Energy-Based Acoustic Source Localization Methods: A Survey <s> Sequential Algorithm <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> BIB001 </s> Energy-Based Acoustic Source Localization Methods: A Survey <s> Sequential Algorithm <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> BIB002
The sequential projection method, also termed POCS (projection on convex sets), is a cyclic algorithm BIB001 . From Figure 1b , we can see that the data are processed across sensors in sequence. The method is used for the energy-based source localization problem in BIB001 . The update rule of the sequential projection method is given as follows: where {λ(k)} is a sequence of relaxation parameters satisfying for all k, The pseudocode of the sequential projection method is given by Algorithm 1. Algorithm 1: Sequential projection method (1) Initialization: Set θ(0) to be an arbitrary value. (2) Iteration step: where is a predefined threshold. From the above theorem, we can see that the POCS algorithm has good convergence performance in the consistent case of CFP. However, the convergence behavior of POCS in the inconsistent case is generally unsatisfactory. A new relaxation sequence {λ(k) (k≥0) } needs to be used, which is given by: By using the relaxation sequence stated above, the POCS will converge to a point in G, which has been verified by simulation in BIB001 , but no theoretical analysis is provided. Incremental subgradient optimization method For the sequential distributed algorithms, in BIB002 , the authors proposed an incremental optimization algorithm called the normalized incremental subgradient algorithm. Since in their algorithm, a simple subgradient operator is adopted, hence it is hard to avoid the local optima solutions. The localization performance is low even though the authors claim and prove the convergence of the proposed algorithm. One issue of the sequential distributed algorithm is that a specified data transmitting path is required. How to do path planning in a large sensor network is challenging. Furthermore, the convergence rate is low when the sensor density is high, and data transmission becomes unreliable when some of the nodes fail. Hence, the robustness of such networks is low.