T L Kaiser
email: tkaiser@imft.fr

G Ôztarlik

L Selle

T Poinsot

Impact of symmetry breaking on the Flame Transfer Function of a laminar premixed flame

Keywords: Laminar fl ame, Symmetry breaking, Interference, Laminar ONS, Rotating cylinder

This work presents a numerical study of the acoustic response of a laminar flame with tunable asymmetry. A V-shaped prernixed flame is stabilised in the wake of a cylindrical fl ame holder that can be rotated. The configuration is symmetric when the flame holder is fixed but increasing its rotation rate breaks the symmetry of the flow. This configuration is submitted to acoustic forcing to measure the effect of rotation of the flame holder on the Flame Transfer Functions. lt appears that the asymmetry of the two flame branches changes their respective time delays, resulting in interference in the global unsteady heat release rate fl uctuations. Consequently, the Flame Transfer Function exhibits dips and bumps, which are studied via larninar Direct Numerical Simulation. Potential applications for the control of combustion instabilities are discussed.

Introduction

Combustion instabilities caused by a coupling of heat release fl uctuations and acoustic modes of the combustion chamber are one of the key un certainties in the development process of gas tur bines. Their consequences may range from limiting the operating conditions to the destruction of the combustion chamber.

Combustion instabilities involve various com plex physical mechanisms. Therefore, canonical test cases are often used to isolate certain phenom-ena. ln this context, several configurations involv ing laminar flames in various shapes have been studied in the past. Besides V-flames and conical flames [1,2], the case of a flame anchoring in the wake of a bluff body in laminar cross flow, has been studied using numerical and analytical ap proaches. Using larninar Direct Numerical Simu lations (ONS) validated by experiments, Brebion et al. [3] investigated the influence of the cylin drical flame holder's temperature on the flame an choring mechanism. ln their follow up study, Mejia et al. [4] investigated the influence of the cylin der's temperature on the Flame Transfer Function (FTF). The effect of rotation of the cylinder on the fl ame anchoring mechanism was explained by Xavier et al. [5] and Mejia et al. [6]. Their results showed that rotating the cylinder breaks the sym metry of both flame branches. Since the asymme try of both flame branches is tunable by the ro tation rate, it is a variable parameter to adapt the flame topology, without changing the flow velocity or the cold gas mixture. The dynamic response of both flame branches and their interaction neverthe Jess remain unknown.

Upper flamc branch

This article focuses on infl uence of the cylinder rotation rate on the dynamic response of the flames to harmonie acoustic forcing. ln Section 2 the setup and numerical strategy are laid out. Section 3.1 de scribes the influence of the rotation rate on the un perturbed flame topology and Section 3.2 discusses the FTF of the case without cylinder rotation. This FTF is compared in Section 3.3 to FTFs for vari ous cylinder rotation rates. Section 3.4 discusses the flame branches topology in case of monochromatic forcing for several rotation rates. Finally, the results are concluded and discussed in Section 4.

Configuration and simulation framework

The configuration corresponds to that of Xavier et al. [5,7): it consists of a constant-section rect angular channel (width W = 34 mm and depth 94 mm) at the center of which a stainless-steel cylin der (diameter d = 8 mm) serves as a flame holder (see Fig. 1). The cylinder is mounted on an elec tric motor with a controlled rotation speed ranging from 600 to 20,000 rpm. The normalised rotation rate is defined as

wd a= - 2ub ( 1
)
where Ub is the bulk velocity of the premixed gases in the channel.

As shown in previous studies on this exper iment [3,5,7), because of its geometry and the Jaminar conditions, two-dimensional simulations are adequate. Laminar ONS are performed using the AVBP code [START_REF] Poinsot | The AVBP Handbook[END_REF], jointly developed by CERFACS and IFP Energies nouvelles, which solves the com pressible Navier-Stokes equations for multicom ponent reacting flows on unstructured grids. The numerical integration is performed by a two-step Taylor-Galerkin scheme called TTGC [START_REF] Colin | [END_REF], which is of third-order in both space and time. ln the current work, the combustion of methane and air is modelled via an analytically reduced skeletal mechanism accounting for 19 transported species and 11 quasi-steady species [10). This mechanism, referred to as LUJ9, is based on the GRJ-mech3.0 mechanism [11) and reproduces the adiabatic tem perature and flame speed over a wide range of operating conditions. It also includes Jow temperature-chemistry, required to describe flame wall interaction at the flame holder interface.

The Jateral walls are water-cooled and mod elled by a no-slip iso-thermal boundary condi tion (T wa u = 292K). The computational domain extends from x = -35 mm to x = 97 mm, where x = 0 corresponds to the center of the cylinder.

The inlet and outlet boundary conditions are mod elled via the Navier-Stokes characteristic bound ary conditions (NSCBC) [12). The inlet is fed by a methane-air mixture at equivalence ratio <l> = O. 75 and temperature T. = 292 K. The velocity profile is given by experimental measurements. The bulk speed is ub = 0.8 mis. A pressure of p = 100, 700 Pa is imposed at the outlet. This operating point is far away from flame detachment of the cylinder.

