The Social Force PHD Filter for Tracking Pedestrians

The probability hypothesis density (PHD) filter is a practical alternative to the optimal Bayesian multi-target filter based on random finite sets. It propagates the PHD function, the first order moment of the posterior multi-target density, from which the number of targets as well as their individual states can be extracted. Furthermore, the sequential Monte Carlo (SMC) approximation of the PHD filter (also known as particle-PHD filter) is available in the literature in order to overcome its intractability. However, the PHD filter keeps no track of the target identities and hence cannot produce track-valued estimates of individual targets. This work consider the use of an improved implementation, of the particle-PHD filter that gives the track-valued estimates of individual targets and propose a novel way for doing so. The improved PHD filter combines the particles approximation of the posterior PHD function and the peak extraction from the posterior PHD particles to create the target identities of the individual estimates. The improved PHD filter does not affect the convergence results of the particle-PHD filter

The probability hypothesis density (PHD) filter is a first moment approximation to the evolution of a dynamic point process which can be used to approximate the optimal filtering equations of the multiple-object tracking problem. We show that, under reasonable assumptions, a sequential Monte Carlo (SMC) approximation of the PHD filter converges in mean of order \(p \geq 1\), and hence almost surely, to the true PHD filter. We also present a central limit theorem for the SMC approximation, show that the variance is finite under similar assumptions and establish a recursion for the asymptotic variance. This provides a theoretical justification for this implementation of a tractable multiple-object filtering methodology and generalises some results from sequential Monte Carlo theory.

Multi-target tracking is a common problem with many applications. In most of these the expected number of targets is not known a priori, so that it has to be estimated from the measured data. Poisson point processes (PPPs) are a very useful theoretical model for diverse applications. One of those is multi-target tracking of an unknown number of targets, leading to the intensity filter (iFilter) and the probability hypothesis density (PHD) filter. This chapter presents a sequential Monte Carlo (SMC) implementation of the iFilter. In theory it was shown that the iFilter can estimate a clutter model from the measurements and thus does not need it as a priori knowledge, like the PHD filter does. Our studies show that this property holds not only in simulations but also in real world applications. In addition it can be shown that the performance of the PHD filter decreases substantially if the a priori knowledge of the clutter intensity is chosen incorrectly.

The Bernoulli filter is an exact Bayesian filter for target tracking under the framework of random finite sets. It has been implemented by using Gaussian mixture (GM) and sequential Monte Carlo (SMC) techniques for joint target detection and tracking. However, a single model is not enough to accommodate a maneuvering target that switches between a set of motion models. In this paper, we extend the Bernoulli filter to switching motion models, and propose a multiple-model (MM) Bernoulli filter for a maneuvering target. Then we give a GM implementation of the MM Bernoulli filter for linear Gaussian models. Finally, a numerical example is presented to verify the effectiveness of the GM-MM Bernoulli filter for a maneuvering target.

<div class="section abstract"><div class="htmlview paragraph">In the domain of advanced driver assistance systems and autonomous vehicles, precise perception and interpretation of the vehicle's environment are not merely requirements they are the very foundation upon which every aspect of functionality and safety is constructed. One prevalent method of representing the environment is through the use of an occupancy grid map. This map segments the environment into distinct grid cells, each of which is evaluated to determine if it is occupied or free. This evaluation operates under the assumption that each grid cell is independent of the others. The underlying mathematical structure of this system is the binary Bayes filter (BBF). The BBF integrates sensor data from various sources and can incorporate measurements taken at different times. The occupancy grid map does not rely on the identification of individual objects, which allows it to depict obstacles of any shape. This flexibility is a key advantage of this approach. Traditional occupancy grid maps fall short when it comes to predicting dynamic environments due to their lack of a process model. A notable enhancement to this static model is the Bayesian Occupancy Filter (BOF), which, unlike its predecessor, estimates a velocity distribution for each grid cell's occupancy using a histogram filter. However, the BOF's computational demands are high. To address this, research propose representing the dynamic state of grid cells using particles. This method enables the computation of dynamic grid maps in real-time applications, even with larger grid cell sizes and higher resolution. Despite these advancements, dynamic occupancy grid maps remain a relatively new field of study, especially when compared to more established object-tracking approaches. Until now, the BOF has been treated as a distinct research area with minimal overlap with other tracking methodologies. This methodology aims to bridge that gap and foster a more integrated approach to dynamic environment estimation.</div><div class="htmlview paragraph">This study introduces a novel approach to dynamic grid mapping, conceptualized as an approximation of a Random Finite Set (RFS) filter. An RFS is a probabilistic representation of a finite, random collection of objects and their respective states. Finite Set Statistics (FISST) provide a framework for Bayesian filtering of these random finite sets and form the foundation for several multi-object tracking methodologies, such as the Probability Hypothesis Density (PHD) filter. By characterizing the grid as an RFS, we can apply sophisticated concepts from the well-established domain of RFS filtering to dynamic grid mapping. The research develops a filter known as the Probability Hypothesis Density/Multi-Instance Bernoulli (PHD/MIB) filter. This filter alternately represents and propagates the dynamic grid map as a PHD and as multiple instances of Bernoulli filters, thereby offering a more integrated and efficient approach to dynamic environment estimation.</div><div class="htmlview paragraph">Furthermore, this research introduces a Sequential Monte Carlo (SMC) implementation of the PHD/MIB filter, as well as an approximation within the Dempster-Shafer framework, termed the Dempster-Shafer PHD/MIB (DS-PHD/MIB) filter. This DS-PHD/MIB filter necessitates fewer particles than the original PHD/MIB filter, thereby enhancing computational efficiency. The study provides a comprehensive description of an efficient, massively parallel implementation of the DS-PHD/MIB filter. The algorithm's pseudo code is also outlined, offering a clear and concise understanding of its workings. This approach further strengthens the integrated and efficient methodology for dynamic environment estimation.</div><div class="htmlview paragraph">In conclusion, the research delineates the attributes of the DS-PHD/MIB filter and debates its pros and cons in comparison to object-based tracking methodologies, using practical examples for illustration. A quantitative assessment using real-world data demonstrates that the DS-PHD/MIB filter yields consistent state estimation outcomes. It effectively models both the stochastic multi-object transition process and the stochastic multi-object observation process. Moreover, the evaluation affirms the real-time capability of the parallelized implementation of the DS-PHD/MIB filter. It validates its utility for state estimation in dynamic vehicle environments, thereby underscoring its potential for practical applications in dynamic environment estimation. This comprehensive approach offers a promising avenue for future research and development in this field.</div></div>

