Adrien Ickowicz

Track estimation with binary derivative observations

Keywords: parameter estimation, binary sensors, maximum likelihood, single index model, tracking algorithm

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Track estimation with binary derivative observations

I. INTRODUCTION

Sensor networks are systems made of many small and simple sensors deployed over an area in an attempt to sense events of interest within that particular area. In general, the sensors have limited capacities in terms of say range, precision, etc. The ultimate information level for a sensor is a binary one, referring to its output. However, it is important to make a distinction according to the nature of this binary information. Actually, it can be related to a 0 -1 information (non-detection or detection) or to relative {-, +} motion information. For example, if the sensors are getting sound levels, instead of using the real sound level (which may cause confusion between loud near objects and quieter close objects), the sensor may simply report whether the Doppler frequency is suddenly changing, which can be easily translated in whether the target is getting closer or moving away. Moreover, low-power sensors with limited computation and communication capabilities can only perform binary detection. We could also cite video sensors, with the intuitive reasoning: the target is getting closer if its size is increasing. The need to use that kind of sensor networks leads to the development of a model for target tracking in binary sensor networks.

We consider a sensor network, made with N sensors (e.g. video),with (known) positions. Each sensor can only gives us a binary {-, +} information [START_REF] Aslam | Tracking a moving object with a binary sensor network[END_REF], i.e. whether the target-sensor distance is decreasing (-) or increasing (+). This "choice" can result from severe communication requirements or from the difficulties from fusing inhomogeneous data. Even if many important works deal with proximity sensors [START_REF] Wang | Multitarget Tracking Using Virtual Measurements of Binary Sensor Networks[END_REF], [START_REF] Lazos | Probabilistic Detection of Mobile Targets in Heterogeneous Sensor Networks[END_REF], we decide here to focus on the binary {-, +} information [START_REF] Aslam | Tracking a moving object with a binary sensor network[END_REF]. Here, the aim is to estimate the parameters defining the target trajectory. Even if our methods can be rather easily extended to more complex models of target motion, we decide to focus here on a constant velocity movement. Actually, this framework is sufficiently general to present the main problems we have to face, as well as the foundations of the methods we have to develop for dealing with these binary data. See fig. I for an example. In a first time, the observability requirements are considered. Then, we turn toward the development of specific estimation Fig. 1. A view of a getting closer car methods. Especially, the new concept of the velocity plane is introduced as an exhaustive representation of the spatio-temporal sequence of binary data. It is then used both in a separation-oriented framework (SVM) and in a projection pursuit regression (PPR) one. The corresponding methods are carefully presented and analyzed.

In the following part we release the assumption of (piecewise) constant velocity motion, and we try to follow both position and velocity in real time. In particular, it is shown that it is the trajectory "diversity" which renders this possible.

A. Ickowicz, CEREMADE, University of Paris-Dauphine, Tel.: +33-1-45465874, e-mail: ickowicz@ceremade.dauphine.fr Obviously, tracking a diffusive Markovian target widely differs from the (batch) estimation of deterministic parameters. However, both problems present strong similarities. Indeed, the geometrical properties remains the same at each instant. Once the target motion model has been introduced, the most important properties we used to perform the tracking are presented. Then, the method which allows us to perform adapted corrections for tracking the target is presented. It is the main contribution of this part of the paper.

Simulation results illustrate the behavior of the estimators, as well as the performances of the tracking algorithm. We conclude on further works about the tracking in binary sensor networks.

II. BINARY SENSOR NETWORK OBSERVABILITY PROPERTIES

Let us denote s i a sensor whose position is represented by the the vector t i Similarly, the vector x t represents the position vector of the target at the time-period t. Let us denote d i (t) the (time-varying) distance from sensor s i to the target at time t. Then, we have that:

d i (t) ց⇐⇒ ḋi (t) < 0 , or: x t -t i , v t < 0 , (1)
where v t is the instantaneous target velocity. We thus have the following lemma.

Lemma 1. Let s i (resp. s j ) a sensor whose the target distance is decreasing (resp. increasing) at the time-period t, then we have:

t j , v t < x t , v t < t i , v t . (2)
If we restrict to binary motion information, we consider that the output s i (t) of a sensor (at time t) is +1 or -1 according to the distance d i (t) is decreasing or increasing, so that we have:

s i (t) = +1 if ḋi (t) < 0 , s j (t) = -1 if ḋj (t) > 0 . (3)
Let us denote A the subset of sensor whose output is +1 and B the subset of sensors whose output is -1, i.e. A = {s i | s i (t) = +1} and B = {s j | s j (t) = -1} and C(A) and C(B) their convex hulls, then we have [START_REF] Aslam | Tracking a moving object with a binary sensor network[END_REF]:

Proposition 1. C(A) C(B) = ∅ and x t / ∈ C(A) C(B).
Proof: The proof is quite simple is reproduced here only for the sake of completeness. First assume that C(A) C(B) = ∅, this means that there exists an element of C(B), lying in C(A). Let s be this element (and t its associated position), then we have (t ∈ C(B)): t = j∈B β j t j , β j ≥ 0 and j∈B β j = 1 so that we have on the first hand:

t, v t = j∈B β j t j , v t < x t , v t (see eq. 2),
and, on the other one (t ∈ C(A)):

t, v t = i∈A α i t i , v t ≥ i∈A α i min i { t i , v(t) } > x t , v t . (4)
Thus a contradiction which shows that C(A) C(B) = ∅. For the second part, we have simply to assume that x(t) ∈ C(A)

( x t = i∈A α i t i , α i ≥ 0), which yields: x t , v t = i∈A α i t i , v t ≥ min i∈A t i , v t , (5)
which is clearly a contradiction, idem if X(t) ∈ C(B).

So, C(A) and C(B) being two disjoint convex subsets, we know that there exists an hyperplane (here a line) separating them. Then, let s k be a generic sensor, we can write t k = λ v t + µ v ⊥ t , so that:

t k , v t = λ v t 2 > 0 ⇐⇒ λ > 0 . ( 6
)
This means that the line spanned by the vector v ⊥ t separates C(A) and C(B). Without considering the translation and considering again the v t , v ⊥ t basis , we have :

t k ∈ A ⇐⇒ λ v t 2 > x t , v t , t k ∈ B ⇐⇒ λ v t 2 < x t , v t . (7)
Thus in the basis (v t , v ⊥ t ), the line passing by the point x t , v t v t 2 , 0 and whose direction is given by v ⊥ t is separating C(A) and C(B). We have now to turn toward the indistinguishability conditions for two trajectories. Two trajectories are said indistinguishable if they induce the same outputs from the sensor network. We have then the following property [START_REF] Aslam | Tracking a moving object with a binary sensor network[END_REF].

