L Weynans
email: lisl.weynans@math.u-bordeaux1.fr

Y Gorsse

A Iollo

A second order finite-volume scheme on cartesian grids for Euler equations

. In the fluid domain, away from the boundary, we use a classical finite-volume method based on an approximate Riemann solver for the convective fluxes. At the cells located on the boundary, we solve an ad hoc Riemann problem taking into account the relevant boundary condition for the convective fluxes by an appropriate definition of the contact discontinuity speed. To avoid pressure oscillations near the solid we balance the boundary condition with an extrapolation of the fluid values, as a function of the angle between the normal to the solid and the normal to the cell. Our objective is to device a method that can easily be implemented in existing codes and that is suitable for massive parallelization.

Resolution in the fluid domain

Governing equations

The compressible Euler equations are:

∂ρ ∂t + ∇ • ρu = 0 ( 1
)
∂ρu ∂t + ∇ • (ρu ⊗ u + pn) = 0 (2) ∂E ∂t + ∇ • ((E + p)u) = 0 (3) (4)
where E denotes the total energy per unit volume. For a perfect gas

E = p γ -1 + 1 2
ρu 2 and p = ρRT (5)

Discretization far in the fluid domain and boundary representation

We focus on a two-dimensional setting. We consider a cartesian grid, with indexes i and j and spacings h x and h y in the x and y direction, respectively. Far from the boundary with the solid, we use a classical finite-volume scheme based on an approximate Riemann solver, and a second-order Runge-Kutta scheme for the time integration.

In order to improve accuracy at the solid walls crossing the grid cells we need additional geometric information. This information, mainly the distance from the wall and the wall normal, is provided by the distance function. The level set method, introduced by Osher and Sethian is used to implicitly represent the interface of solid in the computational domain. The zero isoline of the level set function represents the boundary Σ of the immersed body. The level set function is defined here by:

ϕ(x) = dist Σ (x) outside of the solid -dist Σ (x) inside of the solid (6)
A useful property of this level set function is:

n(x) = ∇ϕ(x) (7)
where n(x) is the outward normal vector of the isoline of φ passing on x. In particular, this allows to compute the values of the normal to the interface, represented by the isoline ϕ = 0.

The boundary condition that we have to impose is u A .n A = 0, where u A is the speed of the fluid at the boundary, and n A the outward normal vector of the body. We want to impose the boundary condition on what we call the interface points: the intersections between the interface (ϕ = 0) and the segment connecting the two cell centers concerned by the sign change (for example the points A, B and C on Fig. 1(a)). To this purpose, a fictitious state is created for instance between the cells (i, j) and (i + 1, j) on Fig. 1(a).

B A (i,j) (i+1,j) φ=0 n A n cell (a) Geometric configuration (u .n )n (u . ) u u i-1 j n * i-1 j i-1 j n T φ=0 (i, j) (i+1, j) (i-1, j) T T B B B B B B (u . ) = u T T -B B * T (b) Construction of vector
•    u w v w p w c w    = α    u * v * p - c -    + (1 -α)    u f v f p f c f   ; • α = n A .n cell ; • (u f , v f , p f , c f
) is computed as a linear extrapolation of the fluid variables to the point x i+1/2 j using the variables and the slopes of the closest upstream fluid cell. It plays a role when the scalar product between the normal to the physical boundary and the normal to the mesh side is close to zero, because in this case the boundary condition only weakly affects the numerical flux.

We determine the value of the contact discontinuity speed u * , relative to a Riemann problem defined in the direction normal to the cell side through x i+1/2, j , consistent at second order accuracy with u A • n A = 0 in A. The vector u * is determined as follows:

u * .n A = u * n = u A .n A + 1 2 -d s n A u * .τ A = u * τ = u -.τ A    ⇒ u * = u * n n x + u * τ τ x u * n n y + u * τ τ y
Where u A is the velocity of the obstacle (= 0 for a steady body), n A = n x , n y t and τ A = τ x , τ y t are respectively the normal and tangential vectors to the boundary at point A and the slope s n A is defined as:

s n A = u A .n A -u i .n A + 1 -d 1 + d (u A .n A -u i-1 .n A ) . ( 8
)

Numerical illustrations

The Ringleb flow The Ringleb flow refers to an exact solution of Euler equations. The solution is obtained with the hodograph method. In our test case, the computational domain is [-0.5; -0.1] × [0; 0.6] and we numerically solve the flow between the streamlines Ψ 1 = 0.8 and Ψ 2 = 0.9. The inlet and outlet boundary condition are supersonic for y = 0 and y = 0.6 respectively.

The convergence orders are calculated in L 2 norm on four different grids 32 × 48, 64 × 96, 128 × 192 and 256 × 384. The results for the L 2 norm of our method are compared to a simple symmetry technique, the ghost-cell CCST method [START_REF] Dadone | Ghost-cell method for analysis of inviscid three dimensionnal flows on cartesian grids[END_REF], that relies on a local isoentropic flow model at the wall, and a standard finite-volume scheme case with a body-fitted meshs in Fig. 2.

The overall results show that the classical symmetry scheme is first order accurate in the L 2 norm. The other schemes have overall comparable accuracy, although the amplitude of the error is lower for the present scheme compared to the ghost-cell CCST method.

For the same test case, Coirier and Powell [START_REF] Coirier | An accuracy assessment of cartesian mesh approaches for the Euler equations[END_REF] observed also a convergence order between one and two in the case of their own cartesian method.

Mach 10 shock over three spheres The computation of a planar shock reflecting over three spheres is performed. The spheres are located at (0,0,0), (-0.75, 1.4,-1.4),

Figure 1 :

 1 Figure 1: Fig. 1(a): Example of geometric configuration at the interface. A is the interface point located between (i, j) and (i + 1, j). The flux on cell interface (i + 1/2, j) is modified to enforce the boundary condition on A. Fig. 1(b): Graphical illustration of the construction of the u * vector.According to Fig.1(b) and considering the (i + 1/2, j)-flux, we compute the left state primitive variables u -, p -, c -relative to the Riemann problem at the concerned numerical interface by a standard MUSCL reconstruction. The left state will be and U -= (u -, v w , p -, c -) and the right state U + = (-u -+ 2u w , v w , p w , c w ), where:

Figure 2 :

 2 Figure 2: Comparison of the L 2 accuracy of the present scheme with several methods. The convergence orders are detailed in the legend.

Figure 3 :

 3 Figure 3: Mach 10 planar shock reflecting on a 3D sphere. 20 isopressure surfaces.


Tiago Staudt

Frédéric Wurtz

Laurent Gerbaud

Nelson Jhoe Batistela

Patrick A Kuo-Peng

New

A New Approach on the Design and Optimization of Brushless Doubly-Fed Reluctance Machines

Keywords: Design methodology, Brushless machines, Finite element methods, Electromagnetic analysis, Wind energy

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The Brushless Doubly Fed Reluctance Machine (BDFRM) is being particularly considered as a viable alternative to the Doubly Fed Induction Machine (DFIG) in variable speed wind energy conversion systems (WECS). It keeps the cost advantages of the DFIG by allowing the use of a converter power rating of around 30% of the generator capacity. Additionally, the BDFRM has the advantage of maintenance-free brushless operation [START_REF] Dorrell | Improvements in Brushless Doubly Fed Reluctance Generators Using High-Flux-Density Steels and Selection of the Correct Pole Numbers[END_REF]. As evidenced in literature, although a promising solution, there is still a demand for new researches on the electromagnetic design so that this machine could be used in industrial scale. Most papers analyze existent machine designs rather than focusing on the development of new ones [START_REF] Knight | Design and Analysis of Brushless Doubly Fed Reluctance Machines[END_REF]. Similar conclusions are inferred in [START_REF] Xu | Design and control of a high-efficiency Doubly-Fed Brushless machine for wind power generator application[END_REF], where it is pointed out a series of fundamental issues and challenges with respects to the BD-FRM design and control such as requirements to maximize the torque and power density. It is also stressed the need for a systematic design procedures to obtain optimal designs considering different sizes, power ratings and applications to meet a specific market demand. In this sense, it is proposed a new structured method on the modeling, design and optimization of the BD-FRM. The main goal is to provide means to the designer to take pertinent decisions in all development phases based on fast to obtain and sufficiently accurate information, which will lead to an application specific optimal design.

GENERAL BACKGROUND

It is out of scope of this paper to explain in details the operating principles of the BDFRM. A brief discussion on them, considering aspects relevant to this paper, is presented in this section based on [START_REF] Liang | D-q analysis of a variable speed doubly AC excited reluctance motor[END_REF][START_REF] Xu | Transient model of a doubly excited reluctance motor[END_REF][START_REF] Betz | Introduction to the Space Vector Modeling of the Brushless Doubly Fed Reluctance Machine[END_REF]. The reader is referred to these references for further information on the BDFRM theory.

The BDFRM has two, three-phase, windings with different number of poles. In this paper, the windings are named "grid" (W g ) and "control" (W c ). Fig. 1 and Fig. 2 show the winding's configuration for a salient pole reluctance rotor. The basic condition to electromechanical conversion on this machine is that the reluctance rotor must satisfy the following condition :

P r = |P g ± P c | 2 (1)
Only the positive case will be considered on the rest of the analysis. Windings such as the ones in Fig. 1 and Fig. 2 leads to a reluctance rotor of 4 poles for the BDFRM. If (1) holds, there are torque production and the machine synchronous speed in steady state is then given by :

ω rm = ω g + ω c P r (2)
where P means number of poles, ω is the angular speed in radians per second [rad/s] and the indexes g and c refers to the grid and control windings and r is related to the rotor.

It is important to notice that P g and P c are different and, if a smooth rotor is considered, there is no direct coupling between the windings to induce useful electromotive force (EMF) in the other winding. The main role of the rotor is modulate the MMF's from W g and W c in order to induce voltage with correct frequency in each winding. In other words, if conditions (1) and (2) hold, the rotor allows that a current circulating in W g (I g ) with frequency ω g induces a voltage in W c with frequency ω c which is the basis for electromechanical conversion [START_REF] Xu | Design and control of a high-efficiency Doubly-Fed Brushless machine for wind power generator application[END_REF]. The dual situation is also valid.

