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7.0

Jun 30, 2018
06/18

by
Habib Ammari; Hai Zhang

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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...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1405.2513

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10.0

Jun 29, 2018
06/18

by
Habib Ammari; Faouzi Triki

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The multifrequency electrical impedance tomography is considered in order to image a conductivity inclusion inside a homogeneous background medium by injecting one current. An original spectral decomposition of the solution of the forward conductivity problem is used to retrieve the Cauchy data corresponding to the extreme case of perfect conductor. Using results based on the unique continuation we then prove the uniqueness of multifrequency electrical impedance tomography and obtain rigorous...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1609.07091

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6.0

Jun 29, 2018
06/18

by
Habib Ammari; Hai Zhang

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We derive an effective medium theory for acoustic wave propagation in bubbly fluid near Minnaert resonant frequency. We start with a multiple scattering formulation of the scattering problem of an incident wave by a large number of identical small bubbles in a homogeneous fluid. Under certain conditions on the configuration of the bubbles, we justify the point interaction approximation and establish an effective medium theory for the bubbly fluid as the number of bubbles tends to infinity. The...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1604.08409

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4.0

Jun 30, 2018
06/18

by
Habib Ammari; Hai Zhang

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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.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1412.2214

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Jun 26, 2018
06/18

by
Habib Ammari; Dehan Chen; Jun Zou

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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...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1502.06803

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8.0

Jun 30, 2018
06/18

by
Habib Ammari; Alden Waters; Hai Zhang

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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...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1409.5138

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7.0

Jun 30, 2018
06/18

by
Habib Ammari; Francisco Romero; Matias Ruiz

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In this paper we use layer potentials and asymptotic analysis techniques to analyze the heat generation due to nanoparticles when illuminated at their plasmonic resonance. We consider arbitrary-shaped particles and both single and multiple particles. For close-to-touching nanoparticles, we show that the temperature field deviates significantly from the one generated by a single nanoparticle. The results of this paper open a door for solving the challenging problems of detecting plasmonic...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1703.00422

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Jun 26, 2018
06/18

by
Habib Ammari; Simon Boulier; Pierre Millien

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We provide a mathematical analysis and a numerical framework for magnetoacoustic tomography with magnetic induction. The imaging problem is to reconstruct the conductivity distribution of biological tissue from measurements of the Lorentz force induced tissue vibration. We begin with reconstructing from the acoustic measurements the divergence of the Lorentz force, which is acting as the source term in the acoustic wave equation. Then we recover the electric current density from the divergence...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1501.04803

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4.0

Jun 30, 2018
06/18

by
Habib Ammari; Jin Keun Seo; Liangdong Zhou

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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 Fr\'echet derivatives of the least-squares discrepancy functional with respect to the shear modulus and shear viscosity. The proposed iterative reconstruction method using this Fr\'echet derivative does not require any differentiation of the displacement data for the...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1409.2932

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5.0

Jun 30, 2018
06/18

by
Habib Ammari; Loc Hoang Nguyen; Laurent Seppecher

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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.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1405.2679

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Jun 28, 2018
06/18

by
Habib Ammari; Jin Keun Seo; Tingting Zhang

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We are aiming to identify the thin insulating inhomogeneities and small conductive inhomogeneities inside an electrically conducting medium by using multi-frequency electrical impedance tomography (mfEIT). The thin insulating inhomogeneities are considered in the form of tubular neighborhood of a curve and small conductive inhomogeneities are regarded as circular disks. Taking advantage of the frequency dependent behavior of insulating objects, we give a rigorous derivation of the potential...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1510.08494

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Jun 26, 2018
06/18

by
Giovanni S. Alberti; Habib Ammari

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The main focus of this work is the reconstruction of the signals $f$ and $g_{i}$, $i=1,...,N$, from the knowledge of their sums $h_{i}=f+g_{i}$, under the assumption that $f$ and the $g_{i}$'s can be sparsely represented with respect to two different dictionaries $A_{f}$ and $A_{g}$. This generalizes the well-known "morphological component analysis" to a multi-measurement setting. The main result of the paper states that $f$ and the $g_{i}$'s can be uniquely and stably reconstructed...

