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https://doi.org/10.1016/j.inffus.2015.10.001

Multi-Hypotheses Tracking using the Dempster-Shafer

Theory

Application to ambiguous road context

Dominique Gruyera, Sébastien Demmelb,∗, Valentin Magniera, RachidBelaroussia

aCOSYS-LIVIC (IFSTTAR), 77 rue des Chantiers, 78000 Versaillles - France

bCentre for Accident Research nad Road Safety - Queensland, 130 Victoria Park Road,Kelvin Grove QLD 4059 - Australia

Abstract

This paper presents a Multi-Hypotheses Tracking (MHT) approach that allowssolving ambiguities that arise with previous methods of associating targets andtracks within a highly volatile vehicular environment. The previous approachbased on the Dempster-Shafer Theory assumes that associations between tracksand targets are unique; this was shown to allow the formation of ghost trackswhen there was too much ambiguity or conict for the system to take a meaning-ful decision. The MHT algorithm described in this paper removes this unique-ness condition, allowing the system to include ambiguity and even to preventmaking any decision if available data are poor. We provide a general introduc-tion to the Dempster-Shafer Theory and present the previously used approach.Then, we explain our MHT mechanism and provide evidence of its increased per-formance in reducing the amount of ghost tracks and false positive processedby the tracking system.

Keywords: Tracking; association; ambiguity; Dempster-Shafer Theory

1. Introduction

Many existing or in development Intelligent Transportation Systems (ITS) ap-plications perform tasks that provide some degree of perception of the road'senvironment, and use this perception to achieve their goal, for example the

∗Corresponding author; phone: +61 7 3138 7783Email addresses: dominique.gruyer@ifsttar.fr (Dominique Gruyer),

sebastien.demmel@qut.edu.au (Sébastien Demmel), valentin.magnier@ifsttar.fr(Valentin Magnier), rachid.belaroussi@ifsttar.fr (Rachid Belaroussi)

Preprint submitted to Information Fusion 29th May 2015

Table 1: Association methods and their characteristics

MethodManage

Multi-targetsProcess.

appear./disappear. load

Nearest neighbour No No LowPDA No No LowJPDA No Yes Average

NNJPDA No Yes LowMHF Yes Yes Heavy

Dempster-Shafer Yes Yes Average

detection of impending collisions between vehicles. To understand their envir-onment, they rely on sensors that gather information about their surroundingsat a given frequency. Because of hardware/software limitations, or because oneuses multiple dierent sensors, the information about the vehicle's environmentcan be highly asynchronous. For the ITS applications to perform their taskproperly, a mechanism is needed to reconstruct the evolution of the scene overtime, taking into account those gaps and also imperfections in the known dataarising from, for example, sensors defects; this mechanism is known as tracking.

Tracking an object such as a vehicle on the road is a three step process, with thestages: (1) synchronisation, (2) association and (3) fusion. The synchronisationtask is to predict the evolution of the known objects to the current timestampk, knowning information on their behaviour at time k − 1. The classical wayto predict this evolution is to use a Kalman Filter estimator [1, 2]. Predictedobjects are called tracks, while observations from the sensor(s) are called targets.The association step consist in nding which tracks correspond to which targetsbefore they can be fused together in a last step to obtain a more accuratedescription of the scene at time k. In this paper, we will focus on the secondassociation step, which is the most complex of the three.

Many dierent association methods exist (a summary of their advantages anddisadvantages is given in Table 1), some fairly straighforward; for example thenearest neighbour method that simply considers the distance between tracks andtargets and associate the objects that fall closest to each others. The distancecan be computed using the Euclidean distance, the Mahalanobis distance, etc.Unfortunately, this method is inappropriate for complex problems [3]. Somemore complex methods are based on probabilistic approaches: probabilistic dataassociation (PDA), joint probabilistic data association (JPDA and JPDAM),nearest neighbour JPDA, etc. [4, 5] Overall, these methods compute associationprobabilities between the tracks and targets, but cannot manage the appareanceor dissappearance of tracks.

The Multi-hypotheses Filter (MHF) [6] can manage new tracks. It looks for theprobabilities associated with three specic hypotheses for targets, whether (1)they associate with known tracks, (2) they associate with new tracks, and (3)they are false positives. Unfortunately, the MHF is relatively computationally

2

heavy.

An alternative approach growing in popularity is uses the Dempster-Shafer The-ory [7, 8], also known as the Belief Theory. It is advantageous in an automotivecontext because it can handle imprecision and incertitude in a more suitableway than probabilistic theories, as well as manage ignorance and conicts. Aframework for association using Dempster-Shafer was proposed in [3, 9], but thisapproach has some aws. Notably, it will not use all the available informationbecause it is forced to make a choice when associating tracks and target, loosingall the potentially useful information contained within ambiguities and conicts.This leads to the formation of false positives, dubbed ghost tracks. In this pa-per, we propose a solution to this latter problem using a Multi-HypothesesTracking (MHT) algorithm.

In the remainder of this paper we will at rst introduce the Dempster-ShaferTheory in general terms, notably the notion of belief functions (section 2). Then,after having led out the formulation of our association problem (section 3), wewill present the principles for generating basic belief assignments (BBA) fromsensors data (section 4) and their combination so that the system can take adecision on which tracks to associate with each targets (section 5). Then, wewill present our MHT algorithm (section 6) and provide a demonstration of itsperformance compared to the PDAF and the classical Dempster-Shafer tracking(section 7). The chosen examples for this comparison focus on pedestrianswalking in front of a laserscanner; this situation is particularly complex due tothe location of the scanning plane at legs level, creating multiple ambiguities.Another example more linked to the automotive context is also proposed throughthe study of a car overtaking maneuvre observed by a laserscanner sensor.

