带限信号压缩感知重建及其应用

项目名称： 带限信号压缩感知重建及其应用

项目编号： No.61271012

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 无线电电子学、电信技术

项目作者： 渠刚荣

作者单位： 北京交通大学

项目金额： 88万元

中文摘要： 香浓采样定理它告诉我们，带限可由所有的时域采样频率大于带宽2倍等间隔采样点精确重建。离散Foureir变换的压缩感知说，有限长度的稀疏信号可由频域上的少量随机均匀采样用复的1-范数等式约束极小重建。本项目研究用实的1-范数等式约束极小化方法重建稀疏离散Fourier变换的压缩感知问题，改进用复的1-范数等式约束极小重建需要更多采样点的问题；研究由时间区间均匀（随机和非随机）采样有限个点重建一般带限信号的方法，对于知道带限信号频域支集时，建立重建问题相应离散线性方程组条件数与采样区间的内在关系以及有效重建的关系，对于不知道带限信号频域支集时，建立相应离散格式的压缩感知重建问题的限制等距性质与采样区间的关系和有效重建的关系，用实的1-范数等式约束极小化方法进行重建。由此建立由带限信号有限个均匀采样重建该带限信号的理论和算法，并应用于MR CS重建以及有限角图像重建。

中文关键词： 带限信号；随机均匀采样；压缩感知；磁共振成像；有限角图像重建

英文摘要： The Shannon sampling theorem tell us that a band-limited signal can be reconstructed exactly by its all sampling points whose sampling frequency is bigger than the band width. Compressed sensing (CS) of discrete Fourier transform (DFT) says that a finite sparse discrete signal can be reconstructed by complex 1-norm minimization subject to equality constrain from its uniform sampling Fourier transform at random on small set of frequencies. In the subject, we study CS of DFT with the real 1-norm minimization with equality constrain and so improve that with the complex 1-norm minimization with equality constrain in which reconstructing signal needs more samples. We also study to reconstruct a band-limited signal from its uniformly sampling points in interval [?T, T]. For the support of the signal in frequent domain is known, we establish the relation between the conditioned number of corresponding discrete linear system and the sampling interval and so available reconstruction. For the support of the signal in frequent domain is not known, we establish the relation between RIP of corresponding discrete DFT CS and the sampling interval and so available reconstruction. Therefore, we establish the theory and algorithm of reconstruction a signal from finite samples in time interval and apply them to MR CS reconstructi

英文关键词： band-limited signal；uniform samples at random;compressed；compressed sensing；Magnetic Resonance Imaging；angle-limited image reconstruction