Proposition 2. Assume that the sensor network is dense, then two target trajectories (say x t and y t ) are indistinguishable iff the following conditions hold true:

ẏt = λ t ẋt (λ t > 0) ∀t , x t -y t , ẋt = 0 ∀t . ( 8
)
Proof: First, we shall consider the implications of the indistinguishability. Actually, the two trajectories are indistinguishable iff the following condition holds:

t j -t i , ẋt ≤ 0 ⇐⇒ t j -t i , ẏt ≤ 0 ∀t ∀(t i , t j ) . (9)
We then choose t jt i = α ẋ⊥ t (i.e. t i and t j both belongs to the line separating A and B) and consider the following decomposition of the ẏt vector:

ẏt = λ t ẋt + µ t ẋ⊥ t ,
so that we have:

t j -t i , ẏt = αµ t ẋ⊥ t 2 ≤ 0 . (10)
Now, it is always possible to choose a scalar α of the same sign than µ t . So, we conclude that the scalar µ t is necessarily equal to zero. Thus , if the trajectories x t and y t are indistinguishable we have necessarily:

ẏt = λ t ẋt , ∀t .
Furthermore, the scalar λ t is necessarily positive (see eq. 9). Then, the lemma 1 inequalities yield:

t j -t i , ẋt < x t -y t , ẋt < t i -t j , ẋt . (11)
Choosing once again t jt i = α ẋ⊥ t , we deduce from eq. 11 the second part of prop. 2, i.e. x ty t , ẋt = 0 ∀t . Considering now the distance between the two indistinguishable trajectories, we have ( ẏt = λ t ẋt ) :

d dt x t -y t 2 = 2 x t -y t , ẋt -ẏt = 0 , (12)
so that we have x ty t = cst. Reciprocally, assume that the two conditions ẏt = λ t ẋt and x ty t , ẋt = 0 hold true ∀t, are the two trajectories then indistinguishable? It is sufficient to remark that:

y t , ẏt = x t + (y t -x t ), ẏt = x t , ẏt = λ t x t , ẋt , t i , ẏt = λ t t i , ẋt . (13)
Since the scalar λ t is positive this ends the proof.

Let us now consider the practical applications of the above general results.

Rectilinear and uniform motion

Admitting now that the target motions are rectilinear and uniform (i.e. x t = x 0 + t ẋ). Then prop. 2 yields ẏ = λ ẋ (λ > 0) and:

y t -x t , ẋ = y 0 -x 0 , ẋ + t (1 -λ) ẋ 2 = 0 ∀t . (14)
Then, from eq. 14 we deduce that λ = 1 and y 0 = x 0 + α ẋ⊥ . So that, the target velocity is fully observable while the position is uniquely determined modulo a α ẋ⊥ translation.

leg-by-leg trajectory

Consider now a leg-by-leg trajectory modeling. For a 2-leg one, we have for two indistinguishable trajectories:

x t = x 0 + t 1 v 1 x + (t -t 1 ) v 2 x , y t = y 0 + t ′ 1 v 1 y + (t -t ′ 1 ) v 2 y , (15)
where v i x is the velocity of the x(t) trajectory on the i-th leg and t i is the epoch of maneuver. Furthermore, we can assume that t 1 < t ′ 1 . Considering the implications of prop. 2 both for t < t 1 and for t > t ′ 1 , we know that if the trajectories are indistinguishable we must have:

v 1 x = v 1 y and: v 2 x = v 2 y . (16)
So, our objective is now to prove that we have also t 1 = t ′ 1 . Considering prop. 2, we thus have the following system of equations :

   y 0 -x 0 + (t -t 1 ) v 1 x -v 2 x , v 1 x = 0 for : t 1 < t < t ′ 1 (a), y 0 -x 0 + (t -t 1 ) v 1 x -v 2 x , v 2 x = 0 for : t 1 < t < t ′ 1 (b), y 0 -x 0 + (t ′ 1 -t 1 ) v 1 x -v 2 x , v 2 x = 0 for : t ′ 1 < t (c) . (17)
Now, on the 1-st leg we have also y 0x 0 , v 1 x = 0 (see prop. 2 for t = 0), so that eqs 17a,b yield:

v 1 x -v 2 x , v 1 x = v 1 x -v 2 x , v 1 x = 0 . ( 18
)
This means that v 1 x and v 2 x are both orthogonal to the same vector (v 1 xv 2 x ), so they are collinear, and we straightforwardly deduce from eq. 18 that v 1 x = v 2

x . Finally, it has thus been proved that t 1 = t ′ 1 and this reasoning can be extended to any leg number. The observability requirements having been considered, we turn now toward the development of the algorithmic approaches. Let us first introduce the following functional.

III. THE STAIRWISE FUNCTIONAL

Our fist aim is to estimate the target velocity, within a batch processing framework. We assume that N binary ({-, +} sensors are uniformly distributed on the field of interest (see fig. 2). Each sensor will be coupled with a counter, that will be increased by a unity each time-period the sensor gives us a {+}, an will keep its value each time the sensor gives us a {-}. Then, at the end of the trajectory, each sensor has a entire value representing the number of periods the target was approaching. Within a given batch, the outputs of the sensor counters can be represented by a stairwise functional (see fig. 3). Then, once this stair is built, we can define what we call the velocity plane. This plane is the tangent plane of the stairwise functional, which means that its direction gives the direction of the stair, while its angle θ gives the slope. The direction of the plane gives us the target heading, while the target speed v is given by:

v = 1 tan(θ) . ( 19
)
Thus, estimating the velocity is equivalent to estimating the velocity plane parameters. Mathematical justifications are then

( 3 ) ( 3 ) ( 3 ) ( 2 )
(

( 2 ) presented. The target moves with a constant velocity v. Considering the results of section V, its starting position is given by the following equation:

( 1 ) ( 1 )
x(0) = x 0 + λ v ⊥ , λ ∈ R , so, that : x(t) = x 0 + λ v ⊥ + t v (20)
This means that at each time period t ∈ R + , the possible positions x(t) define a (moving) straight line, whose direction is v ⊥ . Let us consider now the scalar product x(t), v , then we have:

∂ ∂ t x(t), v = v 2 . ( 21
)
This is clearly constant, which means that the surface is a plane. The conclusion follows: the stairwise plane is an exhaustive information for the velocity vector. We provide in the next section two solutions to estimate the velocity plane from the observed data, and give some asymptotic results about the estimation.

IV. STATISTICAL METHODS TO ESTIMATE THE VELOCITY PLANE

We showed that estimating the velocity plane allows us to estimate the velocity vector. Wile there exists several methods to do that, we shall focus on two of them.

A. The Support Vector Machine (SVM) approach [3]

As seen previously, the problem we have to face is to optimally separate the two classes of sensors (i.e. the + and -). So, we can use the general framework of SVM, widely used in the classification context. The set of labeled patterns {(y 1 , x 1 , • • • , y l , x l } (y i ∈ {-1, 1} and x i sensor positions) is said to be linearly separable if there exists a vector w and a scalar b such that the following inequalities hold true:

w, x i + b ≥ 1 if : y i = 1 , w, x i + b ≤ -1 if : y i = -1 . (22)
Let H(w, b) ∆ = {x| w, x + b = 0} (w: normal vector) be this optimal separation plane. and define the margin (marg) as the distance of the closest point x i to H, then it is easily seen that marg = 1 w . Thus, maximizing the margin lead to consider the following problem: min

w,b τ (w) δ = w 2 , s.t. : y i ( w, x i + b) ≥ 1 ∀ i = 1, • • • , l y i = ±1 . ( 23
)
Denoting Λ the vector of Lagrange multipliers, dualization of eq. 23 leads to consider again a quadratic problem, but with more explicit constraints [START_REF] Cortes | Support-Vector Networks[END_REF], i.e. :

max Λ W (Λ) = - 1 2 Λ T D Λ + Λ T 1 , s.t. : Λ ≥ 0 , Λ T Y = 0 , (24)
where 1 is a vector made of 1 and Y T = (y 1 , • • • , y l ) is the l-dimensional vector of labels, and D is the Gram matrix:

D i,j = y i x i , y j x j . (25)
The dualized problem can be efficiently solved by classical quadratic programming methods. The less-perfect case consider the case when data cannot be separated without errors and lead to replace the constraints of eq. 23 by the following ones:

y i ( w, x i + b) ≥ 1 -ξ i , ξ i ≥ 0 , i = 1, • • • , l . (26)
Consider now a multiperiod extension of the previous analysis. Let us restrict first to a two-period analysis, we shall consider two separating hyperplanes (say H 1 , H 2 ) defined by:

w, x 1 l + b 1 ≷ ±c 1 according to: y 1 l = ±1 , w, x 2 l + b 2 ≷ ±c 2 according to: y 2 l = ±1 . ( 27
)
It is also assumed that these two separating planes are associated with time periods T and T + ∆T , ∆T known. It is easily seen that the margin for the separating plane H 1 is c1 w , while for the plane H 2 it is c2 w . Thus, the problem we have to solve reads:

min w,c1,c2,b1,b2 max 1,2 w 2 c 2 1 , w 2 c 2 2 , s.t.: y 1 l w, x 1 l + b 1 ≥ c 1 , y 2 l w, x 2 l + b 2 ≥ c 2 ∀l. (28)
At a first glance, this problem appears as very complicated. But, without restricting generality, we can assume that c 1 < c 2 .

