Eric Lefèvre
email: eric.lefevre@univ-artois.fr

Zied Elouedi
email: zied.elouedi@gmx.fr

How to preserve the conflict as an alarm in the combination of belief functions?

Keywords: Belief function theory, Data fusion, Alarm of singular source, Conflict management

In the belief function framework, a unique function is induced from the use of a combination rule so allowing to synthesize all the knowledge of the initial belief functions. When information sources are reliable and independent, the conjunctive rule of combination, proposed by Smets, may be used. This rule is equivalent to the Dempster rule without the normalization process. The conjunctive combination provides interesting properties, as the commutativity and the associativity. However, it is characterized by having the empty set, called also the conflict, as an absorbing element. So, when we apply a significant number of conjunctive combinations, the mass assigned to the conflict tends to 1 which makes impossible returning the distinction between the problem arisen during the fusion and the effect due to the absorption power of the empty set.

The objective of this paper is then to define a formalism preserving the initial role of the conflict as an alarm signal announcing that there is a kind of disagreement between sources. More exactly, that allows to preserve some conflict, after the fusion by keeping only the part of conflict reflecting the opposition between the belief functions. This approach is based on dissimilarity measures and on a normalization process between belief functions. Our proposed formalism is tested and compared with the conjunctive rule of combination on synthetic belief functions.

Introduction

Since many years, the belief function theory [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF][START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] has known an increasing interest from scientific community since it allows to deal with imperfect data (imprecise and uncertain) and to combine them using a combination rule. One of the classical combination rules is the conjunctive rule. This latter, introduced by Smets [START_REF] Smets | The combination of evidence in the Transferable Belief Model[END_REF][START_REF] Smets | The Transferable Belief Model[END_REF], is equivalent to the Dempster rule of combination [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF][START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] without the normalization process. Properties and also hypotheses that sources should satisfy before being combined by this rule are well established.

This rule has an orthogonal behavior which is very precious because it permits a fast and clear convergence towards a solution, but in return, the empty set is an absorbing element. Smets supports that the existence of this mass on the empty set, called also conflict, can play a role of alarm. So, contrary to Dempster's rule where the conflict is reallocated proportionally to the other masses of the focal elements, this conflict must not be redistributed since it may be at the origin of important information concerning the progress of the fusion process and show the disagreement between sources. In fact, if the conflict is small, it means that the joint bba fits with the opinions given by the sources to fuse and consequently try to reinforce them, whereas when the conflict is high, it means that the induced bba is largely in contradiction with the previous opinions. Nevertheless, due to its absorbing conjunctive effect, a series of combinations aims at getting the empty set equal to 1, making impossible the distinction between a real problem between sources to fuse and an effect caused by the absorbing of the empty set.

In addition to the conflict definition of Smets, other works have been dealt with the conflict definition namely Liu [START_REF] Liu | Analyzing the degree of conflict among belief functions[END_REF] proposes a quantitative measure taking into account the mass on the empty set induced from the combination of two or more bbas and the distance between betting commitments of these same bbas after applying the pignistic transformation. However, this mass on the empty set remains not sufficient to exactly express the conflict. On the other hand, in [START_REF] Osswald | Understanding the large family of Dempster-Shafer theory's fusion operators -a decision-based measure[END_REF], Osswald and Martin present another interpretation of the conflict by defining the auto-conflict as the amount of intrinsic conflict of a belief function, in other words it is the conflict generated by such a function relative to one information source.

Besides in [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF], Destercke and Burger present some properties that a mea-sure of extrinsic conflict should satisfy. They defined conflict as the inconsistency arising from a conjunctive combination, and based on properties, they also proposed conflict measurements making no a priori assumptions regarding the dependence between sources. Thus, two types of conflict can be defined:

• The conflict which allows the estimation of the confusion rate of a source and which will be called intrinsic conflict [START_REF] Daniel | Conflicts within and between belief functions[END_REF][START_REF] George | Quantification of conflict in dempster-shafer framework: A new approach[END_REF][START_REF] Osswald | Understanding the large family of Dempster-Shafer theory's fusion operators -a decision-based measure[END_REF][START_REF] Schubert | The internal conflict of a belief function[END_REF],

• The conflict which evaluates the discordance between two bodies of evidence and will be labeled extrinsic conflict [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF][START_REF] Ristic | The TBM global distance measure for the association of uncertain combat ID declaration[END_REF][START_REF] Tessem | Approximations for efficient computation in the theory of evidence[END_REF].

In this paper, we characterize the opposition between belief functions by means of a measure of dissimilarity. This measure is then used in our proposed approach named Combination With Adapted Conflict (CWAC) providing an adaptive weighting between Dempster's rule and conjunctive rule, allowing to keep the initial meaning of the conflict obtained during the combination and so to restore its initial role of alarm. Thus, it permits to the conflict to take back its initial sense by only mentioning that there is a problem somewhere and reducing its absorbing power. Our proposal is not a conflict measure but a combination rule preserving the main role of a conflict as a signal making aware of this opposition between sources. A preliminary work of this approach has been proposed in [START_REF] Lefevre | Towards an alarm for opposition conflict in a conjunctive combination of belief functions[END_REF].

This paper is organized as follows. Section 2 presents the basics of the belief function theory. Combination rules, proposed in the belief function framework, are detailed in Section 3. The definition and properties of our CWAC rule are exposed in section 4. Section 5 brings to light our proposed approach by comparing its behavior with that of the conjunctive combination in the case of synthetic data. Section 6 concludes our study and presents some future works.

Belief function theory: background

The belief function theory is considered as a useful theory for representing and managing uncertain knowledge. In this Section, we shall briefly recall some basics of this theory. More details can be found in [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF][START_REF] Smets | The combination of evidence in the Transferable Belief Model[END_REF][START_REF] Smets | The Transferable Belief Model[END_REF].

Representing information

Let Ω be a finite non empty set including all the elementary events related to a given problem. These events are assumed to be exhaustive and mutually exclusive. Such set Ω is named frame of discernment.

The impact of a piece of evidence on the different subsets of the frame of discernment Ω is represented by the so-called basic belief assignment (bba), called initially by Shafer [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] basic probability assignment.

The bba m is a function m : 2 Ω → [0, 1] that satisfies:

A⊆Ω m(A) = 1 (1)
The basic belief mass m(A), expresses the part of belief exactly committed to the event A of Ω given a piece of evidence. Due to the lack of information, this quantity cannot be apportioned to any strict subset of A.

Shafer [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] has initially proposed a normality condition expressed by:

m(∅) = 0 ( 2
)
Such bba is called a normalized basic belief assignment. Smets [START_REF] Smets | The combination of evidence in the Transferable Belief Model[END_REF][START_REF] Smets | Belief functions: The disjunctive rule of combination and the generalized Bayesian theorem[END_REF] relaxes this condition by considering m(∅) as the amount of conflict between the pieces of evidence or as the part of belief given to the fact that none of the hypotheses in Ω is true. All the subsets A of Ω such that m(A) is strictly positive, are called the focal elements of m.

Associated with m is the belief function, denoted bel, corresponding to a specific bba m, assigns to every subset A of Ω the sum of masses of belief committed to every subset of A by m [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]. This belief function, bel, represents the total belief that one commits to A without being also committed to A. The belief function bel : 2 Ω → [0, 1] is defined so that:

bel(A) = ∅ =B⊆A m(B), ∀A ⊆ Ω (3) bel(∅) = 0 (4)
The plausibility function pl : 2 Ω → [0, 1] quantifies the maximum amount of belief that could be given to a subset A of Ω. It is equal to the sum of the masses given to subsets B compatible with A:

pl(A) = A∩B =∅ m(B), ∀A ⊆ Ω (5) pl(∅) = 0 (6)

Special belief functions

In this subsection, we propose some belief functions used to express particular situations related generally to uncertainty. A vacuous bba is defined as follows [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]:

m(Ω) = 1 and m(A) = 0 ∀A = Ω (7)
Such function quantifies the state of total ignorance by having only Ω as a focal element.

A categorical bba is a normalized bba defined as follows:

m(A) = 1 ∀A ⊂ Ω and m(B) = 0 ∀B ⊆ Ω, B = A (8)
This function has a unique focal element different from the frame of discernment Ω.

A certain bba is a particular categorical bba such that its focal element is a singleton. A certain bba is defined as follows:

m(A) = 1 and m(B) = 0 ∀B = A and B ⊆ Ω and |A| = 1 (9)
where A is a singleton event of Ω. This function represents a state of total certainty on the focal element.

A simple support function (ssf ) if it has at most one focal element different from the frame of discernment Ω. A simple support function is defined as follows [START_REF] Smets | The canonical decomposition of a weighted belief[END_REF]:

m(X) =    w if X = Ω 1 -w if X = A ∀A ⊆ Ω 0 otherwise ( 10
)
where A is the focal element and w ∈ [0, 1]. It presents a belief function induced by a piece of evidence supporting A (with 1w) and leaving the remaining beliefs for Ω. This bba can also be noted A w . A Bayesian bba is a particular case of belief functions where all the focal elements are singletons. The corresponding bba is defined as follows: m(A) > 0 only when |A| = 1 [START_REF] Destercke | Toward an axiomatic definition of conflict between belief functions[END_REF] In this case, bel = pl and they are considered as a probability distribution.

A consonant bba is a bba when all its focal elements (A

1 , A 2 , . . . , A n ) are nested, that is A 1 ⊆ A 2 ⊆ . . . ⊆ A n . It is a special case of possibilities.
A dogmatic belief function is defined such that m(Ω) = 0. Inversely a non-dogmatic belief function is defined such that m(Ω) > 0 [START_REF] Smets | The canonical decomposition of a weighted belief[END_REF].

The discounting operation

Handling evidence given by experts requires to take into account the level of expertise of each information source. Indeed, reliability differs from one expert to another and a discounting method is imperative to update experts' beliefs based on weighting most heavily the opinions of the best experts and conversely for the less reliable ones.

Let α ∈ [0, 1], be the discounting rate, and consequently 1α is regarded as a degree of confidence on the expert. Updating the expert's opinions leads to:

m α (A) = (1 -α)m(A), ∀A ⊂ Ω, m α (Ω) = (1 -α)m(Ω) + α , (12)
The larger α, the closer m α is from the vacuous belief function. So, when α = 1, it means that the expert is not reliable at all, and his opinions should be totally ignored. On the other hand, when α = 0, it means that the expert is totally reliable.