In the BDFRM, torque is given by (3).

T em = -3 2

P g + P c 2 L gcmax I g I c sin(φ T orque ) (3)
where L gcmax is the maximum value of the mutual inductance between windings, I g and I c are the current's amplitudes on each winding and φ T orque is the torque angle.

PROPOSED DESIGN METHODOLOGY

In general, the idea behind modeling a device has two main goals that are not necessarily satisfied simultaneously : (i) formulate the specification, i.e. to define system parameters (inputs, outputs) and constraints and (ii) solve the problem by calculating parameters with enough precision, e.g. using the Finite Element Analysis (FEA). The objective (i) aims to help the spe- cialist to understand and define the problem. Several parameters are unknown and many combinations among them may, a priori, match the preliminary specification. At this stage, FEA may be useful to help in the comprehension of the device electromagnetic behavior. However, to refine the model and find an optimal solution, the use of this technique may cause delays in the development process due to large computational times and large search space. During this phase, it is usually more important to provide faster answers with a macroscopic model rather than obtaining high precision results. Finally, the goal (ii) searches to verify if the defined parameters will satisfy the specifications and accurate results are essentials for it.

A complex model, such as one of an electrical machine, may involve several different parameters and phenomenons like iron saturation and losses, thermal constraints, harmonics, definition of slots and turns number and many others. Consequently, to assemble all constraints in a single thorough model is usually impracticable considering that many of the parameters are unknown or undefined when the design process starts. To address this issue it is proposed in the present work a methodology which uses the concepts of sizing (SM), optimization (OM) and validation (VM) models [START_REF] Carlson | The concepts of sizing and optimization model : Applied to the optimal design of a Permanent Magnet Generator[END_REF]. Fig. 3 illustrates the idea with two different models approaches.

The SM aims to estimate device parameters from few specifi-

Complex Analytic Models:

-Nonlinearities; -Magnetic Losses.

Sizing Optimization

↑ Fast; ↑ Provide inital solution; ↓ Low precision.

↑ Analytical or semi-analytical; ↑ Optimization algorithms; ↑ No limits for sub-models. cations known by the designer, typically three or four. It is clear that the SM relies on designer expertise because several parameters shall be calculated from few specifications. Some of them are estimated by knowledge and experience and the outputs calculated as a function of these assumptions. This might result in imprecisions, but the main goal of a SM is to provide a good overview of the problem. It allows to test parameters variations very quickly, which helps to restrict the search space for the optimization problem. Ultimately, this approach shall provide at least one realizable machine, that will be the starting point to the optimization model.

The OM, on the other hand, is a direct model where the specification contents (which often includes performance parameters such as voltages, power etc.) are not necessarily considered as inputs. Its goal is to solve an objective function with constrained input and output parameters. The limited search space may be obtained from the SM results or defined by the designer expertise. Moreover, it allows the integration of, a priori, an unlimited number of sub-models to describe other phenomenons like losses, thermal and application-specific constraints. The great difficulty behind the OM is that some specification requirements are usually outputs. Thus, to solve this model, the OM shall be coupled to an optimization algorithm.

The VM takes place after the SM and OM definitions. It is related to the objective (ii) and it is used to verify if the former models provided coherent and accurate results. The VM model is most often a fine and accurate model, e.g. a FEA model. After that, the next step in the development process would be to build a prototype.

BUILDING THE MODELS

The OM and the SM are complementary, but this do not mean necessarily that one requires the other to implement the methodology. This depends on the application and also on the designer expertise. In general, the SM is very useful to understand the basic principles, requirements and to find the constraints. To obtain an optimized solution, it is the OM which plays a dominant role.

Usually, the BDFRM sizing model relies on the classical electrical machine theory. In [START_REF] Knight | Design and Analysis of Brushless Doubly Fed Reluctance Machines[END_REF], for example, it is shown an interesting procedure considering analytical approaches to design a BDFRM with an ideal ducted rotor. They provide a set of design equations and an analysis of the rotor and stator windings poles combination. It is an approach that could be used on the SM definition. This paper focuses in the OM development rather than describes the SM in details. As the SM is not presented, the initial machine main dimensions, the starting point in the optimization, will be based on one machine referenced in the literature [START_REF] Scian | Assessment of Losses in a Brushless Doubly-Fed Reluctance Machine[END_REF].

Optimization Model

In order to build the OM, some methods may be used to represent the machine such as Finite Element Method (FEM), Boundary Element Method (BEM) or Reluctance Networks (RN). There are applications in which one is more suitable than the other. In general, the FEM is not very well adapted for optimization problems in early development stages since the search space and the computation time may be large. This may limit the number of parameters and constraints to be used [START_REF] Delale | Automatic Generation of Sizing Models for the Optimization of Electromagnetic Devices Using Reluctance Networks[END_REF].

In this work it is proposed the utilization of the Reluctance Network approach to build the OM. The RN technique is very interesting because it allows a good compromise between precision and computational time. Furthermore, it may help in the comprehension of the machine electromagnetic behavior and it also allows the integration of ferromagnetic non-linearities with analytical models.

One utilization example of this technique in the literature is shown in [START_REF] Raminosoa | Time-Stepping Simulation of Synchronous Reluctance Motors Using a Nonlinear Reluctance Network Method[END_REF], where it is found a good agreement between FEA and the model with a calculation time among 7 and 9 times faster then FEA for the synchronous reluctance machine. Similar strategy has already been used on the design of electromagnetic devices [START_REF] Dogan | Multistatic Reluctance Network Modeling for the Design of Permanent-Magnet Synchronous Machines[END_REF][START_REF] Ibala | Accounting for the Armature Magnetic Reaction and Saturation Effects in the Reluctance Model of a New Concept of Claw-Pole Alternator[END_REF] and it is now applied to the BDFRM.

One criticism for the use of the RN method is that it could be very onerous to implement. It implies in building the network topology and also in the solution of a set of equations numerically. There are in literature solutions which allow to overcome this difficulty. This paper, for example, utilizes the software package Cades/RelucTool [START_REF] Delale | Automatic Generation of Sizing Models for the Optimization of Electromagnetic Devices Using Reluctance Networks[END_REF][START_REF] Du Peloux | Automatic generation of sizing static models based on reluctance networks for the optimization of electromagnetic devices[END_REF][START_REF] Vestasystems | Cades Solutions[END_REF] to develop the OM. One advantage of these tools is that they are able to consider nonlinear behaviors due to saturated materials [START_REF] Delale | Automatic Generation of Sizing Models for the Optimization of Electromagnetic Devices Using Reluctance Networks[END_REF]. Additionally, they also calculate the partial derivatives between the outputs and inputs, finding parameter sensitivities.

The input and output parameters can then be constrained in a range and an objective function can be optimized by gradientbased algorithms [START_REF] Du Peloux | Automatic generation of sizing static models based on reluctance networks for the optimization of electromagnetic devices[END_REF] (e.g. Sequential Quadratic Programming (SQP) optimization algorithm), which is significantly faster than using FEM for optimization purposes.

The optimization model developed in this work is shown in the diagram represented in Fig. 4. The input parameters are defined considering the initial machine. The RN (Fig. 5) is built as a function of the machine dimensions to calculate the reluctances in each part where the flux lines are significants.

The RN is used to calculate the fluxes and flux densities around the machine considering ferromagnetic non-linearities. These quantities are used to calculate performance parameters such as torque, voltages and self and mutual inductances. Several others equations may be included to describe different aspects in the machine such as power, total mass, current densities and so on.

When compiling the model, the Cades system generates the Jacobian matrix with the partial derivatives of all outputs Y 1 (x 1 , . . . , x n ), . . . , Y m (x 1 , . . . , x n ) with respect to the inputs in the form :

CADES SYSTEM Reluctance Network • Fluxes • Flux Densities Inputs: • Currents • Dimensions • Windings Results: • Power • Mass • Current Densities • ... Results: • Torque • Inductances • Voltages
J =       ∂Y 1 ∂x 1 • • • ∂Y 1 ∂x n . . . . . . . . . ∂Y m ∂x 1 • • • ∂Y m ∂x n       . (4)
The equation system is then coupled to the SQP optimization algorithm for system solution.

CASE STUDY

5.1.

Machine Modeling Since the main goal of the case study is to illustrate method capabilities rather than propose a ready to fabricate machine, the salient pole rotor has been chosen due to its simpler analytical modeling. Similar approach with minor modifications could be used for different rotor types. It has been chosen a combination of P r = 4 rotor poles and P g = 2 and P c = 6 poles in the grid and control windings, respectively, as shown in Fig. 1 and Fig. 2. Additionally, Fig. 5 shows a quarter of the RN implemented as part of the OM. To develop this network, a first FEA has been performed in the machine considering a DC excitation only in phase A of the grid winding (i gA ). This procedure allows to define the most significant flux paths where the reluctances should be located in the network and also to calculate λ gAA , the phase A flux linkage, to verify the initial RN model accuracy.

The machine torque is calculated using equation [START_REF] Xu | Design and control of a high-efficiency Doubly-Fed Brushless machine for wind power generator application[END_REF]. Based on the salient pole analytical model [START_REF] Betz | Introduction to the Space Vector Modeling of the Brushless Doubly Fed Reluctance Machine[END_REF] and in order to simplify the RN construction, some idealized assumptions have been considered to find a simplified torque expression. This parameter can be shown to be proportional to :

T e ∝ n c n g λ gAA i gA (5)
where n c and n g are the total number of turns per phase per pole pair divide by 2 in W c and W g respectively.

