Keywords: bubbly flow, oscillating jet, depth of penetration, jet impact, aeration, coherent jet

 for a non-oscillating jet tilted relative to the normal of the liquid surface, provided an effective inclination angle is introduced to account for the jet translation velocity. In the case of jets with a Reynolds number smaller than 10 4 , we find that the penetration depth of the oscillated jet can be increased of up to 30% by moderate oscillation velocities. This increase of the penetration is due to the decrease of the mixing layer angle when the symmetry around the bubble cloud is broken by the oscillation. We finally discuss the shape of the overall region impacted by the oscillating bubble cloud. We show that there is a regime for which the penetration is maximum at the center of this region, and a regime for which the penetration is maximum at the periphery. We propose a cartography to predict when each regime is relevant.

Introduction

Air entrainment due to high-speed liquid structures such as drops, jets or waves impacting a free surface is commonly encountered in nature, e.g. cascades or breaking waves [START_REF] Deike | Air entrainment and bubble statistics in breaking waves[END_REF]), as well as in industry [START_REF] Chanson | Bubble entrainment and dispersion in plunging jet flows : Freshwater vs. seawater[END_REF]; [START_REF] Pagliara | Scour process due to symmetric dam spillways crossing jets[END_REF]; [START_REF] Duarte | An enhanced physically based scour model for considering jet air entrainment[END_REF]; [START_REF] Descloux | Efficiency of the nam theun 2 hydraulic structures on water aeration and methane degassing[END_REF]). It is notably present in environmental engineering (e.g. aeration), for process control (e.g. foam destruction) or in energy production (dams, turbines). Air 5 entrainment involves various elementary processes such as bubble formation mechanisms, bubble transport mechanisms beneath the free surface which have been investigated in diverse situations. Extensive reviews of our current understanding of air entrainment by plunging jets have been proposed by [START_REF] Biń | Gas entrainment by plunging liquid jets[END_REF], [START_REF] Kiger | Air-entrainment mechanisms in plunging jets and breaking waves[END_REF] and more recently [START_REF] Miwa | Experimental investigation of air entrainment by vertical plunging liquid jet[END_REF].

In this article, we focus on the penetration depth reached by the bubble cloud entrained by a cylindrical liquid jet plunging into a pool of the same fluid. Many semi empirical relations or empirical relations have been proposed to predict this penetration depth. The most popular are reported in table 1, and include those of [START_REF] Mckeogh | Air entrainment rate and diffusion pattern of plunging liquid jets[END_REF], [START_REF] Falvey | Behaviour of turbulent water jets in the atmosphere and in plunge pools[END_REF], [START_REF] Nakasone | Study of aeration at weirs and cascades[END_REF] and [START_REF] Ohkawa | Flow and oxygen transfer in a plunging water jet system using inclined short nozzles and performance characteristics of its system in aerobic treatment of wastewater[END_REF] among others. Note that the dimensional correlations proposed by [START_REF] Mckeogh | Air entrainment rate and diffusion pattern of plunging liquid jets[END_REF], and by [START_REF] Ohkawa | Flow and oxygen transfer in a plunging water jet system using inclined short nozzles and performance characteristics of its system in aerobic treatment of wastewater[END_REF] are given in S.I. units. Yet, and as discussed in the above-mentioned reviews, there is no clear consensus on the prevailing parameters and on the relevant scaling laws for predicting the penetration depth. This situation is probably due to the variety of configurations to be accounted for in terms of jet topology at impact which range from smooth to rough and up to fully destabilized depending on the size and the velocity of the jet. The nature of the initial disturbances (including the turbulence in the liquid, secondary flows etc.) and the distance between the injector and the free surface are also expected to impact the bubble cloud dynamics. In this context, our objective is to investigate the influence of a specific disturbance consisting in an oscillation of the liquid jet parallel to the free surface. Such oscillations are indeed observed to occur along falling jets, notably when the distance between the injector and the free surface is not too small [START_REF] Guyot | Experimental study of large scale plunging[END_REF]). In order to isolate the effect of oscillations from other kinds of perturbations, we consider turbulent water jets of small diameter and with moderate interfacial corrugations. These types of jets have been investigated by [START_REF] Clanet | Depth of penetration of bubbles entrained by a plunging water jet[END_REF] for jet diameters up to a few millimeters and using long needles as injectors (length to diameter ratios larger than 50) in order to ensure fully developed turbulent flows. Moreover, these authors have established a predictive model for the penetration depth which proves valid over a large range of injection velocities. The control parameters in this model are the jet diameter, the jet velocity at the nozzle, the terminal bubble velocity, the spread angle of the bubble cloud and the jet inclination with respect to the free surface. We will investigate similar jets but with a vertical needle oscillating above the free surface and parallel to it. We will show how the [START_REF] Clanet | Depth of penetration of bubbles entrained by a plunging water jet[END_REF] model can be adapted when the liquid jet is mobile. We first present the experimental method in the next section. The experimental results and their interpretation will be presented in the following section.

Experimental method

A round water jet issued from a nozzle of diameter D at a mean outlet velocity V 0 falls into an overflowing cubic reservoir (edge 50 cm). The distance h between the tip of the nozzle and the water surface is constant for a given nozzle diameter : as in [START_REF] Clanet | Depth of penetration of bubbles entrained by a plunging water jet[END_REF], the ratio h/D was kept constant and equal to 20. The ratio of the maximum penetration depth to the total depth never exceeded 0.30, in order to minimize the effect of the finite size of the pool.

All nozzles are stainless steel needles with length to diameter ratios L 0 /D larger than 50. The exit velocity profile can then be assumed to be fully developed. A water circuit delivers a steady flow rate at the nozzle via a pressure regulator.

A 10 microns water filter has been inserted upstream of the circuit to clear the water. The flow rate is measured with a flowmeter located just above the nozzle. The flow rate range for our experiments is from 6.10 -4 l/s to 3.10 -2 l/s, with an overall uncertainty of 5%. The nozzle, which remains vertical in all our experiments, is attached to a vibration table controlled by a computer. This lab-made vibration table consists in a metal plate driven by a linear motor, guided by bearings. The jet behavior under the water surface can be captured by two cameras : a Nikkon D200 (3872 pixels by 2592 pixels) or a Phantom V10 (1280 pixels by 720 pixels, 1016 frames per seconds) monitored by the same computer.

