Vincent Richard

Renaud Loison
email: renaud.loison@insa-rennes.fr

Raphael Gillard

Herve Legay

Maxime Romier

J.-P Martinaud

Daniele Bresciani

Fabien Delepaux

Jean-Paul Martinaud

Spherical Mapping of the Second-Order Phoenix Cell for Unbounded Direct Reflectarray Copolar Optimization

HAL is

INTRODUCTION

Passive ReflectArrays (RA) are promising antennas for applications requiring high gain, low profile, and low mass [START_REF] Huang | Reflectarray Antennas[END_REF]. Preliminary demonstrations have been published for directive beams [START_REF] Encinar | Design of two-layer printed reflectarrays using patches of variable size[END_REF], contoured beams [START_REF] Karimipour | Accurate method for synthesis of shaped-beam non-uniform reflectarray antenna[END_REF][START_REF] Encinar | Three-layer printed reflectarrays for contoured beam space applications[END_REF][START_REF] Zhou | High-performance curved contoured beam reflectarrays with reusable surface for multiple coverages[END_REF][START_REF] Legay | Measurement of a 1.3 m reflectarray antenna in at panels in ku band[END_REF] or for dual-polarization [START_REF] Hasani | Single-layer quad-band printed reflectarray antenna with dual linear polarization[END_REF][START_REF] Florencio | Reflectarray antennas for dual polarization and broadband telecom satellite applications[END_REF]. Most of them rely on a quite simple design process where only the phase of the main polarization at one frequency is controlled over the radiating aperture (Phase Only Synthesis, POS). Typically, this is done by selecting the geometry of each elementary cell on the reflecting panel so that it provides the desired local phase. Generally, the tuning of a single geometrical parameter is sufficient to obtain an almost 360 • phase range and, then, a simple one-to-one correspondence between cell geometry and local phase can be established.

However, more advanced synthesis approaches [START_REF] Bucci | A new approach to the power pattern synthesis of reflectarrays[END_REF][START_REF] Zhou | Direct optimization of printed reflectarrays for contoured beam satellite antenna applications[END_REF] are needed when specifications get more stringent, as for dual and multiple frequency RA or large bandwidth contoured beams for space communications. In this case, RA could advantageously compete with shaped reflectors whose fabrication is both complex and expensive. Nevertheless, their design has to address several simultaneous goals, directly at the radiation pattern level. The cell selection process is then the result of a complex trade-off. Naturally, stringent specifications also require more complex cells, with several independent geometrical parameters, in order to be able to meet the different goals. This contributes further to increased complexity. Indeed, required advanced synthesis approaches have to handle multiple goals for the array radiation by tuning multiple degrees of freedom (DoF) at each cell. For a realistic RA made of several thousands of cells, this really becomes a challenging task. On the other hand, it is believed that the successful management of this task is a key point in RA design since the availability of so many DoF may offer unique opportunity to outperform classical reflectors.

At this stage, a deeper insight in RA design approaches is necessary to better understand the associated challenges. As full-wave EM optimization of RA is not possible yet, design approaches usually involve a 2-step process. The first step aims at providing a full characterization of the chosen cell, by computing its complete reflection matrix [START_REF] Wan | Efficient computation of generalized scattering matrix for analyzing multilayered periodic structures[END_REF] for any set of geometrical parameters. To do so, full-wave simulations are carried out, considering a single cell in a periodic environment (local-periodicity assumption). The resulting data is then used all along the optimization process and a very rapid access mechanism is thus essential. Most approaches rely on lookup tables with appropriate interpolating schemes [START_REF] Legay | Measurement of a 1.3 m reflectarray antenna in at panels in ku band[END_REF]. Equivalent circuits can also be used [START_REF] Al-Joumayly | A generalized method for synthesizing low-profile, band-pass frequency selective surfaces with non resonant constituting elements[END_REF]. Combined with filter synthesis techniques [START_REF] Liang | An impedance surface-based method for designing wideband reflectarrays[END_REF], they have demonstrated good capabilities for FSS [START_REF] Zhou | Design of aperiodic frequency selective surfaces for compact quasioptical networks[END_REF] and even RA synthesis [START_REF] Hum | Equivalent circuit modeling for reflectarrays using oquet modal expansion[END_REF], especially regarding bandwidth optimization. Unfortunately, they are usually restricted to canonical geometries and normal incidence. Behavioral models based on Artificial Neural Networks (ANN) are more general [START_REF] Robustillo | Ann characterization of multi-layer reflectarray elements for contoured-beam space antennas in the ku-band[END_REF][START_REF] Freni | Neural network characterization of reflectarray antennas[END_REF]. They have been used in [START_REF] Robustillo | Design of a contoured-beam reflectarray for a eutelsat european coverage using a stacked-patch element characterized by an artificial neural network[END_REF] to tune the geometry of each RA cell of a narrow band RA. Here, in continuation to [START_REF] Richard | Optimized artificial neural network for reflectarray cell modelling[END_REF] and [START_REF] Richard | Loss analysis of a reflectarray cell using anns with accurate magnitude prediction[END_REF], we use 2 different ANN, one for magnitude and one for phase, in order to improve the prediction accuracy of the full reflection matrix. Efficient approaches based on ordinary kriging and support vector machines have also been proposed recently [START_REF] Salucci | Efficient prediction of the em response of reflectarray antenna elements by an advanced statistical learning method[END_REF]. Whatever the used model, the second step in the design approach is to select the best geometry for all cells, in order to meet the specifications for the array radiation. Theoretically, this requires an iterative process in which all degrees of freedom are varied and the radiation pattern of the array is re-assessed continuously. Due to the huge number of DoF, this step is very time-consuming, even if rapid-access models are used. Here, we propose to use model-reduction techniques based on spline as it provides an efficient way to approximate a problem with a large number of DoF. This is classically done to optimize the complex geometry of shaped reflectors [START_REF]User's Manual[END_REF] and even the geometrical evolution of cells over RA panels [START_REF] Zhou | Efficient optimization of large reflectarrays using continuous functions[END_REF].

Finally, it is well known [START_REF] Milon | Analysis of mutual coupling for the simulation of reflectarrays radiating cells[END_REF] that successive cells must not differ too much in their geometry, and a smooth evolution has to be guaranteed all over the RA surface. This is the condition for complying with the local-periodicity assumption. To do so, an additional goal can be added in the optimization process to enforce some kind of similarity between neighbor cells [START_REF] Marnat | Comparison of synthesis strategies for a dual-polarized reflectarray[END_REF]. However, this makes the process even more complex. Moreover, it does not prevent from the abrupt variation encountered after a complete 360 • phase cycle has been achieved. A more relevant approach to address this issue is to use Phoenix cell. Thanks to its rebirth property [START_REF] Moustafa | The Phoenix cell: A new reflectarray cell with large bandwidth and rebirth capabilities[END_REF], this cell has the unique capability to come back to its initial geometry after a complete 360 • cycle has been reached, thus naturally preventing for any abrupt variation. Unfortunately, no RA synthesis has been reported yet, that fully takes benefit of the capabilities of the Phoenix cell. Indeed, most publications involve first order Phoenix cell [START_REF] Makdissy | Phoenix: Reflectarray unit cell with reduced size and inductive loading[END_REF] (i.e., with only one tuning parameter), whose capabilities are of course limited regarding multiple-purpose optimization. On the other hand, higher order Phoenix [START_REF] Legay | Measurement of a 1.3 m reflectarray antenna in at panels in ku band[END_REF] cell offer additional DoF but the rebirth property is not so easy to handle since the variation domain for the geometry is now multi-dimensional.

The main contribution of this paper is to propose an original spherical mapping of the second order Phoenix cell that intrinsically handles its rebirth mechanism, whatever the variation in the geometrical parameters. This mapping is associated with ANN modeling for a fast characterization of the cell. It is then integrated into an optimization process, which automatically enables to guarantee a smooth evolution of the geometry with no need to use additional constraints. Furthermore, the mapping transforms the initial optimization domain into an unbounded periodic space, and it also reduces its dimension. As a result, we arrive to a quite powerful tool dedicated to the full exploitation of the second order Phoenix cell. This is a first step before addressing the case of higher order cells with even more capabilities.

The paper is organized as follows. Section 2 is dedicated to the presentation of the spherical mapping of the second order Phoenix cell. Section 3 shows its efficient application in RA synthesis. A design example focusing on co-polarization illustrates the good performance for a 83 × 71 cell RA in section 4 with a full-wave simulation to valid the process. Section 5 concludes the paper.

A SPHERICAL MAPPING OF THE 2ND ORDER PHOENIX CELL

The main objective of this section is to present the new spherical mapping of the 2nd order Phoenix cell that will be later used to facilitate the RA synthesis.

