Energy-Based Acoustic Source Localization Methods: A Survey <s> Single Source Localization: Algorithms and Analysis <s> A novel sensor network source localization method based on acoustic energy measurements is presented. This method makes use of the characteristics that the acoustic energy decays exponentially with respect to the distance from an omni-directional acoustic source. By comparing energy readings measured at surrounding acoustic sensors during the same time interval can be accurately estimated. We show that the potential target location is restricted to a hyper-sphere in the sensor field given the acoustic energy reading at a pair of sensors. Given multiple sensor acoustic energy readings, the target location is solved as the location that is closest (in the least square sense) to all the corresponding hyper-spheres. We further simplified this nonlinear least square problem to an unconstrained linear least square problem that yields a closed form solution. Experiment results using military vehicle acoustic data show great promise of this novel approach. <s> BIB001 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Single Source Localization: Algorithms and Analysis <s> This paper describes a new approach to the source localization and tracking problem in wireless sensor networks. A fast, easy-toimplement algorithm for localizing a source using received signal strength measurements is presented. The algorithm is based on incremental subgradient optimization methods. Using theory on the convergence rates of these methods we characterize the amount of in-network communication required to achieve an accurate estimate of the source’s location. In comparison to other localization and tracking algorithms described in the literature, the amount of communication (and thus energy and bandwidth) used by our algorithm is much lower than that used by other schemes, especially as network size grows. <s> BIB002 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Single Source Localization: Algorithms and Analysis <s> We present a new approach to localizing an isotropic energy source using measurements from distributed sensors based on kernel averaging techniques. The location estimate is easily and efficiently calculated in a decentralized fashion. Statistical properties are derived for a very general measurement model. Experiments suggest that the proposed estimator is much more robust and exhibits better performance characteristics than the popular least squares estimator under a variety of conditions. <s> BIB003 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Single Source Localization: Algorithms and Analysis <s> This correspondence addresses the problem of locating an acoustic source using a sensor network in a distributed manner, i.e., without transmitting the full data set to a central point for processing. This problem has been traditionally addressed through the maximum-likelihood framework or nonlinear least squares. These methods, even though asymptotically optimal under certain conditions, pose a difficult global optimization problem. It is shown that the associated objective function may have multiple local optima and saddle points, and hence any local search method might stagnate at a suboptimal solution. In this correspondence, we formulate the problem as a convex feasibility problem and apply a distributed version of the projection-onto-convex-sets (POCS) method. We give a closed-form expression for the projection phase, which usually constitutes the heaviest computational aspect of POCS. Conditions are given under which, when the number of samples increases to infinity or in the absence of measurement noise, the convex feasibility problem has a unique solution at the true source location. In general, the method converges to a limit point or a limit cycle in the neighborhood of the true location. Simulation results show convergence to the global optimum with extremely fast convergence rates compared to the previous methods <s> BIB004 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Single Source Localization: Algorithms and Analysis <s> The localization of an acoustic source can be based on the energy measurements at a number of spatially separated microphones. This is because the amount of source energy attenuation at a microphone is proportional to the square of the distance between the source and the microphone. This paper develops an algebraic closed-form solution for the acoustic source localization problem using energy measurements, under the condition of direct line-of-sight and free space propagation. First-order analysis is applied to the proposed solution to study its performance, where only the linear noise terms are kept in obtaining the mean-square localization error. The first-order analytical results show that the proposed solution reaches the Cramer-Rao lower bound (CRLB) accuracy for Gaussian noise as the signal-to-noise ratio tends to infinity. In addition, the proposed solution provides much better accuracy than other closed-form solutions available in literature. Improvement on the proposed solution that extends its operating range beyond the threshold noise level was made by imposing nonnegative constraints. Simulations are included to corroborate the performance of the proposed method. <s> BIB005 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Single