Pignistic transformation

To make a decision, Smets proposes to transform beliefs to a probability measure, denoted BetP [START_REF] Smets | The Transferable Belief Model[END_REF]. The link between these two measures is achieved by the pignistic transformation such that:

BetP (A) = B⊆Ω |A ∩ B| |B| m(B) 1 -m(∅) , ∀A ∈ Ω. ( 13
)
BetP can be extended as a function on 2 Ω as:

BetP (A) = ω∈A BetP (ω) ∀A ⊆ Ω and ω ∈ Ω. ( 14
)

Distance between bbas

Several researchers have proposed distances under a belief function framework. Some of them are based on the pignistic transformation [START_REF] Bauer | Approximations algorithm and decision making in the Dempster-Shafer theory of evidence -an empirical study[END_REF][START_REF] Elouedi | Assessing sensor reliability for multisensor data fusion with the Transferable Belief Model[END_REF][START_REF] Tessem | Approximations for efficient computation in the theory of evidence[END_REF][START_REF] Zouhal | An evidence-theoretic k-NN rule with parameter optimization[END_REF], where an unavoidable step is the pignistic transformation of the bbas. On the other hand, some distances based directly on bbas, are also proposed [START_REF] Fixsen | The modified Dempster-Shafer approach to classification[END_REF][START_REF] Jousselme | A new distance between two bodies of evidence[END_REF] defined on the power set of the frame of discernment.

In this paper, we will focus on the Jousselme distance [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF] considered as one of the most known distances.

Let m 1 and m 2 two bbas, the Jousselme distance [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF] is defined as follows:

d J (m 1 , m 2 ) = 1 2 (m 1 -m 2 ) t D(m 1 -m 2 ) ( 15
)
where D is the Jaccard index defined by:

D(A, B) = 0 if A = B = ∅ |A∩B| |A∪B| ∀A, B ∈ 2 Ω . ( 16
)

Combination of pieces of evidence

In the case of imperfect data (uncertain, imprecise and incomplete), data fusion becomes an interesting solution allowing to obtain more relevant information. The belief function theory offers appropriate tools for ensuring fusion. The objective of the combination is then to synthesize the information issued of a set of belief functions in a unique function. Based on the fusion of bbas induced from several sources, the conflict between these sources may be arisen. The conflict is defined as a measure of a disagreement between the sources. In the belief function framework, several combination approaches were developed. They can be classified in two categories according whether sources are reliable or not.

Reliable sources

When information sources, inducing belief functions, are considered as reliable, the used operators are based on the conjunctive combination. Hence, the induced result of the conjunctive combinations of two bbas m 1 and m 2 , is noted m ∩ and is defined by [START_REF] Smets | The Transferable Belief Model[END_REF]:

m ∩ (A) = B∩C=A m 1 (B)m 2 (C) ∀A, B, C ⊆ Ω. ( 17
)
This rule considers m ∩ (∅) as the discord between sources implied in the fusion process and called it conflict. Another kind of conjunctive combination is Dempster's rule [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF], based on the orthogonal sum, and is considered as the standard fusion rule used in the case of reliable sources. The induced bba is noted m ⊕ and is defined by:

m ⊕ (A) = 1 1 -m ∩ (∅) m ∩ (A) ∀A = ∅ and m ⊕ (∅) = 0 ( 18
)
where ) is the normalization factor making m ⊕ (∅) = 0. These two combination rules namely the conjunctive and the Dempster rules are commutative and associative but not idempotent. Therefore, they cannot be used when the sources are not independent 1 .

1 1-m ∩ (∅
In addition, Denoeux [START_REF] Denoeux | Conjunctive and disjunctive combination of belief functions induced by non distinct bodies of evidence[END_REF] has developed the cautious conjunctive rule having also a conjunctival behavior. This rule is idempotent, and can consequently be applied when information sources are dependent. It is defined by the following relation:

m 1 ∧ m 2 = ∩ A⊂Ω A ω 1 (A)∧ω 2 (A) ( 19
)
where ∧ is the minimum operator and ω i are the weights obtained from the decomposition based on the conjunctive combination of m i in simple support functions such as:

m i = ∩ A⊂Ω A ω i (A) . ( 20
)
It is useful to note that these various combinations can also be applied even in the case of unreliable information sources. In such a case, a discounting is then necessary to be applied before the combination. This approach was used in several works [START_REF] Mercier | Decision fusion for postal address recognition using belief functions[END_REF][START_REF] Veremme | Evidential calibration process of multi-agent based system: An application to forensic entomology[END_REF]. The difficulty remains then in the assessment of the degrees of reliability. There are several strategies to estimate this reliability. One of them is based on the calculation of a measure of dissimilarity between the belief functions to be combined. This measure is used as a degree of discounting. Hence, contradictory belief functions are discounted before fusion phase. This approach was studied by several authors [START_REF] Klein | Automatic discounting rate computation using a dissent criterion[END_REF][START_REF] Liu | Combination of sources of evidence with different discounting factors based on a new dissimilarity measure[END_REF][START_REF] Liu | Sequential adaptative combination of unreliable sources of evidence[END_REF][START_REF] Martin | Conflict measure for the discounting operation on belief functions[END_REF][START_REF] Schubert | Conflict management in Dempster-Shafer theory using the degree of falsity[END_REF][START_REF] Yong | Combining belief functions based on distance of evidence[END_REF]. Another technique consists in using further information on the belief functions to define the discounting degree coefficients [START_REF] Elouedi | Discountings of a belief function using a confusion matrix[END_REF][START_REF] Elouedi | Assessing sensor reliability for multisensor data fusion with the Transferable Belief Model[END_REF].

Unreliable sources

If one source in the fusion process is not reliable and there is no possible adjustment then combinations based exclusively on the conjunctive rule cannot be used. As a first rule dealing with such a case, is the disjunctive rule of combination proposed by Dubois and Prade [START_REF] Dubois | A set theoretic view of belief functions: Logical operation and approximations by sets[END_REF]. Then, in the case of two bbas m 1 and m 2 , this rule is defined by:

m ∪ (A) = B∪C=A m 1 (B)m 2 (C) ∀A, B, C ⊆ Ω. ( 21
)
This kind combination is the dual of the conjunctive combination. It is discussed within the Generalized Bayes Theorem by Smets [START_REF] Smets | Belief functions: The disjunctive rule of combination and the generalized Bayesian theorem[END_REF]. As the conjunctive one, this rule is also associative and commutative but not idempotent. In the case where sources are dependent, Denoeux [START_REF] Denoeux | Conjunctive and disjunctive combination of belief functions induced by non distinct bodies of evidence[END_REF] suggests using the bold disjunctive rule. For combining two bbas m 1 and m 2 , this rule is defined in the following way:

m 1 ∨ m 2 = ∪ A =∅ A ν 1 (A)∧ν 2 (A) (22)
where ν 1 and ν 2 represent the disjunctive decomposition2 of m 1 and m 2 based on simple generalized masses such that:

m i = ∪ A =∅ A ν i (A) . (23)
Besides, rules having intermediate behavior between the conjunctive and disjunctive rules of combination were developed these last years [START_REF] Delmotte | Using trust in uncertainty theories[END_REF][START_REF] Dubois | Representation and combination of uncertainty with belief functions and possibility measures[END_REF][START_REF] Dubois | Reliability Data Collection And Analysis, chapter On the combination of evidence in various mathematical frameworks[END_REF][START_REF] Florea | Robust combination rules for evidence theory[END_REF]. Among these rules, we mention the one introduced by Florea [START_REF] Florea | Robust combination rules for evidence theory[END_REF] which allows to define a family of rules having an intermediate behavior between conjunctive and disjunctive operators according to two functions γ 1 and γ 2 dependent on the conflict m ∩ (∅). This rule is defined in the following way:

m F (A) = γ 1 (m ∩ (∅))m ∩ (A) + γ 2 (m ∩ (∅))m ∪ (A) ∀A = ∅ (24)
Several definitions for the functions γ 1 and γ 2 were proposed. However, the authors recommend the use of logarithmic forms which better correspond to the non-symmetric distribution of the conflict. This rule is called robust rule of combination. For other combination rules, it is a question to redistribute the partial conflict [START_REF] Daniel | Associativity in combination of belief functions; a derivation of minC combination. Soft computing A Fusion of foundation[END_REF][START_REF] Florea | Adaptative combination rule and proportional conflict redistribution rule for information fusion[END_REF][START_REF] Martin | Toward a combination rule to deal with partial conflict and specificity in belief functions theory[END_REF][START_REF] Yamada | A new combination of evidence based on compromise[END_REF]. Finally, we can also note the other rules, less used, which allow too to combine bbas [START_REF] Josang | The consensus operator for combining beliefs[END_REF][START_REF] Josang | Cumulative and averaging fusion of beliefs[END_REF][START_REF] Murphy | Combining belief functions with evidence conflicts[END_REF][START_REF] Yager | On the Dempster-Shafer framework and new combination rules[END_REF][START_REF] Zhang | Representation, independence and combination of evidence in the Dempster-Shafer theory[END_REF]. The objective of all these rules is to redistribute the conflict induced during the combination.

Discussion

Some tricky situations may often happen, when it is necessary to combine a set of sources where some of them are contradictory. The term of singular sources can be defined for sources that supply different information of what the other sources propose [START_REF] Klein | Singular sources mining using evidential conflict analysis[END_REF]. So, being singular or not depends on the fact to be in agreement with the majority. This discord with the majority can result [START_REF] Lefevre | Belief function combination and conflict management[END_REF][START_REF] Smets | Analyzing the combination of conflicting belief functions[END_REF]:

• That the source at the origin of the discord is not reliable. For example, a failing sensor or working outside his range of functioning can disagree with the other information sources. In that case, the information induced from this source is not relevant.

• Or what this source possesses as information not perceived by the others. For example in the case of the detection of target, several sensors cannot see a target hidden by an obstacle while a source positioned in a different way sees perfectly this target. This source is then very instructive.

So, for a given application, the selection of a combination operator, among those proposed, is relatively sensitive to handle. In fact, in most of the situations, it is impossible to know if some sources are (or will become) reliable or not reliable. The use of a strategy of combination of reliable sources (for example conjunctive rule) containing one or several non-informative singular sources is so going to end in a process of less robust fusion. In the same manner, the use of a combination of unreliable sources with informative singular sources means, generally, redistributing some conflict. Redistribution which may be likened to a loss of information.

To identify the situation and thus select the appropriate combination, recent works [START_REF] Daniel | Non-conflicting and conflicting parts of belief functions[END_REF][START_REF] Klein | Singular sources mining using evidential conflict analysis[END_REF][START_REF] Liu | Analyzing the degree of conflict among belief functions[END_REF][START_REF] Martin | Conflict measure for the discounting operation on belief functions[END_REF][START_REF] Schubert | Specifying nonspecific evidence[END_REF][START_REF] Schubert | Conflict management in Dempster-Shafer theory using the degree of falsity[END_REF] introduced measures of conflict, other than the value of m(∅), allowing to quantify the opposition between belief functions. However, these measures are decoupled by the combination belief functions. So, these solutions produce a set of measures. Each of these measures reflecting the opposition between a source and the other sources involved in the process of fusion. This set of measures remains still difficult to interpret and thus the identification of the situation (reliable or unreliable sources) is tricky. In the following section, a formalism, which allows to protect the part of opposition between belief functions during the combination, is presented. This formalism supplies an alarm to the decision maker, concerning his process of fusion, allowing him to choose adequate measures as for example to strengthen the weight of the singular source if it is informative or to treat it as an erroneous source in the opposite case. This approach allows to obtain the result of the combination and a measure of the quality of the fusion process.

