Distributing the Kalman Filter for Large-Scale Systems

This paper derives a \emph{distributed} Kalman filter to estimate a sparsely connected, large-scale, $n-$dimensional, dynamical system monitored by a network of $N$ sensors. Local Kalman filters are implemented on the ($n_l-$dimensional, where $n_l\ll n$) sub-systems that are obtained after spatially decomposing the large-scale system. The resulting sub-systems overlap, which along with an assimilation procedure on the local Kalman filters, preserve an $L$th order Gauss-Markovian structure of the centralized error processes. The information loss due to the $L$th order Gauss-Markovian approximation is controllable as it can be characterized by a divergence that decreases as $L\uparrow$. The order of the approximation, $L$, leads to a lower bound on the dimension of the sub-systems, hence, providing a criterion for sub-system selection. The assimilation procedure is carried out on the local error covariances with a distributed iterate collapse inversion (DICI) algorithm that we introduce. The DICI algorithm computes the (approximated) centralized Riccati and Lyapunov equations iteratively with only local communication and low-order computation. We fuse the observations that are common among the local Kalman filters using bipartite fusion graphs and consensus averaging algorithms. The proposed algorithm achieves full distribution of the Kalman filter that is coherent with the centralized Kalman filter with an $L$th order Gaussian-Markovian structure on the centralized error processes. Nowhere storage, communication, or computation of $n-$dimensional vectors and matrices is needed; only $n_l \ll n$ dimensional vectors and matrices are communicated or used in the computation at the sensors

A new high-resolution sea surface temperature blended analysis

The National Oceanic and Atmospheric Administration’s (NOAA) office of National Environmental Satellite, Data, and Information Service (NESDIS) now generates a daily 0.05° (∼5 km) global high-resolution satellite-based sea surface temperature (SST) analyses on an operational basis. The new analysis combines SST data from U.S., Japanese, and European geostationary infrared imagers, and low-Earth-orbiting infrared (United States and Europe) SST data, into a single high-resolution 5-km product. An earlier version produced a 0.1° (∼11 km) resolution, a resolution chosen to approximate the Nyquist sampling criterion for the midlatitude Rossby radius (∼20 km), in order to preserve mesoscale oceanographic features such as eddies and frontal meanders. Comparison between the two analyses illustrates that the higher-resolution grid spacing has more success in this regard. The analysis employs a rigorous multiscale optimum interpolation (OI) methodology that approximates the Kalman filter, together with a data-adaptive correlation length scale, to ensure a good balance between detail preservation and noise reduction. The product accuracy verified against globally distributed buoys is ∼0.02 K, with a robust standard deviation of ∼0.25 K. The new analysis has proven a significant success even when compared to other products that purport to have a similar resolution. This analysis forms the basis for other operational environmental products such as coral reef bleaching risk and ocean heat content for tropical cyclone prediction. Forthcoming enhancements include the incorporation of microwave SST products from low-Earth-orbiting platforms [e.g., Global Change Observation Mission for Water-1 (GCOM-W1)] in order to improve the resolution of SST features in areas of persistent cloud and correct for diurnal effects via a turbulence model of upper-ocean heating

