Local Tomography and Microlocal Analysis

这个项目的主要目标是解决断层成像中的问题。Quinto教授将开发和测试局部重建算法，并将其应用于电子显微镜（ET）、单光子发射断层扫描（SPECT）、声纳和x射线断层扫描。昆托教授正在开发他的全三维断层扫描算法，sidc和斯德哥尔摩卡罗林斯卡研究所的合作者正在对生物标本的电子显微镜数据进行测试。他正在调整和测试算法在新的扫描几何形状（获取数据的方式）在ET和SPECT。与Andreas Rieder和Thomas Schuster一起，他正在开发三维声纳的局部算法，并将其应用于更一般的数据采集几何。Quinto教授正在开发新的x射线源曲线的x射线断层扫描算法，并将其应用于医学断层扫描。在每种情况下，他将发展纯数学来分析算法对边界或奇点的影响，并利用这些信息改进算法。PI将开发物理模型和他的算法背后的微局部分析。他将与汉斯·鲁加德一起量化物体奇点的强度及其层析成像数据。首席研究员将解决具体的数学问题，并将其应用于工业科学家的实际问题。他的电子显微镜项目是与生物学家和数学家共同进行的，他的算法被用来帮助科学家更好地成像和理解单个分子的结构。x射线断层扫描（CT）项目的动机是一种用于手术室的新型x射线CT扫描仪，他的算法可以用来在医生手术时对身体内部进行成像。他对从氡数据中测量奇点强度的研究可以帮助研究人员量化从层析成像数据中可见的内容，就像首席研究员之前对定性奇点检测的研究表明这些奇点位于体内的位置一样。本提案的工作也支持昆图教授学生的数学教育。他将利用部分国家科学基金会的资助，与至少两名本科生一起完成他提出的研究项目。

The main goal of this project is to solve problems in tomography. Professor Quinto will develop and test local reconstruction algorithms and apply them to electron microscopy (ET), single photon emission tomography (SPECT), Sonar, and X-ray tomography. Professor Quinto is developing his fully three-dimensional tomography algorithm, and collaborators at Sidec and the Karolinska Institute, Stockholm, are testing it on electron microscope data of biological specimens. He is adapting and testing the algorithms on new scanning geometries (ways of taking the data) in ET and SPECT. With Andreas Rieder and Thomas Schuster, he is developing local algorithms for three-dimensional Sonar, and he is applying it to more general data acquisition geometries. Professor Quinto is developing X-ray tomography algorithms for novel X-ray source curves and applying them to medical tomography. In each case, he will develop pure mathematics to analyze what the algorithms do to boundaries or singularities, and he will improve the algorithms using this information. The PI will develop the microlocal analysis behind the physical models and his algorithms. Along with Hans Rullgard, he will quantify the strength of singularities of objects and their tomographic data. The principal investigator will solve specific mathematical problems and apply them to practical problems with scientists in industry. His electron microscopy project is undertaken jointly with biologists and mathematicians, and his algorithm is being used to help scientists better image and understand the structure of individual molecules. The X-ray tomography (CT) project is motivated by a novel X-ray CT scanner for the operating room, and his algorithm can be used to image the inside of the body as doctors operate. His research on measuring strength of singularities from Radon data could help researchers quantify what is visible from tomographic data, much as the principal investigator's previous research on qualitative singularity detection shows where these singularities are located in the body. The work in this proposal also supports the mathematical education of Prof. Quinto's students. He will use some of his NSF support to work with at least two undergraduate students on his proposed research projects.