The boundary condition at the cylinder's sur face is a no-slip rotating wall, which was shown by Mejia et al. [7] to accurately account for the rota tion of the flame holder. An estimation of the Biot number of the present cylinder is 0.01, meaning that the assumption of spatially uniform temper ature is valid. Consequently, the cylinder tempera ture, Tc y 1, can be obtained by solving [5):

dTc y 1 s . dt= C ppV (Qcond -Qrad ), (2)
where t denotes time, S the cylinder surface, p and V the density and the volume of the cylinder, and C p the mass specific heat capacity. Qeond and Qrad are the conductive and radiative heat fluxes, respectively. Qcond is obtained by integrating the laminar ONS heat flux long the cylinder surface. Radiation is governed by the Stefan-Boltzmann radiation law [13):

• ( 4 4
)

Qrad = E<l T,,yl -T wall • ( 3
)
where <1 is the Stefan-Boltzmann constant, and the emissiency coefficient, E, is set to 0.9. The gas is assumed to be fully transparent and the tempera ture of the surroundings of the cylinder is supposed to be equal to the temperature of the cooled walls Tw all = 292 K. Since the time scales of the chemical reactions and that of the preheating of the cylinder are distinctively different, the transient heat up of the experiment can be accelerated following, for ex ample the strategy presented in [14). Also, because of the Jow Biot number and the high rotation rate, once in the permanent regime, the temperature of the cylinder is homogeneous and constant. Finally, regarding spatial resolution, the coars est cell size in the domain is l:ix = 500 µ,m, while it is l:ix = 70 µ,m close to the flame, which corre sponds to 8 points in the flame [START_REF] Brebion | Joint Numerical and Experimental Study of Thermo-acoustic Instabilities[END_REF]. Based on a mesh convergence study focusing on one dimen sional flame head on quenching, the cell size at all solid walls is set to lix = 25 µm. The setup of the mesh convergence study is presented in [START_REF] Ghani | [END_REF]. This numerical strategy was successfully com pared to experimental results in many previous studies [3,5,7], including investigations on flame wall interaction [START_REF] Ghani | [END_REF] and it is used here without modification.

Results

In this section, the influence of rotation of the flame holder on the response of the flame to acous tic waves is studied. This analysis is conducted in the framework of FTF, i.e., in the linear regime cor responding to low amplitude perturbations.

Steady-state ana/ysis

First, the steady flame shapes are presented in Fig. 2, for all of which the fluctuations in heat re lease are smaller than 0.05% of the mean heat re lease value. As expected, without rotation the flame is symmetric (Fig . 2a) but this symmetry is broken by the rotation of the flame holder. With increas ing rotation rates, the root of the upper branch de taches from the cylinder, but moves upstream again for very large values of et (cf. Fig. 2e). The tip of the upper branch moves downstream until et = l .25 but then goes back upstream, even past its loca tion for the symmetric flame. Regarding the lower branch, it always remains attached to the cylinder and its tip is monotonously pushed downstream. A thorough investigation of the effect of bluff body rotation on the flame root is provided by Xavier et al. [5].

From the simple observation of the flame shape in the steady state, one can anticipate that the two branches will have a very different response to acoustic waves. lndeed, one can show from first principles [17,18] that the flame response is driven by its length and the dynamics of its root. While the length greatly impacts the delay, the dynamics of the flame root drives the gain of the FTF [4,6]. Moreover, the rotation of the cylinder breaks the symmetry of the flow so that the bulk velocities on both sides of the flame holder are different. This will change the speed at which perturbations travel on the flame front, also impacting the flame de lay. Finally, because of the complex flow pattern around the cylinder, the temperature of the flame holder changes with the rotation rate (see right side of Fig. 2). lt first decreases significantly at low ro tation rates and increases again, past the reference value without rotation. The quantitative measure ment of these changes is reported in Table 1, which gives the cylinder temperature, the flame root and tip positions and the bulk velocities for the two branches for various rotation rates.

Ali these changes in the flame front topology and the flow field point towards a modification of the FTF, which will be studied in the next sections.

