New technological environments: new constraints, new opportunities for the teacher by Luc Trouche

Equipe ERES, université Montpellier II, Place Eugène Bataillon, 34000 Montpellier, France trouche@math.univ-montp2.fr

French experiments carried out over the last three years on mathematics teaching in complex calculator environments --in particular with TI-92 --allow to draw some lessons:

-the introduction of this kind of calculators does not simplify the teacher's work, nor the pupils' ; -it requires a new teaching organization of learning space and a new management style of time.

We shall argue that, under these conditions, work in complex calculator environments can lead to a transformation of students' relationships to mathematics (giving priority to the creative side of work). Moreover, one can observe a transformation of students' collective relationships to knowledge (giving greater importance to social aspects).

About a common idea

New technological environments make teachning and learning easier or more interesting : it's a belief appearing in some papers about experiments with computers or calculators. [START_REF] Shoaf | Using the total power of the TI-92 ! From discovery explorations to compel lab reports[END_REF] describes students's behaviour in such environments as « a process in which the student is silently conversing with himself through the calculator, asking questions of himself as he manipulates the concrete screen image. This leads him to have the knowledge to conjecture not only what is actually occuring with the image, but what it is happening. By using the graphic calculator students are more likely to construct their own mathematical understanding through conscious reflection ».

Such point of view can also appear from inquiries among students. I teach mathematics for five years in the last level of a french highschool (students 18 years old) in an experimental environment : each student has a TI-92 which is given by the school (from september to june). He can use it both at school and at home, in normal courses as in practicals [START_REF] Trouche | A propos de l'apprentissage des limites de fonctions dans un environnement calculatrice, étude des rapports entre processus d'instrumentation et processus de conceptualisation[END_REF]. We can see below these students'answers at four questions :

does this calculator environment help you for understanding mathematics ?

what do you think about doing practicals in pairs ?

does this kind of environment modify your point of view on calculators ?

does this kind of environment modify your point of view on mathematics ? calculator and what they precisely take off their own activity. [START_REF] Trouche | Seeing is reality : how graphic calculators may influence the conceptualisation of limits[END_REF] The first lesson of our experiment is about he perception of mathematical objects throughout calculators. As [START_REF] Noss | Windows on Mathematical Meanings[END_REF] wrote, « these tools wrap up some of the mathematical ontology of the environment and form part of the web of ideas and actions embedded in it ».

Seeing is reality

An example : the students had to find the equation of a quadratic curve tangential to three given lines.

They try various coefficients and find an approximate solution. To check this solution, they made successive zooms.

The teacher asked : « What is your definition of a line tangential ? ».

One student answered : « A line is all the more tangential to a curve that it shares the higher number of points with it ». This isn't a definition possible in a paper and pencil environment… But in a calculator environment, it's a definition created by the zooming action.

Frequent students'use of calculators bears on the conceptualization process :it can lead to a confusion between mathematical objects and their representations by the tool.

Particular environment, particular action.

The second lesson of ou experiment is the influence of the environment on the students'mathematical actions. This kind of influence exists also for the teacher, or the mathematician himself… 3. a. Auto-observation.

The following question comes from a personnal problem, arrived during the ICTM4 (Plymouth, August 1999). My bag was lost by the airline compagny and arrives at Plymouth four days after me (quite at the end of the conference). I imagined that an airline compagny had a lot of ways to bring a bag from Montpellier to Plymouth :

a direct way (that means carrying my bag in the same airplane than me, from Montpellier to Plymouth) ;

an indirect way (that was the way choosed by the airline compagny : Montpellier-Paris-London-Philadelphia USA-London-Plymouth…) ;

a random way. That is the way I would like to study in an TI-92 environment, and in the same time, I would like to look at me solving this kind of problem, in order to analyse the influence of this kind of environment.

What is for me a random way ? Look at the figure 1 below.

1 2 3 4 n ❏ ❏ ❏ ❏………….❏……..❏ ❏ ❏ ❏ ❏ Montpellier Plymouth Figure 1
Montpellier is at the square 0. Plymouth at the square n. Each time I throw a dice. my bag can progress (between one and six squares obviously). I ask (to me) three questions :

what is, in such conditions, the probability to reach Plymouth ?

if I call P(n) the probability to reach the square n, what is the limit of P(n) when n tends to infinity ?

has this problem something to do with technology (why and how) ? . To reach the square 2, two possibilities : obtaining 2 with a dice or obtaining twice 1 (when throwing two dices). Then P 2 = 1 6 + 1 36 . To reach the square 3, one can obtain 3 (with only one dice), or 1-2, or 2-1 (with two dices), or 1-1-1 (with three dices)

; then P 3 = 1 6 + 2 6 2 + 1 6 3 ;
It becomes more and more complicated. Here takes place, in a calculator environment, a second movement, a movement of abstraction : it is necessary to give a mathematical form to the problem if one wants to be helped by the calculator. So I'll try to detect a regularity in the first results, to find a general formula for P(n). For a numerical reason first. We can tranform our result by using the Newton binomial formula :

P n = C n-1 p-1 6 p p=1 n ∑ = 1 6 C n -1 p -1 1 6 p-1 = 1 6 C n -1 k 1 6 k = 1 6 (1 + 1 6 ) n -1 k =0 n-1 ∑ p=1 n ∑
. The exponential increasing of our result is now obvious… For a theoritical reason also ! It's always usefull to think about the validity domain of a result. Our formula for P n suppose than we can reach the square n with 1 dice, or 2 dice… or n dice. From the square 7, our formula is false, because 7 cannot be obtained with a dice alone ! Then

P 7 = C 6 p-1 6 p p=2 7 ∑
. For the same reason, the square 13 can be reached with ad less three dices. One can see here how the two movements of abstraction and concretisation are linked, and how it's is difficult to come out the calculator and engage a theoritical reflexion.

I have now two possibilities :

-looking for another strategy ;

-trying to modify my first formula.

For some « economical thinking reasons », I'll choose the second possibility. I have a good formula for the first values, I spent a lot of time to write this formula for my calculator, I have understood the origine of my mistake, so I can hope doing a good « reparation » of it.

My sum has to be made from the integer part of (n-1)/6. I obtain a new formula : [START_REF] Duval | Semiosis et pensée humaine[END_REF]. This movement of diversification is not natural in any mathematical environment. It's even more difficult in a calculator environment : one can often observe in such environment a movement of fixation on a same application (graphical for example) and then on the same form of a given problem.

P n = C n-1 p-1
So, about our problem. Let us try to come back on our walk : because we know the first results, could we find a « recurrence » (I need help from reviewers, I don't know the english word !) formula ? We suppose known the results till square (n-1), let us calculate P n . If we are on square n, where were we just before ? Of course, on square n-1, or on square n-2, or on square n-3, or on square n-4, or on square n-5, or at last on square n-6. Each of these possibilities has the same probability (if our dice is not fixed…). Then P n = 1 6 (P n-1 + P n -2 + P n-3 + P n -4 + P n -5 + P n-6 ). To go further, it's necessary to follow both a movement of abstraction and diversification :

-our sequence is a classical linear sequence, with classical ways of studies (study of the carateristic polynomial, research of its roots, etc…). It would be too long to do this here, but it would need both theoritical calculation and calculator use ;

-the apparent sequence limit, about 0,28571429, can also arouse some thinking. If it was a rational number, it could be 2/7. Why such a limit ? It's quite easy to understand (after asking to himself the question !). With a dice, we can obtain 1, 2, 3, 4, 5 or 6, with an equal probability. So the arithmetical mean of a step is

1+ 2 + 3 + 4 + 5 + 6 6 = 7 2
. It means that, at the limit, we can reach two squares among seven squares. This is our limit for P(n) ! End of my bag's random walk and of a mathematician thinking about it… One can see, through this example, both the virtualities of a symbolic calculator and the difficulties to stand back the first results of the screen. He ends this exercice by interpretation of the first results (for n=0 and n=1).

The differences between these two students' kind of work are very significant. It shows a real danger : a complex calculator environment may increase disparities among pupils' behaviour, favouring the better students, hampering the weakers ones.

Some theoritical elements

In order to understand what kind of knowledge can be learnt in these computer environment, it is usefull to give due consideration to cognitive ergonomy. As langage and thought are related, Vygotsky (1962) points out the fundamental relationship between gestures and thought. Verillon and [START_REF] Rabardel | Cognition and artifacts : A contribution to the study of thought in relation to instrumented activity[END_REF] studies focusing on learning processes in the area of cognitive ergonomy are based on this idea. If cognition evolves through interaction with the environment, accomodating to artefacts may have an effect on cognitive development, knowledge construction and processing, and the nature itself of the knowledge generated. They suggest models to analyse the instrumented activity of students confronted with tasks involving artefacts :

an instrument does not exist in itself, an artefact (like a calculator) becomes an instrument when the subject has been able to appropriate it for himself and has integrated it with his activity ;

this appropriation is not only an individual process ; as in each human activity, it's a social process, in wich both the teacher 's responsability and the peers'one are engaged.