Numerical Analysis

To verify the methodology, two optimization analysis have been performed using CADES at the operating point I g = I c = 7.5 A : "Max T" aims to maximize the torque and "Min M" to minimize the mass whereas keeping the mean torque maximum found in "Max T". In both calculations the input physical parameters have been constrained in specific ranges, but left free to vary within the range. The minimum air gap length constraint has been set to the value considered in the initial machine. The external diameter and the stack length have been kept constant to match a specific machine frame. The thermal constraints are indirectly taken into account by limiting the current density in the windings at 5 A/mm 2 .

At this point, the validation model (VM) based on FEA takes place in order to verify the optimization output parameters. It is important to notice that the FEA in this strategy is also very important, because the optimization results are verified using it. The advantage of the aforementioned procedure is to reduce the number of FEA calculations (so saving time) to obtain an optimized design. The resulting machines are verified using multistatic non linear FEA for different levels of current I c . The current I g = 5 A and angular mechanical speed ω rm = 750 rpm are kept constants. The VM results are summarized in Table 1. The estimated iron mass calculated analytically has decreased approximately 20 % compared to the initial design. The mean torque curves of the optimized machines "Max T" and "Min M" are shown in Fig. 6. It can be seen that the torque has significantly increased in all operating range. The different designs can be seen in Fig. 7.

CONCLUSIONS

This paper addresses a new BDFRM design method focusing on its electromagnetic structure optimization using a Reluctance Network approach and the concepts of sizing and optimization models. As stated by many authors, there is still much work to be done in order to establish the BDFRM as a widely used ma- chine. The described methodology aims to help reaching this objective. Although the fact that the OM has given indeed good results for the optimized design, it is worth to mention that the RN used in the case study as example is oversimplified. This may eventually result in low accuracies in the determination of some performance parameters when compared to FEA depending on the operating point. Further work shall be made to increase its robustness. Nevertheless, it does satisfies its objective by showing an "appropriated direction" in order to build an optimized design. The provided example which maximizes the torque whereas keeping the mass to a minimum illustrates the interesting possibilities on using this procedure.

Fig. 1 .

 1 Fig. 1. Winding's example with 2 poles.

Fig. 2 .

 2 Fig. 2. Winding's example with 6 poles.

Fig. 3 .

 3 Fig. 3. Sizing and Optimization models characteristics.

Fig. 4 .

 4 Fig. 4. Optimization model implemented in CADES.

Fig. 5 .

 5 Fig. 5. One quarter of the implemented BDFRM Reluctance Network model.

Tableau 1 .

 1 Torque and mass comparison between designs.

Fig. 6 .

 6 Fig. 6. Mean torque comparison between different designs as a function of current Ic. Current Ig is fixed in 5 A and the angular mechanical speed in 750 rpm.

Fig. 7 .

 7 Fig. 7. Machine initial and optimized ("Min M") designs.

  Torque [Nm] T Initial T M axT T M inM

	T Ic=05A	4.0	8.4	8.2
	T Ic=10A	7.7	13.6	13.3
	T Ic=20A	13.3	16.1	15.8
	T Ic=30A	16.0	16.4	16.2
	Mass [kg]	51.7	42.0	41.6


Sophie Pujol

Marc Berthillier

Jérôme Defrance

Joseph Lardies

Jean-Pierre Levain

Rémy Petit

Hélène Houot

Frédéric Mauny
email: frederic.mauny@univ-fcomte.fr

Indoor noise exposure at home: a field study on urban schoolchildren

Keywords: noise exposure, indoor noise sources, children, dwelling, multilevel model, field study Practical implications

To evaluate indoor noise levels and to investigate the potential factors that may be related to, an eight-day noise measurement campaign was conducted in the homes of 44 schoolchildren attending the public primary schools of Besançon (France). The presence of the inhabitants in the dwelling and the noisy events occurring indoors and outdoors were daily collected using a time-location-activity diary (TLAD); 902 time periods were analysed. The indoor noise level increased significantly with the outdoor noise level, along with the duration of presence or level of activity of the inhabitants at home. However, this effect may vary according to the period of day and the day of the week. Moreover, a significant part of the day and evening indoor noise level variability was explained by factors collected by the TLAD: 46% and 45% in the bedroom, 54% and 39% in the main room, respectively. Our results highlight the complexity of the indoor environment in the dwellings of children living in an urban area. Combining the inhabitant presence and indoor noise source descriptors with outdoor noise levels and other dwelling or inhabitant characteristics could improve large-scale epidemiological studies. However, additional efforts are still needed, particularly during the night period.

This is the pre-peer reviewed version of the following article: Pujol, S., Berthillier, M., Defrance, J., Lardies, J., Levain, J.-P., Petit, R., Houot, H., Mauny, F., 2014. Indoor noise exposure at home: a field study in the family of urban schoolchildren. Indoor Air n/a-n/a. doi:10.1111/ina.12094, which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1111/ina.12094/abstract.

Introduction

The relationship between noise pollution and human health has been the subject of numerous studies over the last two decades. Researchers have recently focused increased attention on the relationship between noise and non-auditory effects, such as annoyance, hypertension, cardiovascular diseases, sleep disturbance, and cognitive impairments [START_REF] Clark | The Effect of Transportation Noise on Health and Cognitive Development:A Review of Recent Evidence[END_REF][START_REF] Passchier-Vermeer | Noise exposure and public health[END_REF][START_REF] Paunovic | Epidemiological studies on noise and blood pressure in children: Observations and suggestions[END_REF].

People are exposed to environmental noise of various origins: transport (road, rail, or air traffic), construction and industry, community sources (neighbourhood, bars and restaurants, discotheques), and social or leisure sources (World Health Organization, 2011).

The assessment of exposure to noise requires the consideration of many factors, including measured or modelled exposures, choice of noise indicators, population distribution, timeactivity patterns of the exposed population and combined exposures to multiple sources of noise (World Health Organization, 2011).

Advances have been made in assessing the actual exposure of populations to noise sources, such as air traffic and road traffic (World Health Organization, 2011). Thus, noise exposure mapping is a commonly adopted step in the process of estimating the noise exposure of a population (European Commission, 2002;[START_REF] Murphy | Strategic environmental noise mapping: Methodological issues concerning the implementation of the EU Environmental Noise Directive and their policy implications[END_REF][START_REF] Seong | Modeling of road traffic noise and estimated human exposure in Fulton County, Georgia, USA[END_REF][START_REF] Xie | Mapping Urban Environmental Noise: A Land Use Regression Method[END_REF]. However, only outdoor exposure is estimated, typically in front of either the most exposed facade or the bedroom facade of the dwelling in which the participant subjects reside.

According to the United States Environmental Protection Agency (US EPA), children spend approximately 90% of their time indoors and over 60% of their time within their own residence (US EPA, 2009). Generally speaking, noise pollution within the indoor environment is a complex mixture of agents migrating from outdoors, in addition to agents generated by indoor sources [START_REF] Le Cann | Indoor e i o e t a d hild e s health: 'e e t de elop e ts i he i al, iologi al, physical and social aspects[END_REF]. [START_REF] Pirrera | Nocturnal road traffic noise: A review on its assessment and consequences on sleep and health[END_REF] recommend to record the indoor noise level in the bedroom of each participant to provide the most exact and reliable noise pollution estimates. Thus, several epidemiological studies have used indoor noise measurements to assess the relationship between noise and health. However, these studies are typically conducted using limited human samples [START_REF] Aasvang | A field study of effects of road traffic and railway noise on polysomnographic sleep parameters[END_REF][START_REF] Graham | Habitual traffic noise at home reduces cardiac parasympathetic tone during sleep[END_REF][START_REF] Pirrera | Nocturnal road traffic noise assessment and sleep research: The usefulness of different timeframes and in-and outdoor noise measurements[END_REF] or are based on short-term noise level measurements [START_REF] Babisch | Blood pressure of 8-14 year old children in relation to traffic noise at home -Results of the German Environmental Survey for Children (GerES IV)[END_REF][START_REF] Evans | Environmental Justice, Cumulative Environmental Risk, and Health Among Low-and Middle-Income Children in Upstate New York[END_REF]. Alternative methods for determining the indoor noise levels consider the indoor noise level to be the difference between the outdoor noise level and the facade insulation [START_REF] Amundsen | The Norwegia Fa ade I sulaio "tud : The ei a of fa ade i sulaio i edu i g oise a o a e due to oad t ai[END_REF][START_REF] Ohrström | Longitudinal surveys on effects of changes in road traffic noiseannoyance, activity disturbances, and psycho-social well-being[END_REF][START_REF] Pirrera | Nocturnal road traffic noise: A review on its assessment and consequences on sleep and health[END_REF]World Health Organization, 2009). This attractive method can be applied to a large number of people and allows researchers to calculate the proportion of subjects exposed to harmful noise levels. However, this method introduces uncertai t of the i ha ita ts oise e posu e assessment on an individual level and is still under investigation.

A previous noise measurement campaign conducted within the places of residence of schoolchildren examined the variability of weekly indoor and outdoor noise levels [START_REF] Pujol | Urban ambient outdoor and indoor noise exposure at home: A population-based study on schoolchildren[END_REF]. The aims of the present study are to quantify the daily indoor noise exposure of children living in an urban area and to analyse the factors that may influence noise level variability. Specifically, day-to-day variabilit fa to s ithi the sa e d elli g ithid elli g a ia ilit a d a ia ilit fa to s that diffe et ee sepa ate d elli gs et ee -d elli g a ia ilit a e e a i ed.

Methods

The population characteristics and the methods used in this study have been previously reported by [START_REF] Pujol | Urban ambient outdoor and indoor noise exposure at home: A population-based study on schoolchildren[END_REF]. The major points are described below.

Population

This study population consisted of 8-and 9-year-old school-children who were randomly selected from among the 900 pupils attending one of the 35 public primary schools in key stage 2, year 4 in the French city of Besançon in 2006-2007. The parents of the children were contacted by telephone for consent to participate in the study and to determine each hild s i lusio eligi ilit , hi h i luded the following characteristics:

eside e ithi the it at the sa e eside e fo at least o e ea , the hild s ed oo being located either higher than the ground floor or at ground level with a private garden or courtyard, and a bedroom window of an appropriate size to affix the outdoor microphone.

