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Coupled-physics imaging methods and geodesic X-ray transforms

耦合物理成像方法和测地 X 射线变换

基本信息

批准号：
1712790
负责人：
Francois Monard
金额：
$ 6.74万
依托单位：
University of California-Santa Cruz
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-08-01 至 2019-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712790&HistoricalAwards=false
关键词：
Coupled physics imaging methods geodesic

项目摘要

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.

改进特征检测是医学和地球物理成像领域不断面临的挑战，对社会有巨大的好处，如对医疗条件或自然资源的充分监测，以及对以前“看不见”的特征的成像。本项目提出了导致这种改进的两种方法。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，从而提高成像策略的对比度和分辨率，并提供新的特征，如肌肉纤维的各向异性特性。第二种方法是提供无扭曲的重建，并精确定位人体或地壳中的内含物。这是通过使用更现实的描述（特别是空间变化）来实现的，用于探测感兴趣的介质的波的传播速度。所考虑的模型被新的技术所增强，其中可能出现额外的现象（例如，焦散），并且存在大量开放的数学问题。本项目为这些新问题提供了理论和实际的成像答案。该项目侧重于对以下方面的理论和数学理解：(i)耦合物理反问题，这些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；（ii）测地线x射线变换（著名的Radon变换在二维中的推广），应用于可变折射率介质的x射线断层扫描。在前一个主题中，来自分析和偏微分方程的方法被用于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系统的过渡。在后一个主题中，研究者使用积分几何、傅立叶分析和偏微分方程的工具，为黎曼曲面上的张量层析成像问题提供了关于注入性/稳定性（或缺乏注入性/稳定性）、重建公式及其数值验证的陈述。推广到更高的维度，衰减变换和其他类型的流动（例如，磁性和恒温流）也被考虑。