Gradient Projection with Approximate L₀ Norm Minimization for Sparse Reconstruction in Compressed Sensing.

Ziran Wei,Jianlin Zhang,Xiangsuo Fan,Yongmei Huang,Zhiyong Xu,Yong Liu

doi:10.3390/s18103373

Abstract

In the reconstruction of sparse signals in compressed sensing, the reconstruction algorithm is required to reconstruct the sparsest form of signal. In order to minimize the objective function, minimal norm algorithm and greedy pursuit algorithm are most commonly used. The minimum L1 norm algorithm has very high reconstruction accuracy, but this convex optimization algorithm cannot get the sparsest signal like the minimum L0 norm algorithm. However, because the L0 norm method is a non-convex problem, it is difficult to get the global optimal solution and the amount of calculation required is huge. In this paper, a new algorithm is proposed to approximate the smooth L0 norm from the approximate L2 norm. First we set up an approximation function model of the sparse term, then the minimum value of the objective function is solved by the gradient projection, and the weight of the function model of the sparse term in the objective function is adjusted adaptively by the reconstruction error value to reconstruct the sparse signal more accurately. Compared with the pseudo inverse of L2 norm and the L1 norm algorithm, this new algorithm has a lower reconstruction error in one-dimensional sparse signal reconstruction. In simulation experiments of two-dimensional image signal reconstruction, the new algorithm has shorter image reconstruction time and higher image reconstruction accuracy compared with the usually used greedy algorithm and the minimum norm algorithm.

Highlights

Compressed sensing theory was put forward in 2006 by Donoho and Candès et al The main concept is that the sampling and compression process of the signal are completed by one measurement process with a lesser number of measurements than Nyquist sampling, and the original signal is recovered directly from the measured signal by a corresponding reconstruction algorithm.The transmission and storage costs of signals are saved, and the computational complexity is reduced [1]
The greatest advantage of compressed sensing is that the amount of data obtained by signal measuring is much smaller than that obtained by conventional sampling methods, which breaks through the limitation of sampling frequency in the Nyquist sampling theorem and makes it possible for one to compress and reconstruct high resolution signals
The remainder of this paper is organized as follows: in Section 2, we present a theoretical analysis of the new algorithm and introduce the basic algorithm framework

Summary

Introduction

Compressed sensing theory was put forward in 2006 by Donoho and Candès et al The main concept is that the sampling and compression process of the signal are completed by one measurement process with a lesser number of measurements than Nyquist sampling, and the original signal is recovered directly from the measured signal by a corresponding reconstruction algorithm. The non-convex optimization algorithm by the minimum L0 norm method can reconstruct the sparsest expression of the signal, which requires less measurement times. Both Equations (4) and (5) are convex optimization problem, is a quadratic program (QP) [15] Both Equations (4) and (5) are convex optimization problem, they they reach the global optimal value, but they cannot guarantee the signal to be the sparsest. For non-convex L0 norm optimization, the theoretical usually has a large amount of computation.

Best in in

Formulation from

The Gradient Projection Implementation in the Approximating L0 Norm Algorithm

Experimental Analysis and Discussion

Conclusions