Multi-Object Association Decision Algorithms with Belief Functions

Transcription

1 ulti-object Association Decision Algorithms with Belief Functions Jérémie Daniel and Jean-Philippe Lauffenburger Université de Haute-Alsace UHA) odélisation Intelligence Processus Systèmes IPS) laboratory - EA rue des frères Lumière, ulhouse Cedex, France namesurname@uhafr Abstract ulti-object Association OA) consists in determining, at each processing cycle, the best associations set lining the detected objects to the already nown ones The aim is then to determine if the objects are propagated, appearing or disappearing This association is relevant when the data imperfections imprecision, inaccuracy, etc) are considered As the Transferable Belief odel T B) helps to consider these imperfections, it represents an interesting framewor for OA The focus is here placed on T B-based OA decision-maing, ie the selection of the most relevant associations among the possible ones In this context, the comparison of existing decision algorithms is provided Based on the analysis of their performance, two decision approaches are proposed Simulations, performed considering a literature example, show the differences between the algorithms and the interests of the proposed solutions I INTRODUCTION One of the main challenges of mechatronic systems is to perform their safe operations in real-time considering their own limits as well as those involved by their environment For example, an autonomous vehicle has to trac static and dynamic objects which may correspond to pedestrians, vehicles, cyclists, etc The detection of these objects is usually performed by sensors which help to give the objects location, but which are not directly tracing them To cope with this limitation, a ulti-object Association OA) algorithm is often used OA originates from the probabilistic domain related to ulti-target Tracing T T ) with the noticeable wor of Reid [], Blacman [2] and Bar-Shalom [3] mostly dedicated to military applications Usually, T T s trac objects regarding inaccurate, incomplete and conflicting information, ie to contexts in which the limits of the probability theory, used in the aforementioned references, are reached A solution to handle the information imperfections can be the Transferable Belief odel T B) proposed by Smets [4] This framewor, close to the Evidence Theory introduced by Dempster [5] and formalized by Shafer [6], is well suited for the data imperfections and conflict modeling The goal of OA is to lin detected objects by sensors with already nown objects By performing this process over time, objects tracing is possible The question raised is then: how to perform the association between the detected objects and the nown ones? A straightforward solution, used in [], is to perform a single direction association from the detected to the nown objects However, often, a two-direction solution detected nown and nown detected), is preferred and particularly for T B-based OA) The association process is depicted in Fig with X the set of perceived objects and Y the set of nown ones A two-direction association is particularly useful for the determination of appearing and disappearing objects and the detection of sensor incoherences and/or conflict The consideration of this information leads to a safer and coherent association decision Several studies dedicated to OA in the T B framewor, essentially centered on decision-maing algorithms, have been proposed The first, initiated by Rombaut [7] is focused on the combination of masses expressing the association confidence between nown and newly detected objects It has been exted by Gruyer et al [8] through a basic belief assignment bba) based on similarity measures In both papers, the combination allocates mass only on singletons, on the ignorance and the conflict In [8], the decision is performed considering the Hungarian coupling algorithm which gives the best association according to the maximal global belief criterion However, this algorithm leads to counter-intuitive and hazardous local associations as mentioned in [9] That is why, in this latter wor, the authors proposed a time-propagation OA algorithm considering two decision methods: Cascade and Threshold The solution of the Cascade method is defined based on a