Topics: Mathematics, Analysis of PDEs, Numerical Analysis

Source: http://arxiv.org/abs/1502.04540

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5.0

Jun 28, 2018
06/18

by
Habib Ammari; Matias Ruiz; Sanghyeon Yu; Hai Zhang

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In this paper we use the full Maxwell equations for light propagation in order to analyze plasmonic resonances for nanoparticles. We mathematically define the notion of plasmonic resonance and analyze its shift and broadening with respect to changes in size, shape, and arrangement of the nanoparticles, using the layer potential techniques associated with the full Maxwell equations. We present an effective medium theory for resonant plasmonic systems and derive a condition on the volume fraction...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1511.06817

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Jun 27, 2018
06/18

by
Habib Ammari; Pierre Millien; Matias Ruiz; Hai Zhang

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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...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1506.00866

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10.0

Jun 30, 2018
06/18

by
Habib Ammari; Youjun Deng; Pierre Millien

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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...

Topics: Mathematics, Analysis of PDEs, Mathematical Physics

Source: http://arxiv.org/abs/1412.3656

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Jun 26, 2018
06/18

by
Habib Ammari; Hongjie Dong; Hyeonbae Kang; Seick Kim

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We study an elliptic equation with measurable coefficients arising from photo-acoustic imaging in inhomogeneous media. We establish Holder continuity of weak solutions and obtain pointwise bounds for Green's functions subject to Dirichlet or Neumann condition.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1502.01804

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6.0

Jun 30, 2018
06/18

by
Habib Ammari; Kyungkeun Kang; Kyounghun Lee; Jin Keun Seo

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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...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1409.3650

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4.0

Jun 29, 2018
06/18

by
Habib Ammari; Thomas Widlak; Wenlong Zhang

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Electropermeabilization is a clinical technique in cancer treatment to locally stimulate the cell metabolism. It is based on electrical fields that change the properties of the cell membrane. With that, cancer treatment can reach the cell more easily. Electropermeabilization occurs only with accurate dosage of the electrical field. For applications, a monitoring for the amount of electropermeabilization is needed. It is a first step to image the macroscopic electrical field during the process....

Topics: Analysis of PDEs, Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1603.00764

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9.0

Jun 30, 2018
06/18

by
Habib Ammari; Jin Keun Seo; Tingting Zhang; Liangdong Zhou

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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...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1405.4582

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Jun 30, 2018
06/18

by
Habib Ammari; Laure Giovangigli; Loc Hoang Nguyen; Jin-Keun Seo

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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.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1403.5708

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20

Jun 30, 2018
06/18

by
Giovanni S. Alberti; Habib Ammari; Kaixi Ruan

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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...

Topics: Mathematics, Numerical Analysis, Analysis of PDEs

Source: http://arxiv.org/abs/1410.4119

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4.0

Jun 30, 2018
06/18

by
Habib Ammari; Brian Fitzpatrick; Hyundae Lee; Sanghyeon Yu; Hai Zhang

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The aim of this paper is to show both analytically and numerically the existence of a subwavelength phononic bandgap in bubble phononic crystals. The key is an original formula for the quasi-periodic Minnaert resonance frequencies of an arbitrarily shaped bubble. The main findings in this paper are illustrated with a variety of numerical experiments.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1702.05317

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6.0

Jun 29, 2018
06/18

by
Habib Ammari; Brian Fitzpatrick; David Gontier; Hyundae Lee; Hai Zhang

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The aim of this paper is to provide a mathematical and numerical framework for the analysis and design of bubble meta-screens. An acoustic meta-screen is a thin sheet with patterned subwavelength structures, which nevertheless has a macroscopic effect on the acoustic wave propagation. In this paper, periodic subwavelength bubbles mounted on a reflective surface (with Dirichlet boundary condition) is considered. It is shown that the structure behaves as an equivalent surface with Neumann...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1608.02733

