Abstract
In this paper, we propose a fast sparse recovery algorithm based on the approximate l0 norm (FAL0), which is helpful in improving the practicability of the compressed sensing theory. We adopt a simple function that is continuous and differentiable to approximate the l0 norm. With the aim of minimizing the l0 norm, we derive a sparse recovery algorithm using the modified Newton method. In addition, we neglect the zero elements in the process of computing, which greatly reduces the amount of computation. In a computer simulation experiment, we test the image denoising and signal recovery performance of the different sparse recovery algorithms. The results show that the convergence rate of this method is faster, and it achieves nearly the same accuracy as other algorithms, improving the signal recovery efficiency under the same conditions.
Highlights
Compressed sensing (CS) [1,2,3,4] is a theory proposed in recent years
We focus on proposing a sparse recovery algorithm for CS
The main computational complexity of FAL0 lies in the computation of the initial solution and the iteration, which involves the product of the matrix and vector
Summary
Compressed sensing (CS) [1,2,3,4] is a theory proposed in recent years. Based on this theory, the original signal can be recovered with sampling values much lower than the Nyquist sampling rate. The direct l0 norm minimization problem is NP-hard [6,7,8], so there are many proposed algorithms to approximate its solution These methods can be classified into three categories:. The minimization problem can be converted to the following equation: x min exp Algorithms of this category directly solve the linear equations in the process of iteration and do not need to use linear programming. This algorithm has fast convergence speed and guarantees global optimum. Based in the idea of AL0, we adopt a simple fractional function to approximate the l0 norm, and, to avoid jaggies in the iterative process, we propose a modified Newton method, which can make the algorithm converge faster. We compare the performance of the proposed algorithm with several typical sparse recovery algorithms in the field of signal recovery and image processing to prove the advantages of FAL0
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