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The availability of multichannel ground-penetrating radar systems capable of gathering multiview, multistatic, multifrequency data provides novel chances to improve subsurface imaging results. However, customized data processing techniques and smart choices of the measurement setup are needed to find a trade-off between image quality and acquisition time. In this paper, we adopt a Born Approximation-based full 3D approach, which can manage multiview-multistatic, multifrequency data and faces the imaging as a linear inverse scattering problem. The inverse problem is solved by exploiting the truncated singular value decomposition as a regularization scheme. The paper presents a theoretical study aimed at assessing how the reconstruction capabilities of the imaging approach depend on the adopted measurement configuration. In detail, the performance achievable in the standard case of multimonostatic, multifrequency data is compared with that provided by a multiview-multistatic, multifrequency configuration, where the data are gathered by considering a progressively increasing number of transmitting antennas. The comparison of the achievable imaging performance is carried out by exploiting the spectral content and the point spread function, which are general tools to foresee the achievable reconstruction capabilities. Reconstruction results related to a numerical experiment based on full-wave data are also provided. [ABSTRACT FROM AUTHOR]

Microwave sensing is envisioned to be a reliable imaging technique for detecting and monitoring brain tumours. It uses low-power, nonionized microwave signals to visualize the interior of the living tissues. This study automatically exploits categorization of brain tumours using a deep neural network called the YOLOv5 l algorithm in non-invasive microwave images. A modified antipodal structure (including parasitic patch between flares) that operates over the wideband is proposed for the collection of scattering parameters. The staircase parasitic patch located at the centre of the radiator provides the improved radiation performance at low frequencies. Subsequently, a simulation environment was modelled using CST Microwave Studio 2020, which contained a data acquisition system using a designed microwave antenna and head phantom. Next, the image reconstruction is implemented MATLAB software. The detection and classification of tumours were performed using the YOLOv5l model. A dataset with 500 image samples was formed by pre-processing, which was divided into training (70%), testing (15%) and validation (15%). Then, augmentation is used to create final dataset with 1000 training images. The model was successful in locating an early-stage brain tumour with a size of 5 mm. The YOLOv5 model automatically detected tumour (s) and categorized them as benign tumour or malignant tumour based on a predicted bounding box with an accuracy of 91.5%. [ABSTRACT FROM AUTHOR]

In this short note, we present a probabilistic perspective on Schiffer's problem in inverse scattering theory, which asks whether one can uniquely determine the shape of an unknown obstacle by a single far-field measurement. It is a longstanding problem and has received considerable studies in the literature. We show that this conjecture holds true in more general settings in the probabilistic sense. Our new perspective has important implications from the practical viewpoint and also points an interesting direction of research for broader inverse problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse scattering problem associated with an anisotropic medium with a conductive boundary. We will assume that the corresponding far–field pattern is known/measured and we consider two inverse problems. First, we show that the far–field data uniquely determines the boundary coefficient. Next, since it is known that anisotropic coefficients are not uniquely determined by this data we will develop a qualitative method to recover the scatterer. To this end, we study the so–called monotonicity method applied to this inverse shape problem. This method has recently been applied to some inverse scattering problems but this is the first time it has been applied to an anisotropic scatterer. This method allows one to recover the scatterer by considering the eigenvalues of an operator associated with the far–field operator. We present some simple numerical reconstructions to illustrate our theory in two dimensions. For our reconstructions, we need to compute the adjoint of the Herglotz wave function as an operator mapping into H 1 of a small ball. [ABSTRACT FROM AUTHOR]