8
8.0

Jun 29, 2018
06/18

by
Habib Ammari; Matias Ruiz; Wei Wu; Sanghyeon Yu; Hai Zhang

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In this paper we derive an impedance boundary condition to approximate the optical scattering effect of an array of plasmonic nanoparticles mounted on a perfectly conducting plate. We show that at some resonant frequencies the impedance blows up, allowing for a significant reduction of the scattering from the plate. Using the spectral properties of a Neumann-Poincare type operator, we investigate the dependency of the impedance with respect to changes in the nanoparticle geometry and...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1602.05019

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10.0

Jun 29, 2018
06/18

by
Habib Ammari; Brian Fitzpatrick; David Gontier; Hyundae Lee; Hai Zhang

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Through the application of layer potential techniques and Gohberg-Sigal theory we derive an original formula for the Minnaert resonance frequencies of arbitrarily shaped bubbles. We also provide a mathematical justification for the monopole approximation of scattering of acoustic waves by bubbles at their Minnaert resonant frequency. Our results are complemented by several numerical examples which serve to validate our formula in two dimensions.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1603.03982

4
4.0

Jun 28, 2018
06/18

by
Habib Ammari; Yat Tin Chow; Jun Zou

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In this work we shall review the (phased) inverse scattering problem and then pursue the phaseless reconstruction from far-field data with the help of the concept of scattering coefficients. We perform sensitivity, resolution and stability analysis of both phased and phaseless problems and compare the degree of ill-posedness of the phased and phaseless reconstructions. The phaseless reconstruction is highly nonlinear and much more severely ill-posed. Algorithms are provided to solve both the...

Topics: Numerical Analysis, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1510.03999

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7.0

Jun 30, 2018
06/18

by
Habib Ammari; Pol Grasland-Mongrain; Pierre Millien; Laurent Seppecher; Jin-Keun Seo

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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...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1401.2337

7
7.0

Jun 30, 2018
06/18

by
Habib Ammari; Hyeuknam Kwon; Yoonseop Lee; Kyungkeun Kang; Jin Keun Seo

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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 ($\sigma$) and permittivity ($\epsilon$) distributions inside the tissues through Maxwell's equations. The corresponding inverse problem consists of...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1409.6095

5
5.0

Jun 29, 2018
06/18

by
Habib Ammari; Mihai Putinar; Matias Ruiz; Sanghyeon Yu; Hai Zhang

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We prove by means of a couple of examples that plasmonic resonances can be used on one hand to classify shapes of nanoparticles with real algebraic boundaries and on the other hand to reconstruct the separation distance between two nanoparticles from measurements of their first collective plasmonic resonances. To this end, we explicitly compute the spectral decompositions of the Neumann-Poincar\'{e} operators associated with a class of quadrature domains and two nearly touching disks. Numerical...

Topics: Spectral Theory, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1602.05268

5
5.0

Jun 30, 2018
06/18

by
Habib Ammari; Matias Ruiz; Sanghyeon Yu; Hai Zhang

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This paper is concerned with the inverse problem of reconstructing a small object from far field measurements. The inverse problem is severally ill-posed because of the diffraction limit and low signal to noise ratio. We propose a novel methodology to solve this type of inverse problems based on an idea from plasmonic sensing. By using the field interaction with a known plasmonic particle, the fine detail information of the small object can be encoded into the shift of the resonant frequencies...

Topics: Spectral Theory, Mathematical Physics, Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1704.04870

5
5.0

Jun 30, 2018
06/18

by
Habib Ammari; Han Wang

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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.

Topics: Mathematics, Numerical Analysis, Computing Research Repository, Computer Vision and Pattern...

