Elliptical Radon Transforms in Image Reconstruction

该项目的主要研究人员和他的合作者研究椭圆Radon变换(ERT)，它在近场超声层析成像、合成孔径雷达、地球物理勘探成像和声纳等各种成像方式的双基地模型中发挥着重要作用。双站数据采集结构涉及使用声波或电磁波等信号对介质进行成像的独立发射器和接收器。这种方法有几个优点。在雷达成像的情况下，接收器是被动的；因此，将它们的运动与有源发射器分开，使接收器能够单独在不安全的环境中飞行。在声纳中，数据收集涉及船后拖曳的接收器传输声源。在超声断层成像中，分开的发射器和接收器可以提供例如高角度分辨率的数据集。假设波在介质中的传播速度恒定，所有这些振型的数学模型都通过椭球或椭圆上未知函数的积分所定义的ERT来表示。研究人员开发了基于这种椭圆Radon变换的几种成像方式的图像重建技术。他们解决的主要数学问题包括：在各种实际重要的数据采集几何中发展ERT的微局部分析，发展ERT的精确和近似反演公式和算法，注入性集的描述，以及ERT的范围表征，即上述成像模式的数据一致性条件的表征。研究人员在放射学和生物医学工程领域的合作者进行的实验室超声实验中检验他们的研究结果。主要研究人员和他的合作者解决图像重建问题，这些问题在医学超声反射率成像、地球物理勘探、声纳和合成孔径雷达成像等领域具有重要意义。这些问题中的许多都有类似的数学表示，研究人员为这些问题开发解决方案并推导出数值算法。该项目的目标和新成果对医疗保健技术(例如癌症的早期发现)、国家安全(例如改进的雷达系统)、经济(例如改进的地球物理勘探或工业无损检测)以及现代生活的许多其他领域产生了积极影响，在这些领域，遥感和成像已成为不可或缺的工具。研究人员与他们的合作者一起在实验室超声实验中测试他们的研究结果。研究人员就与该提案直接相关的研究课题向学生提供建议和指导，并培训该领域的未来专家。他们致力于研究生和本科教学，并以入门讲习班、教程和短期课程的形式吸引来自代表性不足群体的学生。调查人员通过在选定的科学期刊上发表出版物和在会议上发言来广泛传播该项目的成果。

The principal investigator of this project and his collaborator study elliptical Radon transforms (ERT), which play an important role in bi-static models of various imaging modalities such as near-field ultrasound tomography, synthetic aperture radar, geophysical exploration imaging, and sonar. Bi-static data acquisition geometries involve separate emitter and receiver to image a medium using signals such as acoustic or electromagnetic waves. This approach offers several advantages. In the case of radar imaging the receivers are passive; hence separating their motion from the active emitters allows receivers alone to be flown in an unsafe environment. In sonar, data collection involves the receivers towed behind the boat transmitting the sound source. In ultrasound tomography separate emitter and receiver can provide data sets, for example, of high angular resolution. Assuming constant speed of wave propagation in the medium, the mathematical model for all these modalities is expressed through the ERT defined by integrals of an unknown function over ellipsoids or ellipses. The investigators develop image reconstruction techniques for several imaging modalities based on such elliptical Radon transforms. The principal mathematical problems they address include: development of microlocal analysis of ERT in various practically important data acquisition geometries, development of exact and approximate inversion formulas and algorithms for ERT, description of injectivity sets, and range characterization of ERT, i.e. the characterization of consistency conditions for the data for the aforementioned imaging modalities. The investigators test the results of their research in laboratory ultrasound experiments performed by their collaborators in radiology and biomedical engineering.The principal investigator and his collaborator solve image reconstruction problems that are of great importance in several fields including ultrasonic reflectivity imaging in medicine, geophysical exploration, sonar and synthetic aperture radar imaging. Many of these problems have similar mathematical representation, and the investigators develop solutions as well as derive numerical algorithms for such problems. The goals and novel results of this project provide a positive impact on healthcare technologies (e.g. early detection of cancer), national security (e.g. improved radar systems), economics (e.g. improved geophysical exploration, or industrial nondestructive testing), and for many other fields of modern life, where the use of remote sensing and imaging has become an indispensable tool. The investigators work with their collaborators to test the results of their research in laboratory ultrasound experiments. The investigators advise and mentor students on research topics directly related to this proposal, and train future specialists of the field. They are committed to graduate and undergraduate teaching and engage students from underrepresented groups in the form of introductory workshops, tutorials and short courses. The investigators broadly disseminate the results of the project through publications in selected scientific journals and presentations at conferences.