Source Localization: Algorithms and Analysis <s> We propose a novel approach to the source localization and tracking problem in wireless sensor networks. By applying minimax approximation and semidefinite relaxation, we transform the traditionally nonlinear and nonconvex problem into convex optimization problems for two different source localization models involving measured distance and received signal strength. Based on the problem transformation, we develop a fast low-complexity semidefinite programming (SDP) algorithm for two different source localization models. Our algorithm can either be used to estimate the source location or be used to initialize the original nonconvex maximum likelihood algorithm. <s> BIB006 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Single Source Localization: Algorithms and Analysis <s> In this paper, energy-based localization methods for source localization in sensor networks are examined. The focus is on least-squares-based approaches due to a good tradeoff between performance and complexity. A suite of methods are developed and compared. First, two previously proposed methods (quadratic elimination and one step) are shown to yield the same location estimate for a source. Next, it is shown that, as the errors which perturb the least-squares equations are nonidentically distributed, it is more appropriate to consider weighted least-squares methods, which are observed to yield significant performance gains over the unweighted methods. Finally, a new weighted direct least-squares formulation is presented and shown to outperform the previous methods with much less computational complexity. Unlike the quadratic elimination method, the weighted direct least-squares method is amenable to a correction technique which incorporates the dependence of unknown parameters leading to further performance gains. For a sufficiently large number of samples, simulations show that the weighted direct solution with correction (WDC) can be more accurate with significantly less computational complexity than the maximum-likelihood estimator and approaches Cramer-Rao bound (CRB). Furthermore, it is shown that WDC attains CRB for the case of a white source. <s> BIB007 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Single Source Localization: Algorithms and Analysis <s> A new incremental optimization algorithm called normalized incremental subgradient (NIS) algorithm is proposed in this letter, which can be used for distributed maximum likelihood estimation (MLE). Its convergence with a diminishing stepsize has been proved and analyzed theoretically. We then apply the NIS algorithm to the energy-based sensor network source localization problem where the decay factor of the energy decay model is unknown. Simulation results show it can achieve very high estimation performance, which is only somewhat lower than that of the centralized localization method based on global optimization techniques, but with hundreds of times lower computational complexity than the centralized method. <s> BIB008 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Single Source Localization: Algorithms and Analysis <s> This paper addresses the energy-based localization problem in wireless sensor networks. The maximum likelihood (ML) location estimation problem is a difficult optimization problem due to the non-convexity of the objective function, and finding an exact solution is difficult. In this work, an approximate solution to the ML localization is presented, by relaxing the minimization problem into semidefinite programming form. Simulation results show that the proposed algorithm outperforms the existing solutions. <s> BIB009 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Single Source Localization: Algorithms and Analysis <s> In this paper, the energy-based localization problem in wireless sensor networks is addressed. The maximum likelihood (ML) location estimation problem is a difficult optimization problem due to the non-convexity of the objective function, and an exact solution is difficult to obtain. An approximate solution to the ML problem is proposed, by relaxing the minimization problem into semidefinite programming form. Simulation results show that the proposed algorithm outperforms the existing solutions. <s> BIB010 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Single Source Localization: Algorithms and Analysis <s> Source localization based on energy measurements is an important problem in wireless sensor networks (WSNs). It is well known that the associated objective function is not convex and may have multiple local optima and saddle points. Most of the existing algorithms can not achieve global optima. In this paper, we formulate the source localization as a convex feasibility problem (CFP) and propose a diffusion based projection method as a solution. In the proposed method, no fusion center is required; the sensor nodes need to communicate only with their closest neighbors and all the sensors update their estimations simultaneously and finally they are able to achieve consensus on a possible minimizer asymptotically. The proposed method has low complexity and achieves global optimality. Theoretical analysis and simulation results show that the proposed method has good estimation performance whether or not the CFP is consistent or inconsistent. <s> BIB011 