This means that max

1,2 w 2 c 2 1 , w 2 c 2 2 = w 2 c 2 1 . Making the changes 1 c1 w → w ′ and b1 c1 → b ′ 1 then leads to consider the classical problem: min w ′ ,b ′ 1 ,b ′ 2 w 2 s.t. : y 1 l w ′ , x 1 l + b ′ 1 ≥ 1 , y 2 l w ′ , x 2 l + b ′ 2 ≥ 1 ∀l. ( 29
)
Let w * be the (unique) solution of eq. 29, then a straightforward calculation yields the distance d(H * 1 , H * 2 ) between the two separating planes, i.e.:

d(H * 1 , H * 2 ) = |b * 1 -b * 2 | w * .
Finally, we deduce that the estimated velocity vector v is given by:

v = α w * and: v = 1 ∆T d(H * 1 , H * 2 ) . ( 30
)
The previous analysis can be easily extended to an arbitrary number of periods, as long as the target trajectory remains rectilinear. Another definite advantage is that it can be easily extended to multitarget tracking.

1) 3D-SVM:

We can also mix the SVM ideas with that of section III. Indeed, instead of focusing on a 2-D dataset, we can consider a 3-dimensional dataset (sensor coordinates and values of the sensor counters). The second 3-D dataset is the same, but the value of the counter is increased with unity. So, the separation plane is 2-D, and will be as closed to the velocity plane as the sensor number can allow. See fig. 4 for a more explicit understanding. The results of the SVM estimation of the velocity plane are discussed in the Simulation Results section.

B. Projection Pursuit Regression

The projection pursuit methods have first been introduced by Friedman and Tuckey [START_REF] Friedman | A Projection Pursuit Algorithm for Exploratory Data Analysis[END_REF]. Then, they have been developed for regression with the projection pursuit regression (PPR) by Friedman and Stuetzle [START_REF] Friedman | Projection Pursuit Regression[END_REF]. PPR is mainly a non-parametric method to estimate a regression, with however a certain particularity. Indeed, instead of estimating a function f such as Y i = f (X i )+ε i , where X i and Y i are known, and ε i assuming to follow a certain law, PPR estimates g such as Y i = g(X i θ) + ε i . The first step of the algorithm is to estimate the direction θ, and then ĝ. In our specific case, θ will represent the direction of the target, and ĝ will give us the value of the velocity. 1) Modeling : Let Y i be the value of the i-th sensor counter. X i are the sensor coordinates. If n(X i θ) is the value of the counter i at the end of the track, and p the probability to have the right {+, -} decision, we then have (B: binomial):

L(Y i |X i θ) = B(n(X i θ), p) (31)
Assuming in a first time that p = 1, the two parameters we would like to estimate are the θ parameter and the n(.) function.

2) The PPR method in the network context: We have some additional constraints on n(.). First of all, it only takes integer values. Then, it is an increasing function (because p = 1). The optimization problem we have to solve is the following:

θ = arg min θ (n(X i θ) -Y i ) 2 , ( 32
)
where n is calculated in a quite special way. First, we define a non parametric estimation of a function f, via:

f (u) = Y i K h (X i θ -u) K h (X i θ -u) . (33)
Then, we sort (X θ) i into a vector (Xθ) (i) from the smallest to the biggest. After which we define n(.) via:

n(X θ (i) ) = f (X θ (i) ) if f (X θ (i) ) ≥ f (X θ (i-1) ) , n(X θ (i) ) = f (X θ (i-1) ) otherwise . (34)
Sometimes, due to the integer value of the estimated n(.) function, we have to deal with many possible values of θ. Then, in this case, we choose the mean value of θ. Due to the specific behavior of our target and our modeling, we know in addition that the general form of n (say ñ) is given by:

ñ(u) = iI [(Xθ) ⊥ +(i-1)v,(Xθ) ⊥ +iv] (u) . ( 35
)
The next step is then to estimate v. Such an estimation is given by the following optimization program:

v = arg min v ( n(X θ i ) -Y i ) 2 (36)
3) Convergence: We will study if the estimation is good with an infinite number N of sensors. Assuming we have an infinite number of sensors in a closed space, this means that each point of the space gives us an information {+, -}. We then will have the exact parameters of the stairwise functional. To that aim, we will show in the following paragraph that the probability of having a sensor arbitrary close to the limits of each stair steps is 1. We assume that the sensor positions are randomly distributed, following an uniform law. Then, y being fixed:

L(X|Z) = U [B inf ;Bsup] (37)
If the velocity vector v is denoted with [a; b], then:

B inf = -b a y -c inf a , B sup = -b a y - csup a , (38)
where (c inf , c sup ) only depends on v and x 0 , which means that they are deterministic, and independent from X. It is quite obvious that B inf represents the smaller x-limit of a step, when B sup represents its higher x-limit. Then, considering the velocity plane, B inf and B sup both belong to the plane. Denote u = inf i (X i ), then:

∀ε > 0 P (|u -B inf | < ε) = P (u -B inf < ε) , (39) = P (u < ε + B inf ) .
where we note A = |u -B inf | < ε. We know that:

P (inf X i ≤ t) =    0 if t ≤ B inf , 1 -( Bsup-t Bsup-B inf ) N if t ∈ [B inf ; B sup ] , 1 if t > B sup . (40)
Then, we have the following probability calculations:

P (A) = P (|u -B inf | < ε) , = P (u < ε + B inf ) , = 1 -( B sup -(ǫ + B inf ) B sup -B inf ) N 1 [B inf ;Bsup] ((ε + B inf )) , =    0 if ε ≤ 0 1 -(1 - ε Bsup-B inf ) N if ǫ ∈]0; B sup -B inf ] 1 if ε > B sup -B inf .
Given the above equation, 1-ε Bsup-B inf is smaller than one, which means that (1-ǫ Bsup-B inf ) N converges to 0 as N increases to infinity. Thus, we have finally:

∀ε > 0 lim N →∞ P (|u -B inf | < ε) = 1 . (41)
ending the proof.

V. NON-LINEAR TRAJECTORY ESTIMATION

A. Target Motion Model

The target is assumed to evolve with a Markov motion, given by:

x k |x k-1 ∼ N (F k x k-1 , Q k ) (42)
for k = 1, 2... where N (µ, σ 2 ) is a gaussian distribution with mean µ and variance σ 2 . The starting position is assumed to be unknown.

B. Sensor Measurement Model and Analysis

At each time period, each sensor gives us a {+, -} information, meaning that the target is getting closer or moving away. Given all the sensors reports at the time-period t, we can easily define a space where the target is assumed to be at this time-period. This is the fundamental uncertainty we have at t, and the area of this domain is, of course, directly related to the network parameters (sensor number, network geometry, etc.).

C. Velocity Estimation

We can estimate the direction of the target based on the simple information given by the sensors. Obviously, that estimator will only be precise if the number of sensors is significantly great. To perform that estimation, we can use several methods, such as the Projection Pursuit Regression Method, or the Support Vector Machine Method. The SVM method chosen for our algorithm as a most common method, and is presented in the next paragraphs.

