Lipschitz stable estimates on the shear modulus distributions from internal measurements of displacement fields. These estimates imply convergence of a numerical scheme known as the Landweber iteration scheme for reconstructing the shear modulus distributions.

math.AP↗
arXiv2014

Electrical impedance tomography-based pressure-sensing using conductive membrane

This paper presents a mathematical framework for a flexible pressure-sensor model using electrical impedance tomography (EIT). When pressure is applied to a conductive membrane patch with clamped boundary, the pressure-induced surface deformation results in a change in the conductivity distribution. This change can be detected in the current-voltage data ({\it i.e.,} EIT data) measured on the boundary of the membrane patch. Hence, the corresponding inverse problem is to reconstruct the pressure distribution from the data. The 2D apparent conductivity (in terms of EIT data) corresponding to the surface deformation is anisotropic. Thus, we consider a constrained inverse problem by restricting the coefficient tensor to the range of the map from pressure to 2D-apparent conductivity. This paper provides theoretical grounds for the mathematical model of the inverse problem. We develop a reconstruction algorithm based on a careful sensitivity analysis. We demonstrate the performance of the reconstruction algorithm through numerical simulations to validate its feasibility for future experimental studies.

math.AP↗
arXiv2014

Time-domain multiscale shape identification in electro-sensing

This paper presents premier and innovative time-domain multi-scale method for shape identification in electro-sensing using pulse-type signals. The method is based on transform-invariant shape descriptors computed from filtered polarization tensors at multi-scales. The proposed algorithm enjoys a remarkable noise robustness even with far-field measurements at very limited angle of view. It opens a door for pulsed imaging using echolocation and induction data.

math.NA↗
arXiv2014

Viscoelastic modulus reconstruction using time harmonic vibrations

This paper presents a new iterative reconstruction method to provide high-resolution images of shear modulus and viscosity via the internal measurement of displacement fields in tissues. To solve the inverse problem, we compute the Fréchet derivatives of the least-squares discrepancy functional with respect to the shear modulus and shear viscosity. The proposed iterative reconstruction method using this Fréchet derivative does not require any differentiation of the displacement data for the full isotropic linearly viscoelastic model, whereas the standard reconstruction methods require at least double differentiation. Because the minimization problem is ill-posed and highly nonlinear, this adjoint-based optimization method needs a very well-matched initial guess. We find a good initial guess. For a well-matched initial guess, numerical experiments show that the proposed method considerably improves the quality of the reconstructed viscoelastic images.

math.AP↗
arXiv2014

Detection and classification from electromagnetic induction data

In this paper we introduce an efficient algorithm for identifying conductive objects using induction data derived from eddy currents. Our method consists of first extracting geometric features from the induction data and then matching them to precomputed data for known objects from a given dictionary. The matching step relies on fundamental properties of conductive polarization tensors and new invariants introduced in this paper. A new shape identification scheme is introduced and studied. We test it numerically in the presence of measurement noise. Stability and resolution capabilities of the proposed identification algorithm are quantified in numerical simulations.

math.AP↗
arXiv2014

Mathematical modeling in full-field optical coherence elastography

We provide a mathematical analysis of and a numerical framework for full-field optical coherence elastography, which has unique features including micron-scale resolution, real-time processing, and non-invasive imaging. We develop a novel algorithm for transforming volumetric optical images before and after the mechanical solicitation of a sample with sub-cellular resolution into quantitative shear modulus distributions. This has the potential to improve sensitivities and specificities in the biological and clinical applications of optical coherence tomography.

math.OC↗
arXiv2014

Electrical impedance spectroscopy-based nondestructive testing for imaging defects in concrete structures

An electrical impedance spectroscopy-based nondestructive testing (NDT) method is proposed to image both cracks and reinforcing bars in concrete structures. The method utilizes the frequency-dependent behavior of thin insulating cracks: low-frequency electrical currents are blocked by insulating cracks, whereas high-frequency currents can pass through the conducting bars without being blocked by thin cracks. Rigorous mathematical analysis relates the geometric structures of the cracks and bars to the frequency-dependent Neumann-to-Dirichlet data. Various numerical simulations support the feasibility of the proposed method.

math.AP↗
arXiv2014

Mathematical modeling of mechanical vibration assisted conductivity imaging

This paper aims at mathematically modeling a new multi-physics conductivity imaging system incorporating mechanical vibrations simultaneously applied to an imaging object together with current injections. We perturb the internal conductivity distribution by applying time-harmonic mechanical vibrations on the boundary. This enhances the effects of any conductivity discontinuity on the induced internal current density distribution. Unlike other conductivity contrast enhancing frameworks, it does not require a prior knowledge of a reference data. In this paper, we provide a mathematical framework for this novel imaging modality. As an application of the vibration-assisted impedance imaging framework, we propose a new breast image reconstruction method in electrical impedance tomography (EIT). As its another application, we investigate a conductivity anomaly detection problem and provide an efficient location search algorithm. We show both analytically and numerically that the applied mechanical vibration increases the data sensitivity to the conductivity contrast and enhances the quality of reconstructed images and anomaly detection results. For numerous applications in impedance imaging, the proposed multi-physics method opens a new difference imaging area called the vibration-difference imaging, which can augment the time-difference and also frequency-difference imaging methods for sensitivity improvements.

physics.med-ph↗
arXiv2014

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.

math.AP↗
arXiv2014

A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators

A rigorous mathematical theory is developed to explain the super-resolution phenomenon observed in the experiment by F.Lemoult, M.Fink and G.Lerosey (Acoustic resonators for far-field control of sound on a subwavelength scale, Phys. Rev. Lett., 107 (2011)). A key ingredient is the calculation of the resonances and the Green function in the half space with the presence of a system of Helmholtz resonators in the quasi-stationary regime. By using boundary integral equations and generalized Rouche's theorem, the existence and the leading asymptotic of the resonances are rigorously derived. The integral equation formulation also yields the leading order terms in the asymptotics of the Green function. The methodology developed in the paper provides an elegant and systematic way for calculating resonant frequencies for Helmholtz resonators in assorted space settings as well as in various frequency regimes. By using the asymptotics of the Green function, the analysis of the imaging functional of the time-reversal wave fields becomes possible, which clearly demonstrates the super-resolution property. The result provides the first mathematical theory of super-resolution in the context of a deterministic medium and sheds light to the mechanism of super-resolution and super-focusing for waves in deterministic complex media.

math.AP↗