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>

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

<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>

Multitarget tracking in the presence of false alarm is a difficult problem to consider. The objective of multitarget tracking is to estimate the number of targets and their states recursively from available observations. At any given time, targets can be born, die and spawn from already existing targets. Sensors can detect these targets with a defined threshold, where normally the observation is influenced by false alarm. Also if the targets are with low signal to noise ratio (SNR) then the targets may not be detected. The Random Finite Set (RFS) filters can be used to solve such multitarget problem efficiently. Specially, one of the best and most widely used RFS based filter is the Probability Hypothesis Density (PHD) filter. The PHD filter approximates the posterior probability density function (PDF) by the first order moment only, where the targets SNR assumed to be much higher. The PHD filter supports targets die, born, spawn and missed-detection by using the well known implementations including Sequential Monte Carlo Probability Hypothesis Density (SMC-PHD) and Gaussian Mixture Probability Hypothesis Density (GM-PHD) methods. The SMC-PHD filter suffers from the well known degeneracy problems while GM-PHD filter may not be suitable for nonlinear and non-Gaussian target tracking problems. It is desirable to have a filter that can provide continuous estimates for any distribution. This is the motivation for the use of B-Splines in this thesis. One of the main focus of the thesis is the B-Spline based PHD (SPHD) filters. The Spline is a well developed theory and been used in academia and industry for more than five decades. The B-Spline can represent any numerical, geometrical and statistical functions and models including the PDF and PHD. The SPHD filter can be applied to linear, nonlinear, Gaussian and non-Gaussian multitarget tracking applications. The SPHD continuity can be maintained by selecting splines with order of three or more, which avoids the degeneracy-related problem. Another important characteristic of the SPHD filter is that the SPHD can be locally controlled, which allow the manipulations of the SPHD and its natural tendency for handling the nonlinear problems. The SPHD filter can be further extended to support maneuvering multitarget tracking, where it can be an alternative to any available PHD filter implementations. The PHD filter does not work well for very low observable (VLO) target tracking problems, where the targets SNR is normally very low. For very low SNR scenarios the PDF must be approximated by higher order moments. Therefore the PHD implementations may not be suitable for the problem considered in this thesis. One of the best estimator to use in VLO target tracking problem is the Maximum-Likelihood Probability Data Association (ML-PDA) algorithm. The standard ML-PDA algorithm is widely used in single target initialization or geolocation problems with high false alarm. The B-Spline is also used in the ML-PDA (SML-PDA) implementations. The SML-PDA algorithm has the capability to determine the global maximum of ML-PDA log-likelihood ratio with high efficiency in terms of state estimates and low computational complexity. For fast passive track initialization, search and rescue operations the SML-PDA algorithm can be used more efficiently compared to the standard ML-PDA algorithm. Also the SML-PDA algorithm with the extension supports the multitarget tracking. / Thesis / Doctor of Philosophy (PhD)

La poursuite multi-cibles a pour objet le suivi d'un ensemble de cibles mobiles à partir de données obtenues séquentiellement. Ce problème est particulièrement complexe du fait du nombre inconnu et variable de cibles, de la présence de bruit de mesure, de fausses alarmes, d'incertitude de détection et d'incertitude dans l'association de données. Les filtres PHD (Probability Hypothesis Density) constituent une nouvelle gamme de filtres adaptés à cette problématique. Ces techniques se distinguent des méthodes classiques (MHT, JPDAF, particulaire) par la modélisation de l'ensemble des cibles comme un ensemble fini aléatoire et par l'utilisation des moments de sa densité de probabilité. Dans la première partie, on s'intéresse principalement à la problématique de l'application des filtres PHD pour le filtrage multi-cibles maritime et aérien dans des scénarios réalistes et à l'étude des propriétés numériques de ces algorithmes. Dans la seconde partie, nous nous intéressons à l'étude théorique des processus de branchement liés aux équations du filtrage multi-cibles avec l'analyse des propriétés de stabilité et le comportement en temps long des semi-groupes d'intensités de branchements spatiaux. Ensuite, nous analysons les propriétés de stabilité exponentielle d'une classe d'équations à valeurs mesures que l'on rencontre dans le filtrage non-linéaire multi-cibles. Cette analyse s'applique notamment aux méthodes de type Monte Carlo séquentielles et aux algorithmes particulaires dans le cadre des filtres de Bernoulli et des filtres PHD.

La poursuite multi-cibles a pour objet le suivi d’un ensemble de cibles mobiles à partir de données obtenues séquentiellement. Ce problème est particulièrement complexe du fait du nombre inconnu et variable de cibles, de la présence de bruit de mesure, de fausses alarmes, d’incertitude de détection et d’incertitude dans l’association de données. Les filtres PHD (Probability Hypothesis Density) constituent une nouvelle gamme de filtres adaptés à cette problématique. Ces techniques se distinguent des méthodes classiques (MHT, JPDAF, particulaire) par la modélisation de l’ensemble des cibles comme un ensemble fini aléatoire et par l’utilisation des moments de sa densité de probabilité. Dans la première partie, on s’intéresse principalement à la problématique de l’application des filtres PHD pour le filtrage multi-cibles maritime et aérien dans des scénarios réalistes et à l’étude des propriétés numériques de ces algorithmes. Dans la seconde partie, nous nous intéressons à l’étude théorique des processus de branchement liés aux équations du filtrage multi-cibles avec l’analyse des propriétés de stabilité et le comportement en temps long des semi-groupes d’intensités de branchements spatiaux. Ensuite, nous analysons les propriétés de stabilité exponentielle d’une classe d’équations à valeurs mesures que l’on rencontre dans le filtrage non-linéaire multi-cibles. Cette analyse s’applique notamment aux méthodes de type Monte Carlo séquentielles et aux algorithmes particulaires dans le cadre des filtres de Bernoulli et des filtres PHD. / The problem of multiple-object tracking consists in the recursive estimation ofthe state of several targets by using the information coming from an observation process. The objective of this thesis is to study the spatial branching processes andthe measure-valued systems arising in multi-object tracking. We focus on a class of filters called Probability Hypothesis Density (PHD) filters by first analyzing theirperformance on simulated scenarii and then by studying their properties of stabilityand convergence. The thesis is organized in two parts: the first part overviewsthe techniques proposed in the literature and introduces the Probability Hypothesis Density filter as a tractable approximation to the full multi-target Bayes filterbased on the Random Finite Sets formulation. A series of contributions concerning the numerical implementation of PHD filters are proposed as well as the analysis of their performance on realistic scenarios.The second part focuses on the theoretical aspects of the PHD recursion in the context of spatial branching processes. We establish the expression of the conditional distribution of a latent Poisson point process given an observation process and propose an alternative derivation of the PHD filter based on this result. Stability properties, long time behavior as well as the uniform convergence of a general class of stochastic filtering algorithms are discussed. Schemes to approximate the measure valued equations arising in nonlinear multi-target filtering are proposed and studied.