Forty-four dwellings were selected and equipped with microphones in the order in which the families agreed to participate, taking into account the availability of the inhabitants and the measurement equipment and avoiding long holiday periods and unusual living conditions.

Dwellings and family characteristics

Before the beginning of the measurement session, the presence of indoor noise sources in the dwelling (radio, television, musical instrument, computer, or others) and the number of children sleeping in the bedroom were recorded by the operator. Standardised questionnaires were distributed to the families to collect the household socio-economic characteristics (single parenthood and parental occupation, employment status, and educational level); family size; the number of residents; residency duration; the hild s age, sex, and birth order; and dwelling characteristics (number of rooms, floor level, type of dwelling, and type of windows). The families were also asked to record the following information for the duration of the measurement session in a time-location-activity diary (TLAD) by periods of 30 minutes: the presence of adults and children in the dwelling, noisy events occurring indoors (use of television, radio, musical instruments, or household appliances), noisy events occurring outdoors, and opening of the windows.

Noise measurement

The study was conducted from December 2006 to July 2007 using three class 1 acoustic equipment chains, each composed of a sound level meter (Blue Solo®, 01dB-Metravib) and a front-end acquisition equipment (Harmonie® or Symphonie®, 01dB-Metravib). Three microphones were used in each dwelling: one microphone was used in the room where the child spent most of the time, i.e. the main room; one microphone was used i the hild s ed oo ; and one microphone was used outdoors (2 m i f o t of the hild s bedroom window). The microphone location was chosen carefully, considering acoustic and family constraints and avoiding locations close to walls, windows, and doors. The microphone was placed 1.15 m above the floor, corresponding to the approximate height of a hild s ea .

Equivalent continuous A-weighted sound levels (L Aeq , in dB) were measured every second during an eight-day period, according to the French standard (NF S 31-010, 1996).

The measurement chains were calibrated at the beginning and at the end of the measurement sessions.

In parallel, theoretical outdoor noise levels in front of the main room facade were calculated using a strategic noise map (Pujol et al., 2009) that was built in accordance with the European environmental noise directive 2002/49/CE (European Commission, 2002).The noise prediction software MITHRA [START_REF] Cstb | MITHRA 5.0, manuel technique[END_REF] was used to position virtual receivers were on the floor of the dwelling, at 2m in front of the facade of the main room.

Data processing

For each measurement location, L Aeq, day (6:00-18:00), L Aeq, evening (18:00-22:00) and L Aeq, night (22:00-6:00) were calculated. Data recorded during the first day of measurement and during the following unfavourable measurement conditions were excluded from the analyses: rainfall or strong wind (wind speed higher than 5 m.s -1 ), unusual outdoor or indoor sound events reported by families (fairs, demonstrations, and open-air concerts) and periods for which acoustic data were available for less than half the time.

Three classes of socio-economic status were defined using the pa e ts o upatio s, according to the French National Institute of Statistics and Economic Studies classification.

The class of the more privileged member of the couple was used to determine the household socio-economic status. Crowding was defined based on the number of people per room. Both apartment buildings and semi-detached houses were defined as collective dwellings.

The data olle ted i the TLAD e e used to ua tif the i ha ita ts p ese e i the d elli g, the o u e e of ois e e ts i the hild s ed oo , out of the hild s ed oo , outdoors), and the opening of the i do s i the hild s ed oo a d the main room for each period (day, evening, and night). Periods were excluded from the analysis if the data were not fulfilled during the totality of the period. The occurrence was defined for each period as the number of 30-min periods associated with an event, as reported by the family, divided by the total number of 30-min periods (i.e. 24 in the day, 8 in the evening and 16 in the night). The occurrence values were multiplied by 100 for convenience. The school calendar was used to determine school days, in addition to the evenings and nights before a school day.

Statistical analysis

Descriptive statistics are presented as the means and standard deviations (SD). To take into account the hierarchical structure of the data, multilevel linear regression models [START_REF] Goldstein | Multilevel Statistical Models[END_REF] were used to assess the associations between the indoor noise level and independent factors, including the outdoor noise level, the TLAD variables, and the dwelling or family characteristi s. T o le els e e defi ed, i ludi g da of easu e e t le el I a d d elli g le el II , to pa titio the o e all a ia ilit i to ithi -d elli g a ia ilit

(day-to-da a ia ilit , le el I a d et ee -d elli g a ia ilit f o one dwelling to another, level II). The variables that were associated with the indoor noise level at P ≤ . i a univariate analysis were then included in a multivariate analysis using a backward step-bystep elimination procedure. These analyses were only performed on the time periods for which both the TLAD variables and the noise level were available. The proportion of the a ia e e plai ed a odel as al ulated usi g a do effe t a ia es of the ull model (containing only an intercept term) and those of the considered model. Multilevel analyses were also used to test for the period effect on TLAD variables and noise levels. A Pvalue of 0.05 was used as a threshold for statistical significance. The SYSTAT 12.02 (SYSTAT Software, Inc., Chicago, IL) and MLwiN 2.1 (University of Bristol, UK) [START_REF] Rasbash | Bristol University | Centre for Multilevel Modelling | Manual supplement for MLwiN 2[END_REF] software programs were used to perform the analyses.

Ethics

Permission to conduct this study was given by the French National Committee for the Treatment of Information in Health Research (CCTIRS) and the French National Computing and Freedom Committee (CNIL).

Results

Sample characteristics

Among the 44 dwellings included in the analysis, 80% were in a collective building.

The ie f o the hild s ed oo i do as eithe a ou t a d o a g ass a ea i % of the dwellings, whereas the view was a street in 66% of the main room windows. Most of the windows were double-glazed windows, both in the bedroom (75%) and in the main room (82%). Forty-one per cent of the participant children shared their bedroom with another child or two other children. A TV set was present in 32% of the hild e s ed oo s. The main room was a living room in 89% of the dwellings (n = 39), a separate bedroom in 7% of the dwellings (n = 3), and a kitchen in 4% of the dwellings (n = 2). The number of inhabitants ranged between 2 and 6 (mean = 4.2), and each family had on average 2.4 children (range = 1-4 children). A majority (57%) of the families had an intermediate socio-economic status, whereas 34% had a privileged socio-economic status, and 9% had an underprivileged socioeconomic status.

Living conditions

A total of 902 time periods were retained from the 1077 TLAD data. On average, the participant children spent approximately 17 hours per day (67%) at home. A majority of the evening and night periods were spent at home (77% and 90%, respectively), whereas only 48% of the day period was spent at home (Table 1). Unsurprisingly, the inhabitants were significantly more present in the dwelling during the evening and the night periods compared with the day period (all P < 10 -3 ). Noisy indoor events were significantly more frequent during the evening period (P < 10 -3 ). Window-opening behaviour was prone to both considerable day-to-day and between-dwelling variability. The windows were open every da fo : hou s i the hild s ed oo a d : hou s i the main room on average.

Dwelling noise exposure

Measurements of noise level using L Aeq,day , L Aeq,evening , and L Aeq,night in the main room, i the hild s ed oo , a d outdoo s a e sho i Ta les a d . During the evening, the indoor L Aeq in the main room was significantly higher than outdoors (+ 4.1 dB, P < 10 -3 ).

" hool da a d Da efo e s hool da

During school days, both children and adults spent significantly less time at home than when there was no school: 37.1% of the day period vs. 58.3 for the participant child (P < 10 -3 ), 36.8% vs. 48.8% for the other children (P < 10 -3 ) and 60.0% vs. 67.6 for the adults (P = 0.01). The occurrence of the noisy events declared by the family was similar in the hild s ed oo . % of the da pe iod o s hool days vs. 4.1% on days without school). In the main room, the noisy events were less frequent on days of school (20.6% vs. 24.5% on days without school, P = 0.01). The indoor L Aeq, day was significantly lower during school days compared with days without school (-3.1 dB in the bedroom and -2.1 dB in the main room, both P = 10 -3 ) (Table 2).

Before a school day, both the participant children and the adults spend more time at home than before a day without school (i.e. during the evening 84.0% vs. 68.7%, P < 10 -3 for the participant child and 88.4% vs. 80.3%, P = 0.01 for the adults, respectively and during the night 95.8% vs. 84.1%, P < 10 -3 for the participant child and 98.4% vs. 94.5%, P = 0.03, for the adults, respectively). The indoor L Aeq, evening was significantly higher than before a day without school (+ 2.3 dB in the bedroom and + 2.7 dB in the main room, P = 0.02 and 0.01, respectively) (Table 3). Conversely, on a night before a school day, the L Aeq, night was significantly lower than before a day without school i the hild s ed oo -3.9 dB, P < 10 - 3 ) but not in the main room (-1 dB, P = 0.10).

The outdoor L Aeq was statistically significantly higher during the day periods of school days (+ 1.2 dB, P < 10 -3 ) and slightly lower on a night before a school day (-0.5 dB, P = 0.02).

No difference was observed during the evening period.

Multilevel analysis results

Among the variables selected during the univariate analyses, four were no more significant at the issue of the backward step-by-step procedure: type of windows, crowding, noisy outdoor event and the presence of a musical instrument in the main room. The ultile el odels fo the hild s ed oo a d the ai oo a e p ese ted i Ta les a d 5.

Child s ed oo

A significant and positive association was found between the bedroom L Aeq and the outdoor L Aeq during the day and the evening (P < 10 -3 and P = 0.04, respectively), whereas a marginally significant association was found during the night (P = 0.06) (Table 4). Each time the outdoor L Aeq, day increased by 10 dB, the bedroom L Aeq, day increased on average by 3.6 dB.