Flarne response without rotation

This section focuses on the FTF of the var ious flames, starting with the case without rota tion. Under the assumption that the system is linear and invariant, it has been shown in many studies that the Wiener-Hopf Inversion (WHI) is a pow- erful methodology for the determination of FTFs [START_REF] Giauque | 14th AIAA/CEAS Aeroacoustics Conference[END_REF][START_REF] Gentemann | Proceedings of ASME Turbo Expo[END_REF][START_REF] Polifke | [END_REF]. ln this method, the transfer fonction of a system is estimated by broad band forcing of the system and subsequent correlation analysis of its input and output signal. A detailed description of the method is provided in [START_REF]): Theory for the User[END_REF][START_REF] Huber | [END_REF]. ln order to re main in the linear regime, the forcing amplitude for the estimation of the FTF is kept at 1 % of the bulk velocity. The reference point for the velocity per turbations is at x = -JO mm and y = O. Figure 3 shows the FTF of the symmetric reference flame (i.e., o: = 0), obtained by a WHI analysis of Jaminar ONS data. As seen in previous studies of this con figuration, this flame only responds under 250 Hz and shows a strong gain ( > 2) at Jow frequencies, here around 80 Hz. The phase of the FTF is linear, corresponding to a constant lime delay r defined as the slope of the phase. For the present operat ing point, r = 35.5 ms. The accuracy of the WHI was tested by perfonning FTF evaluation at vari ous discrete frequencies (see circles in Fig. 3). We now investigate the motion of the flame front under harmonie forcing at 80 Hz, i.e., close to the frequency of the peak in the FTF. Figure 4 presents the mean and perturbed flame front and shows the definition of the normal displacement 5 between the two. ln order to elucid.ate the dynam ics of the flame, the evolution of 5 along the flame front is presented in Fig. 5 at four ditferent phases of the harmonie excitation. lt appears that the am plitude of the displacement increases weakly along the flame front but shows non-Jinear deformation close to the flame tip. lt is also qui te clear that a lot of the variation in the flame surface is caused by the flame-wall interaction at the flame tip. The second observation is that for many rota tion rates, the gains exhibit strong dips (see markers in Fig. 6). It is anticipated that these dips could stem from the symmetry breaking of the flame, result ing in interference between the responses of the two flame branches. Regarding the phase of the FTF, it is linear in ail cases and the corresponding delay monotonously increases with the rotation rate. ln order to investigate the dips in the FTF gain, the two flame branches are now analysed sepa rately. The underlying assumption is that their dy namics are not, or sufficiently weakly coupled.

Flarne response with rotation

The FTF of the upper (y> 0, Jabelled y + re spectively Jower (y< 0, labelled y-) branches are determined by integration of the heat release rate over one half of the domain. The reference point for the velocity is the same as for the global FTF and the heat release rate fluctuations are nor malised by the total flame power. The resulting FTFs are displayed in Fig. 7a for the y + -branch and in Fig. 7b for the y--branch.

For the upper branch, the variations in max imum gain are limited but the cutoff frequency clearly decreases with a. lt goes from 250 Hz with out rotation down to around 130 Hz for a> 2.5. The delay of the flame response first increases and then decreases with a maximum around a = 1.25, which is consistent with the evolution of the steady upper-flame length shown in Fig. 2.

The evolution of the FTF of the Jower branch is very different. First there is no decrease in the cutoff frequency but the striking feature is that the maximum gain increases drastically past a = 2.5 over a broad frequency range. The delay of the FTF monotonously increases with a, replicating the in crease in flame Jength concomitant with the bulk velocity increase for y< O. Despite the high gains at a = 3.75, results of monochromatic forcing, which are illustrated by the circles in Fig. 7, and WHI are in good agreement.

We now investigate the hypothesis that the breaking of the flame symmetry results in an inter ference between the two branches. A practical ap plication for this modification of the FTF via ro tation of the flame holder could be the control of combustion instabilities by tuning the location of the dips in the FTF. A simple mode) can be built to predict the frequency at which this dip occurs. lt is based on the idea that the dip is the conse quence of the interference between the responses in heat release fluctuation of the two branches. The total heat release fluctuation is equal to the sum of the heat release fluctuation of both branches, q;0I = <f,,+ + <1,,-. lntroducing the FTF using the notation of Crocco, the fluctuation in heat release can be re lated to a harmonie velocity fluctuation, ri, which yields:

q'� I = �(n y + + n,-exp (iwM)) exp (iwry+ ) . (4) q u
Here, n are the gains of the respective branches and the difference in time delays is depicted as Lh = r ,-r y +. From Eq. ( 4) it is evident, that the gain of the global FTF is depending on the phase angle between the two branches, given by l:i</J = wl:J.r.

(

) 5
The two extreme cases are ideal constructive in terference and ideal destructive interference. For the former case, the phase angle between the both branches' heat release signais is O or an even mul tiple of ,r. Theo the gain of the global heat release reduces to n,o, = ny+ + ny-. ln the other case, the difference in phase angles is an odd multiple of ,r, so that the gain of the global heat release is equal to n101 = ln 1 + -ny-1 • Therefore, the frequency at which destructive interference is maximal can be deduced from Eq. ( 5):

l:i</J /d,,, = 2,rt:J.r ' l:i</J = (2n-l),r, nEZ. (6)
The values, extracted from the laminar DNS, of the delays of the two branches (ry+ and r 1 -for the upper and lower branch, respectively), their differ ence, l:J. r, and the first two frequencies of construc tive and destructive interference lfc,n and /d,n for n = 1 and n = 2) are given in Table 2. For a = 0.625 and a = 1.25, the predicted values of /d, 1 corre spond very well to the FTF of Fig. 6. For a = 2.5, the first destructive interference occurs at a fre quency where the gain of the upper-branch FTF is close to zero (cf. Fig. 7a). Consequently, the inter ference is not meaningful and no dip is observed in the FTF. Finally for the higher rotation rate (a = 3.75), the value of /d, 1 is again very low and a dip is observed.