We present an algorithm for the computation of scattering poles for an impenetrable obstacle with Dirichlet or Robin boundary conditions in acoustic scattering. This paper builds upon the previous work of Cakoni et al. (2020) titled 'A duality between scattering poles and transmission eigenvalues in scattering theory' (Cakoni et al. 2020 Proc. A.476, 20200612 (doi:10.1098/rspa.2020.0612)), where the authors developed a conceptually unified approach for characterizing the scattering poles and interior eigenvalues corresponding to a scattering problem. This approach views scattering poles as dual to interior eigenvalues by interchanging the roles of incident and scattered fields. In this framework, both sets are linked to the kernel of the relative scattering operator that maps incident fields to scattered fields. This mapping corresponds to the exterior scattering problem for the interior eigenvalues and the interior scattering problem for scattering poles. Leveraging this dual characterization and inspired by the generalized linear sampling method for computing the interior eigenvalues, we present a novel numerical algorithm for computing scattering poles without relying on an iterative scheme. Preliminary numerical examples showcase the effectiveness of this computational approach. [ABSTRACT FROM AUTHOR]

A novel image-reconstruction method is proposed for the processing of data acquired at random spatial positions. The images are reconstructed and updated in real time concurrently with the measurements to produce an evolving image, the quality of which is continuously improving and converging as the number of data points increases with the stream of additional measurements. It is shown that the images converge to those obtained with data acquired on a uniformly sampled surface, where the sampling density satisfies the Nyquist limit. The image reconstruction employs a new formulation of the method of scattered power mapping (SPM), which first maps the data into a three-dimensional (3D) preliminary image of the target on a uniform spatial grid, followed by fast Fourier space image deconvolution that provides the high-quality 3D image. [ABSTRACT FROM AUTHOR]

Quantitative inversion of GPR data opens the door to precise characterization of underground environments. However, in order to make the inverse scattering problem solution easier from a computational viewpoint, simplifying assumptions are often applied, i.e., two-dimensional approximations or the consideration of idealized field probes and electromagnetic sources. These assumptions usually produce modeling errors, which can degrade the dielectric reconstruction results considerably. In this article, a processing step based on long short-term memory cells is proposed for the first time to correct the modeling error in a multiantenna GPR setting. In particular, time-domain GPR data are fed into a neural network trained with couples of finite-difference time-domain simulations, where a set of sample targets are simulated in both realistic and idealized configurations. Once trained, the neural network outputs an approximation of multiantenna GPR data as they are collected by an ideal two-dimensional measurement setup. The inversion of the processed data is then accomplished by means of a regularizing Newton-based nonlinear scheme with variable exponent Lebesgue space formulation. A numerical study has been conducted to assess the capabilities of the proposed inversion methodology. The results indicate the possibility of effectively compensating for modeling error in the considered test cases. [ABSTRACT FROM AUTHOR]

We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in R 2 . The obstacle is the polygonal shape and the surface impedance parameter is non-zero constant. We establish the stability results using a single far-field pattern, constituting a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. The stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. Several technical novelties and developments in the mathematical strategy developed for establishing the aforementioned stability results exist. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships between the obstacle's geometric configurations and the wave field's vanishing order at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner with the impedance boundary condition. [ABSTRACT FROM AUTHOR]

The inverse scattering problem for obstacles and cavities with Neumann boundary conditions is studied by using the factorization method. The data from near-field measurements is used to deal with both inverse scattering for obstacles and cavities without making any asymptotic assumption about the distance between the measured surface and the scatterers. Firstly, mathematical model for the inverse scattering problem is given, and the results of the estimation of the double-layer operator are given by using the jump relation of the double-layer potential and Green's formula. Secondly, the decomposition of the near-field operators of obstacles and cavities is studied separately. Finally, the imaging function of obstacles and cavities under Neumann boundary conditions is constructed. [ABSTRACT FROM AUTHOR]

