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任何时候都更丰富的信息，但这种能力也带来了一些挑战。首先，这些系统提供的测量结果通常无法直接解释。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，可以用于以不同的方式收集信息。在任何特定场景中，哪些配置是最好的，这可能事先并不明显。为了应对上述挑战，该项目的重点是（i）开发新的理论和算法来处理测量和提取底层信息，以及（ii）提供如何优化设计此类测量系统并在测量过程中自适应重新配置它们的指导。本项目的大部分研究是通用的，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都具有重大的潜在利益。该项目还包括一个综合的外展和教育计划，重点是通过虚拟和面对面的外展活动，促进K-12学生对STEM概念的可访问性和认识。推动物理世界可测量的前沿需要远远超出直接的线性测量。相反，由于设计或必要性，许多现代传感器仅获取感兴趣的某些对象的间接非线性或概率观测。这个项目的重点是三个激励应用，涉及这种间接测量：（i）相位成像，其中简单的成像系统使用散焦转换（经典不可测）相位信息转化为（可测量的）强度变化;（ii）稀疏超分辨率成像，其中直接成像（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;以及（iii）量子状态和量子过程层析成像，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。结合这些应用，该项目集中在两个首要的研究方向：（i）结构化推理，其中开发新的表示和算法，用于有效地从间接测量中恢复低秩矩阵，结构化因子和其他感兴趣的参数;以及（ii）自适应测量设计，其旨在回答诸如如何最佳地设计和配置测量系统，如何结合结构化推理算法，以及如何基于所收集的测量自适应地调整测量系统状态。使用原子范数最小化作为基础，该项目开发了新的分析的非凸几何的低秩矩阵恢复概率测量和新技术的稀疏密度估计，允许推理的结构化因素和其他参数的概率测量。该项目还利用现代机器学习和基于优化的信号恢复技术开发了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，并最终重建具有增强信噪比，提高分辨率，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的学术价值和更广泛的影响评审标准。

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.