Recursive Bayesian filtering in circular state spaces

Abstract

For recursive circular filtering based on circular statistics, we introduce a general framework for estimation of a circular state based on different circular distributions, specifically the wrapped normal distribution and the von Mises distribution. We propose an estimation method for circular systems with nonlinear system and measurement functions. This is achieved by relying on efficient deterministic sampling techniques. Furthermore, we show how the calculations can be simplified in a variety of important special cases, such as systems with additive noise as well as identity system or measurement functions. We introduce several novel key components, particularly a distribution-free prediction algorithm, a new and superior formula for the multiplication of wrapped normal densities, and the ability to deal with non-additive system noise. All proposed methods are thoroughly evaluated and compared to several state-of-the-art solutions. system measurement publication distribution model noise model noise Azmani, Reboul, Choquel, Benjelloun [9] von Mises identity additive identity additive Markovic, Chaumette, Petrovic [14] von Mises-Fisher identity additive identity additive Kurz, Gilitschenski, Julier, Hanebeck [21] Bingham identity additive identity additive Kurz, Gilitschenski, Hanebeck [15] wrapped normal/von Mises nonlinear additive identity additive Kurz, Gilitschenski, Hanebeck [16] wrapped normal nonlinear additive nonlinear any this paper wrapped normal/von Mises nonlinear any nonlinear anyTable 1: Circular filters based on directional statistics.There has been some work on filtering algorithms based on circular statistics by Azmani et al. [9], which was further investigated by Stienne et al. [10]. Their work is based on the von Mises distribution and allows for recursive filtering of systems with a circular state space. However, it is limited to the identity with additive noise as the system equation and the measurement equation. The filter from [9] has been applied to phase estimation of GPS signals [11], [12] as well as map matching [13]. Markovic et al. have published a similar filter [14] based on the von Mises-Fisher distribution, a generalization of the von Mises distribution to the hypersphere.We have previously published a recursive filter based on the wrapped normal distribution allowing for a nonlinear system equation [15]. The paper [16] extends this approach to make a nonlinear measurement update possible. Both papers rely on a deterministic sampling scheme, based on the first circular moment. This kind of sampling is reminiscent of the well-known unscented Kalman filter (UKF) [2]. We have extended this sampling scheme to the first two circular moments in [17], so the proposed filters are, in a sense, circular versions of the UKF.The developed filters have been applied in the context of constrained tracking [18], bearings-only sensor scheduling [19], as well as circular model predictive control [20].Furthermore, we proposed a recursive filter based on the circular Bingham distribution in [21]....

Visual recency bias is explained by a mixture model of internal representations

“…Since the predictive prior distribution resulting from the mixture components is non-parametric we used a discrete circular filter to approximate the distributions in this simulation. The discrete filter is based on a grid of weighted Dirac components equally distributed along the circle (Kurz et al, 2016;Kurz, Gilitschenski, & Hanebeck, 2013) and was implemented with libDirectional toolbox for Matlab (Kurz, Gilitschenski, Pfaff, & Drude, 2015). Because in the unidimensional circular state space of orientations the quality of approximation is only given by the number of components, we felt that 10,000 Dirac components can adequately approximate a distribution of a circular variable.…”

“…A detailed derivation of the circular filter using Dirac mixtures can be found in Kurz et al (2013Kurz et al ( , 2016. Briefly, a wrapped Dirac mixture with L components and Dirac positions β 1 , .…”

On orientation estimation using iterative methods in Euclidean space

“…For the nonlinear dynamics (14), there is no exact expression and approximate methods, like Euler approximations, must be used.…”

“…Some of them still use rotation states in the filtering but explicitly take into account that these states reside in SO (3). Examples include the Lie Algebra-EKF [13] and approaches making use of distributions on manifolds [14,15]. In this paper, however, we consider a quite natural modification of the above-mentioned EKF implementations.…”

Joint Smoothing and Tracking Based on Continuous-Time Target Trajectory Function Fitting

“…This has formed the majority of extensions of the model-driven framework, e.g., a huge number of works for noise covariance estimation [17], but also formed new challenges to the real time implementation due to escalated computational requirement. More importantly, it is unclear how to optimally use some important but fuzzy information such as a linguistic context that the target moves close to a straight line, which might not be easily defined as constraints [14], [15], [35]- [37] and [2,Chapter 6]. This class of information is very common and useful to targets like aircraft, satellites, large cruise ships, and trains, which are supposed to move on pre-defined runways.…”