Tracking multiple targets with uncertain target dynamics is a difficult problem, especially with nonlinear state and/or measurement equations. Random finite set theory provides a rigorous foundation to multitarget tracking problems. It provides a framework to represent the full multitarget posterior in contrast to other conventional approaches. However, the computational complexity of performing multitarget recursion grows exponentially with the number of targets. The Probability Hypothesis Density (PHD) filter, which only propagates the first moment of the multitarget posterior, requires much less computational complexity. This thesis addresses some of the essential issues related to practical multitarget tracking problems such as tracking target maneuvers, stealthy targets, multitarget tracking in a distributed framework. With maneuvering targets, detecting and tracking the changes in the target motion model also becomes important and an effective solution for this problem using multiple-model based PHD filter is proposed. The proposed filter has the advantage over the other methods in that it can track a timevarying number of targets in nonlinear/ non-Gaussian systems. Recent developments in stealthy military aircraft and cruise missiles have emphasized the need to t rack low SNR targets. The conventional approach of thresholding the measurements throws away potential information and thus results in poor performance in tracking dim targets. The problem becomes even more complicated when multiple dim targets are present in the surveillance region. A PHD filter based recursive track-before-detect approach is proposed in this thesis to track multiple dim targets in a computationally efficient way. This thesis also investigates multiple target tracking using a network of sensors. Generally, sensor networks have limited energy, communication capability and computational power. The crucial consideration is what information needs to be transmitted over the network in order to perform online estimation of the current state of the monitored system, whilst attempting to minimize communication overhead. Finally, a novel continuous approximation approach for nonlinear/ non-Gaussian Bayesian tracking system based on spline interpolation is presented. The resulting filter has the advantages over the widely-known discrete particle based approximation approach in that it does not suffer from degeneracy problems and retains accurate density over the state space. The filter is general enough to be applicable to nonlinear/non-Gaussian system and the density could even be multi-modal. / Thesis / Candidate in Philosophy

<div>The space around the Earth is becoming increasingly populated with a growth in number of launches and proliferation of debris. Currently, there are around 44,000 objects (with a minimum size of 10cm) orbiting the Earth as per the data made publicly available by the US strategy command (USSTRATCOM). These objects include active satellites and debris. The number of these objects are expected to increase rapidly in future from launches by companies in the private sector. For example, SpaceX is expected to deploy around 12000 new satellites in the LEO region to develop a space-based internet communication system. Hence in order to protect active space assets, tracking of all the objects is necessary. Probabilistic tracking methods have become increasingly popular for solving the multi-target tracking problem in Space Situational Awareness (SSA). This thesis studies one such technique known as the GM-PHD filter, which is an algorithm which estimates the number of objects and its states when non-perfect measurements (noisy measurements, false alarms) are available. For Earth orbiting objects, especially those in Geostationary orbits, ground based optical sensors are a cost-efficient way to gain information.In this case, the likelihood of gaining target-generated measurements depend on the probability of detection (p_D) of the target.An accurate modeling of this quantity is essential for an efficient performance of the filter. p_D significantly depends on the amount of light reflected by the target towards the observer. The reflected light depends on the relative position of the target with respect to the Sun and the observer, the shape, size and reflectivity of the object and the relative orientation of the object towards Sun and the observer. The estimation of the area and reflective properties of the object is in general, a difficult process. Uncontrolled objects, for example, start tumbling and no information regarding the attitude motion can be obtained. In addition, the shape can change because of disintegration and erosion of the materials. For the case of controlled objects, given that the object is stable, some information on the attitude can be obtained. But materials age in space which changes the reflective properties of the materials. Also, exact shape models for these objects are rare. Moreover,, area can never be estimated with optical measurements or any other measurements, as it is always albedo-area i.e., reflectivity times area that can be measured.</div><div> The purpose of this work is to design a variation of the GM-PHD filter which accounts for the uncertainty in p_D as the original GM-PHD filter designed by Vo and Ma assumes p_D as a constant. It is validated that the proposed method improves the filter performance when there is an uncertainty in area(hence uncertainty in p_D) of the targets. In the tested cases, the uncertainty in p_D was modeled as an uncertainty in area while assuming that the targets are spherical and that the reflectivity of the targets is constant. It is seen that a model mismatch in p_D affects the filter performance significantly and the proposed method improves the performance of the filter in all cases.</div>

