基于BFGS拟牛顿法的压缩感知SL0重构算法

平滑l₀范数(SL0)算法是一种基于近似l₀范数的压缩感知信号重构算法，采用最速下降法和梯度投影原理，通过选择一个递减序列来逐步逼近最优解，具有匹配度高、计算量低、不需要已知信号稀疏度等优点。但是，其迭代方向为负梯度方向，使得在迭代过程中产生“锯齿现象”，导致在最优解附近收敛速度较慢。牛顿法具有较快的收敛速度，但是对初值的要求较高，并且需要计算Hesse矩阵。拟牛顿法则克服了这个缺点，利用BFGS公式计算Hesse矩阵的近似矩阵，只需要计算1阶导数信息。该文在SL0算法的基础上，结合BFGS拟牛顿法，提出一种改进的压缩感知信号重构算法。首先采用最速下降法迭代得到信号的某个估计值，然后将此估计值作为拟牛顿法的初值继续迭代，直至得到最优解。计算机仿真结果表明，在相同的条件下，该算法在重构精度、峰值信噪比和重建匹配度等方面均有较大提高。

SL0 Reconstruction Algorithm for Compressive Sensing Based on BFGS Quasi Newton Method

Smoothed l₀ norm (SL0) algorithm is a compressive sensing reconstruction algorithm based on approximate l₀ norm, which uses the steepest descent method and gradient projection principle, by selecting a decreasing sequence to get the optimal solution. It has the advantages of high matching degree, low computational complexity and without knowing the signal sparsity. However, the iterative direction of steepest descent method is negative gradient direction, which leads to the " sawtooth phenomenon” and the slower convergence speed in the vicinity of the optimal solution. The Newton method has a good convergence speed but has higher requirement of the initial value and needs to calculate the Hessian matrix. The quasi Newton method overcomes this shortcoming and uses BFGS formula to calculate the approximate matrix of the Hessian matrix, it only needs the first derivative information. On the basis of SL0 algorithm and BFGS quasi Newton method, an improved reconstruction algorithm for Compressed Sensing (CS) signal is proposed. The steepest descent method is first used to get an estimated value, and then is taken as the initial value of quasi Newton method, using BFGS method to update the iterative direction until retaining the optimal solution. The simulation results show that the proposed algorithm has great improvement in reconstruction accuracy, peak signal to noise ratio and reconstruction matching degree.