However, this increase was lower during the evening (+ 2.3 dB) and the night (+ 1.7 dB). In the odels da , e e i g , ight , the p ese e of people i the d elli g was associated with an increased bedroom L Aeq . However, the influence of the presence of different people in the dwelling depended on the time period. Only the participant hild s p ese e as statistically significant during each time period. Compared with the bedrooms without a TV set, the L Aeq in the bedrooms with a TV set was higher during the day [+ 2.6 dB (P = 0.03)] and the evening periods [+ 4.2 dB (P = 0.01)]. The indoor L Aeq, night was significantly lower when the participant child was sleeping alone in his bedroom (-2.7 dB, P = 0.05) or before a day of school (-4.1 dB, P <10 -3 ).

The proportions of the a ia e e plai ed the da , e e i g a d ight models reached 46%, 45%, and 17%, respectively. When excluding the TLAD variables, the proportions decreased to 16%, 14% and 9%, respectively.

Main room

In the main room, the presence of adults was significantly and positively associated with indoor L Aeq , regardless of the time of day (Table 5). The other events described in the TLAD (presence of the participant child, presence of other children, or a noisy event in the dwelling) were also significantly associated with an increased indoor L Aeq i the da a d e e i g odels. A sig ifi a t a d positi e asso iatio et ee i doo a d outdoo L Aeq as fou d du i g the da β = 3.8; 95% CI = [0.7; 6.8]; P = 0.02). Regarding the evening and ight pe iods, this elatio ship as ot sig ifi a t β = 0.9; 95% CI = [-2.1; 3.9]; P = 0.55 and β = 1.2; 95% CI = [-4.3; 6.7]; P = 0.67, respectively). The main room L Aeq was, on average, 1.9 dB lower on a night before a school day than on a night that was not before a school day.

The p opo tio of a ia e e plai ed the da , e e i g a d ight odels reached 54%, 39% and 3%, respectively. When excluding the TLAD variables, the proportion of the variance that was explained by the models fell to 27%, 10% and 1%, respectively.

Discussion

This study on noise exposure in schoolchildren aimed to explore the between-and within-dwelling variability of noise levels at home. In addition to the expected betweendwelling variability, the within-dwelling variability of noise levels from the three time periods was significant, as was the course of the week for both indoor and outdoor noise levels.

Among the identified factors that influence noise variability, dwelling characteristics and ho e i ha ita ts p ese e o a ti ities appeared to play a predominant role.

These results were obtained from a large data set based on a six-month acquisition campaign that was conducted at the places of residence of randomly sampled schoolchildren. To insure the quality of the data, approved noise measurement equipment, an adapted standardised measurement protocol and questionnaires were used [START_REF] Pujol | Urban ambient outdoor and indoor noise exposure at home: A population-based study on schoolchildren[END_REF]. Furthermore, data control and data processing measures were carefully conducted, including the identification and exclusion of invalid or incomplete time periods and careful verification of the TLAD with the family at the end of each session. However, biases due to the underestimation or underreporting of noisy events due to the absence of parents or wakefulness periods cannot be excluded. Similar to [START_REF] Díaz | Sound exposure during daily activities[END_REF], whose study relies on a moderate sample size but a long measurement period, the sample size was optimised to include an entire week to quantify fluctuations in family life, including days of school vs. days without school and working days vs. days off. The multilevel multivariate analyses were conducted to control for the repeated structure of the data, to better quantify the day-to-day variability and the factors influencing the day-to-day variability, and to control for confounding effects.

The indoor noise levels increased with the presence or activity of the inhabitants at home, although the influence of these parameters may vary according to the period of the day and the day of the week. Individuals, as well as household items in the dwelling, can be considered to be indoor sound sources and can help to explain a large part of the level and the variability of the indoor noise. This observation was especially true during the day and evening periods, when the inhabitants were awake. The notion of indoor noise sources has already been tackled by several authors [START_REF] Aasvang | A field study of effects of road traffic and railway noise on polysomnographic sleep parameters[END_REF][START_REF] Pirrera | Nocturnal road traffic noise assessment and sleep research: The usefulness of different timeframes and in-and outdoor noise measurements[END_REF]. Our results demonstrate the crucial necessity of quantifying noisy events and inhabitants presence for indoor noise level assessment.

A complex relationship between indoor and outdoor noise levels was underlined.

Indeed, the indoor noise level was found to be associated with outdoor noise levels or opening of the windows during the day or the evening periods. Fortunately, this finding strengthens the results of numerous studies or noise regulation policies that are based upon outdoor noise to assess human exposure or evaluate health effects [START_REF] Belojevic | Urban road-traffic noise and blood pressure and heart rate in preschool children[END_REF]European Commission, 2002;[START_REF] Mehdi | Spatio-temporal patterns of road traffic noise pollution in Karachi, Pakistan[END_REF]World Health Organization, 2011, 2009).

However, the correlation between increasing bedroom L Aeq and increasing outdoor L Aeq was greater during the day than during the evening; and, the correlation was over two times less during the night and only marginally significant. The same trend can be observed in the main room results. Several factors could explain the nonlinearity of this relationship across the different time periods. First, the concurrence of a higher outdoor noise level, a lower indoor noise source emission and a longer opening time during the day should be considered. Thus, the relationships between the outdoor noise levels and the window opening behaviour (European Commission Working Group on Health and Socio-economic Aspects, 2004) or indoor noise levels [START_REF] Amundsen | The Norwegia Fa ade I sulaio "tud : The ei a of fa ade i sulaio i edu i g oise a o a e due to oad t ai[END_REF] should be taken into consideration. Secondly, the low contribution of the outdoor L Aeq to the indoor L Aeq during the evening and night periods is consistent with the moderate outdoor noise levels. As described in [START_REF] Pujol | Urban ambient outdoor and indoor noise exposure at home: A population-based study on schoolchildren[END_REF], the main noise source in this city of 120, 000 inhabitants (INSEE, 2011) is ground transport, including road and rail traffic; however, no motorways crossed the inhabited districts. Due to the facade insulation, the amount of outdoor noise that enters the dwelling is relatively low, especially when the windows are closed. Therefore, indoor noise source emissions become the major contributors to the ambient noise level inside the dwelling. This finding is particularly true in the evening period, when both indoor L Aeq and indoor noisy events, as declared by the inhabitants, are the highest.

The World Health Organization (2009) proposed a default reduction of 21 dB(A) to convert a theoretical outside night noise level on the most exposed facade to an inside night noise level, thereby taking into account the possibility that the windows may be open a large part of the year. Applied to our results, this default reduction index provides an average indoor night noise level that is very close to the observed one (32.5 vs. the observed 33.5 dB(A), after using the +6 dB(A) correction proposed by [START_REF] Amundsen | The Norwegia Fa ade I sulaio "tud : The ei a of fa ade i sulaio i edu i g oise a o a e due to oad t ai[END_REF] to convert a bedroom facade to a most exposed facade noise level). When the default reduction index was applied to the main room, the predicted results diverged from the observed results, at, respectively, 33.0 vs. the observed 41.5 dB(A). Thus, whereas the default reduction index a e of i te est i a hild s ed oo oise le el assess e t, this parameter does not have a lea p edi ti e alue i othe oo s, i ludi g the ai oo i ou stud , o a adult s bedroom [START_REF] Pirrera | Nocturnal road traffic noise assessment and sleep research: The usefulness of different timeframes and in-and outdoor noise measurements[END_REF].

Many factors have been identified as independent sources of variability, including the presence of people in the dwelling, noisy event occurrences, measurement before a day of school, presence of a TV set i the hild s ed oo , the location of the dwelling in a detached house, a child sleeping alone in a bedroom, a child being the eldest child or the number of children living in the dwelling. However, the influence of these parameters may vary according to the time period. When the parameters included in the day and evening models were relatively similar, night-time appeared to be a specific period with its own variability factors. A small number of TLAD parameters were included in the night period multivariate analysis; the night period model explained only a fraction of the noise level variability. Certain permanent noise sources, including ventilation, refrigerator, freezer, electronic equipment, and low power setting, may have not been reported in the TLAD because their contributions may have been masked by the other sources present during the diurnal period. Noises created by the participant child, including body movements during sleep (motility) and respiration, may also increase the ambient noise; these factors were only taken into account by the i ha ita ts presence information. Furthermore, the presence and activities of the inhabitants in the dwelling differ among days because the hild s a ti ities, ti e of a ake i g, a d p ese e a e egulated s hool ti eta les. Additio all , s hool a also i flue e the hild s a ti ities du ing the night period. For example, a child may have an earlier bedtime on a school night. Conversely, when there is no school the following day, the child may be allowed to watch more TV. [START_REF] Pirrera | Nocturnal road traffic noise assessment and sleep research: The usefulness of different timeframes and in-and outdoor noise measurements[END_REF] did not observe any influence of the weekday on indoor nocturnal noise effects. However, the experimental design of this adult-focused study was not able to compare the working day with the days off as the measurements were performed excluding weekends.

In the risk assessment process, the exposure is the result of the pollutant presence and the time over which a person is in contact with that pollutant [START_REF] Morawska | Indoor aerosols: from personal exposure to risk assessment[END_REF].

TLADs have already been used to describe personal exposure to several pollution sources in epidemiological studies [START_REF] Meng | Influence of ambient (outdoor) sources on residential indoor and personal PM2.5 concentrations: Analyses of RIOPA data[END_REF][START_REF] Saborit | Measurement of personal exposure to volatile organic compounds and particle associated PAH in three UK regions[END_REF][START_REF] Viel | Variability of radiofrequency exposure across days of the week: A population-based study[END_REF][START_REF] Wallace | Continuous weeklong measurements of personal exposures and indoor concentrations of fine particles for 37 healthimpaired North Carolina residents for up to four seasons[END_REF]. A self-reported survey recently provided annual individual noise exposure duration of urban adult residents [START_REF] Neitzel | Exposures to Transit and Other Sources of Noise among New York City Residents[END_REF]. In our study, the TLAD variables allowed us to assess the effect of the presence of the inhabitants at home on noise occurrence and to quantify the i doo oise sou e a ti it . Ou esults o a hild s p ese e in the dwelling were consistent with the statistics of the US EPA regarding time spent at the residence (US EPA, 2009). Our use of TLAD variables in our multivariate models enhanced the understanding of the indoor noise levels and highly improved the fit of the models.

