Mathematical Analysis of Super-Resolution via Nonconvex Optimization and Machine Learning

衍射对光学系统的分辨率施加了基本的限制。因此，在显微镜、天文学和医学成像等领域，从可用测量中辨别细胞结构、遥远的恒星或肿瘤通常是具有挑战性的。这个问题也出现在电子成像中，其中散粒噪声限制了最小像素大小，以及其他应用，包括信号处理、光谱学、雷达和地震学。超分辨率的目标就是迎接这一挑战，从粗略的数据中发现精细的结构。在这个项目中，研究人员将分析超分辨率技术，根据所获得的见解设计新的方法，并将该方法应用于荧光显微镜，这已成为生物学中必不可少的成像工具。拟议的教育和研究活动综合计划将通过培训数据科学、信号处理和机器学习的交叉学科的学生来影响劳动力发展。这将有助于解决工业界和学术界对数据科学家和工程师日益增长的需求。基于非凸优化的现代超分辨技术提供了模型灵活性、计算效率和良好的实证结果。然而，缺乏理论分析来表明这些技术在什么条件下保证有效，或者可能失败。此外，最近的工作表明，基于神经网络的基于学习的方法可以被训练成有效和高效地执行超分辨率。校准这些模型需要最小化一个高度非凸的成本函数。所提出的研究活动将为超分辨率的非凸优化以及相关问题，如线谱估计和盲反卷积提供理论基础。该项目将侧重于点源的超分辨率，这些点源可能代表显微镜中的荧光粒子、天文学中的星体、信号处理中的光谱线或神经科学中的神经元动作电位。研究人员将对使用非凸成本函数拟合点源时出现的几何景观进行数学分析，并研究基于学习的方法在点源信号模型上的特性。为了补充他们的理论研究，他们将把这些技术应用于真实的荧光显微镜数据。这一奖项反映了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。

Diffraction imposes a fundamental limit on the resolution of optical systems. Consequently, in fields such as microscopy, astronomy, and medical imaging, it is often challenging to discern cellular structures, far-away stars, or tumours from the available measurements. The issue also arises in electronic imaging, where shot noise constrains the minimum pixel size, and in other applications, including signal processing, spectroscopy, radar, and seismology. The goal of super-resolution is to meet this challenge, uncovering fine-scale structure from coarse-scale data. In this project the investigators will analyze super-resolution techniques, design new methodology based on the resulting insights, and apply the methodology to fluorescence microscopy, which has become an essential imaging tool in biology. The proposed integrated program of educational and research activities will impact workforce development by training students at the intersection of data science, signal processing, and machine learning. This will contribute to address the rising demand for data scientists and engineers in industry and academia.Modern super-resolution techniques based on nonconvex optimization provide model flexibility, computational efficiency, and yield good empirical results. However, theoretical analysis showing under what conditions these techniques are guaranteed to work, or may fail, is lacking. In addition, recent works show that learning-based methods based on neural networks can be trained to perform super-resolution effectively and efficiently. Calibrating these models requires minimizing a highly nonconvex cost function. The proposed research activities will advance the theoretical underpinnings of nonconvex optimization for super-resolution and related problems such as line-spectra estimation and blind deconvolution. The project will focus on the super-resolution of point sources, which may represent fluorescent particles in microscopy, astral bodies in astronomy, spectral lines in signal processing, or neuron action potentials in neuroscience. The investigators will perform a mathematical analysis of the geometric landscapes that arise when using nonconvex cost functions to fit point sources, and also study the properties of learning-based approaches when deployed on point-source signal models. To complement their theoretical investigations, they will apply these techniques to real fluorescence-microscopy data.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.