Conical Radon transforms and their applications in tomography

圆锥氡变换及其在层析成像中的应用

• 批准号: 1616564
• 负责人: Gaik Ambartsoumian
• 金额: $ 19.76万
• 依托单位: University of Texas at Arlington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1616564&HistoricalAwards=false
• 关键词: Conical; Radon; transforms; their; applications

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.

这个项目解决了几个新兴成像领域中的图像重建的一些数学问题。具体主题包括单次散射光学断层扫描和伽马射线发射断层扫描，它们被用于医学上对各种疾病的诊断和治疗监测。该项目还考虑了康普顿相机成像，它用于国土安全和射电天文学中的放射源检测。该项目的成果对现代医疗技术、国家安全、太空探索和几个工业应用具有潜在的影响。该项目的重点是培训来自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)和圆锥(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，它将定义在平面上的函数映射到沿某一族折线的积分。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值，而图像重建过程取决于这些变换的稳定反转的可能性。研究人员解决的中心问题包括：BRT/CRT的内射性集的描述及其反演公式，这些变换的微局部性质的研究，它们的范围的描述，以及不完全数据问题的调查。解决这类问题的方法使用了傅立叶分析、偏微分方程组、积分方程式和微局部分析的各种技术组合。