Further studies could evaluate solely the amount of time a child spends at home instead of using conventional noise indicators calculated on standardised time periods. Data obtained i this a e ould p o ide a o e a u ate assess e t of a hild s oise e posu e i everyday life.

The indoor environment within the dwelling of a child living in an urban area appears to be complex and cannot be easily elucidated from any single factor, including the outdoor noise level and the facade insulation values. Many variability factors differ based on the time of day. In contrast to the outdoor noise level, which can be calculated at a city scale, indoor noise level assessment requires the simultaneous consideration of dwelling and inhabitant characteristics, in addition to outdoor noise levels and noise sources within the dwelling. The use of a TLAD allo s the e o di g of the i ha ita ts p ese e a d a ti ities ithout resorting to the acoustical constraints of the equipment, the long-term recording, and the data processing. The TLAD is a refined alternative that could be distributed to numerous people and combined with the other identified variability factors to improve large-scale indoor home noise exposure assessment. In the child's bedroom 12.0 (22.1) 0-100 10.1 (24.2) 0-100 3.4 (16.4) 0-100 <10 -3 In the main room 14.0 (25.0) 0-100 13.6 (28.7) 0-100 6.6 (23.6) 0-100 <10 -3 The results are expressed as the percentage of time during the considered time period. As an example, at least one adult is present in the dwelling during 63.7 percent of the day period. SD: Standard deviation P: Difference between day, evening or night P-value (multilevel analysis) * Use of television, radio, musical instruments, household appliances, or other noisy event or activity

Table 1 :

 1 Living conditions during day, evening and night

		Day		Evening		Night
		(n = 304)	(n = 297)		(n = 301)
	TLAD variables	Mean (SD) Range	Mean (SD) Range	Mean (SD) Range	P
	Presence in the dwelling
	At least one adult	63.7 (28.6) 0-100	84.5 (26.5)	0-100	96.4 (16.5)	0-100 <10 -3
	Child participant	47.5 (27.1) 0-100	76.6 (33.4)	0-100	90.1 (28.4)	0-100 <10 -3
	At least one other child 42.7 (35.3) 0-100	63.6 (42.9)	0-100	72.7 (44.0)	0-100 <10 -3
	Noisy events*
	In the child's bedroom	4.2 (9.3)	0-66.7	8.7 (21.2)	0-100	0.7 (4.5)	0-50 <10 -3
	In the main room	22.5 (20.3) 0-87.5	48.2 (37.0)	0-100	12.2 (15.5)	0-100 <10 -3
	Outdoors	2.9 (9.0)	0-66.7	2.8 (10.7)	0-100	1.3 (7.4)	0-100 0.03
	Window's opening

Table 2 :

 2 L Aeq,day during days of school and days without school a (dB) Saturday, Sunday, legal holiday, and other school vacation SD: standard deviation b Difference between days of school and days without school: multilevel analysis P-value

	Time and place of measurement		Total		Days of school	Days without school	a
	Day (6:00-18:00)	n	Mean (SD)	Range	n	Mean (SD)	n	Mean (SD)	P	b
	Child's bedroom	298	48.2 (5.1)	23.6-66.2	153	46.7 (5.8)	145	49.8 (6.5)	10 -3
	Main room	289	55.2 (5.4)	33.0-75.8	151	54.1 (6.5)		56.2 (5.5)	10 -3
	Outdoors (bedroom)	273	55.4 (5.6)	40.5-68.5	140	55.9 (5.5)	133	54.7 (5.2)	< 10 -3
	Outdoors (main room)	44	55.8 (3.4)	49.9-65.2	-	-	-	-	-
	a Wednesday,

Table 3 :

 3 L Aeq,evening and L Aeq,night during evenings and nights before a day of school c (dB) Friday, Saturday, day before a legal holiday or other school vacation SD: standard deviation d Difference between before a day of school and before a day without school: multilevel analysis P-value

	Total	Before a day of school	Before a day without school c

Acknowledgements

We are grateful to the families who cooperated in this research. We would also like to thank Laurence Tilatti and Jean-Marc Cote for their dedication to this project and Sophie Muraccioli for the graphic design. This research was supported by the French Ministry of Environment and Sustainable Development (MEDD) within the third research programme P edit GO .

54 39 3 β: the esti ated ha ge in the indoor noise level; CI: confidence interval; P: P-value. * The outdoor noise level was calculated in front of the main room facade using a strategic noise map.


Chiara Troiani

Agostino Martinelli

Christian Laugier

Davide Scaramuzza

2-Point-based Outlier Rejection for Camera-Imu Systems with applications to Micro Aerial Vehicles

This paper presents a novel method to perform the outlier rejection task between two different views of a camera rigidly attached to an Inertial Measurement Unit (IMU). Only two feature correspondences and gyroscopic data from IMU measurerments are used to compute the motion hypothesis. By exploiting this 2-point motion parametrization, we propose two algorithms to remove wrong data associations in the featurematching process for case of a 6DoF motion. We show that in the case of a monocular camera mounted on a quadrotor vehicle, motion priors from IMU can be used to discard wrong estimations in the framework of a 2-point-RANSAC based approach. The proposed methods are evaluated on both synthetic and real data.

I. INTRODUCTION

Egomotion estimation is a core problem in the framework of many robotic applications. The estimation of the motion of a vehicle from the input of onboard cameras is known as Visual Odometry. In recent years, the coupling of vision and inertial sensing has received great attention by the mobile robotics community. It has been proved [START_REF] Weiss | Monocular-slam-based navigation for autonomous micro helicopters in gps-denied environments[END_REF], [START_REF] Achtelik | Visual-inertial slam for a small helicopter in large outdoor environments[END_REF], [START_REF] Scaramuzza | Vision-controlled micro flying robots: from system design to autonomous navigation and mapping in gps-denied environments[END_REF] that exploiting the complementarity of these sensors, it is possible to build a system capable to perform Inertial-Aided visual odometry and mapping in unknown environments.

One of the primary problems in Visual Odometry is wrong data associations. Matched features between two different camera views are usually affected by outliers. This is due to the fact that changes in viewpoint, occulsions, image noise, illumination changes and image noise are not modeled by feature-matching techiniques. To perform a robust motion estimation, it is essential to remove the outliers. The outlier detection task is usually very expensive from a computational point of view and is based on the exploitation of the geometric constraints induced by the motion model.

The standard method to process datasets contaminated by outliers is RANSAC (RANdom SAmple Consensus) [START_REF] Fischler | Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography[END_REF]. It is an iterative approach consisting in the generation of hypothesis by using the minimum number of data points required to estimate the model. The generated hypothesis is than verified on the remaining subset of data and the hypothesis with the highest consensus is selected as solution.

The number of iterations, N , necessary to guarantee that a solution free of outliers is found is [START_REF] Scaramuzza | Visual odometry [tutorial[END_REF]:

1 C.
N = log(1 -p) log(1 -(1 -ε) s ) ( 1
)
where s is the number of data points from which the model can be computed, ε is the percentage of outliers in the dataset, p is the requested probability of success.

Table I shows the number of iterations, N , with respect to the number of points necessary to estimate the model, s. The values are computed for p = 0.99 and ǫ = 0.5.

From Table I, we can see that the number of iterations is exponential in the minimum number of points necessary to estimate the model. For this reason, it is very important to have a minimal parametrization of the model.

In this paper, we presented two methods to remove outliers between two different views of a camera rigidly attached to an Inertial Measurement Unit (IMU). This paper extends our previous contribution in [START_REF] Troiani | 1-pointbased monocular motion estimation for computationally-limited micro aerial vehicles[END_REF] by including the general case of a 6DoF motion. In [START_REF] Troiani | 1-pointbased monocular motion estimation for computationally-limited micro aerial vehicles[END_REF] the analysis only regarded the case of a planar motion. We also show that in the case of a monocular camera mounted on a quadrotor vehicle, motion priors from IMU can be used to discard wrong estimations in the framework of a 2-point RANSAC based approach.

The paper is organized as follows. In Section II we illustrate the related works. In Section III we describe the camera motion model and the computation of the essential matrix. Section IV provides the description of the proposed algorithms for outlier rejection. Section V describes the quadrotor motion model. Section VI illustrates the performance of our methods while Section VI draws the conclusions.

II. RELATED WORKS

The minimum number of feature correspondences needed to infer a 6 degrees of freedom (6DoF) camera motion is five. This problem was solved for the first time by Kruppa [START_REF] Kruppa | Zur ermittlung eines objektes aus zwei perspektiven mit inner orientierung[END_REF] in 1913. Later on, many different five-point minimal solvers have been implemented [START_REF] Faugeras | Motion from point matches: multiplicity of solutions[END_REF], [START_REF] Philip | A non-iterative algorithm for determining all essential matrices corresponding to five point pairs[END_REF], [START_REF] Triggs | Routines for relative pose of two calibrated cameras from 5 points[END_REF], but the six [START_REF] Pizarro | Relative pose estimation for instrumented, calibrated imaging platforms[END_REF], seven and eight-point solvers were still the most used. The more efficient five-point minimal solver was proposed by Nister [START_REF] Nistér | An efficient solution to the five-point relative pose problem[END_REF] in 2003 and improved by Stewenius later [START_REF] Stewénius | Recent developments on direct relative orientation[END_REF]. The advantage of the five-point algorithm is that it is suitable also for planar scenarios.

In the last twenty years, there has been a significant interest in minimal motion parametrizations in order to decrease the minimum number of correspondences necessary to recover the rotation and the translation (up to a scale factor) between two different camera views.

