[Show abstract][Hide abstract] ABSTRACT: The Kalman filter has been the work horse in model based filtering for five decades, and basic knowledge and understanding of it is an important part of the curriculum in many Master of Science programs. It is therefore important to combine theoretical studies with practical experience to allow the students to deepen their understanding of the filter. We have developed a lab where the students implement a Kalman filter in a real-time MATLAB framework, to which data are streamed from the smartphone over WiFi. The goal of the lab is to estimate the orientation of the smartphone, which can be nicely visualized graphically and also be compared to the built-in filters in the smartphone. The filter can accept any combination of sensor data from accelerometers, gyroscopes, and magnetometer, with different performance. Different tunings and tricks in the Kalman filter are easily evaluated on-line. The smartphone app is also a stand-alone tool to visualize the sensor data graphically. So far the lab seems to have been successful in reaching the pedagogic goals and to engage the students.
The 19th World Congress of the International Federation of Automatic Control (IFAC), Cape Town, South Africa; 08/2014
[Show abstract][Hide abstract] ABSTRACT: Traffic monitoring using low-cost two-axis magnetometers is considered. Although detection of metallic vehicles is rather easy, detecting the driving direction is more challenging. We propose a simple algorithm based on a nonlinear transformation of the measurements, which is simple to implement in embedded hardware. A theoretical justification is provided, and the statistical properties of the test statistic are presented in closed form. The method is compared with the standard likelihood ratio test on both simulated data and real data from field tests, where very high detection rates are reported, despite the presence of sensor saturation, measurement noise, and near-field effects of the magnetic field.
IEEE Transactions on Intelligent Transportation Systems 08/2014; 15(4):1405-1418. · 2.47 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Nonlinear Kalman filter adaptations such as extended Kalman filters (EKF) or unscented Kalman filters (UKF) provide approximate solutions to state estimation problems in nonlinear models. The algorithms utilize mean values and covariance matrices to represent the probability densities in the otherwise intractable Bayesian filtering equations. As a consequence, their estimation performance can show significant dependence on the choice of state coordinates. The here considered problem of tracking maneuvering targets using coordinated turn (CT) models is one practically relevant example: The velocity in the target state can either be formulated in Cartesian or polar coordinates. We extend a previous study to a broader range of CT models that allow for changes in target speed and turn rate, and investigate UKF as well as EKF variants in terms of their performance and sensitivity to noise parameters. The results advocate for the use of polar CT models.
17th International Conference on Information Fusion, Salamanca, Spain; 07/2014
[Show abstract][Hide abstract] ABSTRACT: Random set-based methods have provided a rigorous Bayesian framework and have been used extensively in the last decade for point object estimation. In this article, we emphasize that the same methodology offers an equally powerful approach to estimation of so-called extended objects, i.e., objects that result in multiple detections on the sensor side. Building upon the analogy between Bayesian state estimation of a single object and random finite set (RFS) estimation for multiple objects, we give a tutorial on random set methods with an emphasis on multiple-extended-object estimation. The capabilities are illustrated on a simple yet insightful real-life example with laser range data containing several occlusions.
[Show abstract][Hide abstract] ABSTRACT: In this paper, the measurements of individual wheel speeds and the absolute position from a global positioning system are used for high-precision estimation of vehicle tire radii. The radii deviation from its nominal value is modeled as a Gaussian random variable and included as noise components in a simple vehicle motion model. The novelty lies in a Bayesian approach to estimate online both the state vector and the parameters representing the process noise statistics using a marginalized particle filter (MPF). Field tests show that the absolute radius can be estimated with submillimeter accuracy. The approach is tested in accordance with regulation 64 of the United Nations Economic Commission for Europe on a large data set (22 tests, using two vehicles and 12 different tire sets), where tire deflations are successfully detected, with high robustness, i.e., no false alarms. The proposed MPF approach outperforms common Kalman-filter-based methods used for joint state and parameter estimation when compared with respect to accuracy and robustness.
IEEE Transactions on Intelligent Transportation Systems 04/2014; 15(2):663-672. · 2.47 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A marginal version of the enumeration Bayesian Cram´er-Rao Bound (EBCRB) for jump Markov systems is proposed. It is shown that the proposed bound is at least as tight as EBCRB and the improvement stems from better handling of the nonlinearities. The new bound is illustrated to yield tighter results than BCRB and EBCRB on a benchmark example.
IEEE Signal Processing Letters 03/2014; PP(99):1-1. · 1.64 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The dependence of radio signal propagation on the environment is well known, and both statistical and deterministic methods have been presented in the literature. Such methods are either based on randomised or actual reflectors of radio signals. In this work, we instead aim at estimating the location of the reflectors based on geo-localized radio channel impulse response measurements using methods from synthetic aperture radar (SAR). Radio channel measurements from 3GPP E-UTRAN have been used to verify the usefulness of the proposed approach. The obtained images show that the estimated reflectors are well-correlated with the aerial map of the environment. Also, trajectory segment contributions to different reflectors have been estimated with promising results.
