CIF: Small: Collaborative Research: Compressed Sensing for High-Resolution Image Inversion

摘要一个人所做的测量或形成的图像是由什么物理、化学或生物构型产生的？这是一个为产生它的领域反转图像的问题，它出现在几乎所有的科学和工程领域。新兴的压缩传感方法在成像科学、信号处理和网络等领域开辟了许多应用领域。然而，它在高分辨率图像反演中的适用性还没有得到证实。本研究的目的是对压缩感知作为一种图像反演原理的性能进行全面的分析。该程序是跨学科的，信号处理构成了成像科学和数学之间的桥梁。压缩传感理论表明，如果图像在已知的基础上是稀疏的，对物理场图像的次采样对图像反转具有可管理的后果。但在现实中，没有物理场在已知的基础上是稀疏的，因此任何稀疏性的假定基础总是与问题的物理选择的实际稀疏性基础不匹配。这就是所谓的模型不匹配。这项研究建立了压缩感知对模型失配的敏感性界限，并将其性能与更成熟的图像反演原理进行了比较。这项研究的目标是在基数过拟合、压缩采样率和对失配的稳健性之间建立定量的权衡。这项研究开发了压缩传感原理，即使在不匹配的情况下也能保持反演的保真度。它将压缩感知理论从一阶建模理论扩展到用于稀疏协方差和频率波数谱估计的二阶理论。

AbstractWhat physical, chemical, or biological configuration produced the measurements one has made or the images one has formed? This is a question of inverting an image for the field that produced it, and it arises in almost all fields of science and engineering. The emerging methodology of compressed sensing has opened up many applications in imaging science, signal processing, and networking. However, its applicability to high-resolution image inversion is as yet unproven. The objective of this research is to provide a comprehensive analysis of the performance of compressed sensing as an image inversion principle. The program is interdisciplinary, with signal processing forming the bridge between imaging science and mathematics.The theory of compressed sensing suggests that sub-sampling of an image of a physical field has manageable consequences for image inversion, provided that the image is sparse in a known basis. But in reality, no physical field is sparse in a known basis and therefore any presumed basis for sparsity is always mismatched to the actual sparsity basis chosen by the physics of the problem. This is called model mismatch. This research establishes bounds on the sensitivities to model mismatch of compressed sensing and compares its performance to more established principles of image inversion. The goal of the research is to establish quantitative trade-offs between basis over-fitting, compressed sampling rate, and robustness to mismatch. The research develops principles for compressed sensing that preserve the fidelity of inversions, even under conditions of mismatch. It extends the theory of compressed sensing from a first-order theory of modeling to a second-order theory for sparse covariance and frequency-wave-number spectrum estimation.