Studied Cell: The 2nd Order Phoenix Cell

Phoenix cells are specific RA cells that have the property to provide a 360 • phase range while their geometry evolves in a cycling way, coming back to the initial geometry at the end of the cycle. This so-called rebirth capability theoretically enables designing RA with smooth geometrical variations on the reflecting panel. The concept of Phoenix cells has been proposed originally in [START_REF] Moustafa | The Phoenix cell: A new reflectarray cell with large bandwidth and rebirth capabilities[END_REF] and different possible geometries have then been studied such as [START_REF] Salti | A single layer stub-patch phoenix cell for large band reflectarrays[END_REF][START_REF] Zhang | Design of broadband, low cost single layer reflectarray using phoenix cell[END_REF]. A new step has been passed in [START_REF] Makdissy | Phoenix: Reflectarray unit cell with reduced size and inductive loading[END_REF] where two different geometries, one capacitive and the other inductive, are combined successively to form the Phoenix cycle. The main advantage is to provide two complementary phase ranges and hence to avoid any steep resonance, thus preventing for very dispersive behaviors [START_REF] Makdissy | Phoenix: Reflectarray unit cell with reduced size and inductive loading[END_REF]. Figure 1(a) shows a representation of such a typical Phoenix cycle. Here, the capacitive part (n = 0) simply consists of a square patch with increasing size (C 1 ). The inductive part (n = 1) is the complementary configuration with the patch replaced by a square aperture. Figure 1(b) gives a wrapped representation of this cycle, clearly showing it provides an unbounded and continuous way of varying the geometry, as a direct consequence of the rebirth property. In the following, we refer to this cycle as a first order one since there is only one geometrical parameter (C 1 ). In this paper, we use the second order of this Phoenix cell by introducing a second geometrical parameter, C 2 . To do so, the square patch (or aperture) is replaced by a square ring, as shown in Figure 2. The additional DoF provides more flexibility to control the reflection properties of the cell. On the other hand, it also makes trickier the management of the rebirth capabilities since this one has now to be described in a multi-dimensional space. Indeed, in order to fully benefit from the possibilities offered by the second order Phoenix cell, the representation shown in Figure 1 has to be generalized. This is the objective of the next section.

n = 0 n = 1 C 1 (a) (b) n = 0 (c) n = 1

Spherical Representation of the 2nd Order Phoenix Cell

The basic rules defining this generalized representation are straightforwardly derived from those observed in Figure 1. Firstly, the representation must include all possible configurations of the cell, whatever its type (n) and its geometrical parameters (C 1 and C 2 ). Secondly, it must preserve unicity, in the sense that any configuration of the cell is only represented once. Thirdly, it must guarantee a smooth evolution of the cell geometry all over the representation. Finally, it must not be bounded, i.e., any cell in the representation has to be surrounded by neighbors with slightly-perturbed geometries and none of them should be a stopping boundary.

C 1 C 2 1 1 C 1 C 2 (a) (b)
As seen in Figures 3(a) and (b), a quite natural way to define such a scheme is to use a spherical representation, similar to a world map with a 360 • periodicity for θ and φ. For the ease of understanding, Figure 3(c) also gives a planisphere version of this representation. The North and South Poles correspond to non-metallized and fully-metallized cells respectively. The 0 • and 180 • meridians match the capacitive (n = 0) and inductive (n = 1) parts of first order cycle (Figure 1). They are characterized by C 2 = 0. All other cells in the spherical representation are second order cells (C 2 = 0) also defining continuous cycles in any direction. More precisely, one parallel in the sphere is characterized by cells that all have the same metal rate. This parameter represents the percentage of metal along the length l of the cell. It continuously varies from 0% at the North Pole to 100% at the South Pole.

This representation scheme defines a bijective correspondence between the angular position (θ, φ), in degrees, on the sphere and the cell configuration

(C 1 , C 2 , n): ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ θ(C 1 , C 2 , n) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 180 1 - C 2 -C 1 + l l if n = 0 180(C 2 -C 1 + l) l if n = 1 φ(C 1 , C 2 , n) = ⎧ ⎪ ⎨ ⎪ ⎩ 180C 2 C 2 -C 1 + l if n = 0 180C 2 C 2 -C 1 + l + 180 if n = 1 (1) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ C 1 (θ, φ, n) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ω(θ) + (l -ω(θ)) φ 180 if n = 0 l -ω(θ) + ω(θ) (φ -180) 180 if n = 1 C 2 (θ, φ, n) = ⎧ ⎪ ⎨ ⎪ ⎩ (l -ω(θ)) φ 180 if n = 0 ω(θ) (φ -180) 180 if n = 1 ω(θ) = lθ 180 (2)
In these equations, l is the fixed periodicity in the array. As it will be seen in following sections, this representation proves to be particularly convenient in the RA synthesis process. Indeed, it provides a very powerful tool to deal with the tricky selection of adjacent cells when designing a RA panel. By moving over the sphere, variations in the cell configuration can be achieved with only two tuning parameters (θ and φ) in a continuous and periodic domain. Practically, this means the synthesis process can be advantageously carried out by optimizing θ and φ instead of dealing with true geometrical parameters. The interest is threefold. Firstly, the number of DoF per cell is reduced from 3 (i.e., C 1 , C 2 , n) to 2 (i.e., θ and φ). Secondly, the structure of the new optimization domain naturally prevents for any abrupt transition in the succession of cells on the panel. Thirdly, unconstrained optimization is possible since the new optimization parameters are not bounded as they evolve periodically. Note that the reduction in the number of DoF per cell does not mean the solution space itself is shrunk. It is just organized in a more convenient and compact way that enforces smoothness and continuity over the RA panel. As such, it can be seen as some kind of preconditioning for the optimization process.

Spherical Mapping of the Main Characteristics

Before going into the details of the synthesis process, we show here that the spherical mapping first offers a very complete and synthetic way to observe the performance of the cell. A cell is classically characterized by its reflection matrix:

R = R TE-TE R TE-TM R TM-TE R TM-TM (3)
A rapid-access model is of course required to compute this matrix, whatever the geometry of the cell, the direction of the illuminating wave and the frequency. In this work, we use the ANN model we developed in [START_REF] Richard | Optimized artificial neural network for reflectarray cell modelling[END_REF] and [START_REF] Richard | Loss analysis of a reflectarray cell using anns with accurate magnitude prediction[END_REF] to predict both the phase and magnitude for all matrix coefficients. A fast assessment with high accuracy (typically 1.6 • RMS error for the phase parameters) was demonstrated compared to full-wave simulation. Combined with the proposed spherical mapping, it enables an overview of the overall performance attainable by the chosen cell. Moreover, as the representation domain naturally guarantees smooth variations in the geometry, it directly shows whether this translates into smooth variations in the matrix coefficients themselves.

As an example, Figure 4 presents the mapping of the R TE-TE phase calculated using the ANN model over the sphere for all possible cell configurations. The studied cell is printed on a Honeycomb substrate with thickness h = 20 mm, dielectric constant = 1.03 and loss tangent tan δ = 0.003. The periodicity is l = 25.6 mm (λ/3 at 3.9 GHz). The mapping is carried out at normal incidence and for 3 different frequencies. In order to better see the transition in the phase when moving from a capacitive cell to an inductive one (or conversely), the φ angle in the planisphere mapping is varied from 0 • to 385 • . In other words, the [360 • , 385 • ] interval on the right of the plot is a repetition of the [0 • , 25 • ] interval on the left.

The first output from this plot is that any phase value from -180 • to 180 • can be reached by the cell. Moreover, most of them can be obtained with several different geometries, which provides additional flexibility in the synthesis process. Globally, the plot also shows the phase variations are quite regular. However, one can detect a slight discontinuity (less than 15 • ) when moving from a capacitive to an inductive cell (at φ = 180 • ) and a more significant one (up to 50 • ) when passing the converse transition (at φ = 360 • ). An optimal trajectory on the sphere would be a closed path passing through all possible phase values while minimizing the discontinuities when n is changed. A closer examination of the plot shows it is achieved by choosing the equator (dotted line in Figure 4(b)). This results from the fact that the discontinuity at the first transition (φ = 180 • ) gets higher when approaching the South Pole while that at the second transition (φ = 360 • ) becomes larger close to the North pole. The R TE-TE phase evolution (at f = 3.9 GHz) along this equatorial cycle is presented in Figure 5. As we will see later on, this trajectory will play an important role when choosing the initial conditions for the synthesis process.

Finally, the modification of the plot versus frequency is an indicator about dispersion, described in Eq. (4) in • /GHz. A more direct information is given by Figure 6 that plots phase dispersion over the bandwidth. It confirms each phase value can be obtained with various possible dispersion levels. The minimum value, 0 • /GHz, is obtained close to the South Pole where the fully-metallized cell is equivalent to a PEC boundary. The maximum value (absolute value), 340 • /GHz, is obtained in the back area, where a resonant phenomenon associated with large and narrow slot-rings is encountered.

ΔΦ TE-TE Δf = Φ fmax -Φ f min f max -f min (4)

OPTIMIZATION ALGORITHM

Goal and Used Cost Function

In the context of space antennas, the goal of the RA optimization process can be to reach an objective radiated field E obj associated with a defined coverage on Earth. This field is specified at N sta station distributed over the Earth and at N freq frequencies f in a given band. To this aim, a min/max optimization process is used [START_REF] Hald | Combined lp and quasi newton methods for minimax optimization[END_REF]. For example, in the case of a co-polar optimization, the cost function is defined as the maximum of the difference between the co-polar components of the objective field and the field E rad radiated by the RA under optimization:

= max f max k E obj co (k, f ) -E rad co (k, f ) (5)
with k = 1, . . . , N sta . In the optimization process, E rad is calculated using geometrical optics where the reflection on each cell is obtained from its full scattering matrix (ANN model) with the real incidence angle and far-field illumination from primary horn. The modeled radiation pattern of this horn is derived from measurements.