This model gives us some guiding ideas to control the calculator appropriation process :

the teacher is the designer of activities where mathematics is the central focus. [START_REF] Dreyfus | Didactic design of computer-based learning environments[END_REF] stresses the importance of the choice and the way activities are promoted by the teacher for making an effective learning tool. Situations have to be carefully designed in order to take advantage of the constraints and discrepancies caused by calculators, which may be considered as new learning potential ;

the choice of the applications involved in the activity and their articulation aims to improve investigation by enhancing varying points of view (to favour the movement of diversification) ;

one must develop situations aiming to foster experimental work with interaction between graphic observations and theoritical calculation, in order to tackle the distorsion between the paper and machine environment (to favour the articulation of the calculator concretisation movement and of the abstraction one) ;

it is important to buid classroom devices to allow the control by the teacher of the pupils'instrumented activity (in order to limit the movement of distraction, the random research way, to correct the approximate syntax) ; -it's at last important to buid classroom collectives work divices, to favour social learning process : students' various behaviours can compensate each to each (for example it can favour a diversification of the action, a combination of calculator concretisation and abstraction).

So new environments require a new teaching organization of learning space and a new management style of time.

5. Some ideas from our experimental work with a TI-92 highschool class.

We shall present here only two elements of this new organization [START_REF] Guin | The Complex Process of Converting Tools into Mathematical Instruments. The Case of Calculators[END_REF].

A new organization of the ordinary learning space

As the figure 3 shows it, the teacher, througout his courses, combines both blackboard and screen to enhance interactions between paper/pencil and screen/keyboard classwork ; the data from one student's calculator are overhead projected on the class screen : it gives a particular go-between role to him (and it allows the control by the teacher and by the other students of this student's instrumented activity).

  answers, very positives, do'nt allow to know what they have actualy learnt in these environment. Knowing it needs to study what they precisely do with a

  Let's try to answer to the first question. We could modelize this walk with the command « random » of a TI-92 (see first screen below) ; we could so realize a simulation of a lot of walks and have a statistic answer for each value of n. It's a « tentation » in this kind of environment to use immediatly the powerfull of the calculator and see what happens. I call this first movement a movement to calculator concretisation : to try, to calculate something is easy with a calculator, so we do it, even if it is not the most usefull… It is also possible to use sequences syntax : we can have quicker some « random walks » (see screens below). Little problem : althought these sequences are (from their definition) growing, the table of values shows some irregular phenomenas… It happens often in a calculator environment : an amazing event arises, and attracts our attention. It is perhaps an interesting thing, but it is far from our problem… I call this a movement of distraction. I suppose that I don't follow this movement. So I leave my calculator and I calculate the first values of P(n). It is quite obvious than P(0) = 1 (my bag start surely from Montpellier…). The only possibility to reach the square 1 is obtaining 1 when throwing the dice : P 1 = 1 6

  It's easy to observe at the denominators the successive powers of 6 (because one throws one dice, then two dices, etc…) ; it's necessary to think a little more to remarque at the numerators the « Pascal triangle» : 1, puis 1-1, puis 1-2-1… Then I have two possibilities : -if I follow the movement of calculator concretisation, I'll try immediatly to enter the new formula in my calculator « to see what happens » ; -if I follow the movement of abstraction, I'll try to understand the reasons of such a formula. It needs to think again : to reach the square n with p throwing of a dice, I need to share n in p non empty pieces. It means I have to choose (p-1) « partitions» among the (n-1) spaces separating the n squares. In fact, I do obtain C nll see what happens on my calculator : The first results are well convenient : we obtain the results already known. Let's see further : the sequence graph (see below) is a little surprising. The probabilities overtake 1… So our formula is perhaps true for the first values of n, then it's false. Why ?

  away this new formula. It's OK for n=2, n=7. Let's try further. Alas (see below) ! The probabilities become also bigger than 1… Thinking again : for n=9 (for example), if one shares the walk in two parts, one can obtain a part of 1 (possible with a dice) and a part of 8 (impossible with a dice !). Our new formula is true till 7, false further. Again two possibilities, looking for a new strategy or trying to treat our poor formula… Our choice : changing our point of view. It's very important for a scientific thinking (cf. Robert et Tenaud 1989). Many authors have showed how difficult it was to change a representation register, even to change, inside a same representation register, a problem form (cf.

  Let's have a glance on such a defined sequence (I use the TI-92 sequences editor, which provides approximate values ). The graphic representation, with always the same window, seems to show a convergent phenomena. It seems quite « normal », because of the formula barycentric form. The opposite screen shows a « stabilisation » of the sequence at about 0,28571429. We are at the end of the movement to calculator concretisation. We can see the sequence, know the differentes values of P(n), have a quite precise idea of the sequence limit.

  The movements of distraction (to follow each new calculator suggestion), of concretisation (to see at once something) and of fixation on a single application (by trying and trying again, by zooming and zooming again) are very strong in a calculator environment. Following a movements of abstraction or diversification requieres a conscious effort of a mathematician. Seeing and thinking is absolutely not the same thing. And what is true for a teacher is all the more true for a learner ! 3.b. A second example, about students' behaviour Let's look at various students work about a particular problem solving. The students have to find, according to the different values of n, the number of solutions of to give the problem to the calculator, before thinking about. But what is the unknown ? He tries x, then n, without any success. At last, he repeats the same question, in the approximate mode. No more success… We recognize here a movement of calculator concretisation (to see what happens with the calculator) and of fixation in the same application, but in a quasi random way (it's quite similar as throwing a dice…) : trying to translate the question for the calculator on each possible (or imaginative) form. Artigue (1995) calls this kind of behaviour a « fishing behaviour ». After this first attempt, a second one, after changing the problem form. This student remember it's possible to solve a problem by studying some functions variations. He tries to derivate the exponential function with the calculator. (I ask to him : « you don't know this derivative ? », he answers : « yes, but I prefer to check it ! »). Alas, he uses a false syntax, and obtains some answers he can'nt understand (false syntax for the derivative, false syntax for the exponential function-he uses the letter e). Again the same movements of calculator concretisation and of fixation one a same application, but in an approximative form. On the contrary, a « good » student of the same class follows first a movement of abstraction. With a small drawing (cf. figure 2 below), he sets the differents elements of the problem. He reminds the theoritical result : the exponential function increases faster than each power function. Then he supposes the existence of three solutions (we can observe the deformation of the curve in order to obtain the third solution).

Figure 2 After

 2 Figure 2

-

  Figure 3


Andrea De Maio
email: andrea.de-maio@laas.fr

Simon Lacroix
email: simon.lacroix@laas.fr

Towards a versatile framework to integrate and control perception processes for autonomous robots

I. INTRODUCTION

Perception is at the heart of autonomous robots, as it is the way through which the decision-making processes get information on the environment and the robot itself. In the last years, the availability for different types of sensors in robotics has greatly improved, allowing the integration of a significant variety of sensors on the same platform. Jointly, the state of the art of data processing and data fusion became much richer, offering a broad choice of solutions. Despite this richness, data fusion and perception processes are still tailored, by the robotic engineers, to the task they are needed for, with the sequences of data processes being defined by initially hardcoded scripts. Yet, perception is an active process: deciding which data to acquire and how to process them yields the possibility to optimize the throughput of perception, that is the relevance and quality of the information provided to the decision making processes. It is indeed short sighted to think that the configuration of the perception processes does not have to be changed regardless of the task a robot is carrying out, or the context within which the task is executed. This principle of actively controlling perception becomes even more relevant when thinking of the vast perception possibilities nowadays robots can be endowed with. The fact that perception is active has been early identified by the robotics community [START_REF] Bajcsy | Active perception[END_REF], which has mostly treated the problem as to purposefully change the sensors state parameters according to sensing strategies. Contributions on active perception often turned into active vision, due to the predominant diffusion of visual sensors in robotic platforms [START_REF] Chen | Active vision in robotic systems: A survey of recent developments[END_REF]). In the many works published in this field, one particular task related to computer vision is considered, e.g. inspection [START_REF] Trucco | Model-based planning of optimal sensor placements for inspection[END_REF], grasping [START_REF] Motai | Hand-eye calibration applied to viewpoint selection for robotic vision[END_REF], or object modeling [START_REF] Chen | Vision sensor planning for 3-d model acquisition[END_REF][START_REF] Pito | A solution to the next best view problem for automated surface acquisition[END_REF]).

Even though there have been multiple efforts in active perception, there is the lack of a system abstracting from the type of sensor or from the nature of the task. To the best of our knowledge, there exist no work on defining a generic architecture modeling perception process being sensor and task agnostic. In this paper, we present our ongoing work on the definition of an approach to design an architecture controlling a broad perception layer, that allows to dynamically assemble, sequence and control perception processes on board an autonomous robot.

II. COMPOSING PERCEPTION TASKS

In the context of autonomous navigation, Visual Odometry, SLAM and the generation of a Digital Elevation Map (DEM) describe processes which are composed of several signal processes that organized together return the desired output. Let us consider the Visual Odometry example. This process can be considered as a perception compound (PC), composed of atomic signal processing and data fusion functions, refined to a lower level of granularity that can be named Perception Nodes (PN): the PC Visual Odometry can be summarized as the composition of three main PNs: (1) a keypoint extractor, which takes as input a view of the world (e.g. through a camera) and outputs a set of keypoints (KP); (2) a data association algorithm matches two different sets of KP after a small motion is performed; (3) a motion estimator which computes motion thanks to the matches and the absolute 3D position of each KP w.r.t. to a given frame. The final output of the overall process is a 3D transformation between the two time steps at which the features were collected, along with an uncertainty measure of the transform. This "composition approach" is applicable to all the PCs a given robot must be endowed with -note though they do not always follow a pipeline model: some atomic processes can be asynchronously invoked, e.g. a loop closure algorithm, and some feedback sequences can be defined.

