This paper proposes a novel sparsity adaptive simulated annealing algorithm to solve the issue of sparse recovery. This algorithm combines the advantage of the sparsity adaptive matching pursuit (SAMP) algorithm and the simulated annealing method in global searching for the recovery of the sparse signal. First, we calculate the sparsity and the initial support collection as the initial search points of the proposed optimization algorithm by using the idea of SAMP. Then, we design a two-cycle reconstruction method to find the support sets efficiently and accurately by updating the optimization direction. Finally, we take advantage of the sparsity adaptive simulated annealing algorithm in global optimization to guide the sparse reconstruction. The proposed sparsity adaptive greedy pursuit model has a simple geometric structure, it can get the global optimal solution, and it is better than the greedy algorithm in terms of recovery quality. Our experimental results validate that the proposed algorithm outperforms existing state-of-the-art sparse reconstruction algorithms.
In recent years, the research of compressed sensing (CS) [
The core idea of CS is that sampling and compression are done simultaneously. CS technology can reduce the hardware requirements, further reduce the sampling rate, improve the signal restoration quality, and save the cost of signal processing and transmission. Currently, CS has been widely used in wireless sensor networks [
CS theory is mainly divided into three aspects: (1) sparse representation; (2) uncorrelated sampling; (3) sparse reconstruction. However, how to reconstruct sparse signals is crucial. It is a huge challenge for the researchers to propose a reconstruction algorithm with reliable accuracy. Since reconstruction methods need to recover the original signal from the low-dimensional measurements, the signal reconstruction requires solving an underdetermined equation. That may include infinite solutions. Mathematically, the issue can be regarded as the
At present, some scholars have proposed many reconstruction algorithms. Mostly, the nonconvex
In this paper, we propose a novel sparsity adaptive simulated annealing method (SASA) to reconstruct the sparse signal efficiently. This new method combines the advantage of the SAMP [
The main contributions of this paper are listed as follows: We take advantage of the sparsity adaptive simulated annealing algorithm in global searching to guide the sparse reconstruction The proposed algorithm can reconstruct a sparse signal accurately, and it does not require sparsity as a priori information The proposed algorithm has both the simple geometric structure of the greedy algorithm and the global optimization ability of the SA algorithm
The remainder of this paper is organized as follows. Section
In this section, we introduce the compressive sensing model and the sparse signal approximation method.
Sparse reconstruction is to recover the high dimensional original signal
The above model is a convex optimization problem. It can resort to linear programming methods or Bregman split algorithm. And linear programming methods have shown to be effective in solving such problems with high accuracy. Theoretically, the CS has explained that the sparse signal
Generally speaking, if
There are many kinds of sensing matrix
In this section, we give some notations and matrix operators to facilitate the summary of the proposed algorithm.
Let
It can be seen from equation (
The residue of the projection vector
Suppose that
As can be seen from Figure
In each iteration, a
The greedy pursuit algorithms for sparse recovery have low computational complexity and perfect reconstruction quality. Thus, how to design a stable and efficient CS reconstruction algorithm is a meaningful research issue. But some existing greedy algorithms often get a local optimization solution to the reconstruction problem. To better solve this issue, we introduce an intelligent optimization algorithm to get better reconstruction result.
Simulated annealing (SA) is a famous heuristic optimization algorithm, and it has been empirically indicated that SA has superior performance in finding global optimization solutions. As a special optimization reconstruction problem, we can directly solve the problem (2) by using the SA algorithm. However, due to the high computational complexity, it is not practical to directly use SA to solve the sparse recovery problem.
As we all know, the greedy algorithms have low complexity and simple geometric interpretation. But greedy algorithms do not have global optimization capabilities, and it is easy to get a suboptimal solution. This means that the greedy strategy and SA can complement each other. We combine the advantage of the SAMP algorithm and the SA algorithm to recover a sparse signal from only a few measurements
Inspired by this, we propose a new method to solve the sparse reconstruction issue. The proposed algorithm not only does not require sparsity as a necessary prior but also has superior optimization performance. This paper proposes a novel sparse signal recovery algorithm called sparsity adaptive simulated annealing matching pursuit algorithm (SASA).
Kirkpatrick proposed the SA algorithm in 1983. SA was firstly proposed to solve thermodynamic problems. The SA algorithm can escape from a local optimization solution. Specially, since it can add a few interference items with a certain probability, SA is more likely to find global optimums or approximate optimization solutions. So far, the SA algorithm is still receiving great attention due to its special strategy.
The detail procedure of the standard SA is summarized in Algorithm
Input: Initial solution S; Initial the temperature Cooling rate The outer-loop iteration The inner-loop iteration Iteration: While for if Take a random if
In this part, we introduce the processing of the SASA algorithm in detail. In real signal processing, the sparsity is usually unavailable for restoration task. We cannot know the signal sparsity in advance. If the sparsity level
Suppose the support set
From theorem 1, we can define the cost function as
Input: Measurement signal Sensing matrix Initialize Iteration: Estimated sparsity level Initialize the Initialize
for Choose Calculate Choose the Calculate else Calculate Take a random else Calculate end for Update End while
In this section, a series of simulations are explained to evaluate the superior performance of the proposed SASA algorithm. To demonstrate the efficiency of the proposed algorithm, the SASA is compared with those existing state-of-the-art reconstruction methods. Those methods include SP, StOMP, CoSaMP, SWOMP [
All the simulations were performed on the MATLAB (R2017b) on Windows 10 system with a 3.20 GHz Intel Core i7 processor and 8 GB memory.
In the experiment, the Gaussian random sparse signal length is set to
In Figure
Curve of the exact reconstruction rate under different measurement numbers
In Figure
Curve of the exact reconstruction rate under different sparsity level
In Figure
Curve of the exact reconstruction rate under different signal length
In Figure
Curve of the exact reconstruction rate under different measurement numbers.
Through the above various simulation experiments, we can demonstrate that the proposed SASA algorithm has superior reconstruction performance under different reconstruction conditions.
To validate the performance of the proposed SASA algorithm for real data, we use standard test images with a size of
Images reconstructed by different algorithms.
All methods are evaluated on three different images, and those images include the Lena, Barche, and Camer. As shown in Figure
Conclusively, various experiments show that the proposed SASA algorithm achieves superior performance for different types of data. Compared with other algorithms, the proposed SASA got higher exact reconstruction rate and PSNR.
In this paper, we propose a new sparsity adaptive simulated annealing algorithm. It can solve the sparse reconstruction issue efficiently. The proposed algorithm considers both adaptive sparsity and global optimization for sparse signal reconstruction. A series of experimental results show that the proposed SASA algorithm achieves the best reconstruction performance and is superior to the existing state-of-the-art methods in terms of reconstruction quality. For those advantages, the proposed SASA algorithm has broad application prospects and a higher guiding significance for the sparse signal reconstruction.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The research was supported by the National High-Tech Research and Development Program (National 863 Project, No. 2017YFB0403604) and the project supported by the National Nature Science Foundation of China (No. 61771262). Yangyang would also like to thank the Chinese Scholarship Council.