Optimization DoF and Initial Condition

As already mentioned, we propose to optimize the RA panel directly on the θ and φ parameters instead of the true geometrical ones (C 1 , C 2 , n). For a RA constituted of N x and N y cells along x and y axes respectively, the array is entirely defined by θ ij and φ ij for i ∈ [1, N x ] and j ∈ [1, N y ]. The initial number of DoF is thus given by:

N DoF init = 2N x N y (6)
Classically, the optimizer starts from an initial condition defined by θ 0 ij and φ 0 ij . To derive this one, the equator cycle discussed in Section 2.3 is used. It provides a one-to-one correspondence between cell geometry and local phase while minimizing the phase discontinuities along the cycle. The initial layout is built by only satisfying the required co-polarization phase over the panel (accounting for local field incidence) and at the central frequency with classical Phase Only Synthesis techniques [START_REF] Legay | A multi facets composite panel reflectarray antenna for a space contoured beam antenna in ku band[END_REF][START_REF] Encinar | A transmit-receive reflectarray antenna for direct broadcast satellite applications[END_REF]. However, there is no reason for it to work properly for other frequencies and for the cross-polarization. Accordingly, the RA panel performance must be optimized. To vary θ ij and φ ij from the initial layout within the optimization process, Δθ ij and Δφ ij are introduced:

θ ij = θ 0 ij + Δθ ij φ ij = φ 0 ij + Δφ ij (7)
Δθ ij and Δφ ij are the new N DoF init parameters to be optimized. For space applications, N DoF init can be huge reaching up to several thousands. For computation time and convergence reasons, the complexity is reduced by expanding the unknowns on approximation functions so that the required information is compressed without sacrificing accuracy:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Δθ ij = N S x l=1 c x l S x l (x i ) N S y m=1 c y m S y m (y j ) Δφ ij = N S x l=1 d x l S x l (x i ) N S y m=1 d y m S y m (y j ) (8)
In this equation, S x l and S y m are bi-cubic spline basis functions [START_REF] De Boor | A Practical Guide to Splines[END_REF] along x and y respectively, N S x and N S y are the numbers of these functions, x i and y j are the center coordinates of the (i, j) cell and c

x/y l/m and d

x/y l/m are the spline expansion coefficients. This approach is largely inspired by the techniques used to optimize shaped reflectors for space applications [START_REF]User's Manual[END_REF] and has also been used for RA [START_REF] Zhou | Efficient optimization of large reflectarrays using continuous functions[END_REF]. The main issue consists in choosing the appropriate number of expansion functions so that the approximation is precise enough. Furthermore, the choice of the resolution also enables controlling the smoothness of geometrical variations over the panel. In practise, it may be increased gradually until a sufficient accuracy is reached, as it is classically done when designing shaped reflectors. Finally, the number of optimization DoF N DoF is compressed to:

N DoF = 2N S x N S y (9)
Strictly speaking, this is not really a reduction in the number of available DoF but a compressed representation of them.

Optimization Algorithm

Based on the min/max algorithm, the objective is to minimize the cost function (Eq. ( 5)) by modifying the spline coefficients. To achieve this, an off-the-shelf conventional iterative gradient descent is used. In order to get the best orientation in the DoF space, the following gradient vector G is calculated numerically at each iteration:

G = ∂ ∂c x 1 . . . ∂ ∂c x N S x ∂ ∂c y 1 . . . ∂ ∂c y N S y ∂ ∂d x 1 . . . ∂ ∂d x N S x ∂ ∂d y 1 . . . ∂ ∂d y N S y (10)

APPLICATION TO A C-BAND MISSION

Mission Specifications

To illustrate the algorithm performances, we consider a C-band space telecommunication mission. The coverage zones are shown in Figure 8 and the co-polarization (co-polar) gain and cross-polarization discrimination (XPD) requirements are summarized in Table 1. The purpose of the mission is to provide service to Europe, North Africa and Middle East (FC1 zone) while minimizing illumination in South America (FC2 zone). The antenna operates in left-handed circular polarization in the limited transmission C-band [3.6, 4.2] GHz.

Antenna and Optimization Parameters

This work concerns only the layout design, and the antenna system parameters such as the feeding antenna type and position and the RA dimensions are fixed and given in Figure 9.

The RA panel is 28λ along x and 24λ along y at the central frequency f = 3.9 GHz. The array periodicity is λ/3 along x and y resulting in N x = 83 and N y = 71 RA cells, the initial number of optimization DoF is thus N DoF init = 11786.

In order to drastically compress the information and insure smooth geometrical variations around the initial condition, the numbers of spline coefficients N S x and N S y are both chosen to 20. The final In this paper, the developed mapping is restricted to the 2nd order cell. Such a cell intrinsically offers no simple means of controlling cross-polar because it has a square symmetric geometry. Then in this example, we decided to focus only on co-polarization and try to optimize it on a quite large bandwidth (16%). Consequently, no goal was given for the cross.

Introducing new geometrical parameters, XPD could be optimized as it is done in [START_REF] Zhou | Design and optimization of a single-layer planar transmitreceive contoured beam reflectarray with enhanced performance[END_REF] by converting square Phoenix cells into rectangular ones or by the use of parallelogram or trapezoid shaped elements as in [START_REF] Legay | A multi facets composite panel reflectarray antenna for a space contoured beam antenna in ku band[END_REF]. Cross-polarization could also theoretically be optimized at the array level with symmetrical cells [START_REF] Encinar | Reduction of cross-polarization in contoured beam reflectarrays using a three-layer configuration[END_REF]. Once again, this was not possible here as all available DoF were already necessary to address the co-polar specifications (for both the illuminated and isolated zones) over the bandwidth.

Initial Layout

The required co-polarization phase distribution on the RA at the central frequency f = 3.9 GHz is presented in Figure 10. Using the equator cycle and its one-to-one correspondence between cell geometry and reflected co-polar phase, the initial layout is easily designed. As shown in Figure 11, this layout presents smooth geometrical variations. This property tends to respect the expected local periodicity for the cells.

The simulated radiation patterns of the initial layout are presented in Figure 12(a), and the achieved performances are synthesized in Table 2. As expected, the performances respect the specifications only at the central frequency f = 3.9 GHz and for the radiated co-polar.

Optimization Run

Figure 13 presents the evolution of the cost function with iterations. Clearly, the optimization converges since the cost function decreases with the iteration number and tends to zero. The optimization was carried out using a workstation with a 2.5-GHz dual-core AMD Opteron processor with 128 GB RAM. The computation time is around 12 hours. Note that the simulation time for one cell is 4 µs using ANN. It would be around 1 min for full-wave simulation, which is definitely intractable within an optimization process. The total computation time could be reduced further by improving the present min/max routine and the radiation pattern calculation (to be done several times at each iteration in order to calculate gradients given by Eq. ( 10)). Such improvements were not addressed in this study, whose main scope was the spherical mapping. Figure 12(b) presents the radiation patterns of the optimized RA. Compared to the initial RA ones, we clearly observe the benefits of the optimization process at the lateral frequencies. Table 3 confirms this observation since the co-polar gain specifications are now reached at 3.6 GHz and 3.9 GHz. Only a small 0.06 dBi residual error is noted at 4.2 GHz. Note that the gain improvement compared to an initial POS is up to 1.62 dB. Concerning the XPD, the specifications are obviously not reached since it has not been considered in the cost function. However, the XPD has not been too much affected by the co-polar optimization process.

Validation by Full-Wave Simulation

For validation purpose, a full-wave simulation of the whole antenna system is proposed in this section.

A homemade solver based on integral equations in frequency domain is used [START_REF] Martinaud | Domain decomposition techniques for the scattering by partly bor objects application to the antenna-radome modeling[END_REF]. Figure 14 shows the comparison between the full-wave solver results and the one obtained with array theory under local periodicity assumption.

It must be highlighted that the agreement is very good. This definitely shows that the reached geometrical continuity over the panel is a key point to comply with the local periodicity assumption. Thanks to that, conventional array theory can be used with confidence in the optimizing process.

CONCLUSION

This paper describes an original spherical mapping for the 2nd order Phoenix cell. It appears as an essential tool to fully benefit from this versatile but complex cell. It first provides a convenient way to visualize its overall properties. Moreover, it leads to an optimization domain that directly inherits its continuous and cyclic evolution. This makes it fully suitable with spline that leads to a compressed formulation of the problem to be optimized. In this paper, the mapping is combined with ANN modeling and encapsulated in a min/max optimization process, thus resulting in a general optimization package. It is successfully applied to optimize a contoured beam coverage for space communication on a 16% frequency bandwidth. The gain at the upper frequency is improved by 1.62 dB compared to the initial POS. The excellent agreement with full-wave simulation confirms that the proposed methodology automatically generates a regular RA layout that naturally complies with the implicit local-periodicity assumption.

Further work is now needed to extend the mapping to higher order or asymmetrical cells, which will offer additional DoF, especially for the management of the cross-polarization.

Figure 1 .

 1 Figure 1. 1D geometrical cycle of the 1st order Phoenix cell. (a) Unwrapped version. Wrapped version -(b) front side and (c) back side.

Figure 2 .

 2 Figure 2. Two different types of cells. (a) Capacitive (n = 0). (b) Inductive (n = 1).

Figure 3 .

 3 Figure 3. Spherical representation with three overviews. (a) Front side view (θ = 90 • , φ = 0 • ). (b) Top view (θ = 0 • whatever φ). (c) Planisphere view.

Figure 4 .

 4 Figure 4. R TE-TE phase (degrees) on the spherical representation and at different frequencies.

Figure 5 .