45

The lens used with these cameras is a Nikon AF Nikkor 50 mm f/1.8D. Figure 1 The displacement imposed by the vibration table along the axis of its linear motor is a sinusoid A/2 sin(2π f t). The metal plate entrained by the motor, to which the nozzle is attached, moves parallel to the reservoir surface. The values of peak-to-peak amplitude A and frequency f for our experiments are shown in table 3. The motion of the vibration table was recorded with high speed imaging before each measurement series in order to check i) that the difference between the actual displacement amplitude and the prescribed amplitude never exceeded 5% ii) that the motion was harmonic, and that the frequency of the motion was close enough to the prescribed frequency (the maximum measured difference was 0.1 Hz). The values of D, V 0 , A and f in tables 2 and 3 correspond to a total of 848 different experimental conditions.

A minimum of 50 pictures were recorded for each velocity/diameter/amplitude/frequency condition, at a low sampling frequency of 1 Hz : the time between two images is therefore much larger than the typical time of flight H/V 0 , and this ensures that two consecutive images are relatively independent, which accelerates convergence of average quantities. The penetration depth of the bubble cloud was experimentally measured, applying a grey gradient analysis on each sample image (such as figure 2) : starting from the bottom of the image i) For a given line on the image a vertical gradient of gray level is computed for each horizontal position ; ii) The maximum value of this gradient is retained for the given line. The values of these maxima are plotted on figure 2 for each line. It can be seen on figure 2 that the lower limit of the first bubble corresponds to a very steep increase of the maximum gradient ; iii) The depth 70 where this maximum gradient becomes larger than a fixed threshold is retained as the bubble cloud edge. The distance from this edge to the location of the free surface at rest (marked by a dashed line on figure 2) is computed for each set of 50 pictures, and given V 0 , D, A and f : we note H the average of these 50 values. The chosen number of 50 pictures is largely sufficient to ensure convergence of H for our conditions. The overall uncertainty for these penetration depth measurements is 10%.

Experimental results and discussion

Bubble penetration depth for a non-oscillating jet

First of all, our experimental results were compared to the model proposed by [START_REF] Clanet | Depth of penetration of bubbles entrained by a plunging water jet[END_REF] for a steady plunging jet normal to a flat liquid surface. This simple model is based on the idea that bubbles will be entrained downwards by the liquid jet until the liquid jet velocity reaches U T , the terminal bubble velocity. The liquid velocity U as a function of depth z can be estimated from momentum conservation, and by assuming that the liquid jet widens with a constant half angle α. As demonstrated in [START_REF] Maxworthy | Experiments on the rise of air bubbles in clean viscous[END_REF] there is a minimum terminal velocity U T = 0.22 m/s for bubbles over 1 mm diameter in water. The size of the bubble cloud is therefore directly deduced from the depth where liquid velocity reaches U T . Writing that U(z) = V 0 D/D z where D z is the diameter of the liquid jet at depth z, we find that velocity U T is reached at a depth V 0 D/(2U T tan α). The depth H 1 predicted by this model is then :

H 1 D = 1 + tan α 2 tan α V 0 U T (1)
Clanet & Lasheras (1997) propose a constant value of α = 12.5 • for all their measurements : this assumption is obviously a strong one, because α is actually expected to vary spatially as the liquid jet slows down and bubbles escape the jet near its bottom. We estimate the value of this (assumed constant) angle α for our experiments by deducing it 80 from the width of the entrained bubble cloud at the penetration depth (see figure 2). The average α measured for our experimental conditions is 11.8 • with a standard deviation of 1.2 • : this value is consistent with the value of 12.5 • proposed by [START_REF] Clanet | Depth of penetration of bubbles entrained by a plunging water jet[END_REF], but slightly smaller than the value of α = 14 • proposed by [START_REF] Ervine | Pressure fluctuations on plunge pool floors[END_REF].

0 1 2 3 Re 10 4 0.7 0.8 0.9 1 1.1 1.2 b) a) c) Zoom x3
We compare in figure 3a the result of measurements of the average penetration depth for a non-oscillating jet, H, to the model penetration depth H 1 as a function of the Reynolds number of the liquid jet Re. All diameters and outlet velocities of table 2 were tested. The error bars correspond to the amplitude of the temporal fluctuations of H, which are of the order of the jet width. These fluctuations, which arise possibly because of turbulence and because of the interaction with ascending bubbles, are the main cause of uncertainty, and largely overcome the uncertainty introduced by the image analysis. They are estimated at 10% of H. We observe on figure 3a that the ratio H/H 1 is close to one for all conditions, in spite of the strong assumption on the mixing layer angle. There is however an influence of the Reynolds number : for Re > 10 4 , the model overestimates the measured length by 10%-20%. The correctness of the prediction of equation ( 1) indicates that considering the linear spreading of a single phase jet [START_REF] Pope | Turbulent Flows[END_REF]; [START_REF] Horn | Angle of spread of free jets[END_REF]) is an acceptable approximation even in the presence of entrained bubbles. ) and it strongly decays radially. These features imply that the void fraction is significant only in a relatively small portion of the jet beneath the free surface. This feature is confirmed by the two-fluid simulations of [START_REF] Ma | A quantitative sub-grid air entrainment model for bubbly flows -plunging jets[END_REF] that account for momentum transfer including drag, added mass, lift as well as turbulent dispersion contribution and that properly recover the experimental void fraction profiles at different distances beneath the free surface.

Bubble penetration depth for an oscillating jet

We now present measurements of H when the nozzle is oscillated parallel to the interface with an amplitude A and a frequency f . On figure 4, the dimensional bubble penetration depth is plotted as a function of f for all the conditions of tables 2 and 3. The D = 0. mm, for all 848 experimental points. For most series, bubble penetration depth decreases when frequency is increased at constant D, V 0 and A. However, a maximum of penetration depth is measured for low frequencies (between 1 and 5 Hz) for several series : this occurs in particular for the smaller V 0 , and is relatively more pronounced for the smaller diameter D (see close-up on figure 5). When the oscillation amplitude A is increased, the penetration depth generally 7 decreases : an increase of penetration depth with amplitude is nonetheless observed for a few cases, namely just before the maximum in frequency when this maximum is sharp (see e.g. case V 0 = 15 m/s and A = 8 mm to 16 mm for f < 2.5 Hz on figure 5).