Combination with adapted conflict (CWAC)

In this paper, belief functions are considered as outcomes of independent information sources. In this frame, the conflict m(∅) obtained during the combination of belief functions allows to draw the attention on a possible problem like bad modelling, to an unreliable source now and more generally the presence of a singular source. Most of the combinations proposed in the literature (see Section 3) try to redistribute this conflict and not to use it as indicator. Only the conjunctive combination allows to preserve the mass on the empty set.

However, when the conjunctive combination is applied on a large number of belief functions, the conflict can take important proportions without reflecting a real problem. In such a case, this value does not represent a real opposition. This phenomenon is due to the absorbing character of the empty set.

Based on this analysis, we intend to develop a method allowing to transform the mass on the empty set as a real indicator of problems even if the number of sources to be combined is important. Our approach is named Combination With Adapted Conflict (CWAC).

Considering that there is a problem when sources produce strongly different belief functions, the conflict must be kept during the fusion. On the contrary, in the case of the combination of information sources for which the distributions of masses are equivalent, the conflict does not have to exist. The spirit of CWAC is to keep the conflict exclusively reflecting the disagreement between sources without taking into account the part that could be called auto-conflict is defined for only one source [START_REF] Martin | Human experts fusion for image classification[END_REF][START_REF] Osswald | Understanding the large family of Dempster-Shafer theory's fusion operators -a decision-based measure[END_REF]. To define CWAC rule, a measure allowing to determine the resemblance between bbas is necessary.

So, one of the interests of our CWAC rule is to detect the unreliable source(s). In fact, an information source is not usually reliable and its reliability may change from time to time. Two solutions are then possible. The first one consists in checking before fusing if each source is reliable or not. It is then necessary to compute the reliability of each source at each time and then fuse them (if sources are identified enough as reliable). The second solution has only one phase, using CWAC for combination, so get at the meantime, the fusion result and the disagreement measure (so potentially the unreliability of a source). According to the value of this measure, the expert may accept or not the results of the fusion. If he does not accept it, he should then look for the unreliable source(s) (like the first solution). Hence, our proposed solution needs less time than the first one.

So, through CWAC, we can discover this through the mass on the empty set, whereas as mentioned using the conjunctive rule, it is not so obvious due to the absorbing behavior of the empty set and also especially when the number of the sources is important.

With two belief functions

At first, only the case of two bbas m 1 and m 2 is considered. The dissimilarity notion may be obtained from a measure of distance. This distance may be calculated by the method proposed in section 2.5 and is noted d(m 1 , m 2 )3 . The borders of the function d are:

• d(m 1 , m 2 ) = 0: m 1 and m 2 are similar (and consequently in agreement)

and their combination should not generate a conflict. In this case, the conflict must be redistributed in the same manner as the combination rule of Dempster.

• d(m 1 , m 2 ) = 1: m 1 and m 2 are antinomic (i.e. m 1 ({ω j }) = 1 and m 2 ({ω i }) = 1 with ω i = ω j , ω i ∈ Ω and ω j ∈ Ω). Their combination produces a conflictual mass expressing their opposition. This value must be kept in the same way as with the conjunctive combination.

CWAC combination is defined by an adaptive weighting between the conjunctive and Dempster's rules. This adaptive weighting allows to obtain a behavior similar to that of the conjunctive rule when belief functions are antinomic and equivalent to that of the Dempster rule when belief functions are similar. Between these two extremes, a gradual passage may be considered. The combination rule which we propose noted ↔ can be written then in the following way:

m ↔ (A) = γ 1 m ∩ (A) + γ 2 m ⊕ (A) ∀A ⊆ Ω (25)
with:

m ⊕ (A) = (m 1 ⊕ m 2 )(A) ∀A ⊆ Ω (26) m ∩ (A) = (m 1 ∩ m 2 )(A) ∀A ⊆ Ω (27)
where γ 1 and γ 2 are dependent functions of distance d(m 1 , m 2 ). These functions should satisfy the following constraints:

γ 1 = f 1 (d(m 1 , m 2 )) with f 1 (0) = 0 and f 1 (1) = 1 (28) γ 2 = f 2 (d(m 1 , m 2 )) with f 2 (0) = 1 and f 2 (1) = 0 (29)
with γ 1 + γ 2 = 1. However other functions are possible, we can choose at first, linear functions such that:

γ 1 = d(m 1 , m 2 ) (30) γ 2 = 1 -d(m 1 , m 2 ). (31)
Hence, our rule can be written ∀A ⊆ Ω and m ∩ (∅) = 1:

m ↔ (A) = m 1 ↔ m 2 (A) = d(m 1 , m 2 )m ∩ (A)+(1 -d(m 1 , m 2 )) m ⊕ (A). ( 32
)
When m ∩ (∅) = 1, then we have m ↔ (∅) = 1.

General case

The generalization of this approach arises when more than two bbas are to be merged. Indeed, in a classic way the measures of dissimilarity are defined only between two bbas functions. Let N bbas noted m 1 , . . . , m i . . . , m N which should be fused. The measure of dissimilarity between these functions, which it is necessary to use in the case of the proposed approach, may be a synthesis of the distances between bbas. The objective of the proposed approach is to offer to the decision maker, after the fusion, a measure of conflict which allows to distinguish the presence of a singular source among bbas. It seems then natural to make this synthesis by using, for example, the maximal value of the set of distances 4 . In that case, the value of D may be defined by:

D = max i,j [d(m i , m j )] (33)
with i ∈ [1, N ] and j ∈ [1, N ]. Our rule can be generalized as the following manner ∀A ⊆ Ω and m ∩ (∅) = 1:

m ↔ (A) = ↔ i m i (A) = Dm ∩ (A) + (1 -D)m ⊕ (A) (34)
and

m ↔ (∅) = 1 when m ∩ (∅) = 1 (35)
with:

m ∩ (A) = ∩ i m i (A) and m ⊕ (A) = + i m i (A) ∀i ∈ [1, N ]. ( 36
)

Properties

Here are some properties characterizing our CWAC rule:

• Commutativity: CWAC is commutative, for all m 1 , m 2 :

m 1 ↔ m 2 = m 2 ↔ m 1 .
• Associativity: CWAC is not associative, for all m 1 , m 2 , m 3 :

m 1 ↔ (m 2 ↔ m 3 ) = (m 1 ↔ m 2 ) ↔ m 3 .
• Neutral element: The neutral element of the CWAC is Ω. Let m 0 be the vacuous bba. So for all m:

m ↔ m 0 = m.
• Absorbing element: The absorbing element of the CWAC is ∅. Let m e be a bba having ∅ has the unique focal element (m(∅) = 1). So, for all m: m ↔ m e = m e .

• Idempotent: CWAC is not idempotent, for all m: m ↔ m = m.

Experimental results

In this Section, the proposed combination is compared to the conjunctive combination on synthetic data. The first experimentation more emphasizes on the study of the behavior of the different operators in the case of a reduced number of belief functions. Then, the second and the third experimentations allow to study the behavior of the operators according to the number of bbas and to the similarity between them.

Example n • 1

For this first experimentation, we consider a frame of discernment Ω = {ω 1 , ω 2 , ω 3 }. At first, we consider 5 bbas defined in Table 1. We notice that these bbas are relatively similar. In this Table, we find the results of the combination of these 5 sources with the conjunctive operator (m 1,...,5 ∩ ) and our CWAC operator ( J m 1,...,5 ↔ with Jousselme's distance). The conflict induced by the conjunctive combination is relatively important which is not the case with our approach (0.9317 against 0.1118 if we use the Jousselme distance). Now, let us consider the case, where a 6th source is added. This source is singular with regard to 5 first ones. Results of the combination of these 6 bbas are also presented in Table 1. These results allow to deduce that there is an important increase of the value of the conflict in the case of our proposed approach. This value is, according to the used distance, from 5 to 6 times more important than during the first combination. On the other hand, in the conjunctive combination case, the increase is only 6.4 % (that is only 1.06 more important). So, the detection of the presence of a singular source in the fusion process is more easier with our CWAC operator. These conclusions are consolidated by the experimentations presented in the following sections.

Example n • 2

For this experimentation, we are going to study the evolution of the conflict, according to the maximal distance, in a set of N belief functions. The focal elements as well as degrees of beliefs are randomly generated according to a normal law. The focal elements are altogether selected in 2 Ω with the cardinal of Ω equal to 3. Results presented on Figure 1 are the average of 100 experiences. In that case, the behavior of our proposal is almost linear according to the maximal distance. This conclusion is usually true for any chosen distance and also for any number of belief functions involved in the fusion process. In case with more than 10 belief functions and when the maximum distance is greater than 0.5, the conjunctive combination produces already a conflictual mass close to 1, this is not the case for our proposal. In this situation, the proposed approach yet allows us to distinguish the opposition between the belief functions. We can also note that when the maximal distance is low, the conflict is almost null. On the contrary, and in the same case, the use of the conjunctive combination induces a significant conflict (superior or equal to 0.5 according to the number of the belief functions combined) while bbas are similar. Furthermore, the convergence towards a maximal value of conflict is fast as the number of bbas is important.

While both approaches have the same behavior, the dynamics obtained by CWAC function is more important. Indeed, the combination CWAC, the conflicting mass ranges from 0 (no conflict) to 0.8 (conflict situation). In this case, the dynamics is then 0.8. With the conjunctive combination in the best case, this value is only 0.5. With a variation range greater, it is more easily to define the threshold at which the system can be questioned (unreliable source or singular source). Finally, for a given combination rule, the results vary according to the number of sources (N = 10, 20 or 30). However, these variations are more important in the case of conjunctive combination and remain very limited with CWAC.

Example n • 3

In this experimentation, two sets of belief functions S 1 and S 2 are considered. Within every set, bbas are identical. On the other hand, between both sets, bbas are randomly generated so as to obtain maximal distances according to 3 categories:

• D ′ category: where the maximal distance between bbas of the two sets is included in [0; 0.3]

• D ′′ category: where the maximal distance between bbas of the two sets is included in ]0.3; 0.6]

• D ′′′ category: where the maximal distance between bbas of the two sets is included in ]0.6; 0.9]

The generation of bbas is realized in the following way. We generate randomly, according to a normal law, the focal elements of the bba belonging to the set S 1 . Then, we generate, according to a normal law, the degree of belief to be assigned to each of these focal elements. The same process will be repeated for bbas belonging to the set S 2 by verifying the constraint imposed on the maximal distance. Once these bbas were obtained, we duplicate them in each of the sets according to the wished proportion. For that purpose, we note r the proportion of the set S 1 which expresses itself in the following way:

r = |S 1 | N (37)
Where On these figures, when the proportion is equal to 0 or to 1, the involved bbas in the fusion process belong all to the same set. In that case, the conjunctive combination gives nevertheless an important conflicting mass 5 .

