1. Object association in the TBM framework, application to vehicle driving aid D. Mercier, E. Lefèvre, D. Jolly Univ. Lille Nord de France, F-59000 Lille, France UArtois, LGI2A, F-62400 Béthune, France Prospects Object association algorithm TBM in a nutshell Example Input: Objects (vehicles) detected at time step t Objects (vehicles) detected at time step t +1 X i : a perceived object Y1 Y2 Y3 Y4 Y5 Y j : a known object X1 X2 X3 X4 Modeling in the belief function framework (Transferable Belief Model – TBM -) Association problem description Objective: Example: Object X 1 cor- responds to object Y 2 with some degree of belief… Find the best possible association between perceived objects {X 1, X 2, …, X N,*} and known objects {Y 1, Y 2, …, Y M,*} under the following constraints:  each perceived object X i is associated with at most one known object;  each known object Y j is associated with at most one perceived object;  object * can be associated with any objects. Object “*” means: “an object not present in the scene”. Contribution: Uncertain and imprecise information regarding the association of each object X i and with each object Y j. Where do perceived objects X i come from? What are known objects Y j become? Questions to be solved Framework for reasoning with partial (imprecise, uncertain) knowledge. Two levels: Credal level: information represented by belief functions to be manipulated. Decision level: probability transformation when a decision has to be made.  m   BetP  and the expected utility is maximized 1 perceived object X 1 and 2 known objects Y 1, Y 2 Frames of discernment involved:   i,j = {y i,j, n i,j }: the two possible answers (yes or no) to the question “Is the perceived object X i associated with the known object Y j ?”;   Xi = { Y 1, Y 2, …, Y M, *} = {1, …, M, *}: answers to the question “Which known object is associated with the perceived object X i ?”   Yj = {X 1, X 2, …, X N, *} = {1, …, N, *}: answers to the question “Which perceived object is associated with the known object Y j ?” N×M belief mass functions m  i,j = m i,j regarding each association (X i, Y j ) Input Algorithm:  Express each piece of information m  i,j on a common frame  Xi (or  Yj ): m  i,j  Xi (vacuous extension operation)  Combine conjunctively BBAs m  i,j  Xi = m j  Xi. Let us denote m  Xi this result.  Chosen decision = the association maximizing the probability BetP  X 1 ×  X 2 × … ×  X N and verifying the constraints expressed in the objective section. Refining  i,j allowing one to transport the information m  i,j on  Xi Vacuous extension Pignistic transformation  Investigation on conflicting decisions between perceived and known objects points of view.  Decomposition of the BBAs (cf Denœux’s works).  Introducing information from the tracking of the vehicles. By expressing this information on  X1 (X 1 point of view: with which known object Y j, the perceived object X 1 is associated?): CISIT project (Campus International pour la Sécurité et l'Intermodalité des Transports). These works have been financed by the French region Nord-Pas de Calais. The conjunctive combination m  X 1 of m 1  X 1 and m 2  X 1, and the pignistic probability BetP  X 1 are given by: Conclusion from X 1 point of view: 1.The singleton maximizing BetP  X 1 is {2}, so X 1 is associated with Y 2. 2. Y 1 is not associated, Y 1 has disappeared (or is hidden). On the other hand, it is also possible to express the available information on  Y1 and  Y2 : As there is only one perceived object X 1, no combination is necessary: (Y 1, Y 2 ) is then associated with (*,1): Y 1 has disappeared and Y 2 is associated with X 1. The decision coming from X 1 and the decision coming from Y 1 and Y 2 are the same. Unfortunately this not always the case… (in practice a reduce number of cases) Conclusion from Y 1 and Y 2 points of view:   = {  1, …,  K }: finite set of the possible answers to a given question Q of interest (frame of discernment)  Information held by a rational agent regarding the answer to question Q can be quantified by a mass function or BBA m  such that m  : 2   [0,1] and:  m(A) represents the part of the unit mass allocated to the hypothesis: “ The answer to question Q is in the subset A of  ”.