H (mm) D=2.4mm V 0 =7.2m/s D=2.4mm V 0 =6.8m/s D=2.4mm V 0 =5m/s D=2.4mm V 0 =3.9m/s D=2.4mm V 0 =2.9m/s D=2.4mm V 0 =1.9m/s D=1.3mm V 0 =18.6m/s D=1.3mm V 0 =15.6m/s D=1.3mm V 0 =12.2m/s D=1.3mm V 0 =8.4m/s D=1.3mm V 0 =5.7m/s D=1.3mm V 0 =1.9m/s D=0.3mm V 0 =20.1m/s D=0.3mm V 0 =15m/s D=0.3mm V 0 =11.8m/s D=0.3mm V 0 =9.8m/s

Non dimensionalization of results

We now want to non-dimensionalize the results of figure 4. We first carry out a simple dimensional analysis : for the non-oscillating jet, the physical parameters of the problem are H, V 0 , D, U T . These parameters must be associated to two dimensionless numbers. Equation (1) gives H/D as a function of V 0 /U T , and as a function of α which is itself dimensionless. Figure 3 suggests that the Reynolds number (via the kinematic viscosity) should be introduced as a fourth dimensionless number. The oscillation of the jet introduces two additional physical parameters A and f . The amplitude can be non dimensionalized as A/D : the effect of A for fixed f will be discussed in section 3.2.4. In order to non dimensionalize frequency, we have to introduce a time scale. When frequency is increased, the horizontal velocity of the liquid jet relative to the surface increases. In a frame moving with the liquid jet, the situation is similar to that of a liquid jet entraining bubbles in an oscillating cross flow of characteristic velocity of order A f (the horizontal velocity is more precisely πA f with the definition of A introduced above). We therefore non dimensionalize frequency by introducing the velocity ratio associated to this situation, namely the ratio of horizontal velocity A f to vertical velocity. There are two vertical velocities in this problem : the initial jet velocity V 0 , and the terminal velocity U T reached by the liquid at the bottom of the bubble cloud, and therefore two possible Strouhal numbers, A f /V 0 and A f /U T . We compare on figure 6 how these two Strouhal numbers, A f /V 0 and A f /U T succeed in capturing the location of the maximum of H observed at low frequencies for small diameters and lower V 0 series. In this graph, the measured bubble penetration depth is made non dimensional with the length H 1 of equation (1). A first observation on figure 6a 135 is that values of A f /V 0 for our experiments are always smaller than 0.03 : horizontal velocity is negligible compared to V 0 , meaning that the liquid jet remains almost vertical when it hits the water surface. Note also that while maxima on figure 6a are clearly scattered over a wide range of A f /V 0 , the same maxima occur at a constant A f /U T ≈ 0.1 on figure 6b.

H (mm) V 0 =15m/s A=2.5mm V 0 =15m/s A=4mm V 0 =15m/s A=8mm V 0 =15m/s A=12mm V 0 =15m/s A=16mm V 0 =11.8m/s A=2.5mm V 0 =11.8m/s A=4mm V 0 =11.8m/s A=8mm V 0 =11.8m/s A=12mm V 0 =11.8m/s A=16mm V 0 =9.8m/s A=2.5mm V 0 =9.8m/s A=4mm V 0 =9.8m/s A=8mm V 0 =9.8m/s A=12mm V 0 =9.8m/s A=16mm
The ratio between the (maximum) horizontal jet velocity and the terminal vertical liquid velocity within the bubble cloud, A f /U T , is directly related to the (maximum) angle of inclination of the bubble cloud at its bottom. The Clanet & Lasheras (1997) work precisely includes the case of a jet inclined of an angle θ relative to the vertical : their experimental penetration depth exhibits a maximum when θ is varied, and this maximum is in the same proportions as the one we observe when f is varied (up to plus 30% in depth on figure 6b). Instead of A f /U T , we therefore introduce as a new parameter the angle β = atan(πA f /2U T ), which can be interpreted as the angle of the bubble cloud relative to the vertical when its vertical velocity is 2U T . This angle is expected to be more representative of the average inclination of the bubble cloud than the maximum angle built with vertical velocity U T , only reached at the tip of the cloud. We plot on figure 7 the ratio H/H 1 as a function of β. The scattered data of figure 4 is now relatively gathered around ratios H/H 1 close to one. While the existence of a maximum of H with the inclination angle θ is not explained in the work 0 0.01 0.02 0.03 0 0.5 of [START_REF] Clanet | Depth of penetration of bubbles entrained by a plunging water jet[END_REF], they propose a model for the decrease of H observed at large θ . This model, which is simply a geometrical variation of the model behind equation ( 1) when the jet is tilted, predicts a depth H 2 such that :

1 1.5 Af/V 0 H/H 1 D=1.3mm V 0 =1.9m/s A=8mm D=1.3mm V 0 =1.9m/s A=12mm D=1.3mm V 0 =1.9m/s A=16mm D=0.3mm V 0 =20.1m/s A=16mm D=0.3mm V 0 =15m/s A=8mm D=0.3mm V 0 =15m/s A=12mm D=0.3mm V 0 =15m/s A=16mm D=0.3mm V 0 =11.8m/s A=8mm D=0.3mm V 0 =11.8m/s A=12mm D=0.3mm V 0 =11.8m/s A=16mm D=0.3mm V 0 =9.8m/s A=8mm D=0.3mm V 0 =9.8m/s A=12mm D=0.3mm V 0 =9.8m/s A=16mm
H 2 D = (1 + tan α) cos θ + tan α sin θ 2 tan α cos(θ -α) cos α V 0 U T (2)
In this model it is assumed that the bubble cloud remains axisymmetric relative to the jet axis, namely that α does not depend on the azimuthal angle around the jet. The ratio H 2 /H 1 is plotted as a dashed black line on figure 7, based on equation (2) with θ equated to β. Note that even though there is still a large dispersion around the prediction, the widely scattered data of figure 4 is relatively well gathered around the black curve. The model captures the decrease in penetration depth observed at larger horizontal velocities (equivalent to larger angles). The large dispersion of the experimental data around β = 0.2 ± 0.1 rad (corresponding to 11 • ± 6 • ) results from the fact that in this region some of the curves exhibit a maximum of H, while others decrease monotonically. Something is obviously missing in the model, and this is discussed in the next section.

Existence of a maximum of penetration

As mentioned above, a maximum in penetration depth of up to 30% is also observed by [START_REF] Clanet | Depth of penetration of bubbles entrained by a plunging water jet[END_REF] in their experimental data when they vary the angle of inclination θ of the jet, but it is not explained by their model. We first discuss this maximum observed when a non-oscillating inclined jet is inclined. Observation of the dynamics of the bubble cloud shows that when the jet is vertical and not oscillated, the bubbles released from the jet ascend all around the jet, figure 8a ; when the jet is slightly tilted, or equivalently when it is moving relative to the liquid bath because of an oscillation, the symmetry is broken and the bubbles ascend in a confined region farther from the plunging jet, see figure 8b. This has a strong impact on the angle α of the mixing layer : it can be seen on figure 8 that the value of α is larger when the bubbles ascend all around the jet. The angle α measured from these images is equal to 17 8a, and to 12 • for figure 8b. We interpret the larger α of figure 8a as resulting from the annular upwards flow generated by ascending bubbles. When the jet is tilted, the ascending flow is displaced away from the plunging jet, α therefore decreases, which in turn leads to a larger penetration depth via momentum conservation (plus 30% on figure 8b).