The evolution of r only modifies very slightly this value especially in the case of a low distance. In the case of our CWAC operator, when all the bbas arise from the same set (r = 0 or r = 1), the conflicting mass is null.

As soon as a singular source is involved in the fusion process, our rule allows to reach a maximal value which will remain unchanged. This maximum stays in reasonable proportions when the maximal distance is included between [0; 0.3], that is when bbas of both sets are considered as relatively similar. This remark is not verified in case of the conjunctive combination.

Conclusion

In this paper, we have proposed an approach with adapted conflict preserving the initial role of the conflict which is an alarm signal for the combination of belief functions. CWAC presents an adaptive weighting between Dempster's and conjunctive rules based on Jousselme et al.'s distance. Experimentations have shown that through CWAC, the absorbing power of the conflict is reduced compared with the conjunctive rule of combination. As future works, we intend to deal with the interpretation of the conflict when sources are dependent. Besides, the dual of CWAC will be proposed in the disjunctive behavior where the frame of discernment Ω is the absorbing element.

Figure 1 :

 1 Figure 1: Behavior of conflict when the distance between bbas varies.

  N = |S 1 | + |S 2 | and |.| represent the cardinal of the considered set. In this example, we consider N = 20 and |Ω| = 3. This experiment is 100 times realized and then the results are averaged and represented in Figure 2 with Jousselme's distance.

Figure 2 :

 2 Figure 2: Behavior of conflict when the proportion of conflictual bbas varies. The CWAC operator is obtained with Jousselme's distance.

Table 1 :

 1 Behavior of the different combinations

	bba	∅	{ω 1 }	{ω 2 } {ω 1 , ω 2 }	{ω 3 }	{ω 1 , ω 3 } {ω 2 , ω 3 } Ω
	m 1	0	0.55	0	0	0.40	0	0.05	0
	m 2	0	0.50	0	0	0.30	0	0.2	0
	m 3	0	0.60	0	0	0.25	0.	0.15	0
	m 4	0	0.52	0	0	0.25	0	0.23	0
	m 5	0	0.59	0	0	0.22	0	0.19	0
	m 1,...,5 ∩	0.9317 0.0506	0	0	0.0176	0	0.0001	0
	J m 1,...,5 ↔	0.1118 0.6580	0	0	0.2294	0	0.0009	0
	m 6	0	0	0.5	0	0.30	0	0.2	0
	m 1,...,6 ∩	0.9911 0.0001	0	0	0.0088	0	0	0
	J m 1,...,6 ↔	0.5628 0.0016	0	0	0.4349	0	0.0006	0

The notion of independence is relatively delicate to define. We consider here the idea of cognitive independence[START_REF] Smets | Belief functions: The disjunctive rule of combination and the generalized Bayesian theorem[END_REF].

The function ν i is the disjunctive counterpart of function ω, see[START_REF] Denoeux | The cautious rule of combination for belief functions and some extensions[END_REF][START_REF] Denoeux | Conjunctive and disjunctive combination of belief functions induced by non distinct bodies of evidence[END_REF] for a detailed definition of function ν.

However, the other measures of dissimilarity could be used[START_REF] Cuzzolin | Two new Bayesian approximation of belief functions based on convex geometry[END_REF][START_REF] Cuzzolin | Geometric conditioning of belief functions[END_REF]. Details of the measures of dissimilarity are given in[START_REF] Jousselme | On some properties of distance in evidence theory[END_REF][START_REF] Jousselme | Distances in evidence theory: Comprehensive survey and generalizations[END_REF]. Besides, the use of measures of conflict, as that proposed in[START_REF] Klein | Singular sources mining using evidential conflict analysis[END_REF], can be considered. The objective of this paper is not to compare these approaches but simply to use one of these measures to supply to the decision maker a unique belief function, induced from the combination, containing a part of belief on the revealing empty set of the existing opposition.

According to the wished behavior, the decision maker can use the other approaches (such as the average for example) to synthesize the measures of dissimilarity.

This value is lesser when the distance is high because in that case the generated bbas are more categorical.


Yann Lemeret

Eric Lefevre

Daniel Jolly

Improvement of an association algorithm for obstacle tracking

Keywords: Data Fusion, Evidence Theory, Association, Tracking

This article describes a modication of an association algorithm for object tracking based on the evidence theory. This association algorithm was rst developed by M. Rombaut and subsequently improved in a general way by D. Gruyer. This algorithm has been modied here in order to obtain better results when data reliability is poor. This article presents the basic concepts of the evidence theory. Then, the association algorithm developed by M. Rombaut is explained, and some examples are given to show that this algorithm fails to give the proper decision when data reliability decreases. Finally, the new algorithm is presented and the two algorithms are compared using synthetic data. In order to test the robustness of the two algorithms, they were also tested using real data coming from a CCD camera and these data can be qualied as very noisy with a reliability ranging from good to very bad.

Introduction

This article relates to intelligent vehicles and more precisely to making an algorithm for obstacle detection and tracking. Many European projects on intelligent vehicles are currently in progress, and before explaining the work done, here, a short description will be given of the work achieved in some of these projects.

In the Carsense project [START_REF] Gruyer | Credibilistic multi sensor fusion for real time application. application to obstacle detection and tracking[END_REF], the car is tted with six sensors: two cameras in stereovision, another for lane following, a long-range radar, a short-range radar and a laser. Except for lane-keeping camera, all the sensors are dedicated to obstacle detection in front of the car. The algorithm must be able to detect many objects simultaneously. Data fusion is carried out using the evidence theory and an association algorithm for tracking is described. The aim of this project is to establish, the car environment as reliably as possible.

The aim of the Argo project [START_REF] Broggi | Visual perception of obstacles and vehicles for platooning[END_REF] is to provide a driver assistance system capable of driving the car at the driver's request. The experimental car is tted with just two stereovision cameras and one speedometer. The wheel is motorized to allow automatic drive. Some tests were performed to compare the detection results between a single camera and two in stereovision [START_REF] Bensrhair | A cooperative approach to vision based vehicle detection[END_REF].

The Autonomes Fahren project [START_REF] Stiller | Multi-sensor obstacle detection and tracking[END_REF] is based on a car integrating a DGPS, a digital map, three lasers (one at the rear and two in the front), several shortrange radars on both sides, a long-range radar and a stereovision system. Objects are detected and tracked in a Kalman lter. The car is tted with actuators to make the driving task fully automatic.

Also worth mentioning is the development of the Adaptative Cruise Control (ACC), named Stop and Go ACC [START_REF] Kato | An obstacle detection method by fusion of radar and motion stereo[END_REF]. This system works with a millimeter wave radar and a lane-keeping camera. Its aim is to maintain automatically a safety distance between two vehicles in cruise control mode.

Finally, other research work on intelligent vehicles using stereovision systems can be found in [START_REF] Bertozzi | Vision-based intelligent vehicles : state of the art and perspectives[END_REF].

The project discussed here is a part of is RaViOLi: "Radar, Vision Orientable et Lidar". In this project, three sensors are used: a long-range radar, a stereovision system and a Lidar. The interest of this project is to make a long-range detection system. Given that reliability decreases with increasing in ranging distance, a fusion algorithm for detection and tracking of moving objects has been developed that is still able to work when data reliability is poor.

Taking directly into account the imperfections due to each item of information and each sensor, the algorithm is based on the evidence theory [START_REF] Dempster | Upper and lower probabilities induced by multivalued mapping[END_REF][START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]. The procedure is as follows: rst, belief masses are created for each sensor, with the dierent measurements they provide. Then, the belief functions made from the measurements are combined and then, those from each sensor are also combined. Finally, an association algorithm gives a decision: a new object appears, the object disappears, or the object had already been detected before.

This article is organized as follows: section 2, describes the basic concept of the evidence theory. Section 3 presents the method for creating the mass sets for the dierent sensors and the fusion steps. Section 4, describes the association algorithm and the modications made to improve it when detected objects are a long way ahead of the sensors. Section 5 compares the results obtained with this algorithm using synthetic and real data. Finally, possible future developments of this algorithm are indicated in section 6.

Evidence theory

The evidence theory was initially introduced by Dempster [START_REF] Dempster | Upper and lower probabilities induced by multivalued mapping[END_REF] in relation to his work on lower and upper bounds of a distribution probability family. Using this mathematical formalism, Shafer [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] shows the benets of belief functions for modelling uncertain knowledge. The usefulness of Belief functions, as an alternative to subjective probabilities, was subsequently proved in an axiomatic way by Smets [START_REF] Smets | What is dempster-shafer's model ?[END_REF][START_REF] Smets | The transferable belief model[END_REF] in the Transferable Belief Model thus giving a clear and coherent interpretation of the subjacent concept of this theory. In this section, some mathematical elements of belief functions will be introduced. Not all of the concepts of this theory will be introduced but just those needed to understand this article. Then, an approach will be explained for using the evidence theory in the case of object association.

Basic concepts

The evidence theory is based on the assumption that, from the beginning, a set Ω called the frame of discernment is known and which is dened as follows:

Ω = {ω 1 , ω 2 , . . . ω N } (1)
This set is composed of N exhaustive and exclusive hypotheses. From this frame of discernment, a set noted 2 Ω can be built, including the 2 N proposals A of Ω:

2 Ω = {A/A ⊆ Ω} (2)
A belief function can be mathematically dened by a mass function (or allocation (1) ), noted m dened by 2 Ω in [0, 1], and that veries:

A⊆Ω m(A) = 1 (3)
Each subset A ⊆ Ω such that m(A) = 0 is called a focal element of Ω. Thus, mass m(A) represents the degree of belief allocated to the proposal A and that cannot, in the present state of knowledge, be attributed to a subset more specic than A. The belief function for which m(∅) = 0 is called normal. In the Transferable Belief Model, the condition ∅ =A⊆Ω m(A) = 1 is not imposed and it is possible to have m(∅) = 0. This can introduce the notion of open world while assuming that the belief cannot be attributed to a subset of Ω. In this case, ∅ can be interpreted as a proposal which is not in the frame of discernment Ω and that it is likely to be the solution to the problem as opposed to the closed world where the set Ω is assumed to be exhaustive.

Dual functions

From the belief mass, other belief functions such as plausibility can be dened:

pl(B) = A∩B =0 m(A) ∀B ⊆ Ω (4)
This function can be interpreted as the part of belief that can potentially be allocated to B taking into account the elements that do not discredit this proposal.