sequential pignistic redistribution, while the second, less complex, identifies the associations providing pignistic probabilities higher than a predefined conflict-depent threshold A further study done by ercier et al presents two decisionmaing algorithms looing for the globally satisfying solution, similarly to [8] but in a simplified way, by maximizing the Joint Pignistic Probability [0] computed over all potential associations In these aforementioned wors, the information sources/experts are assumed to express themselves over a particular objects association, ie two sources cannot express themselves over the same association This assumption is raised in [] and [2] but clustering is then necessary before the fusion to group the bbas focused on the same association This approach avoids the generation of inappropriate conflict which may be generated during the combination of sources dedicated to different objects association The current paper is centered on the decision-maing algorithms selecting, among the possible solutions, the objects 669

2 X4 X5 Detected objects X X3 X6 X2 Object appearance X* Association X Y Y Y3 Known objects Y4 Y2 m : 2 Θ [0,], A 2 Θ,mA) 0 ma) = A 2 Θ A is a focal element Once the different sources bbas are generated, the combination can be processed The latter gathers the information obtained from the sources S using a combination operator Considering the sources as reliable and distinct, the conjunctive operator is defined as [4]: 2) * Y Fig ulti-object Association Object disappearance m H) = m S H) m S2 H) m S H) = H H =H i= m Si H i ) 3) associations, appearances or disappearances After a brief recall of the T B in Section II and its application to OA in Section III, the decision algorithms proposed in [9] and in [0] are described in Section IV They constitute the latest solutions proposed for T B-based OAs Previous wors such as [3] or [4] and which have already been updated or replaced, are not treated within the context of this paper A literature example shows the limits of these solutions providing hazardous local associations when evidence conflicts To cope with these limits, two algorithms are proposed in SectionV and evaluated on the basis of the same example Finally, Section VI concludes this paper II THE TRANSFERABLE BELIEF ODEL The T B is a model of uncertain reasoning and decisionmaing based on two levels: the credal level in which belief masses are used to represent the pieces of information, and the pignistic level or decision level) where the belief masses are transformed into probability measures This section only presents the basic elements of the T B Additional details can be found for example in [4] Let Θ be the discernment frame, ie the finite set of all the problem solutions also called hypotheses H j, j =,2,,, with the number of possible hypotheses In the T B framewor, the propositions of Θ are exclusive but not necessarily exhaustive The referential subset 2 Θ defines all the 2 propositions A which can arise regarding the hypotheses of Θ such that: Θ = {{H },{H 2 },,{H }} 2 Θ = {A/A Θ} = {/0,{H },,{H },{H,H 2 },,Θ} ) /0 represents the impossible proposition which characterizes the conflict between the information and Θ represents the ignorance, ie the union of all the hypotheses H j For clarity reasons the rigorous notation of singletons {H i } and proposition disjunctions { H i,h j } is respectively replaced by Hi and H i H j The veracity of a proposition A of 2 Θ is characterized by its basic belief assignment bba) or belief mass m defined as follows: This operator allows the belief mass on the conflict m/0) to be non-null, illustrating, under the assumption of reliable sources observing the same situation, either a non exhaustiveness of the discernment frame or conflicting bbas This copes with the classical Dempster rule of combination based on the product of each combined mass m A) by the conflictrelated factor κ /0 defined such as: κ /0 = m /0) = H H =/0 i= m Si H i ) with m /0) = 0 after redistribution The selection of the problem solution, ie the decision, aims at eeping the most relevant hypothesis H j regarding the combination results This is a critical step as a selection involves the rejection of the other propositions, ie a loss of information Usually in the Evidence