Source: http://arxiv.org/abs/1409.3714

5
5.0

Jun 29, 2018
06/18

by
Habib Ammari; Francisco Romero; Cong Shi

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This paper aims at imaging the dynamics of metabolic activity of cells. Using dynamic optical coherence tomography, we introduce a new multi-particle dynamical model to simulate the movements of the collagen and the cell metabolic activity and develop an efficient signal separation technique for sub-cellular imaging. We perform a singular-value decomposition of the dynamic optical images to isolate the intensity of the metabolic activity. We prove that the largest eigenvalue of the associated...

Topics: Medical Physics, Numerical Analysis, Physics, Mathematics

Source: http://arxiv.org/abs/1608.04382

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Jun 30, 2018
06/18

by
Habib Ammari; Yat Tin Chow; Jun Zou

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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...

Topics: Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1410.1253

5
5.0

Jun 30, 2018
06/18

by
Habib Ammari; Faouzi Triki; Chun-Hsiang Tsou

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The multifrequency electrical impedance tomography consists in retrieving the conductivity distribution of a sample by injecting a finite number of currents with multiple frequencies. In this paper we consider the case where the conductivity distribution is piecewise constant, takes a constant value outside a single smooth anomaly, and a frequency dependent function inside the anomaly itself. Using an original spectral decomposition of the solution of the forward conductivity problem in terms...

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1704.04878

5
5.0

Jun 30, 2018
06/18

by
Habib Ammari; Lingyun Qiu; Fadil Santosa; Wenlong Zhang

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In this paper we present a mathematical and numerical framework for a procedure of imaging anisotropic electrical conductivity tensor by integrating magneto-acoutic tomography with data acquired from diffusion tensor imaging. Magneto-acoustic Tomography with Magnetic Induction (MAT-MI) is a hybrid, non-invasive medical imaging technique to produce conductivity images with improved spatial resolution and accuracy. Diffusion Tensor Imaging (DTI) is also a non- invasive technique for...

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1702.05187

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10.0

Jun 29, 2018
06/18

by
Habib Ammari; Hyeuknam Kwon; Seungri Lee; Jin Keun Seo

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This paper presents a static electrical impedance tomography (EIT) technique that evaluates abdominal obesity by estimating the thickness of subcutaneous fat. EIT has a fundamental drawback for absolute admittivity imaging because of its lack of reference data for handling the forward modeling errors. To reduce the effect of boundary geometry errors in imaging abdominal fat, we develop a depth-based reconstruction method that uses a specially chosen current pattern to construct reference-like...

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1607.05893

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14

Jun 29, 2018
06/18

by
Giovanni S. Alberti; Habib Ammari; Francisco Romero; Timothée Wintz

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This paper provides a mathematical analysis of ultrafast ultrasound imaging. This newly emerging modality for biomedical imaging uses plane waves instead of focused waves in order to achieve very high frame rates. We derive the point spread function of the system in the Born approximation for wave propagation and study its properties. We consider dynamic data for blood flow imaging, and introduce a suitable random model for blood cells. We show that a singular value decomposition method can...

Topics: Probability, Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1604.04604

9
9.0

Jun 29, 2018
06/18

by
Tasawar Abbas; Habib Ammari; Guanghui Hu; Abdul Wahab; Jong Chul Ye

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The concept of scattering coefficients has played a pivotal role in a broad range of inverse scattering and imaging problems in acoustic, and electromagnetic media. In view of their promising applications in inverse problems related to mathematical imaging and elastic cloaking, the notion of elastic scattering coefficients of an inclusion is presented from the perspective of boundary layer potentials and a few properties are discussed. A reconstruction algorithm is developed and analyzed for...

