CIF: Small: Super-Resolution for Imaging Science

使用依赖硬件的传统方法获得的图像受到可实现的分辨率水平的限制。然而，对高图像分辨率的需求不断增加，以增强医疗诊断、地震危险监控和相机节能，这些都受到硬件支持的限制，而不会增加成本。为了打破硬件造成的分辨率瓶颈，依赖于机器学习的基于软件的超分辨率技术正在开发中，并取得了前所未有的性能。然而，即使是这一类别中最复杂的技术，也缺乏关于如何选择调整参数或在什么条件下可以保证准确的恢复和稳定性的坚实的理论指导。该项目将探讨一类技术背后的理论，称为用于高维传感几何的凸超分辨率技术，并寻求确定精确和稳定恢复等问题的理论极限和性能保证。这一项目的成功结果将有助于推动成像科学中的超分辨率技术。时间反转等成像技术受到成像模式物理条件所决定的分辨率瓶颈，包括有限的源波长、捕获几何中的小方位角以及接收器阵列的带限特性。所提出的研究位于压缩感知和反问题领域的交叉点。超分辨率是反问题和成像科学中长期存在的需求，而稀疏性的参与是最近才出现的。虽然稀疏性只被严格地证明在直接傅立叶测量下对一维信号的超分辨率有用，但该理论有可能扩展到高维环境和偏微分方程测量。在这个项目中，首席调查员的目标是建立这样一个理论扩展，并证明由此产生的高维反演方案可以超过现有一维反演算法对数据矩阵每一列的一系列特别应用。更明确地说，将分析以下两类导致超分辨率的优化问题：1)具有物理驱动的粒子微分方程约束的逆源问题的L1范数正则化优化框架；以及2)有限比特预算下图像量化的全变差范数正则化优化框架。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。

Images obtained using traditional approaches relying on hardware are limited by the level of resolution achievable. However, there is an increasing demand for high image resolutions to enhance medical diagnosis, seismic hazard monitoring, and camera power saving that are limited by what hardware can support without increasing cost. In order to break the resolution bottleneck imposed by hardware, software-based super-resolution techniques that rely on machine learning are being developed that achieve unprecedented performance. However, even the most sophisticated techniques in this category lack solid theoretical guidance about how to choose tuning parameters or about under what conditions exact recovery and stability can be guaranteed. This project will address the theory behind a certain class of techniques, called convex super-resolution techniques for high-dimensional sensing geometries, and seek to determine theoretical limits and performance guarantees for problems such as exact and stable recovery. The successful outcome of this the project will help advance super-resolution techniques in imaging science.Imaging techniques such as time-reversal suffer from a resolution bottleneck determined by the physical conditions of the imaging modality including limited source wavelengths, small azimuth in the acquisition geometry, and band-limited characteristics of the receiver arrays. The proposed research lies at the intersection of the fields of compressed sensing and inverse problems. Super-resolution is a long-standing demand in inverse problems and imaging science, while the involvement of sparsity is quite recent. Although sparsity is only rigorously shown to be useful to the super-resolution of 1D signals under direct Fourier measurements, the theory has the potential to be extended to high dimensional settings and to partial differential equation measurements. In this project, the Principal Investigator aims to build such a theoretical extension and to demonstrate that the resulting high dimensional inversion scheme can outperform a series of ad-hoc applications of the existing 1D inversion algorithms to each column of the data matrix. More explicitly, the following two types of optimization problems that lead to super-resolution will be analyzed: 1) the L1 norm regularized optimization framework for inverse source problems with physics-driven particle differential equation constraints; and 2) the total variation norm regularized optimization framework for image quantization under a finite bit 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.