III. MODELLING PERCEPTION ACTIVITIES

In order to allow the dynamic control of perception activities, we need to model them, so as to explicit concepts such as composability, performance and cost, upon which one can defined a principled reasoning formalism. As first step to structure the perception activities, it is important to build a taxonomy of PNs which is as vast as possible in terms of inclusion of signal processes and data fusion functions. The importance of a taxonomy is twofold: on a high level of abstraction it is crucial to understand the nature of perception processes, to classify them under the output they produce and to structure them in a way that eases the use of perception and data fusion processes (PNs) as building blocks of perception-driven tasks (PCs). On a more functional level, such classification will be an element of guidance for the search in the process of autonomously building PCs, given a certain availability of PNs.

The proposed model of a PN x is as follows:

x = U, Y, Σ, Θ, Ψ, H (1)
where U and Y are respectively the input(s) and the output(s), Σ is a set of figures of merit related the PN, that qualifies the produced output, Θ is the set of controllable parameters of the PN, Ψ represents the context information, that is the uncontrollable parameters influencing the performance of x (which are not necessarily fully observable), and H is a set of measurements on how x impacts on available resources (CPU, Memory, Power, etc: these are cost information). The proposed model can generally fit any perception processes and can be exploited to compare them. In particular, we aim to taxonomize PNs over the output they produce: this implies that Σ should characterize the quality of the output so to make processes comparable over Y . In perception, performance criteria are mostly related to uncertainty due to the nature of sensors robotic systems: for instance it is common to asses the quality of an estimation process by its covariance matrix [START_REF] Mihaylova | Active robotic sensing as decision making with statistical methods[END_REF]. Whilst estimation PNs and PCs are easily evaluated with such a criterion, there are many more figures of metric which can be applied to other kind of processes. They can either be generic (e.g. Precision/Recall or F1 score for a data association process), or tailored to the process itself. For instance, the quality of the output generated from a feature detector can be described by the number of generated features, possibly weighted by a quality assessment of the features.

IV. PLANNING PERCEPTION ACTIVITIES

To select and configure the perception processes to be executed to achieve a given perception task, one shall take into account the possibility to vary Θ in order to increase the quality of the output. One shall also reason on Ψ and explore two main possibilities: (1) given Ψ, x , tune Θ to maximize/minimize Σ, (2) given Ψ, x s , select an alternative PN x c so that ∃ σ i (x c ) > σ i (x s ) 1 , with σ i ∈ Σ. All these choices shall consider the availability of on-board resources (CPU, Memory, Power) and time. The system must be able to sacrifice performances in order to satisfy external constraints dictated by H. This impacts on both cases shown before: the change of Θ may influence H as well as the selection of an alternate PN. One of the crucial challenges is going to be the formalization of Ψ, which more than H shall be the driving factor of the search. The information extrapolated from the perceivable world can be vary and hardly representable but the system shall be able to identify the knowledge usable by the reasoning module and to formalize it.

V. A TAXONOMY OF PERCEPTION NODES

As introduced in Sec.III, a taxonomy of perception nodes is fundamental to obtain a deep understanding of the perception layer. Our aim is to classify and characterize a broad set of data fusion and signal processes, addressing the problem in 1 If we are trying to maximize Σ. σ i (xc) < σ i (xs) otherwise. the vastest possible manner. One of the necessities originating from this approach involves the modeling of sensors. In fact, to enable autonomous composition and reconfiguration of PCs, it is necessary to treat sensors as real PNs outputting data structures. By doing so the only thing that matters in terms of composability is the produced data format, while the sensor type loses importance. For instance, in some PCs a Time-of-Flight camera and a Stereo Camera (or a LiDAR if color is not necessary) are completely interchangeable as they are both able to produce point clouds. Sensors shall often represent the start node for any PC. A large set of relevant sensory data structures have been identified: Image Data, Depth Image Data, Depth Data, Planar Depth Data, Force Data, Torque Data, Angular Data, Angular Magnetic Data, Angular Velocity Data, Angle of Acceleration Data. Every data type is produced by one or more sensors which can be considered as PN requiring "0 inputs". Perception processes having only one input are taxonomized as signal processing nodes. These PNs usually can either produce output having the same semantic as its input, e.g. filters such as a pass-through filter or a stastical outlier removal filter; or output different data types by just having one input. This is the case of feature detectors which can be further subdivided in several categories (edge detection, feature extraction, corner detection). Finally there are the PNs dealing with more than two inputs. These processes fall into the data fusion category and are classified through the relation lying between their input and output. Three metacategories have been identified: Same-Input Same-Output, Same-Input Different-Output and Different-Input Different-Output. The first one features two main categories of data fusion techniques. Data accretion concerns the increase in volume of same type of data, like map/image stitching algorithms; while estimation filters normally fuse two different data sources to reduce the information uncertainty relative to the data. A typical example of estimation filters is represented by the Extended Kalman Filter. Same-Input Different-Output and Different-Input Different-Output nodes are respectively represented by data association, like feature matchers, and data cooperation techniques. A partial tree view of this taxonomy which does not show instances but only categories is present in Fig. 1.

All these techniques are generically combined together so as to form PCs. At the end of such compounds we usually find dedicated data structures which are usable by high-level decision making functions. These are the final products of every PC, a global path planner will make use of a digital elevation map, while a local path planner may need to read pose information incoming from visual odometry.

The proposed model is still preliminary, and in particular lacks the definition of the geometric dimensions of some perception nodes (in particular for nodes of Sensor type, for which the notion of field of view and resolution must be explicited).

Finally, a major challenge in the configuration of perception compounds will be to have a rich representation of the world. The information modeled shall be the main driving factor for tuning parameters relative to algorithms and for on-board reconfiguration of perception compounds.

See[START_REF] Mikolajczyk | A performance evaluation of local descriptors[END_REF]

See[START_REF] Mihaylova | Active robotic sensing as decision making with statistical methods[END_REF][START_REF] Nistér | Visual odometry for ground vehicle applications[END_REF]

Perception Nodes

How many inputs?

Data Fusion

What is the relation between input and output?

VI. A USE CASE

Let us instantiate a PC into the proposed model. For this example we will consider again the Visual Odometry case. We already introduced in Sec.II its nature and its processing steps but it is now possible to instantiate the PNs composing it, in the context of the proposed taxonomy. Moreover, as mentioned in Sec. V sensors (and their data processing algorithms) are PNs too. A stereo computation algorithm, which we will consider as step 0, has to be added to the PC in case. Therefore, we will consider 4 PNs which are here depicted and instantiated. Note some part of their model are not definite (as shown by the numerous "..."), these 4 instanciations being preliminary: the purpose of presenting them is to better grab the generic PN model definition.

A stereo camera requires no data in input and produces an image and a point cloud. Recalling the model described in Eq.1, its instantiation is as follows:

For the first computation step, a keypoint extractor is used. These kind of nodes compute descriptors for selected regions of interest which are reliably recognizable under changes in scale, light and distortion. One of the possible choice could be the SIFT descriptor.

Type: Keypoint extractor

This process takes in input a black and white image and produces a set of keypoints at a given time instant t. The KPs are expressed in the sensor reference frame. In the case of SIFT, parameters are the peak threshold and the edge threshold.

The second step involves a data association algorithm. Its task is to match KPs extracted at different time steps.

2 An acceptable feature is a feature that has been evaluated under some utility/quality terms, see for instance [START_REF] Hartmann | Predicting matchability[END_REF] for feature matchability Type: Data association Instance: SIFT Matcher

Since we used the SIFT descriptor to extract keypoints from the images, we will use an algorithm that matches SIFT features which can be called SIFT Matcher. The algorithm receives in input two different sets of KPs extracted at time t and t+1, then proceeds to find all the possible matches in these two sets. The output is a set of pair of KPs where every pair represent the same 3D point in a world reference frame. Finally, the last step of visual odometry requires a motion estimation algorithm to estimate the motion between the two considered time instants.

Type: Motion estimation

Least Square Method takes in input the set of matches computed by the chosen matcher and produces two pieces of output: an homogeneous transformation minimizing the reprojection error and an uncertainty measure expressed as a covariance matrix [START_REF] Howard | Real-time stereo visual odometry for autonomous ground vehicles[END_REF].

VII. CONCLUSIONS

We have presented on-goig work on an approach to model and taxonomize perception processes. We have shown how task oriented perception processes can be subdivided in several sub-processes that we refer to as perception nodes. The composition of such nodes can be generalized and is driven by the matching of inputs and outputs. Although this is only a formalization, this modeling will enable autonomous reconfigurability of perception processes thanks to the use of a decision making framework, for instance planning approaches: planning generally works well with well defined rules which permit to abstract from the specific domain of the problem. This is reflected in the abstraction from the nature of the sensors or of the perception nodes. Other decisional frameworks may be considerd, such as decision theory, constraint-based or resource-based programming, optimisation...


Deise S Maia

Arnaldo De A Araujo

Jean Cousty

Laurent Najman

Benjamin Perret

Hugues Talbot

Evaluation of combinations of watershed hierarchies

Keywords: Watershed hierarchies, Quasi-flat zones hierarchies, Combination of saliency maps

The main goal of this paper is to evaluate the potential of some combinations of watershed hierarchies. We also propose a new combination based on merging level sets of hierarchies. Experiments were performed on natural image datasets and were based on evaluating the segmentations extracted from level sets of each hierarchy against the image ground truths. Our experiments show that most of combinations studied here are superior to their individual counterparts, which opens a path for a deeper investigation on combination of hierarchies.

