Multidimensional and Compressive Super-Resolution: Theory, Computation, and Fundamental Limits

产生高分辨率图像的新型成像方法是实现重要科学发现和技术进步不可或缺的工具。成像设备具有基本的分辨率限制，而超分辨率技术是通过利用有关成像对象的先验信息来绕过这些限制的计算方法。在遥感、超分辨率荧光显微镜和量子信息论等许多应用中，可以将成像物体建模为点源的集合，所收集的信息是正弦波的叠加。该项目开发新的超分辨率算法，将用于成像和信号处理，并提供其性能保证。纽约城市学院是美国最多元化的大学之一，该项目将通过研究机会支持学生。现有的超分辨理论和计算方法主要针对一维均匀采样情况。另一方面，许多应用程序本质上是多维的，收集样本可能很昂贵，而新的成像技术允许获取专门的信息。本项目围绕三个主题展开，旨在将理论与实践相结合。(1)引入新颖高效的多维超分辨算法，分析其稳定性和分辨极限。(2)提出了需要较少样本的新型压缩超分辨率算法，并严格推导了其采样复杂度和稳定性。(3)荧光分子的使用彻底改变了生物样品的成像，本项目制定了一个具有预期最小-最大误差的新模型，研究候选算法以寻找最优算法。伴随这些方法的理论性能保证将为实践者提供有价值的指导。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Novel imaging methods that produce high resolution images are indispensable tools that have enabled important scientific discoveries and technological advancements. Imaging devices have fundamental resolution limits, and super-resolution techniques are computational methods that are used to bypass such limits by leveraging prior information about an imaged object. For many applications including remote sensing, super-resolution fluorescence microscopy and quantum information theory, the imaged objects can be modeled as a collection of point sources and the collected information is a superposition of sinusoidal waves. This project develops novel super-resolution algorithms that will be used for imaging and signal processing and provides their performance guarantees. The City College of New York is one of the most diverse universities in the United States, and this project will support students through research opportunities.Existing super-resolution theory and computational methods primarily pertain to the one-dimensional uniform sampling case. On the other hand many applications are inherently multidimensional and collecting samples may be expensive while new imaging technology allows for the acquisition of specialized information. This project focuses on three themes with the goal of bridging theory and practice. (1) It introduces novel and efficient multidimensional super-resolution algorithms and analyzes their stability and resolution limits. (2) It formulates novel compressive super-resolution algorithms that require fewer samples and rigorously derives their sampling complexities and stability. (3) Use of fluorescence molecules has revolutionized imaging of biological samples and this project formulates a novel model with an accompanying expected min-max error, studying candidate algorithms in search of an optimal one. The theoretical performance guarantees that accompany these methods will provide valuable guidelines for practitioners.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.