The frequency of the second dip,/d, 2, (Table 2) is much higher than the cutoff frequency of the flame response for all rotation rates except for a = 3.75, which exhibits a dip around the expected value of 93 Hz. The frequencies /d, n are illustrated by the markers in Fig. 6, showing very good agree ment with the location of the dips in the corre sponding gains. lt seems that the simple mode! of Eq. ( 6) gives an accurate prediction of this phe nomenon, giving a posteriori confirmation of the hypothesis that there is indeed interference between the two flame branches.

Given the strong impact of symmetry breaking on the shape of the FTF, one may use the rotation rate of the flame holder to control combustion in stabilities. However, it can be anticipated that only the low rotation rates, say a < 1.25 can be of prac tical use for the present flame, first because of the strong decrease of the cutoff frequency of one of the branches, but also because of the large increase in gain of the other branch, which drives the gain of the total FTF and hinders the destructive inter ference.

Flame front dynamics

We pursue the analysis of the flame response by plotting the flame displacement along the flame front for the two branches in Fig. 8. As for the non rotating case, the frequency of interest is 80 Hz, which lies close to the peak in the gain of the FTF for all rotation rates.

We start the analysis for the upper flame branch. For ail rotation rates a growth of the flame displace ment along the flame front is observed. With in creasing rotation rate from a = 0 to a = 2.5, the flame displacement amplitude at the flame tip de creases moderately. It appears that the response in flame displacement amplitude increases again be yond this rotation rate. The same trend can be ob served regarding the non-linear deformation close to the flame tip. This effect seems to decrease un til a rotation rate of a = 2.5 and increase again for higher rotation rates. Nevertheless, only a small variation of the flame displacement amplitude at the flame tip is observed. At the Jower flame branch, Finally, the flame tip movement is analysed. Figure 9 shows the flame tip position of the per turbed upper and lower flame branches with re spect to their positions in the unperturbed case for a cylinder rotation rate of a = 0.625. The pul sation amplitude is at 2% of the bulk speed and its frequency is 132 Hz, i.e., at the frequency of ideal destructive interference. It appears that the tip movements are perfectly out of phase, which is expected since at these frequencies the FTFs show a phase difference of ,r and most of the heat re lease fluctuation is produced at the interaction of the flame tip and the lateral walls. Together with Fig. 6, Fig. 9 shows that introducing rotation of a flame holder can be used to build a flame which be cornes almost insensitive to acoustic perturbation, suggesting that this could be used for combustion instability control. Note however that this result is obtained only in a limited frequency range.

Conclusion

The FTF of a laminar flame stabilised behind a rotating cylindrical bluff body is investigated for various rotation rates of the cylinder by the means of laminar DNS. The rotation causes a symmetry break between the two flame branches, where es pecially the flame tip positions vary significantly, especially for high rotation rates. FTFs are esti mated for each branch separately, assuming that they can be described as two independent Linear Time Invariant (LTI)-systems. The results show a difference in time delays between the upper and the lower flame branch. This is mainly due to an in dependent modification of the flame tip positions of both branches, caused by two effects: a detach ment of the upper branch from the cylinder and a modification of the bulk speeds in the two half domains due to the cylinder's viscous boundary layer. The resulting difference in time delay causes interference of the flames' heat release signais. At frequencies where destructive interference is max imal, the global FTFs, which include both flame branches, as a consequence show deep gaps in the gain. The difference in time delay and therefore the frequency at which destructive interference of both flame branches occurs can be tuned by adapting the rotation rate of the cylinder. ln this context, the analysis suggests, that the rotation rate is a sui table and easily adaptable measure of open loop con trot of combustion instabilities. It is shown that for cylinder rotation rates, a � 2.5, the flame displace ment considerably grows along the flame front of the lower flame branch, leading to gains, that by far exceed the gain of the non-rotating confi guration. This effect therefore limits the applicability of the method to moderate rotation rates and frequencies /� I 00 Hz. Most practical flames are highly turbu lent, and the existence of a corresponding effect in the turbulent regime remains unknown. Neverthe less, the present paper underlines the asset of de creasing the gain of the FTF significantly around a specific frequency by being able to tune the asym metry of a combustion system which leads to de structive interference.

Fig. 1 .

 1 Fig. 1. Transverse eut of the computationaJ domain.

Fig. 2 .

 2 Fig.2. Field of velocity component in main flow direction, u, with stream lines highlighted as thin black solid lines, iso surface of heat release as thick solid line and cylinder temperature for various cylinder rotation rates, a.

Fig. 3 .

