The benefits of compressive sensing (CS) on counting and localization in wireless sensor networks have already been demonstrated in previous works. However, most existing solutions usually become infeasible in the presence of off-grid targets, which is inevitable for the traditional CS-based localization methods. To mitigate the error caused by off-grid targets, in this paper, a more accurate sparse approximation model is applied where the true and unknown sparsifying dictionary is approximated with its first-order Taylor expansion around a known dictionary. Based on the model, the counting and localization problem is formulated as a sparse reconstruction problem that recovers two sparse signals with related supports. To solve the problem, we develop an iterative algorithm under a variational Bayesian inference framework. Moreover, in practice, some faulty prior information (e.g., coarse positions) is usually available. To incorporate the information, a three-level hierarchical prior model is introduced and imposed on the sparse signals to be recovered, where the supports of sparse signals are learned in the third level. Additionally, a grid pruning mechanism and matrix inversion lemma are leveraged to accelerate the inference process. Extensive simulation results demonstrate that, compared with state-of-the-art algorithms, the proposed algorithm can achieve counting and localization with higher accuracy as well as lower cost.

Localization technologies play an increasingly important role in pervasive applications of wireless sensor networks. Since the number of targets is usually limited, localization benefits from compressed sensing (CS): measurements number can be greatly reduced. Despite many CS-based localization schemes, existing solutions implicitly assume that all targets fall on a fixed grid exactly. When the assumption is violated, the mismatch between the assumed and actual sparsifying dictionaries can deteriorate the localization performance significantly. To address such a problem, in this paper, we propose a novel and iterative multiple target counting and localization framework. The key idea behind the framework is to dynamically adjust the grid to alleviate or even eliminate dictionary mismatch. The contribution of this paper is twofold. First, we consider the off-grid target issue in CS-based localization and formulate multiple target counting and localization as a joint sparse signal recovery and parameter estimation problem. Second, we solve the joint optimization problem using a variational Bayesian expectation-maximization algorithm where the sparse signal and parameter are iteratively updated in the variational Bayesian expectation-step and variational Bayesian maximization-step, respectively. Extensive simulation results highlight the superior performance of the proposed framework in terms of probability of correct counting and average localization error.