Collaborative Research: CIF: Medium: Structured Inference and Adaptive Measurement Design in Indirect Sensing Systems

现代传感器和成像系统可以捕获比以往更丰富的信息，但这种能力也带来了一些挑战。首先，这些系统提供的测量通常不能直接解释。一个专门设计的算法必须处理它们，以提取感兴趣的信息，并创建，例如，最干净的图像。其次，测量系统本身通常具有不同的配置，可用于以不同的方式收集信息。在任何特定的场景中，可能事先并不清楚哪种配置是最好的。为了应对上述挑战，本项目侧重于(i)开发处理测量和提取底层信息的新理论和算法，以及（ii）为如何优化设计此类测量系统并在测量过程中自适应地重新配置它们提供指导。这个项目中的大部分研究都是一般性的，旨在适用于广泛的测量系统。因此，该项目的进展在生物医学成像、天文学、计算和通信等不同领域具有重大的潜在效益。该项目还包括一项综合外展和教育计划，重点是通过虚拟和面对面的外展活动，促进K-12学生对STEM概念的了解和认识。要拓展物理世界的测量范围，需要的远不止是直接的线性测量。相反，由于设计或需要，许多现代传感器只能获得一些感兴趣对象的间接非线性或概率观察。该项目侧重于涉及间接测量的三个激励应用：(i)相位成像，其中简单的成像系统使用离焦将（传统上不可测量的）相位信息转换为（可测量的）强度变化；（ii）稀疏超分辨率成像，将直接成像（光子计数）与各种类型的量子测量相结合，克服“瑞利诅咒”，实现多点源场景的超分辨率；（iii）量子态和量子过程断层扫描，其中使用概率测量技术来推断量子系统的状态和作用于这些系统的量子过程。与这些应用程序相结合，该项目集中在两个总体研究重点上：(i)结构化推理，其中开发了新的表示和算法，用于有效地从间接测量中恢复低秩矩阵、结构化因素和其他感兴趣的参数；（ii）自适应测量设计，旨在回答如何优化设计和配置测量系统，如何纳入结构化推理算法，以及如何根据收集到的测量自适应调整测量系统状态等问题。以原子范数最小化为基础，该项目开发了对概率测量中低秩矩阵恢复的非凸几何的新分析，以及允许从概率测量中推断结构化因素和其他参数的稀疏密度估计的新技术。该项目还利用现代机器学习和基于优化的信号恢复技术来开发一个有凝聚力的通用框架，使间接测量系统能够自适应地配置测量系统，并最终重建信号，提高信噪比，提高分辨率，减少总体传感预算。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。

Modern sensors and imaging systems can capture far richer information than ever before, but with this capability comes certain challenges. First, the measurements provided by these systems are often not directly interpretable. A specially designed algorithm must process them to extract the information of interest and create, for example, the cleanest possible image. Second, the measurement systems themselves often have different configurations that can be used to collect information in different ways. It may not be evident in advance which configurations are best in any particular scenario. To address the above challenges, this project focuses on (i) developing new theories and algorithms for processing measurements and extracting the underlying information and (ii) providing guidance on how to optimally design such measurement systems and adaptively reconfigure them during the measurement process. Much of the research in this project is general and intended to be applicable across broad classes of measurement systems. Therefore, the advances in this project have significant potential benefits across diverse areas such as biomedical imaging, astronomy, computation, and communications. The project also involves an integrated outreach and education plan, focusing on promoting accessibility and awareness of STEM concepts for K-12 students through virtual and in-person outreach events.Pushing the frontiers of what can be measured about the physical world requires going far beyond direct linear measurements. Instead, by design or necessity, many modern sensors acquire only indirect nonlinear or probabilistic observations of some object of interest. This project focuses on three motivating applications that involve such indirect measurements: (i) phase imaging, in which simple imaging systems use defocus to convert (classically unmeasurable) phase information into (measurable) intensity variations; (ii) sparse super-resolution imaging, in which direct imaging (photon counting) is combined with various types of quantum measurements to overcome "Rayleigh's curse" and achieve super-resolution of scenes with multiple point sources; and (iii) quantum state and quantum process tomography, in which probabilistic measurement techniques are used to infer the states of quantum systems and the quantum processes that act on these systems. Integrated with these applications, the project is focused in two overarching research thrusts: (i) structured inference, in which new representations and algorithms are developed for efficiently recovering low-rank matrices, structured factors, and other parameters of interest from indirect measurements; and (ii) adaptive measurement design, which aims to answer questions such as how to optimally design and configure the measurement system, how to incorporate structured inference algorithms, and how to adaptively adjust the measurement system state based on the collected measurements. Using atomic norm minimization as a foundation, the project develops new analysis of the nonconvex geometry of low-rank matrix recovery from probabilistic measurements and new techniques for sparse density estimation that allow inference of structured factors and other parameters from probabilistic measurements. The project also leverages modern machine learning and optimization-based signal recovery techniques to develop a cohesive and general framework that enables indirect measurement systems to adaptively configure the measurement system and ultimately reconstruct signals with enhanced signal-to-noise ratio, improved resolution, and reduced overall sensing budget.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.