In this paper we study the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a real, positive, multi-humped, fairly smooth but not necessarily analytic potential decaying at infinity. We provide the rigorous semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues, the norming constants, and the reflection coefficient. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse scattering of the rough surfaces with multi-frequency phase and phaseless measurements. We present a high-order recursive iteration method based on the homotopy iteration technique to reconstruct the rough surfaces. The convergence for the multi-frequency homotopy iteration method is obtained under some conditions. Some numerical experiments show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we present a framework for the solution of inverse scattering problems that integrates traditional imaging methods and deep learning. The goal is to image piece-wise homogeneous targets and it is pursued in three steps. First, raw-data are processed via orthogonality sampling method to obtain a qualitative image of the targets. Then, such an image is fed into a U-Net. In order to take advantage of the implicitly sparse nature of the information to be retrieved, the network is trained to retrieve a map of the spatial gradient of the unknown contrast. Finally, such an augmented shape is turned into a map of the unknown permittivity by means of a simple post-processing. The framework is computationally effective, since all processing steps are performed in real-time. To provide an example of the achievable performance, Fresnel experimental data have been used as a validation. [ABSTRACT FROM AUTHOR]

The phase-encoded linear sampling method (PE-LSM) is an inverse scattering technique for reconstructing the shape of a conducting target from scattered electric fields. It is a variant of the well-known linear sampling method (LSM), which solves the nonlinear shape reconstruction problem using linear optimization. The PE-LSM mitigates the primary obstacle to practical imaging via LSM-based processing – its need for copious multistatic-multiview transmit-receive channels. In this study, we evaluate the PE-LSM using experimental data. We collect synthetic aperture data in an anechoic chamber using only a single transmit-receive channel. With the aid of a monostatic-to-multistatic transform, we generate reconstructions of each target via the PE-LSM. The results evince significant improvements in fidelity to the true target geometries compared to imagery generated by both conventional LSM processing and a conventional backprojection-based radar approach.

We derive a new system of integrable derivative non-linear Schrödinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac–Moody algebra A2(1). The construction of the fundamental analytic solutions of L is outlined and they are used to introduce the scattering data, thus formulating the scattering problem for the Lax pair L,M.

This article investigates the extraction of complex refractive indices from the amplitude and phase of the transmitted electric field. In the first step, an incident plane wave has been assumed and the amplitude and phase of the transmitted plane wave is calculated analytically. In this calculation, different values of the complex refractive index have been assumed for the non-magnetic material under test. In fact, the real part and imaginary part of the refractive index are assumed in the range of [1–10] and [0–1], respectively. Furthermore, a general study is made by an assumption of the material thickness to simulation wavelength ratio in the range of [0.01–20]. Due to examining the measurement noise, noisy data are produced for different values of signal-to-noise ratio in the range of [25–40] dB. Due to the difficulties of estimating the refractive index confronted in the theoretical or iterative methods, a Long short-term memory (LSTM) network is proposed and used for the estimation of complex refractive index based on the amplitude and phase of the transmitted electric field. It is shown that the estimation accuracy of about 97% can be achieved in the trained network. Furthermore, the estimation accuracy as a function of thickness-to-wavelength ratio, signal-to-noise ratio, and the values of real and imaginary parts of the refractive index are studied in detail and shown that higher estimation accuracy can be achieved. The simulated results have been confirmed by the measurement for the thickness-to-wavelength ratio below 0.1 and a good agreement has been found. Therefore, the proposed method can replace analytical or repetitive methods as an optimal and more accurate method.

We consider the inverse random potential scattering problem for the two- and threedimensional biharmonic wave equation in lossy media. The potential is assumed to be a microlocally isotropic Gaussian rough field. The main contributions of the work are twofold. First, the unique continuation principle is proved for the fourth order biharmonic wave equation with rough potentials, and the well-posedness of the direct scattering problem is established in the distribution sense. Second, the correlation strength of the random potential is shown to be uniquely determined by the high frequency limit of the second moment of the backscattering data averaged over the frequency band. Moreover, we demonstrate that the expectation in the data can be removed and the data of a single realization is sufficient for the uniqueness of the inverse problem with probability one when the medium is lossless. [ABSTRACT FROM AUTHOR]

We derive a new system of integrable derivative non-linear Schrödinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac–Moody algebra A 2 (1) . The construction of the fundamental analytic solutions of L is outlined and they are used to introduce the scattering data, thus formulating the scattering problem for the Lax pair L , M . [ABSTRACT FROM AUTHOR]