2. Belief functions

Belief functions were introduced by Dempster [7] and further rened by Shafer[8], taking the name of the Dempster-Shafer Theory. It is also sometimes referredto as the Belief Theory. A further extension was undertaken by Smets [10, 11]to create the Transferable Belief Model (TBM). Let us at rst dene Ω, theuniversal set that represents the various possible states for the system underconsideration, i.e. the frame of discernment. The possible states are thesimple (singletons) acceptable propositions Hi, so that:

Ω = H1, H2, . . . ,Hn (1)

Hi ∩ Hj = ø, ∀i 6= j (2)

From this universal set, we can dene the power-set 2Ω that is the set of allsubsets of Ω, including the empty set ∅. The power-set includes all the com-binations based on the hypotheses from the universal set. Proposition A can be

3

a singleton hypothesis or a complex hypothesis, which includes more than onehypotheses. ∅Ω represents impossible propositions (conicts) and Ω the totalignorance, since it includes all existing hypotheses.

2Ω = A/A ⊆ Ω = øΩ, H1, H2, . . . ,Hn, H1 ∪H2, . . . ,Ω (3)

In [12, 3, 13], the authors used an extended universal set Θ, the extended open

world, by creating a new hypothesis labelled ∗ which represents any new hypo-thesis that is not initially modelled in Ω. This approach allows discriminatingbetween conict and new hypotheses, which is not possible in the general ap-proach.

Θ = H1, H2, . . . ,Hn, ∗ (4)

2Θ = øΘ, H1, H2, . . . ,Hn, H1 ∪H2, . . . , ∗,Ω (5)

The Dempster-Shafer Theory allows to evaluate the likelihood of a propositionA through its belief mass mΩ (A), the mass of elementary probability on thesaid proposition A, a function dened as:

mΩ : 2Ω → [0, 1]A → mΩ (A)

(6)

The set of belief masses constitutes the basic belief assignment (BBA), whichveries:

mΩ (øΩ) = 0 (7)∑A∈2Ω

mΩ (A) = 1 (8)

mΩ (A) is the degree of belief assigned to proposition A, more precisely it ex-presses the proportion of all relevant and available evidence that supports theclaim that the actual state belongs to A but to no particular subset of A.mΩ (Ω) represents the mass of ignorance. If A is a complex hypothesis (i.e.not a singleton one), it means that given the current state of knowledge on thesystem, no mass could be assigned to a more specic proposition. This repres-ents a partial ignorance of the system's state. Total ignorance is represented bythe following masses set: mΩ (Ω) = 1 and mΩ (A) = 0, ∀A 6= Ω. The massmΩ (øΩ) is the mass of conict (the BBA is labelled as normal if mΩ (øΩ) = 0),and propositions A that have a non-null mass (mΩ (A) > 0) are called focal

elements. When focal elements are composed only of singleton hypotheses, themasses are linkable to probabilities, creating a set of Bayesian masses. If the

4

extended open world Θ is used, then, mΘ (øΘ) +mΘ (∗) = mΩ (øΩ). See section4 for the details on how masses are assigned within our approach.

After the sets of mass assignments have been obtained, the problem becomeshow to combine two independent sets of mass assignments, in other words, howto combine evidence from dierence sources (such as dierent sensors)? Thereare a number of dierent rules to do so, the principal one being the Dempster-Shafer (DSR) rule. Let us consider S information sources ∀A ⊂ Ω, whichresulting BBA is mΩ

1,...,S , called the joint mass. Their respective focal elementsare B1, . . . , BS . The DSR is computed so that the nal mass of conict is null(mΩ

1,...,S (øΩ) = 0). The joint mass is given by:

mΩ1,...,S (A) =

1

K

∑B1∩...∩BS=A

mΩ1 (B1) . . .mΩ

S (BS) (9)

where K is a normalisation constant measuring the amount of conict betweenthe sets, and given by:

K = 1−∑

B1∩...∩BS=øΩ

mΩ1 (B1) . . .mΩ

S (BS) (10)

The normalisation is necessary as it distributes the mass of conict on all theother masses, to maintain the sum at its expected value of 1. If the informationsources are in agreement, K tends toward 1; on the other hand, if they are intotal conict, K tends toward 0, making coecient 1

K very large. Use of thatrule has come under serious criticism when signicant conict in the informationis encountered [14, 15], for the normalisation process destroy any informationthat we had on conicts. In fact, conict is a kind of information in itself, andthe origin of this conict becomes an issue, especially when the DSR makesa conscious assumption to ignore conict. A solution to this problem will beoutlined in section 5.

3. Association problem formulation

Let us have objects Oi,k which have their (position, speed) state vector at timek as follows:

Oi,k =

xi,kyi,kVx,i,kVy,i,k

We consider a Markovian Gaussian states model similar to the one used inthe Kalman Filter equations: at each iteration of k, sensors provide raw data,

5

Figure 1: Targets and tracks: the association problem

which are then processed to obtain a set of detected object Xi,k referred to asobservations or targets. The behaviour of the known objects is predicted to thesame time through a Kalman Filter, giving a set of known predicted objectsYj,k called the tracks. The number of tracks nk and targets mk are variableover time; objects can be obstructed from view, appear and disappear veryquickly between two measurements. There is also the possibility of false alarms,and association ambiguities when targets are located in close proximity to eachother. The fact that objects can appear and disappear justies the use of anelaborated multi-target tracking approach such as MHT. As shown in Fig. 1,our problem is now to associate targets with tracks.

For the remainder of this paper, we will use the following notations: from thepoint of view of target Xi, Hj is the hypothesis that Xi is associated withYj (in other words, that track Yj,k and target Xi,k describe the same object);the mass for Hj is noted mi,j (Hj). If one has two hypotheses Hj that Xi isassociated with Yj and Hk that Xi is not associated with Yk, then we canwrite the joint massmi,jk (Hj); this allows tracking the information sources thathave been used to build a particular joint mass. The joint mass for N sourcesof information (sensors) is noted mi,. (Hj), with the dot meaning all.