Theory, the decision is taen considering the maximum of credibility or plausibility which are respectively qualified as pessimist and optimist solutions [6] These two techniques are used when the decision is done over either a singleton or a union proposition In the T B framewor, a decision in the pignistic level) can be taen after information combination with the Pignistic or Bet) Probability BetP) [4] The latter is based on the redistribution of the union masses even the ignorance) over the singletons which compose the union cf 5)) Consequently, only singletons, representing specific pieces of information, remain after the pignistic transformation This transformation, built as a probabilistic rule, is an interesting solution for the OA context cf Section III) ma) BetPH i ) = A m/0)) A 2 Θ H i A It is worth noting that the pignistic transformation involves a normalization of the conflict required if a decision must be taen cf [5]) In Section III-C, this point will be further discussed 4) 5) 670

3 III TB ULTI-OBJECT ASSOCIATION A Belief ass odeling Let us consider, as for most of the T B-based OA [7], [9], [0]), specialized sources expressing themselves, for each of them, only over one hypothesis of Θ This corresponds to a bba performed over one hypothesis H i ), its contrary H i ) and the ignorance Θ) For a source j expressing itself over the association of a detected object X i with a nown object Y j, this leads to a 3 masses bba: m Θ X i j Y j ): object X i is associated with object Y j, m Θ X i m Θ X i j j Y j ): object X i is not associated with object Y j, Θ Xi ): ignorance regarding both previous facts Θ Xi and Θ Yj are the discernment frames referring respectively to the detected nown objects and to the nown detected objects association Hence, Θ Xi is composed of the nown objects Y j and the hypothesis representing an object appearance In the same way, Θ Yj is composed of the N detected objects X i and the object disappearance hypothesis cf 6) Θ Xi Θ Yi = {Y,Y 2,,Y, } = {X,X 2,,X N, } 6) m H j ) = m j H j ) m H j H l ) a= α a = m j Θ)m l Θ) γ a a= a l and for union combinations of 2 to - hypotheses: m H j H l ) m ) = m Hj ) m Θ) = m /0) = a= = m j Θ)m l Θ) γ a a= a= a l ) = m j Hj m a Θ) α a + a= a= a= m a Θ) m a H a ) β b b= b a with α a = m a H a )), β b = m b H b )) and γ a = ma Ha )) γ a 7) The present OA loos for the best association regarding the following constraints [0]: a detected object can only be associated with one nown object, a nown object can only be associated with one detected object, appearance and disappearance ) is possible for multiple detected or nown objects In the T B-based OA literature, the bba is usually performed regarding information provided by sensors For example, distance and angle information extracted from image processing is used in [0] while in [6], corresponding data are extracted from a GPS-based Geographical Information System Here, the focus is placed on the decision-maing algorithms evaluation in iso-conditions, ie using a literature case study cf IV-D) Therefore, the bba is not treated B Combining Belief asses The conjunctive operator introduced in Section II, is used It is associative and commutative, so does not force the definition of a fusion order over the sources As specialized sources are considered, 3) can be formalized for each proposition regarding the initial mass distribution as [7]: C Pignistic Transformation and Association Tables In a two-direction OA, a situation is observed with two different points of view: one is dedicated to the association of the detected objects X i to the nown objects Y j while the second is focused on the association of the different Y j to the X i After the combination thans to the generalized rules depicted above, masses are placed either on singletons or on unions of hypotheses In order to face with the one-to-one association constraint, a pignistic transformation is operated, transferring the masses of disjunctions on simple hypotheses This leads to two pignistic probabilities matrices shown in TableI and TableII Each line defines the probabilities of the associations of X i with Y,,Y and vice versa For the sae of clarity, the complete notation BetP Θ X i Y j) or BetP Θ Y j X i)) has been simplified as BetP Xi Y j ) or BetP Yj X i )) It can be noticed that these matrices are obtained without conflict normalization contrary to 5)) Indeed, this information can be useful for the final decision and is therefore conserved Consequently, the pignistic probabilities are given by: ma) BetPH i ) = A A 2 Θ H i A BetP/0) = m/0) For each object X i, the pignistic probability from 8) is given 8) 67

