Showing papers by "Uwe D. Hanebeck published in 2015"

Minimum Covariance Bounds for the Fusion under Unknown Correlations

Abstract: One of the key challenges in distributed linear estimation is the systematic fusion of estimates. While the fusion gains that minimize the mean squared error of the fused estimate for known correlations have been established, no analogous statement could be obtained so far for unknown correlations. In this contribution, we derive the gains that minimize the bound on the true covariance of the fused estimate and prove that Covariance Intersection (CI) is the optimal bounding algorithm for two estimates under completely unknown correlations. When combining three or more variables, the CI equations are not necessarily optimal, as shown by a counterexample.

On Wasserstein Barycenters and MMOSPA Estimation

Abstract: The two title concepts have been evolving rather rapidly, but independent of each other. The Wasserstein barycenter, on one hand, has mostly made its appearance in image processing as it can describe a measure of similarity between images. Its minimization might, for example, suggest the best match in image alignment. On the other hand, MMOSPA estimation has been applied largely to multi-target tracking. The Optimal Sub-Pattern Assignment (OSPA) measures the distance between two sets and the Mean OSPA (MOSPA) can be minimized to give the Minimum MOPSA (MMOSPA), which improves MMSE estimation of the target locations when the labeling of the targets in the set is not important. Approximate and exact algorithms have evolved for both Wasserstein barycenters and MMOSPA estimation. Here, we draw connections between the two perspectives and elaborate how they can benefit from each other.

Partial likelihood for unbiased extended object tracking

Abstract: An extended object gives rise to several measurements that originate from unknown measurement sources on the object. In this paper, we consider the tracking and parameter estimation of extended objects that are modeled as a curve in 2D such as a circle or an ellipse. A standard model for such extended objects is to assume that the unknown measurement sources are uniformly distributed on the curve. We argue that the uniform distribution may not be the best choice in scenarios where the true distribution of the measurements significantly differs from a uniform distribution. Based on results from curve fitting and errors-in-variables models, we develop a partial likelihood that ignores the distribution of measurement sources and can be shown to outperform the likelihood for a uniform distribution in these scenarios. If the true measurement sources are in fact uniformly distributed, our new likelihood results in a slightly slower convergence but has the same asymptotic behavior.

TrackSort: Predictive tracking for sorting uncooperative bulk materials

Abstract: Optical belt sorters are a versatile, state-of-the-art technology to sort bulk materials that are hard to sort based on only nonvisual properties. In this paper, we propose an extension to current optical belt sorters that involves replacing the line camera with an area camera to observe a wider field of view, allowing us to observe each particle over multiple time steps. By performing multitarget tracking, we are able to improve the prediction of each particle's movement and thus enhance the performance of the utilized separation mechanism. We show that our approach will allow belt sorters to handle new classes of bulk materials while improving cost efficiency. Furthermore, we lay out additional extensions that are made possible by our new paradigm.

A closed-form likelihood for Particle Filters to track extended objects with star-convex RHMs

Abstract: Modeling 2D extended targets with star-convex Random Hypersurface Models (RHMs) allows for accurate object pose and shape estimation. A star-convex RHM models the shape of an object with the aid of a radial function that describes the distance from the object center to any point on its boundary. However, up to now only linear estimators, i.e., Kalman Filters, are used due to the lack of a explicit likelihood function. In this paper, we propose a closed-form and easy to implement likelihood function for tracking extended targets with star-convex RHMs. This makes it possible to apply nonlinear estimators such as Particle Filters to estimate a detailed shape of a target.We compared the proposed likelihood against the usual Kalman filter approaches with tracking pose and shape of an airplane in 2D. The evaluations showed that the combination of the Progressive Gaussian Filter (PGF) and the new likelihood function delivers the best estimation performance and can outperform the usually employed Kalman Filters.