A resampling scheme is proposed for use with Sequential Monte Carlo (SMC)-based Probability Hypothesis Density (PHD) filters. It consists of two steps, first, regions of interest are identified, then an evolutionary resampling is applied for each region. Applying resampling locally corresponds to treating each target individually, while the evolutionary resampling introduces a memory to a group of particles, increasing the robustness of the estimation against noise outliers. The proposed approach is compared to the original SMC-PHD filter for tracking multiple targets in a deterministically moving targets scenario, and a noisy motion scenario. In both cases, the proposed approach provides more accurate estimates.

Multiple target tracking requires data association that operates in conjunction with filtering. When multiple targets are closely spaced, the conventional approaches (as, e.g., MHT/assignment) may not give satisfactory results. This is mainly because of the difficulty in deciding what the number of targets is. Recently, the probability hypothesis density (PHD) filter has been proposed and particle filtering techniques have been developed to implement the PHD filter. In the particle PHD filter, the track labeling problem is not considered, i.e., the PHD is obtained only for a frame at a time, and it is very difficult to perform the multipeak extraction, particularly in high clutter environments. A track labeling method combined with the PHD approach, as well as considering the finite resolution, is proposed here for multitarget tracking, i.e., we keep a separate tracker for each target, use the PHD in the resolution cell to get the estimated number and locations of the targets at each time step, and then perform the track labeling ("peak-to-track" association), whose results can provide information for PHD peak extraction at the next time step. Besides, by keeping a separate tracker for each target, our approach provides more information than the standard particle PHD filter. For example, in group target tracking, if we are interested in the motion of a specific target, we can track this target, which is not possible for the standard particle PHD filter, since the standard particle PHD filter does not keep track labels. Using our approach, multitarget tracking can be performed with automatic track initiation, maintenance, spawning, merging, and termination

The probability hypothesis density (PHD) filter is an effective means to track multiple targets in that it avoids explicit data associations between the measurements and targets. However, the target birth intensity as a prior is assumed to be known before tracking in a traditional target-tracking algorithm; otherwise, the performance of a conventional PHD filter will decline sharply. Aiming at this problem, a novel target birth intensity scheme and an improved measurement-driven scheme are incorporated into the PHD filter. The target birth intensity estimation scheme, composed of both PHD pre-filter technology and a target velocity extent method, is introduced to recursively estimate the target birth intensity by using the latest measurements at each time step. Second, based on the improved measurement-driven scheme, the measurement set at each time step is divided into the survival target measurement set, birth target measurement set, and clutter set, and meanwhile, the survival and birth target measurement sets are used to update the survival and birth targets, respectively. Lastly, a Gaussian mixture implementation of the PHD filter is presented under a linear Gaussian model assumption. The results of numerical experiments demonstrate that the proposed approach can achieve a better performance in tracking systems with an unknown newborn target intensity.