4. Generation of BBA

Given the targets X1, . . . , Xm and the tracks Y1, . . . , Yn, we need to com-pute the belief mass sets for all hypotheses. Our fundamental postulate is thatan information source cannot assign mass to both Hj and Hj simultaneously(it cannot say that Xi is associated with Yj and Xi is not associated withYj at the same time). In other words, there cannot be any intrinsic conict insources, limiting the total amount of conict when multiple sources are present.

6

The belief mass sets are computed as follows, for a hypothesis Hj :

mi,j (Hj) =

0 if Ic,i,j ∈ [0, τ2]

Φ1 (Ic,i,j) if Ic,i,j ∈ [τ2, 1](11)

mi,j

(Hj

)=

Φ2 (Ic,i,j) if Ic,i,j ∈ [0, τ1]

0 if Ic,i,j ∈ [τ1, 1](12)

mi,j (Θ) =

1− Φ2 (Ic,i,j) if Ic,i,j ∈ [0, τ1]

1 if Ic,i,j ∈ [τ1, τ2]

1− Φ1 (Ic,i,j) if Ic,i,j ∈ [τ2, 1]

(13)

In these equations, the Ic,i,j index provides the similarity value between targetXi and track Yj , which is based on a weighting of the Mahalanobis distancebetween the target and tracks by their respective condence. The only con-straint to use this similarity indicator is the ability to handle the uncertaintyon the data, so dierent similarity functions could be used like the Gruyer'sdistance, or the Bhattacharyya distance. Each function is generated in two orthree intervals where the thresholds are τ1 and/or τ2. τ1 and τ2 are used to alterthe impartiality of the association. For example, for a given value of mi,j (Hj),the association is optimistic if a low value of Ic,i,j is necessary, and pessimisticif a high value of Ic,i,j is required. A simplied situation exists when τ1 = τ2,in which case if τ1 = τ2 = 0.5 the association process is impartial, optimisticfor τ1 = τ2 < 0.5 and pessimistic for τ1 = τ2 > 0.5. The higher the τ value, thelarger the initial mass assigned to the assumption Hj . This conguration can beapplied for an reliable and accurate sensor or data source. In the other case (alow value of τ), the generation of the masses will be more careful. The functionsΦ1 and Φ2 are dened according to Eq. (14) and (15), where τ = τ1 = τ2 andα is a reliability index expressing the reliability of the information source usedto assess Hj . These equations take into account at same time the similarityvalue, the reliability level of the sources, and the threshold value(s), allowing togenerate the three curves shown in Fig. 2. In this gure, the α value is 0.8 soit translates a strong reliability of that data source. Moreover, τ1 = τ2 and isxed to 0.3 for optimistic behaviour of the basic masses generator. Note thatthe gure is generated with a "dissimilarity" index, simply equal to 1− Ic,i,j

Φ1 (Ic,i,j) =α0

2

[1− cos

(πIc,i,j − τ

1− τ

)](14)

Φ2 (Ic,i,j) =α0

2

[1 + cos

(πIc,i,jτ

)](15)

7

Figure 2: Example of functions for the generation of Basic Belief Assigments (τ is 0.3 and αis 0.8)

5. BBA combination with conict management

As we noted in section 2, using the DSR when there is a signicant amount ofconict between the various information sources leads to the loss of any inform-ation regarding this conict. This is due to the normalisation process requiredto adhere to the condition that the mass of conict is null, mΩ

1,...,S (øΩ) = 0. Asolution to this issue, using the extended open world Θ was proposed in [3] andlater extended in [16]. They formulated a way to compute the nal joint massesdirectly from the initial mass distributions, for N sources:

mi,. (Hj) = mi,j (Hj)

N∏a = 1a 6= j

(1−mi,a (Ha)) (16)

mi,. (Hj ∪ ∗) = mi,j (Θ)

N∏a = 1a 6= j

mi,a

(Ha

)(17)

...

8

mi,. (Hj ∪ . . . ∪Hm ∪ ∗) = mi,j (Θ) . . .mi,m (Θ) (18)

×N∏

a = 1a 6= j...

a 6= m

mi,a

(Ha

)

mi,.

(Hj

)= mi,j

(Hj

) N∏a = 1a 6= j

mi,a (Θ) (19)

mi,. (Θ) =

N∏a = 1

mi,a (Θ) (20)

mi,. (∗) =

N∏a = 1

mi,a

(Ha

)(21)

mi,. (ø) = 1−

N∏a = 1

(1−mi,a (Ha)) (22)

+

N∑a=1

mi,a (Ha)

N∏b = 1b 6= a

(1−mi,b (Hb))

Then, we perform a Pignistic step to share the masses of disjunction hy-potheses (any of Hj ∪ . . . ∪ Hm ∪ ∗) to the singleton hypotheses [16], as thedisjunctions need to be removed for the tracking process which is based, so far,on the assumption that a target can be associated with only one single track.Pignistic probabilities have been coined by Smets [11] to describe the assigna-tion of regular probabilities to the possible options that can be taken for a givendecision, such as placing a bet.

At rst, let us consider two observers looking at the same scene. The rst ob-server looks at the potential associations between observed objects and known

9

objects (Xi is associated with Yi) and uses the extended universal set Θx =X1, . . . , Xm, ∗; the second observer looks at the potential associations betweenthe known objects and the observed objects (Yi is associated with Xi) and usesthe extended universal set Θy = Y1, . . . , Yn, ∗. Note that the number of ob-servers is not related to the number of sensors or sources of information. Indeed,if we have only a single sensor, we will still consider two observers that are go-ing to look for associations between objects as explained before. From thesetwo observers, we can create two belief matrices with the Pignistic probabilit-ies of association between tracks and targets. The Pignistic probabilities areconstructed as such:

P : 2Θ → [0, 1] (23)

P (A) =∑

B ∈ 2Θ

A ⊂ B

|A ∩B||B|

×m (B) (24)

P (ø) = m (ø) (25)

where |S| is the number of singletons elements in a set S, here A and B. Thebelief matrices are:

Mpgi,. X1 · · · Xm

Y1 P1,. (Y1) · · · Pm,. (Y1)...