This explains the existence of a maximum in the [START_REF] Clanet | Depth of penetration of bubbles entrained by a plunging water jet[END_REF] measurements when the jet inclination is increased. We believe the same mechanism explains the maximum observed in our experimental results. In other terms, for low but finite β, i.e. moderate horizontal velocities, the jet motion breaks the symmetry of the upwards flow and leads to a penetration larger than that for a vertical jet. If β is further increased (i.e. horizontal velocity further increased), the vertical penetration eventually decreases because a larger fraction of the jet momentum is injected in the horizontal direction, as modelled in equation 2.

As pointed out before, the maximum in penetration depth is not observed for all cases in figure 7. This maximum exists when D = 0.3 mm with all the velocities and Reynolds tested, whereas it only appears for velocities under 5.63 m/s (Re= 7345) for a nozzle diameter equal to D = 1.3 mm and only for the lowest velocity V 0 = 1.88 m/s (Re=4524) in the D = 2.4 mm case. Our interpretation is that when the momentum of the jet is larger, the impact of the bubble cloud on the liquid flow is smaller. We check this assumption by plotting on figure 9 the ratio of the maximum penetration depth when A and f are varied to the penetration depth H 0 when A and f are zero (•) as a function of the Reynolds number. There is a clear trend, namely this ratio decreases almost monotonically to one when

Re is increased. The conditions for which no maximum is observed roughly correspond to Reynolds numbers larger than 10 4 . Hence, an increase of the penetration depth with the oscillation of the jet can only be observed for low values of Re, when the jet momentum is weaker. This increase can be of up to 30% for the smallest jet Reynolds numbers investigated, namely Re ≈ 2500 -2900. These values being close to the end of the transition to turbulence region, we can conclude that, for a oscillated fully turbulent jet, the maximum depth of penetration when Re is varied is 30% larger than the Clanet et al prediction. Note that we have not investigated laminar nor transition conditions for which the air entrainment process is changed.

We have also included on figure 9 three points corresponding to the data of [START_REF] Clanet | Depth of penetration of bubbles entrained by a plunging water jet[END_REF] : for these points the ratio H max /H 0 corresponds to the ratio of the maximum penetration depth when the jet is inclined to the penetration depth when it is vertical for the same D and V 0 . For a given Re the ratio H max /H 0 observed in their experiment is larger than for our oscillated jet, but this is expected since the impact of the ascending secondary flow on H will be stronger when the jet is stationary than when its "effective inclination" oscillates. Interestingly, a decrease of H max /H 0 with Re is also observed in their experiments.

Impact of mixing layer angle α on the penetration length

As mentioned in the preceding subsection, the existence of a maximum of penetration when the jet is oscillated is due to the impact of the oscillation on the mixing layer angle α. The model of [START_REF] Clanet | Depth of penetration of bubbles entrained by a plunging water jet[END_REF] is based on the assumption that α, the angle of the mixing layer, is constant. While this is mostly true for the case of the fixed jet, α is expected to exhibit strong variations when the jet is oscillated. We discuss in this paragraph these variations of α during the course of an oscillation of the liquid jet. We present some pictures of the bubble cloud along its oscillation on figure 10, for two distinct conditions : on the bottom line is represented the case where one of the highest maximum this case results from the larger momentum of the liquid jet for these conditions. On the contrary, when Re is smaller as in the bottom line, the mixing layer angle of the jet is strongly impacted by the oscillation. Namely, the effect of the oscillation is to remove the jet from its ascending bubble cloud, leading to smaller values of α at the position of maximum amplitude.

Via the conservation of momentum, these local variations of α are expected to strongly impact the local penetration depth H loc , defined as the instantaneous penetration depth observed at a given position : table 4 shows how a variation of α in the range 10-14 • leads to variations of the model penetration depth H 1 of about 30%, via equation (1). A local decrease of the mixing layer angle at the outermost positions will therefore logically induce a larger local penetration depth at this location.

We believe the dispersion of the average penetration depth in figure 7 is caused by these variations of α, which are not accounted for in the [START_REF] Clanet | Depth of penetration of bubbles entrained by a plunging water jet[END_REF] inertia (much smaller Re), it will display a much larger local penetration depth at its outermost positions, where its mixing layer angle is the smallest, and therefore a larger average penetration depth H.

Shape of the bubble cloud

While results of figures 4 to 9 were devoted to measurements of the average penetration depth H of the bubble cloud, a consequence of the previous discussion is that the local penetration depth H loc is actually not uniform over 220 the region covered by the oscillating jet. This can be illustrated by superposing instantaneous images of the clouds, see figure 11. The depth of the bubble cloud exhibits strong spatial variations for V 0 = 2.9 m/s and f = 10 Hz, figure 11a, The fact that the minimum at the center of the cloud disappears when frequency is lowered can be interpreted via the impact of the effective angle β, illustrated in figure 7 : for conditions of large frequency and hence large β, H loc will decrease when the jet reaches its maximum velocity, namely at the center. Patterns such as those of figures 11a to c are therefore expected to occur for large values of β. A second necessary condition is that the amplitude of the motion must be larger than the width of the instantaneous cloud itself. The width of the cloud at its bottom can be estimated from DV 0 /U T , the condition is then simply that dimensionless number AU T /DV 0 must be large enough. In the dimensional analysis, this parameter replaces parameter A/D introduced in section 3.2.1, and whose impact had not yet been discussed. We represent on figure 12 a histogram of all our data points, with axes β and AU T /DV 0 . Conditions for which there is a maximum at the center (e.g. figure 11d) are colored in blue, and conditions for which there is on the contrary a minimum at the center (e.g. figure 11a) are colored in orange. We consider that there is a minimum at the center when H loc (max)/H loc (center) > 1.05. Figure 12 shows that there is very little overlap of the blue/orange regions, and that the two dimensionless numbers we propose are relevant to predict the shape of the bubble cloud generated by an oscillating jet : spatial heterogeneity of the penetration depth clearly corresponds to larger AU T /DV 0 and β values.

Note that this cartography has been obtained by varying independently A, f , V 0 , and D, i.e. all the physical parameters introduced in both dimensionless groupings except U T .