We propose an approach for tracking a varying number of simultaneously active acoustic wideband signal sources in an acoustic enclosure. Relying on the assumption of W-disjoint orthogonality, the method uses narrowband position estimates of the sources for the targets. The instantaneous position estimates form clusters rather than single points, as would be required for a conventional Probability Hypothesis Density (PHD) filter. Therefore, we model the position estimates as extended targets and use a special form of the PHD filter, the extended target Gaussian mixture PHD filter, for tracking the targets. This allows to model target birth and death, which correspond to speech onset and end of utterance, respectively. With this model and by using the well-developed theory of Finite Set Statistics (FISST)-based multi-target tracking, we provide a comprehensive, strictly Bayesian treatment of the problem of tracking wideband acoustic sources using narrowband position estimates. We validate the results by tracking a varying number of targets in an enclosure simulated with the image-source method.

We evaluate the use of a probability hypothesis density (PHD) filter in a bearings-only tracking application. The main feature of a PHD filter is that it propagates the first-order statistical moment of a multisource posterior distribution. Multisource estimation using a PHD filter has been shown to reliably track multiple simulated targets in the bearings-only case. In this paper we evaluate the utility of the sequential Monte-Carlo PHD filter for tracking surface ships using bearings-only data acquired from a Bluefin-21 unmanned underwater vehicle in Boston Harbor. The unmanned underwater vehicle was equipped with a rigidly mounted planar hydrophone array that measures the bearing angle to sources of acoustic noise, of which shipping traffic is the dominant source. We further evaluate several target maneuvering models, including clockwise and counter-clockwise coordinated turns. The combination of the coordinated turn models with a constant velocity model is used in a multiple model PHD filter. The results of the multiple model PHD filter are compared to the results of a PHD filter using only a constant velocity model.

In the extended target tracking using Gaussian process (GP) or other methods to estimate the extension state, all of them is necessary to obtain the measurement model, and the deviation from the true kinematic state affects the extension result easily. Meanwhile, with the growth of the shape complexity, the corresponding covariance matrix dimension increases, leading to a very high computational burden. To this end, we present an extended target tracking algorithm based on deep learning, which adopts Gaussian mixture (GM) probability hypothesis density (PHD) filter for extended target (ET) tracking, called the ET-GM-PHD filter, to estimate the kinematic state, and uses convolutional conditional neural process (ConvCNP) to estimate the extension state (i.e., the shape of the target) avoiding the complex computation and dependence on priors. Therefore, the proposed algorithm is named as the ConvCNP-GM-PHD filter. In our algorithm, ConvCNP lies on the neural network (NN) and GP, where the former can learn the rules from a large amount of data without any priors, while the latter uses the covariance function to express the distribution over possible functions according to a small number of context observations. Due the proposed algorithm can effectively combine the advantages of both, it is reliable to get a model via training which is able to realize function regression flexibly. Thus, ConvCNP is able to estimate the shape function by putting the angles and radial value between the kinematic state and associated measurements into the pre-trained model. The results show that the accuracy of the extension state estimation of the proposed ConvCNP-GM-PHD filter is improved and its computational complexity is also reduced. It implies that the proposed algorithm is more real-time and effective, and has a good application prospect.

The probability hypothesis density (PHD) filter is an attractive approach to tracking an unknown and time-varying number of targets in the presence of data association uncertainty, clutter, noise, and detection uncertainty. The PHD filter admits a closed-form solution for a linear Gaussian multi-target model. However, this model is not general enough to accommodate maneuvering targets that switch between several models. In this paper, we generalize the notion of linear jump Markov systems to the multiple target case to accommodate births, deaths, and switching dynamics. We then derive a closed-form solution to the PHD recursion for the proposed linear Gaussian jump Markov multi-target model. Based on this an efficient method for tracking multiple maneuvering targets that switch between a set of linear Gaussian models is developed. An analytic implementation of the PHD filter using statistical linear regression technique is also proposed for targets that switch between a set of nonlinear models. We demonstrate through simulations that the proposed PHD filters are effective in tracking multiple maneuvering targets.

We propose a filtering framework for multitarget tracking that is based on the probability hypothesis density (PHD) filter and data association using graph matching. This framework can be combined with any object detectors that generate positional and dimensional information of objects of interest. The PHD filter compensates for missing detections and removes noise and clutter. Moreover, this filter reduces the growth in complexity with the number of targets from exponential to linear by propagating the first-order moment of the multitarget posterior, instead of the full posterior. In order to account for the nature of the PHD propagation, we propose a novel particle resampling strategy and we adapt dynamic and observation models to cope with varying object scales. The proposed resampling strategy allows us to use the PHD filter when a priori knowledge of the scene is not available. Moreover, the dynamic and observation models are not limited to the PHD filter and can be applied to any Bayesian tracker that can handle state-dependent variances. Extensive experimental results on a large video surveillance dataset using a standard evaluation protocol show that the proposed filtering framework improves the accuracy of the tracker, especially in cluttered scenes.

Sequential Monte Carlo methods for complexity-constrained MAP equalization of dispersive MIMO channels

The multiple model multi-Bernoulli filter based track-before-detect using a likelihood based adaptive birth distribution