A new sensor selection scheme for Bayesian learning based sparse signal recovery in WSNs

Abstract

In this paper, we address the issue of sparse signal recovery in wireless sensor networks (WSNs) based on Bayesian learning. We first formulate a compressed sensing (CS)-based signal recovery problem for the detection of sparse event in WSNs. Then, from the perspective of energy saving and communication overhead reduction of the WSNs, we develop an optimal sensor selection algorithm by employing a lower-bound of the mean square error (MSE) for the MMSE estimator. To tackle the nonconvex difficulty of the optimum sensor selection problem, a convex relaxation is introduced to achieve a suboptimal solution. Both uncorrelated and correlated noises are considered and a low-complexity realization of the sensor selection algorithm is also suggested. Based on the selected subset of sensors, the sparse Bayesian learning (SBL) is utilized to reconstruct the sparse signal. Simulation results illustrate that our proposed approaches lead to a superior performance over the reference methods in comparison.

Introduction

Wireless sensor networks (WSNs) usually consist of a large number of sensor nodes (SNs), which are spread out over fields of interest, where each SN is capable of sensing, processing, and communicating to other nodes or fusion center (FC). WSNs have been extensively studied for environmental monitoring, habitat monitoring, prediction and detection of natural calamities, medical monitoring and structural health monitoring [1]. As SNs are generally small devices with limited battery power, memory, computational capability, and physical size [2], it is important to design WSNs such that the implementation and the associated traffic overhead are as small as possible while preserving the same level of information acquisition and transmission.

In some real-world applications, a physical scenario or event of interest may be described by a sparse signal in the spatial domain. This can be found in a broad range of monitoring applications, such as tracking multiple sources or multiple targets, sensing the underutilized spectrum in a cognitive radio network, and monitoring civil structural health conditions [3], [4], [5]. In these applications, conventional lossy compression techniques such as distributed source coding (DSC) and entropy coding could be exploited [6]. However, these techniques require heavy computational load and communication overhead which are not suitable for implementation in WSNs. As such, compressive sensing (CS) [7] has been employed in WSNs for data acquisition with a small number of measurements [8], [9], [10]. In [8], Haupt et al. investigated CS for networked data in WSNs through considering distributed data sources and their sampling, transmission, and storage. Fazel et al. [9] proposed a compressive sensing based random access scheme with the assumption that the measured signal at sensor nodes is spatially sparse in some basis domain, making it possible to reconstruct the data from under-sampled measurements at FC. In [10], Mamaghanian et al. investigated CS for energy-efficient signals gathering in a wireless body sensor network. All of these works can be viewed as sparse signal recovery problem, and thus, a special effort has been made to develop efficient sparse signal reconstruction methods for CS-based WSNs applications.

Several techniques have been developed based on convex optimization methods [11], [12] and iterative greedy search algorithms [13] for reconstructing the sparse signal from few measurements. Another different strategy to solve the problem is to relax the cost function of the minimization by replacing the l₀-norm with l_p-norm for some p ∈ (0, 1], and an example is the focal underdetermined system solver (FOCUSS) [14]. Recently, a method called approximate message passing (AMP) [15] has been shown to be an efficient algorithm for solving the optimization problem. Some techniques developed under the probabilistic framework for sparse signal reconstruction have also received a great deal of attention, which exploit priors on the regression coefficients for regularizing the under-determined problem [16]. One method that has attractive properties is so-called sparse Bayesian learning (SBL). It was developed for the relevance vector machine (RVM) [17] and adapted for basis selection from over-complete dictionaries [18]. Later, it was further developed to improve the estimation performance from sparse information [19], [20], [21], and has been shown to outperform many of other schemes in terms of the recovery accuracy [21]. The applications of SBL in WSNs could be found in [22], [23], [24]. In [22], the authors proposed a distributed sparse Bayesian learning (dSBL) algorithm for sparse regression in WSNs based on loopy belief propagation (LBP). In [23], the authors applied a temporal sparse Bayesian learning (T-SBL) algorithm to solve the drift observation equation for estimating sensor drift. In our recent work [24], we investigated multi-target localization using SBL in WSNs from a small number of measurements. However, these measurements are randomly selected. In this paper, we extend our previous work in order to operate in a more general framework with optimal selection of measurements.