The Smart Sampling Kalman Filter with Symmetric Samples

Abstract: Nonlinear Kalman Filters are powerful and widely-used techniques when trying to estimate the hidden state of a stochastic nonlinear dynamic system. In this paper, we extend the Smart Sampling Kalman Filter (S2KF) with a new point symmetric Gaussian sampling scheme. This not only improves the S2KF's estimation quality, but also reduces the time needed to compute the required optimal Gaussian samples drastically. Moreover, we improve the numerical stability of the sample computation, which allows us to accurately approximate a thousand-dimensional Gaussian distribution using tens of thousands of optimally placed samples. We evaluate the new symmetric S2KF by computing higher-order moments of standard normal distributions and investigate the estimation quality of the S2KF when dealing with symmetric measurement equations. Finally, extended object tracking based on many measurements per time step is considered. This high-dimensional estimation problem shows the advantage of the S2KF being able to use an arbitrary number of samples independent of the state dimension, in contrast to other state-of-the-art sample-based Kalman Filters.

Stochastic sampling of the hyperspherical von mises–fisher distribution without rejection methods

Abstract: We propose a novel sampling algorithm for the von Mises–Fisher distribution on the unit hypersphere. Unlike previous works, we show a solution for an arbitrary number of dimensions without requiring rejection sampling. As a result, the proposed algorithm has a deterministic runtime. The key idea consists in applying the inversion method to a one-dimensional subproblem and analytically calculating the integral occurring in the distribution function. The proposed method is most efficient for odd numbers of dimensions. We compare the algorithm to a state-of-the-art rejection sampling method in simulations.

Recursive Bayesian pose and shape estimation of 3D objects using transformed plane curves

Abstract: We consider the task of recursively estimating the pose and shape parameters of 3D objects based on noisy point cloud measurements from their surface. We focus on objects whose surface can be constructed by transforming a plane curve, such as a cylinder that is constructed by extruding a circle. However, designing estimators for such objects is challenging, as the straightforward distance-minimizing approach cannot observe all parameters, and additionally is subject to bias in the presence of noise. In this article, we first discuss these issues and then develop probabilistic models for cylinder, torus, cone, and an extruded curve by adapting related approaches including Random Hypersurface Models, partial likelihood, and symmetric shape models. In experiments with simulated data, we show that these models yield unbiased estimators for all parameters even in the presence of high noise.

The Smart Sampling Kalman Filter with Symmetric Samples

Abstract: Nonlinear Kalman Filters are powerful and widely-used techniques when trying to estimate the hidden state of a stochastic nonlinear dynamic system. In this paper, we extend the Smart Sampling Kalman Filter (S2KF) with a new point symmetric Gaussian sampling scheme. This not only improves the S2KF's estimation quality, but also reduces the time needed to compute the required optimal Gaussian samples drastically. Moreover, we improve the numerical stability of the sample computation, which allows us to accurately approximate a thousand-dimensional Gaussian distribution using tens of thousands of optimally placed samples. We evaluate the new symmetric S2KF by computing higher-order moments of standard normal distributions and investigate the estimation quality of the S2KF when dealing with symmetric measurement equations. Finally, extended object tracking based on many measurements per time step is considered. This high-dimensional estimation problem shows the advantage of the S2KF being able to use an arbitrary number of samples independent of the state dimension, in contrast to other state-of-the-art sample-based Kalman Filters.

Multimodal circular filtering using Fourier series

Abstract: Recursive filtering with multimodal likelihoods and transition densities on periodic manifolds is, despite the compact domain, still an open problem. We propose a novel filter for the circular case that performs well compared to other state-of-the-art filters adopted from linear domains. The filter uses a limited number of Fourier coefficients of the square root of the density. This representation is preserved throughout filter and prediction steps and allows obtaining a valid density at any point in time. Additionally, analytic formulae for calculating Fourier coefficients of the square root of some common circular densities are provided. In our evaluation, we show that this new filter performs well in both unimodal and multimodal scenarios while requiring only a reasonable number of coefficients.

Polynomial-Time Algorithms for the Exact MMOSPA Estimate of a Multi-Object Probability Density Represented by Particles

Abstract: In multi-object estimation, the traditional minimum mean squared error (MMSE) objective is unsuitable: a simple permutation of object identities can turn a very good estimate into what is apparently a very bad one. Fortunately, a criterion tailored to sets—minimization of the mean optimal sub-pattern assignment (MMOSPA)—has recently evolved. Aside from special cases, exact MMOSPA estimates have seemed difficult to compute. But in this work we present the first exact polynomial-time algorithms for calculating the MMOSPA estimate for probability densities that are represented by particles. The key insight is that the MMOSPA estimate can be found by means of enumerating the cells of a hyperplane arrangement, which is a traditional problem from computational geometry. Although the runtime complexity is still high for the general case, efficient algorithms are obtained for two special cases, i.e., (i) two targets with arbitrary state dimensions and (ii) an arbitrary number of one-dimensional targets.