......

Yn P1,. (Yn) · · · Pm,. (Yn)∗ P1,. (∗) · · · Pm,. (∗)ø P1,. (ø) · · · Pm,. (ø)

Mpg.,j Y1 · · · YnX1 P.,1 (X1) · · · P.,n (X1)...

......

Xm P.,1 (Xm) · · · P.,n (Xm)∗ P.,1 (∗) · · · P.,n (∗)ø P.,1 (ø) · · · P.,n (ø)

Then, we need to solve the possible contradictions or ambiguities that can befound between these two matrices. If there is no contradictions, we have Pi,. (Yj)and P.,j (Xi) with the largest values in their respective columns, and so on.Contradictions arise when the two matrices do not associate the same objectstogether, such as when in Mpg

i,. object X1 is associated with Y1 (the probabilityP1,. (Y1) has the highest value) whereas in Mpg

.,j , object Y1 is associated with

10

Xm (the probability P.,1 (Xm) has the highest value). Without ambiguities, allthe probabilities in each column are unique (in a single matrix). If ambiguitiesarise, there are two equal probabilities in a same column; for example, in Mpg

i,. ,if P1,. (Y1) = P1,. (Yn).

To solve any of these two problems, in previous research [9] we used a globalPignistic matrix and the Hungarian Algorithm [17, 18] to nd the best optimalglobal association, according to the maximum global belief criteria. Note thatthe global Pignistic matrix is only computed if there are some ambiguities orcontradictions in the two belief matrices. Otherwise, the associations can beextracted from them directly (a decision is taken). If it is required, the globalmatrix which is constructed from the two belief matrices is:

Mpgi,j Y1 · · · Yn ∗1 · · · ∗mX1 M1,1 · · · M1,n P1,. (∗) · · · 0...

......

...Xm Mm,1 · · · Mm,n 0 · · · Pm,. (∗)∗1 P.,1 (∗) · · · 0 0 · · · 0...

......

...∗n 0 · · · P.,n (∗) 0 · · · 0

with:

Mi,j =Pi,. (Yj) + P.,j (Xi)

2(26)

If we have three targets X1, X2, X3 and four tracks Y1, Y2, Y3, Y4, the naldecision process can be visualised as such:

X1 X2 X3 ∗ ∗↓ ↓ ↓ ↓ ↓Y1 ∗ Y2 Y3 Y4

In that case,X1 is associated to Y1 andX3 is associated with Y2. X2 is associatedwith the ∗ hypothesis, which mean it is a new object; a new track Y5 will haveto be created in consequence (added to the universal set Θ); as we will showwith the example from section 6.3, Y5 is likely to be a false positive that couldunecessarily clutter the tracking process. The ∗ hypothesis is associated withY3 and Y4, which means that the two tracks have disappeared. They will notbe deleted from the system's map immediately, indeed targets that correspondto them might re-appear later because the objects are only temporarily out ofview. The system will keep them (the action is referred to as propagation) andcontinue to process them for a certain time, by using a condence value. A newtrack has its condence set to a value depending on the incertitude on its initial

11

Figure 3: Tracks' condence update process

detection. The condence belongs to [0, 1], it increases when there is a successfulassociation, and decreases when no targets can be associated with a track; whenit reaches zero, the track is taken out of the system. The lifespan of a trackcan be modulated depending on the function that is used to increase/decreasecondence. The simplest approach is to use a linear function (see Fig. 3). Wealso proposed to use Bezier curves to modulate the pessimism or optimism ofthe tracking strategies [3], allowing it to maintain longer or, on the other hand,quickly remove a track that is not associated with the targets anymore.

6. Multi-Hypotheses Tracking approach

Multi-Hypotheses Tracking (MHT) removes the previous assumption that atarget can be associated with only one track and vice versa. Indeed, this as-sumption can force a decision that is incorrect. Abandoning the assumptionof uniqueness has several benets. Now, the system will be able to includemultiple hypotheses in its decision (the equivalent of doubt), or, on the con-trary, reject any decision because the available data is of poor quality. This willprove useful in a highly volatile environment such as on the road. MHT canbe achieved using two distinct methods: the Cascade and Threshold methods,each with their specic advantages and disadvantages.

6.1. Cascade method

Let us call the k-level hypotheses all the complex hypotheses (disjunctions)composed of k singletons. We build belief matrices but using the normalisationshown in Eq. (27) and (28); in that case the information on conict is lost.

12

P : 2Θ → [0, 1] (27)

P (A) =∑

B ∈ 2Θ

A ⊂ B

|A ∩B||B|

× m (B)

1−m (ø)(28)

The Cascade method operates a recursive Pignistic repartition of masses fromthe k-level hypotheses to the (k − 1)-level hypotheses. It looks for the rsthypothesis (complex or singleton) that has a mass greater than 0.5 (if the beliefmass on a hypothesis is greater than 0.5, it will represent the belief maximum,since the sum of all masses is always equal to 1). Starting at the highest k-level(i.e. Θ), the Cascade method spreads the masses from a k-level to the underlyinglevel, as long as it cannot nd a mass greater than 0.5. If a hypothesis has amass greater than 0.5, it is propagated further down the processing chain. TheCascade method must run for all targets currently available to the associationprocess.

However, if the propagated hypothesis is complex, the Cascade method contin-ues its spreading of masses to obtain the weights that will be aected on thatpropagation. Indeed, if the method returns the hypothesis H1 ∪ ∗, we need toknow in which proportions the system believes H1 or ∗ are true.

