scholarly journals Holomorphic Extensions Associated with Fourier–Legendre Series and the Inverse Scattering Problem

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1009
Author(s):  
Enrico De Micheli
Keyword(s):  
Spectral Density ◽  
Inverse Scattering ◽  
Scattering Amplitude ◽  
Laguerre Polynomials ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Holomorphic Extension ◽  
Legendre Series ◽  
Partial Waves ◽  
Legendre Expansion

In this paper, we consider the inverse scattering problem and, in particular, the problem of reconstructing the spectral density associated with the Yukawian potentials from the sequence of the partial-waves fℓ of the Fourier–Legendre expansion of the scattering amplitude. We prove that if the partial-waves fℓ satisfy a suitable Hausdorff-type condition, then they can be uniquely interpolated by a function f˜(λ)∈C, analytic in a half-plane. Assuming also the Martin condition to hold, we can prove that the Fourier–Legendre expansion of the scattering amplitude converges uniformly to a function f(θ)∈C (θ being the complexified scattering angle), which is analytic in a strip contained in the θ-plane. This result is obtained mainly through geometrical methods by replacing the analysis on the complex cosθ-plane with the analysis on a suitable complex hyperboloid. The double analytic symmetry of the scattering amplitude is therefore made manifest by its analyticity properties in the λ- and θ-planes. The function f(θ) is shown to have a holomorphic extension to a cut-domain, and from the discontinuity across the cuts we can iteratively reconstruct the spectral density σ(μ) associated with the class of Yukawian potentials. A reconstruction algorithm which makes use of Pollaczeck and Laguerre polynomials is finally given.

Inverse scattering problem for transparent obstacles

1982 ◽  
Vol 92 (2) ◽  
pp. 361-367 ◽  
Author(s):  
Vesselin Petkov
Keyword(s):  
Inverse Scattering ◽  
Convex Domain ◽  
Scattering Amplitude ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Normal Derivative ◽  
Index Of Refraction ◽  
Image Position ◽  
Compact Boundary ◽  
Limiting Values

Let K ⊂ ℝ3 be an open bounded strictly convex domain with smooth connected compact boundary ∂K. SetWe wish to study the filtered scattering amplitude, related to the transmission problemHere w± are the limiting values of w on ∂Ω; from the Ω;± side, while ∂w±/∂n are the corresponding limiting values of the normal derivative ∂w?/∂n on ∂Ω;. The function α(x) ∈ C∞(K¯), called the index of refraction, has the property α(x) > 0 for x ∈ K¯.

Inverse scattering for three-dimensional quasi-linear biharmonic operator

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Markus Harju ◽  
Jaakko Kultima ◽  
Valery Serov
Keyword(s):  
Inverse Scattering ◽  
Space Variable ◽  
Scattering Amplitude ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Regularity Conditions ◽  
Biharmonic Operator ◽  
Essential Information ◽  
Complex Valued

Abstract We consider an inverse scattering problem of recovering the unknown coefficients of a quasi-linearly perturbed biharmonic operator in the three-dimensional case. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove Saito’s formula and uniqueness theorem of recovering some essential information about the unknown coefficients from the knowledge of the high frequency scattering amplitude.

scholarly journals SEMI-ANALYTIC EQUATIONS TO THE COX–THOMPSON INVERSE SCATTERING METHOD AT FIXED ENERGY FOR SPECIAL CASES

2008 ◽  
Vol 22 (23) ◽  
pp. 2191-2199 ◽  
Author(s):  
TAMÁS PÁLMAI ◽  
MIKLÓS HORVÁTH ◽  
BARNABÁS APAGYI
Keyword(s):  
Inverse Scattering ◽  
Scattering Problem ◽  
Identical Particle ◽  
Inverse Scattering Problem ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Fixed Energy ◽  
Angular Momenta ◽  
Partial Waves ◽  
Special Cases

Solution of the Cox–Thompson inverse scattering problem at fixed energy1–3 is reformulated resulting in semi-analytic equations. The new set of equations for the normalization constants and the nonphysical (shifted) angular momenta are free of matrix inversion operations. This simplification is a result of treating only the input phase shifts of partial waves of a given parity. Therefore, the proposed method can be applied for identical particle scattering of the bosonic type (or for certain cases of identical fermionic scattering). The new formulae are expected to be numerically more efficient than the previous ones. Based on the semi-analytic equations an approximate method is proposed for the generic inverse scattering problem, when partial waves of arbitrary parity are considered.