Topics: Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1606.08070

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6.0

Jun 29, 2018
06/18

by
Robert Plato

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We study quadrature methods for solving Volterra integral equations of the first kind with smooth kernels under the presence of noise in the right-hand sides, with the quadrature methods being generated by linear multistep methods. The regularizing properties of an a priori choice of the step size are analyzed, with the smoothness of the involved functions carefully taken into consideration. The balancing principle as an adaptive choice of the step size is also studied. It is considered in a...

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1604.08703

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8.0

Jun 30, 2018
06/18

by
Habib Ammari; Elie Bretin; Pierre Millien; Laurent Seppecher; Jin-Keun Seo

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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...

Topics: Mathematics, Optimization and Control

Source: http://arxiv.org/abs/1405.4987

4
4.0

Jun 29, 2018
06/18

by
Robert Plato

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This paper deals with Lavrentiev regularization for solving linear ill-posed problems, mostly with respect to accretive operators on Hilbert spaces. We present converse and saturation results which are an important part in regularization theory. As a byproduct we obtain a new result on the quasi-optimality of a posteriori parameter choices. Results in this paper are formulated in Banach spaces whenever possible.

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1607.04879

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11

Jun 29, 2018
06/18

by
Giovanni S. Alberti; Habib Ammari; Bangti Jin; Jin-Keun Seo; Wenlong Zhang

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This paper provides an analysis of the linearized inverse problem in multifrequency electrical impedance tomography. We consider an isotropic conductivity distribution with a finite number of unknown inclusions with different frequency dependence, as is often seen in biological tissues. We discuss reconstruction methods for both fully known and partially known spectral profiles, and demonstrate in the latter case the successful employment of difference imaging. We also study the reconstruction...

Topics: Numerical Analysis, Mathematics

Source: http://arxiv.org/abs/1602.04312

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35

Jul 26, 2018
07/18

by
Plato; Kyriakos Zambas

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Δ

Topics: State, The -- Early works to 1800, JC, Political science -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36284

53
53

Jul 26, 2018
07/18

by
Plato; Kyriakos Zambas

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Ε

Topics: JC, State, The -- Early works to 1800, Political science -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36262

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48

Jul 26, 2018
07/18

by
Plato; Kyriakos Zambas

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Γ

Topics: JC, Political science -- Early works to 1800, State, The -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36191

53
53

Jul 26, 2018
07/18

by
Plato; Kyriakos Zambas

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Α

Topics: Political science -- Early works to 1800, JC, State, The -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36088

36
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07/18

by
Kyriakos Zambas; Plato

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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος B

Topics: Political science -- Early works to 1800, JC, State, The -- Early works to 1800

Source: https://www.gutenberg.org/ebooks/36116

83
83

Jul 26, 2018
07/18

by
I. N. (Ioannes N.) Grypares; Plato

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Book from Project Gutenberg: Πολιτεία, Τόμος 2

Topics: Utopias -- Early works to 1800, Justice -- Early works to 1800, Political science -- Early works to...

Source: https://www.gutenberg.org/ebooks/39493

114
114

Jul 25, 2018
07/18

by
I. N. (Ioannes N.) Grypares; Plato

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Book from Project Gutenberg: Πολιτεία, Τόμος 3

Topics: JC, Political science -- Early works to 1800, Utopias -- Early works to 1800, Justice -- Early...

Source: https://www.gutenberg.org/ebooks/39524

78
78

Jul 25, 2018
07/18

by
Plato; I. N. (Ioannes N.) Grypares

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Book from Project Gutenberg: Πολιτεία, Τόμος 4

Topics: Utopias -- Early works to 1800, JC, Justice -- Early works to 1800, Political science -- Early...

Source: https://www.gutenberg.org/ebooks/39530

108
108

Jul 26, 2018
07/18

by
Plato; I. N. (Ioannes N.) Grypares

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Book from Project Gutenberg: Πολιτεία, Τόμος 1

Topics: Utopias -- Early works to 1800, Political science -- Early works to 1800, JC, Justice -- Early...

Source: https://www.gutenberg.org/ebooks/39476