CIF: Small: Super-Resolution for Imaging Science

使用依赖于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日益增加，以增强医疗诊断、地震灾害监测和相机节能，这些都受到硬件在不增加成本的情况下可以支持的限制。为了打破硬件带来的分辨率瓶颈，正在开发基于软件的超分辨率技术，这些技术依赖于机器学习，实现了前所未有的性能。然而，即使是最复杂的技术在这一类缺乏坚实的理论指导如何选择调谐参数或在什么条件下可以保证准确的恢复和稳定性。该项目将解决某类技术背后的理论，称为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限制和性能保证。该项目的成功将有助于推进成像科学中的超分辨率技术。成像技术（如时间反演）受到成像模态的物理条件（包括有限的源波长、采集几何中的小方位角以及接收器阵列的带限特性）所决定的分辨率瓶颈的影响。 拟议的研究在于在压缩传感和逆问题领域的交叉点。超分辨率是反问题和成像科学中的一个长期需求，而稀疏性的参与是最近的。虽然稀疏性仅被严格证明对直接傅立叶测量下的1D信号的超分辨率有用，但该理论有可能扩展到高维设置和偏微分方程测量。在这个项目中，主要研究者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优于现有的一维反演算法的一系列ad-hoc应用程序的数据矩阵的每一列。 更明确地说，以下两种类型的优化问题，导致超分辨率将被分析：1）L1范数正则化优化框架的反源问题的物理驱动的粒子微分方程约束;和2）有限比特预算下的图像量化的总变差范数正则化优化框架。该奖项反映了NSF的法定使命，并被认为值得支持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。

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.