This dissertation presents solutions to two open problems in estimation theory. The first is a tractable analytical solution for problems in multi-target filtering which are too complex to solve using traditional techniques. The second explores a new approach to the nonlinear filtering problem for a general class of models. The approach to the multi-target filtering problem which involves jointly estimating a random process of the number of targets and their state, developed using the probability hypothesis density (PHD) filter alleviates the intractability of the problem by avoiding explicit data association. Moreover, the notion of linear jump Markov systems is generalized to the multiple target case to accommodate births, deaths and switching dynamics to derive a closed form solution to the PHD recursion for this so-called linear Gaussian jump Markov multi-target model. The proposed solution is general enough to accommodate a broad class of practical problems which are deemed intractable using traditional techniques. Based on this closed form solution, an efficient method is developed for tracking multiple maneuvering targets that switch between multiple models without the need for gating, track initiation and termination, or clustering for extracting state estimates. The approach to the nonlinear filtering problem explores the framework of the virtual linear fractional transformation (LFT) model which localizes the nonlinearity to the feedback with a simple and sparse structure. The LFT is an exact representation for any differentiable nonlinear mapping and therefore amenable to a general class of problems. An alternative analytical approximation method is presented which avoids linearization of the state space model. The uncorrelated structure of the feedback connection gives of the state space model. The uncorrelated structure of the feedback connection gives better second-order moment approximation of the nonlinearly mapped variables. By arranging the unscented transform in the feedback, the prediction and estimation steps are derived in closed form. The proposed filters for the discrete-time model and continuous-time dynamics with sampled-data measurements respectively are shown to be robust under highly nonlinear and uncertain conditions where standard analytical approximation based filters diverge. Moreover, the LFT based filters are efficient for online implementation. In addition, the LFT framework is applied to extend the closed form solution of the PHD recursion to the nonlinear jump Markov multi-target model.

abstract: Neural activity tracking using electroencephalography (EEG) and magnetoencephalography (MEG) brain scanning methods has been widely used in the field of neuroscience to provide insight into the nervous system. However, the tracking accuracy depends on the presence of artifacts in the EEG/MEG recordings. Artifacts include any signals that do not originate from neural activity, including physiological artifacts such as eye movement and non-physiological activity caused by the environment. This work proposes an integrated method for simultaneously tracking multiple neural sources using the probability hypothesis density particle filter (PPHDF) and reducing the effect of artifacts using feature extraction and stochastic modeling. Unique time-frequency features are first extracted using matching pursuit decomposition for both neural activity and artifact signals. The features are used to model probability density functions for each signal type using Gaussian mixture modeling for use in the PPHDF neural tracking algorithm. The probability density function of the artifacts provides information to the tracking algorithm that can help reduce the probability of incorrectly estimating the dynamically varying number of current dipole sources and their corresponding neural activity localization parameters. Simulation results demonstrate the effectiveness of the proposed algorithm in increasing the tracking accuracy performance for multiple dipole sources using recordings that have been contaminated by artifacts. / Dissertation/Thesis / Masters Thesis Electrical Engineering 2014

Multi-target tracking (MTT) methods estimate the trajectory of targets from noisy measurement; therefore, they can be used to handle the pedestrian-vehicle interaction for a moving vehicle. MTT has an important part in assisting the Automated Driving System and the Advanced Driving Assistance System to avoid pedestrian-vehicle collisions. ADAS and ADS rely on correct estimates of the pedestrians' position and velocity, to avoid collisions or unnecessary emergency breaking of the vehicle. Therefore, to help the risk evaluation in these systems, the MTT needs to provide accurate and robust information of the trajectories (in terms of position and velocity) of the pedestrians in different environments. Several factors can make this problem difficult to handle for instance in crowded environments the pedestrians can suffer from occlusion or missed detection. Classical MTT methods, such as the global nearest neighbour filter, can in crowded environments fail to provide robust and accurate estimates. Therefore, more sophisticated MTT methods should be used to increase the accuracy and robustness and, in general, to improve the tracking of targets close to each other. The aim of this master's thesis is to improve the situational awareness with respect to pedestrians and pedestrian-vehicle interactions. In particular, the task is to investigate if the GM-PHD and the GM-CPHD filter improve pedestrian tracking in urban environments, compared to other methods presented in the literature.  The proposed task can be divided into three parts that deal with different issues. The first part regards the significance of different clustering methods and how the pedestrians are grouped together. The implemented algorithms are the distance partitioning algorithm and the Gaussian mean shift clustering algorithm. The second part regards how modifications of the measurement noise levels and the survival of targets based on the target location, with respect to the vehicle's position, can improve the tracking performance and remove unwanted estimates. Finally, the last part regards the impact the filter estimates have on the tracking performance and how important accurate detections of the pedestrians are to improve the overall tracking. From the result the distance partitioning algorithm is the favourable algorithm, since it does not split larger groups. It is also seen that the proposed filters provide correct estimates of pedestrians in events of occlusion or missed detections but suffer from false estimates close to the ego vehicle due to uncertain detections. For the comparison, regarding the improvements, a classic standard MTT filter applying the global nearest neighbour method for the data association is used as the baseline. To conclude; the GM-CPHD filter proved to be the best out of the two proposed filters in this thesis work and performed better also compared to other methods known in the literature. In particular, its estimates survived for a longer period of time in presence of missed detection or occlusion. The conclusion of this thesis work is that the GM-CPHD filter improves the tracking performance and the situational awareness of the pedestrians.

