Tomography and Microlocal Analysis

断层扫描是一门数学和工程学，用于从间接数据中对物体的内部结构进行成像。断层扫描算法从这些间接数据产生物体内部的图像（重建）。重要的断层扫描类型包括x射线、声纳、雷达和地震成像。标准的层析成像算法可以很好地处理完整的数据——足够的数据来准确和稳定地成像物体（例如，来自身体周围视图的x射线数据提供完整的x射线层析成像数据）。然而，许多层析成像问题的数据有限——一些数据缺失，标准重建算法通常不能对物体的所有特征进行成像，并且可能在重建中产生伪影。添加条纹和其他伪影在有限的数据重建中很常见，本项目的结果将描述伪影，分析它们发生的原因，并确定如何抑制它们。这种对人工制品的表征将使研究人员能够在重建中区分物体的特征和人工制品。抑制伪影的研究将提供算法，产生更清晰的重建，其中伪影不太明显。该研究的直接应用包括x射线断层成像、同步加速器成像、地震成像和雷达。该项目侧重于断层扫描中的逆问题，这使得研究人员能够从间接数据中对物体的内部结构进行成像。有许多有效的重建算法利用完整的层析成像数据（例如，来自身体周围视图的x射线数据）；然而，许多层析问题涉及的数据有限，其中一些数据缺失。本计划的目标是发展微局部分析的新数学方法，以理解和解决有限的数据断层扫描问题。目前，这种范式仅对具有特定方向的平滑边界的有限数据集有效。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，并开发正常算子的微局部分析和推广，包括计算其符号。此计算将用于改进运算符，以便重建突出显示对象边界并在整个对象中保持一致。符号表示在重建中哪些对象奇点是可见的，哪些是不可见的。该算子将使用近似逆来实现，并在波动方程和实际数据生成的数据上进行测试。这些方法还将扩展到双基地雷达应用，其特点是发射器和接收器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.