4 by: BetP Xi Y j ) = m Θ X i j Y j ) A Θ Xi Y j A A > A a= b= Y b A m Θ ) X i a Y a ) + m Θ X i b Θ Xi ) ) b= Y b / A m Θ X i ) ) 9) b Yb with the pignistic probability of the * hypothesis being BetP Xi ) = a= A Θ Xi A A > m Θ X i ) ) a Ya + A m Θ ) X i b Θ Xi ) m Θ X i ) ) b Yb 0) b= b= Y b A Y b / A and the conflicting pignistic probability defined by BetP Xi /0) = a= ξ a + a= m a Θ Xi Y a ) η b b= b a ) with ξ a = m Θ ) X i a Y a ) and η b = m Θ ) X i b Y b ) The pignistic probability BetP Yj X i ) for the association of an object Y j with X i is defined similarly The matrices help to detect contradictions and/or ambiguities in the associations cf [9]) An ambiguity occurs when, on a given line of a matrix, several probabilities are equal, ie several associations are equally possible On the other hand, a contradiction occurs when the matrices conclude on different associations for the same object Consequently, ambiguity highlights an intra-matricial association problem whereas contradiction highlights an inter-matricial one TABLE I DETECTED TO KNOWN OBJECTS PIGNISTIC PROBABILITIES BetP Xi ) Y Y /0 X BetP X Y ) BetP X Y ) BetP X ) BetP X /0) X 2 BetP X2 Y ) BetP X2 Y ) BetP X2 ) BetP X2 /0) X N BetP XN Y ) BetP XN Y ) BetP XN ) BetP XN /0) IV OA DECISION-AKING In this section, 3 algorithms from the literature are compared on behalf of the same case study originally proposed by Gruyer and reused by ercier et al in [0] cf Table III) This study will highlight that, even if the algorithms have comparable performance, they can lead, when contradictions/ambiguities occur, to hazardous local associations, by looing for the best global association This conducted the authors to propose two algorithms described in Section V and evaluated using the same example A Joint Pignistic Probability The Joint Pignistic Probability JPP) has been introduced by ercier et al [0], exting previous wors from Lemeret et al [3] Among the + ) N possible associations for the X N objects, this algorithm rejects those non satisfying the predefined constraints cf Algorithm ) For the retained set, the products of the pignistic probabilities BetP Xi ) are computed cf 2)) and the association maximizing the product is finally selected cf 3)) Based on BetP Yj ), the same approach is followed to define the joint law BetP Y Y and find the suitable Y j X i association BetP X X N = BetP X Y a ) BetP X2 Y b ) BetP XN Y l ) 2) with a,b,,l and a b l Solution = argmax{betp X X N } 3) Algorithm : JPP Algorithm for detected objects X i Data: BetP Xi Y j ), i I = {,,N}, pignistic probabilities of each detected object X i Result: the decision maximizing the joint pignistic probability BetP X X N and satisfying the association constraints begin Remove the associations which do not verify the association constraints Compute the products of the pignistic probabilities over the remaining associations Select the association maximizing the products TABLE II KNOWN TO DETECTED OBJECTS PIGNISTIC PROBABILITIES BetP Yj ) X X N /0 Y BetP Y X ) BetP Y X N ) BetP Y ) BetP Y /0) Y 2 BetP Y2 X ) BetP Y2 X N ) BetP Y2 ) BetP Y2 /0) Y BetP Y X ) BetP Y X N ) BetP Y ) BetP Y /0) The pignistic matrices constitute the common basis for the decision done using one of the algorithms presented in the next section In this case, the best solution is the one giving a global satisfaction, indepently of the local pignistic probabilities For example, a solution based on average probabilities will be preferred to a solution composed of several strong pignistic probabilities indicating a strong belief) weighted by low probabilities B Joint Classified Pignistic Probability The maximum of Joint Classified Pignistic Probability JCPP), also introduced by ercier et al [0], behaves similarly to the JPP The difference lies in a preliminary raning of 672