IEEE Transactions on Antennas and Propagation 03/2014; 62(4). · 2.46 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Stochastic dynamical systems are fundamental in state estimation, system
identification and control. System models are often provided in continuous
time, while a major part of the applied theory is developed for discrete-time
systems. Discretization of continuous-time models is hence fundamental. We
present a novel algorithm using a combination of Lyapunov equations and
analytical solutions, enabling efficient implementation in software. The
proposed method circumvents numerical problems exhibited by standard algorithms
in the literature. Both theoretical and simulation results are provided.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we address the problem of multi-target detection and tracking over a network of separately located Doppler-shift measuring sensors. For this challenging problem, we propose to use the probability hypothesis density (PHD) filter and present two implementations of the PHD filter, namely the sequential Monte Carlo PHD (SMC-PHD) and the Gaussian mixture PHD (GM-PHD) filters. Performances of both filters are carefully studied and compared for the considered challenging tracking problem. Simulation results show that both PHD filter implementations, successfully tracks multiple targets using only Doppler shift measurements. Moreover, as a proof-of-concept, an experimental setup consisting of a network of microphones and a loudspeaker was prepared. Experimental study results reveal that it is possible to track multiple ground targets using acoustic Doppler shift measurements in a passive multi-static scenario. We observed that the GM-PHD is more effective, efficient and easy to implement than the SMC-PHD filter.
Digital Signal Processing 01/2014; · 1.50 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: With the electromagnetic theory as a basis, we present a sensor model for three-axis magnetometers suitable for localization and tracking as required in intelligent transportation systems and security applications. The model depends on a physical magnetic dipole model of the target and its relative position to the sensor. Both point target and extended target models are provided as well as a heading angle dependent model. The suitability of magnetometers for tracking is analyzed in terms of local observability and the Cramér-Rao lower bound as a function of the sensor positions in a two sensor scenario. The models are validated with real field test data taken from various road vehicles which indicate excellent localization as well as identification of the magnetic target model suitable for target classification. These sensor models can be combined with a standard motion model and a standard nonlinear filter to track metallic objects in a magnetometer network.
IEEE Transactions on Signal Processing 01/2014; 62(3):545-556. · 3.20 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider robust geolocation in mixed line-of-sight (LOS)/non-LOS (NLOS) environments in cellular radio networks. Instead of assuming known propagation channel states (LOS or NLOS), we model the measurement error with a general two-mode mixture distribution although it deviates from the underlying error statistics. To avoid offline calibration, we propose to jointly estimate the geographical coordinates and the mixture model parameters. Two iterative algorithms are developed based on the well-known expectation-maximization (EM) criterion and joint maximum a posteriori-maximum likelihood (JMAP-ML) criterion to approximate the ideal maximum-likelihood estimator (MLE) of the unknown parameters with low computational complexity. Along with concrete examples, we elaborate the convergence analysis and the complexity analysis of the proposed algorithms. Moreover, we numerically compute the Cramér-Rao lower bound (CRLB) for our joint estimation problem and present the best achievable localization accuracy in terms of the CRLB. Various simulations have been conducted based on a real-world experimental setup, and the results have shown that the ideal MLE can be well approximated by the JMAP-ML algorithm. The EM estimator is inferior to the JMAP-ML estimator but outperforms other competitors by far.
IEEE Transactions on Signal Processing 01/2014; 62(1):168-182. · 3.20 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this contribution, we present an online method for joint state and
parameter estimation in jump Markov non-linear systems (JMNLS). State inference
is enabled via the use of particle filters which makes the method applicable to
a wide range of non-linear models. To exploit the inherent structure of JMNLS,
we design a Rao-Blackwellized particle filter (RBPF) where the discrete mode is
marginalized out analytically. This results in an efficient implementation of
the algorithm and reduces the estimation error variance. The proposed RBPF is
then used to compute, recursively in time, smoothed estimates of complete data
sufficient statistics. Together with the online expectation maximization
algorithm, this enables recursive identification of unknown model parameters.
The performance of the method is illustrated in simulations and on a
localization problem in wireless networks using real data.
IEEE Transactions on Signal Processing 12/2013; · 3.20 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: It is well-known that the motion of an acoustic source can be estimated from Doppler shift observations. It is however not obvious how to design a sensor network to efficiently deliver the localization service. In this work a rather simplistic motion model is proposed that is aimed at sensor networks with realistic numbers of sensor nodes. It is also described how to efficiently solve the associated least squares optimization problem by Gauss-Newton variable projection techniques, and how to initiate the numerical search from simple features extracted from the observed frequency series. The methods are demonstrated on real data by determining the distance to a passing propeller driven aircraft and by localizing an all-terrain vehicle. It is concluded that the processing components included are fairly mature for practical implementations in sensor networks.