Another belief function that is often found in the literature is the credibility function which is obtained as follows:

bel(B) = A⊆B m(A) ∀B ⊆ Ω (5)
This function represents the belief brought to the elements of that proposal. Plausibility and credibility functions are dual measures. Those measures can be viewed as lower and upper bounds of function m. In addition, the dierent functions pl, bel and m represent the same information but expressed in dierent ways. Moreover, they can be translated from one to the other thanks to the Mbius transform [START_REF] Kennes | Uncertainty in Articial Intelligence, chapter Computational aspect of the Möbius transformation[END_REF][START_REF] Kennes | Computational aspects of the möbius transformation of graphs[END_REF].

Discounting

When the information that results in the belief function is not totally reliable, it may be useful to discount this belief. In order to do that, a coecient α is used, which represents knowledge of the source reliability. This coecient will allow the transfer of the belief to the set Ω when the information is not totally reliable. The discounting belief function m α , dened by a reliability coecient α can then be deduced from m by means of the following expression:

m α (A) = αm(A) m α (Ω) = 1 -α + αm(Ω). (6)

Fusion of belief functions

In the evidence theory, data from distinct sources are fused using the rule of Dempster combination, also known as the orthogonal sum. This sum, which is commutative and associative, is dened by:

∀ A ∈ 2 Ω m(A) = m 1 (A) ⊕ . . . ⊕ m Q (A) (7)
where ⊕ represents the combination operator. In cases of two sources noted S i and S j , giving respectively belief functions noted m i et m j , the combination can be written as follows:

m(C) = 1 1 -K A∩B=C m i (A).m j (B) (8)
where K is dened by:

K = A∩B=∅ m i (A).m j (B). (9)
In equation ( 8), the coecient K reects the existing conict between two sources S i and S j . When this factor equals 1, the sources are in total conict and the information cannot be fused. On the other hand, when K equals 0, the two sources totally agree. This fusion rule, deduced from the conditioning rule [START_REF] Smets | The combination of evidence in the transferable belief model[END_REF], has been criticized in several publications such as [START_REF] Smets | The combination of evidence in the transferable belief model[END_REF][START_REF] Zadeh | Fuzzy sets and information granularity[END_REF][START_REF] Yager | On the dempster-shafer framework and new combination rules[END_REF], particularly for the case of two totally conicting sources. In order to avoid this disadvantage, Dubois and Prade [START_REF] Dubois | A set-theoric view of belief functions: Logical operations and approximations by fuzzy sets[END_REF] have dened a disjunctive and conjunctive fusion operator. Finally, another form of combination has been proposed in order to gather all the fusion operators that can be used in the evidence theory [START_REF] Lefevre | Informations et combinaison : les liaisons conictuelles[END_REF][START_REF] Lefevre | Belief function combination and conict management[END_REF].

When comparing it to the probability theory, the evidence theory shows some advantages, the most important of which is the possibility of expressing the degree of uncertainty. In this theory, the mass assignment on a subset does not mean that the remainder automatically goes to the complement.

Measures of uncertainty

It follows from the nature of the evidence theory that it subsumes two distinct types of uncertainty [START_REF] Klir | Measures of Uncertainty in the Dempster-Shafer Theory of Evidence[END_REF]. One of them is called the nonspecicity. This measure is dened by the following formula:

N (m) = A⊂Ω m(A) • log 2 | A | (10)
where | A | denotes the cardinality of the focal element A. This measure, which was proven to be unique under appropriate requirements [START_REF] Ramer | Uniqueness of information measure in the theory of evidence[END_REF], expresses the uncertainty of the belief function m. It ranges from:

0 ≤ N (m) ≤ log 2 | Ω |. N (m) = 0 when m(A) = 1 (A ⊂ Ω and A is a singleton), corresponding to the full certainty. N (m) = log 2 | Ω | when m(Ω) = 1 (total ignorance).
The second type of uncertainty captured by the evidence theory is called the discord. This measure was dened as follows by [START_REF] Klir | Uncertainty in the dempster-shafer theory : a critical re-examination[END_REF]:

D(m) = - A⊂Ω m(A) • log 2 [1 - B⊂Ω m(B) • | B -A | | B | ] (11)
This measure gives the mean conict (expressed by the logarithmic transformation) among evidential claims within each given body of evidence. Note that the following part:

B⊂Ω m(B) • | B -A | | B | (12)
expresses the sum of individual conicts of evidential claims with respect to a particular set A.

After this introduction to the basic concepts of the evidence theory, the next section deals with the creation of masses, adapted to the present application, using information from the dierent sources.

Creation of masses

In an intelligent vehicle, objects are detected by sensors installed in the vehicle. From these sensors used for RaViOLi, the kind of information is known and is expressed in terms of (distance;angle): (ρ; Θ). Moreover, it is considering that using a radar, the speed can be obtained independently of the angle and the distance, by Doppler eect. In addition, the dimensions of the object re also available.

The aim is to track vehicles. Therefore, it must be possible to nd them from one moment to another. All the available information is used to create several mass sets indicating the relations that exist between perceived objects at time t and those known at time t -1. The term X i denote the objects Perceived (i = 1 : N bP ) at time t by the sensors and Y j the Known objects (j = 1 : N bK). The known objects are perceived objects from the previous sample time that were stored in memory.

In a rst stage, The frame of discernment must be dened as a basis for the work. Two hypotheses have been chosen: Ω = {(X i RY j ); (X i RY j )}. Either the perceived object is the same as the known one (in relation:(X i RY j )), or it is not in relation :((X i RY j )) [START_REF] Rombaut | Decision making in data fusion using dempster-shafer's theory[END_REF].

The rst step consists in creating belief functions using the information. A dierent mass set is built for each usable item of information from each sensor.

The mathematical equations of the mass set, that can be used to determine whether objects are in relation or not, is based on a negative exponential introduced by Denoeux [START_REF] Denoeux | A k-nearest neighbour classication rule based on dempstershafer theory[END_REF]:

m i,j (X i RY j ) = α 0 • exp (-e 2 i,j ) m i,j (X i RY j ) = α 0 • (1 -exp (-e 2 i,j ) ) m i,j (Ω i,j ) = 1 -α 0 (13)
where e i,j is the variation between two items of information (distance, angle, or speed) from perceived X i and known Y j objects. α 0 is a coecient that characterizes the sensor reliability. There is then one mass set for distance (Fig. 1) and one for angle for each sensor. There will be another set, based on speed, with the Doppler radar.

The next step consists in combining the distance, angle and speed mass sets for each sensor, and then combining the various sensors. For data fusion, the evidence theory is used with Dempster's conjunctive operator and normalisation.

At the end of those two fusion steps, the mass sets obtained are the number of possible relations between objects. It is now necessary to decide which are the recognized objects, those that appear and those that disappear. This stage is described in the next section.

Object association

When associating objects the decision is taken by means of a fusion algorithm developed by M. Rombaut [START_REF] Rombaut | Decision in multi-obstacle matching process using the theory of belief[END_REF]. This algorithm was improved by D. Gruyer [START_REF] Gruyer | Credibilist multi-sensors fusion for the mapping of dynamic environnement[END_REF].

Mathematical formulation

The algorithm is based on the calculation of certain masses. The relation mass for a given object i to all the objects j is computed. These masses are noted m i,. . The next operation consists in computing the relation masses for each object j to all other i. These masses are noted m .,j . The mathematical formulation adopted for m i,. is:

m i,. (X i RY j ) = K i,. • m i,j (X i RY j ) • N bK k=1 k =j (1 -m i,k (X i RY k )) m i,. (X i R * ) = K i,. • N bK j=1 m i,j (X i RY j ) m i,. (Ω i,. ) = K i,. • [ N bK j=1 (m i,j (Ω i,j ) + m i,j (X i RY j )) - N bK j=1 m i,j (X i RY j )] K i,. = 1 ( N bK j=1 (1-mi,j (XiRYj )))•(1+ N bK j=1 ( m i,j (X i RY j ) 1-m i,j (X i RY j ) )) (14)
Then, m .,j can be deduced:

m .,j (Y j RX i ) = K .,j • m i,j (X i RY j ) • N bP k=1 k =i (1 -m k,j (X k RY j )) m .,j (Y j R * ) = K .,j • N bP i=1 m i,j (X i RY j ) m .,j (Ω .,j ) = K .,j • [ N bP i=1 (m i,j (Ω i,j ) + m i,j (X i RY j )) - N bP i=1 m i,j (X i RY j )] K .,j = 1 ( N bP i=1 (1-mi,j (XiRYj )))•(1+ N bP i=1 ( m i,j (X i RY j ) 1-m i,j (X i RY j ) )) (15
) Note that the non relation masses are not computed. However, this information is not necessary in order to make the decision: deciding that X i is not in relation with Y j does not give any information about the other associations.

Two new hypotheses m i,. (X i R * ) and m .,j (Y j R * ) are introduced. In these hypotheses, * represents the fact that one object is in relation with none of the others. Thus, if a perceived object X i is not associated, it is a new object and if a known object is not associated, it has disappeared (masked by another object, out of range, ...). The decision is taken according to the two sets of masses m i,. and m .,j . The couples that have the maximum credibility in the two mass set are chosen.