A three-point minimal solver was presented in 2010 by [START_REF] Fraundorfer | A minimal case solution to the calibrated relative pose problem for the case of two unknown orientation angles[END_REF]. The authors considered a monocular camera rigidly attached to a gravity sensor. The gravity sensor can provide two out of three angles characterizing the relative orientation between two consecutive camera frames. This approach was later enhanced in [START_REF] Naroditsky | Two efficient solutions for visual odometry using directional correspondence[END_REF], where the authors demonstrated the possibility of using a three-point minimal solver in a fourpoint RANSAC framework. The fourth point (a point distant from the scene) was used to fix the two angles of the orientation.

If the motion is constrained on a plane, its degrees of freedom decrease to three and only two points are enough to parameterize it [START_REF] Ortin | Indoor robot motion based on monocular images[END_REF]. The authors in [START_REF] Scaramuzza | Real-time monocular visual odometry for on-road vehicles with 1-point ransac[END_REF], [START_REF] Scaramuzza | 1-point-ransac structure from motion for vehiclemounted cameras by exploiting non-holonomic constraints[END_REF] demonstrated that the motion of wheeled vehicles can be considered locally circular and planar, and, consequently, characterized by two parameters. This leads to a one-point minimal solver. They proposed as well a single iteration outlier detection technique based on histogram voting.

A comparison between five, two, and one-point RANSAC algorithms for Visual Odometry can be found in [START_REF]Performance evaluation of 1-point-ransac visual odometry[END_REF].

III. EPIPOLAR GEOMETRY

When a camera is calibrated, it is always possible to project the feature coordinates onto a unit sphere. This allows us to make our approach independent of the camera model.

Let p 1 = (x 1 , y 1 , z 1 ) and p 2 = (x 2 , y 2 , z 2 ) be the image coordinates of a point feature seen from two camera positions and back projected onto the unit sphere (i.e., p 1 = p 2 = 1) (Figure 1).

The image coordinates of point features relative to two different unknown camera positions must satisfy the epipolar constraint (Figure 1) [START_REF] Hartley | Multiple view geometry in computer vision[END_REF].

p 2 T Ep 1 = 0 ( 2
)
where E is the essential matrix, defined as

E = [T] × R. R and T = [T x , T y , T z ]
T describe the relative rotation and translation between the two camera positions, and [T] X is the skew symmetric matrix: [

T] × =   0 -T z T y T z 0 -T x -T y T x 0   (3)
According to equation ( 2), the essential matrix can be computed given a set of image coordinate points. E can then be decomposed into R and T [START_REF] Hartley | Multiple view geometry in computer vision[END_REF].

The minimum number of feature correspondences needed to estimate the essential matrix is function of the degrees of freedom of the camera's motion. In the case of a monocular camera performing a 6DoF motion (three for the rotation and three for the translation), considered the impossibility to recover the scale factor, a minimum of five correspondences is needed.

Let us consider a camera rigidly attached to an Inertial Measurement Unit (IMU) consisting of three orthogonal accelerometers and three orthogonal gyroscopes. The transformation between the camera reference frame {C} and the IMU frame {I} can be computed using [START_REF] Lobo | Relative pose calibration between visual and inertial sensors[END_REF]. Without loss of generality, we can therefore assume that these two frames are coincident ({I} ≡ {C}). The ∆φ, ∆θ and ∆ψ angles characterizing the relative rotation between two consecutive camera frames can be calculated by integrating the high frequency gyroscopic measurements, provided by the IMU. This measurement is affected only by a slowly-changing drift term and can safely be recovered if the system is in motion.

If R x (∆), R y (∆), R z (∆) are the orthonormal rotation matrices for rotations of ∆ about the x-, y-and z-axes, the matrix

C0 R C1 = (R x (∆φ) • R y (∆θ) • R z (∆ψ)) T (4)
allows us to virtually rotate the first camera frame {C 1 } into a new frame {C 0 } (Figure 1) having the same orientation of the second one {C 2 }. The matrix C0 R C1 allows us to express the image coordinates relative to C 1 into the new reference frame C 0 :

p 0 = C0 R C1 • p 1 .
(

) 5
At this point, the transformation between {C 0 } and {C 2 } is a pure translation

T = ρ[s(β) • c(α) -s(β) • s(α) c(β)] T R = I 3 , (6)
which depends only on the angles α and β and on the scale factor ρ. The essential matrix results therefore simplified:

E = [T] × R = ρ   0 -c(β) -s(β) • s(α) c(β) 0 -s(β) • c(α) s(β) • s(α) s(β) • c(α) 0   .
(7) With s(•) and c(•) we denote the sin(•) and cos(•) respectively. At this point, being p 0 = [x 0 y 0 z 0 ] T and p 2 = [x 2 y 2 z 2 ] T , the coordinates of a feature matched between two different camera frames and backprojected onto the unit sphere, we impose the epipolar constraint according to (2) and we obtain the homogeneous equation that must be satisfied by all the point correspondences.

x 2 (y 0 c(β) + z 0 s(α)s(β)) -y 2 (x 0 c(β) -z 0 c(α)s(β))+ -z 2 (y 0 c(α)s(β) + x 0 s(α)s(β)) = 0. ( 8
) Equation ( 8) depends on two parameters (α and β). This means that the relative vehicle motion can be estimated using only two image feature correspondences that we will identify as p A and p B , where p i j = [x ij y ij z ij ] T with i = A, B and j = 0, 2 indicate the direction of the feature i in the reference frame j.

At this point, we can recover the angles α and β solving (8) for the features p A and p B :

α = -tan -1 c4c2-c1c5 c4c3-c1c6 , β = -tan -1 c1 c2c(α)+c3s(α) , (9)
where

c 1 = x A2 y A0 -x A0 y A2 , c 2 = -y A0 z A2 + y A2 z A0 , c 3 = -x A0 z A2 + x A2 z A0 , c 4 = x B2 y B0 -x B0 y B2 , c 5 = -y B0 z B2 + y B2 z B0 , c 6 = -x B0 z B2 + x B2 z B0 . (10)
Finally, without loss of generality, we can set the scale factor ρ to 1 and estimate the essential matrix according to [START_REF] Kruppa | Zur ermittlung eines objektes aus zwei perspektiven mit inner orientierung[END_REF].

IV. OUTLIER REJECTION

Starting from the above-mentioned camera-motion parametrization, we propose two different approaches to reject outliers from a set of feature correspondences.

The former is based on estimating the angles α * and β * by selecting the pick in a Hough Space and discarding the outliers by using the reprojection error. We call this method "Hough". The second one is a 2-point RANSACbased method with an extension to the particular case of a camera mounted on a quadrotor. The algorithm takes into account the relations between the orientation and the translation of the vehicle to remove wrong estimations. The inliers are identified using the reprojection error.

A. Hough

The angles α and β are computed according to ( 9) from all the feature pairs matched between two consecutive frames and distant from each other more than a defined threshold (see VI). A distribution {α i , β i } with i = 1, 2, . . . , N is obtained, where N is a function of the position of the features in the environment.

To estimate the best angles α * and β * , we build a Hough Space (Figure 3) which bins the values of {α i , β i } into a grid of equally spaced containers. Considering that the angle β is defined in the interval [0, π] and that the angle α is defined in the interval [0, 2π], we set 360 bins for the variable α and 180 bins for the variable β. The number of bins of the Hough Space encodes the resolution of the estimation.

The angles α * and β * are therefore computed as

< α * , β * >= argmax{H},
where H is the Hough Space.

The factors that influence the distribution are the error on the estimation of the relative rotation, the image noise, and the percentage of outliers in the data. The closer we are to ideal conditions (no noise on the IMU measurements), the narrower will be the distribution. The wider is the distribution, the more uncertain is the motion estimate.

To detect the outliers, we calculate the reprojection error relative to the estimated motion model.

The camera motion estimation can be then refined processing the remaining subset of inliers with standard algorithms [START_REF] Stewénius | Recent developments on direct relative orientation[END_REF], [START_REF] Hartley | Multiple view geometry in computer vision[END_REF].

B. 2-point RANSAC

Using (6) we compute the motion hypothesis that consists of the translation vector T and the rotation matrix R = I 3 by randomly selecting two features from the correspondence set. To have a good estimation, we check that the distance between the selected features is below a defined threshold (see VI). If it is not the case, we randomly select another pair of features. Constraints on the motion of the camera can be exploited to discard wrong estimations (see Section V). The inliers are than computed using the reprojection error. The hypothesis that shows the highest consensus is considered to be the solution.

V. QUADROTOR MOTION MODEL

We consider a quadrotor equipped with a monocular camera and an IMU.

The vehicle body-fixed coordinate frame {B} has its Z Baxis pointing downward (following aerospace conventions [START_REF] Corke | Robotics, Vision & Control: Fundamental Algorithms in Matlab[END_REF]). The X B -axis defines the forward direction and the Y B -axis follows the right-hand rule.

Without loss of generality we can consider the IMU reference frame {I} coinciding with the vehicle body frame {B}.

The modelization of the vehicle rotation in the World frame {W } follows the Z -Y -X Euler angles convention: being φ, θ, ψ respectively the Roll, P itch and Y aw angles of the vehicle, to go from the World frame to the Body frame, we first rotate about z W axis by the angle ψ, then rotate about the intermediate y-axis by the angle θ, and finally rotate about the X B -axis by the angle φ.

The transformation between the camera reference frame {C} and the IMU frame {I} can be computed using [START_REF] Lobo | Relative pose calibration between visual and inertial sensors[END_REF]. Without loss of generality, we can therefore assume that also these two frames are coincident ({I} ≡ {C} ≡ {B}).

A quadrotor has 6DoF, but its translational and angular velocity are strongly coupled to its attitude due to dynamic constraints. If we consider a coordinate frame {B 0 } with the origin coincident with the one of the vehicle's body frame {B} and the X B0 and Y B0 axes parallel to the ground, we observe that, in order to move in the X B0 direction, the vehicle must rotate about the y-axes axis (P itch angle), while, in order to move in the Y B0 direction, it must rotate about the x-axis (Roll angle) (Figure 5).