In this paper, we provide an analytical study of the transmission eigenvalue problem with two conductivity parameters. We will assume that the underlying physical model is given by the scattering of a plane wave for an isotropic scatterer. In previous studies, this eigenvalue problem was analyzed with one conductive boundary parameter whereas we will consider the case of two parameters. We prove the existence and discreteness of the transmission eigenvalues as well as study the dependence on the physical parameters. We are able to prove monotonicity of the first transmission eigenvalue with respect to the parameters and consider the limiting procedure as the second boundary parameter vanishes. Lastly, we provide extensive numerical experiments to validate the theoretical work. [ABSTRACT FROM AUTHOR]

We analyze an approximate interior transmission eigenvalue problem in R d for d = 2 or d = 3 , motivated by the transmission problem of a transformation optics-based cloaking scheme and obtained by replacing the refractive index with its first order approximation, which is an unbounded function. Using the radial symmetry we show the existence of (infinitely many) complex transmission eigenvalues and prove their discreteness. Moreover, it is shown that there exists a horizontal strip in the complex plane around the real axis, that does not contain any transmission eigenvalues. [ABSTRACT FROM AUTHOR]

Reconstructing 3D refractive index profile of scatterers using optical microscopy measurements presents several challenges over the conventional microwave and RF domain measurement scenario. These include phaseless and polarization-insensitive measurements, small numerical aperture, as well as a Green's function where spatial frequencies are integrated in a weighted manner such that far-field angular spectrum cannot be probed and high spatial frequencies that permit better resolution are weighed down. As a result of these factors, the non-linearity and the ill-posedness of the inverse problem are quite severe. These limitations have imposed that inverse scattering problems in the microscopy domain largely consider scalar wave approximations and neglect multiple scattering. Here, we present first inverse scattering results for optical microscopy setup where full-wave vectorial formulation and multiple scattering is incorporated. We present (a) how three popular inverse scattering solvers from microwave domain can be adapted for the present inverse problem, (b) the opportunities and challenges presented by each of these solvers, (c) a comparative insight into these solvers and contrast with the simpler Born approximation approach, and (d) potential routes to improve the performance of these solvers for the hard inverse problem of optical microscopy. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem of multidimensional quasilinear hyperbolic systems of diagonal form without self-interaction. In both L 1 and L ∞ frameworks, we will first show the global existence of classical solutions with small initial data, and then prove that the global solution will scatter to free linear waves and study the rigidity aspect of the scattering problem. We also show the inverse scattering result: The scattering field can determine the global solution uniquely, in the L 1 framework. [ABSTRACT FROM AUTHOR]

Inverse obstacle scattering is the recovery of an obstacle boundary from the scattering data produced by incident waves. This shape recovery can be done by iteratively solving a PDE-constrained optimization problem for the obstacle boundary. While it is well known that this problem is typically non-convex and ill-posed, previous investigations have shown that in many settings these issues can be alleviated by using a continuation-in-frequency method and introducing a regularization that limits the frequency content of the obstacle boundary. It has been recently observed that these techniques can fail for obstacles with pronounced cavities, even in the case of penetrable obstacles where similar optimization and regularization methods work for the equivalent problem of recovering a piecewise constant wave speed. The present work investigates the recovery of obstacle boundaries for impenetrable, sound-soft media with pronounced cavities, given multi-frequency scattering data. Numerical examples demonstrate that the problem is sensitive to the choice of iterative solver used at each frequency and the initial guess at the lowest frequency. We propose a modified continuation-in-frequency method which follows a random walk in frequency, as opposed to the standard monotonically increasing path. This method shows some increased robustness in recovering cavities, but can also fail for more extreme examples. An interesting phenomenon is observed that while the obstacle reconstructions obtained over several random trials can vary significantly near the cavity, the results are consistent for non-cavity parts of the boundary. [ABSTRACT FROM AUTHOR]