 3 Fig. 3. FTF without rotation (a = 0) based on WHI and monochromatic forcing.

Fig. 4 .

 4 Fig. 4. Flame shapes behind a non-rotating cylinder; dashed line: unperturbed flame; solid line: flame forced at 80 Hz; Insert: definition of flame angle, a, and flame normal displacement, �.

Fig. 5 . 5 _Fig. 6 .

 556 Fig. 5. Flame displacement, �, along the flame front with out rotation for various phases at f = 80 Hz. Forcing am plitude: 5%ub,

Figure 6

 6 Figure 6 shows the FTFs for various rotation rates, up to o: = 3.75, including the case without rotation as the reference. The first observation is that the maximum gain is marginally atfected up to o: = 2.5. However, between o: = 2.5 and o: = 3.75, it increases by at Jeast a factor of 2. The location of the maximum gain remains in the Jow-frequency range (between 50 Hz and 80 Hz), but without clear trend.The second observation is that for many rota tion rates, the gains exhibit strong dips (see markers

  Upper flame branch (y + ). Lower flame branch (y-).

Fig. 7 .

 7 Fig. 7. FTF of individuaJ ftame branches versus rotation rate a; the circles mark results of monochromatic forcing ata = 3 .75.

  •y-for the upper and lower branch, respe<> tively) and vaJues of the first two frequencies for destructive and constructive interferenoe (cf. Eq.

fd, 1 (Fig. 8 .Fig. 9 .

 189 Fig. 8. Flame displacement, �, along the flame front at a velocity forcing of amplitude of 5%ub and/= 80 Hz for various phase angles, <p, and rotation rates a.

Table 1

 1 Properties taken from the laminar ONS solution for diff erent rotation rates, a.

a Tc

y l (K) Xroot,y- Xroot,y+

			(mm)	(mm)
	0	758.6	1.9	1.9
	0.625 697.6 1.25 757.7	2.4 2.4	7.8 12.1
	2.5 3.75	804.7 791.2	3.4 4.3	14.8 8.3

Acknowledgments

The research leading to these results has re ceived funding from the European Research Coun cil under the European Union's Seventh Frame work Programme (FP/2007-2013)/ERC Grant Agreement ERC-AdG 319067-INTECOCIS.

This work was granted access to the high performance computing resources of CINES un der the allocation A0032B07036 made by Grand Equipement National de Calcul Intensif.


Amine Kacete

Thomas Wentz

J Ér Ôme Royan

[POSTER] Decision Forest For Efficient and Robust Camera Relocalization

Keywords: Camera relocalization, pose estimation, Random Forest, SLAM

To robustly estimate the pose, classical methods assume some geometrical and temporal assumptions (SfM: Structure from Motion, SLAM: Simultaneous Localization and mapping). These approaches take a pair of images as input and establish correspondences based on global strategy (using the whole image information) or sparse strategy (using key-points features). These correspondences allow solving a set of linear equations related to the 3D information and camera pose in that environment. These past years, machine learning has been considered as an efficient way to tackle different problems in image processing and computer vision fields. To handle the task in hand, we propose to learn directly the mapping function between the acquired information from the camera and its pose using sparse decision forest. We achieved state-of the-art results on public and on our databases.

INTRODUCTION

Camera pose estimation is the process of determining the position and the orientation of the camera in a given scene using the acquired information (rgb-image, depth-image, rgb-d image, infraredimage,point cloud..). Knowing the pose information is crucial in various fields such Augmented Reality, indeed, to add additional synthetic objects on the environment in a realistic way, the camera pose is required to render optimally these objects.

Common approaches use global or sparse features matching between two images to estimate the camera continuous pose. With sparse strategy, a set of key-points are detected such as SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF], SURF [START_REF] Bay | Speeded-up robust features (surf)[END_REF] or ORB [START_REF] Rublee | Orb: An efficient alternative to sift or surf[END_REF]. Using the descriptors of these 2D key points, a matching can be established across key-frames descriptors corresponding to different poses to retrieve the optimal one as performed in [START_REF] Dong | Keyframe-based real-time camera tracking[END_REF].

In [START_REF] Lepetit | Keypoint recognition using randomized trees[END_REF], the 2D-2D correspondences are robustly learned using randomized trees. [START_REF] Lepetit | Keypoint recognition using randomized trees[END_REF] exploits directly the key-points to establish 2D-3D correspondences and to optimize the camera pose calculation. In [START_REF] Eade | Unified loop closing and recovery for real time monocular slam[END_REF][START_REF] Mur-Artal | Orb-slam: a versatile and accurate monocular slam system[END_REF][START_REF] Mur-Artal | Orb-slam2: an open-source slam system for monocular, stereo and rgb-d cameras[END_REF][START_REF] Williams | Automatic relocalization and loop closing for real-time monocular slam[END_REF], key-points are used as a primordial component in a global relocalization system based on visual SLAM. One of the most important limitation of sparse camera pose estimation is related to the key-points detection and matching efficiency. Indeed, in unconstrained of textureless environments, matching key-points yields outliers which impacts considerably the accuracy and robustness of the pose estimation. To ensure sparsity without keypoint detection [START_REF] Engel | Direct sparse odometry[END_REF] samples directly pixel intensities to establish correspondences.