Conclusion

We have measured the penetration depth of the bubble cloud generated by an oscillating liquid jet. We have shown that the order of magnitude of the average penetration depth can be well estimated by the simple model of Clanet 11d) and in orange conditions for which there is a local minimum in penetration depth at the center (e.g. figure 11a). A minimum of penetration at the center is observed for larger values of AU T /DV 0 and β.

& Lasheras (1997) for a non-oscillating tilted jet, with a constant mixing layer angle α = 12.5 • . The effect of an oscillation of amplitude A and frequency f can be captured via an effective inclination angle β = atan(πA f /2U T ).

When frequency is varied, a maximum of penetration depth can be observed if the inertia of the jet is low enough, namely if Re = DV 0 /ν < 10 4 : this maximum is caused by the modification of the ascending flow around the bubble cloud, which impacts the mixing layer angle α. The existence of this maximum, which is not captured by the [START_REF] Clanet | Depth of penetration of bubbles entrained by a plunging water jet[END_REF] Examination of the dynamics of the oscillated jet shows that for lower inertia jets, the angle α varies strongly during the course of an oscillation : α is larger at the central position, and smaller at the outermost positions. The local penetration depth is therefore smaller at the central position, and larger at the edges. A consequence of these variations of α, is that for these conditions the global shape of the bubble cloud itself displays a minimum at the central position for large β and AU T /DV 0 . The cartography we provide on figure 12 can be used to predict this shape.

The distance h between the nozzle and the surface is equal to 20D for all our measurements. A variation of this distance is expected to affect the amplitude of perturbations on the liquid jet surface when it hits the pool, and therefore the amount of gas entrained below the surface. Preliminary measurements have shown that the quantity of bubbles had little effect on the penetration depth, and only played a part via the secondary flow of ascending bubbles. However, this will have to be quantified more precisely in future experiments.

The penetration depths measured in the present study were obtained for turbulent jets, but the scale of the liquid jets remained small compared to the scales involved in hydraulic systems. As mentioned in the introduction, for larger systems the jet is expected to exhibit natural oscillations related to the development of instabilities along its fall, and it may also fragment. The model presented in this work for the penetration depth is expected to apply as well for such systems provided the oscillation and jet structure at impact are adequately quantified. This will have to be demonstrated in future works.

Figure 1 :

 1 Figure 1: Sketch and view of the experimental apparatus. The nozzle is oscillated parallel to the interface with a peak to peak amplitude A and frequency f .

  illustrates the set-up : note in particular on figure 1 right that the reservoir is slightly tilted to the right, so that it overflows over a single lateral wall without perturbing visualization. Three nozzle diameters were tested, D =2.4 mm, 1.3 mm, and 0.3 mm. The velocity range is 1.9 m/s to 20.1 m/s. We introduce the Reynolds number Re = V 0 D/ν and Weber number We = ρV 2 0 D/σ, where ν is water kinematic 50 viscosity, ρ the water density and σ the surface tension of water. For our experiments at T = 20 • C, values of Re and We are computed with ν = 10 -6 m 2 /s, ρ=1000 kg/m 3 and σ = 0.072 N/m.

Figure 2 :

 2 Figure 2: Sample picture of a plunging jet and corresponding values of the maximum vertical gray gradient for each depth. The angle α is the angle of the mixing layer of the liquid jet, which is assumed to be constant in the model.

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Figure 3 :

 3 Figure 3: a) Ratio of measured penetration depth to the model of Clanet and Lasheras (1997), for the case of a non-oscillating jet. The error bar corresponds to a 10% uncertainty on the measurement of H. b) Image of the bubble cloud for D 0 = 1.3 mm, V 0 = 1.86 m/s (top : same scale as in image c) ; bottom : magnified x3) : the predicted penetration depth H 1 = 3.0 cm is in agreement with the size of the bubble cloud ; c) Same nozzle, but V 0 = 18.6 m/s. Predicted H 1 = 30 cm is again in agreement with the size of the bubble cloud.

  3 mm nozzle corresponds to the orange and red lines, the D = 1.3 mm nozzle to the yellow and green lines and the D = 2.4 mm nozzle to the the cyan and dark blue lines. For each D, five amplitude values are tested which is why five lines are represented for each series. A close-up of the data for D = 0.3 mm and lower H values is shown on figure 5. The first observation is that H spans a large range of values, from 9 mm to 271

Figure 4 :

 4 Figure 4: Bubble penetration depth H as a function of frequency f , for different D, V 0 and A. The orange and red lines correspond to the D = 0.3 mm nozzle, the yellow and green ones to the D = 1.3 mm nozzle and the cyan and dark blue ones to the D = 2.4 mm nozzle. Solid lines correspond to A = 16 mm, dashed lines to A = 12 mm, dotted lines to A = 8 mm, dash-dotted lines to A = 4 mm, and solid lines with symbol to A=2.5 mm.

Figure 5 :

 5 Figure 5: Bubble penetration depth H as a function of frequency f . Close-up on the data of figure 4 for diameter D = 0.3 mm and three smaller values of V 0 . A maximum in penetration depth is observed when f is varied.

Figure 6 :

 6 Figure 6: Comparison of two possible Strouhal numbers in scaling the maximum in penetration depth : maxima occur for variable AF/V 0 (a), but for a constant AF/U T ≈ 0.1 (b).

Figure 7 :

 7 Figure 7: Non dimensional bubble penetration depth as a function of β = atan(πA f /2U T ). Same legend as in figure 4. Dashed black line corresponds to the model of Clanet and Lasheras (1997) for a jet tilted of an angle θ = β, equation (2).

Figure 8 :

 8 Figure 8: Penetration cloud for D = 1.3 mm, V 0 = 15.64 m.s -1 , Re=20331 ; a) The nozzle is vertical ; b) The nozzle is tilted with an angle θ equal to 15.5 • .

Figure 9 :

 9 Figure 9: Ratio of maximum penetration depth H max when A and f are varied, to penetration depth H 0 when A = 0 and f = 0, for fixed D and V 0 , as a function of the Reynolds number : • present measurements, • Clanet and Lasheras (1997) measurements. The maximum is enhanced for low values of Re.

  model, only based on the assumption that α = 12.5 • . Namely we believe the difference between the red (small diameter D) and blue (large D) curves on figure7are the same as the difference between the top and bottom line on figure 10 : for a given amplitude/frequency if the jet has a much smaller

Figure 10 :

 10 Figure 10: Top line : D = 1.3 mm, V 0 = 18.6 m/s, A = 16 mm, f = 1 Hz, Re = 24137. Bottom line : D = 0.3 mm, V 0 = 9.8 m/s, A = 12 mm, f = 1 Hz, Re = 2950. Left column shows the bubble cloud when the jet reaches its leftmost position. Center column shows the jet reaching its maximum velocity while travelling rightwards. Right column shows the bubble cloud when the jet has reached the rightmost position. The angle αis illustrated for each bubble cloud : this angle is constant for the conditions of the top line, but varies strongly for the conditions of the bottom line, leading to a decrease in the instantaneous penetration depth when horizontal velocity is the largest.

