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Explicit Methods for Linear and Non-Linear Tomography

线性和非线性层析成像的显式方法

基本信息

批准号：
1814104
负责人：
Francois Monard
金额：
$ 16.37万
依托单位：
University of California-Santa Cruz
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1814104&HistoricalAwards=false
关键词：
Explicit Methods Linear Non Tomography

项目摘要

Several inverse problems with applications to medical imaging, geophysical imaging and non-destructive material testing are modeled as integral geometric problems, which consist of reconstructing internal features of materials from their cumulated integrals along a given family of trajectories. Examples include the X-ray/Radon transform used in Computerized Tomography, the travel-time tomography problem in seismology or geophysical prospection, and the Neutron Spin Tomography problem, where a magnetic field inside a material must be recovered from its "non-abelian" integrals. In this project, the investigator and his collaborators focus on integral geometric problems where the complexity lies in the type of object to be reconstructed, in the fact that the geometry of propagation of information is curved, and in the nonlinear character of some of these problems. For each of the problems considered, the task is to put in mathematical terms the answer to the following questions: (i) Is the unknown reconstructible from the given measurements and, if yes, how stable is the inversion? (ii) How to reconstruct the unknown in practice? (iii) How to deal with imperfections in the model and in the measurements (due to instrumental noise for example) and how to quantify the uncertainty induced on the proposed reconstructions?This project focuses on some linear and nonlinear integral geometric problems where unknowns are modeled as functions, tensors, connections over bundles or sections of these bundles, and the measurements are integral functionals of the unknowns. For various "solvable" settings, explicit reconstruction algorithms will be derived and implemented whenever possible, assessing injectivity, stability, implementation, and uncertainty quantification aspects of the inverse problems at play. The proposed methods combine deep theoretical tools (microlocal analysis, harmonic analysis, Clifford analysis and partial differential equations on manifolds) with a concern for implementability, to produce explicit answers, some of which already exist in non-explicit form. Numerical validations will be provided to confirm the implementability of the derivation and uncover the next challenges toward real-life applications.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.

应用于医学成像、地球物理成像和无损材料检测的几个反问题被建模为积分几何问题，其中包括从材料的累积积分沿沿着给定的轨迹族重建材料的内部特征。例子包括X射线/Radon变换用于计算机断层扫描，地震学或地球物理勘探中的走时断层扫描问题，以及中子自旋断层扫描问题，其中材料内部的磁场必须从其“非阿贝尔”积分中恢复。在这个项目中，研究者和他的合作者专注于积分几何问题，其中复杂性在于要重建的对象的类型，事实上，信息传播的几何形状是弯曲的，以及其中一些问题的非线性特征。对于每一个问题的考虑，任务是把在数学术语的答案，以下列问题：（一）是未知的重建从给定的测量，如果是，如何稳定的反演？(ii)如何在实践中重构未知？(iii)如何处理模型和测量中的缺陷（例如由于仪器噪声）以及如何量化拟议重建中引起的不确定性？这个项目的重点是一些线性和非线性的积分几何问题，其中未知数被建模为函数，张量，连接在丛或这些丛的部分，和测量是未知数的积分泛函。对于各种“可解”的设置，显式重建算法将推导和实施，尽可能，评估注入性，稳定性，实施和不确定性量化方面的反问题在发挥作用。所提出的方法结合了联合收割机深的理论工具（微局部分析，调和分析，Clifford分析和偏微分方程的流形上）与关注的可实施性，以产生明确的答案，其中一些已经存在于非明确的形式。将提供数值验证，以确认推导的可实施性，并揭示对现实生活中的应用的下一个挑战。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。