Practical Compressed Sensing

点击翻译按钮获取中文摘要

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.