Unlike sparse strategy, global matching uses the whole image information to perform a geometrically dense mapping as * e-mail:amine.kacete@b-com.com † e-mail:thomas.wentz@b-com.com ‡ e-mail:jerome.royan@b-com.com in [START_REF] Engel | Lsd-slam: Large-scale direct monocular slam[END_REF][START_REF] Newcombe | Kinectfusion: Real-time dense surface mapping and tracking[END_REF][START_REF] Whelan | Elasticfusion: Dense slam without a pose graph[END_REF]. Some methods define the global matching as a learning task. [START_REF] Gee | 6d relocalisation for rgbd cameras using synthetic view regression[END_REF] uses a global regression over a set of synthetic views based on Nadaraya-Watson estimator [START_REF] Nadaraya | On estimating regression[END_REF]. [START_REF] Kendall | Posenet: A convolutional network for real-time 6-dof camera relocalization[END_REF] traines a convolutional neural network to regress the camera pose from a single rgb image. These methods perform a robust camera pose estimation but present a considerable training and testing complexity. The work from [START_REF] Shotton | Scene coordinate regression forests for camera relocalization in rgb-d images[END_REF] and [START_REF] Cavallari | On-the-fly adaptation of regression forests for online camera relocalisation[END_REF] presents the closest work to our approach. Indeed, to regress the camera pose, we train a decision forest on labeled training rgb-d patches. Unlike these approaches which use dense prediction from the image pixels, we extract a set patches based on a highly sparse key-points extraction. Our approach considerably reduces training complexity and enhances discrimination during trees learning. Figure 1 shows an example of an automatic camera pose estimation using our approach.

SPARSE CAMERA RELOCALIZATION LEARNING

In this section we explain in details our approach. In 2.1 and 2.2, we describe our camera pose estimation forest training and testing steps respectively.

Training

Random forest is a powerful technique to map highly dimensional input spaces into discrete or continuous output spaces. Introduced in [START_REF] Breiman | Random forests[END_REF], Random forest is a collection of randomly trained trees with high ability of generalization compared to the standard version. They have been successfully used for classification problems [START_REF] Lepetit | Randomized trees for real-time keypoint recognition[END_REF][START_REF] Shotton | Real-time human pose recognition in parts from single depth images[END_REF] and regression [START_REF] Fanelli | Real time head pose estimation with random regression forests[END_REF][START_REF] Gall | Class-specific hough forests for object detection[END_REF][START_REF] Kacete | Unconstrained gaze estimation using random forest regression voting[END_REF]. Figure. 2 gives an illustrative example of a decision forest applied for an automatic camera pose estimation.

To train our decision forest T = {χ k } to automatically estimate camera pose from input rgb-d images, each tree χ k is learned in a Concretely, in every rgb-d sample, we extract a set of fixed size patches {Pi = (I c i , yi)} centred on the detected SURF key-points. I c i defines the appearance channel (ie., c = 0 corresponds to the grayscale intensities and c = 1 to the depth). yi represents the label which is the camera pose ground truth encapsulated as quaternion and translation as (Qi, Ti) = (qw, qx, qy, qz, tx, ty, tz).

We provide, for each tree χ k , 50% of the total training set. We train the trees in a greedy way, at each node, a simple binary test t is performed. According to the result of the test, a data sample is directed towards the left or the right child node. The tests are selected to achieve an optimal clustering. The terminal nodes of the tree called leaves, store the estimation models which approximate the best the desired output.

To achieve robustness, we formulate a differential binary test as performed in [START_REF] Shotton | Real-time human pose recognition in parts from single depth images[END_REF] as follows:

f θ (P) = P c i (x, y) -P c i (x , y ) (1)
Where θ = ((x, y), (x , y ), c) represents the parameters encapsulating the two 2D pixels locations and the appearance channel in the processed patch Pi respectively. Supervising the training consists in finding at each non-leaf node the optimal separation which maximizes the purity of the data clustering. We performed the training following the standard Random forest pipeline:

1. Generate a set of binary test parameters Θ = (θ, τ ) 2. For each Θ, split the parent set of patches P into two child subsets PL and PR as follows:

PL(Θ) = {P|f θ ≤ τ } (2) PR(Θ) = {P\PL(Θ)} (3)
3. Evaluate the discrimination quality E for each splitting as follows:

E(Θ) = i∈P L ||yi -ȳL|| 2 2 + i∈P R ||yi -ȳR|| 2 2
(4)

ȳ = i∈P yi |P| (5)
E defines the global euclidean distance between all the elements yi to the centrod ȳ in each cluster.

