Novel Kalman filters for non-linear and non-Gaussian systems : algorithms and applications

Abstract

Kalman filters (KFs) are classical state estimation algorithms with myriad of applications ranging from fault/event detection, process monitoring, digital communications, object tracking/positioning, navigation, to system/robotic control, etc. This thesis is devoted to the development of novel KFs for state estimation of nonlinear and non-Gaussian systems and their practical applications. This thesis first focuses on two important problems in KFs for linear dynamical systems, namely non-Gaussian state and measurement noises and their estimation. Based on a recently proposed Bayesian KF (BKF) with Gaussian mixture simplification (GMS) algorithm, a novel BK smoother (BKS) using the concept of optimal Bayesian smoothing formula and the Rauch-Tung-Striebel smoother is proposed. Moreover, a new adaptive fading (AF) approach for adaptive scalar correction of the state/measurement noise covariance in KF is proposed. The proposed AF BKF/BKS algorithm is applied to automatic estimation of muscle fiber orientation in ultrasound images. It models the motion of muscle fiber as a linear and non-Gaussian system and employs an efficient simplification method to approximate the non-Gaussian state densities. Experimental results show that the proposed algorithms outperform conventional approaches and achieve a performance close to those obtained by experienced operators. The proposed AF-BKF is further extended to include variable number of measurements (VNM) for better tracking of nonstationary signals. The resultant AF-BKF-VNM algorithm is applied to estimate the time varying autoregressive (TVAR) model for spectrum analysis of biomedical signals. A data-driven method is also introduced to determine the variable number of measurements. Its performance and effectiveness is evaluated using simulated and real EEG signals. Moreover, we also consider the TVAR estimation problem from observations with additive Gaussian noise. Two novel algorithms, namely, two-stage AF variational BKF (TS-AF-VBKF) and its variant TS-AF-VBKF with iterative smoothing (TS-AF-VBKF-IS) are proposed. Experimental results show that the performance of the proposed algorithms are better than most conventional algorithms and are comparable to that of particle filter (PF)-based methods, which usually require higher computational complexity. The second part of this thesis extends the concept of AF-BKF to the nonlinear settings. Particularly, an AF Bayesian unscented Kalman filter with simplified Gaussian mixtures (AF-BUKF-SGM) algorithm for state estimation is proposed. In the AF-BUKF-SGM, the state and noise densities are approximated as Gaussian mixtures, where the mean and covariance for each component are recursively estimated using the UKF. Moreover, the AF concept is successfully incorporated to update the Gaussian components. Experimental results on simulated and real small unmanned aircraft data show that the proposed algorithm achieves better performance than conventional algorithms. Finally, we investigate the application of the BUKF to the problem of model-based multi-view human body tracking. Specifically, the proposed system first reconstructs an initial 3D textured model of the subject offline using Kinect depth cameras or from a general 3D model. The pose of the subject is then tracked with multi-view videos using an Annealed Particle Filter-based tracker with a new color-based likelihood function and the proposed BUKF filter. The superiority of the proposed algorithm is demonstrated using the public HumanEva dataset and our captured RGB-D multi-view dataset.