Data-driven reduced order models (ROMs) have recently emerged as an efficient tool for the solution of inverse scattering problems with applications to seismic and sonar imaging. One requirement of this approach is that it uses the full square multiple-input/multiple-output (MIMO) matrix-valued transfer function as the data for multidimensional problems. The synthetic aperture radar (SAR), however, is limited to the single-input/single-output (SISO) measurements corresponding to the diagonal of the matrix transfer function. Here we present a ROM-based Lippmann-Schwinger approach overcoming this drawback. The ROMs are constructed to match the data for each source-receiver pair separately, and these are used to construct internal solutions for the corresponding source using only the data-driven Gramian. Efficiency of the proposed approach is demonstrated on 2D and 2.5D (3D propagation and 2D reflectors) numerical examples. The new algorithm not only suppresses multiple echoes seen in the Born imaging but also takes advantage of their illumination of some back sides of the reflectors, improving the quality of their mapping. [ABSTRACT FROM AUTHOR]

The availability of multichannel ground-penetrating radar systems capable of gathering multiview, multistatic, multifrequency data provides novel chances to improve subsurface imaging results. However, customized data processing techniques and smart choices of the measurement setup are needed to find a trade-off between image quality and acquisition time. In this paper, we adopt a Born Approximation-based full 3D approach, which can manage multiview-multistatic, multifrequency data and faces the imaging as a linear inverse scattering problem. The inverse problem is solved by exploiting the truncated singular value decomposition as a regularization scheme. The paper presents a theoretical study aimed at assessing how the reconstruction capabilities of the imaging approach depend on the adopted measurement configuration. In detail, the performance achievable in the standard case of multimonostatic, multifrequency data is compared with that provided by a multiview-multistatic, multifrequency configuration, where the data are gathered by considering a progressively increasing number of transmitting antennas. The comparison of the achievable imaging performance is carried out by exploiting the spectral content and the point spread function, which are general tools to foresee the achievable reconstruction capabilities. Reconstruction results related to a numerical experiment based on full-wave data are also provided.

Consider three electromagnetic scattering models, namely, electromagnetic scattering by an elastic body, by a chiral medium, and by a cylinder at oblique incidence. We are concerned with the corresponding inverse problems of determining the locations and shapes of the scatterers from phaseless far-field patterns. There are certain essential differences from the usual inverse electromagnetic scattering problems, and some fundamental conclusions need to be proved. First, we show that the phaseless far-field data are invariant under the translation of the scatterers and prove the reciprocity relations of the scattering data. Then, we justify the unique determination of the scatterers by utilizing the reference ball approach and the superpositions of a fixed point source and plane waves as the incident fields. The proofs are based on the reciprocity relations, Green's formulas, and the analyses of the wave fields in the reference ball. [ABSTRACT FROM AUTHOR]

The modified transmission eigenvalue (MTE) problem was introduced in [S. Cogar, D. Colton, S. Meng and P. Monk, Modified transmission eigenvalues in inverse scattering theory, Inverse Problems 33 2017, 12, Article ID 125002] and used as a target signature for nondestructive testing. In this paper, we study the inverse spectral problem to reconstruct the modified transmission eigenvalues using Cauchy data. We propose a reciprocity gap functional method and show that the MTEs can be determined by solving some linear ill-posed integral equations. Numerical examples for both absorbing and non-absorbing media are presented to validate the effectiveness and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

A hybrid method which is suitable to find numerical solutions for two-dimensional inverse scattering problems connected with cylindrical bodies buried in a half-space with the known orientation is established. The mentioned hybrid method is based on the classical moment method in connection with vertical coordinate while it involves the horizontal coordinate through a spectral expansion. The structure of the spectral expansion which is imposed by the Green's function contributes to improvement of the ill-posedness of the problem along with Tikhonov regularization. By taking this particular structure into consideration one determines the basis functions such that the effect of the ill-posedness is reduced to a minimum. Someillustrative examples show the applicability and accuracy of the method. [ABSTRACT FROM AUTHOR]