New formulation

The rst remark that can be made about these formulae concerns the framework. At the beginning, when the masses are built, Ω i,j = {(X i RY j ); (X i RY j )}. With the association formulae, it can be seen that Ω changes from a closed world to an extended open world but also that the hypotheses are not the same too:

Ω i,. = {(X i RY j ); (X i R * )}.
Then, it has been found that in some situations, they did not work well. Indeed, the masses m i,j (X i RY j ), are added on m i,. (Ω i,. ) and m .,j (Ω .,j ). While this is not important for the decision because deciding that X i is not in relation with Y j does not give any decision, the manner in which the formulae are established adds these masses to Ω and can lead to the choice of a bad decision. For example with one perceived and two known objects the following mass sets are obtained:

m 1,1 (X 1 RY 1 ) = 0.2 m 1,2 (X 1 RY 2 ) = 0.45 m 1,1 (X 1 RY 1 ) = 0.45 m 1,2 (X 1 RY 2 ) = 0.15 m 1,1 (Ω 1,1 ) = 0.35 m 1,2 (Ω 1,2 ) = 0.4
Therefore:

M i,. cr = X 1 Y 1 0.121 Y 2 0.396 * 0.073 Ω 1,.
0.41

M .,j cr = X 1 * Ω .,j Y 1
0.2 0.45 0.35 Y 2 0.45 0.15 0.4

From the rst matrix (m i,. ), it can be deduced that the relation on X 1 is unknown: m 1,. (Ω 1,. ) = 0.41 while seeing that m 1,. (X 1 RY 2 ) = 0.396. From the second matrix, it can be decided that Y 1 disappears and that X 1 is in relation with Y 2 . So there is a problem with deciding on the couples. To solve this problem, it was decided to modify the equations for computing the masses m i,. and m .,j . This modication consists in dening a unique framework:

Ω i,. = {(X i RY 0 ); . . . ; (X i RY N bK ); (X i R * )} and noting (X i RY j ) = [ N bK k=1 k =j (X i RY k ) ] ∪ (X i R * ).
Thus, the masses are created as follows:

m i,j (X i RY j ) = α 0 • exp (-e 2 i,j ) m i,j (X i RY j ) = α 0 • (1 -exp (-e 2 i,j ) ) m i,j (Ω i,. ) = 1 -α 0 (16)
The masses of relation (X i RY j ) are increased using the mass on star combined with the complementary masses (X i RY j ) that do not contradict the rela-tion hypothesis. The new m i,. values are dened thus:

m i,. (X i RY j ) = K i,. • [m i,j (X i RY j ) • N bK k=1 k =j (1 -m i,k (X i RY k )) + N bK k=1 k =j 1 N bK • m i,k (X i RY k )• N bK p=1 p =k m i,p (Ω i,. ) + N bK-1 k=1 k =j N bK l>k l =j 1 N bK-1 • m i,k (X i RY k ) • m i,l (X i RY l )• N bK p=1 p =k,p =l m i,p (Ω i,. ) + • • • + 1 2 • m i,j (Ω i,. ) • N bK p=1 p =j m i,p (X i RY p )] m i,. (X i R * ) = K i,. • [ N bK j=1 m i,j (X i RY j ) + N bK k=1 1 N bK • m i,k (X i RY k )• N bK p=1 p =k m i,p (Ω i,. ) + N bK-1 k=1 N bK l>k 1 N bK-1 • m i,k (X i RY k ) • m i,l (X i RY l )• N bK p=1 p =k,p =l m i,p (Ω i,. ) + • • • + N bK j=1 1 2 • m i,j (Ω i,. ) • N bK p=1 p =j m i,p (X i RY p )] m i,. (Ω i,. ) = K i,. • N bK j=1 m i,j (Ω i,.
)

K i,. = 1 ( N bK j=1 (1-mi,j (XiRYj )))•(1+ N bK j=1 ( m i,j (X i RY j ) 1-m i,j (X i RY j ) )) (17)
and the m .,j values:

m .,j (Y j RX i ) = K .,j • [m i,j (X i RY j ) • N bP k=1 k =i (1 -m k,j (X k RY j )) + N bP k=1 k =i 1 N bP • m k,j (X k RY j )• N bP p=1 p =k m p,j (Ω .,j ) + N bP -1 k=1 k =i N bP l>k l =i 1 N bP -1 • m k,j (X k RY j ) • m l,j (X l RY j )• N bP p=1 p =k,p =l m p,j (Ω .,j ) + • • • + 1 2 • m i,j (Ω .,j ) • N bP p=1 p =i m p,j (X p RY j )] m .,j (Y j R * ) = K .,j • [ N bP i=1 m i,j (X i RY j ) + N bP k=1 1 N bP • m k,j (X k RY j )• N bP p=1 p =k m p,j (Ω .,j ) + N bP -1 k=1 N bP l>k 1 N bP -1 • m k,j (X k RY j ) • m l,j (X l RY j )• N bP p=1 p =k,p =l m p,j (Ω .,j ) + • • • + N bP i=1 1 2 • m i,j (Ω .,j ) • N bP p=1 p =i m p,j (X p RY j )] m .,j (Ω .,j ) = K .,j • N bP i=1 m i,j (Ω .,j ) K .,j = 1 ( N bP i=1 (1-mi,j (XiRYj )))•(1+ N bP i=1 ( m i,j (X i RY j ) 1-m i,j (X i RY j ) )) (18)
In the terms of relation, coecients appear: ( 1 N bK , . . . , 1 2 ) for m i,. and ( 1 N bP , . . . , 1 2 ) for m .,j . These correspond to an equal distribution of the mass when there are one or more masses on Ω multiplied with the non relation masses. Indeed, with one perceived and two known objects, if m 1,2 (X

1 RY 2 ) = (m 1,1 (X 1 RY 1 ) ∪ m 1 (X 1 R * )) is multiplied by m 1,1 (Ω 1,.
), this means that it is certain that X 1 is not in relation with Y 2 . So as the framework is exhaustive and closed, the result goes on m 1,. (X 1 RY 1 ) and m 1,. (X 1 R * ) (multiplied by 1 2 ). In the case of three known objects, if there are two masses on Ω in the product

(m 1,1 (X 1 RY 1 ) • m 1,2 (Ω 1,. ) • m 1,3 (Ω 1,.
)), it is not possible to decide where the mass should go, so 1 3 goes to the rst relation (m 1,. (X 1 RY 2 )), 1 3 to the second (m 1,. (X 1 RY 3 )) and 1 3 to the mass on star (m 1,. (X 1 R * )), and so on until divided by N bK (or N bP ). Considering the example again, this gives:

M i,. = X 1 Y 1 0.15 Y 2 0.495 * 0.201 Ω 1,. 0.154 M .,j = X 1 * Ω .,j Y 1 0.2 0.45 0.35 Y 2 0.45 0.15 0.4
This time, there is no ambiguity about the best decision which is: (X 1 RY 2 ) and Y 1 disappears. The next section compares the two formulae using synthetic and real data.

Experimental results

The aim of this section is to validate the association formulae made using results obtained from simulated or real data. The rst part illustrates the use of the formulae (2) with synthetic data and a limited uncertainty, and then with a strong uncertainty. The second part shows the results obtained with real data.

5.1

Synthetic data

With limited uncertainty

In this section, the data were articially created and the association matrices are not shown. The relevant masses are displayed in order to be able to decide the association as the results are known in advance. The rst step was to dene a trajectory for the objects (Fig. 2). The objects move for 10s, with a sample time of 0.1s, according to the following equations (which introduce strong constraints while remaining realistic): From these data, the distances between perceived and known objects (Section 3) are computed at each sample time, so as to plot the variation of the relation masses between those objects (Fig. 3). As the relative distances are not very high, a reliability α 0 = 0.8 was chosen for the sensor 0 and α 1 = 0.8 for sensor 1. As shown in this gure, the objects are well separated until the moment object 1 approaches object 0 (5.8s < t < 6.8s). During this time, m 0,1 (X 0 RY 1 ) falls whereas m 0,1 (X 0 RY 1 ) increases. However, m 0,1 (X 0 RY 1 ) is still higher than m 0,1 (X 0 RY 1 ).

Object 0 → d = 1 2 • 0.1 • t 2 + 5 • t + 19.3 Object 1 → d = 45 1+exp (-2•t+10) + 10 (19)
m 0,1 (X 0 R Y 1 ) m 0,1 (X 0 R Y 1 ) m 0,1 (Ω 0,1 ) m 0,0 (X 0 R Y 0 ) m 0,0 (X 0 R Y 0 ) m 0,0 (Ω 0,0 )
Figure 3: Variation in relation masses between X 0 and Y 0 , X 0 and Y 1 .

Figure 4 shows the variation of the association masses using M. Rombaut's formulae. These formulae give good results in this case. In fact, they work well in most cases.

Note that on gure 5 the changes have made to these formulae also give good results. A peak can be seen on m 0,. (Ω 0,. ) (Fig. 5). This variation of mass does not come from the m i,j (Ω i,j ) (because these masses are constant with time (Fig. 3)) but from the variation of the conict coecient.

The next part shows the advantage of the modied formulae in the case of low reliability.

With high uncertainty

This is a particular case (Fig. 6) where the temporal equations are:

Object 0 → d = 1 2 • 0.1 • t 2 + 5 • t + 59.3 Object 1 → d = 45 1+exp (-2•t+10) + 50 (20)
For this simulation, distances were greater than 50m, α 0 = 0.6 and α 1 = 0.7 were xed. The variation of relation masses is represented on gure 7.

Figure 8 shows the evolution of association masses using M. Rombaut's formulae. Normally, it should be decided that X 0 is in relation with Y 0 . But in this case, it can be seen on the curves that till the beginning the decision of

m 0,1 (X 0 R Y 1 ) m 0,1 (X 0 R Y 1 ) m 0,1 (Ω 0,1 ) m 0,0 (X 0 R Y 0 ) m 0,0 (X 0 R Y 0 )
m 0,0 (Ω 0,0 ) association is debatable. Indeed, the relation mass m 0,. (X 0 RY 0 ) is only slightly higher than m 0,. (Ω 0,. ) for t < 5.8s and becomes slightly lower when t > 5.8s. Now, looking at the variation in masses obtained with the modied formulae (Fig. 9), it can be seen that the decision is not ambiguous: X 0 is in relation with Y 0 , and this decision is not debatable at any time even when the two objects are very close.

Variation in relation masses according to reliability

For this test, two objects are considered side by side with the same speed and 1m apart. Those two vehicles move away progressively from the sensor (from 20 to 100m with a constant speed of 5m.s -1 ) which leads to a reduction in the reliability coecient (from 0.9 at 20m to 0.3 at 100m). Figure 10 represents the evolution of the decision reliability according to the sensor reliability for M. Rombaut's algorithm and for the modied one. The decision is not considered to be completely reliable when the variation between the mass of a good decision and another one is lower than a threshold which have been set to 0.1 in this case.

When the reliability is greater than 0.69, the two algorithms give the same results. However, the modied algorithm is able to give a reliable decision down to a data reliability value of 0.49. For Rombaut's algorithm, the decision is not reliable below 0.69 and is false if the data reliability value falls below 0.62.

Tests were conducted with a large number of threshold values, always resulting in the same conclusion: the modied algorithm is better when the reliability decreases. When real tests could be made, it would be interesting to plot the percentage of good decisions of the two algorithms with various data reliability values from several scenarios. Figure 11 shows the variation in non specicity according to data reliability. This gure shows that the modied formulae give more specic mass sets, so there is less uncertainty on the best decision. These results prove that the modied formulae presented lead to a better decision.

Real data

The association algorithm was initially tested by using a simulator developed by the authors [START_REF] Lemeret | Simulator of obstacle detection and tracking[END_REF]. This simulator provided the only data that could be used. It was able to create various scenarios thanks to adjustable parameters such as: number and speed of vehicles, number of sensors, sweeping time, angle, range, etc. The driver's behaviour was governed by the driving law. The problem with this simulator was that data were synthetic, so not representative of the driver's behaviour and not noisy. In order to improve the association algorithm and to test its robustness, it was decided to use real data. These data were supplied by a simple DV camera placed behind the windshield of a car. This DV camera has a CCD sensor. The resolution is 720x576 pixels, the angle ranges from -0.5 to +0.5 radians (almost ±30 • ) and works at 25 images per second (∆ t = 0.04s). For calibration purposes, a sequence was lm in a car park where a reference car was parked. A pocket laser telemeter was used to measure the distance between the camera and the car. The measurements were taken from 0 to 60m, because for greater distances, it was dicult to point the laser on the car. From the calibration sequence, while assuming that detected objects are cars, two equations were computed by two dierent interpolation methods (Fig. 12) giving the distance according to object height and width in the image. The angle is extracted from the centre of gravity position in the image. In this way, information (ρ, Θ), for two dierent sensors is obtained (3) .