Optimal Reduction of Multivariate Dirac Mixture Densities

Abstract: This paper is concerned with the optimal approximation of a given multivariate Dirac mixture, i.e., a density comprising weighted Dirac distributions on a continuous domain, by an equally weighted Dirac mixture with a reduced number of components. The parameters of the approximating density are calculated by minimizing a smooth global distance measure, a generalization of the well-known Cramer-von Mises Distance to the multivariate case. This generalization is achieved by defining an alternative to the classical cumulative distribution, the Localized Cumulative Distribution (LCD), as a characterization of discrete random quantities (on continuous domains), which is unique and symmetric also in the multivariate case. The resulting approximation method provides the basis for various efficient nonlinear state and parameter estimation methods.

Treatment of Dependent Information in Multisensor Kalman Filtering and Data Fusion

Abstract: Distributed and decentralized processing and fusion of sensor data are becoming increasingly important. In view of the Internet of Things and the vision of ubiquitous sensing, designing and implementing multisensor state estimation algorithm have already become a key issue. A network of interconnected sensor devices is usually characterized by the idea to process and collect data locally and independently on the sensor nodes. However, this does not imply that the data are independent of each other, and the state estimation algorithms have to address possible interdependencies so as to avoid erroneous data fusion results. © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business.

Exploiting clutter: Negative information for enhanced extended object tracking

Abstract: When tracking an extended object, traditional approaches exploit information only from measurements that are assumed to stem from the target, and discard observations assumed to have been generated elsewhere However, the fact that these observations were received contains valuable information about where the target is not This information, which is usually treated as clutter with little value, can also be exploited in order to improve estimation results This becomes particularly important in situations with low measurement quality or occlusions, where positive observations from the target may be scarce In these cases, negative observations, which show where the target cannot be, become highly valuable In this paper, we introduce Silhouette Models, which are able to incorporate information from both types of observations The benefits of this approach, which include more robust results and resistance to occlusion, are confirmed in the evaluation

GPU-accelerated progressive Gaussian filtering with applications to extended object tracking

Abstract: Since the last years, Graphics Processing Units (GPUs) have massive parallel execution capabilities even for non-graphic related applications. The field of nonlinear state estimation is no exception here. Particle Filters have already been successfully ported to GPUs. In this paper, we propose a GPU-accelerated variant of the Progressive Gaussian Filter (PGF). This allows us to combine the advantages of the particle flow with the ability to process thousands of measurements at once in order to improve state estimation quality. To get a meaningful comparison between its CPU and GPU variants, we additionally propose a likelihood for tracking a sphere and its extent in 3D based on noisy point measurements. The likelihood considers the physical relationship between sensor, measurement, and sphere to best exploit the information of the received measurements. We evaluate the GPU implementation of the PGF using the proposed likelihood in combination with tens of thousands of measurements. Although the CPU implementation fully exploits parallelization techniques such as SSE and OpenMP, the GPU-accelerated PGF reaches speedups over 20 and real-time tracking can nearly be achieved.

Association-free direct filtering of multi-target random finite sets with set distance measures

Abstract: We consider association-free tracking of multiple targets without identities. The uncertain multi-target state and the uncertain measurements cannot be described by a random vector as this would imply a certain order. Instead, they are described by an unordered random finite set (RFS). Particle-based random finite set densities are used for characterizing the RFS in a simple and natural way. For recursive Bayesian filtering, optimal multi-target state estimates are calculated by systematically minimizing an appropriate set distance measure while directly operating on the particles. Although methods for calculating point estimates of random finite set densities based on appropriate distance measures are available in literature, the proposed recursive filtering is a novel contribution.

Toroidal information fusion based on the bivariate von Mises distribution

Abstract: Fusion of toroidal information, such as correlated angles, is a problem that arises in many fields ranging from robotics and signal processing to meteorology and bioinformatics. For this purpose, we propose a novel fusion method based on the bivariate von Mises distribution. Unlike most literature on the bivariate von Mises distribution, we consider the full version with matrix-valued parameter rather than a simplified version. By doing so. we are able to derive the exact analytical computation of the fusion operation. We also propose an efficient approximation of the normalization constant including an error bound and present a parameter estimation algorithm based on a maximum likelihood approach. The presented algorithms are illustrated through examples.

