Hybrid fuzzy probabilistic data association filter and joint probabilistic data association filter

Abstract

Multitarget tracking problems are theoretically interesting because, unlike other estimation problems, the origins of the measurements are not identified. This involves hypothesis generation and their evaluation in terms of degree of agreement between the given measurements and the underlying tracks. Typical algorithms to deal with such problems are the probabilistic data association filter (PDAF) in the case of single target tracking and joint probabilistic data association filter (JPDAF) in the case of multiple target tracking proposed by Bar-Shalom and his team. The basis of JPDAF is the calculus of the joint probabilities over all targets and hits. The algorithm assigns weights for reasonable hits and uses a weighted centroid of those hits to update the track. In this paper, we propose a new weight assignment based on fuzzy c-means methodology. Particularly, in order to take account for the false alarms (clutter) where none of the measurements is target originated, a new noisy fuzzy c-means algorithm is elaborated. The latter contrasts with that provided by Dave regarding the location of the noise prototype as well as the meaning of the universality of the noise class. The treatment of conflictual situations where, for instance, more than one hit fail in a target extension gate is accomplished using some weighted based procedure with respect to all feasible joint matrices involved in the construction of joint probabilities in JPDAF. In the meantime, the general methodology of PDAF and JPDAF remains unchanged. This leads to Hybrid Fuzzy PDAF in the case of single target tracking and Hybrid Fuzzy JPDAF in the case of multiple target tracking. This investigation shows a fruitful combination between fuzzy and probabilistic approaches in order to accomplish target tracking tasks.

Introduction

Typically, target tracking in dense environments where probability of presence of “clutter” or false alarms is not neglected is a difficult problem. Indeed, in addition to the noisy measurements (hits) supplied by the source, there is an additional uncertainty concerning the origin of the measurement. In other words, we do not know exactly from which sensor or source (a clutter is also a source) the given measurement is originated from. This induces a risk of updating the target model by a wrong measurement, which obviously leads to a wrong estimation of a target state vector (position, velocity and so on). Consequently, this leads to a generation of a set of measurement/target associations and, thereby, evaluation of such hypotheses, usually in terms of joint probability values. Such problems arise in applications like air traffic control, ocean/battlefield surveillance, positioning of enemy targets in military context.

The field of multiple target tracking has been investigated intensively since the 1970s and the pioneer works of Singer and Sea [16] on tracking in clutter environment, see also recent papers in [1], [6].

A significant problem in multiple target tracking is the hit-to-track data association. The aim is to perform the estimation of some unknown parameters, described in terms of the state vector and its associated variance–covariance matrix. This estimation is performed such that each hit is related to its appropriate target including a clutter for false alarms with a high value of joint probability or confidence factor in more general terms. Bar-Shalom and Tsee [2] have first proposed “Probabilistic Data Association Filter” (PDAF) as a method for associating hits where only one target is available. That is, it assumes that all hits are in particular target extension gate and originated either from a target or clutter. If another target is persistently in this target extension gate, the results are poor and may be wrong. To account for the problem of more than one target in the cluttered environment, an extension of PDAF called “joint probabilistic data association filter” (JPDAF) was investigated by Fortmann et al. [12]. Their proposal acts as a natural extension of PDAF where all the feasible joint events are considered. The process was simplified by considering just the feasible joint events, those for which there is at most one hit per target and no two tracks associated with the same hit. These algorithms, among others, have successfully demonstrated their feasibility in high cluttered situations including air traffic management and military applications [3].

However, the calculation of joint probabilities in JPDAF seems complicated even if the used formulas are well established in probability theory, while the non-optimality still occurs regarding the number of hypotheses and restrictions governing the construction of PDAF or JPDAF. This motivates some authors to investigate simplifications of these formulations. This includes, for instance, Fitzgerald's ad hoc proposal [11], which seems to work well in some cases of crossing targets.

The class of algorithms, so-called fuzzy ISODATA algorithms developed mainly by original works of Dunn and Bezdek (see [4] and references therein), permits performing an unsupervised classification of a set of data given in multidimensional space into a given c number of classes. The algorithm assigns at each datum a grade of membership to which it deemed to be in agreement with each of the c classes such that the datum belongs certainly to all the classes in the probabilistic sense. The latter arises from the constraint, which asserts that, for each datum, the sum of the membership grades over the c classes is equal to one. Dave [9] introduced a robust clustering by incorporating an additional imaginary class that takes account for all noisy data. The idea developed in this paper is the possibility of incorporating the membership grades supplied by the fuzzy algorithm as a direct counterpart of the joint probabilities in PDAF or JPDAF while keeping the main structure of both PDAF and JPDAF. This leads to Hybrid Fuzzy PDAF in the case of single target tracking and Hybrid Fuzzy JPDAF in the case of multiple target tracking. The fuzzy classification algorithm is inspired from Dave's approach. Particularly, in order to take account for the false alarms (clutter) where none of the measurements is target originated, a new noisy fuzzy c-means algorithm is elaborated. The latter contrasts with that provided by Dave regarding the location of the noise prototype as well as the meaning of the universality of the noise class. Strictly speaking, the fact of substituting joint probabilities by other values even outside the framework of probability in the same process is not quite strange as it may sound at first glance. Indeed, the justification of such substitution can be carried out from different viewpoints:

• The existence of different proposals like Fitzgerald's ad hoc formulations for the joint probabilities proves the non-uniqueness of the solution.

• The interpretation of the joint probability quantities as degrees of agreement between measurement and targets means that both joint probabilities and membership grades have the same interpretative setting.

• The general constraints governing the construction of the joint probabilities are kept preserved in the fuzzy setting.

This induces a fruitely combination of probabilistic and fuzzy approaches. Section 2 of this paper deals with a short description of PDAF and JPDAF algorithms including their physical constraints, requirements and formulations. In Section 3 the methodology of fuzzy c-means is detailed. Then, a modified approach, which takes account for PDAF/JPDAF requirements, is investigated. Section 4 emphasizes some simulation results where the performances of the elaborated hybrid fuzzy PDAF/JPDAF are compared with the original version of PDAF/JPDAF as well as with the Fitzgerald's ad hoc approach. The simulations consist of a constant velocity moving targets with the same or different directions. Single target moving makes a link to PDAF (and alternative approach) and multiple targets build a bridge to JPDAF (and alternative approach).