FRG: Collaborative Research: Integrated Mathematical Methods in Medical Imaging

该基金支持亚利桑那州立大学和巴罗神经学研究所凯勒成像创新中心的研究人员组成的合作团队。研究小组由五名数学科学家组成，他们在纯谐波分析和应用谐波分析、计算数学和统计学方面具有互补的专业知识，另外还有两名磁共振成像（MRI）专家。该项目强调了MR数据采集和重建领域的科学挑战，包括从稀疏和/或非均匀收集的数据中生成图像，恢复因系统缺陷而损坏的数据，以及从多个接收器获取的数据中快速和稳健地构建图像。研究团队的横切专业知识正在开发应对这些挑战的严格工具，为可证明的结果、可量化的性能度量和有效的算法提供基础。该团队的方法需要对MRI中使用的数据收集程序进行深入的数学研究，包括必须进行的物理限制。这种理解为概念、分析、实施和验证准确、高效和有效的处理成像数据的实用算法提供了基础。该项目特别强调的是MRI数据收集和图像生成方面的统一，即使是知情的研究人员也经常独立考虑这两个方面。正在寻求方法，以便能够联合设计能够最佳地满足医疗需求的采集和后处理技术。特别是，本研究旨在促进数据后处理方法的设计，充分了解原始传感器数据的特征，例如其非均匀采样的光谱性质和潜在的统计变化。来自亚利桑那州立大学和巴罗神经学研究所（BNI）的合作研究小组的活动预计将对加强磁共振成像（MRI）的数学基础产生重大影响。该项目将应用数学的几个领域集中在一个应用问题的循环上，其中改进的数学技术的引入提供了实质性改进性能的潜力。这些，反过来，最终将通过提高医疗诊断的保真度，降低目前非常昂贵的医疗工具的成本，并通过减少成像时间和患者长时间保持不动以获得准确成像的需要来减轻患者的不适，从而产生广泛的社会影响。通过计划研究与其他应用领域的连接，将实现更广泛的影响，例如合成孔径雷达，其在采集和后处理方面的算法挑战与MRI相似。与研究项目相一致的是一个整合教育组件的计划，包括对本科生和研究生的支持，以及新课程的设计。初级学员将接受现代数学训练，这是他们日后追求跨学科前沿专业生涯所需要的。

This grant supports a collaborative team of researchers from Arizona State University and the Keller Center for Imaging Innovation at the Barrow Neurological Institute. The research team consists of five mathematical scientists, representing complementary expertise in pure and applied harmonic analysis, computational mathematics, and statistics, and two experts in Magnetic Resonance Imaging (MRI). The project highlights scientific challenges in the domain of MR data acquisition and reconstruction, including image formation from sparse and/or non-uniformly collected data, restoration of data corrupted by system imperfections, as well as rapid and robust image construction from data acquired by multiple receivers. The crosscutting expertise of the research team is enabling development of rigorous tools for addressing these challenges, providing the underpinnings for provable results, quantifiable measures of performance, and efficient algorithms. The team's approach entails in-depth mathematical study of data collection procedures utilized in MRI, including the physical constraints under which they must be undertaken. This understanding provides the basis for conceptualization, analysis, implementation, and validation of accurate, efficient and effective practical algorithms for processing imaging data. A particular emphasis of the project is unification of the data collection and image generation aspects of MRI, which are often considered independently even by informed researchers. Methodology is sought to enable joint design of acquisition and post-processing techniques that can optimally serve medical requirements. In particular, this research aims to facilitate design of data post-processing methods that are fully informed about the characteristics of the raw sensor data, such as its non-uniformly sampled spectral nature and underlying statistical variations.The activities of the collaborative research team from Arizona State University and the Barrow Neurological Institute (BNI) are expected to have significant impact on strengthening the mathematical foundations of magnetic resonance imaging (MRI). The project focuses several areas of applicable mathematics on a circle of application problems where the introduction of improved mathematical techniques offers potential for substantial performance improvements. These, in turn, will ultimately have broad social impact by improving the fidelity of medical diagnoses, decreasing the cost of medical tools that are presently very expensive, and alleviating patient discomfort by decreasing imaging time and the need for patients to remain motionless for extended periods for accurate imaging. Broader impact will be realized through the connections of the planned research to other application areas, such as synthetic aperture radar, where algorithmic challenges in acquisition and post-processing are similar to those in MRI. Aligned with the research program is a plan for integration of educational components which includes support of undergraduate and graduate students, as well as the design of new courses. Junior participants will be provided with the modern mathematical training which is needed for their later pursuit of cross-disciplinary cutting-edge professional careers.