Explicit Methods for Linear and Non-Linear Tomography

一些应用于医学成像、地球物理成像和无损材料检测的逆问题被建模为积分几何问题，这些问题包括沿着给定的轨迹族从累积积分中重建材料的内部特征。例子包括计算机断层扫描中使用的x射线/氡变换，地震学或地球物理勘探中的旅行时间断层扫描问题，以及中子自旋断层扫描问题，其中必须从其“非阿贝尔”积分中恢复材料内部的磁场。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，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。对于所考虑的每一个问题，任务是用数学术语来回答以下问题：(i)未知是否可以从给定的测量中重建，如果是，反演有多稳定？（二）实践中如何重构未知？（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？本项目主要研究一些线性和非线性积分几何问题，其中未知量被建模为函数、张量、束上的连接或这些束的部分，测量是未知量的积分泛函。对于各种“可解”设置，将推导并尽可能实现显式重建算法，评估反问题的注入性、稳定性、实施和不确定性量化方面。所提出的方法将深层理论工具（微局部分析、谐波分析、Clifford分析和流形上的偏微分方程）与可实施性相结合，产生明确的答案，其中一些已经以非显式形式存在。将提供数值验证，以确认推导的可实现性，并揭示面向现实应用的下一个挑战。该奖项反映了美国国家科学基金会的法定使命，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。

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.