Coupled-physics imaging methods and geodesic X-ray transforms

改进特征检测是医学和地球物理成像领域的一项持续挑战，给社会带来了巨大的好处，如对医疗条件或自然资源进行充分监测，以及对以前看不见的特征进行成像。该项目提出了两种导致这种改进的方法。第一种方法是设计和理论分析成像方法，以新的方式利用物理现象，导致成像策略具有更高的对比度和分辨率，并提供访问新特征的途径，如肌肉纤维的各向异性特性。第二种方法是提供无失真重建，并准确定位人体或地壳中的包裹体。这是通过使用对用于探测感兴趣介质的波的传播速度的更逼真的描述(具体地说，空间变化)来实现的。所考虑的模型增加了新的技术细节，其中可能会出现额外的现象(例如焦散)，并且有大量悬而未决的数学问题。本项目为其中一些新问题提供了理论和实际的成像答案。本项目侧重于以下方面的理论和数学理解：(I)耦合物理反问题，通常包括使用内部测量的偏微分方程(PDE)中的参数识别问题；和(Ii)测地线X射线变换(著名的二维Radon变换的推广)，及其在可变折射率介质中的X射线层析成像中的应用。在前一个主题中，使用分析和偏微分方程组的方法来推导各种日益复杂的模型的重建算法，包括与医学成像相关的附加本构参数，以及从标量到偏微分方程组的过渡。在后一个主题中，研究人员使用积分几何、傅立叶分析和偏微分方程组的工具，为黎曼曲面上的张量层析成像问题提供关于注入性/稳定性(或缺乏)、重建公式及其数值验证的陈述。还考虑了对高维、衰减变换和其他类型的流动(例如，磁流和恒温流)的概括。

Improving feature detection is a constant challenge in medical and geophysical imaging, with tremendous benefits to society such as the adequate monitoring of medical conditions or natural resources, and the imaging of previously "invisible" features. This project presents two approaches leading to such improvements. A first approach is the design and theoretical analysis of imaging methods exploiting physical phenomena in new ways, leading to imaging strategies with both improved contrast and resolution, and providing access to new features such as anisotropic properties of muscle fibers. A second approach is to provide distortion-free reconstructions and accurately located inclusions in human bodies or the Earth's crust. This is achieved by using more realistic descriptions (specifically, spatially varying) of the propagation speed of the waves used to probe the medium of interest. The model considered becomes augmented with new technicalities, where additional phenomena (e.g., caustics) can occur and open mathematical questions abound. The present project provides theoretical and practical imaging answers to some of these new problems. This project focuses on the theoretical and mathematical understanding of: (i) coupled-physics inverse problems, which often consist of parameter identification problems in partial differential equations (PDE) using internal measurements; and (ii) geodesic X-ray transforms (a generalization of the famous Radon transform, in two dimensions), with applications to X-Ray Tomography in media with variable index of refraction. In the former topic, methods from analysis and PDEs are used to derive reconstruction algorithms for various models of increasing complexity, including additional constitutive parameters relevant for medical imaging, and including a transition from scalar to systems of PDEs. In the latter topic, the investigator uses tools of integral geometry, Fourier analysis, and PDEs to provide statements about injectivity/stability (or lack thereof), reconstruction formulas and their numerical validation, for the tensor tomography problem on Riemannian surfaces. Generalizations to higher dimensions, attenuated transforms and other types of flows (e.g., magnetic and thermostat flows) are also considered.