Figure 11 :

 11 Figure 11: Global shape of the bubble cloud obtained by superposing images below the oscillating jet. Fixed D = 2.4 mm and oscillation amplitude A = 16 mm.

Figure 12 :

 12 Figure12: Histogram of all data points as a function of AU T /DV 0 and β (in • ), showing in blue conditions for which there is a maximum in penetration depth at the center of the bubble cloud (e.g. figure11d) and in orange conditions for which there is a local minimum in penetration depth

  model, is the main cause of discrepancy between experiments and model, up to 30% for β in the range [0.15-0.25].

Table 1 :

 1 Examples of correlations for the penetration depth H

	McKeogh & Ervine (1981)	H = 2.6(V 0 D) 0.7	V 0 : velocity at injection
			D : nozzle diameter
	Falvey & Ervine (1987)	H/d i = 3.1Q/Q i	Q : water discharge

Table 3 :

 3 Values of amplitude A and frequency f tested in our experiments.

	Amplitude A (mm)					Frequency f (Hz)
		0 1 2 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 32.5
	2.5	x x x x	x	x	x	x	x	x	x	x	x	x	x
	4	x x x x	x	x	x	x	x	x	x	x	x
	8	x x x x	x	x	x	x	x	x
	12	x x x x	x	x	x	x	x
	16	x x x x	x	x
		h
	H			distance to the surface (pix)
						0		100		200
						Maximum gray gradient

Table 4 :

 4 Impact of mixing layer angle α on model penetration depth H 1 .

Acknowledgements

The authors acknowledge the help of Jean Berard on the experimental set-up. The laboratory LEGI is part of the LabEx Tec 21 (Investissements d'Avenir-Grant Agreement No. ANR-11-LABX-0030).


Wassim Swaileh
email: wassim.swaileh2@univ-rouen.fr

Yann Soullard

Thierry Paquet

A Unified Multilingual Handwriting Recognition System using multigrams sub-lexical units

We address the design of a unified multilingual system for handwriting recognition. Most of multilingual systems rests on specialized models that are trained on a single language and one of them is selected at test time. While some recognition systems are based on a unified optical model, dealing with a unified language model remains a major issue, as traditional language models are generally trained on corpora composed of large word lexicons per language. Here, we bring a solution by considering language models based on sub-lexical units, called multigrams. Dealing with multigrams strongly reduces the lexicon size and thus decreases the language model complexity. This makes possible the design of an end-to-end unified multilingual recognition system where both a single optical model and a single language model are trained on all the languages. We discuss the impact of the language unification on each model and show that our system reaches state-of-the-art methods performance with a strong reduction of the complexity.

Introduction

Offline handwriting recognition is a challenging task due to the high variability of data, as writing styles, the quality of the input image and the lack of contextual information. The recognition is commonly done in two steps. First, an optical character recognition (OCR) model is optimized for recognizing sequences of characters from input images of handwritten texts [START_REF] Plötz | Markov models for offline handwriting recognition: a survey[END_REF]; [START_REF] Singh | Optical character recognition techniques: a survey[END_REF]). Second, a language model is used to model language constraints [START_REF] Mikolov | Recurrent neural network based language model[END_REF]; [START_REF] Croft | Language modeling for information retrieval[END_REF]; [START_REF] Mousa | Sub-word based language modeling of morphologically rich languages for LVCSR[END_REF]). At decoding step, both models are combined to get a prediction. Recent improvements in deep learning techniques [START_REF] Graves | Speech recognition with deep recurrent neural networks[END_REF]; [START_REF] Lecun | Deep learning[END_REF]; [START_REF] Schmidhuber | Deep learning in neural networks: An overview[END_REF]) have enhanced the capacity of optical models. In contrast, only few works have been dedicated to language models for handwriting recognition. They are generally trained to model sequences of words and sequences or characters [START_REF] Messina | Over-generative finite state transducer ngram for out-of-vocabulary word recognition[END_REF]).

Most recognition systems are defined to recognize texts written in a specific language and only few works proposed mul-tilingual recognition systems. A multilingual recognition system allows to process documents written in various languages, without prior knowledge about the language. They are of two types. The first one consists in defining one specific system per language and selecting one of them to get a prediction [START_REF] Miguel | Character-based handwritten text recognition of multilingual documents[END_REF]; [START_REF] Mathew | Multilingual ocr for indic scripts[END_REF]). The second one consists in defining a unified recognition system where at least a part of the system is trained on several languages. Recent works proposed unified OCR models, by merging the character sets of every languages, as in [START_REF] Bluche | Gated convolutional recurrent neural networks for multilingual handwriting recognition[END_REF]; [START_REF] Moysset | The a2ia multi-lingual text recognition system at the second maurdor evaluation[END_REF]; Kozielski et al. (2014b). Some characters are often shared by languages of the same origin, which may be beneficial to improve recognition performance. For instance, Latin languages share at least 21 characters [START_REF] Diringer | The alphabet: a key to the history of mankind[END_REF]) while 14 Arabic and Persian languages share at least 28 characters [START_REF] Märgner | Guide to OCR for Arabic scripts[END_REF]).

Designing a unified language model is less straightforward than designing a unified optical model. This is due to the word lexicons sizes which are often large (to model as much as possible each language). Thus, combining word lexicons from various languages strongly increases the model complexity and may become intractable. One solution is to consider a lexicon of characters but language models based on characters perform often poorly [START_REF] Plötz | Markov models for offline handwriting recognition: a survey[END_REF]).

Designing a unified language model is attractive for taking into account similarities between languages of the same origin that exist at the lexical, morphological or syntactic levels [START_REF] Kalindra | Some lexical, morphological and syntactical similarities and differencies in lithuanian, italian and english languages[END_REF]). In this respect, language models based on sub-lexical units have been recently proposed for handwritten recognition [START_REF] Swaileh | handwriting recognition with multigrams[END_REF], [START_REF] Swaileh | A unified french/english syllabic model for handwriting recognition[END_REF]). Dealing with sub-lexical units has a number of advantages: on the one hand, it allows to significantly reduce the lexicon size and, on the other hand, to improve the recognition rate, as it partly tackles the out-of-vocabulary words problem.