1) The effect of target acceleration: To illustrate the effect of velocity change for estimating the target position, let us consider a very simple example. Assume that the target motion is uniformly accelerated, i.e. :

x t = x 0 + t ẋ0 + t 2 ẍ0 . (43)
We have now to deal with the following question: Is the target trajectory fully observable? To that aim, we first recall the following result. Considering a dense binary network, two target trajectories are said indistinguishable iff they provide the same (binary) information which is equivalent to the following conditions:

ẋt = ẏt , y t -x t , ẏt = 0 ∀t . ( 44
)
Expliciting the second condition of eq 44, with the target motion model 43, we obtain that the following condition holds (∀t):

y 0 -x 0 , ẏ0 + t ẏ0 -ẋ0 , ẏ0 + 1 2 t 2 ÿ0 -ẍ0 , ẏ0 , +t y 0 -x 0 , ÿ0 + t 2 ẏ0 -ẋ0 , ÿ0 + 1 2 t 3 ÿ0 -ẍ0 , ÿ0 = 0 . (45)
Thus, y tx t , ẋt is a zero polynomial, which means that all its coefficients are zero. For the t 3 coefficients we obtain the condition ÿ0 -ẍ0 , ÿ0 = 0. Similarly with the y tx t , ẋt = 0, we obtain ÿ0 -ẍ0 , ẍ0 = 0. Subtracting these two equalities yield ÿ0 -ẍ0 = 0 , or ẍ0 = ÿ0 .

Quite similarly, we obtain the equality ẋ0 = ẏ0 and the last equality:

y 0 -x 0 , ẏ0 + tÿ 0 = 0 ∀t . ( 46
)
Assuming that the couple { ẏ0 , ÿ0 } spans the sensor space then we deduce that x 0 = y 0 . So, it has been shown that it was the target acceleration which render the problem fully observable. This reasoning can be extended to a wide variety of target modeling.

D. Tracking algorithm

The main issue with the SVM estimation is that it only provides us the general direction of the target within a deterministic framework. Moreover, it is highly desirable to develop a reliable algorithm for target tracking (velocity and position). To solve this problem, we build a two-step algorithm. In the first step, we perform a correction through the estimated unitary velocity vector at each time-period t, called λ t . Then, in a second time, we perform a correction through the orthogonal-estimated (unitary) velocity vector, also at each time-period, called θ t . These two corrections give us a better estimation of both the velocity and the position of the target. We refer to fig. 5 for the presentation of the rationale of the two correction factors.

1) The λ factor: To build that correction factor, we started with a very simple assumption. At each period t, the sensors provide binary motion information. Thanks to the first part of this article, we know that the target is in the (special) set lying between the two same-sign-sensors set. Then, starting from the previous estimated position of the target, we move the estimated target through the estimated velocity vector direction until it stands in that special set. We now define this operator in a mathematical way: Let vt the estimated normalized velocity vector at time t. Moreover, let {t (-) i } i (respectively {t (+) i } i ) the coordinates of the sensors (s i ) giving a {-} (respectively a {+}) at time t. We sort vs

(-) i = vt , t (-) i (respectively vs (+) i = vt , t (+) i
). Then, following a very simple geometrical reasoning, we note that vt ; Xt should be between vs (-) max and vs (+) min . To ensure that property, we define the following correction factor:

λ t = vs (+,-) moy -vt, xt-1 vt, vt-1 ,
with the following definition of vs (+,-) moy :

vs (+,-) moy = vs (-) max +vs (+) min 2 , (47)
To calculate this factor, we consider the projection equality:

vt , (x t-1 + λ t vt-1 ) = vs (+,-) moy ( 48
)
which means that the projection of the corrected value is equal to the mean value of the projection. Geometrically, this means that the position of the target is estimated to be in the center of the special set defined by the sensors. The value of the correction factor λ t (see eq. 47) is then straightforwardly deduced from eq. 48. Similarly, the target position is updated via:

xcorr t = xt-1 + λ t vt-1 . ( 49
)
Here the correction factor λ t has been calculated via the average value of the projection. This is an arbitrary choice and we can consider the lower or the upper bound of the projection with no significant difference on the results of the algorithm. Obviously, if the estimation of the position is not very good, the estimated velocity value (clearly based on λ t ) will be quite different from the real value of the velocity. The next correction factor is based on the assumption that the target velocity changes are upper and lower bounded. 2) The θ correction factor: We assume that the velocity of the target has bounded acceleration. Then, if the velocity estimated at a certain time t is too different from the velocity estimated at time t -1, this means that the estimated position of the target is far from the right one. Then, in that precise case, we consider an orthogonal correction, through vt ⊥ .

For that deterministic algorithm we decided to perform a very simple modeling of the velocity. Indeed, we take as a right value for the velocity the simple mean of the k previous values of the estimated velocity (m t,k ). We calculate in addition the variance (σ t,k ), and the factor θ t can be non-zero iff the estimated value of the velocity at time t is not in the interval given by [m t,kσ t,k ; m t,k + σ t,k ]. We then look for θ t such that:

xcorr t + θ t v⊥ t -(x t-1 + θ t v⊥ t-1 ); vt-1 = m t,k . (50)
The previous equation needs some explanation. Given that xt is the estimated target position at time t, we would like to correct the value to be closer to the right position. The only way we can deal with it, is to correct the estimated value of the velocity. xcorr t xt-1 is the previous calculated correction. If the difference between that estimation and the value m t,k is too important, we try to reduce that difference with a translation of the positions at time periods t and t -1. As we want the positions to stay in the special set defined by the sensors, the direction of that translation is given by v⊥ t for the position at time t, and v⊥ t-1 for the position at time t -1. Performing straightforward calculation, leads to consider the following correction factor:

θ t = m t,k -λ t v⊥ t ; vt-1 . (51)
Obviously, as we could expect when presenting the method, if the target motion is rectilinear and uniform , no correction factor can be calculated. Then, the final estimated position is given by:

xfin t = xcorr t + θ t v⊥ t . (52)
3) The final correction step: Noticeably the most important step of the algorithm, i.e. the θ correction factor, is based on the estimation of the velocity change. Indeed, the best the estimation of the velocity is, the best we can estimate the position. Then, our aim is to perform a better analysis of the target motion. Considering that from time to time, the estimation of the position increases in quality, a promising way should be to perform a feedback of the newest corrector to the oldest position estimation. We denote ẑt the updated estimated position of the target at time t. Then, according to the previous paragraph, the estimated position is updated via:

∀j < t : ẑj = xfin j + t i=j+1 θ i v⊥ i . (53)
With this new estimator we will be able to perform a better analysis of the target motion (position and velocity).

4) The final algorithm: With the definition of the correction factors, the theoretical part of the algorithm is finished. Then, it is presented as follows, at time period t:

1) Get the binary information of each sensor, and then the target position set.

2) Estimate the velocity direction at time t via a SVM method 3) Perform the λ calculation, and add that correction to the estimated velocity at time t -1. The time-t-position is then updated.

4) Check if the estimated velocity at time t -1 is too different from the modeled value, and in this case, calculate θ t .

5) Update the position at time t, and in this case, the velocity at time t -1 with the correction θ.

Step 2 and 3 can be inverted with no damage in the process. This is the main part of the algorithm. However, there is no mention in that enumeration of the initialization. There are two main state vectors that have to be initialized. The position and the velocity. The position is assumed to be unknown, but thanks to the sensors, we can have a space where the target is assumed to be at first. We use here a uniform law for the initialization, given that we have no further information about where the target can start. The initialization of the velocity is not far from that solution. Indeed, with the binary information, we can provide a convenient estimate of the velocity direction. Even if we don't have a precise idea of the speed value, we can then start the algorithm.

