Mathematical inverse problems arising in acoustic imaging

声学成像中出现的数学反问题

• 批准号: RGPIN-2022-04547
• 负责人: Gibson, Peter
• 金额: $ 1.31万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758975
• 关键词: Mathematical; inverse; problems; arising; acoustic

This research program concerns mathematical inverse problems motivated by a natural question from acoustics. To what extent can we "see" with sound? In other words, to what extent can physical properties of a medium be inferred by transmitting a sound pulse toward it and measuring the resulting echoes? Can we do more than create a rough picture (as is currently done with ultrasound medical scans, for example) and determine the actual values of physical parameters such as density of acoustic impedance? Recent mathematical research suggests that the latter, while not currently feasible, may in fact be possible. The propagation of acoustic waves is modelled by a special class of what are known as partial differential equations (PDE). The physical parameters that characterize a body or physical medium through which sound is propagating occur as specific terms in these PDE called coefficients. In medical imaging, for example, when one records acoustic echoes with an ultrasonic sensor, one is in effect recording part of the solution to a PDE, and one wants to determine the coefficients of the equation, which are unknown, from the recorded partial solution. This is known in mathematics as an inverse problem (as opposed to the classical forward problem of computing a solution to a given PDE). The theory of inverse problems is under rapid development, but we still do not understand many basic questions in the subject, and in particular, it is not currently understood how best to compute the coefficient of the PDE governing sound propagation from partial solutions to the equation---in other words, how to make the most accurate possible picture from recorded acoustic echoes! The overall goals of the proposed research program are to develop: (1) novel mathematical methods for solving inverse problems; (2) new computational techniques related to wave phenomena of interest to the broader scientific community, including physicists and engineers; (3) to translate mathematical insights into practical real-world imaging technologies. The research involves both pure mathematics, including the analysis of PDE and related fields, as well as computational methods, including the design, implementation and analysis of algorithms, and will involve a diverse team of talented graduate students, postdoctoral researchers and international collaborators. Achieving the program's objectives will not only yield new mathematics and improve our scientific understanding of wave phenomena, but it offers the possibility of an enhanced capacity for diagnostic imaging, and non-destructive testing of the buildings and structures we rely on.

本研究计划涉及数学逆问题的动机从声学的一个自然的问题。我们能在多大程度上用声音“看”？换句话说，通过向介质发射声脉冲并测量产生的回波，可以在多大程度上推断出介质的物理特性？除了创建一个粗略的图片（例如，目前超声医学扫描所做的）和确定物理参数的实际值（如声阻抗密度），我们还能做更多的事情吗？最近的数学研究表明，后者虽然目前不可行，但实际上是可能的。 声波的传播由一类特殊的偏微分方程（PDE）来模拟。表征声音传播通过的物体或物理介质的物理参数作为这些PDE中的特定项出现，称为系数。例如，在医学成像中，当利用超声传感器记录声学回波时，实际上记录PDE的解的一部分，并且想要从所记录的部分解确定方程的未知系数。这在数学中被称为逆问题（与计算给定PDE的解的经典正问题相反）。反问题的理论正在迅速发展，但我们仍然不了解许多基本问题的主题，特别是，它是目前还不知道如何最好地计算偏微分方程的系数控制声传播从部分解决方案的方程-换句话说，如何使最准确的可能图片从记录的声学回波！拟议的研究计划的总体目标是：（1）解决逆问题的新数学方法;（2）与更广泛的科学界（包括物理学家和工程师）感兴趣的波动现象相关的新计算技术;（3）将数学见解转化为实际的现实世界成像技术。该研究涉及纯数学，包括PDE和相关领域的分析，以及计算方法，包括算法的设计，实现和分析，并将涉及一个由才华横溢的研究生，博士后研究人员和国际合作者组成的多元化团队。 实现该计划的目标不仅将产生新的数学和提高我们对波动现象的科学理解，而且它提供了增强诊断成像能力的可能性，以及我们所依赖的建筑物和结构的无损检测。