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中的一个持续挑战，对社会有巨大的好处，例如对医疗条件或自然资源的充分监测，以及对以前“不可见”特征的成像。该项目提出了两种导致这种改进的方法。第一种方法是以新的方式利用物理现象的成像方法的设计和理论分析，从而导致具有改进的对比度和分辨率的成像策略，并提供对诸如肌肉纤维的各向异性特性的新特征的访问。第二种方法是提供无失真的重建和精确定位人体或地壳中的夹杂物。这是通过使用用于探测感兴趣介质的波的传播速度的更真实的描述（具体地，空间变化）来实现的。所考虑的模型随着新的技术细节而增强，其中额外的现象（例如，焦散线）可能发生，并且开放的数学问题比比皆是。本项目为这些新问题提供了理论和实践的成像答案。该项目侧重于以下方面的理论和数学理解：（i）耦合物理逆问题，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;以及（ii）测地X射线变换（着名的Radon变换的推广，在二维空间中），应用于具有可变折射率的介质中的X射线层析成像。在前一个主题中，从分析和偏微分方程的方法被用来获得重建算法的各种模型的复杂性不断增加，包括额外的本构相关的医学成像参数，并包括从标量系统的偏微分方程的过渡。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的工具来提供有关注入性/稳定性（或缺乏），重建公式及其数值验证的陈述，用于黎曼曲面上的张量层析成像问题。推广到更高维度、衰减变换和其他类型的流（例如，磁流和恒温器流）也被考虑。

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.