 5 Figure 5. R TE-TE phase evolution along the equatorial cycle at the central frequency 3.9 GHz.

Figure 6 .

 6 Figure 6. Dispersion over the bandwidth of the R TE-TE phase (degrees/GHz) on the spherical representation and at the central frequency 3.9 GHz.

Figure 7 Figure 7 .

 77 Figure 7 summarizes the algorithm.

Figure 8 .

 8 Figure 8. Coverage zones of the telecommunication mission. FC1: co-polar directivity. FC2: co-polar isolation.

Figure 9 .

 9 Figure 9. Fixed antenna architecture.

Figure 10 .

 10 Figure 10.Ideal R TE-TE phase (degrees) requirement at 3.9 GHz.

Figure 11 .Figure 12 .

 1112 Figure 11. Initial layout for the C-band mission.

Figure 13 .

 13 Figure 13. Convergence of the cost function.

Figure 14 .

 14 Figure 14. Comparison of radiation patterns obtained with full-wave solver (blue curves) and homemade analytical tool (red curves) at 3.9 GHz.

Table 1 .

 1 Coverage requirements.

	Zone	Co-polar Co-polar XPD min Gain min Gain max
		[dB]	[dB]	[dB]
	FC1	24.2	X	28.0
	FC2	X	10.0	X

Table 2 .

 2 Initial RA performances.

	Frequency Zone	Co-polar Co-polar XPD min Gain min Gain max
	GHz		[dB]	[dB]	[dB]
	3.6	FC1 FC2	23.03 X	X 9.52	21.6 X
	3.9	FC1 FC2	24.36 X	X 6.7	22.47 X
	4.2	FC1 FC2	22.7 X	X 7.84	21.95 X

Table 3 .

 3 Optimized RA performances.

	Frequency Zone	Co-polar Co-polar XPD min Gain min Gain max
	GHz		[dB]	[dB]	[dB]
	3.6	FC1 FC2	24.65 X	X 7.62	22.52 X
	3.9	FC1 FC2	24.28 X	X 6.46	22.09 X
	4.2	FC1 FC2	24.14 X	X 7.43	20.25 X


Simon Bernard
email: simon.bernard@univ-rouen.fr

Hongliu Cao
email: caohongliu@gmail.com

Robert Sabourin
email: robert.sabourin@etsmtl.ca

Laurent Heutte
email: laurent.heutte@univ-rouen.fr

Random Forest for Dissimilarity-based Multi-view Learning

Many classification problems are naturally multi-view in the sense their data are described through multiple heterogeneous descriptions. For such tasks, dissimilarity strategies are effective ways to make the different descriptions comparable and to easily merge them, by (i) building intermediate dissimilarity representations for each view and (ii) fusing these representations by averaging the dissimilarities over the views. In this work, we show that the Random Forest proximity measure can be used to build the dissimilarity representations, since this measure reflects similarities between features but also class membership. We then propose a Dynamic View Selection method to better combine the view-specific dissimilarity representations. This allows to take a decision, on each instance to predict, with only the most relevant views for that instance. Experiments are conducted on several real-world multi-view datasets, and show that the Dynamic View Selection offers a significant improvement in performance compared to the simple average combination and two state-ofthe-art static view combinations.

Introduction

In many real-world pattern recognition problems, the available data are complex in that they cannot be described by a single numerical representation. This may be due to multiple sources of information, as for autonomous vehicles for example, where multiple sensors are jointly used to recognize the environment [START_REF] Chen | Multi-view 3d object detection network for autonomous driving[END_REF]. It may also be due to the use of several feature extractors, such as in image recognition tasks, often based on multiple representations of features, such as color, shape, texture descriptors, etc. [START_REF] Cao | Random forest dissimilarity based multi-view learning for radiomics application[END_REF] Learning from these types of data is called multi-view learning and each modality/set of features is called a view. For this type of task, it is assumed that the views convey different types of information, each of which can contribute to the pattern recognition task. Therefore, the challenge is generally to carry out the learning task taking into account the complementarity of the views. However, the difficulty with this is that these views can be very different from each other in terms of dimension, nature and meaning, and therefore very difficult to compare or merge. In a recent work [START_REF] Cao | Random forest dissimilarity based multi-view learning for radiomics application[END_REF], we proposed to use dissimilarity strategies to overcome this issue. The idea is to use a dissimilarity measure to build intermediate representations from each view separately, and to merge them afterward. By describing the instances with their dissimilarities to other instances, the merging step becomes straightforward since the intermediate dissimilarity representations are fully comparable from one view to another.

For using dissimilarities in multi-view learning, two questions must be addressed: (i) how to measure and exploit the dissimilarity between instances for building the intermediate representation? and (ii) how to combine the viewspecific dissimilarity representations for the final prediction?

In our preliminary work [START_REF] Cao | Random forest dissimilarity based multi-view learning for radiomics application[END_REF], the first question has been addressed with Random Forest (RF) classifiers. RF are known to be versatile and accurate classifiers [START_REF] Breiman | Random forests[END_REF][START_REF] Fernández-Delgado | Do we need hundreds of classifiers to solve real world classification problems?[END_REF] but they are also known to embed a (dis)similarity measure between instances [START_REF] Englund | A novel approach to estimate proximity in a random forest: An exploratory study[END_REF]. The advantage of such a mechanism in comparison to traditional similarity measures is that it takes the classification/regression task into account for computing the similarities. For classification for example, the instances that belong to the same class are more likely to be similar according to this measure. Therefore, a RF trained on a view can be used to measure the dissimilarities between instances according to the view, and according to their class membership as well. The way this measure is used to build the per-view intermediate representations is by calculating the dissimilarity of a given instance x to all the n training instances. By doing so, x can be represented by a new feature vector of size n, or in other words in a n-dimensional space where each dimension is the dissimilarity to one of the training instances. This space is called the dissimilarity space [START_REF] Pekalska | The Dissimilarity Representation for Pattern Recognition: Foundations And Applications[END_REF][START_REF] Costa | The dissimilarity approach: a review[END_REF] and is used as the intermediate representation for each view.

As for the second question, we addressed the combination of the view-specific dissimilarity representations by computing the average dissimilarities over all the views. That is to say, for an instance x, all the view-specific dissimilarity vectors are computed and averaged to obtain a final vector of size n. Each value in this vector is thus the average dissimilarity between x and one of the n training instances over the views. This is a simple, yet meaningful way to combine the information conveyed by each view. However, one could find it a little naive when considering the true rationale behind multi-view learning. Indeed, even if the views are expected to be complementary to each other, they are likely to contribute in very different ways to the final decision. One view in particular is likely to be less informative than another, and this contribution is even likely to be very different from an instance to predict to another. In that case, it is desirable to estimate and take this importance into account when merging the view-specific representations. This is the goal we follow in the present work.

In a nutshell, our preliminary work [START_REF] Cao | Random forest dissimilarity based multi-view learning for radiomics application[END_REF] has validated the generic framework explained above, with the two following key steps: (i) building the dissimilarity space with the RF dissimilarity mechanism and (ii) combining the views afterward by averaging the dissimilarities. In the present work, we deepen the second step by investigating two methods to better combine the view-specific dissimilarities:

1. combining the view-specific dissimilarities with a static weighted average, so that the views contribute differently to the final dissimilarity representation; in particular, we propose an original weight calculation method based on an analysis of the RF classifiers used to compute the view-specific dissimilarities;

2. combining the view-specific dissimilarities with a dynamic combination, for which the views are solicited differently from one instance to predict to another; this dynamic combination is based on the definition of a region of competence for which the performance of the RF classifiers is assessed and used for a view selection step afterward.

The rest of this chapter is organized as follows. The Random Forest dissimilarity measure is firstly explained in Section 2. The way it is used for multi-view classification is detailed in Section 3. The different strategies for combining the dissimilarity representations are given in Section 4, along with our two proposals for static and dynamic view combinations. Finally, the experimental validation is presented in Section 5.

Random Forest Dissimilarity

To fully understand the way a RF classifier can be used to compute dissimilarities between instances, it is first necessary to understand how an RF is built and how it gives a prediction for each new instance.

Random Forest

In this work, the name "Random Forest" refers to the Breiman's reference method [START_REF] Breiman | Random forests[END_REF]. Let us briefly recall its procedure to build a forest of M Decision Trees, from a training set T . First, a bootstrap sample is built by random draw with replacement of n instances, amongst the n training instances available in T . Each of these bootstrap samples is then used to build one tree. During this induction phase, at each node of the tree, a splitting rule is designed by selecting a feature over mtry features randomly drawn from the m available features. The feature retained for the splitting rule at a given node is the one among the mtry that maximizes the splitting criterion. At last, the trees in RF classifiers are grown to their maximum depth, that is to say when all their terminal nodes (also called leaves) are pure. The resulting RF classifier is typically noted as:

H(x) = {h k (x), k = 1, . . . , M } (1)
where h k (x) is the k th Random Tree of the forest, built using the mechanisms explained above [START_REF] Breiman | Random forests[END_REF][START_REF] Biau | A random forest guided tour[END_REF]. Note however that there exist many other RF learning methods that differ from the Breiman's method by the use of different randomization techniques for growing the trees [START_REF] Rokach | Decision forest: Twenty years of research[END_REF].