Due to the geographical locations of SNs and environmental conditions, the observations from different SNs often differ in their quality and information carried. In order to save communication resources and prolong the network lifetime, it is necessary to select only part of the SNs, identified as active sensor nodes, for transmission of their observations while keeping others, called inactive SNs, sleeping. As such, sensor selection has been actively studied over the last decade. Optimal sensor selection has been a commonly investigated problem, which is generally formulated by selecting the best subset of the available SNs subject to a specific performance constraint. Unfortunately, the optimal sensor selection often turns out to be a combinatorial problem. Therefore, some researchers have resorted to approximation or relaxation to make the problem feasible especially in the case of there being a moderate-to-large number of sensors. A related problem has been studied in [25], where the authors considered a parameter called the volume of the confidence ellipsoid and adopted a convex relaxation technique to simplify the combinatorial problem to a convex one, which can be solved in polynomial time. Liu et al. [26] proposed an optimal node-selection algorithm to select a subset of camera sensors for estimating the location of a target while minimizing the energy cost. Moreover, different optimization criteria can be adopted for optimal sensor selection. For example, the authors in [27] tackled the combinatorial optimization problem of maximizing the mutual information between the chosen sensors and the sensors which are not selected, and provided a polynomial-time approximation algorithm by exploiting the submodularity of mutual information. Furthermore, the authors in [28] have shown that certain structural properties of the sensor selection problem allow recasting it as a maximization of submodular functions over uniform matroids. In [29], the authors selected the sensors with a greedy algorithm by minimizing the coherence between the rows of the sensing matrix.

It is noted that several works have addressed the sensor selection problem in CS-based WSNs [30], [31], [32]. Ling and Tian [30] investigated the problem of monitoring sparse phenomena using a large-scale and distributed WSN, in which compressive data collection is enforced by turning off a fraction of sensors using a simple random node sleeping strategy. In [31], the authors considered a Bayesian compressive sensing approach and proposed two efficient algorithms to decrease the number of active sensor nodes while maintaining a high performance. In [32], the authors formulated the active node selection problem in compressive sleeping WSNs as an optimization problem, which is then approximated by a constrained convex relaxation plus a rounding scheme. However, in these existing literatures, the study of sensor selection problem hinges on the assumption of uncorrelated measurement noises. In general, the noises experienced by the sensors are often correlated, due to the fact that the measurement noises of different sensors may depend on a common parameter [33]. Although the authors in [34] have studied the sensor selection paradigm with correlated measurement noises, yet it is not designed for sparse signal estimation, and thus not suitable for CS-based WSNs. Therefore, it is an important and imperative task to develop sensor selection methods that are applicable to a more practical framework of correlated measurement noises in CS-based WSNs.

Motivated by the observations above, in this paper, we focus on accurate recovery of a sparse signal in WSNs using SBL and a subset of sensory measurements. Our goal is to improve the sparse signal reconstruction performance of the WSNs while using only a subset of sensors. The main contributions of this paper are outlined as follows:

• A new CS-based sparse signal estimation framework is first established with an objective of detecting sparse physical events in WSNs.

• A sensor selection algorithm is then proposed to select the best subset of sensors in the MMSE sense by employing a lower-bound of the MSE. To tackle the nonconvexity of the optimum sensor selection problem, a convex relaxation is introduced, and a reweighted relaxation algorithm is further proposed to obtain a suboptimal solution.

• Both uncorrelated and correlated noises are considered in sensor selection, and an equivalent stochastic optimization method with reduced computational complexity is also developed to solve the sensor selection problem.

• The SBL together with the proposed sensor selection is used for the first time to solve the sparse signal recovery problem in WSNs, which yields an improved signal reconstruction performance as compared with some of the existing methods in literature.

The remainder of this paper is organized as follows. Section 2 describes the problem formulation, and Section 3 presents the new sensor selection methods. Then, in Section 4, SBL is applied to recover the sparse signal in WSNs. Next, we present some experimental results in Section 5. Finally, conclusions are drawn in Section 6.