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

产生高分辨率图像的新颖成像方法是实现重大科学发现和技术进步的不可或缺的工具。成像设备有基本的分辨率限制，而超分辨率技术是通过利用关于成像对象的先前信息来绕过这些限制的计算方法。在遥感、超分辨荧光显微镜和量子信息理论等许多应用中，成像目标可以被建模为点源的集合，所采集的信息是正弦波的叠加。该项目开发了新的超分辨率算法，将用于成像和信号处理，并提供性能保证。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为学生提供支持。现有的超分辨理论和计算方法主要与一维均匀抽样情况有关。另一方面，许多应用程序本质上是多维的，采集样本可能很昂贵，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，目的是在理论和实践之间架起桥梁。(1)介绍了新颖高效的多维超分辨算法，并分析了它们的稳定性和分辨率极限。(2)提出了一种新的压缩超分辨算法，该算法需要较少的样本，并严格推导了其采样复杂性和稳定性。(3)荧光分子的使用使生物样品的成像发生了革命性的变化，本项目建立了一个带有期望最小-最大误差的新模型，研究了寻找最优候选算法的方法。这些方法附带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NSF的法定使命，并通过使用基金会的智力优势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.