VI. SIMULATION RESULTS

A. Constant Velocity movement

We shall now investigate the previous developments via simulations. The first figure (fig. 6) will show the stair built by the previously explained method (N : fixed). The position of the sensors are considered random, following a uniform law on the surveillance set. To evaluate the performance of our methods, we decided to calculate the mean square error of the two Fig. 6. 2D sensor position and velocity's direction projection. N=100 sensors, Velocity Vector is [START_REF] Ickowicz | A new method for target trajectory estimation within a binary sensor network[END_REF][START_REF] Aslam | Tracking a moving object with a binary sensor network[END_REF] estimated parameters, which are the velocity value and the velocity direction. Fig. 7 shows the two MSEs values for both direction and velocity values, assuming the sensor number is growing from 10 to 100, and the velocity vector is the [START_REF] Ickowicz | A new method for target trajectory estimation within a binary sensor network[END_REF][START_REF] Aslam | Tracking a moving object with a binary sensor network[END_REF] vector (m/s). Providing 2000 simulations, the MSEs seems to be unstable. However, the two parameter estimation methods leads to a very different conclusion. In the case of the direction estimation, the PPR method works highly better than the SVM method, and seems quite stable as the sensor number N grows. On the other side, the SVM method is more erratic.One possible explanation is that the PPR method has been first developed for the particular case of direction estimation, while the SVM method is more focused on the margins maximization, which means in our case a simultaneous estimation of both parameters. The conclusions we can make on the velocity value estimation are rather opposite. The MSE becomes reasonable only for the SVM method, and for a number of sensors up to 60. Indeed, we have a 0.05 m/s error on a velocity value estimation for a theoretical value of √ 5. As erratic as the SVM's MSE was in the direction estimation, it was however less erratic than the result we have for the PPR value. One answer to the MSE erratic value for the PPR could be to find a best way to estimate the velocity value. Indeed, in our case, we choose for estimating functional a sum of indicators functions. However, it is not clear that this optimization gives a single minimum solution. There could be a finest functional that could lead to a most robust optimization solution, and this would be the subject of future works.

B. Random walk

We will present in that section the results of the tracking algorithm. We consider here that the target starts from the [100, 100] position and that its initial velocity vector is the [START_REF] Ickowicz | A new method for target trajectory estimation within a binary sensor network[END_REF][START_REF] Ickowicz | A new method for target trajectory estimation within a binary sensor network[END_REF] vector. The number of sensors is equal to 70, in a quite wide space (300mx300m). The variance of the target motion is not very important, and the tracking duration is T = 30 seconds. One simulation is presented in figure 8. In red is represented the real target trajectory, quite diffusive, and in green the estimated successive positions. The initialization is not very bad because the number of sensors is quite important, which means the uniform set is not too large. After the first step, the estimation seems to hang the real trajectory, and follows the target well (less than 10 meter error). However, when the target turns right, we loose some precision, mainly because the correction factors seemed to be "lost". The reason for that behavior is that the SVM method provides us a bad estimation of the velocity vector. Then, the algorithm provides a correction in a bad direction, which moves away from the real trajectory. During a few seconds, the estimation works quite bad, before hanging again the target direction, and then performing a quite good estimation of the velocity. Unfortunately, there is no evidence in that example that increasing indefinitely the tracking duration results in an estimated position closer and closer to the real target position. This is precisely the aim of the two next figures in 9. The first one shows the mean square error of the estimated position of the target through the trajectory. The total time is T = 30 seconds, and we can see an amazing and remarkable decrease of that MSE in the first seconds. It seems however that there is a limit to that decrease. Indeed, the MSE will not converge to a zero value, even if we could perform a long-time tracking. Clearly, the limitation is due to the binary information at first, and certainly to the number of sensors in a second time. Some further work could certainly exhibits a strong link between the number of sensors and the MSE of the position. In the same way, the velocity estimation has some acceptable MSE through the tracking process. Despite the clearly strong decrease at the beginning, the curve then stands to an acceptable but non zero value. The effect is more obvious than in the position case, surely because of the velocity modeling we make in the algorithm, which forces the velocity estimation to very bad evolution. A clue could be to perform a most sophisticated modeling of the velocity, but given the binary information, this won't be easy. This is another work in progress for the evolution of our algorithm.

VII. CONCLUSION

A. CVM

In this paper, we chose to focus on the use of the {-, +} at the level of information processing for a sensor network. Though this information is rather poor, it has been shown that it can provide very interesting results about the target velocity estimation. The theoretical aspects of our methods have been thoroughly investigated, and it has been shown that the PPR method leads to the right velocity plane if the number of sensors increase to infinity. The feasibility of the new concept ("velocity plane") for estimating the target trajectory parameters has been put in evidence. The proposed methods seem to be sufficiently general and versatile to explore numerous extensions like: target tracking and dealing with multiple targets within the same binary context.

B. Random walk

A new method for tracking both position and velocity of a moving target via binary data has been developed. Though the instantaneous data are poorly informative, our algorithm takes benefit of the network extent and density via specific spatio-temporal analysis. This is remarkable since the assumptions we made about target motion are not restrictive. Noticeably also, our algorithm is quite fast and reliable. Furthermore, it is clear that performance can be greatly improved if we can consider that the acquisition frequency is (far) greater than the maneuver frequency. In particular, we can mix the present method with the one we developed in [START_REF] Ickowicz | A new method for target trajectory estimation within a binary sensor network[END_REF].

However, some important questions remain. The first one concerns the velocity modeling. We focused on this paper on the adaptability of the different correction factors, but we didn't pay much attention to that modeling, which can definitely improve the estimation quality. Moreover, our tracking algorithm is basically deterministic even if the target motion modeling is basically probabilistic. Thus, it should be worth to calculate the first correction factor (λ) via a likelihood, such that x corr does not always stays in the mean of the special set. Moreover, that likelihood should be related to all the sources of sensor uncertainty. In addition, the present algorithm gives a slow response to sudden target maneuver. A remedy should be to incorporate a stochastic modeling of such event in our algorithm.

The second correction factor (θ) may also be improved via a stochastic approach. Instead of considering a correction only related to the estimated velocity estimated, we could immerse this correction within a stochastic framework involving both vt and θ. These observations are part of our next work on that very constrained but also quite exciting tracking framework. The last important point is multiple target tracking. Even if our work in this area is quite preliminary, it is our strong belief that our spatio-temporal separation based algorithm should be the natural way to overcome the association problems.

Fig. 2 .

 2 Fig. 2. A scenario of target evolution and sensor network information

(Fig. 3 .

 3 Fig. 3. The theoretical stairway of the trajectory.

Fig. 4 .

 4 Fig. 4. The theoretical stairway of the trajectory.

Step 1 : 1 Fig. 5 .

 115 Fig. 5. Correction scenario.

Fig. 7 .

 7 Fig. 7. Mean Square Error of the Velocity Estimators. Green for the SVM, Blue for the PPR.

Fig. 8 .

 8 Fig. 8. Trajectory Estimation of a target. In Red, the real trajectory, in green the estimated one.


V Tisserand

Recent results from BaBar

Keywords: numbers: 14.40.Pq, 14.40.Lb, 13.20.He I. BOTTOMONIUM SPECTOSCOPY A. Introduction

We present recent results on bottomonium spectroscopy, rare neutral D mesons decays, and semileptonic B s mesons decays. They are based on datasets collected at the Υ(2S), Υ(3S), and Υ(4S) resonances and slightly below and above (up to twice the Λ b mass) by the BaBar detector at the PEP-II storage rings at SLAC from year 2000 to 2008. All the results presented here are preliminary.