For predicting the class of a given instance x with a Random Tree, x goes down the tree structure from its root to one of its leaves. The descending path followed by x is determined by successive tests on the values of its features, one per node along the path. The prediction is given by the leaf in which x has landed. More information about this procedure can be found in the recently published RF reviews [START_REF] Verikas | Mining data with random forests: A survey and results of new tests[END_REF][START_REF] Biau | A random forest guided tour[END_REF][START_REF] Rokach | Decision forest: Twenty years of research[END_REF]. The key point here is that, if two test instances land in the same terminal node, they are likely to belong to the same class and they are also likely to share similarities in their feature vectors, since they have followed the same descending path. This is the main motivation behind using RF for measuring dissimilarities between instances.

Note that the final prediction of a RF classifier is usually obtained via majority voting over the component trees. Here again, there exist alternatives to majority voting [START_REF] Rokach | Decision forest: Twenty years of research[END_REF], but this latter remains the most used as far as we know.

Using Random Forest for measuring dissimilarities

The RF dissimilarity measure is the opposite measure of the RF proximity (or similarity) measure defined in Breiman's work [START_REF] Breiman | Random forests[END_REF][START_REF] Verikas | Mining data with random forests: A survey and results of new tests[END_REF][START_REF] Cao | Random forest dissimilarity based multi-view learning for radiomics application[END_REF], the latter being noted p H (x i , x j ) in the following.

The RF dissimilarity measure is inferred from a RF classifier H, learned from T . Let us firstly define the dissimilarity measure inferred by a single Random Tree h k , noted d k : let L k denote the set of leaves of h k , and let l k (x) denote a function from the input domain X to L k , that returns the leaf of h k where x lands when one wants to predict its class. The dissimilarity measure d k is defined as in Equation 2: if two training instances x i and x j land in the same leaf of h k , then the dissimilarity between both instances is set to 0, else it is equal to 1.

d k (x i , x j ) = 0, if l k (x i ) = l k (x j ) 1, otherwise (2)
The d k measure is the strict opposite of the tree proximity measure p k [START_REF] Breiman | Random forests[END_REF][START_REF] Verikas | Mining data with random forests: A survey and results of new tests[END_REF], i.e. d k (x i , x j ) = 1 -p k (x i , x j ). Now, the measure d H (x i , x j ) derived from the whole forest consists in calculating d k for each tree in the forest, and in averaging the resulting dissimilarity values over the M trees, as follows:

d H (x i , x j ) = 1 M M k=1 d k (x i , x j ) (3)
Similarly to the way the predictions are given by a forest, the rationale is that the accuracy of the dissimilarity measure d H relies essentially on the averaging over a large number of trees. Moreover, this measure is a pairwise function d H : X × X → R + that satisfies the reflexivity property (d H (x i , x i ) = 0), the nonnegativity property (d H (x i , x j ) ≥ 0) and the symmetry property (d

H (x i , x j ) = d H (x j , x i ))
. Note however that it does not satisfy the last two properties of the distance functions, namely the definiteness property (d H (x i , x j ) = 0 does not imply x i = x j ) and the triangle inequality (d

H (x i , x k ) is not necessarily less or equal to d H (x i , x j ) + d H (x j , x k )).
As far as we know, only few variants of this measure have been proposed in the literature [START_REF] Englund | A novel approach to estimate proximity in a random forest: An exploratory study[END_REF][START_REF] Cao | Random Forest For Dissimilarity Based Multi-View Learning: Application To Radiomics[END_REF]. These variants differ from the measure explained above in the way they infer the dissimilarity value from a tree structure. The motivation is to design a finer way to measure the dissimilarity than the coarse binary value given in Equation 2. This coarse value may seem intuitively too superficial to measure dissimilarities, especially considering that a tree structure can provide richer information about the way two instances are similar to each other.

The first variant [START_REF] Englund | A novel approach to estimate proximity in a random forest: An exploratory study[END_REF] modifies the p H measure by using the path length from one leaf to another when two instances land in different leaf nodes. In this way, p k (x i , x j ) does not take its value in {0, 1} anymore but is computed as follows:

p H (x i , x j ) = 1 M M k=1 p k (x i , x j ) = 1 M M k=1 1 exp (w.g ijk ) (4)
where, g ijk is the number of tree branches between the two terminal nodes occupied by x i and x j in the k th tree of the forest, and where w is a parameter to control the influence of g in the computation. When

l k (x i ) = l k (x j ), d k (x i , x j )
is still equal to 0, but in the opposite situation the resulting value is in ]0, 1].

A second variant [START_REF] Cao | Random Forest For Dissimilarity Based Multi-View Learning: Application To Radiomics[END_REF], noted RFD in the following, leans on a measure of instance hardness, namely the κ-Disagreeing Neighbors (κDN) measure [START_REF] Smith | An instance level analysis of data complexity[END_REF], that estimates the intrinsic difficulty to predict an instance as follows:

κDN (x i ) = |x k : x k ∈ κN N (x i ) ∩ y k = y i | κ (5)
where κN N (x i ) is the set of the κ nearest neighbors of x i . This value is used for measuring the dissimilarity dk (x, x i ), between any instance x to any of the training instances x i , as follows:

dk (x, x i ) = M k=1 (1 -κDN k (x i )) × d k (x, x i ) M k=1 (1 -κDN k (x i )) (6)
where κDN k (x i )) is the κDN measure computed in the subspace formed by the sole features used in the k th tree of the forest. Any of these variants could be used to compute the dissimilarities in our framework. However, we choose to use the RFD variant in the following, since it has been shown to give very good results when used for building dissimilarity representations for multi-view learning [START_REF] Cao | Random Forest For Dissimilarity Based Multi-View Learning: Application To Radiomics[END_REF].

The dissimilarity representation for multi-view learning 3.1 The dissimilarity space

Among the different dissimilarity strategies for classification, the most popular is the dissimilarity representation approach [START_REF] Pekalska | The Dissimilarity Representation for Pattern Recognition: Foundations And Applications[END_REF]. It consists in using a set R of m reference instances, to build a n × m dissimilarity matrix. The elements of this matrix are the dissimilarities between the n training instances in T and the m reference instances in R:

D(T, R) =     d(x 1 , p 1 ) d(x 1 , p 2 ) . . . d(x 1 , p m ) d(x 2 , p 1 ) d(x 2 , p 2 ) . . . d(x 2 , p m ) . . . . . . . . . . . . d(x n , p 1 ) d(x n , p 2 ) . . . d(x n , p m )     (7)
where d stands for a dissimilarity measure, x i are the training instances and p j are the reference instances. Even if T and R can be disjoint sets of instances, the most common is to take R as a subset of T , or even as T itself. In this work, for simplification purpose, and to avoid the selection of reference instances from T , we choose R = T . As a consequence the dissimilarity matrix D is always a symmetric n × n matrix.

Once such a squared dissimilarity matrix is built, there exist two main ways to use it for classification: the embedding approach and the dissimilarity space approach [START_REF] Pekalska | The Dissimilarity Representation for Pattern Recognition: Foundations And Applications[END_REF]. The embedding approach consists in embedding the dissimilarity matrix in a Euclidean vector space such that the distances between the objects in this space are equal to the given dissimilarities. Such an exact embedding is possible for every symmetric dissimilarity matrix with zeros on the diagonal [START_REF] Pekalska | The Dissimilarity Representation for Pattern Recognition: Foundations And Applications[END_REF]. In practice, if the dissimilarity matrix can be transformed in a positive semidefinite (p.s.d.) similarity matrix, this can be done with kernel methods. This p.s.d. matrix is used as a pre-computed kernel, also called a kernel matrix. This method has been successfully applied with RFD along with SVM classifiers [START_REF] Gray | Random forest-based similarity measures for multi-modal classification of alzheimer's disease[END_REF][START_REF] Cao | Random forest dissimilarity based multi-view learning for radiomics application[END_REF].

The second approach, the dissimilarity space strategy, is more versatile and does not require the dissimilarity matrix to be transformed into a p.s.d. similarity matrix. It simply consists in using the dissimilarity matrix as a new training set. Indeed, each row i of the matrix D can be seen as the projection of a training instance x i into a dissimilarity space, where the j th dimension is the dissimilarity with the training instance x j . As a consequence, the matrix D(T, T ) can be seen as the projection of the training set T into this dissimilarity space, and can be fed afterward to any learning procedure. This method is much more straightforward than the embedding approach as it can be used with any dissimilarity measurement, regardless its reflexivity or symmetry properties, and without transforming it into a p.s.d. similarity matrix.

In the following, the dissimilarity matrices built with the RFD measure are called RFD matrices and are noted D H for short. It can be proven that the matrices derived from the initial RF proximity measure [START_REF] Breiman | Random forests[END_REF][START_REF] Verikas | Mining data with random forests: A survey and results of new tests[END_REF] are p.s.d and can be used as pre-computed kernels in SVM classifiers [START_REF] Cao | Random forest dissimilarity based multi-view learning for radiomics application[END_REF], following the embedding approach. However, the proof does not apply if the matrices are obtained using the RFD measure [START_REF] Cao | Random Forest For Dissimilarity Based Multi-View Learning: Application To Radiomics[END_REF]. This is the main reason we use the dissimilarity space strategy in this work, as it allows more flexibility.

