CAREER: Integral Geometry: Theory, Implementations, and Applications

积分几何的一个典型问题是，根据已知的物理量在给定曲线族（如沿不同方向的直线射线）上的平均值，确定体内某些物理量（如密度）的分布。成像科学中的几个问题被建模为像刚才提到的那样的积分几何问题。例子包括：计算机断层扫描中使用的x射线/氡变换，应用于医学成像和国土安全；地震学或地球物理勘探中的走时层析成像问题最近，中子自旋层析成像问题，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，首席研究员将从理论和应用两个层面解决积分几何的核心问题。对于这样的问题，其复杂性在于传播的几何形状，未知参数的表示，有时是问题的非线性方面，从现有数据重建的可行性将首先在连续模型的水平上进行评估。在可能进行反演的情况下，理论反演将通过概念验证实现来验证，这将解决噪声数据、采样/分辨率、正则化和不确定性量化等实际问题。该研究项目将为本科生和研究生以及博士后提供培训、研究经验和职业发展的机会。该项目侧重于积分几何问题的理论和应用分析，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。测量值是所述未知数的线性或非线性积分泛函。除了通常的注入性、稳定性和反演评估外，重点将转向提高积分几何算子的映射特性，并获得某些非线性算子的范围表征。所提出的方法结合了深层理论工具（微局部分析、谐波分析、Clifford分析和流形上的偏微分方程），并关注产生对未知的重建的明确路线，其中一些已经以非显式形式存在。为了补充这一理论议程，解决实际问题，如噪声数据，采样和不确定性量化将需要理论统计和信号处理的方法。将提供数值验证来实现理论推导并揭示现实应用中的下一个挑战。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。

A typical problem of integral geometry is to determine the distribution of some physical quantity inside the body (say, density) from the known averages of this quantity over a given family of curves (say, along straight rays of different directions). Several problems in imaging sciences are modeled as integral geometric problems like the one just mentioned. Examples include: the X-ray/Radon transform used in Computerized Tomography, with applications to medical imaging and homeland security; the travel-time tomography problem in seismology or geophysical prospection; and more recently, the Neutron Spin Tomography problem, where a magnetic field inside a material must be recovered from its 'non-abelian' integrals. In this project, the Principal Investigator will address core questions in integral geometry, on both theoretical and applied levels. For such problems where the complexity lies in the geometry of propagation, the representation of the unknown parameters, and sometimes the non-linear aspects of the problem, the feasibility of reconstruction from available data will be first assessed at the level of the continuous models. In cases where inversion is possible, theoretical inversions will be validated by proof-of-concept implementations which will address practical issues of noisy data, sampling/resolution, regularization and Uncertainty Quantification. This research project is integrated with opportunities for training, research experience and career development for undergraduate and graduate students, as well as for a postdoctoral researcher. This project focuses on the theoretical and applied analysis of integral geometric problems, where unknowns are modeled as functions, tensors, connections over bundles, or sections of these bundles. The measurements are linear or non-linear integral functionals of said unknowns. In addition to the usual injectivity, stability and inversion assessments, focus will be turned toward sharpening the mapping properties of integral geometric operators and obtaining range characterizations for certain non-linear operators. The proposed methods combine deep theoretical tools (microlocal analysis, harmonic analysis, Clifford analysis and partial differential equations on manifolds) with a concern to produce explicit routes toward the reconstruction of the unknowns, some of which already exist in non-explicit form. To complement this theoretical agenda, addressing practical issues such as noisy data, sampling and Uncertainty Quantification will require methods from theoretical statistics and signal processing. Numerical validations will be provided to implement the theoretical derivations 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.