16th International Conference on Information Fusion, Istanbul, Turkey; 07/2013
[Show abstract][Hide abstract] ABSTRACT: Knowledge of the noise distribution is typically crucial for the state estimation of general state-space models. However, properties of the noise process are often unknown in the majority of practical applications. The distribution of the noise may also be non-stationary or state dependent and that prevents the use of off-line tuning methods. For linear Gaussian models, Adaptive Kalman filters (AKF) estimate unknown parameters in the noise distributions jointly with the state. For nonlinear models, we provide a Bayesian solution for the estimation of the noise distributions in the exponential family, leading to a marginalized adaptive particle filter (MAPF) where the noise parameters are updated using finite dimensional sufficient statistics for each particle. The time evolution model for the noise parameters is defined implicitly as a Kullback–Leibler norm constraint on the time variability, leading to an exponential forgetting mechanism operating on the sufficient statistics. Many existing methods are based on the standard approach of augmenting the state with the unknown variables and attempting to solve the resulting filtering problem. The MAPF is significantly more computationally efficient than a comparable particle filter that runs on the full augmented state. Further, the MAPF can handle sensor and actuator offsets as unknown means in the noise distributions, avoiding the standard approach of augmenting the state with such offsets. We illustrate the MAPF on first a standard example, and then on a tire radius estimation problem on real data.
[Show abstract][Hide abstract] ABSTRACT: A computational algorithm is presented for the Bayesian Cramer-Rao lower bound (BCRB) in filtering applications with measurement noise from mixture distributions with jump Markov switching structure. Such mixture distributions are common for radio propagation in mixed line- and non-line-of-sight environments. The newly derived BCRB is tighter than earlier more general bounds proposed in literature, and thus gives a more realistic bound on actual estimation performance. The resulting BCRB can be used to compute a lower bound on root mean square error of position estimates in a large class of radio localization applications. We illustrate this on an archetypical tracking application using a nearly constant velocity model and time of arrival observations.
[Show abstract][Hide abstract] ABSTRACT: We consider time-of-arrival based robust geolocation in harsh line-of-sight/non-line-of-sight environments. Herein, we assume the probability density function (PDF) of the measurement error to be completely unknown and develop an iterative algorithm for robust position estimation. The iterative algorithm alternates between a PDF estimation step, which approximates the exact measurement error PDF (albeit unknown) under the current parameter estimate via adaptive kernel density estimation, and a parameter estimation step, which resolves a position estimate from the approximate log-likelihood function via a quasi-Newton method. Unless the convergence condition is satisfied, the resolved position estimate is then used to refine the PDF estimation in the next iteration. We also present the best achievable geolocation accuracy in terms of the Cramér-Rao lower bound. Various simulations have been conducted in both real-world and simulated scenarios. When the number of received range measurements is large, the new proposed position estimator attains the performance of the maximum likelihood estimator (MLE). When the number of range measurements is small, it deviates from the MLE, but still outperforms several salient robust estimators in terms of geolocation accuracy, which comes at the cost of higher computational complexity.
IEEE Transactions on Signal Processing 05/2013; 61(9):2243-2255. · 3.20 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We investigate the unsupervised K-means clustering and the semi-supervised hidden Markov model (HMM) to automatically detect anomalous motion patterns in groups of people (crowds). Anomalous motion patterns are typically people merging into a dense group, followed by disturbances or threatening situations within the group. The application of K-means clustering and HMM are illustrated with datasets from four surveillance scenarios. The results indicate that by investigating the group of people in a systematic way with different K values, analyze cluster density, cluster quality and changes in cluster shape we can automatically detect anomalous motion patterns. The results correspond well with the events in the datasets. The results also indicate that very accurate detections of the people in the dense group would not be necessary. The clustering and HMM results will be very much the same also with some increased uncertainty in the detections.
IEEE Journal of Selected Topics in Signal Processing 02/2013; 7(1):102-110. · 3.63 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The Quantization Theorem I (QT I) implies that the likelihood function can be reconstructed from quantized sensor observations, given that appropriate dithering noise is added before quantization. We present constructive algorithms to generate such dithering noise. The application to maximum likelihood estimation (mle) is studied in particular. In short, dithering has the same role for amplitude quantization as an anti-alias filter has for sampling, in that it enables perfect reconstruction of the dithered but unquantized signal’s likelihood function. Without dithering, the likelihood function suffers from a kind of aliasing expressed as a counterpart to Poisson’s summation formula which makes the exact mle intractable to compute. With dithering, it is demonstrated that standard mle algorithms can be re-used on a smoothed likelihood function of the original signal, and statistically efficiency is obtained. The implication of dithering to the Cramér–Rao Lower Bound (CRLB) is studied, and illustrative examples are provided.