Introduction

In this work, we investigate combinations of watershed hierarchies through their saliency maps. Other approaches can be found in the literature, such as [START_REF] Fehri | Automatic selection of stochastic watershed hierarchies[END_REF][START_REF] Pont-Tuset | Multiscale combinatorial grouping for image segmentation and object proposal generation[END_REF]. Figure 1 provides an example of two different watershed hierarchies, an areabased one and a dynamics-based one, and their combination, which are built from successive filterings of an initial watershed segmentation. Hierarchies are represented thanks to their saliency maps [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF][START_REF] Cousty | Incremental algorithm for hierarchical minimum spanning forests and saliency of watershed cuts[END_REF][START_REF] Cousty | Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency maps[END_REF][START_REF] Kiran | Fusion of ground truths and hierarchies of segmentations[END_REF][START_REF] Najman | Geodesic saliency of watershed contours and hierarchical segmentation[END_REF], i.e., edge-weighted graphs which comprise information of hierarchical contours. We observe that the sky is oversegmented at high levels in the first hierarchy and that the beach ground is oversegmented in the second hierarchy, but this is balanced by their combination. This observation is general, and no hierarchy is optimal for a whole image. We expect combinations of hierarchies to perform better, as illustrated for instance in [START_REF] Cousty | Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency maps[END_REF], where the authors have shown that a simple combination of area and dynamics-based watershed hierarchies produces better visual results than the individual hierarchies.

The main contributions of this paper are:

-Investigation of combinations of hierarchies through their saliency maps using supremum, infimum, linear combination and concatenation functions; and -Evaluation of supervised and unsupervised combinations of watershed hierarchies.

The plan of the paper is the following. We first review basic notions on hierarchies and saliency maps and present several ways for combining hierarchies (section 2). Then we describe our assessment methodology in section 3. The evaluation results and improvement of combinations compared to the individual watershed hierarchies are described in section 4.

Combination of hierarchies

In this section, we give the notations needed to define combinations of hierarchies through saliency maps and the types of combinations investigated here.

Hierarchy of Quasi Flat Zones and Saliency Map

This section presents the formal definitions of graphs, hierarchy of partitions, quasi-flat zones hierarchy and saliency map.

A graph is a pair G = (V, E), where V is a finite set and E is a set of unordered pairs of distinct elements of V , i.e., E ⊆ {{x, y} ⊆ V, x = y}. Each element of V is called a vertex or a point (of G), and each element of E is called an edge (of G).

Let G = (V, E) be a graph. Let X be a subset of V . A sequence π = x 0 , . . . , x n of elements of X is a path (in X) from x 0 to x n if, for any i in {1, . . . , n}, {x i-1 , x i } is an edge of G. The subset X of V is said to be connected if for any x and y in X, there exists a path from

x to y. A subset X of V is a connected component of G if X is connected and, if, for any connected subset Y of V , if X ⊆ Y , then we have X = Y . The set of connected components of a graph G is denoted by C(G).
Let V be a set. A partition of V is a set P of non empty disjoint subsets of V whose union is V . If P is a partition of V , any element of P is called a region of P. Given a graph G = (V, E), the set C(G) of all connected components of G is a partition of V .

Let V be a set and let P 1 and P 2 be two partitions of V . We say that P 1 is a refinement of P 2 if every element of P 1 is included in an element of P 2 . A hierarchy (of partitions) is a sequence H = (P 0 , . . . , P n ) of partitions of V such that P i-1 is a refinement of P i , for any i in {1, . . . , n} and such that P n = {V }. A partition of a hierarchy H is called a level set of the hierarchy.

Let G be a graph, if w is a map from the edge set of G to the set R + of positive real numbers, then the pair (G, w) is called an (edge-)weighted graph. If (G, w) is an edge-weighted graph, for any edge u of G, the value w(u) is called the weight of u (for w).

Important notation. In the sequel of this article, we consider a weighted graph (G, w). We assume that the vertex set of G is connected. We also denote by W the range of w, i.e., the set {w(u) | u ∈ E} and by W • the set W ∪ {k + 1}, where k is the greatest value of W.

Let λ be any element in R. We denote by G λ the graph (V, E λ ) such that

E λ = {e ∈ E | w(e) < λ}. The set C(G λ ) of all connected components of G λ is called the λ-level partition of G. The sequence QFZ(w) = (C(G λ ) | λ ∈ W • ) (1)
is a hierarchy called the Quasi-Flat Zones hierarchy of w.

The saliency map of a hierarchy H = (P 0 , . . . , P n ) is a map from E to {0, . . . , n}, denoted by Φ(H), such that, for any edge e = {x, y} in E, we have Φ(H)(e) is the greatest value i in {0, . . . , n} such that x and y do not belong to the same region of P i .

In [START_REF] Cousty | Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency maps[END_REF], the authors provide a bijection between saliency maps and hierarchies based on quasi-flat zones hierarchies. Hence, a hierarchy is equivalently represented by its saliency map, a property that is particularly useful in the remaining part of this article.

Note also that, for visualization purposes, when the graph G is associated to a digital image [START_REF] Cousty | Incremental algorithm for hierarchical minimum spanning forests and saliency of watershed cuts[END_REF][START_REF] Cousty | Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency maps[END_REF][START_REF] Kiran | Fusion of ground truths and hierarchies of segmentations[END_REF][START_REF] Najman | Geodesic saliency of watershed contours and hierarchical segmentation[END_REF], saliency maps can be visualized with images, called ultrametric contour maps in [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF], in which the contours brightness is proportional to their saliency values.

Generic scheme of combination of hierarchies

Combining partitions and, a fortiori, hierarchies is not straightforward. This problem has been tackled in [START_REF] Cousty | Raising in watershed lattices[END_REF][START_REF] Cousty | Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency maps[END_REF][START_REF] Kiran | Fusion of ground truths and hierarchies of segmentations[END_REF] thanks to the use of saliency maps and we follow the same approach as used in those papers. More precisely, in order to combine two hierarchies H 1 and H 2 , built from the same fine partition, we proceed in three steps [START_REF] Cousty | Raising in watershed lattices[END_REF][START_REF] Kiran | Fusion of ground truths and hierarchies of segmentations[END_REF][START_REF] Cousty | Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency maps[END_REF]: first the saliency maps of H 1 and H 2 are considered, then the two saliency maps are combined to obtain new weights on the edges of G, and, finally, the combination of hierarchies is the quasi-flat zones hierarchy of the new weight function.

Let F be the set of all maps from E into R + . Let n be any positive integer, any map c from F n into F is called a combining n-weight function.

Given a sequence of hierarchies (H 1 , . . . , H n ) and a combining n-weight function c, the combinations of (H 1 , . . . , H n ) by c is the hierarchy H c (H 1 , . . . , H n ) defined by:

H c (H 1 , . . . , H n ) = QFZ(c(Φ(H 1 ), . . . , Φ(H n ))).
(2)

Combining n-weight functions

We consider three classical functions in the instantiation of the combining nweight function (supremum, infimum and linear combination) and we propose a new type of combination called concatenation of hierarchies.

The supremum, infimum and linear combination functions are respectively denoted by , and Θ . Given a sequence (w 1 , . . . , w n ) of n saliency maps and

a sequence Θ = (α 1 , . . . , α n-1 ) of n -1 values in R such that (α 1 + • • • + α n-1 ) ≤ 1 and α i ≥ 0 for i ∈ {1, . . . , n -1}, the linear combination of (w 1 , . . . , w n ) parametrized by Θ is the sum α 1 w 1 + • • • + α n-1 w n-1 + (1 -α 1 -• • • -α n-1 )w n .
We denote by A the case where the linear combination is equal to the average. One example of combination of hierarchies by infimum is shown in Figure 3.

The concatenation of hierarchies is based on merging different level sets of each hierarchy. This type of combination can be useful, for example, when one hierarchy H 1 succeeds at describing the small details of an image at lower level sets, but fails at filtering the small regions to capture the main large objects at higher level sets. Therefore, it can be interesting to concatenate H 1 with another hierarchy H 2 whose high level sets describe well the important regions in the image. This general idea is represented in Figure 2. Given two weight maps w 1 and w 2 and a threshold value λ, the concatenation of w 1 and w 2 consists in: i) setting to zero all weights of w 2 lower than λ; ii) setting to λ all weights of w 1 greater than λ; and iii) computing the supremum of the two maps obtained at steps i) and ii). More generally, given a sequence (w 1 , . . . , w n ) of n weight maps and a series (λ 1 , . . . , λ n-1 ) of n-1 threshold values in R such that λ 1 < λ 2 < • • • < λ n-1 , we define the concatenation of (w 1 , . . . , w n ) parametrized by (λ 1 , . . . , λ n-1 ), thanks to the combining n-weight function Θ , by:

∀e ∈ E, Θ (w 1 , . . . , w n )(e) = max{T (w 1 (e), 0, λ 1 ), . . . , T (w n (e), λ n-1 , ∞)} (3)
where, given a, b, and c ∈ R, we have T (a, b, c) equals 0 if a is lower than b and equals min(a, c) otherwise. Consequently, given a sequence of hierarchies (H 1 , . . . , H n ) and threshold values Θ = (λ 1 , . . . , λ n-1 ), the concatenation of (H 1 , . . . , H n ) with parameter Θ is H Θ (H 1 , . . . , H n ). One example of concatenation of two hierarchies is shown in Figure 3.

Assessment methodology and set-up of experiments

In this section, we present the assessment methodology used to evaluate hierarchies of segmentations.