If no mass greater than 0.5 can be found at any level, the Cascade methodcannot take a decision. However, it has not been designed to reject a decision.If there is no suitable mass obtained from the Pignistic repartition, a reversedcascade is attempted, starting from the smallest k-level and spreading mass tothe higher (k + 1)-level.

6.2. Threshold method

The Cascade method has two issues: (1) it is computationally complex, and(2) it does not manage conict because of the masses normalisation from Eq.(27) and (28). To compensate for these issues, a second less complex method isproposed, dubbed the Threshold method, which is arguably superior (see Table2 for their respective advantages and disadvantages).

This time, the Mpgi,. and Mpg

.,j Pignistic matrices are used. For each line, aweighted threshold t is computed according to Eq. (29), where kt ∈ [0, 1] is theweight.

ti = kt (1− Pi (ø)) (29)

Any associations from the Pignistic matrices where probability is lower than tis ltered out. The value of weight kt can be used to control the number of

13

Table 2: Comparison of the Cascade and Threshold MHT methods

Methods Cascade Threshold

Complexity High LowNumber of generated tracks Accurate InaccurateConict is accounted for? No Yes

Can reject decision? No Yes

hypotheses that pass through the ltering process. The smaller kt, the largerthe number of hypotheses that can pass through the lter; in that case, wehave a cautious lter. This will lead to the creation of virtual tracks, thatwill continue to be processed through future iterations until more informationbecomes available and they are associated more precisely or disappear (theywere false detections).

On the other hand, if kt ≥ 0.5 only a single hypothesis will be accepted at

maximum. This means that with these large values of kt, the ltering processwill not allow any hypotheses through. In other words, it means the associationalgorithm will reject taking any decision on the hypotheses. We recommend thatthis latter case should be used only for computations internal to the associationprocess, without aecting the overall decision taken by the algorithm (uponwhich some critical upper-level systems might depend). The upside of ltering alarge number of hypotheses is that false alarms will be ltered in most situations.

If target Xi is associated to the two tracks Yj and Yk, two virtual tracks Y ∗ijand Y ∗ik are actually created. The system is said to be doubting or hesitating onto which decision to take. These virtual tracks are added to the universal setand will be processed by the system in subsequent iterations, alongside normaltracks, but without being propagated outside of the tracking algorithm, to clientapplications for example. The two tracks are likely to diverge over time, leadingto the association of one with Xi and the other to the ∗ hypothesis. In thelatter case, if the virtual track was representing a false positive, the condenceon the new track will decrease quickly and it will be ltered out. A new trackrepresenting an actual object will have a condence high enough not to be lter.

6.3. Example

As we saw, MHT tolerates a lack of association for some tracks and/or targets,and a level of ambiguity for others. If using the Threshold method and conictis important, it is preferable to reject making a decision rather than makingan erroneous one. Furthermore, the system can associate a target with severaltracks: Xi can be associated with both Yj and Yk, allowing the creation ofvirtual tracks.

Let us now consider an example of the MHT algorithm behaviour. We havethree targets X1, X2, X3 and four tracks Y1, Y2, Y3, Y4 as shown on Fig. 4.

14

Figure 4: Diagram of the scene perceived by the sensor, tracks (blue squares) and targets (redcircles)

At rst, the belief mass sets are computed using the approach described insection 4; the results are shown in the following table:

mi,j (Hj) Y1 Y1 Θ Y2 Y2 Θ

X1 .9 0 .1 0 .6 .4X2 .6 0 .4 .6 0 .4X3 0 .6 .4 .9 0 .1

mi,j (Hj) Y3 Y3 Θ Y4 Y4 Θ

X1 0 .8 .2 0 .9 .1X2 0 .8 .2 0 .9 .1X3 0 .7 .3 0 .9 .1

Then, we compute the two pignistic matrices as described in section 5.

Mpgi,. X1 X2 X3

Y1 .9395 .2894 .012Y2 .0124 .2894 .9382Y3 .0059 .0078 .0088Y4 .0029 .0038 .0028∗ .0395 .0494 .0382ø 0 .36 0

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Mpg.,j Y1 Y2 Y3 Y4

X1 .372 .0040 .0843 .0468X2 .072 .072 .0843 .0468X3 .004 .372 .131 .0468∗ .012 .012 .7003 .8595ø .54 .54 0 0

If one examines Mpgi,. , they can note that there is an ambiguity regarding X2

where P2,. (Y1) = P2,. (Y2). Without MHT, there can only be only one trackassociated with a target. The ambiguity is lifted using the Gobal Pignisticmatrix and the Hungarian Algorithm, yielding the following nal association:

X1 X2 X3 ∗ ∗↓ ↓ ↓ ↓ ↓Y1 ∗ Y2 Y3 Y4

X1 is associated to Y1 and X3 is associated with Y2. X2 cannot be associatedto a known track, so it is a new object associated with the ∗ hypothesis. The ∗hypothesis is associated with Y3 and Y4, which means that the two tracks havedisappeared out of the sensor's eld of view; those tracks will be propagatedaccording to the system's tracking strategy. We now have:

Θ = Y1, Y2, Y3, Y4, Y5, ∗

where Y5 is the new track that has been created because of the association of ∗ toX2. This results maximise the global belief, but it is far from the best solution.Indeed, P2,. (∗) = 0.0494, signicantly less than the mass on conict (0.36) oron the ambiguous associations with Y1 and Y2 (0.2894 each). In that case, thesystem generated a dubious association, creating a track that is not likely torepresent any real independent object from the scene; Y5 is a ghost track, afalse positive. This track will be passed to any higher-level client applicationand might hinder the performance of any such application, particularly if it isperforming some safety tasks (e.g. collision detection).

