A Sparsity Preestimated Adaptive Matching Pursuit Algorithm

Xinhe Zhang,Yufeng Liu,Xin Wang,Yang Li

doi:10.1155/2021/5598180

Abstract

In the matching pursuit algorithm of compressed sensing, the traditional reconstruction algorithm needs to know the signal sparsity. The sparsity adaptive matching pursuit (SAMP) algorithm can adaptively approach the signal sparsity when the sparsity is unknown. However, the SAMP algorithm starts from zero and iterates several times with a fixed step size to approximate the true sparsity, which increases the runtime. To increase the run speed, a sparsity preestimated adaptive matching pursuit (SPAMP) algorithm is proposed in this paper. Firstly, the sparsity preestimated strategy is used to estimate the sparsity, and then the signal is reconstructed by the SAMP algorithm with the preestimated sparsity as the iterative initial value. The method reconstructs the signal from the preestimated sparsity, which reduces the number of iterations and greatly speeds up the run efficiency.

Highlights

Introduction eNyquist sampling theorem specifies that, to avoid losing information when capturing a signal, one must sample at least two times faster than the signal bandwidth. e traditional information collection and compression process is accompanied by a large amount of data waste, resulting in the increase of equipment cost
Research shows that the measurement matrix Φ whose entries are sampled from N(0, σ2), σ2 ≥ 1, is highly likely to satisfy restricted isometry property (RIP)
M is the observation dimension, N is the length of sparse signal, and k1 and k2 are the iteration times of sparsity adaptive matching pursuit (SAMP) and sparsity preestimated adaptive matching pursuit (SPAMP) algorithms, respectively

Summary

Compressed Sensing

Let x be a real signal of length N, that is, x ∈ RN. If a signal x only has K nonzero elements, we can say that x is K-sparse. ΦN] ∈ RM×N. e measurement matrix Φ must allow the reconstruction of the length-N signal x from M measurements (the measurement vector y). If it is known that the signal x only has K nonzero entries, the problem can be solved provided that M > K. Research shows that the measurement matrix Φ whose entries are sampled from N(0, σ2), σ2 ≥ 1, is highly likely to satisfy RIP [22, 23]. E reconstruction of the original signal from the observation signal can be transformed into the smallest l0 norm by solving x􏽢 arg min ‖x‖0. L0 norm optimization is equivalent to l1 norm optimization under the RIP condition; that is, x􏽢 arg min ‖x‖1. L1 norm optimization can use standard convex optimization methods to solve problems. To estimate the sparsity quickly and accurately, sparsity underestimation and overestimation methods are given

Criterion of Sparsity Underestimation

Criterion of Sparsity Overestimation

The Proposed Algorithm

Simulation Experiments

Conclusion