5 the pignistic matrices regarding the decreasing number of zero in each object association possibility, ie in each matrix line This leads to the same results than the JPP saving calculation time The JCPP behavior is recalled in Algorithm 2 Algorithm 2: JCPP Algorithm for detected objects X i Data: BetP Xi Y j ), i I = {,,N}, pignistic probabilities of each detected object X i Result: the decision maximizing the joint pignistic probability BetP X X N and satisfying the association constraints begin Classify BetP Xi Y j ) according to the decreasing number of null probabilities Remove the associations which do not verify the association constraints Compute the products of the pignistic probabilities over the remaining associations Select the association maximizing the products C Threshold ourllion et al proposed other approaches trying to avoid suspicious local associations [9] due to the use of a global association algorithm lie JPP or JCPP These algorithms, named Cascade and Threshold, are characterized by the relaxation of the one-to-one association constraint This can lead either to the non-association of objects or multiple associations In both cases, the system can stay undecided Since in the Cascade method, the masses are normalized, the conflict information is lost oreover, this algorithm is not founded on the use of the pignistic matrices for the associations determination For these reasons, in this paper, only the Threshold algorithm will be studied The latter is based on the elimination of the pignistic probabilities lower than a conflict-related threshold obtained considering a weighting factor s such as: T hreshold = s BetP/0)) 4) The determination of s is crucial as it modulates the threshold level, thus helps to reject or maintain) more or less elements in the association matrices The lower the threshold, the higher the number of retained association hypotheses transferred to the next association cycle For algorithms comparison purposes, the one to one association constraint has been introduced in the Threshold method through the JPP-based decision D Case Study ) Test Scenario: the comparison of the decision algorithms is performed thans to example n o 5 of [0] The latter considers four nown objects Y j and three detected objects X i For this situation, the discernment frames are: Θ Xi = {Y,Y 2,Y 3,Y 4, } Θ Yj = {X,X 2,X 3, } 5) The bbas of the potential associations are given in Table III Under the assumption of reliable sources, as there are more nown than detected objects, at least one object Y j is subject to disappearance From the bbas described by S, a strong confidence is allocated to the association of X with Y 08), but in the same time, equations S 2 show that Y could be associated to X 2 057) The X 2 association is also subject to an ambiguity as S 2 and S 22 show equal confidence 057) in the association to Y and Y 2 The latter also has a second association possibility with X 3 06) given by equations S 32 Finally, it seems that Y 3 and Y 4 will disappear as all association possibilities are rejected 2) Pignistic atrices: using Table III, the combination defining the association matrix BetP Xi ) is performed line wise whereas BetP Yj ) is obtained by a column wise combination using the generalized rules 7) and the pignistic transformation 9), 0) and ) Tables IV and V give respectively the X i Y j and Y j X i associations pignistic probabilities TABLE IV DETECTED TO KNOWN OBJECTS PIGNISTIC PROBABILITIES BetP Xi ) Y Y 2 Y 3 Y 4 /0 X X X TABLE V KNOWN TO DETECTED OBJECTS PIGNISTIC PROBABILITIES BetP Yj ) X X 2 X 3 /0 Y Y Y Y It is interesting to note that both tables are concordant to the a priori analysis done upon the original mass distribution Indeed in Table IV, a strong pignistic probability is allocated to X Y 090) while the association of X 2 reveals an ambiguity: two equal pignistic probabilities are obtained with Y and Y 2 oreover, this ambiguity is coupled to a conflict with an equivalent probability 033) Focusing in Table V on object Y 2, a slightly ambiguous association is reflected for respectively Y 2 X 2 and Y 2 X 3 In this particular case, the highest probability is obtained for the conflict, maing the decision non-trivial The a priori result about X 3 Y 2 is enforced by the combination as the associated pignistic probability is high 077) Finally, as aforementioned, a large probability is allocated to the disappearance of Y 3 and Y 4 These remars are recalled hereafter: X X 2 X 3 Y Y 2 Y 3 Y 4 Y Y Y 2 Y 2 X X 2 X 3 6) since ambiguity is formally defined by equal pignistic values, close values reveal a potential ambiguity, ie difficulty in the decision-maing 673

