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A New Approach to Imaging by Waves

波成像的新方法

基本信息

批准号：
2106255
负责人：
Fioralba Cakoni
金额：
$ 24.9万
依托单位：
Rutgers University New Brunswick
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2106255&HistoricalAwards=false
关键词：
New Approach Imaging Waves

项目摘要

In many areas of national importance, such as the design and manufacturing of advanced and novel materials, public safety, medical imaging, and underground exploration, it is essential to be able to image and perform nondestructive testing of materials using electromagnetic, sound, or elastic waves. Unfortunately, effective methods for testing complicated materials for structural defects or for identifying unknown targets in an efficient way with little a priori information are still in a state of infancy. This is particularly the case for material that exhibits directional properties, nonlinear interaction with interrogating waves, or peculiar geometric structures, all of which are present in contemporary applications in the above areas. In this project, the investigator and her graduate students will develop entirely new techniques in inverse scattering theory that can handle the aforementioned imaging problems, in order to obtain reliable target signatures or usable information about objects being examined in computationally efficient ways. The goal is to minimize dependence on a priori information describing the physics and/or geometry of unknown targets as well as of mathematical and computational complications arising from the complexity of the hosting background. This study will combine practical applications with the mathematical elegance of new imaging techniques that have recently led to establishing a new field in mathematics called the qualitative approach or direct imaging techniques.This research is a multifaceted effort to develop fast imaging methods of advanced and novel materials (including nonlinear materials) based on generalized linear sampling methods and spectral parameters related to non-scattering phenomena. There are three main projects:1) A Spectral Approach to Imaging with Waves: Motivated by the theory of transmission eigenvalues, this project proposes to develop a general framework for modifying the scattering operator in order to provide new eigenvalue problems associated with the scattering by an inhomogeneity. Particularly important is the determination of these (real or complex) eigenvalues from the scattering data, together with their relation to the material properties of the inhomogeneity. A major effort of the investigator is to apply this technique to image complex structures, including anisotropic/absorbing/dispersive media, thin layers, and meta-surfaces. 2) Interior Eigenvalues and Non-scattering Frequencies: A necessary condition for the non-scattering of a particular incident wave is that the wave number (or a specified parameter) is an eigenvalue of an interior eigenvalue problem defined on the support of the scatterer. On the other hand, the converse is true if at an eigenvalue the corresponding eigenfunction is extendable outside the support as a solution to the Helmholtz equation. Currently, only the case of scatterers with a corner is understood. The investigator states a more general conjecture and lays out a proposed approach to prove it. This project is theoretical in nature, but if successful, it would lead to a more general uniqueness result for the support of the scattering object with one incident wave. 3) Initiate the Development of the Qualitative Imaging Approach for Nonlinear Inhomogeneities. Despite the great interest and extensive research on the theory and computation of qualitative (direct) inversion methods, nothing has been done in connection with imaging media that exhibit nonlinear interaction with interrogating waves, which is the case for many contemporary engineered materials. This project aims to pioneer such a study.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在许多具有国家重要性的领域，例如先进和新型材料的设计和制造，公共安全，医学成像和地下勘探，必须能够使用电磁波，声波或弹性波对材料进行成像和无损检测。不幸的是，用于测试复杂材料的结构缺陷或用于以有效的方式识别未知目标的有效方法仍然处于婴儿期。对于表现出方向性、与询问波的非线性相互作用或特殊几何结构的材料尤其如此，所有这些都存在于上述领域的当代应用中。在这个项目中，研究人员和她的研究生将开发全新的逆散射理论技术，可以处理上述成像问题，以获得可靠的目标签名或有用的信息，被检查的对象在计算效率的方式。目标是尽量减少对描述未知目标的物理和/或几何形状的先验信息的依赖，以及对宿主背景复杂性所引起的数学和计算复杂性的依赖。本研究将联合收割机的实际应用与数学优雅的新成像技术，最近导致建立一个新的领域，数学称为定性的方法或直接imaging techniques.本研究是一个多方面的努力，开发快速成像方法的先进和新型材料（包括非线性材料）的基础上广义线性采样方法和光谱参数相关的非散射现象。主要有三个项目：1）波成像的谱方法：受透射本征值理论的启发，该项目提出开发一个修改散射算子的通用框架，以提供与不均匀性散射相关的新本征值问题。特别重要的是从散射数据中确定这些（真实的或复数的）本征值，以及它们与不均匀性的材料性质的关系。研究人员的一个主要努力是将这种技术应用于图像复杂的结构，包括各向异性/吸收/色散介质，薄层和元表面。2)内部本征值和非散射频率：特定入射波非散射的必要条件是波数（或指定参数）是定义在散射体支撑上的内部本征值问题的本征值。另一方面，匡威亦然，如果在一个特征值对应的特征函数是可扩展的支持以外的一个解决方案的亥姆霍兹方程。目前，只有角散射体的情况下被理解。研究人员陈述了一个更一般的猜想，并提出了一种方法来证明它。这个项目在本质上是理论性的，但如果成功，它将导致一个更一般的唯一性结果的散射对象与一个入射波的支持。3)启动非线性不均匀性定性成像方法的开发。尽管对定性（直接）反演方法的理论和计算有很大的兴趣和广泛的研究，但对于表现出与询问波的非线性相互作用的成像介质，还没有做任何事情，这是许多当代工程材料的情况。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。