Cette thèse est consacrée au problème d'estimation bayésienne pour le filtrage statistique, dont l'objectif est d'estimer récursivement des états inconnus à partir d'un historique d'observations, dans un modèle stochastique donné. Les modèles stochastiques considérés incluent principalement deux grandes classes de modèles : les modèles de Markov cachés et les modèles de Markov à sauts conditionnellement markoviens. Ici, le problème est abordé sous sa forme générale dans la mesure où nous considérons le problème du filtrage mono- et multi objet(s), ce dernier étant abordé sous l'angle de la théorie des ensembles statistiques finis et du filtre « Probability Hypothesis Density ». Tout d'abord, nous nous intéressons à l'importante classe d'approximations que constituent les algorithmes de Monte Carlo séquentiel, qui incluent les algorithmes d'échantillonnage d'importance séquentiel et de filtrage particulaire auxiliaire. Les boucles de propagation mises en jeux dans ces algorithmes sont étudiées et des algorithmes alternatifs sont proposés. Les algorithmes de filtrage particulaire dits « localement optimaux », c'est à dire les algorithmes d'échantillonnage d'importance avec densité d'importance conditionnelle optimale et de filtrage particulaire auxiliaire pleinement adapté sont comparés statistiquement, en fonction des paramètres du modèle donné. Ensuite, les méthodes de réduction de variance basées sur le théorème de Rao-Blackwell sont exploitées dans le contexte du filtrage mono- et multi-objet(s) Ces méthodes, utilisées principalement en filtrage mono-objet lorsque la dimension du vecteur d'état à estimer est grande, sont dans un premier temps étendues pour les approximations Monte Carlo du filtre Probability Hypothesis Density. D'autre part, des méthodes de réduction de variance alternatives sont proposées : bien que toujours basées sur le théorème de Rao-Blackwell, elles ne se focalisent plus sur le caractère spatial du problème mais plutôt sur son caractère temporel. Enfin, nous abordons l'extension des modèles probabilistes classiquement utilisés. Nous rappelons tout d'abord les modèles de Markov couple et triplet dont l'intérêt est illustré à travers plusieurs exemples pratiques. Ensuite, nous traitons le problème de filtrage multi-objets, dans le contexte des ensembles statistiques finis, pour ces modèles. De plus, les propriétés statistiques plus générales des modèles triplet sont exploitées afin d'obtenir de nouvelles approximations de l'estimateur bayésien optimal (au sens de l'erreur quadratique moyenne) dans les modèles à sauts classiquement utilisés; ces approximations peuvent produire des estimateurs de performances comparables à celles des approximations particulaires, mais ont l'avantage d'être moins coûteuses sur le plan calculatoire / This thesis focuses on the Bayesian estimation problem for statistical filtering which consists in estimating hidden states from an historic of observations over time in a given stochastic model. The considered models include the popular Hidden Markov Chain models and the Jump Markov State Space Systems; in addition, the filtering problem is addressed under a general form, that is to say we consider the mono- and multi-object filtering problems. The latter one is addressed in the Random Finite Sets and Probability Hypothesis Density contexts. First, we focus on the class of particle filtering algorithms, which include essentially the sequential importance sampling and auxiliary particle filter algorithms. We explore the recursive loops for computing the filtering probability density function, and alternative particle filtering algorithms are proposed. The ``locally optimal'' filtering algorithms, i.e. the sequential importance sampling with optimal conditional importance distribution and the fully adapted auxiliary particle filtering algorithms, are statistically compared in function of the parameters of a given stochastic model. Next, variance reduction methods based on the Rao-Blackwell theorem are exploited in the mono- and multi-object filtering contexts. More precisely, these methods are mainly used in mono-object filtering when the dimension of the hidden state is large; so we first extend them for Monte Carlo approximations of the Probabilty Hypothesis Density filter. In addition, alternative variance reduction methods are proposed. Although we still use the Rao-Blackwell decomposition, our methods no longer focus on the spatial aspect of the problem but rather on its temporal one. Finally, we discuss on the extension of the classical stochastic models. We first recall pairwise and triplet Markov models and we illustrate their interest through several practical examples. We next address the multi-object filtering problem for such models in the random finite sets context. Moreover, the statistical properties of the more general triplet Markov models are used to build new approximations of the optimal Bayesian estimate (in the sense of the mean square error) in Jump Markov State Space Systems. These new approximations can produce estimates with performances alike those given by particle filters but with lower computational cost

