Combination of Masses in .NET

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6.2 Combination of Masses
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The combination is performed using Dempster s rule in unnormalized form [4,6,38]: mA =
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where mi and mj are masses assigned by measures i and j after discounting, and their focal elements Am and B1 B2 Bn , respectively. Since Dempster s rule is commutative and are A1 A2 associative, it can be applied repeatedly regardless of order, until all measures are combined. The measures under combination have to fulfill the condition of being cognitively independent [38], which is weaker than the notion of statistical independence required in the probabilistic case. Measures are cognitively independent if each of them assigns masses without any direct link to the others. This is the case in the proposed model. First, the fusion of measures per sensor is done and internal conflicts (masses of the empty set) of IR and of GPR are analyzed (as mentioned earlier, it does not make sense to do that for MD). If the internal conflict of a sensor is high, its data are corrupted by noise, occlusion, etc., so they should be discounted. We cannot know for sure which information is more affected by this corruption than others. Still, discounting is performed on each measure, since it is modeled in such a way that its influence is proportional to the level of ambiguity and uncertainty of the information, as shown in the previous subsection. Since both GPR sets of measures provide the depth information, this offers a possibility to discount IR in the function of depth. Taking into account knowledge about the IR camera and its reliability as a function of the depth, we have clustered IR separately as soon as the depth extracted by GPR is below 5 cm, and in between, the measures of IR are linearly discounted as a function of the GPR depth by the factor r = 1 + d/5 . Here, d(cm) is the depth extracted by the GPR (and it is negative below the soil surface). After discounting the sensors that have high internal conflicts, the sensors for each associated region are fused. Note that performing the combination in two steps (first for all information extracted from one sensor and then between sensors) is consistent because of the associativity of Dempster s rule: if there is no discounting, the result is the same as for direct global combination. What can still happen, however, is that the conflict after fusion (mass of the empty set) is high. In other words, although it seemed quite possible that they refer to the same object, they actually do not. For example, although IR could reach some depths of 5 cm, where GPR and MD detect something, it actually detects something nonmetallic on the surface. In that case, taking into account the safety and need not to miss mines, we declare that there is more than one object in the region and analyze the object of each of the sensors separately.
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6.3 Decision
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After clustering and combination of sensors, the resulting masses are found for each of the clusters. On the basis of these masses, a final conclusion about the true identity of each object under analysis has to be made for every cluster. Usual decision rules rely on beliefs, Bel, plausibilities, Pl [4], and pignistic probabilities, P [40], defined as follows: for any subset B: Bel B =
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(17.34) (17.35)
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Unfortunately, a decision rule based on any of these three functions does not lead to fruitful conclusions, since there are no focal elements containing mines alone [41]. Namely, after combination of the measures on the second level, the resulting focal elements are: FR, FI, (FR FI), (MR FR), (MI FI) and . Hence, as shown in [35], the following relations are always true: Bel M Bel F Pl M Pl F P M P F (17.37) (17.38) (17.39)
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so the decision would always be made in favor of F . Here, M denotes mines M = MR MI and F denotes friendly objects F = FR FI . Since, in the case of any ambiguity, far more importance has to be given to mines, we propose to define guesses, G A , where A M F 0 , in the following way: GM =
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