A search of η b (1S) and η b (2S) quarkonia is performed in radiative transitions using an experimental technique employing converted γ rays in the Υ(2S) and Υ(3S) decays. The h b (1P) state is searched in the Υ(3S) → h b (1P)π 0 /π + π -decays, and an evidence of signal is seen in the channel with a neutral pion. A search of the rare FCNC D 0 → γγ decay is presented and the channel D 0 → π 0 π 0 is accurately measured. Finally, we present a study of the semi-leptonic branching ratio of the Bs mesons and of the fs fraction, the production of Bs mesons. It is based on the very last BaBar dataset collected in 2008 and corresponding to an energy scan above the Υ(4S) resonances performed at PEP-II.

The bottomonium spectroscopy bellow the B B mass threshold is somewhat richer than in the case of charmonium state [1,2], bellow the D D mass threshold. The measurement of the bottomonium mass states and of the branching ratios (B) are important tests of the heavy q q potential models and set constrains on lattice QCD, as well as on theories such as pNRQCD. Hadronic transitions probe non-perturbative QCD. While bottomonium states with quantum numbers L = 0, 1 and S = 1 have been observed and abundantly studied since 1977, not all the predicted states are yet observed. In particular no spin singlet have been observed until 2008 [START_REF] Aubert | the CLEO Collaboration)[END_REF]. The first Dwave state Υ(1D J=2 ) has only been observed in 2004 by CLEO, in the transition γγΥ(1S) and latter on in 2010 by BaBar, in the channel π + π -Υ(1S) [4] At the end of its operation in 2008, the BaBar experiment collected large datasets of approximately 120M Υ(3S) and 100M Υ(2S) events, creating renewed possibilities for probes on bottomonium system. We present herein a study of the Υ(2S) and Υ(3S) inclusive converted photon spectrum, and the search for the h b (1P) state in both Υ(3S) → π 0 h b (1P)[η b (1S)γ)] and π + π -h b (1P).

B. Radiative transitions using converted γ rays

Following the success encountered in the observation the η b (1S) state in radiative decays of the Υ(3S) and Υ(2S) events [START_REF] Aubert | the CLEO Collaboration)[END_REF], the BaBar collaboration has recently developed a technique to study inclusive converted pho-ton spectrum of these events. The details of the analysis can be foun in Ref. [5] (see also references therein for previous and alternate measurements). The monochromatic γ radiated in the bottomonium transitions are reconstructed through the converted e + e -pair produced in the material of Silicon Vertex Tracker (SVT) [6] and which charged track trajectories are bent in the magnetic field of the axial 1.5 T solenoid. This techniques improves substantially the mass spectrum resolution (E * γ ) wrt to the photons reconstructed in the Electro-Magnetic Calorimeter (EMC) (typically from 25 down to 5 M eV /c 2 ). This accurate measurement helps to resolve overlapping resonances γ rays. The price to pay is a relatively lower efficiency (∼ 1/20) as the material budget BaBar tracking system is quite limited [6].

The various monochromatic γ rays are studied with χ 2 fits to the recoil E * γ spectrum in the Υ(3S) and Υ(2S) events after subtraction on the combinatoric background. The γ spectra presented on Fig. 1 display the rich phenomenology accessible. In these spectra we study the decays: Υ(3S)/Υ(2S) → γη b (1S) and possibly Υ(3S) → γη b (2S). It offers an alternate search of the states η b (1S, 2S) and possibly a more accurate mass measurement. In addition to combinatoric background coming e + e -(

√ s = m Υ(nS) ) → γ ISR Υ(1S)
transitions can more easily be unfolded and also one can study accurately the decays: χ bJ (1P, 2P ) → γΥ(1S), χ bJ (2P ) → γΥ(2S), and Υ(3S) → γχ bJ (1P ).

The recoil E * γ spectra are divided in 4 energy ranges ; for Υ(3S) "low": [180, 300] M eV , "medium": [300, 600] M eV , and "high": [600, 1100] M eV and for Υ(2S): [300, 800] M eV .

In the "low" Υ(3S) region we observe the transitions χ b1,2 (2P ) → γΥ(2S) with more than 12 and 8 statistical standard deviations, while the χ b0 (2P ) → γΥ(2S) is not seen. It is consistent with previous works by CLEO and CUSB (1992) (see Refs. in [5]) and our measurements are more precise.

In the "medium" Υ(3S) region we observe the transitions Υ(3S) → γχ b0,2 (1P ) with more than 7 and 15 statistical standard deviations, while the Υ(3S) → γχ b1 (1P ) is not seen. This work is in agreement with recent measurements performed by CLEO [7]. An upper limit B(Υ(3S) → γη b (2S)) < 1.9 × 10 -3 is set at 90% C.L., for a scan of the resonance in the narrow range: 335 < E * γ < 375 M eV . In the "high" Υ(3S) region we observe the transitions χ b1,2 (2P ) → γΥ(1S) with a much better accuracy than CLEO and CUSB, and confirm an absence of observation of the χ b0 (2P ) → γΥ(1S). For the η b (1S) state a 2.9 statistical standard deviation signal is seen (respectively 2.7 when including systematic uncertainties that are dominated by width assumption for the signal). The fitted mass of the quarkonium state is (9403.6 ± 2.8 ± 0.9) M eV /c 2 and is inconsistent with the PDG average by about 3.1σ deviations [2]. The measured B(Υ(3S) → γη b (1S)) = (5.9 ± 1.6 +1. 4 -1.6 ) × 10 -4 is however consistent with previous measurements [START_REF] Aubert | the CLEO Collaboration)[END_REF].

Finally, for the Υ(2S) data spectrum one observes the transitions χ b1,2 (1P ) → γΥ(1S) with a much better accuracy than CLEO and CUSB, and confirms an absence of observation of the χ (1P ) → γΥ(1S). For the η b (1S) state a non significant 1.7 statistical standard deviation signal when including systematic uncertainties is obtained (2.5σ for statistics only). The fitted mass of the quarkonium state is nevertheless fairly consistent with the PDG average [2]. One sets the upper limit:

B(Υ(2S) → γη b (1S)) < 0.22% at 90% of C.L.
The results of that analysis [5] are the best B(χ bJ (nP ) → γΥ(1S, 2S)) available measurements and in good agreement with theory predictions [8]. We have the most accurate measurements of the transitions Υ(3S) → γχ b0,2 (1P ) and we don't observe the χ b1 (1P ). This is inconsistent with any theory prediction but this is in good agreement with CLEO [7]. Unfortunately very few concluding informations are derived for the η b (1S, 2S) states as initially hoped.

C. Search for Υ(3S) → π 0 h b (1P) and π + π -h b (1P) transitions
The h b (1P) bottomonium state is the axial vector partner of the three Pwave χ bJ (1P ) states and its mass is expected to be at the center of gravity of their masses:

m h b (1P) = Σ J [(2J +1)×m χ bJ (1P ) ]/Σ J (2J +1) = (9900 ± O(3)) M eV /c 2 .
The predicted production mechanisms in Υ(3S) decays are B(Υ(3S) → π 0 h b (1P )) ∼ 10 -3 and B(Υ(3S) → π + π -h b (1P )) ∼ 10 -5 -10 -3 . Such predictions lead to a relative ratio of branching ratios of the 2 decay modes (π 0 /π + π -) ranging from 5 to 20% [9]. The Υ(3S) → γh b (1P ) decay is forbidden by C-Parity.