<div>Multi-target tracking (MTT) is the problem to simultaneously estimate the number of targets and their states or trajectories. Numerous techniques have been developed for over 50 years, with a multitude of applications in many fields of study; however, there are two most widely used approaches to MTT: i) data association-based traditional algorithms; and ii) finite set statistics (FISST)-based data association free Bayesian multi-target filtering algorithms. Most data association-based traditional filters mainly use a statistical or simple model of the feature without explicitly considering the correlation between the target behavior</div><div>and feature characteristics. The inaccurate model of the feature can lead to divergence of the estimation error or the loss of a target in heavily cluttered and/or low signal-to-noise ratio environments. Furthermore, the FISST-based data association free Bayesian multi-target filters can lose estimates of targets frequently in harsh environments mainly</div><div>attributed to insufficient consideration of uncertainties not only measurement origin but also target's maneuvers.</div><div>To address these problems, three main approaches are proposed in this research work: i) new feature models (e.g., target dimensions) dependent on the target behavior</div><div>(i.e., distance between the sensor and the target, and aspect-angle between the longitudinal axis of the target and the axis of sensor line of sight); ii) new Gaussian mixture probability hypothesis density (GM-PHD) filter which explicitly considers the uncertainty in the measurement origin; and iii) new GM-PHD filter and tracker with jump Markov system models. The effectiveness of the analytical findings is demonstrated and validated with illustrative target tracking examples and real data collected from the surveillance radar.</div>

<p>Multiple target tracking (MTT) is a major area that occurs in a variety of real world systems. The problem involves the detection and estimation of an unknown number of targets within a scenario space given a sequence of noisy, incomplete measurements. The classic approach to MTT performs data association between individual measurements, however, this step is a computationally complex problem. Recently, a series of algorithms based on Random Finite Set (RFS) theory, that do not require data association, have been introduced. This thesis addresses some of the main deficiencies involved with RFS methods and derives key extensions to improve them for use in real world systems.\\</p> <p>The first contribution is the Weight Partitioned PHD filter. It separates the Probability Hypothesis Density (PHD) surface into partitions that represent the individual state estimates both spatially and proportionally. The partitions are labeled and propagated over several time steps to form continuous track estimates. Multiple variants of the filter are presented. Next, the Multitarget Multi-Bernoulli (MeMBer) filter is extended to allow the tracking of manoeuvring targets. A model state variable is incorporated into the filter framework to estimate the probability of each motion model. The standard implementations are derived. Finally, a new linear variant of the Intensity filter (iFilter) is presented. A Gaussian Mixture approximation provides more computationally efficient implementation of the iFilter.</p> <p>Each of the new algorithms are validated on simulated data using standard multitarget tracking metrics. In each case, the methods improve on several aspects of multitarget tracking in the real world.</p> / Doctor of Engineering (DEng)

