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

什么样的物理、化学或生物结构产生了我们所做的测量或形成的图像？这是一个为产生图像的领域反转图像的问题，几乎在所有科学和工程领域都存在这个问题。新兴的压缩感知方法在成像科学、信号处理和网络中开辟了许多应用。然而，其适用于高分辨率图像反演尚未得到证实。本研究的目的是提供一个全面的分析性能的压缩感知作为图像反演的原则。该计划是跨学科的，与信号处理形成成像科学和数学之间的桥梁。压缩传感理论表明，一个物理场的图像的子采样有图像反演的可管理的后果，提供了图像是稀疏在一个已知的基础。但在现实中，没有一个物理场在已知的基中是稀疏的，因此任何假设的稀疏基总是与问题的物理学所选择的实际稀疏基不匹配。这被称为模型失配。本研究建立了对压缩感知模型失配的敏感性的界限，并将其性能与更成熟的图像反演原理进行了比较。该研究的目标是建立定量的权衡基础过拟合，压缩采样率，和鲁棒性失配。该研究开发了压缩传感的原理，即使在不匹配的条件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二阶稀疏协方差和频率波数谱估计理论。

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.