This inappropriate association can be ltered by using our MHT algorithm; letus demonstrate the Threshold method solution to this problem. We computethe weighted thresholds ti according to Eq. (29) for each column, with kt = 0.5,and obtain the new pignistic matrices:

Mpgi,. X1 X2 X3

Y1 .9395 .2894 .012Y2 .0124 .2894 .9382

Y3 .0059 .0078 .0088Y4 .0029 .0038 .0028∗ .0395 .0494 .0382ø 0 .36 0t .5 .32 .5

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Mpg.,j Y1 Y2 Y3 Y4

X1 .372 .0040 .0843 .0468X2 .072 .072 .0843 .0468X3 .004 .372 .131 .0468∗ .012 .012 .7003 .8595

ø .54 .54 0 0t .23 .23 .5 .5

The bolded values represent masses that are greater than the threshold t. Afterthe nal association process, the main dierence with the previous approach isthat X2 is not associated with any tracks or hypotheses, being considered a falsealarm that time. However, it can be argued that ignoring X2 is letting someof the information gathered by the sensor out. In order to still use X2 we canlower the threshold to, for instance, kt = 0.4 so that the new pignistic matricesbecome:

Mpgi,. X1 X2 X3

Y1 .9395 .2894 .012Y2 .0124 .2894 .9382

Y3 .0059 .0078 .0088Y4 .0029 .0038 .0028∗ .0395 .0494 .0382ø 0 .36 0t .4 .256 .4

Mpg.,j Y1 Y2 Y3 Y4

X1 .372 .0040 .0843 .0468X2 .072 .072 .0843 .0468X3 .004 .372 .131 .0468∗ .012 .012 .7003 .8595

ø .54 .54 0 0t .184 .184 .4 .4

With the new threshold, the association actually accounts for the existing am-biguity regarding X2, which is now associated with both Y1 and Y2, which isallowed within the context of MHT. Two virtual tracks Y ∗21 and Y ∗22 are createdby the system, since it is doubting whether X2 should be associated with Y1

or Y2. These tracks will be propagated and over time the condence on oneparticular association will increase while the other will be dropped or replacedby a new track (association with ∗); the virtual tracks are not passed to anyclient application, they remain contained within the tracking algorithm. If X2

was an actual false detection resulting from a sensor error, the condence intracks associated with it will quickly decrease and the virtual tracks will quicklybe removed from the tracking process. This approach minimises the problemassociated with the apparition of ghost tracks.

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X1 X2 X3 ∗ ∗↓ ↓ ↓ ↓Y1 Y ∗21 Y

∗22 Y2 Y3 Y4

Θ = Y1, Y2, Y∗21, Y

∗22, Y3, Y4, ∗

7. Demonstration with real data

In this section, we will present an implementation of our MHT algorithm appliedto the detection and tracking of targets in ambiguous road situation. The MHTis applied on two dierent types of scenarios and objects tracking. The rst isdedicated to pedestrians tracking, and the second is focused on vehicles tracking.The goal of the rst set of scenarios (with pedestrians) is to show the capabilityof our approach to manage complex and holonomic manoeuvres. For thesereasons, we have chosen the tracking of pedestrians in close area as a relevantexample.

We will consider three scenarios involving four pedestrians. At rst, a gen-eral situation with the pedestrians walking in front of the sensor with com-plex trajectories and regular occultations to show how multi-hypothesis trackingcan perform in a dicult context. Then, secondly, a rather extreme situationwith two pedestrians almost bumping into each other. At last, two pedestrianspassing each other closely as what could happen on a normal street footpath.

In the pedestrians scenarios, the dataset was collected using a real single-layerlaserscanner, with the pedestrians walking in various fashions in front of thesensor. This situation, which could be encountered on the road while driving,is quite adequate to compare the performance of multi- and single-hypothesistracking (i.e. the type described in section 5). Indeed, the laserscanner's mainscanning plane is located relatively low over the ground, so that most of the timethe pedestrians' legs are in view. This means that at times a single individualcan be perceived by the laserscanner as two closely located but independentobjects formed by each leg. Since the pedestrian itself should be tracked as asingle object, data collected from this scene will display many ambiguity andpotential sources of conict. Note that with recent multi-layers laserscanners,the issue of legs detection is attenuated because higher laser beams are able todetect the rest of the pedestrians' bodies.

In the second type of scenario, focused on vehicle tracking in ambiguous overtak-ing situation, the dataset is generated in simulation from the SiVIC platform.We chose a specic scenario where the overtaking manoeuvre is done with ashort lateral distance between the two vehicles. This conguration will providean erroneous obstacle detection. In this situation, only the MHT could solvethis problem and generate a correct tracks assessment.

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Figure 5: Forward facing camera view of the pedestrians

7.1. Scenarios of pedestrians tracking with real laser scanner data

7.1.1. Pedestrians tracking with multiple hypothesis

In this scenario, there are a number of pedestrians walking in front of the laser-scanner sensor (see Fig. 5). The pedestrians walking in dierent fashions, turnaround, and display a dierent morphology as seen from the sensor (facing it,standing looking to the side, bending down, etc.); these movements create oc-cultations. In Fig. 6a we show the belief on tracks for that scenario, over time (atotal duration of 48 seconds). During occultations the belief on those tracks isshown to fall to zero. Without multi-hypothesis tracking, those tracks would bedeleted and once observations are available again, entirely new tracks would becreated. For example, this would be a signicant problem for track n°1, whichis occulted 11 times in 48 seconds. Without MHT, it would be re-created 11times as a dierent track.

On the other hand, if MHT is used, temporal consistency can be maintainedas the system continues to track those disappearing tracks for some time (usingvirtual tracks). In Fig. 6b we show the condence associated with each tracks.If there are no observations to match with the track for some time, condencedecreases as the track is made virtual. However, it would be only removed com-pletely from the system if condence reaches zero (or any other user denedthreshold). In this scenario, short occultations on the rst track are not a prob-lem anymore: the condence dips slightly but then recovers once new targetsare associated with it again, and the track looses its virtual status.