Figure 12: Characterization function of the two cameras. The distance in meter is obtained from the height and the width in pixels of the object.

Without being specialists in image processing, the measurements extracted were very noisy. Indeed, at 70m, a variation of 20m was possible for the same object from an image to the next. In terms of angle, the variation could be as high as 0.02rd for a measurement of 0.01rd.

Object association

Figure 13 shows the variation in the relation mass for one perceived object. This object is followed for 300 images (12s) and is located almost 20m from the sensor. The mass of relation varies between 0.57 and 1, but is close to 1 most of the time. A value of 0.57 was obtained because, at this time, there is a measurement error of 14.6% on the distance and a simultaneous 8.2% error on the angle. The entire sequence consists of 3230 images (almost 2min10s) and, during this time, 7063 vehicles were perceived (between 0 and 6 per image). Figure 14 shows the variation in mass on * for the entire simulation using both formulae.

In the scenario presented (Fig. 14) it can be seen that there is many appearance and disappearance of objects (763 during the 3230 images). Note that the masses on star with the new formulae are equal to or higher than the one with Rombaut's formulae. The mass of relation (Fig. 13) is plotted between images 450 and 750. Figure 14 shows that during this time there was just one object and it is always tracked (no appearance or disappearance).

Conclusion and future work

This article has presented an improvement to an association algorithm. This algorithm is used to track moving vehicles. Hypotheses to be computed in the association stage have been added in order to reduce the number of ambiguous decisions. This algorithm was tested on synthetic and real noisy data, the maximum variation in distance was about 25% and 200% for the angle from one image to the next. Better results are obtained with the modied formulae especially when the reliability is poor. The number of non-decisions is reduced and the choice of right decision is made easier while increasing the dierence between the mass for the best hypothesis and the others. Improvements could still be made to the detection and tracking of objects by adding a prediction lter to this algorithm. It will thus be possible to predict future associations. This information could then be used to conrm the associations made or reduce the ambiguous decisions on some associations.

Figure 1 :

 1 Figure 1: Association between known and perceived objects using distance information.

Figure 2 :

 2 Figure 2: Variation in object distance from the sensor, for 10s.

Figure 4 :

 4 Figure 4: Variation in m 0,. using Rombaut's formulae.

Figure 5 :

 5 Figure 5: Variation in m 0,. using the modied formulae.

Figure 6 :

 6 Figure 6: Variation in object distance, from the sensor, for 10s.

Figure 7 :

 7 Figure 7: Variation in relation masses between X 0 and Y 0 , X 0 and Y 1 .

Figure 8 :

 8 Figure 8: Variation in m 0,. using Rombaut's formulae.

Figure 9 :

 9 Figure 9: Variation in m 0,. using the modied formulae.

Figure 10 :

 10 Figure 10: Variation in the decision reliability according to data reliability.

Figure 11 :

 11 Figure 11: Variation in the non specicity according to reliability.

Figure 13 :

 13 Figure 13: Variation in the mass of detection for one perceived object.

Figure 14 :

 14 Figure 14: Variation in mass on * during the entire sequence.

The term bba for basic belief assignment is often found in the literature

In order to make the computational stage easier, in this part, i ranges from 0 to N bP -1, and j from 0 to N bK -1

The fact of using two dierent interpolation methods leads to two sources of information. So it can be said that there are two sensors.


S Klemme

M Feldhaus

V Potapkin

M Wilke

M Borchert

M Louvel

A Loges

A Rohrbach

P Weitkamp

E Welter

M Kokh

C Schmidt

D Testemale

A hydrothermal apparatus for X-ray absorption spectroscopy of hydrothermal fluids at DESY

Keywords: high pressure, autoclave, synchrotron, DESY, XAS, X-ray absorption spectroscopy, hydrothermal, ore deposit, hydrothermal fluid, H 2 O, Raman Spectroscopy, Centre for Molecular Water Science

We present a new autoclave, which enables in-situ characterization of hydrothermal fluids at high pressures and high temperatures at synchrotron X-ray radiation sources. The autoclave has been specifically designed to enable X-ray absorption spectroscopy in fluids with applications to mineral solubility and element speciation analysis in hydrothermal fluids in complex compositions.

However, other applications such as Raman spectroscopy in high-pressure fluids are also possible with the autoclave. Preliminary results on scheelite dissolution in fluids of different compositions show that the autoclave is well suited to study the behavior of ore-forming metals at P-T conditions relevant to the Earth's crust.

Introduction

Most ore deposits in the Earth's crust formed via precipitation of metal-rich minerals from hydrothermal fluids. Fluid inclusions in minerals may provide mainly qualitative information about the chemical composition of these fluids (e.g., [1][2][3] ), but fluid inclusion studies do not provide much information about the speciation or complexation of metals or other elements in these hydrous fluids, especially as fluids display often (but not always) large changes upon cooling [4][5][6] . Hence, non-quenchable properties such as the speciation of elements in hydrothermal fluids are best studied experimentally in-situ at high-pressures and high temperatures.

Previously, hydrothermal diamond anvil cells (HDAC) have been used to study fluids at high pressures and high temperatures [7][START_REF] Schmidt | Raman spectroscopy applied to Earth sciences and cultural heritage[END_REF][START_REF] Anderson | [END_REF][10][11][12][13] . However, the application of diamond anvil cells has several limitations: The diamonds are relatively thick which causes significant attenuation of X-rays 14,15 , the sample volume is very small, usually about 0.005-0.1 mm 3 , and the cell design does not permit experiments along an isobar. Furthermore, the pressure in HDAC experiments cannot be accurately determined and precisely controlled 11,16 . Therefore, experiments at synchrotron radiation sources with HDAC equipment must be done by specialists, and this has limited the accessibility and the productivity of such equipment. Other designs of hydrothermal cells permit only the study of dilute, simple, and non-aggressive fluids that do not attack the walls of the autoclave and the high-pressure (HP) windows (e.g., 17 ). A more recently developed hydrothermal autoclave design has enabled routine spectroscopic studies of fluids at high pressures and temperatures, including aggressive Cl-, F-or S-rich fluids 18 .

Here we present a new version of this autoclave, which was tested at DESY in Hamburg. It has been installed at the applied X-ray absorption beamline P65 [START_REF] Welter | AIP Conf Proc[END_REF] and can also be used at the advanced X-ray absorption beamline P64 [START_REF] Welter | AIP Conf Proc[END_REF] . The main aim was to design an autoclave system that is safe and easy to use, and enables experimentation with highly reactive high-temperature high-pressure fluids. This equipment will be an essential part of the new "Centre for Molecular Water Science" initiative at DESY, which focuses on the physics and chemistry of aqueous fluids.

This new autoclave at DESY will complement the existing experimental high-pressure equipment at DESY (e.g., [START_REF] Liermann | [END_REF]21 ), which now encompasses the entire pressure-temperature regimes relevant to the Earth's crust, the mantle, and core.

The hydrothermal apparatus at DESY

Autoclave

The design of the pressure vessel follows Testemale's original design 18 , but some significant adjustments were made to improve the ease of experimentation with the apparatus. The autoclave is designed so that it may be used both for X-ray and infrared-Raman spectroscopy.

The body of the autoclave is made from high-performance steels, which is surrounded by a cooling jacket. The autoclave is closed by a high tensile steel nut, which contains feedthroughs for thermocouple wires and electrical supply wires for the heater. The autoclave is pressurized by He-gas, and has one gas inlet and one outlet. The gas is supplied to the autoclave using a Nova-Swiss (Nova Werke, Effretikon, Switzerland) high performance membrane gas compressor.

Pressure is monitored and controlled using a commercial pressure control unit (Maximator, France).

The autoclave has three windows (Fig. 1) to permit measurements in transmission but also measurements of the fluorescence signal (90° to the incoming synchrotron X-ray beam). The window materials used for X-ray spectroscopy are made of vitreous carbon (Sigradur, HTW Hochtemperatur-Werkstoffe GmbH, Germany), as initial tests with Be-windows showed that the Be itself contained significant amounts of trace elements (Fe, Cu, Ni, Y, W, Ge, REE), often at levels much higher than 50 µg/g each 22 . Windows for IR-Raman spectroscopy are made from commercially available sapphire windows (Thorlabs, USA). The chemical composition of the vitreous carbon was checked by electron microprobe and laser ablation-inductively coupled plasma mass spectrometry (LA-ICPMS) at the University of Münster, Germany. Both analyses did not give quantitative results as standard materials or internal reference elements were not available. However, a wavelength dispersive spectroscopy (WDS) scan over the entire signal band with the electron microprobe and qualitative measurements with LA-ICPMS indicate remarkable low count rates of the aforementioned trace elements which indicates concentrations below the µg/g level.

The internal sample cell for X-ray spectroscopy is also made from vitreous carbon (Sigradur, HTW Hochtemperatur-Werkstoffe GmbH, Germany), with 4 mm internal diameter (I.D.) and 10 mm outer diameter (O.D.). The sample, which in most cases will consist of a liquid together with solids, is placed within the vitreous carbon tube (see Fig 2 ), and it is contained within the tube by two polished vitreous carbon pistons that are sealed with viton O-rings. These pistons can move freely to account for volume changes with increasing or decreasing pressure. In order to minimize absorption of the internal sample cell, a small recess is machined in the center of vitreous carbon tube resulting in a 6 mm O.D. at measurement position (Fig. 2). This is an easy task given the ease and safeness of machining vitreous carbon. The internal sample cell (i.e. the carbon tube with the sample) is heated with resistive wire heaters, made either from stainless steel or W-wire of 0.1 mm thickness. The sample cell and the heater are surrounded with concentric rings of alumina-based ceramics which reduce the gas volume and thermal gradients within the autoclave. Two thermocouples are used to monitor and control temperatures during the experiments. The thermal gradient from the top of the heater to the center of the sample cell is ca. 35K. The sample itself, which consist mainly of hydrous fluids, is expected to convect along the thermal gradient so that the entire fluid is held at constant temperature. The temperature at the measurement position is determined by calibration runs with pure water for a given assembly and pressure. This calibration is based on the density by the equation of state of water 23 and the measured absorption of X-rays at a given energy, using the absorption cross-section of the Elam database 24 . Pressures are automatically measured and controlled using a custom-built pressure regulation system, based on the design of highperformance high-pressure valves 25 . Temperatures are controlled using conventional Eurotherm (Schneider Electric Systems GmbH, Germany) temperature controllers.