6 TABLE III CASE STUDY FRO [0]: 3 PERCEIVED OBJECTS ROWS) VS 4 KNOWN OBJECTS COLUNS) S m Θ X Y ) = 080 m Θ X Y ) = 000 m Θ X Θ) = 020 m Θ X 2 Y ) = 057 S 2 m Θ X 2 Y ) = 000 m Θ X 2 Θ) = 043 S 3 Y ) = 000 Y ) = 099 Θ) = 00 m Θ X 2 Y 2 ) = 000 S 2 m Θ X 2 Y 2 ) = 099 m Θ X 2 Θ) = 00 m Θ X 2 2 Y 2 ) = 057 S 22 m Θ X 2 2 Y 2 ) = 000 m Θ X 2 2 Θ) = 043 S 32 2 Y 2 ) = 06 2 Y 2 ) = Θ) = 039 m Θ X 3 Y 3 ) = 000 S 3 m Θ X 3 Y 3 ) = 097 m Θ X 3 Θ) = 003 m Θ X 2 3 Y 3 ) = 000 S 23 m Θ X 2 3 Y 3 ) = 052 m Θ X 2 3 Θ) = 048 S 33 3 Y 3 ) = Y 3 ) = Θ) = 048 m Θ X 4 Y 4 ) = 000 S 4 m Θ X 4 Y 4 ) = 099 m Θ X 4 Θ) = 00 m Θ X 2 4 Y 4 ) = 000 S 24 m Θ X 2 4 Y 4 ) = 099 m Θ X 2 4 Θ) = 00 S 34 4 Y 4 ) = Y 4 ) = Θ) = 00 This case illustrates the interest of a two-direction association: according to the BetP Xi ) matrix, X is evidentially associated with Y since the corresponding probability is close to Nevertheless, it can be noticed that the Y X i association is controversial since BetP Y ) shows a high conflict and the confidence in Y X is only 037 BetP Y /0) can be explained by equations S 2 concluding that object X 2 could also be associated to Y Focusing on the association of Y 3 and Y 4, if the X i Y j pignistic probabilities do not allow to find the issue, there is no doubt that these objects disappear using the Y j X i results E Discussion on the Global Association Algorithms The results of the three decision algorithms are synthesized in Table VI The latter clearly shows the disparity of the algorithms behaviors regarding similar conditions First, it is worth noting that the JPP and the JCPP algorithms conclude to the same associations In fact, these decision algorithms behave similarly except that JCPP reduces the computation time due to the initial ordering Nevertheless, their conclusions regarding X i Y j and Y j X i are in contradiction Indeed, if X is in both cases associated to Y, Y 2 is associated to X 2 and X 3 respectively for the X i Y j and Y j X i association ore precisely, the JPP and JCPP approaches consider a solution in which one association is almost sure - BetP X Y ) = and two which are quite uncertain - BetP X2 Y 2 ) = 03 and BetP X3 ) = 06 - In addition, it is worth noting that the association of X 2 is subject to an average conflict BetP X2 /0) = 033) revealing a difficult decision-maing For the Threshold method, results obtained with s = 04 depict a non-association for X 2, even with the JPP decision criterion The system stays undecided due to the one-to-one constraint associating initially X to Y and X 3 with Y 2 This is coherent regarding Table IV, as the pignistic repartition shows an ambiguity between Y, Y 2 and /0 It reveals that the decision for this object may be hazardous Nevertheless, the tuning of the threshold is critical, eg a small increase of its value to 05) reduces even more the number of associations Cross-comparing JPP, JCPP and the Threshold method shows that these algorithms do not converge to the same associations and that the last one ts to the expected solution cf 6)) Finally, one can note that the Threshold algorithm is not as eager in computation time 2 as both JPP and JCPP, even if the latter has been proposed as an alternative to the JPP to reduce the calculation time thans to the classification V LOCAL ASSOCIATION ALGORITHS This section highlights two decision algorithms proposed by the authors and providing a solution to the association limitations of a global decision approach adopted by the JPP, JCPP and Threshold methods It will be shown, based on the same case study, that contradictions will be removed, saving also computation time A Local Pignistic Probability The Local Pignistic Probability LPP), is based on the successive selection of the N/ deping on the association direction) local maximums in the pignistic matrices cf Algorithm 3) This algorithm is consequently opposed to the JPP and JCPP looing for a global optimization In addition, the LPP is supposed to be less time consuming compared to these algorithms as the computation of the joint probabilities is avoided Algorithm 3: LPP Algorithm for detected objects X i Data: BetP Xi Y j ), i I = {,,N}, pignistic probabilities of each detected object X i Result: the decision containing the N maximum BetP Xi and verifying the association constraints begin while i N do Select the maximum of pignistic probabilities BetP Xi Y j ) Remove the associations which do not verify the associations constraints 2 The computation time presented in Table VI and Table VII are mean values obtained after 20 runs on a Core 2 Quad computer 233 GHz) with 4Go RA The simulations have been done on atlab 674

