Explicit Methods for Linear and Non-Linear Tomography

将几个应用于医学成像、地球物理成像和材料无损检测的反问题建模为积分几何问题，该问题包括由材料沿给定轨迹族的累积积分重构材料的内部特征。例子包括计算机层析成像中使用的X射线/Radon变换，地震学或地球物理勘探中的旅行时间层析成像问题，以及中子自旋层析成像问题，在该问题中，材料内部的磁场必须从其“非阿贝尔”积分恢复。在这个项目中，研究人员和他的合作者专注于积分几何问题，其中的复杂性在于要重建的对象的类型，在于信息传播的几何是弯曲的，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，任务是用数学术语给出以下问题的答案：(I)未知是否可以根据给定的测量重建，如果是，反演的稳定性如何？(二)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)以及如何量化所建议重建的不确定性？本项目主要研究一些线性和非线性积分几何问题，其中未知数被建模为函数、张量、丛或这些丛的截面上的连接，并且测量值是未知数的积分泛函。对于各种“可解”的设置，只要有可能，显式重建算法将被导出和实现，以评估反问题的可注入性、稳定性、实现和不确定性量化方面。所提出的方法结合了深入的理论工具(微局部分析、调和分析、Clifford分析和流形上的偏微分方程)和对可实现性的关注，以产生显式解，其中一些已经以非显式形式存在。将提供数字验证，以确认推导的可实施性，并揭示实际应用中的下一个挑战。该奖项反映了NSF的法定使命，并已通过使用基金会的智力优势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.