基于分式惩罚函数的块稀疏信号重构

Block Sparse Signal Reconstruction Based on Fractional Penalty Function

摘要: 压缩感知作为一种新颖的信号处理理论突破了传统的采样定理，以信号的稀疏性和可压缩性为基础，实现对信号的高效获取和精确重构，在各个领域均有实际意义。本文以压缩感知为基础，研究基于分式惩罚函数的块稀疏信号重构问题，即以分式函数代替‖Χ‖_2,1，L2,1将范数优化问题转化为FP_B优化问题，同时给出了FP_B-DC算法，并用ADMM算法解决FP_B-DC算法遗留下来的凸子问题。另外我们进行了数值仿真模拟实验，在信噪比、重构准确率等方面与Lasso-ADMM算法、FP-DC算法做了相关比较，实验表明：成功率随稀疏度增加时，新算法的成功率显著高于其余两种算法；SNR在随稀疏度增加时，整体上FP_B-DC算法的SNR高于其余两种算法；FP_B-DC算法在重构性能上确实具有一定优势。

Abstract: As a novel signal processing theory, compressed sensing breaks through the traditional sampling theorem. Based on the sparsity and compressibility of the signal, it can achieve efficient acquisition and accurate reconstruction of the signal, which has practical significance in various fields. Based on compressed sensing, this paper studies the problem of block sparse signal reconstruction based on fractional penalty function. The fractional function is used instead of ‖Χ‖_2,1, the L2,1 norm optimization problem is transformed into FP_B optimization problem, and the FP_B-DC algorithm is given. The algorithm uses the ADMM algorithm to solve the convex problem left over from the FP_B-DC algorithm. In addition, we have carried out numerical simulation experiments, and compared with Lasso-ADMM and FP-DC algorithm in terms of signal-to-noise ratio and reconstruction accuracy. The results show that the success rate of the new algorithm is significantly higher than that of the other two algorithms when the sparsity increases; with the increase of sparsity, the SNR of FP_B-DC algorithm is higher than that of the other two algorithms on the whole; FP_B-DC algorithm does have some advantages in reconstruction performance.