7 TABLE VI JPP, JCPP AND THRESHOLD DECISION RESULTS Algorithm X i Y j X i Y j ean Calculation Time JPP X X Y Y X X 2 Y 3 Y 2 2 Y X ms JCPP T hreshold X Y X 2 Y 2 X 3 X Y X 2 X 3 Y 2 20ms 7ms X Y X Y Remar Contradictions regarding X 3 and Y 2 Contradictions regarding X 3 and Y 2 X 2 not associated B Gradient Classified Pignistic Probability The Gradient Classified Pignistic Probability GCPP) behaves similarly to the LPP but, before the associations selection based on the pignistic tables, a preliminary classification of their probabilities is done This ordering is performed considering decreasing probabilities gradients computed for each line of the matrices This approach is interesting as the pignistic gradient reveals the uncertainty related to the considered objects association For example, a situation in which the gradient is high depicts a strong confidence in an association On the opposite, a small gradient indicates two or more probabilities with close values, ie an ambiguous situation This decision-maing process is summarized in Algorithm 4 Algorithm 4: GCPP Algorithm for detected objects X i Data: BetP Xi Y j ), i I = {,,N}, pignistic probabilities of each detected object X i Result: the decision containing the N maximum BetP Xi and verifying the association constraints begin Ran in decreasing order the matrix regarding maxbetp Xi ) minbetp Xi ) while i N do Select the maximum of pignistic probabilities BetP Xi Y j ) in the matrix Remove the associations which do not verify the associations constraints C Results and Discussion The LPP and GCPP are compared to the JPP, JCPP and T hreshold algorithms in Table VII First, it is worth noting that the LPP and GCPP present concordant results similar association from X i Y j and vice versa) contrary to the JPP and the JCPP algorithms which are giving contradictory associations The adopted solution for the detected objects consists in the selection of two strongly confident associations - BetP X Y ) = 09 and BetP X3 Y 2 ) = and an uncertain one - BetP X2 ) = which represents a solution close to the a priori analysis cf Section IV-D) compared to the one proposed by the JPP and the JCPP algorithms In addition, the LPP and GCPP have the advantage to avoid the calculation of the joint pignistic laws by using decision criteria directly related to the pignistic matrices This allows a reduction of the computation time of four and three times respectively compared to the JPP and the JCPP oreover, the joint pignistic probability is a conservative and sensitive criteria Indeed, it is based on the product of probabilities defined in [0, ] This product ts to zero proportionally to the number of objects considered, especially when low pignistic probabilities are involved in its calculation Contrary to the Threshold algorithm with the conflict related decision, the LPP and GCPP do not require parameter tuning Tuning the threshold can be a tough tas since it strongly modulates the number of retained associations wrt the conflict level cf [9]) It is worth reminding that the algorithm comparison is based on a non-normalization of the conflict during the pignistic transformation cf Section III-C) This maes sense as the normalization would have mased the high conflict present in the association of the X 2, Y and Y 2 objects, characterized by ambiguities and contradictions Ambiguities and contradictions can also be detected thans to the analysis of the gradients obtained with the GCPP algorithm and presented in Table VIII last columns Indeed, the associations of X 2, Y and Y 2, subject to high conflict, present low gradients while the disappearance of Y 3 or the association of X 3, expected through the analysis of the original mass table cf Table III) and not subject to conflict, are characterized by strong gradients TABLE VIII PIGNISTIC PROBABILITIES WITH GRADIENT CLASSIFICATION BetP Xi ) Y Y 2 Y 3 Y 4 /0 Gradient X X X BetP Yj ) X X 2 X 3 /0 Gradient Y Y Y Y

