基于基追踪-Moore-Penrose逆矩阵算法的稀疏信号重构
doi: 10.3724/SP.J.1146.2012.00238
Sparse Signal Reconstruction Based on Basis Pursuit-Moore-Penrose Inverse Matrix
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摘要: 压缩感知(Compressed Sensing, CS)稀疏信号重构其本质就是在稀疏约束条件下求解欠定线性方程组，基于迭代加权Lp(001,p=2)类范数算法减小重构误差成为近来稀疏信号重构热点之一。该文提出了基追踪- Moore-Penrose逆矩阵(Basis Pursuit-Moore-Penrose Inverse Matrix, BP-MPIM)算法：(1)由基追踪(Basis Pursuit, BP)算法得到稀疏信号非零元素位置(亦称支撑集，对应于测量矩阵的列)；(2)通过求解由支撑集所对应测量矩阵的子矩阵和CS测量值组成的超定线性方程组实现稀疏信号重构，并证明了由此重构的稀疏信号是其唯一最小二次范数解。仿真的稀疏信号和实测宽带雷达回波信号脉冲压缩结果表明，和原来算法相比，新算法具有更小的重构误差，且误差只存在于其支撑集内。
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关键词:
- 雷达信号处理 /
- 压缩感知(CS) /
- 信号重构 /
- 基追踪(BP) /
- Moore-Penrose逆矩阵
Abstract: The sparse signal reconstruction with Compressed Sensing (CS) is actually solving a system of underdetermined linear equations within the signal sparsity, of which one focus is to reduce recovery errors by the type of iteratively weightedLp(001,p=2) algorithms recently. The Basis Pursuit-Moore-Penrose Inverse Matrix (BP-MPIM) algorithm is proposed in this paper. First, nonzero element coordinates of the sparse signal are acquired by the basis pursuit algorithm, which are renamed with the sparse signal support set (corresponding with columns of the measure matrix). Then, the sparse signal recovery is solved from a set of superdetermined linear equations, which is composed of the submatrix of the sampling matrix and compressed sensing measurements. At the same time, it is proved that the reconstruction of sparse signals by this new algorithm is the one and only minimize L2 norm. Both simulative sparse signals and pulse compressed data of wideband radar echoes indicate that the new algorithm has less recovery errors than the previous algorithms, which are just in the support set. -
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