Assessment methodology

In order to account for the performance of the different combinations, we use a supervised assessment strategy developed in [START_REF] Perret | Evaluation of hierarchies of watersheds for natural image analysis[END_REF]. This framework evaluates the possibility of extracting a good segmentation from a hierarchy with respect to a given ground-truth, the quality of the extracted segmentation being measured using the Bidirectional Consistency Error (BCE) [START_REF] Martin | An Empirical Approach to Grouping and Segmentaqtion[END_REF]. In order to take account for the hierarchical aspect of the representations, the score of a segmentation is measured against its level of fragmentation, i.e., the ratio between the number of regions in the proposal segmentation compared to the number of regions in the ground-truth segmentation. Two ways of extracting segmentations from a hierarchy are considered. 1) We compute the cut that maximizes the BCE score for each fragmentation level, leading to the Fragmentation-Optimal Cut score curve (FOC). 2) We compute the BCE score of each level set of the hierarchy, leading to the Fragmentation-Horizontal Cut score curve (FHC). A large difference between the FOC and FHC curves, called here fragmentation curves, suggests that the optimization algorithm has selected regions from various levels of the hierarchy to find the optimal cut: the regions of the ground-truth segmentations are thus spread at different levels in the hierarchy. The normalized area under those curves, denoted respectively by AUC-FOC and AUC-FHC, provides an overall performance summary over a large range of fragmentation levels. Since the importance of having high AUC-FOC and AUC-FHC scores varies according to the application, we consider the average of both scores to quantitatively compare hierarchies. The average of AUC-FOC and AUC-FHC will be denoted here by AUC-FOHC.

Set-up of experiments

We describe here the set-up of evaluation of combinations of hierarchies.

In this work, hierarchical watersheds are considered. More precisely, the successive level sets of the considered hierarchies correspond to watershed segmentations of filtered versions of the weight map w, the higher level sets of the hierarchies being associated to the higher level of filterings [START_REF] Cousty | Incremental algorithm for hierarchical minimum spanning forests and saliency of watershed cuts[END_REF]. The successive filtering levels are given by ranking the minima according to extinction values associated with regional attributes: area [START_REF] Meyer | Morphological tools for segmentation: Connected filters and watersheds[END_REF], dynamics [START_REF] Meyer | The dynamics of minima and contours[END_REF], volume [START_REF] Meyer | Morphological tools for segmentation: Connected filters and watersheds[END_REF], topological height [START_REF] Silva | New extinction values from efficient construction and analysis of extended attribute component tree[END_REF], number of minima, number of descendants and diagonal of bounding box [START_REF] Silva | New extinction values from efficient construction and analysis of extended attribute component tree[END_REF]. To shorten the notations, we denote those attributes by Area, Dyn, Vol, Height, Min, Desc and DBB.

The evaluations were performed on the 200 test images of the Berkeley Segmentation Dataset and Benchmark 500 (BSDS500) [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF].

The hierarchies of segmentation are computed from the image gradients obtained from the Structured Edge (SE) detector [START_REF] Dollár | Structured forests for fast edge detection[END_REF], which achieved a high contour detection rate on BSDS500.

Assessment of combinations of hierarchies

In this section we present the experiments with combinations of watershed hierarchies. We then compare the combinations of hierarchies with the individual hierarchies and other two techniques [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF][START_REF] Pont-Tuset | Multiscale combinatorial grouping for image segmentation and object proposal generation[END_REF].

Baseline

Our baseline is the AUC-FOHC scores of individual watershed hierarchies presented in Table 1. The scores were computed over the test set of BSDS500.

Evaluation of parameter-free combinations

The evaluation of parameter-free combinations consisted in computing the AUC-FOHC scores of all combinations of pairs of watershed hierarchies over the test Area DBB Dyn Height Desc Min Vol AUC-FOC 0.603 0.592 0.541 0.560 0.604 0.609 0.617 AUC-FHC 0.423 0.435 0.480 0.493 0.425 0.453 0.465 AUC-FOHC 0.513 0.514 0.510 0.527 0.514 0.531 0.541 Table 1. AUC-FOC, AUC-FHC and AUC-FOHC scores of individual hierarchies computed over the test set of BSDS500. set of BSDS500 using , and A functions. For each pair of watershed hierarchies, H 1 and H 2 , we applied the following combining n-weight functions to their saliency maps: (Φ(H 1 ), Φ(H 2 )), (Φ(H 1 ), Φ(H 2 )) and A(Φ(H 1 ), Φ(H 2 )).

The highest scores achieved by parameter-free combinations is shown in Table 2. We can observe that, for most pairs of watershed hierarchies, the combination with A presents the highest score. In addition, the highest scores are obtained with combinations using A. 2. Combining n-weight functions and highest AUC-FOHC scores obtained from c(Φ(H1), Φ(H2)). For each pair of hierarchies, we have the global combination function which provided the highest AUC-FOHC score and the score obtained from this combination.

H

Evaluation of unsupervised concatenation of hierarchies

We present here the evaluation of concatenation of pairs of watershed hierarchies.

To determine the parameter that should be used in the concatenation of each pair of watershed hierarchies, we analyze their fragmentation curves. For each pair of watershed hierarchies, we check which one presents the highest AUC-FOC and AUC-FHC scores for low and high fragmented segmentations. If one of the hierarchies presents the highest scores for both low and high fragmented segmentations, we do not expect to obtain better results from their concatenation. For example, in the curves of Figure 4, we see that, only for a fragmentation larger than 0 and smaller than approx. 0.65, area outperforms dynamics based watershed hierarchy. Therefore, we conclude that high level sets of area, which are less fragmented, describe an image better than dynamics based hierarchies, and the opposite is true for lower level sets. Hence, the parameters are tuned to concatenate high levels of area to the low levels of dynamics based hierarchy.

In general, the fragmentation curves of concatenations has a smaller difference than the curves of their individual counterparts, which can be seen in Figure 4. This means that the segmentations extracted from the level sets of concatenations are closer to the optimal cuts for each fragmentation level. Also, half of the concatenations tested here presented higher AUC-FOHC scores than the individual watershed hierarchies, as shown in Table 3

H2

Dynamics Height H1

Area DBB Desc Min Vol Area DBB Desc Min Vol AUC-FOC 0.579 0.561 0.586 0.589 0.591 0.579 0.574 0.580 0.582 0.585 AUC-FHC 0.472 0.462 0.462 0.483 0.498 0.472 0.475 0.473 0.485 0.500 AUC-FHCO 0.525 0.511 0.526 0.536 0.545 0.525 0.524 0.527 0.534 0.542 Table 3. AUC-FOC, AUC-FHC and AUC-FHCO scores of Θ (Φ(H1), Φ(H2)), where different values of Θ were used for each concatenation. The AUC-FHCO scores in bold are the ones which are higher than the AUC-FHCO scores of individual H1 and H2 hierarchies.

Evaluation of supervised linear combinations

We present here the evaluation of linear combinations of pairs of watershed hierarchies using learned parameters.

For each pair of watershed hierarchies, we determined the linear combination parameter α that optimizes the AUC-FOHC score on the 300 images of the training set of BSDS500.

From the highest scores reached for each combination, we can see that not all combinations produce relevant results, mainly the ones which do not include either dynamics nor topological height based watershed hierarchies. This is shown in Table 4, that contains the best-fitting parameter for each linear combination and their AUC-FOHC scores. 4. Parameters α and AUC-FOHC scores of each linear combination (α) (Φ(H1), Φ(H2)). The AUC-FOHC scores in bold are the highest scores achieved with linear combination of hierarchies.

One example comparing segmentations extracted from the individual hierarchies based on number of descendants and topological height and their linear combination using the learned parameters is shown in Figure 5. The linear combination computed for this single image presents a higher AUC-FHC score than the individual hierarchies (0.604 versus 0.465 and 0.551) and a slightly higher AUC-FOC score (0.802 versus 0.801 and 0.708). Based on the AUC-FHC score, we expect this combination to have better horizontal cuts than the the individual hierarchies. We can see that the segmentation extracted from the combination separates better the main regions in this image: sky, mountains and the two sea regions.

Comparison with other techniques

In order to have a more complete evaluation, we have performed a comparison with a state-of-the-art method [START_REF] Pont-Tuset | Multiscale combinatorial grouping for image segmentation and object proposal generation[END_REF], called Multiscale Combinatorial Grouping (MCG), and an well-know technique [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF], named Ultrametric Contour Map (UCM), including different assessment measures: Precision-Recall (PR) for boundaries [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF] and the Marked Segmentation [START_REF] Perret | Evaluation of morphological hierarchies for supervised segmentation[END_REF] (Figure 6).

The PR for boundaries score is also assessed on BSDS500 and it evaluates the matching between the boundaries of a given segmentation and the ground-truth segmentation. The PR curves are built from the F-measure scores of the level sets of the hierarchies and are summed up in two measures: Optimal Dataset Scale (ODS) and Optimal Image Scale (OIS).

The Marked Segmentation aims to measure the difficulty of extracting a good segmentation in a hierarchy given sets of background and foreground markers. The markers are generated through erosion, skeletonization and frame of the ground truth, and the score of each segmentation is given by the F-Measure computed over the Weizmann and Grabcut datasets. Each pair of Background-Foreground markers are denoted by the method used to compute them. Figure 6 shows the Marked Segmentation results for three pairs of background and foreground markers: Er-Er, Fr-Sk and Sk-Sk, in which Er, Fr and Sk stand for Erosion, Frame and Skeleton, respectively. The box plots show the quartile distribution of scores on both datasets. The median score of those three pairs of markers is denoted by ODM. Therefore, the best hierarchies in terms of Marked Segmentation correspond to the ones with highest ODM scores and most compressed box plots.

