α-强衰减稀疏信号重构理论研究与算法设计

Compressed sensing is a signal recovery technique that has been proposed in the past 10 years, it is able to stably acquire sparse signals from far fewer samples than required by the Shannon-Nyquist sampling theorem. The α-strongly decaying sparse signals are a special class of sparse signals, which arise from many practical applications including speech communication and audio source separation. Compared with regular sparse signals, it is more difficult and challenging to investigate the recovery of this class of signals. Furthermore, there is less work on them. Based on the theories and techniques of compressed sensing, stochastic matrices, statistics, and algorithm design, etc., this project will design algorithms for acquiring α-strongly decaying and generalized α-strongly decaying sparse signals and study their recovery performance. More specifically, we will first study the properties of α-strongly decaying sparse signals, and then investigate the recovery performance of orthogonal matching pursuit for reconstructing α-strongly decaying sparse signals. Furthermore, we will study the design of recovery algorithms for the α-strongly decaying sparse signals with performance analysis. Finally, we will extend the design and analysis for the α-strongly decaying sparse signals to the generalized α-strongly decaying sparse signals. This project is of essential importance from both theoretical and practical points of view.

压缩感知是最近十余年提出的一种信号重构的技术，它能够在远小于奈奎斯特-香农采样率的条件下可靠地重构稀疏信号。α-强衰减稀疏信号是一类特殊的稀疏信号，它来源于广泛的实际应用，如语音通信、音频源分离等。相比于传统的稀疏信号，α-强衰减稀疏信号的研究难度更大，更具挑战性，已有研究结果更少。本项目拟基于压缩感知、随机矩阵、统计、算法设计等理论和技术，深入研究α-强衰减和广义α-强衰减稀疏信号重构的理论分析与算法设计：1）研究α-强衰减稀疏信号的性质；2）系统分析正交匹配追踪算法对α-强衰减稀疏信号重构的效果；3）α-强衰减稀疏信号快速高效重构算法的设计与性能分析；4）广义α-强衰减稀疏信号快速高效重构算法的设计与性能分析。这些课题的研究具有重要的理论意义和应用价值。

α-强衰减稀疏信号是一类特殊的稀疏信号，它广泛来源于语音通信、音频源分离等领域。相比于传统的稀疏信号，α-强衰减稀疏信号的研究难度更大，更具挑战性，已有研究结果更少。本项目基于压缩感知、随机矩阵、统计、算法设计等理论和技术，深入研究α-强衰减和广义α-强衰减稀疏信号重构的理论分析与算法设计：1)建立了α-强衰减稀疏信号的性质，为稀疏重构算法还原这类信号的性能分析提供帮助; 2）系统分析正交匹配追踪算法对α-强衰减稀疏信号重构的效果，改进了国际数学家大会特邀报告人Gilbert等著名学者的正交匹配追踪算法的重构效果分析,为实际应用中是否采用OMP重构这类信号提供理论依据。理论解释了正交匹配追踪算法重构α-强衰减稀疏信号比重构平稀疏信号、高斯稀疏信号的效果更好；3）设计了广义α-强衰减稀疏信号——二值稀疏信号重构的快速算法二值匹配追踪算法 (BMP) 并分析了其重构性能；设计了广义空间调制活跃用户的快速检测算法并分析了其重构性能；4）首次建立了基于互相干性的通过求解最小化1-范数和2-范数差的优化问题稳健重构稀疏信号的充分条件, 为实际应用中是否可以通过解最小化1-范数和2-范数的差的优化问题重构稀疏信号提供了理论依据。本项目的研究成果可用于医疗成像、信号处理、物联网等领域。