Non-identity measurement models for orientation estimation based on directional statistics

Abstract: We propose a novel measurement update procedure for orientation estimation algorithms that are based on directional statistics. This involves consideration of two scenarios, orientation estimation in the 2D plane and orientation estimation in three-dimensional space. We make use of the von Mises distribution and the Bingham distribution in these scenarios. In the derivation, we discuss directional counterparts to the extended Kalman filter and a statistical-linearization-based filter. The newly proposed algorithm makes use of deterministic sampling and can be thought of as a directional variant of the measurement update that is used in well-known sample-based algorithms such as the unscented Kalman filter.

Infinite-horizon sequence-based networked control without acknowledgments

Abstract: In this paper, we consider infinite-horizon networked LQG control over multipurpose networks that do not provide acknowledgments (UDP-like networks). The information communicated over the network experiences transmission delays and losses that are modeled as stochastic processes. In oder to mitigate the delays and losses in the controller-actuator channel, the controller transmits sequences of predicted control inputs in addition to the current control input. To be able to reduce the impact of delays and losses in the feedback channel, the estimator computes the estimate using the M last measurements. In this scenario, the separation principle does not hold and the optimal control law is in general nonlinear. However, we show that by restricting the controller and the estimator to linear systems with constant gains, we can find the optimal solution. The presented control law is demonstrated in a numerical example.

A stochastic filter for planar rigid-body motions

Abstract: This paper presents a novel algorithm for the estimation of planar rigid-body motions. It is based on using a probability distribution that is inherently defined on the nonlinear manifold representing these motions and on proposing a deterministic sampling scheme that makes consideration of complicated system models possible. Furthermore, we show that the measurement update for a manifold equivalent to noisy direct measurements can be carried out in closed form. Thus, the resulting method avoids errors made due to local linearization and outperforms methods that wrongly assume Gaussian distributions, which we show by comparing the proposed filter to the UKF.

OSPA barycenters for clustering set-valued data

Abstract: We consider the problem of clustering set-valued observations, i.e., each observation is a set that consists of a finite number of real vectors. For this purpose, we develop a k-means algorithm that employs the OSPA distance for measuring the distance between sets. In particular, we introduce a novel alternating optimization algorithm for the OSPA barycenter of sets with varying cardinalities that is required for calculating cluster centroids efficiently. The benefits of clustering set-valued data with respect to the OSPA distance are illustrated by means of simulated experiments in the context of target tracking and recognition.

State estimation for ellipsoidally constrained dynamic systems with set-membership pseudo measurements

Abstract: In many dynamic systems, the evolution of the state is subject to specific constraints. In general, constraints cannot easily be integrated into the prediction-correction structure of the Kalman filter algorithm. Linear equality constraints are an exception to this rule and have been widely used and studied as they allow for simple closed-form expressions. A common approach is to reformulate equality constraints into pseudo measurements of the state to be estimated. However, equality constraints define deterministic relationships between state components which is an undesirable property in Kalman filtering as this leads to singular covariance matrices. A second problem relates to the knowledge required to identify and define precise constraints, which are met by the system state. In this article, ellipsoidal constraints are introduced that can be employed to model a bounded region, to which the system state is constrained. This concept constitutes an easy-to-use relaxation of equality constraints. In order to integrate ellipsoidal constraints into the Kalman filter structure, a generalized filter framework is utilized that relies on a combined stochastic and set-membership uncertainty representation.

Parameter estimation for the bivariate wrapped normal distribution

Abstract: Correlated uncertain angular quantities can be modeled using the bivariate wrapped normal distribution. In this paper, we focus on the problem of estimating the distribution's parameters from a given set of samples. For this purpose, we propose several new parameter estimation methods and compare them to estimation techniques found in literature. All methods are thoroughly evaluated in simulations. One of the novel methods is shown to combine the advantages of maximum likelihood estimation and moment-based methods, thus outperforming current state-of-the-art techniques.

