The Ensemble Kalman Filter: A Signal Processing Perspective

The ensemble Kalman filter (EnKF) is a Monte Carlo based implementation of the Kalman filter (KF) for extremely high-dimensional, possibly nonlinear and non-Gaussian state estimation problems. Its ability to handle state dimensions in the order of millions has made the EnKF a popular algorithm in different geoscientific disciplines. Despite a similarly vital need for scalable algorithms in signal processing, e.g., to make sense of the ever increasing amount of sensor data, the EnKF is hardly discussed in our field. This self-contained review paper is aimed at signal processing researchers and provides all the knowledge to get started with the EnKF. The algorithm is derived in a KF framework, without the often encountered geoscientific terminology. Algorithmic challenges and required extensions of the EnKF are provided, as well as relations to sigma-point KF and particle filters. The relevant EnKF literature is summarized in an extensive survey and unique simulation examples, including popular benchmark problems, complement the theory with practical insights. The signal processing perspective highlights new directions of research and facilitates the exchange of potentially beneficial ideas, both for the EnKF and high-dimensional nonlinear and non-Gaussian filtering in general

A Structurally Informed Data Assimilation Approach for Nonlinear Partial Differential Equations

Ensemble transform Kalman filtering (ETKF) data assimilation is often used to combine available observations with numerical simulations to obtain statistically accurate and reliable state representations in dynamical systems. However, it is well known that the commonly used Gaussian distribution assumption introduces biases for state variables that admit discontinuous profiles, which are prevalent in nonlinear partial differential equations. This investigation designs a new structurally informed non-Gaussian prior that exploits statistical information from the simulated state variables. In particular, we construct a new weighting matrix based on the second moment of the gradient information of the state variable to replace the prior covariance matrix used for model/data compromise in the ETKF data assimilation framework. We further adapt our weighting matrix to include information in discontinuity regions via a clustering technique. Our numerical experiments demonstrate that this new approach yields more accurate estimates than those obtained using ETKF on shallow water equations, even when ETKF is enhanced with inflation and localization techniques

GNSS/LiDAR-Based Navigation of an Aerial Robot in Sparse Forests

Autonomous navigation of unmanned vehicles in forests is a challenging task. In such environments, due to the canopies of the trees, information from Global Navigation Satellite Systems (GNSS) can be degraded or even unavailable. Also, because of the large number of obstacles, a previous detailed map of the environment is not practical. In this paper, we solve the complete navigation problem of an aerial robot in a sparse forest, where there is enough space for the flight and the GNSS signals can be sporadically detected. For localization, we propose a state estimator that merges information from GNSS, Attitude and Heading Reference Systems (AHRS), and odometry based on Light Detection and Ranging (LiDAR) sensors. In our LiDAR-based odometry solution, the trunks of the trees are used in a feature-based scan matching algorithm to estimate the relative movement of the vehicle. Our method employs a robust adaptive fusion algorithm based on the unscented Kalman filter. For motion control, we adopt a strategy that integrates a vector field, used to impose the main direction of the movement for the robot, with an optimal probabilistic planner, which is responsible for obstacle avoidance. Experiments with a quadrotor equipped with a planar LiDAR in an actual forest environment is used to illustrate the effectiveness of our approach

Probabilistic three-dimensional object tracking based on adaptive depth segmentation

Object tracking is one of the fundamental topics of computer vision with diverse applications. The arising challenges in tracking, i.e., cluttered scenes, occlusion, complex motion, and illumination variations have motivated utilization of depth information from 3D sensors. However, current 3D trackers are not applicable to unconstrained environments without a priori knowledge. As an important object detection module in tracking, segmentation subdivides an image into its constituent regions. Nevertheless, the existing range segmentation methods in literature are difficult to implement in real-time due to their slow performance. In this thesis, a 3D object tracking method based on adaptive depth segmentation and particle filtering is presented. In this approach, the segmentation method as the bottom-up process is combined with the particle filter as the top-down process to achieve efficient tracking results under challenging circumstances. The experimental results demonstrate the efficiency, as well as robustness of the tracking algorithm utilizing real-world range information