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(a) Evolution of the belief on tracks

(b) Evolution of the condence associated with tracks

Figure 6: Belief and condence on tracks for pedestrians tracking

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(a) Camera view of the two pedestrians

(b) Laserscanner view of the same scene as in 7a, showing laser impacts and the tracks

Figure 7: Extreme case pedestrian scenario, with near-collision

7.1.2. Extreme case: near-collision and direction switch

In this second case, we take a more limited but extreme situation. Two ped-estrians walk toward each other and get very close (a near-collision). At thatpoint, one reverses direction suddenly. Because the pedestrians are walking veryclosely to each other, only a single target is now detected by the laserscanner(this diers from a simple occultation because of their closeness). Previously,the system was tracking two tracks and repeatedly observed two targets. At thispoint, a single target become associated with two tracks, creating an ambiguity.The MHT algorithm creates two virtual tracks to cope with this ambiguity.

The scenario analysis is shown in Fig. 7. Both the camera (7a) and laserscan-ner (7b) views of the two pedestrians are shown. On Fig. 8, 9, and 10, theresults of the pedestrians tracking is displayed for the three algorithms (PDAF,classical Belief, and MHT) over ve seconds. In these three gures, we can seehow the two pedestrians tracks are getting closer until they reach the ambiguousarea circled in red. Crosses are the detected objects, the colorful circles are thetracks. The yellow circles represent the propagation stage of a track when atrack is not associated with a target anymore. A green circle indicates that thetrack is associated, i.e. it is known by the system without ambiguity and hasbeen updated by regular association with observations. On the other hand, apurple circle indicates that the track is now a virtual track. The 3D temporalvisualisations are useful to understand the timing of movements. If we analysethe behaviour of the three algorithms, we can observe that in the case of PDAF,some disturbances and false assessments of the tracks are visible. This is mainly

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(a) PDAF lter: 2D Temporal view of the scenario, with the evolution of the tracks

(b) PDAF lter: 3D Temporal view of the scenario, with the evolution of the tracks

Figure 8: Result of PDAF in extreme case like near-collision and direction switch

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(a) Classical Belief Filter: 2D Temporal view of the scenario, with the evolution of the tracks

(b) Classical Belief Filter: 3D Temporal view of the scenario, with the evolution of the tracks

Figure 9: Result of classical Belief Filter in extreme case like near-collision and directionswitch

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(a) MHT Filter: 2D Temporal view of the scenario, with the evolution of the tracks

(b) MHT Filter: 3D Temporal view of the scenario, with the evolution of the tracks

Figure 10: Result of MHT Filter in extreme case like near-collision and direction switch

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due to the behaviour of the PDAF which always tries to associate a track to thenearest target. With the second lter (classical belief), we see that the trackingis better. Nevertheless, in the ambiguous area, tracks suer from positioningdrift, and the second track is propagated (because it is not associated). Withthe MHT, we see very clearly the usefulness of the virtual tracks generation incase of ambiguities. In that case, the system does not have enough condencein its association because of the ambiguity due to the near-collision (as seen onthe graph, they come within about 70 cm) and sudden turn of the pedestrian, soit continues to track those virtual tracks internally. Their condence decreasesfor some time as the system is unable to resolve the ambiguity via new observa-tions. However, once the pedestrians are distant enough, the system correctlydistinguishes them again. Note that the outer dots still have the same colours(yellow and red) corresponding to tracks' identier; this conrms the fact thatthe system has correctly maintained temporal consistency. Instead of creatingnew tracks it recognised them again once the ambiguity had disappeared, whichis facilitated by using virtual tracks. Virtual tracks are not outputed to anyhigher-level system.

7.1.3. Normal case: pedestrians overtaking

In this second case with pedestrian, we have a more common situation of twopedestrians passing each other in opposite directions. Their relative distance isnever less than a meter and they do not make any sudden trajectory changes.This situation is much more similar to what a vehicle could be encountering onthe street, for example people walking by a crossing. Note that in Fig. 11a and11b there is a third pedestrian initially leaving the view toward the left, anda fourth one standing motionless in the background; those two pedestrians arenot considered for our analysis.

The ambiguity zone occurs when the two pedestrians are the closest to eachother, and one partially occults the other. In Fig. 12 the black crosses arethe targets (laserscanner observations). We can clearly see that when the twopedestrians are close to each other, the laser impacts are spread on a larger areaand there is a large uncertainity as to which track targets belong to. From theresult presented in Fig. 12, we can observe a real unsolved ambiguity betweenthe two tracks. Moreover a signicative part of the tracks are not taken intoaccount. In Fig. 13, the same positioning devergence behaviour is observed. Tocope with the situation, the MHT system creates virtual tracks (purple circles)as in the previous example (see Fig. 14), and continues to track them correctly.In this case, because the pedestrians' movements are relatively linear, the virtualtracks do not diverge as much as in the previous example, and their condencedoes not decrease as fast. Once again, the virtual tracks are not propagatedto other application layers, so the ambiguity is not experienced by any othersystem; they always have clearly identied non-ambiguous tracks available.

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(a) Camera view of the two pedestrians

(b) Laserscanner view of the same scene as in 11a, showing laser impacts and the tracks

Figure 11: Street-like scenario

7.2. Normal case: vehicle overtaking

As we have explained before, and as we have shown with extreme pedestriansconguration (close manoeuvres with occultations), the MHT aims at limit-ing association mistakes that could lead to a wrong assessment of the object'stracking. The third example presented in this paper takes a conventional roadsituation involving 3 vehicles: the ego-vehicle with an embedded laserscanner,a vehicle being overtaken just ahead of the ego-vehicle, and the vehicle thatundertakes the overtaking maneuvre.