4. Select the optimal splitting binary test which minimizes E:

Θ * = arg min Θ E(Θ). (6)
The learning process finishes when the data reach a predefined maximum depth value of the tree or the number of patches let down a threshold value yielding the creation of the leaves. Each leaf l stores multivariate Gaussian distribution of all the reached poses as p(y) = N (y, ȳl , Σ y l ).

Testing

To estimate camera pose from an unseen rgb-d sample, we extract, following the same strategy as in training, a set of patches. Each patch is passed through all the learned trees in the forest. Using the optimal stored binary test, each tree processes the patch until reaching a leaf. The estimation according to a single tree is given by the reached leaf l in terms of the stored distribution p(y). The pose estimation for a given patch Pj over all the trees is calculated as follows: . All the estimations corresponding to the extracted patches are regrouped as votes. Before performing the clustering of these votes, we discard the estimations from the leaves with high variance considered as non-informative. To locate the centroid of the cluster of the votes, we perform five mean-shift iterations using a Gaussian kernel. Figure . 3a illustrates the knowledge H of the learned forest. It represents all the estimations stored by the leaves. Visualizing H can help in predicting ahead some bad estimation. Figure . 3b shows the non-parametric clustering by mean-shift. For a given sample, the forest casts a set of votes (in red), we can notice the gap between the estimates which is directly linked to the decorrelation (bagging) introduced during learning. Performing mean-shift with some iterations allows to determine more accurate estimation than a simple statistical mean (the green sphere represents the ground truth and the blue one is the final estimation).

p(y|P c i ) = 1 NT j p(y|lj(P c i )) (7)

EXPERIMENTS

In this section, we report the results of our estimations according to different experimentations. Firstly, in 3.1 we describe the data used in our experiments, then we report our decision forest accuracy in 3.2. In 3.3, we compare our method to state-of-the-art methods and evaluate the robustness of the approach on unfavorable scenarios in 3.4.

Datasets

To provide our forest with sufficient annotated training rgb-d samples and evaluate our camera pose accuracy, we used two public datasets RGBDscene.v2 [START_REF] Lai | Unsupervised feature learning for 3d scene labeling[END_REF] and CoRBS [START_REF] Wasenmüller | Corbs: Comprehensive rgb-d benchmark for slam using kinect v2[END_REF]. The first dataset uses the method from [START_REF] Henry | Rgb-d mapping: Using depth cameras for dense 3d modeling of indoor environ-(a) (b) (c) (d) (e) (f) Figure 6: Camera pose estimation results of our decision forests on different databases, the green path represents the ground truth camera pose and the estimations in red. (a),(b) represents two different scenes in RGBDscene.v2 database.(c),(d) and (e) illustrates three different scenes in CoRBS database. The last scene represents one scene of our database[END_REF], which is based on an ICP rigid alignment, to accurately annotate the rgb-d samples. The acquisition is performed using Kinect.v1. For the second dataset, the acquisition is performed with Kinect.v2, the annotation is achieved thanks to a motion capture system. In total, 14 and 4 scenes are reconstructed for the first and second dataset respectively. We also build our own dataset by using Kinect.v2 for the acquisition. To annotate our rgbd samples, we used the method from [START_REF] Newcombe | Kinectfusion: Real-time dense surface mapping and tracking[END_REF] which is a real-time 3D dense reconstruction technique following a dense SLAM pipeline as in [START_REF] Shotton | Scene coordinate regression forests for camera relocalization in rgb-d images[END_REF]. We build a total of 17 RGB-D scenes. We trained a specific forest for each scene on half of the global samples number.

Forest accuracy

Some empirical analysis, based on computational cost and accuracy criteria, allowed us to fix some parameters to their optimal value according to our task. The number of the extracted patches by sample is fixed to 20 with a size of (30 × 30). The forest size is fixed to 15, each tree is learned with a depth of 15. In Figure . 4a and 4b, we report the mean quadratic errors related to translation and rotation respectively on the datasets CoRBS, RGBDscene.v2 and ours. We reported a mean error of 0.047m and 2.46 • for the rotation and translation respectively on CoRBS. On RGBDscene.v2, we achieved best results with errors of 0.029m in terms of translation and 1.34 • for the rotation. This difference in terms of accuracy is directly linked to the nature of the camera trajectory, indeed, in RGBDscene.v2, the camera presents small motions in both translation and rotations (translation according to x and z, rotation only around y) which makes the splitting process inside the trees nodes easier, while CoRBS gives more complicated camera trajectory giving complex camera manifold. On our RGB-D scene, the forest performed a mean error of 0.0501m and 1.95 • for translation and rotation respectively. Figure . 6 gives more visual illustrations of our estimations on different scenes.

