Practical Compressed Sensing

采样，或将连续域现实信号转换为可由计算机操纵的离散数字，是模拟世界和数字世界之间必不可少的桥梁。我们整个数字信息革命都依赖于这个基本过程。然而，在广泛的传感和采样问题中，由于物理限制或时间要求，只能从未知物体获得有限数量的测量。最近在数学上的一个突破，即压缩感知，表明稀疏或可压缩的有限长度离散信号可以从少量的线性、非自适应（即通用）和随机测量中恢复。这很重要，因为许多感兴趣的信号，包括自然图像、诊断图像、视频、语音、音乐，在适当选择的字典中表示时都是稀疏的。本研究将当前的压缩感知方法扩展到具有压倒性实际意义的其他设置。这些扩展包括约束获取、稀疏信号的附加统计先验和无限维情况。本项目的具体目标是：(1)提高信号重构质量；(2)减少达到指定重建质量所需的测量次数；(3)加快重建时间；(4)在实际应用中展示这些成果，特别是在具有挑战性的磁共振成像应用中，包括人类大脑的功能成像。这些目标是通过有效地利用关于未知信号的额外先验来设计优化的采集/传感方案来实现的，从而促进更快、更准确的重建。该项目开发了一系列优化采集设计和信号重建的新算法。在实际数据和应用中对这些方法的性能进行了理论表征和评价。

Sampling, or the conversion of continuous-domain real-life signals to discrete numbers that can be manipulated by computers, is the essential bridge between the analog and the digital world. Our entire digital information revolution relies on this fundamental process. However, in a wide range of sensing and sampling problems, due to physical constraints or timing requirements only a limited number of measurements can be acquired from the unknown object. A recent breakthrough in mathematics under the name compressed sensing shows that sparse or compressible finite length discrete signals can be recovered from small number of linear, non-adaptive (i.e., universal), and random measurements. This is important, because many signals of interest, including natural images, diagnostic images, videos, speech, music are sparse when represented in an appropriately chosen dictionary. This research extends the current methods in compressed sensing to other setups that have overwhelming practical significance. These extensions include constrained acquisition, additional statistical prior on sparse signals, and infinite dimensional cases. The specific goals of this project are to: (1) improve signal reconstruction quality; (2) reduce number of measurements required to achieve a specified reconstruction quality; (3) speed up the reconstruction time; and (4) to demonstrate these gains on real applications, and in particular in challenging magnetic resonance imaging applications, including functional imaging of the human brain. These goals are achieved by effectively exploiting additional priors about unknown signals to design optimized acquisition/sensing schemes that facilitate faster and more accurate reconstruction. The project develops a series of novel algorithms for optimized acquisition design and signal reconstruction. The performance of these methods is theoretically characterized and evaluated in real data and applications.