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，例如高级和新颖材料的设计和制造，公共安全，医学成像和地下探索，必须能够使用电磁，声音或弹性波对材料进行图像和无损测试。不幸的是，用于测试复杂材料的结构缺陷或以有效的方式识别未知目标的有效方法，几乎没有先验信息仍处于婴儿期。对于表现出方向性特性，与询问波的非线性相互作用或特殊的几何结构的材料尤其如此，所有这些相互作用都存在于上述区域的当代应用中。在这个项目中，研究人员和她的研究生将在反向散射理论中开发全新的技术，这些技术可以处理上述成像问题，以获取可靠的目标签名或有关以计算有效方式检查的对象的可用信息。目的是最大程度地减少对描述未知目标物理和/或几何形状的先验信息，以及由托管背景的复杂性引起的数学和计算并发症。这项研究将将实际应用与新成像技术的数学优雅相结合，这些技术最近导致建立了一个名为“定性方法或直接成像技术”的数学领域。这项研究是一项多方面的努力，以开发基于一般的线性采样量和光谱仪的高级和新颖材料（包括非线性材料）的快速成像方法（包括非线性材料（包括非线性材料））。主要项目有三个主要项目：1）通过波动的频谱方法：由传播特征值理论激励，该项目提出了一个通用框架，以修改散射操作员，以便提供与散射相关的新特征值问题。特别重要的是，从散射数据中确定这些（实际或复杂）特征值，以及它们与不均匀性的材料特性的关系。研究人员的一项重大努力是将此技术应用于图像复杂的结构，包括各向异性/吸收/分散介质，薄层和元表面。 2）内部特征值和非散发频率：特定事件波的非散射的必要条件是，波数（或指定的参数）是散射器支持下定义的内部特征值问题的特征值。另一方面，如果在特征值下，相应的特征功能是在支持之外扩展的，则可以作为helmholtz方程的解决方案扩展。目前，只有一个有角落的散点子的情况才能理解。研究人员指出了一个更一般的猜想，并提出了一种证明这一点的方法。该项目本质上是理论上的，但是如果成功的话，它将带来更普遍的独特性结果，从而支持一个事件波。 3）启动针对非线性不均匀性的定性成像方法的发展。尽管对定性（直接）反转方法的理论和计算具有极大的兴趣和广泛的研究，但与成像介质有关的成像介质与询问波的非线性相互作用无关，这对于许多当代工程材料就是这种情况。该项目旨在先驱此研究。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查标准来评估值得支持的。