The expected h b (1P) decay width is less than 1 M eV . The particle decays to 3 gluons (∼ 57%) or to 2 gluons plus a photon (∼ 2%), and for 40 -50% of the time, to γη b (1P ). The latter mode offer an experimental signature that helps to reduce the background and that can be compared to the technique that was employed by CLEO in 2005 and latter on by BES in 2010 to observe the charmonium state h c in the decay

ψ(2S) → π 0 h c [γη c ]. More recently CLEO-c [10] mea- sured the decay e + e -→ π + π -h c .
The existing information for the branching ratios are: [2]. At this conference R. Mizuk [START_REF] Mizuk | La Thuile[END_REF], for the BELLE collaboration, has presented the first observation of the h b (1P) and h

B(Υ(3S) → π + π -h b (1P )) < 1.8 × 10 -3 and B(Υ(3S) → π 0 h b (1P )) < 2.7 × 10 -3 at 90% C.L.
b (1P) states in Υ(5S) → π + π -h b (1P, 2P ) transitions.
We perform the search in the 2 channels Υ(3S) → π 0 h b (1P ) [START_REF] Lees | [END_REF] and Υ(3S) → π + π -h b (1P ) [START_REF] Lees | [END_REF] .

The Υ(3S) → π 0 h b (1P ) channel is reconstructed by requiring a photon with an energy E * γ consistent with the transition h b (1P ) → γη b (1P ) ([420, 540] M eV ). Additional selection criteria are applied. They are based on the number of tracks, event shape, and we photons matching π 0 . The global signal efficiency is about 16%. The number of signal event is extracted from fits to the distribution of the mass recoiling against the pion system in the Υ(3S) rest frame and in a mass region near the predicted h b (1P ) mass (9.9 GeV /c 2 ). The recoil mass window comprises 90 bins of 3 M eV /c 2 width each. Very precise fits to photon pairs are performed and account for accurate effects from re-weighted Monte Carlo simulation to data (the signal region is excluded in the procedure). The average reduced χ 2 of the fit presented in Fig. 2 (left) is 0.98 ± 0.03.

The fit to the recoil mass spectrum yields 9145±2804± 1082 signal events. This is an evidence for the signal at the level of 3.0 standard deviations and this number includes all the sources of uncertainties (statistical and systematic). The systematic uncertainties are dominated by the background and signal line shape models and the m(γγ) fits. The above signal significance is slightly higher (3.2σ) when the systematic uncertainties contributions are omitted . The mass of the h b (1P ) signal is (9902 ± 4 ± 1) M eV /c 2 and is fully compatible with an expected value as the centre of gravity of the χ bJ (1P ) states.

When assuming B(h b (1P ) → γη b (1P )) = (45±5)%, we measure B(Υ(3S) → π 0 h b (1P )) = (3.7±1.1±0.4)×10 -4 . We also set the upper limit to be 5.8×10 -4 at 90% of C.L. It is fully consistent with the prediction by Voloshin [9] and coherent the previous limits.

The Υ(3S) → π + π -h b (1P ) channel is reconstructed by requiring a pair of positively-charged track as the dipion pair. Additional criteria are applied. They are based on the event energy and shape, the number of tracks, and we also veto mainly K 0 S → π + π -decays and we reduce the also less worrying baryon decay Λ → pπ -and converted γ to a e + e -pair. The global signal efficiency is about 42%. Here also a search of a signal peak near 9.9 GeV /c 2 is performed by fitting the recoil mass against the dipion system. The signal resolution is expected to be of the order of 9M eV /c 2 .

The fit of the subtracted combinatoric background spectrum is displayed in Fig. 2 (right). A 1D χ 2 fit is performed to extract the signal and it comprises 7 components: the h b (1P ) signal, the Υ(3S) → π + π -Υ(2S) transition at the Υ(2S) mass, the Υ(2S) → ππΥ(1S) contribution slightly below 9.8 GeV / 2 , the χ b1,2 (2P ) → π + π -χ b1,2 (1P ), the remaining K 0 S → π + π -pollution, and the non-peaking background (including ISR e + e -→ π + π -Υ(1S)). No signal is seen. The fit yields a negative number of signal events: -1106 ± 2432 (statistical uncertainty only is included here). This leads to the upper limit: B(Υ(3S) → π + π -h b (1P )) < 1.2 × 10 -4 at 90% of C.L. The maximum significance over the scanned range is 2 standard deviations at most. The systematic uncertainties are dominated by the decay knowledge in the simulation for the charmless mesons and by the continuum model and residual K 0 S and ISR backgrounds. We also extract the branching ratios of the transitions Υ(3S) → X[χ bJ (2P ) → π + π -χ bJ ′ (1P )], where J = J ′ and are equal to 1 or 2. We measure: B(J = J ′ = 1 or 2) = (1.16±0.07±0.12)×10 -3 or (0.64±0.05±0.08)× 10 -3 . And we improve the PDG [2] accuracy for the B[Υ(3S) → π + π -Υ(2S)] and B[Υ(3S) → X(Υ(2S) → π + π -Υ(1S))] decays. We measure respectively: (3.00 ± 0.02 ± 0.14)% and (1.78 ± 0.02 ± 0.11)%.

Finally it is possible to estimate the ratio of branching ratios B(Υ(3S) → π 0 h b (1P )) over B(Υ(3S) → π + π -h b (1P )) from the above measurements. It is higher that 3.7 -5.8 and so far consistent with predictions from theory [9].

II. RARE D 0 → γγ AND D 0 → π 0 π 0 DECAYS
In the Standard Model (SM) Flavor Changing Neutral Currents (FCNC) are forbidden at tree-level. These decays are allowed at higher order and have been measured in kaons and B mesons. For charm mesons the low mass of the down-type companion quark introduces a large suppression at the 1-loop level from the GIM mechanism. So far no FCNC decays of charm mesons have been observed. The search of rare charmed meson decay such as D 0 → γγ is one possible way to perform that search.

In the SM the process D 0 → γγ is dominated by long distance effects [START_REF] Burdman | [END_REF]. At short range mainly 2-loops contribute and the branching ratio B of D 0 → γγ is expected to be of the order of 3 × 10 -11 . This is several order of magnitude below the sensitivity of current experiments. But in fact, the transition D 0 → γγ is dominated by Vector Meson Dominance processes (VMD), so that the value of B(D 0 → γγ) is enhanced to (3.5 +4.0 -2.6 ) × 10 -8 . This larger value is confirmed in the HQχP T computations that predict B(D 0 → γγ) = (1.0 ± 0.5) × 10 -8 .

Such small values are anyhow still a bit far away from experimental capacities. But possible large enhancements arising from long distance New Physics (NP) effects are such that they can lead values as large as 6 × 10 -6 for B(D 0 → γγ). This is in fact within the reach of present experiments a B-factories. Such effects may for example originate from gluino-exchange within the MSSM framework [15].

The BaBar experiment with about 470 f b -1 of data collected near the Υ(4S) resonance has such a discovery potential. This integrated luminosity corresponds to more than 610 × 10 6 cc quark pairs. The search of the process D 0 → γγ is therefore an appealing, even difficult, mode for NP search. BaBar has effected such an analysis.

The existing upper limit on B(D 0 → γγ) is 2.7 × 10 -5 at 90% of C.L. [2] and was obtained by the CLEO experiment [16]. The measurement of that branching ratio is normalized to the abundant, pure and precisely measured channel D 0 → K 0 S π 0 , which branching ratio is equal to (1.22 ± 0.5) × 10 -2 [2]. When employing that technique some systematic uncertainties cancel in the ratio of branching ratio.