scholarly journals Imaging of bi-anisotropic periodic structures from electromagnetic near-field data

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinh-Liem Nguyen ◽  
Trung Truong
Keyword(s):  
Inverse Scattering ◽  
Maxwell Equations ◽  
Field Data ◽  
Periodic Structures ◽  
Near Field ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Factorization Method ◽  
Unique Determination

AbstractThis paper is concerned with the inverse scattering problem for the three-dimensional Maxwell equations in bi-anisotropic periodic structures. The inverse scattering problem aims to determine the shape of bi-anisotropic periodic scatterers from electromagnetic near-field data at a fixed frequency. The factorization method is studied as an analytical and numerical tool for solving the inverse problem. We provide a rigorous justification of the factorization method which results in the unique determination and a fast imaging algorithm for the periodic scatterer. Numerical examples for imaging three-dimensional periodic structures are presented to examine the efficiency of the method.

scholarly journals Evaluating Performance of Microwave Image Reconstruction Algorithms: Extracting Tissue Types with Segmentation Using Machine Learning

2021 ◽  
Vol 7 (1) ◽  
pp. 5
Author(s):  
Douglas Kurrant ◽  
Muhammad Omer ◽  
Nasim Abdollahi ◽  
Pedram Mojabi ◽  
Elise Fear ◽  
...  
Keyword(s):  
Machine Learning ◽  
Dielectric Properties ◽  
Image Quality ◽  
Inverse Scattering ◽  
Quantitative Assessment ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Malignant Tissue ◽  
Segmentation Technique ◽  
Specific Tissue

Evaluating the quality of reconstructed images requires consistent approaches to extracting information and applying metrics. Partitioning medical images into tissue types permits the quantitative assessment of regions that contain a specific tissue. The assessment facilitates the evaluation of an imaging algorithm in terms of its ability to reconstruct the properties of various tissue types and identify anomalies. Microwave tomography is an imaging modality that is model-based and reconstructs an approximation of the actual internal spatial distribution of the dielectric properties of a breast over a reconstruction model consisting of discrete elements. The breast tissue types are characterized by their dielectric properties, so the complex permittivity profile that is reconstructed may be used to distinguish different tissue types. This manuscript presents a robust and flexible medical image segmentation technique to partition microwave breast images into tissue types in order to facilitate the evaluation of image quality. The approach combines an unsupervised machine learning method with statistical techniques. The key advantage for using the algorithm over other approaches, such as a threshold-based segmentation method, is that it supports this quantitative analysis without prior assumptions such as knowledge of the expected dielectric property values that characterize each tissue type. Moreover, it can be used for scenarios where there is a scarcity of data available for supervised learning. Microwave images are formed by solving an inverse scattering problem that is severely ill-posed, which has a significant impact on image quality. A number of strategies have been developed to alleviate the ill-posedness of the inverse scattering problem. The degree of success of each strategy varies, leading to reconstructions that have a wide range of image quality. A requirement for the segmentation technique is the ability to partition tissue types over a range of image qualities, which is demonstrated in the first part of the paper. The segmentation of images into regions of interest corresponding to various tissue types leads to the decomposition of the breast interior into disjoint tissue masks. An array of region and distance-based metrics are applied to compare masks extracted from reconstructed images and ground truth models. The quantitative results reveal the accuracy with which the geometric and dielectric properties are reconstructed. The incorporation of the segmentation that results in a framework that effectively furnishes the quantitative assessment of regions that contain a specific tissue is also demonstrated. The algorithm is applied to reconstructed microwave images derived from breasts with various densities and tissue distributions to demonstrate the flexibility of the algorithm and that it is not data-specific. The potential for using the algorithm to assist in diagnosis is exhibited with a tumor tracking example. This example also establishes the usefulness of the approach in evaluating the performance of the reconstruction algorithm in terms of its sensitivity and specificity to malignant tissue and its ability to accurately reconstruct malignant tissue.

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