This type of road driving conguration is also very interesting to highlight therelevance of the MHT algorithm. Indeed while overtaking, the two vehiclesin front of the ego-vehicle are so close one to the other that the laserscannerprocessing can only detect one target instead of the two vehicles: the detectionalgorithm does not dierentiate between the vehicles any more. The false targetis generated by the laserscanner processing for 2 seconds. After the overtakingmanoeuvre is completed, the lateral distance between the two vehicles startsagain to increase, thus, in this situation the laserscanner's processing is againable to cluster correctly the two vehicles into two targets. Unfortunately, for ashort time the fact that only one target is detected put the tracking algorithmin a tricky situation as it does not know how to correctly update tracks withone target instead of two.

This third scenario is generated from the SiVIC simulation platform [19, 20, 21,22]. This platform is dedicated to ADAS and autonomous driving prototyping,

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(a) PDAF lter: 2D Temporal view of the scenario, with the evolution of the tracks

(b) PDAF lter: 3D Temporal view of the scenario, with the evolution of the tracks

Figure 12: Result of PDAF in the street-like scenario

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(a) Classical Belief Filter: 2D Temporal view of the scenario, with the evolution of the tracks

(b) Classical Belief Filter: 3D Temporal view of the scenario, with the evolution of the tracks

Figure 13: Result of classical Belief Filter in the street-like scenario

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(a) MHT Filter: 2D Temporal view of the scenario, with the evolution of the tracks

(b) MHT Filter: 3D Temporal view of the scenario, with the evolution of the tracks

Figure 14: Result of MHT Filter in the street-like scenario

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testing, and evaluation. In the SiVIC plateform, all the main embedded sensorsare modeled: cameras, laserscanner, RADAR, GPS, INS, Odometer, etc. Fig.15 shows a bird's-eye view of the scenario and a screenshot of the simulation(from the ego-vehicle point of view) taken during the overtaking manoeuvre. Inthis screenshot, the red square represents the detected target which encompassesthe two vehicles at the same time.

7.2.1. Results without MHT

In the rst trial, the scenario is played with the standard tracking algorithmbased on belief theory. Fig. 16 shows the impact of the detection's mistake.In this case, only one track can be associated with the target. This mistakeyields a strong modication of the track positioning. The right sub-gure onFig. 16 presents the dierent positions of the two tracks. It shows the impactof bad association on both the positioning and the speed vector. As one cansee, track 1 is attracted by the target instead of staying at the same position.This means that this track represents the vehicle being overtaken which hasa null relative velocity compared to the ego-vehicle (it is the reason why thelongitudinal movement of track 1 is small). Track 2 which stands for the othervehicle (which is doing the overtaking) has a greater relative speed.

7.2.2. Results with MHT

The same manoeuvre is replayed, but now with the MHT. In the second case, themultiple hypothesis tracking generates a virtual track in order to assess the speedvector of the new target. While the condence on the virtual track is increasingthe two tracks are propagated. If the speed vector looks similar, the trackingalgorithm will validate the association, else the track will be propagated. On Fig.17 (left sub-gure), the small white square near the centre of the gure representsthe virtual track. The trajectories graph displayed on the same gure (rightpart) clearly shows that no trajectory has been modied due to the appearanceof the new target. The white markers conrm that a virtual track has beencreated in order to retrieve the speed vector of the target. Depending on theMHT lter's tuning, it is possible with this approach to lter false detectionsdue to, for instance, impacts on the road surface.

8. Conclusion

In this paper, we have discussed algorithms to track objects on the road overtime, and in particular solutions to the problem of associating tracks, objectsalready known by the system, to targets, new objects built from the most recentsensor information. We introduced the general concepts of the Dempster-ShaferTheory (also called Belief Theory) before detailing a previous association and

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Figure 15: Scenario with vehicle overtaking

Figure 16: Scenario with vehicle overtaking and with classical belief tracking

Figure 17: Scenario with vehicle overtaking and with MHT lter

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tracking approach that used it. Using a framework based on the Dempster-Shafer Theory has several advantages, such as being able to manage the ap-parition of new objects, conicts, and a better handling of imprecision andincertitude. In this framework, decision on to whether associate a pair of trackand target is taken using a global belief criterion, this criterion is good whenconict is not too important within the data or association process.

However, if conict is important, the presented method allows for the formationof ghost tracks that do not represent any real object and can clutter the trackingprocess and negatively impact higher-level functions. This stems from a previousassumption that associations must respect a uniqueness condition; removing thiscondition allows considering Multi-Hypotheses Tracking.

Multi-Hypotheses Tracking allows the tracking system to account for ambiguityand even to reject making a decision if it has too much doubt, in other wordsif the quality of sensor data is poor. The system generates virtual tracks thatremain entirely internal to its process, which allows limiting the informationthat is passed to higher-level client applications to reliable information only; inthe mean time, the system keeps suspicious targets/tracks and analyses theminternally to try to lter out false positive from actual useful information.

We proposed two methods, the Cascade and Threshold methods, to modify thenormally computed belief masses in an attempt to reduce the impact of conictor incertitude; the Threshold method is then selected over the Cascade methodas it can manage all conictual situations and is less complex.

We demonstrated the improvement of using MHT over a single-hypothesis track-ing using real laserscanner data, as well as a simulated scenario. The real situ-ation considered for this demonstration is the recording of several pedestrianswalking within the laser's eld of view, which provide many occasions for am-biguity and conict: indeed, each leg can be detected individually or occultedand the system must keep track of them as a single pedestrian object. Threecases are presented, with dierent ambiguous situations. MHT performance iscompared with PDAF and classical Belief tracking for two of those cases. Thesimulated scenario represent an overtaking manoeuvre where the laserscannerdetects one target. We have shown how MHT also improves from classical Beliefin this case.

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