The paper is devoted to the inverse scattering problem in order to study the complex media whose electrical and magnetic properties are the tensor functions. The tensor Fourier diffraction theorem, derived in the paper, gives the relations connecting the scattered field and the inherent properties of the scattering medium. The method is based on the first Born approximation. The tensor (dyadic) analysis for derivation of the tensor Fourier diffraction theorem in the coordinate-free form is used. [ABSTRACT FROM AUTHOR]

In this paper, we consider the uniqueness of the phaseless inverse electromagnetic scattering problems. The phaseless data are the modulus of the tangential component of the far‐field pattern measured on the unit sphere and generated by the scatterer as well as the electric dipoles. Based on the superpositions of two electric dipoles with different positions and polarization vectors as the incident fields, the translation invariance property of the phaseless far‐field pattern corresponding to a single incident wave can be broken. A rigorous argument is given to illustrate that the location, shape, and boundary conditions of the obstacle or the refractive index of the medium can be uniquely determined by the phaseless far‐field data. Different from the existing method, we do not need to introduce an additional reference sphere. Thus our method makes the argument of uniqueness much simpler and clearer. [ABSTRACT FROM AUTHOR]

We solve inverse scattering problem for Schrödinger operators with compactly supported potentials on the half line. We discretize S‐matrix: We take the value of the S‐matrix on some infinite sequence of positive real numbers. Using this sequence obtained from S‐matrix, we recover uniquely the potential by a new explicit formula, without the Gelfand–Levitan–Marchenko equation. [ABSTRACT FROM AUTHOR]

The paper is devoted to the electromagnetic inverse scattering problem for a dielectric anisotropic and magnetically isotropic media. The properties of an anisotropic medium with respect to electromagnetic waves are defined by the tensors, which give the relation between the inductions and the fields. The tensor Fourier diffraction theorem derived in the paper can be considered a useful tool for studying tensor fields in inverse problems of electromagnetic scattering. The method is based on the first Born approximation. [ABSTRACT FROM AUTHOR]

Quantitative inversion of GPR data opens the door to precise characterization of underground environments. However, in order to make the inverse scattering problem solution easier from a computational viewpoint, simplifying assumptions are often applied, i.e., two-dimensional approximations or the consideration of idealized field probes and electromagnetic sources. These assumptions usually produce modeling errors, which can degrade the dielectric reconstruction results considerably. In this article, a processing step based on long short-term memory cells is proposed for the first time to correct the modeling error in a multiantenna GPR setting. In particular, time-domain GPR data are fed into a neural network trained with couples of finite-difference time-domain simulations, where a set of sample targets are simulated in both realistic and idealized configurations. Once trained, the neural network outputs an approximation of multiantenna GPR data as they are collected by an ideal two-dimensional measurement setup. The inversion of the processed data is then accomplished by means of a regularizing Newton-based nonlinear scheme with variable exponent Lebesgue space formulation. A numerical study has been conducted to assess the capabilities of the proposed inversion methodology. The results indicate the possibility of effectively compensating for modeling error in the considered test cases.

A hybrid method which is suitable to find numerical solutions for two-dimensional inverse scattering problems connected with cylindrical bodies buried in a half-space with the known orientation is established. The mentioned hybrid method is based on the classical moment method in connection with vertical coordinate while it involves the horizontal coordinate through a spectral expansion. The structure of the spectral expansion which is imposed by the Green's function contributes to improvement of the ill-posedness of the problem along with Tikhonov regularization. By taking this particular structure into consideration one determines the basis functions such that the effect of the ill-posedness is reduced to a minimum. Some illustrative examples show the applicability and accuracy of the method.

