Distributed Maximum Likelihood for Simultaneous Self-Localization and Tracking in Sensor Networks

Abstract

We show that the sensor self-localization problem can be cast as a static parameter estimation problem for Hidden Markov Models and we implement fully decentralized versions of the Recursive Maximum Likelihood and on-line Expectation-Maximization algorithms to localize the sensor network simultaneously with target tracking. For linear Gaussian models, our algorithms can be implemented exactly using a distributed version of the Kalman filter and a novel message passing algorithm. The latter allows each node to compute the local derivatives of the likelihood or the sufficient statistics needed for Expectation-Maximization. In the non-linear case, a solution based on local linearization in the spirit of the Extended Kalman Filter is proposed. In numerical examples we demonstrate that the developed algorithms are able to learn the localization parameters.

Simultaneous localization and tracking via real-time nonparametric belief propagation

“…In the recent state-of-the-art, there are few important methods for SLAT [6][7][8][9]. In [6], the authors propose a Bayesian filtering approach to update a joint probability density functions (PDF) over the sensor positions, the target track, and the calibration parameters of the network.…”

“…A variational approach, based on mean-field method, is used to factorize the joint PDF, which is then approximated with a Gaussian distribution. An extended Kalman filter (EKF) is used in [8] to track the target while recursive maximum likelihood (ML) and expectation maximization (EM) determine the point estimates of the sensors' positions. A fully scalable distributed implementation is provided via message passing based on consensus propagation.…”

“…Most importantly, we note that none of them can provide sensors' posterior PDFs in non-Gaussian form. Since [6] and [7] approximate posterior PDFs with a Gaussian distribution, and [8] and [9] with a point estimate (Dirac delta impulse), it will lead to a suboptimal approximation in case of non-linear models and/or non-Gaussian measurements. This is especially a serious problem in non-rigid graphs, in which the posterior PDFs of a subset of the sensors is multi-modal [10].…”

“…Now, we will approximate (8), which would require us to draw particles from the product of the messages. Since this is in general intractable, we use a proposal distribution, the sum of Gaussian mixtures with reference particles (RPs), and then reweight the particles.…”

Distributed estimation of latent parameters in state space models using separable likelihoods

“…In fusion (or, object tracking) networks, localisation/calibration of sensors in a GPS denying environment using point detections of non-cooperative targets [5][6][7] can be treated as an instance of this problem setting. Another example is the estimation of the orientations and positions of nodes in a camera network based on feature detections [8].…”