Using dissimilarity spaces for multi-view learning

In traditional supervised learning tasks, each instance is described by a single vector of m features. For multi-view learning tasks, each instance is described by Q different vectors. As a consequence, the task is to infer a model h:

h : X (1) × X (2) × • • • × X (Q) → Y (8)
where the X (q) are the Q input domains, i.e. the views. These views are generally of different dimensions, noted m 1 to m Q . For such learning tasks, the training set T is actually made up with Q training subsets:

T (q) = (x (q)
1 , y 1 ), (x

(q) 2 , y 2 ), . . . , (x (q) n , y n ) , ∀q = 1..Q (9)
The key principle of the proposed framework is to compute the RFD matrices

D (q)
H from each of the Q training subsets T (q) . For that purpose, each T (q) is fed to the RF learning procedure, resulting in Q RF classifiers noted H (q) , ∀q = 1..Q. The RFD measure is then used to compute the Q RFD matrices D (q) H , ∀q = 1..Q. Once these RFD matrices are built, they have to be merged in order to build the joint dissimilarity matrix D H that will serve as a new training set for an additional learning phase. This additional learning phase can be realized with any learning algorithm, since the goal is to address the classification task. For simplicity and because they are as accurate as they are versatile, the same Random Forest method used to calculate the dissimilarities is also used in this final learning stage.

Regarding the merging step, which is the main focus of the present work, it can be straightforwardly done by a simple average of the Q RFD matrices:

D H = 1 Q Q q=1 D (q) H (10)
The whole RFD based multi-view learning procedure is summarized in Algorithm 1 and illustrated in Figure 1.

Algorithm 1:

The RFD multi-view learning procedure

Input: T (q) , ∀q = 1..Q: the Q training sets,
1 for q = 1..Q do 2 H (q) = RF (T (q) )
// Build the n × n RFD matrix D (q) H :

3 forall xi ∈ T (q) do 4 forall xj ∈ T (q) do 5 D (q) H [i, j] = RF D(xi, xj|H (q) ) 6 end 7 end
8 end // Build the n × n average RFD matrix DH :

9 DH = 1 Q Q q=1 D (q) H // Train the final classifier on DH : 10 H f inal = RF (DH )
As for the prediction phase, the procedure is very similar. For any new instance x to predict: q) , ∀q = 1..Q, to form Q n-sized dissimilarity vectors for x. These vectors are the dissimilarity representations for x, from each of the Q views.

1. Compute d (q) H (x, x i ), ∀x i ∈ T (

Compute d

H (x, x i ) = 1 Q Q q=1 d (q) H (x, x i ), ∀i = 1.
.n, to form the n-sized vector that corresponds to the projection of x in the joint dissimilarity space.

Combining views with weighted combinations

The average dissimilarity is a simple, yet meaningful way to merge the dissimilarity representations built from all the views. However, it intrinsically considers that all the views are equally relevant with regard to the task and that the resulting dissimilarities are as reliable as each other. This is likely to be wrong from our point of view. In multi-view learning problems, the different views are meant to be complementary in some ways, that is to say to convey different types of information regarding the classification task. These different types of information may not have the same contribution to the final predictions. That is the reason why it may be important to differentiate these contributions, for example with a weighted combination in which the weights would be defined according to the view reliability.

The calculation of these weights can be done following two paradigms: static weighting and dynamic weighting. The static weighting principle is to weight the views once for all, with the assumption that the importance of each view is the same for all the instances to predict. The dynamic weighting principle on the other way, aims at setting different weights for each instance to predict, with the assumption that the contribution of each view to the final prediction is likely to be different from one instance to another.

Static combination

Given a set of dissimilarity matrices {D (1) , D (2) , . . . , D (Q) } built from Q different views, our goal is to find the best set of non-negative weights {w (1) , w (2) , . . . , w (Q) }, so that the joint dissimilarity matrix is:

D = Q q=1 w (q) D (q) ( 11
)
where w (q) ≥ 0 and

Q q=1 w (q) = 1.
There exist several ways, proposed in the literature, to compute the weights of such a static combination of dissimilarity matrices. The most natural one is to deduce the weights from a quality score measured on each view. For example, this principle has been used for multi-scale image classification [START_REF] Li | Combining multi-scale dissimilarities for image classification[END_REF] where each view is a version of the image at a given scale, i.e. the weights are derived directly from the scale factor associated with the view. Obviously, this only makes sense with regard to the application, for which the scale factor gives an indication of the reliability for each view.

Another, more generic and classification-specific approach, is to evaluate the quality of the dissimilarity matrix using the performance of a classifier. This makes it possible to estimate whether a dissimilarity matrix sufficiently reflects class membership [START_REF] Duin | The dissimilarity space: Bridging structural and statistical pattern recognition[END_REF][START_REF] Li | Combining multi-scale dissimilarities for image classification[END_REF]. For example, one can train a SVM classifier from each dissimilarity matrix and use its accuracy as an estimation of the corresponding weights [START_REF] Li | Combining multi-scale dissimilarities for image classification[END_REF]. kNN classifiers are also very often used for that purpose [START_REF] Duin | The dissimilarity space: Bridging structural and statistical pattern recognition[END_REF][START_REF] Li | Survey and experimental study on metric learning methods[END_REF]. The reason is that a good dissimilarity measure is expected to propose good neighborhoods, or in other words the most similar instances should belong to the same class.

Since kernel matrices can be viewed as similarity matrices, there are also few solutions in the literature of kernel methods that could be used to estimate the quality of a dissimilarity matrix. The most notable is the Kernel Alignment (KA) estimate [START_REF] Cristianini | On kerneltarget alignment[END_REF] A(K 1 , K 2 ), for measuring the similarity between two kernel matrices K 1 and K 2 :

A(K 1 , K 2 ) = K 1 , K 2 F K 1 , K 1 F K 2 , K 2 F ( 12
)
where K i is a kernel matrix and where •, • F is the Frobenius norm [START_REF] Cristianini | On kerneltarget alignment[END_REF].

In order to use the KA measure to estimate the quality of a given kernel matrix, a target matrix must be defined beforehand. This target matrix is an ideal theoretical similarity matrix, regarding the task. For example, for binary classification, the ideal target matrix is usually defined as K * = yy T , where y = {y 1 , y 2 , . . . , y n } are the true labels of the training instances, in {-1, +1}. Thus, each value in K * is:

K * ij = 1, if y i = y j -1, otherwise (13)
In other words, the ideal matrix is the similarity matrix in which instances are considered similar (K * ij = 1) if and only if they belong to the same class. This estimate is transposed to multi-class classification problems as follows [START_REF] Camargo | A multi-class kernel alignment method for image collection summarization[END_REF]:

K * ij = 1, if y i = y j -1 C-1 , otherwise (14)
where C is the number of classes.

Both kNN and KA methods presented above are used in the experimental part for comparison purposes (cf. Section 5). However, in order to use the KA method for our problem, some adaptations are required. Firstly, the dissimilarity matrices need to be transformed into similarity matrices by S (q) = 1 -D (q) . The following heuristic is then used to deduce the weight from the KA measure [START_REF] Qiu | A framework for multiple kernel support vector regression and its applications to sirna efficacy prediction[END_REF]:

w (q) = A(S (q) , yy T ) Q h=1 A(S (h) , yy T ) (15)
Strictly speaking, for the similarity matrices S (q) to be considered as kernel matrices, it must be proven that they are p.s.d. When such matrices are proven to be p.s.d, the KA estimates is necessarily non-negative, and the corresponding w (q) are also non-negative [START_REF] Cristianini | On kerneltarget alignment[END_REF][START_REF] Qiu | A framework for multiple kernel support vector regression and its applications to sirna efficacy prediction[END_REF]. However, as it is not proven that our matrices S (q) built from RF D are p.s.d., we propose to use the softmax function to normalize the weights and to ensure they are strictly positive:

w (q) = exp(A(S (q) , K * )) Q h=1 exp(A(S (h) , K * )) (16)
The main drawback of the methods mentioned above is that they evaluate the quality of the dissimilarity matrices based solely on the training set. This is the very essence of these methods, which are designed to evaluate (dis)similarity matrices built from a sample, e.g. the training set. However, this may cause overfitting issues when these dissimilarity matrices are used for classification purposes as it is the case in our framework. Ideally, the weights should be set from the quality of the dissimilarity representations estimated on an independent validation dataset. Obviously, this requires to have additional labeled instances. The method we propose in this section allows to estimate the quality of the dissimilarity representations without the use of additional validation instances.

The idea behind our method is that the relevance of a RFD space is reflected by the accuracy of the RF classifier used to build it. This accuracy can be efficiently estimated with a mechanism called the Out-Of-Bag (OOB) error. This OOB error is an estimate supplied by the Bagging principle, known to be a reliable estimate of the generalization error [START_REF] Breiman | Random forests[END_REF]. Since the RF classifiers in our framework are built with the Bagging principle, the OOB error can be used to estimate their generalization error without the need of an independent validation dataset.

Let us briefly explained here how the OOB error is obtained from a RF: let B denote a Bootstrap sample formed by randomly drawing p instances from T , with replacement. When p = n, n being the number of instances in T , it can be proven that about one third of T , in average, will not be drawn to form B [START_REF] Breiman | Random forests[END_REF]. These instances are called the OOB instances of B. Using Bagging for growing a RF classifier, each tree in the forest is trained on a Bootstrap sample, that is to say using only about two thirds of the training instances. Similarly, each training instance x is used for growing about two thirds of the trees in the forest. The remaining trees are called the OOB trees of x. The OOB error is the error rate measured on the whole training set by only using the OOB trees of each training instance.