In this paper, we proposes an end-to-end unified multilingual recognition system for handwriting recognition, where both the optical model and the language model combine various languages. Both models rest on state-of-the-art methods. On the one hand, we make use of an optical model based on deep recurrent neural networks. On the other hand, our language model uses sequences of sub-lexical units, called multigrams. Dealing with multigrams allows to reduce the lexicon size from each language and thus to build a unified language model of reasonable size. We evaluate our approach on languages of the same origin (French and English), as the similarities between the languages may impact the capacity of the system. In our experiments, we show that a) combining languages of the same origin in a unified framework allows to strengthen the capacity of the optical model; b) combining the languages does not significantly affect the robustness of the language model. This allows to build a language model estimated on all the training corpus, without the need to separate the languages; c) build a unified multilingual system seems better than dealing with specific-language systems, where one of them must be selected to provide a prediction.

The rest of the paper is organized as follows: Section 2 introduces the related works on multilingual handwriting recognition systems; In section 3, we present the framework of the unified multilingual handwriting recognition system that we propose here; Then, we show and discuss experimental results where the English and French languages are combined (section 4) before concluding.

Related works

Multilingual handwriting recognition systems have some similarities with methods proposed for multilingual speech recognition systems, as in [START_REF] Ghoshal | Multilingual training of deep neural networks[END_REF] and [START_REF] Kumar | Multilingual speech recognition: A unified approach[END_REF]. In general the problem can be solved using two types of approaches: the selective approaches and the unified approaches. In the following we only focus on works dedicated to handwriting recognition but the analysis stands for speech recognition as well.

Selective approaches

The selective approaches are based on specialized recognition systems, i.e. each system is dedicated to one language, and the output of one of them is selected at the time of processing one specific sample [START_REF] Peng | Multilingual ocr research and applications: an overview[END_REF]). There are two kinds of selective approaches. The first one consists in applying specialized recognition systems in parallel to get several transcripts. The transcript with the highest confidence score is then selected. A major issue is to compare the confident scores given by each specialized system, as they span over different scales [START_REF] Lee | A unified network-based approach for online recognition of multi-lingual cursive handwritings[END_REF]).

The second approach is based on a language detection module, as in [START_REF] Mathew | Multilingual ocr for indic scripts[END_REF] and [START_REF] Barlas | Language identification in document images[END_REF]. This module aims at detecting the language of the input script. This allows to select the specialized system corresponding to the language detected. For instance, [START_REF] Miguel | Character-based handwritten text recognition of multilingual documents[END_REF] applied first a probabilistic model for language identification and then a specialized language model based on characters to obtain a prediction. However, these approaches are prone to wrong language detections, which has a direct impact on the system performance [START_REF] Moysset | The a2ia multi-lingual text recognition system at the second maurdor evaluation[END_REF]; Kozielski et al. (2014a)).

Unified approaches

Regarding the unified approaches, there is one or more components that model multiple languages in a unified framework. Three types of unified recognition systems have been presented in the literature: there are systems where 1) only the optical model is trained on multiple languages, i.e. in a unified manner, and then specialized language models are used; 2) the optical model is partially-unified as an encoding part is done on multiple languages and then decoding parts are defined for each language; 3) both the optical model and the language model are trained in a unified framework.

Systems with a unified optical model only

Most multilingual handwriting recognition systems are based on a unified optical model and on a specialized language model. For instance, [START_REF] Kessentini | A multi-stream hmm-based approach for off-line multi-script handwritten word recognition[END_REF] defined a multilingual system based on a unified Hidden Markov Model (HMM) for Arabic and Latin handwriting recognition. Similarly, [START_REF] Lee | A unified network-based approach for online recognition of multi-lingual cursive handwritings[END_REF] proposed an interconnected network based on Hidden Markov Models for Hangul and English languages. Kozielski et al. (2014a) presented a comparative study about the use of traditional HMMs and LSTM neural networks to construct a unified optical model for English and French. Specialized language models are used as the authors argue that the languages have different vocabularies (i.e. words). They evaluated the two systems on the Maurdor dataset and showed that the unified LSTM-based system outperforms the HMM-based system. However, any of the two approaches highlights the feasibility of unifying optical models of multiple languages of the same origin (Latin scripts). Similarly, [START_REF] Moysset | The a2ia multi-lingual text recognition system at the second maurdor evaluation[END_REF] built a unified multilingual system for both printed and handwritten recognition. The system is composed of a unified optical model based on a Recurrent Neural Network (RNN) and of a set of specialized language models (for the English, French and Arabic languages).

Systems based on a partially-unified optical model only

Bluche and Messina (2017) proposed a unified multilingual system where an encoder part based on gated convolutional neural network is trained in a unified manner. Then, specialized Bi-directional Long Short Memory RNNs (BLSTM-RNN) are defined for the decoding step of the optical model. Finally, the authors used specialized hybrid words/characters language models to get the final character string predictions.

2.2.3. Systems where both the optical and language models are unified [START_REF] Ray | Ocr for bilingual documents using language modeling[END_REF] introduced a bilingual recognition system consisting of a unified BLSTM optical model and a unified 3-gram language model. The training corpus contains documents written in two languages where each paragraph is related to a specific language. Four unified optical models are trained on various data representations (binarizations, segmentations), which allows to get 4 recognition hypotheses from a single test sequence. Then, a sliding window process is applied to match every 3 consecutive words from a recognition hypothesis with 3-grams estimated by the language model. A cumulative score computed on the entire hypothesis sequence allows to select the most probable hypothesis. This process can be seen as a verification algorithm, which limits the recognition only to the words belonging to the training corpus.

In a previous work [START_REF] Swaileh | A unified french/english syllabic model for handwriting recognition[END_REF]), we introduced a unified bilingual recognition system based on syllables. The proposed system is composed of a unified optical model based on HMMs and a bilingual n-gram language model of syllables. In this work, French and English linguistic ressources were used to decompose words into syllables. Besides, no Out-Of-Vocabulary (OOV) words were considered by the language models. This simplifies the recognition task but this is a serious limitation. This preliminary work showed that a unified language model based on sub-lexical units (syllables) is feasible and can be envisaged as an alternative to word-based language models. A major issue concerns the linguistic expertise which is required to get word decompositions into syllables, and syllables databases are only available for a few number of languages.

This review shows that very few works have proposed multilingual handwriting recognition systems that are completely unified. Such approaches should have the advantage to train a unique system whatever the languages that are considered. This should avoid to face the issues highlighted in the selective approaches, when the system is composed of specialized models.

