Tomography and Microlocal Analysis

层析成像是一门数学和工程学，用于从间接数据中成像对象的内部结构。断层成像算法从这些间接数据中产生物体内部的图像(重建)。重要的层析成像类型包括X射线、声纳、雷达和地震成像。标准的断层成像算法与完整的数据一起工作得很好--足够的数据来准确和稳定地对对象成像(例如，来自全身视图的X射线数据提供完整的X射线断层扫描数据)。然而，许多层析成像问题具有有限的数据--一些数据丢失，标准重建算法通常不会成像对象的所有特征，并且可能在重建中产生伪影。添加的条纹和其他人工产物在有限的数据重建中很常见，本项目的结果将描述这些人工产物的特征，分析它们发生的原因，并确定如何抑制它们。这种伪像的特征将允许研究人员在重建过程中区分物体的特征和伪像。抑制伪影的研究将提供算法，产生更清晰的重建，其中伪影较少可见。这项研究的直接应用包括X射线层析成像、同步辐射成像、地震成像和雷达。这个项目专注于层析成像中的逆问题，它允许研究人员从间接数据中成像物体的内部结构。有许多有效的重建算法可以利用完整的断层扫描数据(例如，来自全身视图的X射线数据)；然而，许多断层扫描问题涉及有限的数据，其中一些数据缺失。该项目的目标是在微观局部分析中开发新的数学方法，以了解和解决有限的数据层析成像问题。目前，该范例仅适用于边界平滑且具有特定方向的有限数据集。由此产生的重建方法将在工业同步加速器数据上进行测试。研究人员还将分析地震成像模型，并开发正常运算符和推广的微局部分析，包括计算其符号。此计算将用于优化操作符，以便高亮显示对象边界并在整个对象中保持一致。该符号指示在重建中哪些对象奇点将可见，哪些对象奇点将不可见。该算子将使用近似逆来实现，并在由波动方程生成的数据和实际数据上进行测试。这些方法还将扩展到双基地雷达应用，其特点是发射机和接收机在相反方向上独立飞行。

Tomography is the mathematics and engineering used to image the internal structure of an object from indirect data. Tomography algorithms produce images (reconstructions) of the inside of objects from this indirect data. Important types of tomography include X-ray, sonar, radar, and seismic imaging. Standard tomographic algorithms work well with complete data--enough data to image the object accurately and stably (for example, X-ray data from views all around the body provides complete X-ray tomography data). However, many tomographic problems have limited data--some data are missing and standard reconstruction algorithms typically do not image all features of the object and might produce artifacts in the reconstruction. Added streaks and other artifacts are common in limited data reconstructions, and the results of this project will characterize the artifacts, analyze why they occur, and determine how to suppress them. This characterization of artifacts will allow researchers to distinguish features of the object from artifacts in the reconstruction. The research to suppress artifacts will provide algorithms that produce clearer reconstructions in which artifacts are less visible. Direct applications of this research include X-ray tomography, synchrotron imaging, seismic imaging, and radar. This project focuses on inverse problems in tomography, which allows researchers to image the internal structure of an object from indirect data. There are many effective reconstruction algorithms that utilize complete tomographic data (for example, X-ray data from views all around the body); however, many tomographic problems involve limited data, in which some data are missing. The goal of this project is to develop new mathematical methods in micro local analysis to understand and solve limited data tomography problems. Currently, this paradigm is valid only for limited data sets with smooth boundaries that have specific orientations. The resulting reconstruction methods will be tested on industrial synchrotron data. The investigator will also analyze models for seismic imaging and develop the microlocal analysis of the normal operator and generalizations, including calculating its symbol. This calculation will be used to refine the operator so that the reconstruction highlights object boundaries and is uniform throughout the object. The symbol indicates which object singularities will be visible and which will be invisible in the reconstruction. The operator will be implemented using the approximate inverse and tested on data generated from the wave equation and real data. These methods will also be extended to bistatic radar applications, characterized by a transmitter and receiver flying independently in opposite directions.