The largest background for D 0 → γγ channel is the decay mode D 0 → π 0 π 0 . Is presently measured B is equal to (8.0 ± 0.8) × 10 -4 [2]. We also perform similarly the measurement of that latter channel using the normalization technique to the D 0 → K 0 S π 0 . Doing that measurement at the same time allows to have a better handling of the D 0 → π 0 π 0 background for the search of the D 0 → γγ mode. The main backgrounds for D 0 → π 0 π 0 are the modes D 0 → K 0 / K0 π 0 and K -π + π 0 .

In order to remove B B backgrounds, we use D * + → D 0 π + tagged events and require P D * > 2.4 -2.85 GeV /c 2 . We remove QED background by requiring at least 4 tracks or neutrals within the BaBar detector acceptance. The channels D * 0 → D 0 π 0 /γ are the largest backgrounds for the normalization channel D 0 → K 0 S π 0 . Finally in the case of the γγ analysis, we perform a veto against photons that can be associated to another photon in the event to build a π 0 candidate. Such a veto is 66% efficient on D 0 signal and removes 95% of the photons originated from π 0 .

For the D 0 → π 0 π 0 analysis the selection efficiencies of the D 0 → π 0 π 0 signal is 15.2% and it is 12.0% for the normalization channel D 0 → K 0 S π 0 . For the D 0 → γγ analysis the selection efficiencies of the D 0 → γγ signal is 6.1% and it is 7.6% for the normalization channel D 0 → K 0 S π 0 . Figure 3 shows the fitted spectra of the D 0 → π 0 π 0 (left) and D 0 → γγ (right) signals.

The π 0 π 0 analysis fitted yield is 26010 ± 304 signal events, while for the normalization channel K 0 S π 0 the yield is 103859 ± 392 events. This corresponds to B(D 0 → π 0 π 0 ) = (8.4 ± 0.1 ± 0.4 ± 0.3) × 10 -4 , where the third uncertainty is related to uncertainty on the the world average for B(D 0 → K 0 S π 0 ) [2]. This measurement is 40% more accurate than the present world average value.

For the γγ analysis the fit yields a negative number: -6 ± 15 events. It corresponds to an upper limit of 25.1 events at 90% of C.L. This number converts into an upper limit, computed from pseudo Monte Carlo experiments. In the case of the γγ analysis the signal is fitted together with the D 0 → π 0 π 0 background component which results in a negative signal (small dash purple curve). The fit is determined from unbinned maximum likelihood but the χ 2 value is determined from binned data and is provided as goodness-of-fit measure. The pull distributions show the differences between the data and the solid blue curve above with values and uncertainties normalized to the Poisson statistics.

That computation includes systematic uncertainties. We set the upper limit: B(D 0 → γγ) < 2.4 × 10 -6 at 90% of C.L. (< 2.06 × 10 -6 without systematic uncertainties). Such a value is already constraining specific NP models [15]. It is an order of magnitude lower than the existing best world limit [16].

III. SEMI-LEPTONIC BRANCHING RATIO OF THE Bs MESONS AND THE FRACTION fs ABOVE THE Υ(4S) RESONANCE

As opposed to the semi-leptonic B(B u,d → Xl -νl ), that are well known and equal to (10.99 -10.33 ± 0.28)%, the existing measurements of the semi-leptonic branching ratio in B s decays are still inaccurate [2]. It is expected to be from 1.5 to 3% lower than that of B d [17]. Its world average is (7.9 ± 2.4)%, from LEP experiment at the Z 0 and they include the information on the fraction P (b → B s ) = (10.5 ± 0.9)%. The alternate measurement (10.2±0.8±0.9)% is from the Υ(5S) data collected by the BELLE experiment [18]. The LHCb experiment has yet already started to contribute by providing measures of ratios of specific semi-exclusive decays to total inclusive semi-leptonic B s decay [19].

Recently BaBar collaboration has performed the measurement of B(B s → Xl -νl ) (B(B s ) SL ) and of f s , the fraction of B ( * ) s mesons produced above the Υ(4S) resonance. For this we use 4.1 f b -1 of data from a final energy scan performed in the last period of the data taking in 2008 [20]. In that energy scan, data were collected every 5 M eV above the Υ(4S) resonance, from which 3.15 f b -1 was taken in the range [2mB s , 2mΛ B ]. These 2 measurements are based on the counting of the yield of produced φ mesons and of φ mesons produced in correlation with a high -momentum lepton. Such signatures are more abundant in B s decays than in B u,d decays. As a function of the center of mass energy in the scan, one can unfold the 2 parameters B(B s ) SL and f s from the 3 observables: the number of produced B hadrons, the φ mesons inclusive rate, and the rate of φ mesons produced in correlation with a high-momentum lepton.

The light q q (q = u, u, s, c) pair of quarks contribution are subtracted by using data collected 40 M eV below the Υ(4S) resonance. The B u,d contributions are computed from data collected at the Υ(4S) resonance. Many quantities derived from the PDG [2] such as B(B s → D s X), B(D s → Xl -νl ), B(D s → φX), B(D s → φXl -νl ) (...) are exploited in the computation of B(B s ) SL and f s from the above enumerated 3 observables. The input B(B s → D s X) is from far the less accurately known of the various input parameters. Its present world average is (93 ± 25)%. We measure B(B s ) SL = (9.9 +2.6 -2.1 (stat.) +1.3 -2.0 (syst))%. This branching ratio is consistent with previously mentioned measurements. The values of f s for bins near the Υ(5S) resonance are fully compatible with those obtained by BELLE: (18.0 ± 1.3 ± 3.2)% and CLEO: (16.8 ± 2.6 +6.7 -3.4 )% in 2007 [21].

FIG. 1 :

 1 FIG. 1: . Fit to the E * γ spectrum in the Υ(3S) and Υ(2S) data. Υ(3S) top left: [180, 300] M eV , top right: [300, 600] M eV , and [600, 1100] M eV . Υ(2S) bottom right: [300, 800] M eV .

FIG. 2 :

 2 FIG. 2: Fitted recoil mass spectra for Υ(3S) → π 0 h b (1P ) (left) and Υ(3S) → π + π -h b (1P ) (right). The right-hand side plot is displayed after that the fitted combinatoric background has been subtracted.

FIG. 3 :

 3 FIG.3: Fitted mass spectra of the D 0 → π 0 π 0 (left) and D 0 → γγ (right) signals. The dots with error bars are the data. On top we superimpose with the long dashed red curve the fitted combinatorial background component, the signal is shown with the solid blue line. In the case of the γγ analysis the signal is fitted together with the D 0 → π 0 π 0 background component which results in a negative signal (small dash purple curve). The fit is determined from unbinned maximum likelihood but the χ 2 value is determined from binned data and is provided as goodness-of-fit measure. The pull distributions show the differences between the data and the solid blue curve above with values and uncertainties normalized to the Poisson statistics.

Figure 4

 4 Figure 4 displays the fitted value of B(B s ) SL with respect to B(B s → D s X) (left) and of f s versus the value of the center of mass energy of the 2008 PEP-II scan (right).We measure B(B s ) SL = (9.9 +2.6 -2.1 (stat.) +1.3 -2.0 (syst))%. This branching ratio is consistent with previously mentioned measurements. The values of f s for bins near the Υ(5S) resonance are fully compatible with those obtained by BELLE: (18.0 ± 1.3 ± 3.2)% and CLEO: (16.8 ± 2.6 +6.7 -3.4 )% in 2007[21].

FIG. 4 :

 4 FIG. 4: Fitted value of B(Bs)SL with respect to B(Bs → DsX) (left) and of fs versus the value of the center of mass energy of the 2008 BaBar scan (right) [20]. The statistical and additional systematic uncertainties are plotted separately.

Acknowledgments

I would like to thank my BaBar colleagues for their help in the preparation of that presentation and their many useful comments and inputs during our discussions.