Comparison to baseline

Table . 1 reports a comparison of our approach to two methods for camera relocalization, PoseNet [START_REF] Kendall | Posenet: A convolutional network for real-time 6-dof camera relocalization[END_REF] and ORB-SLAM [START_REF] Mur-Artal | Orb-slam: a versatile and accurate monocular slam system[END_REF] respectively on RGBDscene.v2. [START_REF] Kendall | Posenet: A convolutional network for real-time 6-dof camera relocalization[END_REF] performed the most important errors, 0.062cm and 4.66 • for translation and rotation respectively. This method uses a convolutional neural network (CNN) learned on the whole image appearance producing a highly smoothy camera pose output. This result can be explained by the provided training set (400 images) producing a suboptimal network weights optimization. In contrast, by extracting a set of patches, our method explicitly extends the training set size which considerably enhance the ability of discrimination of our forest. In addition, by geometrically targeting relevant regions, our method achieves more robustness under noisy scenarios. [START_REF] Mur-Artal | Orb-slam: a versatile and accurate monocular slam system[END_REF] achieves better results compared to the first approach, 0.052cm and 1.94 • respectively. By learning a set of key-frames (on the same training set provided previously), this method optimizes the camera pose trajectory based on sparse key-points matching. Instead of establishing correspondences between successive frame features matching which produces increasing output uncertainty, our approach implicitly reduces considerably temporal uncertainty by processing each patch independently giving more accurate and robust camera relocalization estimation. Our work achieves 0.029cm and 1.34 • for translation and rotation errors respectively.

Method

Translation error (cm) Rotation error ( • ) PoseNet [START_REF] Kendall | Posenet: A convolutional network for real-time 6-dof camera relocalization[END_REF] 0.062 4.66 ORB-SLAM [START_REF] Mur-Artal | Orb-slam: a versatile and accurate monocular slam system[END_REF] 0.052 1.94 Our method 0.029 1.34

Table 1: Comparison of our approach to the methods in [START_REF] Kendall | Posenet: A convolutional network for real-time 6-dof camera relocalization[END_REF] and [START_REF] Mur-Artal | Orb-slam: a versatile and accurate monocular slam system[END_REF].

Robustness to occlusions and illumination variation

One of the most challenging problem in camera relocalization consists in dealing with important image appearance variation. Unfortunately, this problem is frequently present in rgb-d scenarios.

In this section, we evaluate the robustness of our approach with respect to illumination condition variation and strong occlusion in the rgb image. Concretely, we underexpose/overexpose the testing samples to synthetically produce unfavorable lightning conditions.

To model occlusion, we randomly render a dark square with a fixed size of (120 × 120) in the rgb images. We report the results of these experiments in Figure . 5. Figure 5a illustrates the translation and rotation errors with and without illumination variation on RGBDscene.v2 database. The mean error goes from 0.0292m to 0.0570m and 1.34 • to 3.38 • for translation and rotation respectively. These results demonstrate the robustness and the high ability of generalization of our approach on such condition. Figure . 5c gives a visual illustration of the final estimation compared to the ground truth camera trajectory (red and green paths respectively) and typical lightning condition on testing samples. Figure. 5b describes mean errors with respect to occlusions. We notice that errors go from 0.0292m to 0.0502m and 1.34 • to 3.95 • for translation and rotation respectively. Again, this result proves the efficiency of our approach under occlusion presence. Figure . 5d shows the results of our estimation and gives typical examples of the generated occlusions on the testing samples. According to our forest parameters defined previously, our estimation runs in real-time on an Intel Core i7 @2.70GHZ with 8GB of RAM machine.

CONCLUSION

In this paper, we have proposed an approach based on regression decision forest to handle unconstrained camera pose estimation in rgb-d scenarios. Unlike similar approaches, we trained our forest on sparse data represented by a set of extracted patches. Based on SURF key-points detection, we demonstrated that targeting geometrically relevant regions in images increases forest accuracy and generalization. By establishing translation and rotation errors on challenging databases, we validate our approach by achieving stateof-the-art performance.

Figure 1 :

 1 Figure 1: Automatic camera pose estimation performed by our method.

Figure 2 :

 2 Figure 2: A visual example of a decision forest used in this work. Each tree processes a testing sample. By performing a simple learned binary test, each splitting node (white nodes) directs the sample until reaching the leaves (yellow nodes) which store a multivariate distribution of the camera pose (the green frustum represents the camera pose mean). The forest returns a combination of all the outputs.

Figure 3 :

 3 Figure3: (a) shows the estimation space H learned by the tree, which represents all the possible votes than can be cast by the forest (The more the vote is red, the less is confident). (b) illustrates, in red, the votes for a griven testing sample, in green the camera pose ground truth and in blue the final estimation calculated using meanshift clustering.

Figure 4 :

 4 Figure 4: (a) reports the translation errors of our approach on CoRBS, RGBDscene.v2 and our database respectively. (b) indicates the rotation errors on the same databases.

Figure 5 :

 5 Figure 5: (a) reports translation and rotation errors of our method on original and unfavorably illuminated RGBDscene.v2. (c) gives a visual illustration of some under-illuminated frames and the correspondent estimation. (b) gives translation and rotation errors on original and occluded RGBDscene.v2. (d) gives an example of some occluded frames and the correspondent estimation.