Therefore, the method we propose to use consists in using the OOB error of the RF classifier trained on a view directly as its weight in the weighted combination. This method is noted SW OOB in the following.

Dynamic combination

In contrast to static weighting, dynamic weighting aims at assigning different weights to the views for each instance to predict [START_REF] Cruz | Dynamic classifier selection: Recent advances and perspectives[END_REF]. The intuition behind using dynamic weighting in our framework is that the prediction for different instances may rely on different types of information, i.e. different views. In that case, it is crucial to use different weights for building the joint dissimilarity representation from one instance to predict to another.

However, such a dynamic weighting process is particularly complex in our framework. Let us recall that the framework we propose to use in this work is composed of two stages: (i) inferring the dissimilarity matrix from each view, and (ii) combining the per-view dissimilarity matrices to form a new training set. The weights we want to determine are the weights used to compute the final joint dissimilarity matrix in stage (ii). As a consequence, if these weights change for each instance to predict, the joint dissimilarity matrix must be completely recalculated and a new classifier must also be re-trained afterwards. This means that, for every new instance to predict, a whole training procedure has to be performed. This is computationally expensive and quite inefficient from our point of view.

To overcome this problem, we propose to use Dynamic Classifier Selection (DCS) instead of dynamic weighting. DCS is a generic strategy, amongst the most successful ones in the Multiple Classifier Systems literature [START_REF] Cruz | Dynamic classifier selection: Recent advances and perspectives[END_REF]. It typically aims at selecting one classifier in a pool of candidate classifiers, for each instance to predict. This is essentially done through two steps [START_REF] Britto | Dynamic selection of classifiers -a comprehensive review[END_REF]: (i) the generation of a pool of candidate classifiers and (ii) the selection of the most competent classifier in this pool for the instance to predict. The solutions we propose for these steps are illustrated in Figure 2, the first step in the upper part and the second step in the lower part. The whole procedure is also detailed in Algorithm 2 and described in the following.

Generation of the pool of classifiers

The generation of the pool is the first key step of DCS. As the aim is to select the most competent classifier on the fly for each given test instance, the classifiers in the pool must be as diverse and as individually accurate as possible. In our case, the challenge is not to create the diversity in the classifiers, since they are trained on different joint dissimilarity matrices, generated with different sets of weights. The challenge is rather to generate these different weight tuples used to compute the joint dissimilarity matrices. For such a task, a traditional grid search strategy could be used. However, the number of candidate solutions increases exponentially with respect to the number of views. For example, Suppose that we sample the weights with 10 values in [0, 1]. For Q views, it would result in 10 Q different weight tuples. Six views would thus imply to generate 1 million weight tuples and to train 1 million classifiers afterwards. Here again, this is obviously highly inefficient.

The alternative approach we propose is to select a subset of views for every candidate in the pool, instead of considering a weighted combination of all of them. By doing so, for each instance to predict, only the views that are considered informative enough are expected to be used for its prediction. The selected views are then combined by averaging. For example, if a problem is described with six views, there are 2 6 -1 = 63 possible combinations (the situation in which none of the views is selected is obviously ignored), which will result in a pool of 63 classifiers H = {H 1 , H 2 , . . . , H 63 }. Lines 1 to 6 of Algorithm 2 give a detailed implementation of this procedure.

Evaluation and selection of the best classifier

The selection of the most competent classifier is the second key step of DCS. Generally speaking, this selection is made through two steps [START_REF] Cruz | Dynamic classifier selection: Recent advances and perspectives[END_REF]: (i) the definition of a region of competence for the instance to predict and (ii) the evaluation of each classifier in the pool for this region of competence, in order to select the most competent one.

The region of competence Θ t of each instance x t is the region used to estimate the competence of the classifiers for predicting that instance. The usual way to do so is to rely on clustering methods or to identify the k nearest neighbors (kNN) of x t . For clustering [START_REF] Soares | Using accuracy and diversity to select classifiers to build ensembles[END_REF], the principle is usually to define the region of competence as the closest cluster of x t , according to the distances of x t to the centroids of the clusters. As the clusters are fixed once for all, many different instances might share the same region of competence. In contrast, kNN methods give different regions of competence from one instance to another, which allows for more flexibility but at the expense of a higher computational cost [START_REF] De Souto | Empirical comparison of dynamic classifier selection methods based on diversity and accuracy for building ensembles[END_REF].

The most important part of the selection process is to define the criterion to measure the competence level of each classifier in the pool. There are a lot of methods for doing so, that differ in the way they estimate the competence, using for example a ranking, the classifier accuracies, a data complexity measure, etc. [START_REF] Cruz | Dynamic classifier selection: Recent advances and perspectives[END_REF]. Nevertheless, the general principle is most of the time the same: calculating the measure on the region of competence exclusively. We do not give an exhaustive survey of the way it can be done here, but briefly explain the most representative method, namely the Local Classifier Accuracy (LCA) method [START_REF] Woods | Combination of multiple classifiers using local accuracy estimates[END_REF], as an illustration.

The LCA method measures the local accuracy of a candidate classifier H i , with respect to the prediction ŷt of a given instance x t :

w i,t = x k ∈Θ t, ŷt I(H i (x k ) = ŷt ) x k ∈Θt I(y k = ŷt ) (17)
where Θ t = {x 1 , . . . x k , . . . , x K } is the region of competence for x t , and Θ t,ŷt is the set of instances from Θ t that belong to the same class as ŷt . Therefore, w i,t represents the percentage of correct classifications within the region of competence, by only considering the instances for which the classifier predicts the same class as for x t . In this calculation, the instances in Θ t generally come from a validation set, independent of the training set T [START_REF] Cruz | Dynamic classifier selection: Recent advances and perspectives[END_REF].

The alternative method we propose here is to use a selection criterion that does not rely on an independent validation set, but rather relies on the OOB estimate. To do so, the region of competence is formed by the k nearest neighbors of x t , amongst the training instances. These nearest neighbors are determined in the joint dissimilarity space with the RFD measure (instead of the traditional Euclidean distance). This is related to the fact that each candidate classifier is trained in this dissimilarity space, but also because the RFD measure is more robust to high dimensional spaces, contrary to traditional distance measures. Finally, the competence of each classifier is estimated with its OOB error on the k nearest neighbors of x t . Lines 7 to 15 of Algorithm 2 give all the details of this process.

To sum it up, the key mechanisms of the DCS method we proposed, noted DCS RF D and detailed in Algorithm 2, are:

• Create the pool of classifiers by using all the possible subsets of views, to avoid the expensive grid search for the weights generation (lines 4-5 of Algorithm 2).

• Define the region of competence in the dissimilarity space by using the RFD dissimilarity measure, to circumvent the issues that arise from high dimensional spaces (lines 12-13 of Algorithm 2).

• Evaluate the competence of each candidate classifier with its OOB error rate, so that no additional validation instances are required (line 14 of Algorithm 2).

• Select the best classifier for x t (lines 16-17 of Algorithm 2).

These steps are also illustrated in Figure 2 with the generation of the pool of classifiers in the upper part, and with the evaluation and selection of the classifier in the lower part of the figure. For illustration purposes, the classifier ultimately selected for predicting the class of x t is assumed to be the second candidate (in red).

Experiments

Experimental protocol

Both the SW OOB and the DCS RF D methods are evaluated on several realworld multi-view datasets in the following, and compared to state-of-the-art methods: the simple average of the view-specific dissimilarity matrices as a baseline method and the two static weighting methods presented in Section 4.1, namely the 3N N and the KA methods.

The multi-view datasets used in this experiment are described in Table 1. All these datasets are real-world multi-view datasets, supplied with several views of the same instances: NonIDH1, IDHcodel, LowGrade and Progression are medical imaging classification problems, with different families of features extracted from different types of radiographic images; LSVT and Metabolomic are two other medical related classification problems, the first one for Parkinson's disease recognition and the second one for colorectal cancer detection; BBC and BBCSport are text classification problems from news articles; Cal7, Cal20, Mfeat, NUS-WIDE2, NUS-WIDE3, AWA8 and AWA15 are image classification problems made up with different families of features extracted from the images. More details about how these datasets have been constituted can be found in the paper (and references therein) cited in the caption of Table 1.

All the methods used in these experiments include the same first stage, i.e. building the RF classifiers from each view and building then the view-specific Algorithm 2: The DCS RF D method Input: T (q) , ∀q = 1..Q: the Q training sets, each composed of n instances Input: D (q) , ∀q = 1..Q: Q n × n RFD matrices, built from the Q views Input: H (q) , ∀q = 1..Q: the Q RF classifiers used to build the D (q) Input: RF (.): The RF learning procedure Input: RF D(., .|.): the RF D measure Input: k: the number of neighbors to define the region of competence Input: xt: an instance to predict Output: ŷ: the prediction for xt // 1 -Generate the pool of classifiers: 1 {w0, w1, . . . , w 2 Q -1 } = all the possible Q-sized 0/1 vectors 2 H = an empty pool of classifiers 3 for i = 1..2 Q -1 do // The i th candidate classifier in the pool, wi[q] being the q th value of wi, either equal to 1 or 0:

4 Di = 1 Q Q q=1 D (q) .wi[q] 5 H[i] = RF (Di)
6 end // 2 -Evaluate the candidate classifiers for xt 7 for q = 1..Q do // the q th dissimilarity representation of xt 12

D[i] = 1 Q Q q=1 dx (q)
t .wi[q] // The region of competence, Di[j, .] being the j th row of Di : RFD matrices. Therefore, for a fair comparison on each dataset, all the methods use the exact same RF classifiers, made up with the same 512 trees [START_REF] Cao | Random forest dissimilarity based multi-view learning for radiomics application[END_REF]. As for the other important parameters of the RF learning procedure, the mtry parameter is set to √ m q , where m q is the dimension of the q th view, and all the trees are grown to their maximum depth (i.e. with no pre-pruning).