A Systematization of the Unscented Kalman Filter Theory

In this paper, we propose a systematization of the (discrete-time) Unscented Kalman Filter (UKF) theory. We gather all available UKF variants in the literature, present corrections to theoretical inconsistencies, and provide a tool for the construction of new UKF's in a consistent way. This systematization is done, mainly, by revisiting the concepts of Sigma-Representation, Unscented Transformation (UT), Scaled Unscented Transformation (SUT), UKF, and Square-Root Unscented Kalman Filter (SRUKF). Inconsistencies are related to 1) matching the order of the transformed covariance and cross-covariance matrices of both the UT and the SUT; 2) multiple UKF definitions; 3) issue with some reduced sets of sigma points described in the literature; 4) the conservativeness of the SUT; 5) the scaling effect of the SUT on both its transformed covariance and cross-covariance matrices; and 6) possibly ill-conditioned results in SRUKF's. With the proposed systematization, the symmetric sets of sigma points in the literature are formally justified, and we are able to provide new consistent variations for UKF's, such as the Scaled SRUKF's and the UKF's composed by the minimum number of sigma points. Furthermore, our proposed SRUKF has improved computational properties when compared to state-of-the-art methods

Best-first Enumeration Based on Bounding Conflicts, and its Application to Large-scale Hybrid Estimation

With the rise of autonomous systems, there is a need for them to have high levels of robustness and safety. This robustness can be achieved through systems that are self-repairing. Underlying this is the ability to diagnose subtle failures. Likewise, online planners can generate novel responses to exceptional situations. These planners require an accurate estimate of state. Estimation methods based on hybrid discrete/continuous state models have emerged as a method of computing precise state estimates, which can be employed for either diagnosis or planning in hybrid domains. However, existing methods have difficulty scaling to systems with more than a handful of components. Discrete state estimation capabilities can scale to this level by combining best-first enumeration and conflict-directed search. Best-first methods have been developed for hybrid estimation, but the creation of conflict-directed methods has previously been elusive. While conflicts are used to learn from constraint violation, probabilistic hybrid estimation is relatively unconstrained. In this paper we present an approach to hybrid estimation that unifies best-first enumeration and conflict-directed search through the concept of "bounding" conflicts, an extension of conflicts that represent tighter bounds on the cost of regions of the search space. This paper presents a general best-first search and enumeration algorithm based on bounding conflicts (A*BC) and a hybrid estimation method based on this enumeration algorithm. Experiments show that an A*BC powered state estimator produces estimates faster than the current state of the art, particularly on large systems

Assessment of reduced order Kalman filter for parameter identification in one-dimensional blood flow models using experimental data

This work presents a detailed investigation of a parameter estimation approach based on the reduced order unscented Kalman filter (ROUKF) in the context of one-dimensional blood flow models. In particular, the main aims of this study are (i) to investigate the effect of using real measurements vs. synthetic data (i.e., numerical results of the same in silico model, perturbed with white noise) for the estimation and (ii) to identify potential difficulties and limitations of the approach in clinically realistic applications in order to assess the applicability of the filter to such setups. For these purposes, our numerical study is based on the in vitro model of the arterial network described by [Alastruey et al. 2011, J. Biomech. 44], for which experimental flow and pressure measurements are available at few selected locations. In order to mimic clinically relevant situations, we focus on the estimation of terminal resistances and arterial wall parameters related to vessel mechanics (Youngs modulus and thickness) using few experimental observations (at most a single pressure or flow measurement per vessel). In all cases, we first perform a theoretical identifiability analysis based on the generalized sensitivity function, comparing then the results obtained with the ROUKF, using either synthetic or experimental data, to results obtained using reference parameters and to available measurements

Robust Kalman filter-based dynamic state estimation of natural gas pipeline networks

To obtain the accurate transient states of the big scale natural gas pipeline networks under the bad data and non-zero mean noises conditions, a robust Kalman filter-based dynamic state estimation method is proposed using the linearized gas pipeline transient flow equations in this paper. Firstly, the dynamic state estimation model is built. Since the gas pipeline transient flow equations are less than the states, the boundary conditions are used as supplementary constraints to predict the transient states. To increase the measurement redundancy, the zero mass flow rate constraints at the sink nodes are taken as virtual measurements. Secondly, to ensure the stability under bad data condition, the robust Kalman filter algorithm is proposed by introducing a time-varying scalar matrix to regulate the measurement error variances correctly according to the innovation vector at every time step. At last, the proposed method is applied to a 30-node gas pipeline networks in several kinds of measurement conditions. The simulation shows that the proposed robust dynamic state estimation can decrease the effects of bad data and achieve better estimating results.Comment: Accepted by Mathematical Problems in Engineerin