Shape Based Tomographic Inversion for Maximal Geometric Resolution

这项研究涉及开发一套计算工具，用于解决从医学成像和制造部件的非破坏性评估到地下环境修复和民用基础设施监测等领域出现的图像形成问题。这些问题以及许多其他问题的共同点是需要根据沿外围收集的数据来描述空间区域的内部结构。虽然常见的例子，如x射线计算机断层扫描或磁共振成像产生高分辨率图像，但由于与底层传感技术的物理相关的复杂性以及在何处和如何获取数据的限制，这项工作中感兴趣的问题更具挑战性。为了应对这些挑战，本项目的工作利用了这样一个事实，即在许多情况下，处理目标是识别包含缺陷或其他异常的特定兴趣区域。通过工程、数学和计算的多学科努力，研究人员开发和研究了一种强大的新型创新的基于形状的计算工具，它结合了变分活动表面模型、水平集方法、边界元素方法和偏微分方程，直接从基于pde的一般层析模式的原始层析数据中检测/重建感兴趣的几何结构。这项研究有可能在从层析测量中提取几何知识方面产生重大的范式转变，这些知识适用于具有挑战性的物理和限制性传感场景的各种社会重要应用。这里特别感兴趣的是岩土表面波反演、无损检测和环境监测的阻抗层析成像问题。

This research involves the development of a set of computational tools for addressing problems of image formation arising in fields ranging from medical imaging and non-destructive evaluation of the manufactured components to subsurface environmental remediation and civil infrastructure monitoring. Common to these as well as many other problems is the need to characterize the internal structure of a region of space from data collected along the periphery. While common examples such as X-ray Computed Tomography or Magnetic Resonance Imaging yield high resolution images, the problems of interest in this effort are far more challenging due to complications associated with the physics of the underlying sensing technology as well as restrictions concerning where and how data can be acquired.To address these challenges, the work in this project exploits the fact that in many cases, the processing objective is the identification of specific regions of interest containing flaws or other anomalies. Through a multidisciplinary effort in engineering, mathematics, and computation, the investigators develop and study a powerful new class of innovative shaped based computational tools which combine variational Active Surface models, Level Set Methods, Boundary Element Methods, and Partial Differential Equations to detect/reconstruct geometric structures of interest directly from raw tomographic data in a general class of PDE-based tomographic modalities. This study has the potential to yield a substantial paradigm shift in extracting geometric knowledge from tomographic measurements for a diverse collection of societally important applications characterized by challenging physics and restrictive sensing scenarios. Of specific interest here are problems of geotechnical surface wave inversion, non-destructive testing, and impedance tomography for environmental monitoring.