MCG explores hierarchies of images at different resolutions based on combined local contours cues. Single segmentations at different resolutions are aligned and combined into a single saliency map. The UCM described in [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF] is obtained from the watershed transform of the output of a high quality contour detector.

Our best combination does not achieve the PR and fragmentation scores presented by MCG and UCM, but it outperforms UCM in terms of Fragmentation Curves for non-horizontal cuts and presents competitive Marked Segmentation results compared to MCG and UCM.

Discussion and conclusion

This paper shows the potential of combination of hierarchies through the evaluation of supervised and unsupervised combinations of watershed hierarchies. We evaluated combinations of pairs of watershed hierarchies using classical functions (infimum, supremum and linear combination) and a newly proposed method called concatenation of hierarchies. All combinations described here were proved to be useful for at least one pair of watershed hierarchies. However, not all pairs of hierarchies presented significant results when compared to the individual hierarchies. For example, the parameter-free combination of area and volume based hierarchies with highest score did not present a score superior to the individual volume based hierarchy. Also, the best-fitting parameter α for the linear combination of area and volume based hierarchies is equal to zero. This means that none of the combinations of area and volume tested here was superior to both individual hierarchies.

We observed that the combinations with highest scores contained either dynamics or topological height based hierarchies, but not both. Both dynamics and topological height are related to depths. Therefore, this is not surprising that the combination of hierarchies based on those attributes does not bring new interesting results. This is also valid for other similar attributes as area and diagonal of bounding box.

Among all linear combinations with learned parameters, the ones with best performance presented scores close to the combination by average, with the learned parameter α ranging between 0.39 and 0.51. So, the combination of hierarchies by average seems to be a valuable and simple choice.

The framework presented in this article can be used to combine other types of hierarchies, but we have not investigated whether the results could be interesting. The main point is that the contours of watershed hierarchies overlap and this ensures that none of their combinations will present duplicated contours for a same boundary of the input image, which could happen using other hierarchies.

Since we have explored only a few types of combinations, there is still room to find new functions able produce relevant results. The evaluation of combinations performed here also invites us to go a step further in this topic, for example, by learning how to choose the optimal combination parameters for each image.

Fig. 1 .

 1 Fig. 1. First row from left to right: original image, saliency map and the 50 th highest level set of area based hierarchy. Second row from left to right: saliency map and the 50 th highest level sets of dynamics and the combination of area and dynamics based hierarchies.

Fig. 2 .

 2 Fig. 2. Concatenation of low levels of H1 with high levels of H2.

Fig. 3 .

 3 Fig. 3. Illustration of combination by infimum and concatenation of a pair of hierarchies. First row from left to right: H1, Φ(H1), H2 and Φ(H2). Second row from left to right: (Φ(H1), Φ(H2)), QFZ( (Φ(H1), Φ(H2))), (2) (Φ(H1), Φ(H2)) and QFZ( (2) (Φ(H1), Φ(H2)))

Fig. 4 .

 4 Fig. 4. Fragmentation curves of non-horizontal and horizontal cuts of the concatenation of area and dynamics based watershed hierarchies. The 10 th highest levels of area were concatenated to the lower level sets of the dynamics based hierarchy.

  0.92 α = 0.60 α = 0.51 α = 0 α = 0.11 α = 43 α = 0.35 α = 0.19 α = 0.07 α = 0

Fig. 5 .

 5 Fig. 5. From left to right: original image, saliency map and the 5 th highest level set of three hierarchies: number of descendants and topological height based hierarchies, and their linear combination using learned parameters.

  .

	Fragmentation curves of non-horizontal cuts	Fragmentation curves of horizontal cuts
		0.9									0.9
		0.8									0.8
	Optimal Score	0.3 0.4 0.5 0.6 0.7								Optimal Score	0.3 0.4 0.5 0.6 0.7
		0.2									0.2
		0.1									0.1
		0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8
					Fragmentation							Fragmentation
					Area, AUC-FOC: 0.60				Area, AUC-FHC: 0.42
				Dynamics, AUC-FOC: 0.54			Dynamics, AUC-FHC: 0.48
			Concatenation, AUC-FOC: 0.58		Concatenation, AUC-FHC: 0.47

  Comparison of our linear combination of area and topological height with MCG and UCM: Precision-recall (PR) for boundaries, Fragmentation curves of nonhorizontal cuts (plain curve) and horizontal cuts (dashed curve), and Marked Segmentation of three pairs of markers.

			PR-Curve on boundaries				Fragmentation curves		Marked Segmentation
		0.9									0.9								0.9
		0.8									0.8								0.8
	Precision	0.3 0.4 0.5 0.6 0.7								Optimal Score	0.3 0.4 0.5 0.6 0.7								Score	0.3 0.4 0.6 0.7 0.5
		0.2									0.2								0.2
		0.1									0.1								0.1
		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	0.2	0.4	0.6	0.8	1	1.2	1.4	1.6	1.8
						Recall								Fragmentation			Er-Er	Fr-Sk	Sk-Sk
											Area, ODS: 0.66, OIS: 0.70, AUC-FOC: 0.60, AUC-FHC: 0.42, ODM: 0.954
											Height, ODS: 0.71, OIS: 0.73, AUC-FOC: 0.56, AUC-FHC: 0.49, ODM: 0.949
					Linear combination of Area and Height, ODS: 0.71, OIS: 0.74, AUC-FOC: 0.62, AUC-FHC: 0.52, ODM: 0.956
											MCG, ODS: 0.74, OIS: 0.78, AUC-FOC: 0.64, AUC-FHC: 0.55, ODM: 0.955
											UCM, ODS: 0.73, OIS: 0.77, AUC-FOC: 0.61, AUC-FHC: 0.53, ODM: 0.954
	Fig. 6.

This work is partially supported by the Labex Bézout.


Elom Ayih Domlan
email: elom-ayih.domlan@ensem.inpl-nancy.fr

José Ragot
email: jose.ragot@ensem.inpl-nancy.fr

Didier Maquin
email: didier.maquin@ensem.inpl-nancy.fr

Active Mode Estimation for Switching Systems

Switching systems are a particular class of hybrid systems. They are described by several operating regimes, called modes, each of them being active under certain particular conditions. When the switching mechanism is perfectly known, it is easy to handle such systems because the knowledge of the active regime at any moment is available. On the other hand, in the case where there is no information about the switching mechanism, the situation is more complicated. In this configuration, it is difficult to carry out a fault diagnosis scheme or to synthesize a control law. This paper addresses the issue of the determination of the active mode at any moment, using only the system's input/output data. Conditions that guarantee the uniqueness of the determined active mode are also given.

I. INTRODUCTION

The modeling of complex systems often leads to complex non-linear models. To get rid of the obtained model's complexity, a widely used modeling strategy is to represent the system's behavior using a set of models with simple structure, each model describing the behavior of the system in a particular operating zone. Within this modeling framework, hybrid models are very successful in representing such processes.

Hybrid models [START_REF] Heemels | Equivalence of hybrid dynamical models[END_REF], [START_REF] Juloski | Hybrid Systems: Computation and Control[END_REF] characterize physical processes governed by continuous differential equations and discrete variables. The process is described by several operating regimes, called modes, and the transition from one mode to another is governed by a mechanism which depends on the system's variables (input, output, state) or external variables (human operator for instance).

The global behavior obtained for the modeled complex system is strongly related to the nature of the procedure managing the transition from one mode to another. When the transition from one mode to another is abrupt, one obtains a particular, but significant, class of systems namely switching models. This class of models is widely used because tools for analysis and control of linear systems are well mastered and, moreover, many real processes can be represented by models belonging to this class.

Research on switching systems is mainly focused in the fields of identification [START_REF] Juloski | Hybrid Systems: Computation and Control[END_REF]- [START_REF] Bemporad | Data classification and parameter estimation for the identification of piecewise affine models[END_REF], control [START_REF] Bemporad | Control of systems integrating logic, dynamics and constraints[END_REF], stability analysis [START_REF] Branicky | A unified framework for hybrid control: model and optimal control theory[END_REF] and state estimation [START_REF] Ackerson | On state estimation in switching environnements[END_REF], [START_REF] Balluchi | Design of observers for hybrid systems[END_REF]. The knowledge of the mode describing the evolution of the system at any moment, this mode being called active mode, is a crucial information that simplifies the application of the results coming from the fields of identification, control, stability analysis and state estimation. This paper presents a method for the determination of the active mode of a switching system at any instant using the measurement data (input and output) of the system.

Section II introduces the retained model for the representation of switching systems and states the problem under investigation. The results for the determination of the active mode are presented in section III. Conditions guaranteeing the discernability of the various modes are then formulated in section IV. In order to illustrate the proposed methods, an academic example is shown in section VI .

II. POSITION OF THE PROBLEM

Let us consider the system represented by equation (1):

x (k + 1) = A µ k x (k) + Bu (k) y (k) = Cx (k) (1)
x

∈ R n , u ∈ R m , y ∈ R p µ k ∈ {1, 2, . . . , s}, s ∈ N * \{1}
Equation (1) represents a switching system with s operating regimes or modes. The variables u(•), y(•) and x(•) respectively stand for the input, the output and the state of the system. The switchings are introduced by means of the system's state matrix which takes its value in a finite set A = {A 1 , A 2 , . . . , A s } which is a priori known. This formulation does not restrict at all the modeling of the switching system and the results presented in this paper can be extended to the case where the matrices B and C also take different values. The variable µ (•) denotes the active mode at any moment. For example, if one has µ k = i, i ∈ {1, 2, . . . , s}, the system is said to be in the mode i at the instant k. We assume that the switchings are triggered by unknown external variables and then, the mode sequence is arbitrary and independent of the system's variable (input, output and state).