The methods compared in this experiment differ in the way they combine the view-specific RFD matrices afterwards. We recall below these differences:

• Avg denotes the baseline method for which the joint dissimilarity representation is formed by a simple average of the view-specific dissimilarity representations.

• SW 3N N and SW KA both denote static weighting methods for determining Q weights, one per view. The first one derives the weights from the performance of a 3N N classifier applied on each RFD matrix; the second one uses the KA method to estimate the relevancy of each RFD matrix in regards to the classification problem.

• SW OOB is the static weighting method we propose in this work and presented in Section 4.1; it computes the weights of each view from the OOB error rate of its RF classifier.

• DCS RF D is the dynamic selection method we propose in this work and presented in Section 4.2; it computes different combinations of the RFD matrices for each instance to predict based on its k nearest neighbors, with k = 7 following the recommendation in the literature [START_REF] Cruz | Dynamic classifier selection: Recent advances and perspectives[END_REF].

After each method determine a set of Q weights, the joint RFD matrix is computed. This matrix is then used as a new training set for a RF classifier learnt with the same parameters as above (512 trees, mtry = √ n with n the number of training instances, fully grown trees). As for the pre-processing of the datasets, a stratified random splitting procedure is repeated 10 times, with 50% of the instances for training and 50% for testing. The mean accuracy, with standard deviations, are computed over the 10 runs and reported in Table 2. Bold values in this table are the best average performance obtained on each dataset.

Results and discussion

The first observation one can make from the results gathered in Table 2 is that the best performance are obtained with one of the two proposed methods for 13 over the 15 datasets. This is confirmed by the average ranks that place these two methods in the first two positions. To better assess the extent to which these differences are significant, a pairwise analysis based on the Sign test is computed on the number of wins, ties and losses between the baseline method Avg and all the other methods. The result is shown in Figure 3. From this statistical test, one can observe that none of the static weighting methods allows to reach the significance level of wins over the baseline method. It indicates that the simple average combination, when using dissimilarity representations for multi-view learning, is a quite strong baseline. It also underlines that all views are globally relevant for the final classification task. There is no view that is always irrelevant, for all the predictions.

Figure 3 shows also that the dynamic selection method proposed in this work is the only method that predominantly improves the accuracy over this baseline, till reaching the level of statistical significance. From our point of view, it shows that all the views do not participate in the same extent to the good prediction of every instance. Some instances are better recognized when the dissimilarities are computed by relying on some views more than on the others. These views are certainly not the same ones from one instance to another, and some instances may need the dissimilarity information from all the views at some point. Nevertheless, this highlights that the confusion between the classes is not always consistent from one view to another. In that sense, the views complement each others, and this can be efficiently exploited for multi-view learning provided that we can identify the views that are the most reliable for every instance, one by one.

Conclusion

Multi-view data are now very common in real world applications. Whether they arise from multiple sources or from multiple feature extractors, the different views are supposed to provide a more accurate and complete description of objects than a single description would do. Our proposal in this work was to address multi-view classification tasks using dissimilarity strategies, which give an efficient way to handle the heterogeneity of the multiple views.

The general framework we proposed consists in building an intermediate dissimilarity representation for each view, and in combining these representations afterwards for learning. The key mechanism is to use Random Forest classifiers to measure the dissimilarities. Random Forests embed a (dis)similarity measure that takes the class membership into account in such a way that instances from the same class are similar. The resulting dissimilarity representations can be efficiently merged since they are fully comparable from one view to another.

Using this framework, our main contribution was to propose a dynamic view selection method that provides a better way of merging the per-view dissimilarity representations: a subset of views is selected for each instance to predict, in order to take the decision on the most relevant views while at the same time ignoring as much as possible the irrelevant views. This subset of views is potentially different from one instance to another, because all the views do not contribute at the same extent to the prediction of each instance. This has been confirmed on several real-world multi-view datasets, for which the dynamic combination of views has allowed to obtain much better results than static combination methods.

However, in its current form, the dynamic selection method proposed in this chapter strongly depends on the number of candidate classifiers in the pool. To allow for more versatility, it could be interesting to decompose each view into several sub-views. This could be done for example, by using Bagging and Random Subspaces principles before computing the view-specific dissimilarities. In such a way, the dynamic combination could only select some specific part of each view, instead of considering the views as a whole.

Figure 1 :

 1 Figure 1: The RFD framework for multi-view learning.

=

  RF D(xt, xj|H(q) ), ∀xj ∈ T (q) 9 end 10 D = an empty set of dissimilarity representations of xt11 for i = 1..2 Q -1 do //The averaged dissimilarity representation of xt:

13 θt

 13 ,i = the kNN according to RF D(D[i], Di[j, .]|H[i]), ∀j = 1..n // The competence of H(i) on θt,i: 14 St,i = OOBerr(H[i], θt,i) 15 end // 3 -Select the best classifier for xt and predict its class 16 m = arg max i St,i 17 ŷ = H[m](D[m])

Figure 2 :

 2 Figure 2: The DCS RF D procedure, with the training and prediction phases.The best candidate classifier that gives the final prediction for x t is H[START_REF] Cao | Random forest dissimilarity based multi-view learning for radiomics application[END_REF] in this illustration (in red).

Figure 3 :

 3 Figure 3: Pairwise comparison between each method and the baseline Avg. The vertical lines are the level of statistical significance according to the Sign test.

Table 1 :

 1 Real-world multi-view datasets[START_REF] Cao | Random forest dissimilarity based multi-view learning for radiomics application[END_REF]. a Imbalanced Ratio, i.e. the number of instances from the majority class over the number of instances from the

	minority class.
		features instances views classes	IR a
	AWA8	10940	640	6	8	1
	AWA15	10940	1200	6	15	1
	BBC	13628	2012	2	5	1.34
	BBCSport	6386	544	2	5	3.16
	Cal7	3766	1474	6	7	25.74
	Cal20	3766	2386	6	20	24.18
	IDHcodel	6746	67	5	2	2.94
	LowGrade	6746	75	5	2	1.4
	LSVT	309	126	4	2	2
	Metabolomic	476	94	3	2	1
	Mfeat	649	600	6	10	1
	NonIDH1	6746	84	5	2	3
	NUS-WIDE2	639	442	5	2	1.12
	NUS-WIDE3	639	546	5	3	1.43
	Progression	6746	84	5	2	1.68

Table 2 :

 2 Accuracy (mean ± standard deviation) and average ranks

		Avg	SW 3N N	SW KA	SW OOB	DCS RF D
	AWA8	56.22% ± 1.01 56.22% ± 0.99	56.12% ± 1.42	56.59% ± 1.41 57.28% ± 1.49
	AWA15	38.23% ± 0.83 38.13% ± 0.87	38.27% ± 1.05	38.23% ± 1.26 38.82% ± 1.56
	BBC	95.46% ± 0.65 95.52% ± 0.64 95.36% ± 0.74	95.46% ± 0.60	95.42% ± 0.59
	BBCSport	90.18% ± 1.96 90.29% ± 1.83	90.26% ± 1.78	90.26% ± 1.95 90.44% ± 1.89
	Cal7	96.03% ± 0.53 96.10% ± 0.57 96.11% ± 0.60 96.10% ± 0.60	94.65% ± 1.09
	Cal20	89.76% ± 0.80 89.88% ± 0.82	89.77% ± 0.68 90.00% ± 0.71 89.15% ± 0.97
	IDHCodel	76.76% ± 3.59 77.06% ± 3.43	77.35% ± 3.24	76.76% ± 3.82 77.65% ± 3.77
	LowGrade	63.95% ± 5.62 62.56% ± 6.10	63.95% ± 3.57	63.95% ± 5.01 65.81% ± 5.31
	LSVT	84.29% ± 3.51 84.29% ± 3.65	84.60% ± 3.54 84.76% ± 3.63 84.44% ± 3.87
	Metabolomic 69.17% ± 5.80 68.54% ± 5.85	70.00% ± 4.86	70.00% ± 6.12 70.21% ± 4.85
	Mfeat	97.53% ± 1.00 97.53% ± 1.09	97.53% ± 1.09	97.57% ± 1.01 97.63% ± 0.99
	NonIDH1	80.70% ± 3.76 80.47% ± 3.32	80.00% ± 3.15 80.93% ± 4.00 79.77% ± 2.76
	NUS-WIDE2 92.82% ± 1.93 92.86% ± 1.88	92.60% ± 2.12	92.97% ± 1.72 93.30% ± 1.58
	NUS-WIDE3 80.32% ± 1.95 79.95% ± 2.40	80.09% ± 2.07	80.14% ± 2.20 80.77% ± 2.06
	Progression	65.79% ± 4.71 65.79% ± 4.71	65.79% ± 4.99	66.32% ± 4.37 66.84% ± 5.29
	Avg rank	3.67	3.50	3.30	2.40	2.13

Acknowledgement

This work is part of the DAISI project, co-financed by the European Union with the European Regional Development Fund (ERDF) and by the Normandy Region.