Possible applications of the autoclave and first test measurements

The new autoclave at DESY may be used for studies of mineral solubilities in hydrothermal fluids, and for the speciation analysis of metals and other elements in fluids at high pressures and high temperatures. It is especially suited to study mineral solubilities and metal speciation in highly aggressive, chemically reactive fluids with significant amounts of Cl, S, and F, all of which are important ligands in natural ore forming fluids in the Earth's crust [26][27][28][29] .

The pressures and temperatures that can be reached in the autoclave critically depend on the quality of the seals and the window materials. So far, we have run experiments at P<600 bar and T<550°C.

We conducted first test experiments at beamline P65 of Petra III at DESY in Hamburg in transition mode only. In this particular project we aimed to study the solubility of ore metals (e.g., W, Zr, Hf, REE) in hydrothermal fluids and the speciation of these metals in fluids of near-natural compositions. Spectra were recorded in fluids with dissolving scheelite (CaWO 4 ). The element

Cooling water

He gas inlet

content in the fluid is determined using height of the edge-jump in X-ray absorption spectra (e.g. 30 ). Fig. 5A shows the height of the edge-jump at W L3-edge for a single spectrum of each experiment. In the chosen representation, the edge jump is hardly visible in the run with scheelite in pure H 2 O and demonstrates that the measurements resulted in W concentrations of 7*10 -5 mol W / kg H 2 O, which is close to the detection limit that is estimated to be between 1 and 10 µg/g for W. The data show also that the concentration of W in the fluid increases with increasing HCl concentration of the solution to a maximum value of 1.47*10 -2 mol W / kg H 2 O. Tungsten concentrations calculated do only represent concentrations at a given time. Determination of tungsten solubility in these runs is not possible due to incongruent dissolution of scheelite in some of the experiments. the quality of the signal at k > 12 is not sufficient for further analysis. The spectra reflect the wellknown fact that the W-species in solution strongly depend on the pH of the solution (e.g. 31,32 ). A thorough analysis is beyond the scope of this publication and will be provided elsewhere. The results show clearly that the autoclave is well suited to study solubilities and dissolution kinetics of ore forming minerals in fluids of complex compositions. Furthermore, the autoclave can also be used to produce high-quality EXAFS spectra that allow investigating the speciation of these trace elements in fluids at high pressures and high temperatures.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Center:The autoclave (without cooling jacket) fully assembled with top nut, heater, and openings for windows. The autoclave is 46 cm high and 15 cm in diameter and has three windows in the lower third, two in the direction of the X-ray beam to allow transmission measurements, and one at 90° to the incoming X-ray beam to collect the fluorescence signal. Left: B-B; horizontal section through the autoclave at the position of the three windows. Right: Vertical cross section (A-A), together with the top nut, heater, windows, and sample cell. The sample container is depicted as a black tube, together with pistons (for clarity only depicted in grey), all made from vitreous carbon. D: steel body of the autoclave. E: Window seats, C: Gas inlet (supplied from the He-gas compressor and the pressure control unit.

Fig 3 .Fig. 4

 34 Fig 3. Window seats for vitreous carbon windows (right side) and polished sapphire single crystal windows (left side). The windows are mounted in the seats using ceramic or metal guide rings, and O-rings (Viton, DuPont, USA). The carbon windows are used for X-ray spectroscopy, whereas the sapphire windows may be used for IR-Raman spectroscopy.

Fig. 5B shows

  Fig. 5B shows normalized XANES spectra of the same experiments. Numbers given in brackets correspond to the number of single spectra that were collected and summed up. For comparison, normalized XANES spectra of H 2 O and a 0.05 mol/L HCl solutions containing 0.03 mol/L Na 2 WO 4 are presented. In contrast to scheelite, Na 2 WO 4 is highly soluble in aqueous fluids. Thus, high quality XAFS spectra are easily obtained. The comparison between the two kinds of experiments demonstrates the quality of our measurements and highlights the fundamental difference of spectra measured in HCl-solutions and H 2 O in the energy region between 10210 and 10250 eV, which is related to the fundamental difference of W-species. In case of very dilute samples, e.g., in runs on scheelite dissolution in H 2 O, averaging of consecutive measured single spectra results in a reasonably acceptable XANES spectrum. However, for a better spectral quality in such cases and measurement of the EXAFS region measurements need to be taken in fluorescence mode. EXAFS spectra measured in some of the samples are shown in Fig. 6A. A Spectrum taken of a 0.03 mol/L Na 2 WO 4 -solution in H 2 O at 100 bar at beamline P65 (DESY) is compared to one of an identical solution measured at the FAME beamline (ESRF) at 600 bar. For both experiments, Fourier transformed EXAFS spectra (Fig. 6B) show only a single contribution from the first oxygen coordination shell in a very similar manner. Both beamlines and setups allow measuring extended EXAFS spectra up to high k-values. The spectrum of 0.03 mol/L Na 2 WO 4 -solution in hydrochloric acid looks fundamentally different and indicates a more complex first oxygen coordination shell and contribution by W-W scattering paths in the second coordination shell. The spectrum taken on the scheelite dissolution in HCl-solutions resembles the former one to a large extent, although

Fig. 5 .

 5 Fig.5. A: Results from scheelite (CaWO 4 ) dissolution experiments in various fluid compositions at 100°C and 400 bar. W L 3 -edge XAFS transmission spectra were collected at P65 at DESY using continuous scan mode. Counting time per single spectrum is 180 s. The left figure shows single raw XAFS spectra. Tungsten aqueous concentration is calculated using the edge height jump of the X-ray absorption spectra as described in30 . The diagram on the right (B) shows normalized XANES spectra of the same experiments. Numbers given in brackets correspond to the number of single spectra that were collected and summed up. In addition, normalized XAFS spectra on solutions containing 0.03 mol/L Na 2 WO 4 in H 2 O or 0.05 mol/L HCl are presented.

Fig 6 :

 6 Fig 6: A: EXAFS 𝝌(k)×k3 oscillations extracted from transmission spectra of solutions containing 0.03 mol/L Na 2 WO 4 in H 2 O or 0.05 mol/L HCl at 100°C and 100 or 600 bar and from a scheelite dissolution experiment in 0.1 mol/L HCl at 100°C and 400 bar. Two data sets on Na 2 WO 4 in H 2 O are presented in order to compare results from P65 (DESY) and FAME (ESRF). Data collected at P65 were measured in continuous scan mode while at FAME a step-scan mode was used. This is also reflected in much longer acquisition time per spectrum at FAME (ca. 36 minutes per spectrum) compared to DESY. B: Fourier transforms of the W L3-edge EXAFS spectra for the same solutions.

Acknowledgements

We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities and support within the new Center for Molecular Water Science (CMWS). Parts of this research were carried out at DESY beamline P65 and we would like to thank W. Caliebe (P64), M. Hermann and R. Nemausat, for their excellent support, and A.

Ciabano for her generous help and sterling assistance in the chemical laboratories at DESY. Parts of this research were also carried out at FAME at ESRF, France and our sincere thanks go to the FAME staff for excellent support. Further thanks go to H.-P. Liermann for his general support of the project within the CMWS at DESY, and J. Berndt and B. Schmitte for their help with the EPMA and LA-ICPMS measurements at WWU Münster.

Funding

Bundesministerium für Bildung und Forschung (BMBF) grant FKZ 05K16PMA, DESY Center for Molecular Water Science -Early Science Project. A. Loges was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project-ID 387284271 -SFB 1349

Data availability statement

Data available on request from the authors

Author's contributions

All authors contributed equally to the manuscript


Catherine Calzada

M M Boulet

Magali Pettazzoni

Mathilde Di Filippo

Philippe Moulin

D Cheillan

Marie-Caroline Michalski

Postprandial

Postprandial triglyceride-rich lipoproteins from type 2 diabetic women activate platelets

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Postprandial triglyceride-rich lipoproteins from type 2 diabetic women activate platelets Calzada C. 1 , Boulet M.M. 1 , Pettazzoni M. 2 , Di Filippo M. 1,2 , Moulin P. 1,3 , Cheillan D. Increased concentrations of fasting and post-prandial (PP) triglycerides (TG), carried out in the circulation by triglyceride-rich lipoproteins (TGRL), are a risk factor for atherothrombosis in patients with type 2 diabetes (T2D). The objective of our clinical study was to determine the effects of fasting and PP TGRL from T2D patients on platelet activation while assessing the fatty acid (FA) and sphingolipid (SL) composition of TGRL. Thirty women with T2D were randomly assigned a breakfast containing 20 g lipids from a hazelnut-cocoa spread rich in palm oil (Palm Nut) versus butter. Both breakfasts were rich in palmitic acid, and the Palm Nut breakfast also contained 45% oleic acid. Blood samples were collected at fasting and 4h after the breakfast, and TGRL were isolated from the plasma by ultra-centrifugation at a density equivalent to that of chylomicrons (d<1.000). TGRL FA and SL composition were analyzed by gas chromatography and tandem mass spectrometry, respectively. Both breakfasts similarly increased plasma apoB-48 (Palm Nut: +78%, butter: +67%), plasma TG (Palm Nut: +33%, butter: +30%) and TGRL TG concentrations (Palm Nut: 200% and butter: 71%).

TGRL mean diameter (146 ± 5 nm) increased after both breakfasts and was greater after the butter breakfast. The ingestion of the Palm Nut breakfast mainly resulted in increased proportions of oleic acid in TGRL while butter led to increased proportions of palmitic acid. Compared to fasting TGRL, the ingestion of the meals enriched in saturated FA led to increased TGRL total ceramide concentrations, especially C16:0, C24:1 and C24:0 molecular species. Pre-incubation of platelets from healthy donors with fasting and PP TGRL increased collagen-stimulated aggregation. Fasting and PP TGRL similarly increased agonist-induced thromboxane B 2 concentrations. In conclusion, PP TGRL from T2D women after a palm-oil spread or butter-based mixed meal induced similar acute in vitro platelet activation.

  Laboratoire de Biochimie et de Biologie Moléculaire Grand Est, Centre de Biologie et de Pathologie Est, Hospices Civils de Lyon, 69677 Bron.

	1,2 ,
	Michalski M.-C. 1
	1 Univ-Lyon, CarMeN Laboratory, Inserm U.1060, INRAE U.1397, INSA Lyon,
	Université Claude Bernard Lyon 1, 69310 Pierre-Bénite.
	3 Fédération d'endocrinologie, Maladies Métaboliques, Diabète et Nutrition, Hôpital
	Louis Pradel, Hospices Civils de Lyon, 69677 Bron.

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