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Single Source Localization: Algorithms and Analysis <s> Source localization in wireless sensor networks (WSNs) aims to determine the position of a source in a network, given inaccurate position-bearing measurements. This paper addresses the problem of locating a single source from noisy acoustic energy measurements in WSNs. Under the assumption of Gaussian energy measurement errors, the maximum likelihood (ML) estimator requires the minimization of a nonlinear and nonconvex cost function which may have multiple local optima, thus making the search for the globally optimal solution hard. In this work, an approximate solution to the ML location estimation problem is presented by relaxing the minimization problem to a convex optimization problem, namely second-order cone programming. Simulation results demonstrate the superior performance of the convex relaxation approach. More precisely, the new approach shows an improvement of 20 % in terms of localization accuracy when compared to the existing approaches at moderate to high noise levels. Simulation results further show comparable performance of the new approach and the state-of-art approaches at low noise levels. <s> BIB012
This section covers the single source localization case for energy-based methods. Considering the implementation issue, source localization algorithms can be categorized as three types: centralized, sequential and fully-distributed, as shown in Figure 1 . In a centralized algorithm, as shown in Figure 1a , the sensors need to transmit the data to a central point or a fusion center for source location estimation. However, in a sequential or a fully-distributed method, no fusion center is required. As shown in Figure 1b , for the sequential algorithms, the measurements by the sensors are processed sequentially through the networks. A specified data transmitting path is needed. How to do path planning in such networks is a big issue. Furthermore, the convergence rate is low when the sensor density is high and data transmission becomes unreliable when some of the nodes fail; while in a fully-distributed algorithm, as shown in Figure 1c , at each time step, the sensors can exchange their estimates with their one-hop neighbors. All of the sensors update their estimates simultaneously, and finally, they are able to achieve consensus on a possible minimizer asymptotically. The fully-distributed algorithms avoid the path planning problem in the sequential distributed method and improve the robustness of the network. Centralized methods: The first work on energy-based source localization was done by Li and Hu. In their papers  BIB001 , the energy-ratio least-squares (ER-LS) method is proposed. The first step of this method is to obtain the energy-ratio to eliminate the unknown parameter S(t). Then, in the second step, the localization of the source is solved by minimizing a nonlinear least-squares cost function. This method is an approximate solution for maximum likelihood estimation (MLE) for single source localization. Another method reported in their work is two-dimensional CPA (closest point of approach). In general, the CPA method searches for the sensor with the maximum energy reading for the single source situation. However, the estimation accuracy of this method highly depends on the density of the sensors. In BIB005 , a two-stage algebraic closed-form solution was presented. The first stage computes the source location together with an auxiliary variable using the weighted least-squares method. The second stage explores the relationship between the source location and the auxiliary variable to improve the location estimation. In BIB006 , a weighted direct least-squares formulation was presented in BIB007 that provides a tradeoff between performance and computational complexity. In BIB009 BIB012 BIB010 , the source localization problem was formulated using semi-definite programming and second-order cone programming methods. Sequential methods: In BIB002 , Rabbat and Nowak proposed a distributed implementation of the incremental gradient (IG) algorithm to solve the nonlinear least-squares problem. However, in their work, the source energy S(t) is assumed to be known. Later in BIB003 , they propose to use a kernel averaging estimator for robust source localization. In BIB004 , Blatt and Hero first formulated the source localization as a convex feasibility problem (CFP) and proposed to use a projection-onto-convex sets (POCS) method. This method can be implemented in a distributed manner. However, the convergence of the method is verified by simulation, and no rigorous analysis is provided. An incremental optimization algorithm for maximum likelihood-based source localization was proposed in BIB008 . Fully-distributed method: In  BIB011 , the authors formulated the source localization problem as a convex feasibility problem (CFP) and proposed a fully-distributed method for it. In the proposed method, sensor nodes only need to communicate with one-hop neighbors and update their estimates simultaneously based on a projection algorithm. Finally, they are able to achieve consensus on a possible minimizer asymptotically. The proposed method has low complexity and can achieve global optimality fast. In the following, we will give more details about the algorithms for the energy-based single source localization problem.