Distributed implementations of the particle filter with performance bounds

The focus of the thesis is on developing distributed estimation algorithms for systems with nonlinear dynamics. Of particular interest are the agent or sensor networks (AN/SN) consisting of a large number of local processing and observation agents/nodes, which can communicate and cooperate with each other to perform a predefined task. Examples of such AN/SNs are distributed camera networks, acoustic sensor networks, networks of unmanned aerial vehicles, social networks, and robotic networks. Signal processing in the AN/SNs is traditionally centralized and developed for systems with linear dynamics. In the centralized architecture, the participating nodes communicate their observations (either directly or indirectly via a multi-hop relay) to a central processing unit, referred to as the fusion centre, which is responsible for performing the predefined task. For centralized systems with linear dynamics, the Kalman filter provides the optimal approach but suffers from several drawbacks, e.g., it is generally unscalable and also susceptible to failure in case the fusion centre breaks down. In general, no analytic solution can be determined for systems with nonlinear dynamics. Consequently, the conventional Kalman filter cannot be used and one has to rely on numerical approaches. In such cases, the sequential Monte Carlo approaches, also known as the particle filters, are widely used as approximates to the Bayesian estimators but mostly in the centralized configuration. Recently there has been a growing interest in distributed signal processing algorithms where: (i) There is no fusion centre; (ii) The local nodes do not have (require) global knowledge of the network topology, and; (iii) Each node exchanges data only within its local neighborhood. Distributed estimation have been widely explored for estimation/tracking problems in linear systems. Distributed particle filter implementations for nonlinear systems are still in their infancy and are the focus of this thesis. In the first part of this thesis, four different consensus-based distributed particle filter implementations are proposed. First, a constrained sufficient statistic based distributed implementation of the particle filter (CSS/DPF) is proposed for bearing-only tracking (BOT) and joint bearing/range tracking problems encountered in a number of applications including radar target tracking and robot localization. Although the number of parallel consensus runs in the CSS/DPF is lower compared to the existing distributed implementations of the particle filter, the CSS/DPF still requires a large number of iterations for the consensus runs to converge. To further reduce the consensus overhead, the CSS/DPF is extended to distributed implementation of the unscented particle filter, referred to as the CSS/DUPF, which require a limited number of consensus iterations. Both CSS/DPF and CSS/DUPF are specific to BOT and joint bearing/range tracking problems. Next, the unscented, consensus-based, distributed implementation of the particle filter (UCD /DPF) is proposed which is generalizable to systems with any dynamics. In terms of contributions, the UCD /DPF makes two important improvements to the existing distributed particle filter framework: (i) Unlike existing distributed implementations of the particle filter, the UCD /DPF uses all available global observations including the most recent ones in deriving the proposal distribution based on the distributed UKF, and; (ii) Computation of the global estimates from local estimates during the consensus step is based on an optimal fusion rule. Finally, a multi-rate consensus/fusion based framework for distributed implementation of the particle filter, referred to as the CF /DPF, is proposed. Separate fusion filters are designed to consistently assimilate the local filtering distributions into the global posterior by compensating for the common past information between neighbouring nodes. The CF /DPF offers two distinct advantages over its counterparts. First, the CF /DPF framework is suitable for scenarios where network connectivity is intermittent and consensus can not be reached between two consecutive observations. Second, the CF /DPF is not limited to the Gaussian approximation for the global posterior density. Numerical simulations verify the near-optimal performance of the proposed distributed particle filter implementations. The second half of the thesis focuses on the distributed computation of the posterior Cramer-Rao lower bounds (PCRLB). The current PCRLB approaches assume a centralized or hierarchical architecture. The exact expression for distributed computation of the PCRLB is not yet available and only an approximate expression has recently been derived. Motivated by the distributed adaptive resource management problems with the objective of dynamically activating a time-variant subset of observation nodes to optimize the network's performance, the thesis derives the exact expression, referred to as the dPCRLB, for computing the PCRLB for any AN/SN configured in a distributed fashion. The dPCRLB computational algorithms are derived for both the off-line conventional (non-conditional) PCRLB determined primarily from the state model, observation model, and prior knowledge of the initial state of the system, and the online conditional PCRLB expressed as a function of past history of the observations. Compared to the non-conditional dPCRLB, its conditional counterpart provides a more accurate representation of the estimator's performance and, consequently, a better criteria for sensor selection. The thesis then extends the dPCRLB algorithms to quantized observations. Particle filter realizations are used to compute these bounds numerically and quantify their performance for data fusion problems through Monte-Carlo simulations

STATISTICAL PROCESSING OF LARGE IMAGE SEQUENCES (Accepted for publication in IEEE Trans. Image Processing)

Abstract—The dynamic estimation of large-scale stochastic image sequences, as frequently encountered in remote sensing, is important in a variety of scientific applications. However, the size of such images makes conventional dynamic estimation methods, for example the Kalman and related filters, impractical. In this paper we present an approach that emulates the Kalman filter, but with considerably reduced computational and storage requirements. Our approach is illustrated in the context of a 512 × 512 image sequence of ocean surface temperature. The static estimation step, the primary contribution here, uses a mixture of stationary models to accurately mimic the effect of a nonstationary prior, simplifying both computational complexity and modelling. Our approach provides an efficient, stable, positive-definite model which is consistent with the given correlation structure. Thus the methods of this paper may find application in modelling and single-frame estimation. I