<p dir="ltr">Tracking and characterizing the space debris population in Earth orbit is necessary to ensure that space can continue to be used safely. However, because space objects are affected by non-conservative forces like drag and solar radiation pressure, predicting the long-term evolution of their orbits is impossible without knowledge of their attitude profiles. Such knowledge may be unavailable for inactive satellites or objects of which the observer is not the owner or operator. In many cases, attitude cannot be measured directly because resolved images of space objects are unavailable due to the distance between the object and the observer, and the effects of atmospheric seeing. However, the total brightness of objects can still be measured. A set of brightness measurements over time is referred to as a "light curve.'' An object's observed brightness is influenced by its attitude and other factors such as its orbit, shape, and reflective properties. If some of these other factors are known, attitude information may be extracted from a light curve. Existing methods of solving this attitude inversion problem either require a good initial guess for an object's rotational states or do not provide a full state estimate. The work in this thesis avoids both problems and provides a full state estimate without requiring an initial state guess.</p><p><br></p><p dir="ltr">The attitude estimation process assumes that the observation geometry and the observed object's shape, reflection properties, and inertia tensor are known. In this thesis, an initial method of searching for attitudes that could correspond to each measurement using the viewing sphere is described. These possible attitudes or "pseudo-measurements'' are then used to initialize a probability hypothesis density filter that is theoretically capable of representing the multi-modal nature of the attitude estimate using a Gaussian mixture model. However, the probability hypothesis density filter is found to often diverge from the truth because it is necessary to merge and prune components of the Gaussian mixture model to avoid computational intractability. In its place, a particle swarm optimizer method for performing an attitude inversion has been developed. This method uses analytic attitude solutions to quickly propagate a large number of attitude time histories simultaneously. The particle swarm optimizer method is validated using simulated light curves for several objects. A preliminary attempt is made to estimate the attitude of an object using real light curve measurements.</p>

Cette thèse est consacrée au problème d'estimation bayésienne pour le filtrage statistique, dont l'objectif est d'estimer récursivement des états inconnus à partir d'un historique d'observations, dans un modèle stochastique donné. Les modèles stochastiques considérés incluent principalement deux grandes classes de modèles : les modèles de Markov cachés et les modèles de Markov à sauts conditionnellement markoviens. Ici, le problème est abordé sous sa forme générale dans la mesure où nous considérons le problème du filtrage mono- et multi objet(s), ce dernier étant abordé sous l'angle de la théorie des ensembles statistiques finis et du filtre " Probability Hypothesis Density ". Tout d'abord, nous nous intéressons à l'importante classe d'approximations que constituent les algorithmes de Monte Carlo séquentiel, qui incluent les algorithmes d'échantillonnage d'importance séquentiel et de filtrage particulaire auxiliaire. Les boucles de propagation mises en jeux dans ces algorithmes sont étudiées et des algorithmes alternatifs sont proposés. Les algorithmes de filtrage particulaire dits " localement optimaux ", c'est à dire les algorithmes d'échantillonnage d'importance avec densité d'importance conditionnelle optimale et de filtrage particulaire auxiliaire pleinement adapté sont comparés statistiquement, en fonction des paramètres du modèle donné. Ensuite, les méthodes de réduction de variance basées sur le théorème de Rao-Blackwell sont exploitées dans le contexte du filtrage mono- et multi-objet(s) Ces méthodes, utilisées principalement en filtrage mono-objet lorsque la dimension du vecteur d'état à estimer est grande, sont dans un premier temps étendues pour les approximations Monte Carlo du filtre Probability Hypothesis Density. D'autre part, des méthodes de réduction de variance alternatives sont proposées : bien que toujours basées sur le théorème de Rao-Blackwell, elles ne se focalisent plus sur le caractère spatial du problème mais plutôt sur son caractère temporel. Enfin, nous abordons l'extension des modèles probabilistes classiquement utilisés. Nous rappelons tout d'abord les modèles de Markov couple et triplet dont l'intérêt est illustré à travers plusieurs exemples pratiques. Ensuite, nous traitons le problème de filtrage multi-objets, dans le contexte des ensembles statistiques finis, pour ces modèles. De plus, les propriétés statistiques plus générales des modèles triplet sont exploitées afin d'obtenir de nouvelles approximations de l'estimateur bayésien optimal (au sens de l'erreur quadratique moyenne) dans les modèles à sauts classiquement utilisés; ces approximations peuvent produire des estimateurs de performances comparables à celles des approximations particulaires, mais ont l'avantage d'être moins coûteuses sur le plan calculatoire