In this paper, we consider the inverse scattering problem of determining a penetrable coated inhomogeneous dielectric with different kinds of unknown buried objects inside. The variational approach is adapted to show the well‐posedness of the direct scattering problem. For the inverse problem, we intend to reconstruct the shape and location of the coated dielectric from measured far‐field data by developing a modified factorization method. In fact, we shall first prove that the classical factorization method is effective for the case when a Neumann boundary condition is imposed on the buried object. However, for the case when a Dirichlet boundary condition is imposed on the buried object, the classical factorization method is no longer valid due to the loss of the coercivity of the middle operator of the associated factorization of the far‐field operator. In this case, we are expected to develop a modified factorization method to overcome this difficulty. Finally, some numerical examples are carried out to confirm the validity of the developed algorithms. [ABSTRACT FROM AUTHOR]

In recent decades qualitative inverse scattering methods with eigenvalues as target signatures have received much attention. To understand those methods, knowledge of the properties of the related eigenvalue problems is essential. However, even the existence of eigenvalues for such (nonselfadjoint) problems is a challenging question and existing results for absorbing media are usually established under unrealistic assumptions or a smoothing of the eigenvalue problem. We present a technique to prove the existence of infinitely many eigenvalues for such problems under realistic assumptions. In particular we consider the class of scalar and modified Maxwell nonselfadjoint Steklov eigenvalue problems. In addition, we present stability results for the eigenvalues with respect to changes in the material parameters. In distinction to existing results, the analysis of the present article requires only minimal regularity assumptions. By that we mean that the regularity of the domain is not required to be better than Lipschitz, and the material coefficients are only assumed to be piecewise W1,∞. Also the stability estimates for eigenvalues are obtained solely in Lp-norms (p <∞) of the material perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse interaction scattering problem of recovering an interface between the fluid and piezoelectric solid from acoustic measurements. First, the wellposedness of the interaction model is shown in associated function spaces by the variational method. Then new uniqueness results are proved for the inverse problem by taking far-field data at one fixed frequency, based on a uniform a priori estimate of the solutions of the interaction model. With these results, the factorization method is then justified to reconstruct the shape and location of the interface between the fluid and piezoelectric solid. Finally, we investigate an associated interior transmission eigenvalue problem, and show that the set of interior transmission eigenvalues is at most discrete and with no finite accumulation point under a natural assumption on physical coefficients. [ABSTRACT FROM AUTHOR]

We are concerned with the inverse scattering problem of recovering the refractive indices and transmission coefficients by the corresponding acoustic far-field measurement encoded into the scattering amplitude. Our first uniqueness result is to determine a constant refractive index by the fixed incident direction scattering amplitude, the proof of which is mainly based on the discreteness of the corresponding interior transmission eigenvalues. Then motivated by the previous work (Xiang & Yan 2021), the second uniqueness result is established to recover a piecewise constant refractive index from the far-field pattern at a fixed frequency. [ABSTRACT FROM AUTHOR]

The microwave imaging system for breast tumor/cancer detection requires high sensitivity to detect abnormal tissue that has little contrast in high-density breasts. This paper proposes a qualitative microwave imaging system simulation based on inverse scattering using the sensitivity-maps method. This method utilizes two measurement types for system calibration: a reference object as a scatterer-free background and a calibration object to obtain the system’s impulse response. The object under test (OUT) consists of an object with low dielectric contrast and a phantom with multiple low dielectric contrasts (multi-contrast). Reconstruction is carried out on three types of S-parameter measurement data, namely S11, S21, and a combination of both. S-parameters are measured at several frequencies, which are 3, 10, 14, 15, 16, 20 GHz, and the combination of all those frequencies (multifrequency). Reconstructed images show that the system is capable of reconstructing dielectric objects accurately. Quantitatively, the results show that the multifrequency S21 measurement yields the best image quality with relative root mean squared error (RRMSE) values of 0.1272 and structural similarity index (SSIM) of 0.9076. The designed imaging system also successfully reconstructs multi-contrast phantom accurately with RRMSE of 0.1434 and SSIM of 0.4609. [ABSTRACT FROM AUTHOR]