8 TABLE VII DECISION RESULTS Algorithm X i Y j X i Y j ean Calculation Time JPP X X Y Y X X 2 Y 3 Y 2 2 Y X ms JCPP T hreshold LPP GCPP X Y X 2 Y 2 X 3 X Y X 2 X 3 Y 2 X Y X 2 X 3 Y 2 X Y X 2 X 3 Y 2 20ms 7ms 7ms 7ms X Y X Y X Y X Y Remar Contradictions regarding X 3 and Y 2 Contradictions regarding X 3 and Y 2 X 2 not associated X 2 appears X 2 appears VI CONCLUSION This paper has dealt with ulti-object Association OA) based on the Transferable Belief odel T B) ore precisely, the focus was placed on the comparison of three existing decision algorithms Their analysis, performed through a literature example, has shown their limitations when evidence conflicts Indeed, they loo for a globally satisfying solution leading to suspicious local associations To cope with this problem, the authors introduced two algorithms finding the best set of local associations Both algorithms have been compared to the existing ones considering the same literature example The results showed the interests of the local association selection wrt the pignistic probabilities, the solution intuitiveness, and the computation time In this paper, the OA was selecting a set of associations wrt a one-to-one association constraint Removing the latter will allow the system to stay undecided in ambiguous and contradictory situations, thus propagating multiple association hypotheses to the next processing cycle On the other hand, this paper has shown the interest of taing the conflict information into account during the analysis of the results Therefore, conflict-related decision criteria should be investigated REFERENCES [8] D Gruyer and V Berge-Cherfaoui ulti-objects association in perception of dynamical situation In Conference on Uncertainty in Artificial Intelligence UAI), pages , Stocholm, Sweden, 30 July - August 999 [9] B ourllion, D Gruyer, C Royère, and S Théroude ulti-hypotheses tracing algorithm based on the belief theory In IEEE International Conference on Information Fusion, Philadelphia, PA, USA, July 2005 [0] D ercier, E Lefèvre, and D Jolly Object association with belief functions, an application with vehicles Information Sciences, 824): , 20 [] A Ayoun and P Smets Data association in multi-target detection using the transferable belief model International journal of intelligent systems, 6:67 82, 200 [2] B Ristic and P Smets Recursive classification of multiple objects using discordant and non-specific data 2004 [3] Y Lemeret, E Lefevre, and D Jolly Improvement of an association algorithm for obstacle tracing Information Fusion, 92): , 2008 [4] D Gruyer Etude et traitement de données imparfaites pour le suivi multi-objet: application aux situations routières PhD thesis, Université de Technologie de Compiègne, 999 [5] P Smets Decision maing in the tbm: the necessity of the pignistic transformation International Journal of Approximate Reasoning, 38:33 47, 2005 [6] E El Najjar and P Bonnifait Road selection using multicriteria fusion for the road-matching problem IEEE Transactions on Intelligent Transportation Systems, 8: , 2007 [7] C Royère Contribution à la résolution du conflit dans le cadre de la théorie de l évidence : Application à la perception et à la localisation de véhicules intelligents PhD thesis, Université de Technologie de Compiègne, 2002 [] DB Reid An algorithm for tracing multiple targets IEEE Transactions on Automatic Control, 24: , 979 [2] S Blacman and R Popoli Design and Analysis of odern Tracing Systems Artech House, 999 [3] Bar-Shalom Y ulti-target ulti-sensor tracing: Applications and Advances, volume III Artech House, 999 [4] P Smets and R Kennes The transferable belief model Artificial Intelligence, 66:9 234, 994 [5] AP Dempster Upper and lower probabilities induced by a multivalued mapping Annals of athematical Statistics, 38: , 967 [6] G Shafer A mathematical theory of evidence Princeton University Press, 976 [7] Rombaut Decision in multi-obstacle matching process using Dempster-Shafer s theory In Advances in Vehicle Control and Safety AVCS), Amiens, France, 3 July