Igor Gilitschenski

Recursive nonlinear filtering for angular data based on circular distributions

Estimation of circular quantities is a widespread problem that occurs in many tracking and control applications. Commonly used approaches such as the Kalman filter, the extended Kalman filter (EKF), and the unscented Kalman filter (UKF) do not take periodicity explicitly into account, which can result in low estimation accuracy. We present a filtering algorithm for angular quantities in nonlinear systems that is based on circular statistics. The new filter switches between three different representations of probability distributions on the circle, the wrapped normal, the von Mises, and a Dirac mixture density. It can be seen as a systematic generalization of the UKF to circular statistics. We evaluate the proposed filter in simulations and show its superiority to conventional approaches.

Nonlinear measurement update for estimation of angular systems based on circular distributions

In this paper, we propose a novel progressive nonlinear measurement update for circular states. This generalizes our previously published circular filter that so far was limited to identity measurement equations. The new update method is based on circular distributions in order to capture the periodic properties of a circular system better than conventional approaches that rely on standard Gaussian distributions. Besides the progressive measurement update, we propose two additional measurement updates that are obtained by adapting traditional filters to the circular case. Simulations show the superiority of the proposed progressive approach.

Unscented Orientation Estimation Based on the Bingham Distribution

In this work, we develop a recursive filter to estimate orientation in 3D, represented by quaternions, using directional distributions. Many closed-form orientation estimation algorithms are based on traditional nonlinear filtering techniques, such as the extended Kalman filter (EKF) or the unscented Kalman filter (UKF). These approaches assume the uncertainties in the system state and measurements to be Gaussian-distributed. However, Gaussians cannot account for the periodic nature of the manifold of orientations and thus small angular errors have to be assumed and ad hoc fixes must be used. In this work, we develop computationally efficient recursive estimators that use the Bingham distribution. This distribution is defined on the hypersphere and is inherently more suitable for periodic problems. As a result, these algorithms are able to consistently estimate orientation even in the presence of large angular errors. Furthermore, handling of nontrivial system functions is performed using an entirely deterministic method which avoids any random sampling. A scheme reminiscent of the UKF is proposed for the nonlinear manifold of orientations. It is the first deterministic sampling scheme that truly reflects the nonlinear manifold of orientations.

Bivariate angular estimation under consideration of dependencies using directional statistics

Estimation of angular quantities is a widespread issue, but standard approaches neglect the true topology of the problem and approximate directional with linear uncertainties. In recent years, novel approaches based on directional statistics have been proposed. However, these approaches have been unable to consider arbitrary circular correlations between multiple angles so far. For this reason, we propose a novel recursive filtering scheme that is capable of estimating multiple angles even if they are dependent, while correctly describing their circular correlation. The proposed approach is based on toroidal probability distributions and a circular correlation coefficient. We demonstrate the superiority to a standard approach based on the Kalman filter in simulations.

Efficient evaluation of the probability density function of a wrapped normal distribution

The wrapped normal distribution arises when the density of a one-dimensional normal distribution is wrapped around the circle infinitely many times. At first look, evaluation of its probability density function appears tedious as an infinite series is involved. In this paper, we investigate the evaluation of two truncated series representations. As one representation performs well for small uncertainties, whereas the other performs well for large uncertainties, we show that in all cases a small number of summands is sufficient to achieve high accuracy.

The partially wrapped normal distribution for SE(2) estimation.

We introduce a novel probability distribution on the group of rigid motions SE(2) and we refer to this distribution as the partially wrapped normal distribution. Describing probabilities on SE(2) is of interest in a wide range of applications, for example, robotics, autonomous vehicles, or maritime navigation. We derive some important properties of this novel distribution and propose an estimation scheme for its parameters based on moment matching. Furthermore, we provide a qualitative comparison to a recently published approach based on the Bingham distribution, and show that there are complementary advantages and disadvantages of the two approaches.

Efficient deterministic dirac mixture approximation of Gaussian distributions

We propose an efficient method for approximating arbitrary Gaussian densities by a mixture of Dirac components. This approach is based on the modification of the classical Cramér-von Mises distance, which is adapted to the multivariate scenario by using Localized Cumulative Distributions (LCDs) as a replacement for the cumulative distribution function. LCDs consider the local probabilistic influence of a probability density around a given point. Our modification of the Cramér-von Mises distance can be approximated for certain special cases in closed-form. The created measure is minimized in order to compute the positions of the Dirac components for a standard normal distribution.

Non-Parametric Extrinsic and Intrinsic Calibration of Visual-Inertial Sensor Systems

This paper presents a solution for the extrinsic and intrinsic calibration of visual-inertial sensor systems. Calibration is formulated as a joint state and parameter estimation problem of a continuous-time system with discrete-time measurements. A maximum-likelihood estimator is derived to estimate the transform between cameras and inertial sensors, temporal alignment, and inertial sensor intrinsic parameters, such as scale factors, axes misalignment, and sensor noise characteristics. The estimator is simple to implement, consistent, and asymptotically attains the Cramér-Rao lower bound. In contrast to the existing methods, it requires no tuning parameters. Detailed results from repeated calibration experiments with a camera-inertial measurement unit system are reported and compared with the results obtained from a modern, parametric method. We reach a precision of <;1 mm in extrinsic translation, 1 mrad in orientation, and 10μs in time shift-within a calibration window of 20 s.

Efficient Bingham filtering based on saddlepoint approximations

In this paper, we address the problem of developing computationally efficient recursive estimators on the periodic domain of orientations using the Bingham distribution. The Bingham distribution is defined directly on the unit hypersphere. As such, it is able to describe both large and small uncertainties in a unified framework. In order to tackle the challenging computation of the normalization constant, we propose a method using its saddlepoint approximations and an approximate MLE based on the Gauss-Newton method. In a set of simulation experiments, we demonstrate that the Bingham filter not only outperforms both Kalman and particle filters, but can also be implemented efficiently.

Efficient descriptor learning for large scale localization

Many robotics and Augmented Reality (AR) systems that use sparse keypoint-based visual maps operate in large and highly repetitive environments, where pose tracking and localization are challenging tasks. Additionally, these systems usually face further challenges, such as limited computational power, or insufficient memory for storing large maps of the entire environment. Thus, developing compact map representations and improving retrieval is of considerable interest for enabling large-scale visual place recognition and loop-closure. In this paper, we propose a novel approach to compress descriptors while increasing their discriminability and match-ability, based on recent advances in neural networks. At the same time, we target resource-constrained robotics applications in our design choices. The main contributions of this work are twofold. First, we propose a linear projection from descriptor space to a lower-dimensional Euclidean space, based on a novel supervised learning strategy employing a triplet loss. Second, we show the importance of including contextual appearance information to the visual feature in order to improve matching under strong viewpoint, illumination and scene changes. Through detailed experiments on three challenging datasets, we demonstrate significant gains in performance over state-of-the-art methods.