These motion constraints allow us to discard wrong estimations in a RANSAC based outlier detection approach. By looking at the relation between the x and y component of the estimated translation vector and the ∆φ, ∆θ angles provided by the IM U measurements (the same used in ( 4)), we are able to check the consistency of the motion hypothesis. If the estimated motion satisfies the condition

((|∆φ| > ǫ)&(∆φ • T y > 0)) ((|∆θ| > ǫ)&(∆θ • T x < 0)) ((|∆φ| < ǫ)&(|∆θ| < ǫ)), (11)
we count the number of inliers (the number of correspondences that satisfy the motion hypothesis according to a predefined threshold) by using the reprojection error, otherwise we select another feature pair. The condition in ( 11) is satisfied if the x and y components of the motion hypothesis are coherent with the orientation of the vehicle. If both the angles ∆φ and ∆θ are below the threshold ǫ, we cannot infer nothing about the motion and we proceed in the evaluation of the model hypothesis using the reprojection error.

The value of the threshold ǫ (see VI) is a function of the vehicle dynamics and of the controller used.

Using (1) and considering p = 0.99 and ε = 0.5, we calculate the minimum number of iterations necessary to guarantee a good performance to our algorithm and we set it to 16.

VI. RESULTS

To evaluate the performance of our algorithms, we run simulations and experiments on real data. We compared the proposed approaches with the 5-point RANSAC [START_REF] Nistér | An efficient solution to the five-point relative pose problem[END_REF] on synthetic data, and with the 5-point RANSAC [START_REF] Nistér | An efficient solution to the five-point relative pose problem[END_REF] and the 8-point RANSAC [START_REF] Longuet | A computer algorithm for reconstructing a scene from two projections[END_REF] on real data.

A. Experiments on synthetic data

We built a synthetic scenario for our simulation by using the Robotics and Machine Vision Toolbox for Matlab [START_REF] Corke | Robotics, Vision & Control: Fundamental Algorithms in Matlab[END_REF]. We simulated a quadrotor equipped with a downlooking monocular camera and an IMU, moving in an indoor environment (Figure 6). Random features were generated without any assumption on the structure of the environment.

The onboard downlooking monocular camera was simulated as a perspective camera with the same intrinsic parameters of the camera that we used in the experiments. A white gaussian noise with a standard deviation of 0.5 pixels was added to each extracted feature.

We generated a trajectory consisting of a take-off and of a constant-height maneuver. The camera framerate is 15Hz, its resolution is 752 x 480. For the reprojection error in the 2-point RANSAC and in the Hough algorithm, we set a threshold of 0.5 pixels. For the 5-point RANSAC, we set the minimum number to trials to 145 iterations, and the threshold to 0.5 pixels for the reprojection error.

Figure 7 shows the results of a simulation run along the trajectory depicted in Figure 6, in the ideal case of no noisy IMU measurements. The helicopter takes off and performs a constant height maneuver.

In Figure 8, we present the results related to simulations where the quantities ∆φ, ∆θ and ∆ψ are affected by a Gaussian noise with standard deviation of 0.3 degrees. Those errors do not affect the performance of the 5-point algorithm (that does not use IMU readings to compute the motion hypothesis). In this case, the Hough and the 2-point RANSAC approaches can still detect more than half of the inliers. The motion hypothesis can then be computed on the obtained set of correspondences by using standard approaches [START_REF] Stewénius | Recent developments on direct relative orientation[END_REF], [START_REF] Hartley | Multiple view geometry in computer vision[END_REF].

In Figure 9, we present the results related to simulations where the quantities ∆φ and ∆θ are affected by a Gaussian noise with standard deviation of 0.3 degrees and in Figure 10 only the angle ∆ψ is affected by a Gaussian noise with standard deviation of 0.3 degrees. These two plots show that errors on rotations about the camera optical axis (that in our case coincides with rotations about the vehicle Z B axis, i.e. errors on ∆ψ) affects more the performances of both the algorithms than errors on ∆φ and ∆θ.

B. Experiments on real data

The proposed approaches are tested on our nano quadrotor (Figure 11) 1 equipped with a MicroStrain 3DM-GX3 IMU (250 Hz) and a Matrix Vision mvBlueFOX-MLC200w camera (FOV: 112 deg and a resolution of 752 x 480).

The monocular camera calibration has been performed using the Camera Calibration Toolbox for Matlab [START_REF] Bouguet | Camera calibration toolbox for matlab[END_REF].

To extrinsically calibrate the IMU and the camera, we used the Inertial Measurement Unit and Camera Calibration Toolbox [START_REF] Lobo | Relative pose calibration between visual and inertial sensors[END_REF].

1 http://KMelRobotics.com To validate the performance of our methods, we flew the quadrotor in our flying arena, equipped with an Optitrack motion capture system with sub-millimeter accuracy. The trajectory consisted of a take-off and a constant-height maneuver above the ground, as shown in Figure 12 and was generated using the TeleKyb Framework [START_REF] Grabe | The telekyb framework for a modular and extendible ros-based quadrotor control[END_REF]. We recorded a dataset composed of camera images, IMU measurements and ground truth data provided by the Optitrack.

We processed our dataset with SURF features, matching them in consecutive camera frames. We run the 8-point RANSAC method on each correspondences set to have an additional term of comparison.

To evaluate the performance of our methods, we compared the number of inliers detected using the Hough and the 2-point RANSAC methods with 5-point and an 8-point RANSAC. For the 2-point RANSAC we set ǫ = 0.1 deg. The results of this comparison are shown in Figure 13.

Figure 14 shows the error characterizing the estimated relative rotation between two consecutive camera frames obtained by IMU measurements and the ground truth values. Looking at both Figure 13 and Figure 14, we can notice that the smaller are the errors on the angles estimations, the higher is the number of inliers detected by the Hough and the 2-point RANSAC method.

Our algorithms and the algorithms that we used for the comparison, are implemented in Matlab and run on an Intel Core i7-3740QM Processor. We summarize their computation time in Table II. We can notice that the computation time of the 5-point RANSAC is almost 67 times the computation time of the 8-point RANSAC. This is due to the fact that the 5-points returns up to 10 motion solutions for each candidate set. Singular Value Decomposition (SVD) and Groebnerbasis decompositions are involved and this explains the high computation time.

The computation time of the Hough algorithm is function of the number of feature pairs used to compute the distribution in Figure (3). In our experiments, we choose all the feature pairs distant more than a defined threshold one to each other. We experimentally set this threshold to 30 degrees on the unit sphere.

VII. CONCLUSIONS

In this paper, we proposed two algorithms (Hough and 2point RANSAC) to address the outlier rejection task systems equipped with a monocular camera rigidly attached to an IMU. We used a quadorotor micro aerial vehicle as platform to demonstrate the validity of our results. We show that the relations between the vehicle's translational and angular velocity and its attitude can be exploited in order to discard wrong estimations in the framework of a RANSAC-based approach.

Both methods rely on onboard IMU measurements to calculate the relative rotation between two consecutive camera frames and to the reprojection error to detect the inliers. The two algorithms differ in the way to compute the motion hypothesis.

The computation time of the Hough algorithm (Table II) is function of the number of feature pairs used to compute the distribution in Figure [START_REF] Scaramuzza | Vision-controlled micro flying robots: from system design to autonomous navigation and mapping in gps-denied environments[END_REF]. Smart policies for the choice of the pairs of features to use (based for example on the feature positions in the image plane and not only on their relative position) can be used in order to reduce the computational complexity of the approach.

Experimental results show that the 2-point RANSAC algorithm can be a good replacement of the 5-point RANSAC. The motion hypothesis can always be refined by processing the found inliers with classic methods [START_REF] Stewénius | Recent developments on direct relative orientation[END_REF], [START_REF] Hartley | Multiple view geometry in computer vision[END_REF].

Fig. 1 :

 1 Fig. 1: Epipolar constraint. p 1 , p 2 , T and P lie on the same plane (the epipolar plane).

Fig. 2 :

 2 Fig. 2: The reference frame C 0 and C 2 differ only for the translation vector T . ρ = |T | and the angles α and β allow us to express the origin of the reference frame C 2 in the reference frame C 0 .

Fig. 3 :

 3 Fig. 3: Hough Space in α and β computed with real data.

Fig. 4 :

 4 Fig. 4: Notation.

Fig. 5 :

 5 Fig. 5: Motion constraints on a quadrotor relative to its orientation. ∆φ > 0 implies a movement along Y B 0 positive direction, ∆θ < 0 implies a movement along Y B 0 positive direction.

Fig. 6 :

 6 Fig. 6: Synthetic scenario. The red line represents the trajectory and the blue dots represents the simulated features. The green dots are the features in the current camera view.

Fig. 7 :

 7 Fig. 7: The IMU measurements are not affected by noise (ideal conditions).

Fig. 8 :Fig. 9 :

 89 Fig. 8: The angles ∆φ, ∆θ and ∆ψ are affected by noise.

Fig. 10 :

 10 Fig. 10: Only the angle ∆ψ is affected by noise.

Fig. 11 :

 11 Fig. 11: Our nano quadrotor from KMelRobotics: a 150g and 18cm sized platform equipped with an integrated Gumstix Overo board and MatrixVision VGA camera.

Fig. 12 :

 12 Fig. 12: Real scenario. The vehicle body frame is represented in blue, while the red line represents the followed trajectory.

Fig. 13 :

 13 Fig. 13: Number of inliers detected with the Hough approach (red), the 2-point RANSAC (cyan), the 5-point RANSAC (black) and the 8-point RANSAC (blue) along the trajectory depicted in Figure 12.

Fig. 14 :

 14 Fig. 14: Errors between the relative rotations ∆φ (err R ), ∆θ (err P ), ∆ψ (err Y ) estimated with the IMU and estimated with the Optitrack.

TABLE I :

 I Number of iterations of RANSAC

	Number of points (s)	1	2	3	5	8
	Number of iterations (N )	7 16	35	145 1177

TABLE II :

 II Computation time

	Algorithm Hough 2-point	5-points	8-points
	Time [s]	0.498	0.048	2.6869	0.0396