Trigonometric moment matching and minimization of the Kullback-Leibler divergence

Abstract: We show an important property of the von Mises distribution on the unit circle. If we approximate an arbitrary circular distribution using a von Mises distribution, the result obtained by trigonometric moment matching also minimizes the Kullback–Leibler divergence (Theorem 1). This result is a justification for circular filtering algorithms based on trigonometric moment matching as the loss of information is minimized. Furthermore, we show that Theorem 1 does not hold for the wrapped normal distribution.

Heart phase estimation using directional statistics for robotic beating heart surgery

Abstract: Robotic beating heart surgery requires accurate information about the current state of the heart For this purpose, it is of great importance to have a good estimate of the heart's current phase, which in essence corresponds to the percentage of the current heart cycle that has already passed Estimation of the heart phase is a highly nontrivial problem as the heart motion is not exactly periodic On the contrary, it varies slightly from beat to beat and changes in frequency over time In order to derive a robust phase estimation algorithm, we rely on directional statistics, a subfield of statistics that deals with quantities that are inherently periodic, such as the phase of the beating heart The proposed methods are evaluated on a real data set and shown to be superior to the state of the art

Depth sensor calibration by tracking an extended object

Abstract: In this paper, we propose a novel algorithm for automatically calibrating a network of depth sensors, based on a moving calibration object. The sensors may have non-overlapping fields of view in order to avoid interference. Two major challenges are discussed. First, depending on where the object is located relative to the sensor, the number and quality of the measurements strongly varies. Second, a single depth sensor observes the calibration object only from one side. Dealing with these challenges requires a simple calibration object as well as an algorithm that can deal with under-determined measurements of varying quality. A recursive Bayesian estimator is developed that determines the extrinsic parameters by measuring the surface of a moving cube with known pose. Our approach does not restrict the configuration of the network and requires no manual initialization or interaction. Ambiguities that are induced by the rotational cube symmetries are resolved by applying a multiple model approach. Besides synthetic evaluation we perform real data experiments and compare to state-of-the-art calibration.

High-Accuracy Real-Time Whole-Body Human Motion Tracking Based on Constrained Nonlinear Kalman Filtering

Abstract: We present a new online approach to track human whole-body motion from motion capture data, i.e., positions of labeled markers attached to the human body. Tracking in noisy data can be effectively performed with the aid of well-established recursive state estimation techniques. This allows us to systematically take noise of the marker measurements into account. However, as joint limits imposed by the human body have to be satisfied during estimation, first we transform this constrained estimation problem into an unconstrained one by using periodic functions. Then, we apply the Smart Sampling Kalman Filter to solve this unconstrained estimation problem. The proposed recursive state estimation approach makes the human motion tracking very robust to partial occlusion of markers and avoids any special treatment or reconstruction of the missed markers. A concrete implementation built on the kinematic human reference model of the Master Motor Map framework and a Vicon motion capture system is evaluated. Different captured motions show that our implementation can accurately estimate whole-body human motion in real-time and outperforms existing gradient-based approaches. In addition, we demonstrate its ability to smoothly handle incomplete marker data.

Nonlinear stochastic model predictive control in the circular domain

Abstract: In this paper, we present an open-loop Stochastic Model Predictive Control (SMPC) method for discrete-time nonlinear systems whose state is defined on the unit circle. This modeling approach allows considering systems that include periodicity in a more natural way than standard approaches based on linear spaces. The main idea of this work is twofold: (i) we model the quantities of the system, i.e., the state, the measurements, and the noises, directly as circular quantities described by circular probability densities, and (ii) we apply deterministic sampling given in closed form to represent the occurring densities. The latter allows us to make the prediction required for solution of the SMPC problem tractable. We evaluate the proposed control scheme by means of simulations.

Conveying system, plant for sorting bulk goods having a conveying system of this type, and transport method

Abstract: The invention relates to a conveying system for transporting a material stream (M) comprising a plurality of individual objects (O1, O2,...), characterised in that, using the conveying system and by optically detecting individual objects (O1, O2,...) in the material stream (M), the respective local positions (x(t), y(t)) of these objects (O1, O2,...) are determined at several different times (t -4 , t -3 ,...), and on the basis of the local positions (x(t), y(t)) determined for these objects (O1, O2,...) at the different times (t -4 , t -3 ,...), it is possible to calculate their respective position (x b (t b ), y b (t b )) at at least one defined time (t b ) after the respective latest time of the different times (t -4 , t -3 ,...).