Abstract – In this paper, a novel approach for tracking extended objects is presented. The target object is modeled as a circular disc such that the center and extent of the target object can be estimated. At each time step, a finite set of position measurements that are corrupted with stochastic noise may be available. Each position measurement stems from an unknown measurement source on the extended object. In contrast to existing approaches, no statistical assumptions about the distribution of the measurement sources on the extended object are made. As a consequence, it is necessary to deal with stochastic and set-valued uncertainties. For this purpose, a novel combined stochastic and set-theoretic estimator that employs random hyperboloids to express the uncertainties about the true circular disc is derived.

Abstract – Nonlinear fusion of multi-dimensional densities is an important application in Bayesian state estimation. In the approach proposed here, a joint density over all considered densities is build, which is then approximated by means of a Dirac mixture density by partitioning the joint state space into regions that are represented by single Dirac components. This approximation procedure depends on the nonlinear fusion model and only areas relevant to this model are considered. The processing in joint state space has advantages, especially when fusing Dirac mixture densities. Within this approach, degeneration can be avoided and even densities without mutual support can be combined. Thus, this approach gives an alternative to multiplication of Dirac mixtures with a likelihood, as used in the particle filter. Furthermore, a nonlinear Bayesian estimator with filter and prediction step can be formulated, which is able to cope with both discrete and continuous densities.

An algorithm for estimating the pose, i.e., translation and rotation, of an extended target object is introduced. Compared to conventional methods, where pose estimation is performed on the basis of timeof-flight (TOF) measurements between external sources and sensors attached to the object, the proposed approach directly uses the amplitude values measured at the sensors for estimation purposes without an intermediate TOF estimation step. This is achieved by modeling the wave propagation by a nonlinear dynamic system comprising a system and a measurement equation. The nonlinear system equation includes a model of the time-variant structure of the object rotation based on rotation vectors. As a result, the measured amplitude values at the sensors can be processed instantaneously in a recursive fashion. Uncertainties in the measurement process are systematically considered by employing a stochastic filter for estimating the pose, i.e., the state of the nonlinear dynamic system.

Abstract—Target tracking algorithms usually assume that the received measurements stem from a point source. However, in many scenarios this assumption is not feasible so that measurements may stem from different locations, named measurement sources, on the target surface. Then, it is necessary to incorporate the target extent into the estimation procedure in order to obtain robust and precise estimation results. This paper introduces the novel concept of Random Hypersurface Models for extended targets. A Random Hypersurface Model assumes that each measurement source is an element of a randomly generated hypersurface. The applicability of this approach is demonstrated by means of an elliptic target shape. In this case, a Random Hypersurface Model specifies the random (relative) Mahalanobis distance of a measurement source to the center of the target object. As a consequence, good estimation results can be obtained even if the true target shape significantly differs from the modeled shape. Additionally, Random Hypersurface Models are computationally tractable with standard nonlinear stochastic state estimators. Index Terms—Tracking, extended objects, state estimation, random sets I.

Abstract – This paper copes with the problem of nonlinear Bayesian state estimation. A nonlinear filter, the Sliced Gaussian Mixture Filter (SGMF), employs linear substructures in the nonlinear measurement and prediction model in order to simplify the estimation process. Here, a special density representation, the sliced Gaussian mixture density, is used to derive an exact solution of the Chapman-Kolmogorov equation. The sliced Gaussian mixture density is obtained by a systematic and deterministic approximation of a continuous density minimizing a certain distance measure. In contrast to previous work, improvements of the SGMF presented here include an extended system model and the processing of multi-dimensional nonlinear subspaces. As an application for the SGMF, cooperative passive target tracking, where sensors take angular measurements from a target, is considered in this paper. Finally, the performance of the proposed estimator is compared to the marginalized particle filter (MPF) in simulations. Keywords: Bearings-only tracking, nonlinear estimation, Rao-Blackwellization, Gaussian mixture, Dirac mixture 1

Abstract — For the optimal approximation of multivariate Gaussian densities by means of Dirac mixtures, i.e., by means of a sum of weighted Dirac distributions on a continuous domain, a novel systematic method is introduced. The parameters of this approximate density are calculated by minimizing a global distance measure, a generalization of the well–known Cramér– von Mises distance to the multivariate case. This generalization is obtained by defining an alternative to the classical cumulative distribution, the Localized Cumulative Distribution (LCD). In contrast to the cumulative distribution, the LCD is unique and symmetric even in the multivariate case. The resulting deterministic approximation of Gaussian densities by means of discrete samples provides the basis for new types of Gaussian filters for estimating the state of nonlinear dynamic systems from noisy measurements. I.

State estimation for nonlinear systems generally requires approximations of the system or the probability densities, as the occurring prediction and filtering equations cannot be solved in closed form. For instance, Linear Regression Kalman Filters like the Unscented Kalman Filter or the considered Gaussian Filter propagate a small set of sample points through the system to approximate the posterior mean and covariance matrix. To reduce the number of sample points, special structures of the system and measurement equation can be taken into account. In this paper, two principles of system decomposition are considered and applied to the Gaussian Filter. One principle exploits that only a part of the state vector is directly observed by the measurement. The second principle separates the system equations into linear and nonlinear parts in order to merely approximate the nonlinear part of the state. The benefits of both decompositions are demonstrated on a real-world example.

Abstract — Bayesian estimation for nonlinear systems is still a challenging problem, as in general the type of the true probability density changes and the complexity increases over time. Hence, approximations of the occurring equations and/or of the underlying probability density functions are inevitable. In this paper, we propose an approximation of the conditional densities by wavelet expansions. This kind of representation allows a sparse set of characterizing coefficients, especially for smooth or piecewise smooth density functions. Besides its good approximation properties, fast algorithms operating on sparse vectors are applicable and thus, a good trade-off between approximation quality and run-time can be achieved. Moreover, due to its highly generic nature, it can be applied to a large class of nonlinear systems with a high modeling accuracy. In particular, the noise acting upon the system can be modeled by an arbitrary probability distribution and can influence the system in any way. I.