Inverse scattering by the Marchenko or Gelfand–Levitan equations consists of two steps. The first step consists of solving an integral equation to retrieve the wavefield in the interior of a scattering medium from waves recorded outside of the scattering region. The second step consists of estimating the scattering potential from this wavefield. The estimation of the potential can be justified either by taking a high-frequency limit, or by evaluating the wavefield an infinitesimal time before or after the direct wave. Waveforms obtained in experiments are discretized in time with a sampling interval dt. We show an example that this time discretization precludes the extraction of the potential. The reason is that the frequencies of discretized waveforms are below the Nyquist frequency. This limitation precludes taking the high-frequency limit that is used for estimating the potential from the waveforms in the interior of the scattering region. We present an alternative method to estimate the potential from wavefields in the interior of the scattering region. This method can be used even when the waves are in the strong scattering regime. [ABSTRACT FROM AUTHOR]

Electromagnetic imaging is an emerging technology widely applied in many fields, such as medical imaging, biomedical imaging, and nondestructive testing. In this study, we place transmitter and receiver antennas around an unknown object. We can use the direct sampling method (DSM) to reconstruct the material size and shape of the unknown object on the basis of the scattered field. We apply U-Net to reconstruct electromagnetic images of perfect conductors. Perfect conductors in free space are studied by irradiating a transverse magnetic (TM) polarization wave. Using the scattered electric field measured outside the object together with the boundary conditions on the conductor surface, a set of nonlinear integral equations can be derived and further converted into matrix form by the method of moments. Since an iterative algorithm is computationally expensive and time-consuming, a real-time electromagnetic imaging technique combining deep learning neural networks is proposed for reconstructing the perfect conductors. The initial shapes of the conductors are first computed by DSM by using the scattered electric field measured outside the object. The initial shapes of the conductors are then input to U-Net for training. Numerical results show that U-Net is capable of reconstructing accurate conductor shapes. Therefore, artificial intelligence techniques can reconstruct shapes more accurately than iterative algorithms, when combined with DSM. [ABSTRACT FROM AUTHOR]

The modified transmission eigenvalue problem of elastic waves arises from inverse scattering theory. In this work we first analyze the well-posedness of the modified transmission problem and accomplish a rigorous error analysis of conforming finite element methods. Then we establish two types of multigrid schemes. Finally, we numerically research the effect of a flaw in the material on the modified elastic transmission eigenvalues and apply the multigrid schemes to perform parallel computing. Numerical experiments are presented to verify the theoretical convergence order and the efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

Inverse problems for line sources radiating inside a homogeneous magneto-dielectric cylinder are investigated. The developed algorithms concern the determination of the location and the current of each source. These algorithms are mostly analytical and are based on proper exploitation of the moments obtained by integrating the product of the total field on the cylindrical boundary with complex exponential functions. The information on the unknown parameters of the problem is encoded in these moments, and hence all parameters can be recovered by means of relatively simple explicit expressions. The cases of one and two sources are considered and analyzed. Under certain conditions, the permittivity and permeability of the cylinder are also recovered. The results from two types of numerical experiments are presented: (i) for a single source, the effect of noise on the boundary data is studied, (ii) for two sources, the pertinent nonlinear system of equations is solved numerically and the accuracy of the derived solution is discussed. [ABSTRACT FROM AUTHOR]

We study the fixed angle inverse scattering problem of determining a sound speed from scattering measurements corresponding to a single incident wave. The main result shows that a sound speed close to constant can be stably determined by just one measurement. Our method is based on studying the linearized problem, which turns out to be related to the acoustic problem in photoacoustic imaging. We adapt the modified time-reversal method from [P. Stefanov and G. Uhlmann, Inverse Problems, 25 (2009), 075011] to solve the linearized problem in a stable way, and we use this to give a local uniqueness result for the nonlinear inverse problem. [ABSTRACT FROM AUTHOR]