Coming from (1), we wish to recover the active mode (or the value taken by µ (•) ) at any moment, using only the system's input/output data on a finite observation window. We introduce the following definitions:

Definition 1 (Path): A path µ is a finite sequence of modes: µ = (µ 1 • µ 2 • . . . • µ h ). The length of a path µ is denoted |µ| and for µ = (µ 1 • µ 2 • . . . • µ h ), one has |µ| = h. The set of all paths of length h is denoted Θ h . µ [i,j]
is the infix of the path µ between i and j: µ

[i,j] = (µ i • µ i+1 • . . . • µ j ).
Definition 2 (Observability matrix): The observability matrix O µ,h of a path µ ∈ Θ h is defined as :

O µ,h =        C CA µ1 . . . C A µ h-1 A µ h-2 • • • A µ1 h-1        (2)
In order to provide more explanation about definitions (1) and ( 2), let us consider again the model of equation ( 1) with s = 2 (two modes). Table I shows the set Θ 3 of all paths of length 3 for this system.

µ k-3 1 1 1 1 2 2 2 2 µ k-2 1 2 1 2 1 2 1 2 µ k-1 1 1 2 2 1 1 2 2
The path µ 4 (fifth column of table I) describes a mode sequence in which the system is in mode 1 at the time k -3, then switches into mode 2 at k -2 and stays in mode 2 at the instant k -1. In this configuration, the corresponding path is

µ 4 = (1 • 2 • 2). For the path µ 4 = (1 • 2 • 2)
, the observability matrices is defined as:

O µ 4 ,h =     C CA 1 CA 2 A 1 CA 2 A 2 A 1     (3)
Definition 3 (Active path): On a finite observation window [kh, k], the active path µ * is the one describing the true mode sequence on the observation window.

From definitions 1 and 3, the estimation of the active mode at any moment is equivalent to the determination of the path describing the true mode sequence on a finite observation window. For that, throughout the remainder of this paper, we will focus on the recovery of the active path on an observation window.

III. ACTIVE PATH DETERMINATION

The determination of the active path can be formulated as a recursive problem applied to a sliding window. On a time window [kh, k], equation (1) can be written as:

O µ,h x (k -h) =    y (k) . . . y (k -h)    -T µ,h    u (k) . . . u (k -h)    (4)
where T µ,h is a Toeplitz matrix defined by :

T µ,h =        0 0 CB 0 . . . C A µ k-1 . . . A µ k-h+1 h-1 B C A µ k-1 . . . A µ k-h+2 h-2 B . . . 0 0 0 0 . . . . . . . . . CB 0     
(5) Equation ( 4) can be written in a more compact way:

Y k-h,k -T µ,h U k-h,k = O µ,h x(k -h) (6)
where

Y k-h,k = y (k) y (k -1) . . . y (k -h) T and U k-h,k = u (k) u (k -1) . . . u (k -h) T .
The relation ( 6) links on the time window the system's input and output to the initial state x(kh) of the system on the observation window. We introduce the following proposition:

Assumption 1: The observability matrices O µ,h of the paths µ generated on the observation window [k -h, k] are all of full rank: rank (O µ,h ) = dim (x) = n, ∀h ≥ n.
The existence of an integer h, such that assumption 1 holds, has been analyzed in [START_REF] Babaali | Pathwise observability and controllability are decidable[END_REF] and is linked to pathwise observability that have been furthermore shown to be decidable. Moreover, assumption 1 implies that all the pairs (C, A µ h ) are observable for any µ h , µ h ∈ Θ h .

Using assumption 1, one can define a projection matrix1 Ω µ,h in such a way that Ω µ,h O µ,h = 0, i.e. Ω µ,h is selected as a basis for the left null space of O µ,h . Next, residuals r µ,h (•), independent of the initial state x(kh), to be used for detection of the active path on the observation window can be defined as:

r µ,h (k) = Ω µ,h (Y k-h,k -T µ,h U k-h,k ) (7)
The r µ,h (•) residuals depends only on the system's input and output and their calculation requires the preliminary determination of all matrices Ω µ,h . Theorem 1 (Active path determination): The active path µ * describing the true mode sequence on a time window [kh, k] satisfies:

r µ * ,h (k) = Ω µ * ,h (Y k-h,k -T µ * ,h U k-h,k ) = 0 (8)
To recover the true mode sequence µ * from the system's measurements, one can proceed in the following way:

• first, all the possible paths of length h are built on the time window [kh, k]. This is equivalent to finding all the matrices O µ,h .

• knowing the matrices O µ,h , the projection matrices Ω µ,h are easily calculated. • from the matrices O µ,h and Ω µ,h , one can form the residuals r µ,h (•) using the system's measurements. • the active path is recovered from the system's measurements by testing the residuals r µ,h (•) magnitude and it corresponds to the one which residual is equal to zero.

A. On the number of path

It is easy to see that the enumeration of all paths on a time window [k -h, k] introduces a problem of combinative explosion related to the number of modes and the length of the observation window. Indeed, the number of residuals r µ,h (•), µ ∈ Θ h , to be calculated is equal to s h and quickly grows with the length h + 1 of the observation window and the number s of modes. Then, the use of all paths on a time window is awkward and computationally demanding. In practice, all paths µ ∈ Θ h do not have to be considered at every moment. When at a time k 0 , the active path on an observation window [k 0h, k 0 ] is identified, it is not necessary to test the s h residuals at the next instant k 0 + 1.

Only the paths µ

∈ Θ h with infixes µ [k0+1-h,k0-1] identical to the infix µ * [k0+1-h,k0-1]
of the previously recovered path µ * at k 0 are considered at the next instant k 0 + 1. For example, from table I, one can see that if at time k 0 , the recovered active path is the path

µ * = (1 • 2 • 1) (path µ 2 ), then only the paths µ = (2 • 1 • 1) (path µ 5 ) and µ = (2 • 1 • 2) (path µ 7
) have to be considered at the next step k 0 + 1. Moreover, assuming that the minimum sojourn time in a mode is greater than the length of the observation window, one can limit the number of generated paths by only considering paths that describe the mode sequence when the system remains in the same mode all over the duration of the observation window, i.e. µ = (i • i •. . . •i), i ∈ {1, 2, . . . , s}. In table I, the paths µ 1 and µ 8 describe the mode sequence when the system remains respectively in mode 1 and 2 all over the duration of the time window. Nevertheless, the reduction of the number of residuals comes at the expense of a delay in the estimation of the switching time from one mode to another. The recognition of the active path cannot take place as long as the switching instant is in the observation window. Thus, a maximum delay equals to the length of the observation window exists. Prior knowledge on the process such as "prohibited" switching sequences or minimal time between two consecutive switchings, can also help to limit the number of generated residuals or paths to be considered.

IV. PATH DISCERNABILITY

In what follows, we are interested in the conditions guaranteeing the discernability of the various paths enumerated on an observation window. These conditions ensure the uniqueness of the recovered active path µ * during the path recognition process. Discernability guarantees that two different modes never induce the system in the same dynamics on a finite time window. Definition 4 (Path discernability): Two paths µ 1 ∈ Θ h and µ 2 ∈ Θ h are discernible on an observation window [kh, k] if their respective corresponding residuals r µ 1 ,h (•) and r µ 2 ,h (•) are not simultaneously null when one of the two paths is active on the considered observation window.

In order to establish the discernability conditions of two different paths, let us consider two paths µ 1 ∈ Θ h and

µ 2 ∈ Θ h on an observation window [k -h, k]. We denote Y µ 1 k-h,k (respectively Y µ 2 k-h,k
) the output vector related to the system when it undergoes the path µ 1 (respectively µ 2 ). We suppose that at an instant k, the active path on the observation window is the path µ 1 . This information being unknown, we have to analyze the possibilities that the path µ 1 or the path µ 2 are in adequacy with the system's data. From [START_REF] Branicky | A unified framework for hybrid control: model and optimal control theory[END_REF], the expressions of the residuals r µ 1 ,h (•) and r µ 2 ,h (•) are given by :

r µ 1 ,h (k) = Ω µ 1 ,h Y k-h,k -T µ 1 ,h U k-h,k r µ 2 ,h (k) = Ω µ 2 ,h Y k-h,k -T µ 2 ,h U k-h,k (9)
Since µ 1 is assumed to be the active path on the observation window, and by definition 9) can be written as:

Ω µ 1 ,h Y µ 1 k-h,k -T µ 1 ,h U k-h,k = 0, equation (
r µ 1 ,h (k) = 0 r µ 2 ,h (k) = Ω µ 2 ,h Y µ 1 k-h,k -T µ 2 ,h U k-h,k (10)
Adding and taking away Y µ 2 k-h,k from the expression of r µ 2 ,h (•), one obtains:

       r µ 1 ,h (k) = 0 r µ 2 ,h (k) = Ω µ 2 ,h Y µ 1 k-h,k -Y µ 2 k-h,k +Y µ 2 k-h,k -T µ 2 ,h U k-h,k (11)
As by definition

Ω µ 2 ,h Y µ 2 k-h,k -T µ 2 ,h U k-h,k = 0, one has: r µ 1 ,h (k) = 0 r µ 2 ,h (k) = Ω µ 2 ,h Y µ 1 k-h,k -Y µ 2 k-h,k (12)
Equation ( 12) clearly points out that the residual calculated for the path µ 2 (nonactive path) directly depends on the difference between the system's outputs when the mode sequence evolves according to the two paths µ 1 and µ 2 , the system being excited by the same inputs in both cases. If we do not want the residuals r µ 1 ,h (k) and r µ 2 ,h (k) to be simultaneously equal to zero, a necessary and sufficient condition is:

Y µ 1 k-h,k -Y µ 2 k-h,k / ∈ N r (Ω 2 ) (13)
where N r stands for the operator "right null space". Thus, the condition (13) has to be analyzed in order to deduce the discernability conditions of paths µ 1 and µ 2 . According to (4), one has:

Y µ 1 k-h,k -Y µ 2 k-h,k = O µ 1 ,h -O µ 2 ,h x(k -h) + T µ 1 ,h -T µ 2 ,h U k-h,k (14)
where x(k -h) is the value of the system's state at the initial instant of the observation window. One deduces from (14) after multiplication on the left by Ω µ 2 ,h :

Ω µ 2 ,h (Y µ 1 k-h,k -Y µ 2 k-h,k ) = Ω µ 2 ,h O µ 1 ,h x(k -h) +Ω µ 2 ,h T µ 1 ,h -T µ 2 ,h U k-h,k (15) If Y µ 1 k-h,k -Y µ 2
k-h,k belongs to the right null space of Ω 2 , one has:

Ω µ 2 ,h O µ 1 ,h x(k -h) + Ω µ 2 ,h T µ 1 ,h -T µ 2 ,h U k-h,k = 0
(16) The relation is satisfied "for almost every initial state" 2 x(kh) if the following necessary and sufficient condition is satisfied:

Ω µ 2 ,h O µ 1 ,h = 0 Ω µ 2 ,h T µ 1 ,h -T µ 2 ,h U k-h,k = 0 (17)
Therefore, the paths µ 1 and µ 2 are not discernible on a time window [kh, k] if the relations (17) are satisfied. Theorem 2 (Path discernability): Two paths µ 1 and µ 2 of a switching system are discernible on an observation window [kh, k], "for almost every initial state" x(kh), if:

Ω µ i ,h O µ j ,h = 0, i, j ∈ {1, 2} , i = j (18)
or

Ω µ i ,h T µ j ,h -T µ i ,h U k-h,k = 0 i, j ∈ {1, 2} , i = j (19)
where U k-h,k is the vector containing the system's input stacked on the observation window. The proof of this theorem directly comes from the preceding remarks. When the paths µ 1 and µ 2 are of the type (i, i, . . . , i), i ∈ {1, 2, . . . , s}, theorem 2 is equivalent to the modes discernability conditions formulated in [START_REF] Cocquempot | Switching time estimation and fault detection for hybrid systems using structured parity residuals[END_REF].

Remark 1 (Dependency to the initial state): In theorem 2, the expression "for almost every initial state" holds owing to the fact that the discernability of the paths cannot be ensured for any initial state x(kh). In fact, for certain particular values of x(kh), the relation ( 16) is always satisfied independently of the input sequence U k-h,k . For example, in the situation where O µ 1 ,h has full rank, for 16) is satisfied for every input sequence U k,k-h , where Φ belongs to the right null space of Ω µ 2 ,h and O µ 1 ,h † is the pseudo-inverse of O µ 1 ,h .

x(k -h) = O µ 1 ,h † Φ -T µ 1 ,h -T µ 2 ,h U k-h,k , equation (

V. DETERMINATION OF THE ACTIVE MODE IN NOISY

ENVIRONMENT

In section III, the determination of the active mode at any moment was carried out within a deterministic framework, i.e. there were no noise on the system's measurement. Now, we assume the presence of a bounded noise on the output 2 see remark 1 for the explanation of the expression "for almost every initial state" of the system described by equation [START_REF] Heemels | Equivalence of hybrid dynamical models[END_REF]. The only available information on the noise is its maximum magnitude. No probabilistic assumption is formulated on the probability distribution of the measurement noise:

     x (k + 1) = A µ k x (k) + Bu (k) y (k) = Cx (k) + n(k) ∀k, |n(k)| ≤ δ, δ > 0 (20)
where δ is the bound of measurement noise magnitude n(•). In this situation, the residual r µ * ,h (•), defined by [START_REF] Ackerson | On state estimation in switching environnements[END_REF] and which corresponds to the active path µ * on the time window [kh, k], is no more equal to zero. Indeed, the expression of the residual r µ * ,h (•), using equation ( 7), becomes:

r µ * ,h (k) = Ω µ * ,h (Y k-h,k -T µ * ,h U k-h,k + N k-h,k ) (21)
where the values taken by the measurement noise on the observation window

[k -h, k] are stacked in N k-h,k . As Ω µ * ,h (Y k-h,k -T µ * ,h U k-h,k ) = 0, one can write : r µ * ,h (k) = Ω µ * ,h N k-h,k (22)
Using the bound of the measurement noise magnitude, we can define an interval residual [r µ * ,h (k)]:

[r µ * ,h (k)] = [r µ * ,h , rµ * ,h ] (23)
where r µ * ,h and rµ * ,h depends on the bound δ of the measurement noise and are given by : r µ * ,h = -|Ω µ * ,h | Uδ and rµ * ,h = |Ω µ * ,h | Uδ, U being a column vector of length equal to the number of column of Ω µ * ,h and all the elements of U being equal to 1.

In an interval context, the determination of the active path amounts to seeking the path that corresponds to an interval residual including the value zero. This test can be performed by calculating the sign of the product of the upper and lower bounds of each interval residuals [r µ,h (•)]. The interval residual [r µ,h (•)] associated with the active path µ * is the one for which the sign of the product of its upper and lower bound is negative.

Depending on the evolution of the various operating regimes dynamics, it can happen that more than one interval residuals contains the value zero, this situation being linked to the path discernability and the bound of the measurement noise magnitude. In this case, one refrains from making any decision on the active path. We have to consider this situation from a looser point of view and we can only enumerate the set of all possible active paths.

VI. ACADEMIC EXAMPLE

We present here an academic example of a switching system. The simulated system is characterized by three modes and the matrices of the models describing the different T C = 1 2 (24) Figure 1 shows the input u(•), the output y(•), the state x(•) and the mode sequence µ (•) . The vertical dashed lines on the third graphic of figure 1 mark the time instants at which switchings occur. The fourth graphic plots the mode sequence described by the mode selection variable µ (•) . For instance, on the time windows [START_REF] Heemels | Equivalence of hybrid dynamical models[END_REF][START_REF] Ackerson | On state estimation in switching environnements[END_REF] and [START_REF] Balluchi | Design of observers for hybrid systems[END_REF]17], the system is respectively in the modes 1 and 2. As Ω k,h l i O k,h l j = 0, µ i , µ j ∈ Θ 2 , µ i = µ j , Θ 2 being the set of all paths of length 2, the condition (18) of theorem 2 is respected. Condition ( 19) is tested at every moment. If it is not satisfied, no decision is taken concerning the recognition of the active path.

In order to perform the determination of the active path at every moment, we consider an observation window of length 3. The set Θ 2 of all paths of length 2 on the observation window corresponds to the set of the nine paths in table II.

Fig. 1 .

 1 Fig. 1. Input u(•), output y(•), state x(•), mode sequence µ (•)

In fact, the existence of the projection matrix is linked to the observability of the system and to the length of the observation window[START_REF] Gertler | Fault detection and diagnosis in engineering systems[END_REF].

As explained in section VI, in order to reduce the number of residuals to be analyzed during the mode's recognition process, one can consider only the paths describing the mode sequence when the system remains in the same mode all over the duration of the observation window. In this case, only the paths (1 • 1), (2 • 2), and (3 • 3) to be considered. The time evolution of the residuals r

The mode sequence (first graphic of figure 3) and its estimation (second graphic of figure 3) while analyzing the residuals are depicted on figure 3. The figure shows that the mode sequence is exactly reconstructed. The three graphics of figure 4 correspond to the obtained interval residuals in the case of the mode estimation in a noisy environment. The performed simulation takes into account the presence of a bounded measurement noise on the system's output with a magnitude that is at least equal to 10 % of the magnitude of the system's ouput. The bounds of the interval residuals are represented on the three graphics of figure 4. Indeed, one can see that only one of the three interval residuals contains 0 at every moment, the index of this interval residual corresponding to the active path on the time window. The results of the determination of the active mode are illustrated by figure 5. The first graphic of this figure indicates the simulated (true) mode sequence. The second graphic is obtained while analyzing the membership of the interval residuals regarding the value zero. Although the modes are rather well detected, there are situations where it is impossible to provide an estimate of µ (•) because more than one interval residual contains the value zero value or none of the three interval residuals contains the value zero. This kind of situation, due to the measurement noise and to the fact that all possible paths on the observation window are not considered in the analysis, is highlighted by points with ordinate equal to zero on the second graphic of figure 5. The third graphic of figure 5 is obtained while testing the coherence in the succession of the detected active paths at consecutive moments as indicated in section VI. One can noticed a perfect reconstruction of the mode sequence.

VII. CONCLUSION

This paper proposed a method for the determination of the active mode and the switching instants of a switching system, using only the system's input and output data. We have also derived discernability conditions regardless of noise, i.e. conditions that ensure the uniqueness of the recovered mode.

It will be of great interest to consider in future work the situation where all the modes of the system are not previously indexed. In this case, one does not have a complete knowledge of all the operating regimes of the system. Therefore, it is necessary to simultaneously proceed to the detection of not indexed modes and the estimation of their parameters.