Conical Radon transforms and their applications in tomography

该项目解决了一些数学问题的图像重建在几个新兴领域的成像。具体的主题包括单散射光学断层扫描和伽马射线发射断层扫描，它们在医学中用于各种疾病的诊断和治疗监测。该项目还考虑康普顿照相机成像，用于探测国土安全和射电天文学中的辐射源。该项目的成果对现代医疗技术、国家安全、太空探索和一些工业应用具有潜在影响。该项目特别强调培训来自STEM领域历史上代表性不足的群体的学生，包括妇女和少数民族，该项目致力于研究破碎射线（BRT）和锥形（CRT）Radon变换及其在几种现代成像模式中的应用。BRT是一种变换，它将平面上定义的函数映射到它沿沿着某个断裂射线族的积分。CRT将3D空间中定义的函数映射到其在一族圆锥曲面上的积分。这些量通常对应于成像设备的测量，并且图像重建的过程取决于这些变换的稳定反演的可能性。研究人员解决的中心问题包括：BRT/CRT的注入集的描述及其反演公式，这些变换的微局部性质的研究，其范围的描述，以及不完整数据问题的调查。攻击这样的问题的方法利用各种技术的组合，从傅立叶分析，偏微分方程，积分方程和微局部分析。

This project addresses a number of mathematical problems of image reconstruction in several emerging fields of imaging. Specific topics include single scattering optical tomography and gamma-ray emission tomography, which are used in medicine for diagnostics and treatment monitoring of various diseases. The project also considers Compton camera imaging, which is used for detection of radiation sources in homeland security and in radio astronomy. The outcomes of this project have potential impact for modern healthcare technologies, national security, exploration of space, and several industrial applications. A special emphasis in this project is made on training students from groups historically underrepresented in STEM fields, including women and minorities.The project is devoted to the study of broken ray (BRT) and conical (CRT) Radon transforms, and their applications to several modern imaging modalities. BRT is a transform that maps a function defined on a plane to its integrals along a certain family of broken rays. CRT maps a function defined in the 3D space to its integrals over a family of conical surfaces. These quantities often correspond to measurements of an imaging device, and the process of image reconstruction depends on the possibility of stable inversion of these transforms. The central problems addressed by the investigators include: the description of injectivity sets of BRT/CRT and formulas for their inversion, the study of the microlocal properties of these transforms, the description of their range, and the investigation of incomplete data problems. The methodology to attack such problems utilizes various combinations of techniques from Fourier analysis, partial differential equations, integral equations and microlocal analysis.