Design of a unified multilingual system

Unified recognition system proposal

Similarly to standard handwriting recognition system, our proposed system is composed of two components: 1) an optical model which is trained to recognize a sequence of characters from a sequence of observations; 2) a language model which is trained to model language constraints in sequences of words (Figure 1).

Designing a unified optical model can be done easily by combining the character sets. Combining the character sets may have many advantages when the languages are of the same origin, i.e. where some characters are shared between the languages. On the one hand, the system complexity is reduced as the shared characters are modeled only once. In addition, the number of occurrences of the shared characters is larger in the training set and the optical model can take benefit from this.

Moreover, we also expect the optical model to generalize better as the shared characters come from various languages, so they appear in different contexts in the text.

In contrast to optical models, designing a unified language model is less straightforward. This is especially due to the model complexity that increases. Indeed, if one expects the multilingual language model to perform equally well for every language, then the size of the working lexicon is the sum of each lexicon size. This explains why multilingual unified systems are generally based on a unified optical model only and they use several specific language models, as it is the case in Kozielski et al. (2014a), [START_REF] Moysset | The a2ia multi-lingual text recognition system at the second maurdor evaluation[END_REF] and [START_REF] Bluche | Gated convolutional recurrent neural networks for multilingual handwriting recognition[END_REF].

In this paper, we propose a unified language model based on sub-lexical units, called multigrams. Here, we consider multigrams as sequences of characters of variable length. Some examples of word decomposition into multigrams are illustrated in Table 1 and the proposed system architecture is illustrated in Figure 1. Working with a language model based on multigrams has many advantages compared to language model based on words. First, the lexicon size is strongly reduced, as we only consider sub-parts of words i.e. short sequences of characters. This allows to build a unified language model based on a lexicon of reasonable size. Second, sub-lexical units can generate an open-vocabulary (i.e. words that do not belong to the training lexicon can be generated during decoding if the transitions between their multigrams have non zero probabilities). This allows to reduce the Out-Of-Vocabulary words rate and increases the effective coverage rate. Third, compared to other sub-lexical units such as syllables, multigrams do not obey any linguistic rule but statistical rules. Therefore defining the decomposition of a word into multigrams is a statistical data-driven process which does not require any linguistic expertise, and can be apply to any language. Finally, multigrams can be seen as a good trade-off between words and characters to improve the capacity of language models on specific languages [START_REF] Swaileh | handwriting recognition with multigrams[END_REF]). The many advantages of language models based on multigrams led us to design a multilingual language model based on a unified lexicon of multigrams. To our knowledge, this is the first work that proposes an end-to-end unified multilingual system where the language model is based on sublexical units that can be obtained using a data-driven process for any language and which is applied in an open-vocabulary recognition task.

Recognition system architecture

We present the various components which compose our multilingual recognition system.

Optical model

The recognition of a number of different language scripts within a unified optical model rests on three unification steps. The first step is to use the same observation descriptors for all script images, thus obtaining a unified feature vector. The second step is to define a unified character set by combining all the characters of the languages of interest. In the case of languages of the same origin, the unified character set contains fewer characters than the sum of the character sets of the different languages, owing to the shared characters. The third step is to train a statistical model, that relates to the unified optical model of the recognition system, using the unified character set and the unified descriptor set (feature vectors). The training algorithm (initial parameters, estimation scheme, and training strategy) is the same for any example in the multilingual training corpus. Here, we use a BLSTM-RNN network trained using the Connectionist Temporal Classification approach [START_REF] Graves | Bidirectional lstm networks for improved phoneme classification and recognition[END_REF][START_REF] Graves | Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks[END_REF]). Details on our implementation are given in section 4.2.

Language model

Similar to the traditional n-gram language model of words, the n-gram language model of sub-lexical units (made of multigrams) is estimated with back-off coefficients. The idea is to replace words in the traditional language model by their corresponding multigrams sub-lexical units.

Training a multigram-based n-gram language model involves three main steps (as for training any n-gram statistical model): the training corpus tokenization; the lexicon size and type determination; and the language model parameter estimation with a fixed n-gram order. The tokenisation step replaces any word of the training corpus by its corresponding k-multigram decomposition, where k is the multigram order (see section 3.1). Table 1 gives an example of word decomposition into k-multigrams, for various values of k. The training corpus that contains text from two or more languages can be considered as a multilingual language model training corpus.

The second step is to fix the size of the language model lexicon. Due to the compact size of the lexicon of k-multigrams, all words in the training corpus can be considered. The multigram order k defines the type of k-multigram lexicon. The third step is to define the order n of the n-gram language model. Then, the model parameters can be estimated using the defined lexicon and the tokenized training corpus. High values of n are required if one wants to ensure modeling long term dependencies.

Decoding

During the recognition process, the outputs of the unified optical and language models are combined into a search graph. The search graph can be decoded using dynamic programming such as the well-known Viterbi algorithm. Here the search graph is a weighted finite State Transducer (WFST). It is build by the composition of three sub-transducers representing the tokens (characters), the lexicon and the language model (grammar) respectively. The token transducer (T) represents all possible characters that can be produced by the unified optical model, from the input frames. The lexicon transducer (L) represents the possible k-multigrams of the languages of interest that can be produced from the character set. The language model transducer (G) represents the n-gram language model of k-multigrams which is trained from the multilingual training text dataset.

According to equation 1, the composition of the tokens, lexicon and language model transducers is performed after successive application of the minimization and determination processes on the lexicon and language model transducers:

S = T • min(det(L • G)) (1)
where •, det and min denote composition, determinization and minimization respectively and S denote the combined transducer. Both operators aim at reducing the branching factor of the search graph. Decoding consists in applying the Viterbi algorithm on the combined transducer, so as to find the best (or the n best) sequence(s) of multigrams corresponding to the observation (the input observation feature sequence). Two hyper-parameters are used to guide the decoding process: the language model scale parameter γ and the word insertion penalty parameter β that controls the insertion of frequent short words. These two parameters need to be optimized to find an optimum coupling of the optical model with the multigram language model, because these two models are estimated independently from each other during training.

During decoding, we seek for the sentence Ŵ that maximizes the posterior probability P(W|O) among all possible sentences W. Using Bayes' formula and introducing the two hyperparameters mentioned above, we obtain the formula given in equation 2 which governs the decoding process:

Ŵ = arg max w P(W|O) = arg max w P(O|W)P(W) γ β Length(W) (2)
where O is the observation sequence extracted from the image and P(O|W) is the probability of the observation sequence given the sentence W. P(W) is the prior probability of the sentence deduced from the language model.

Deriving multigram sub-lexical units

In this paper, we propose a unified language model based on sub-lexical units called multigrams. We consider multigrams as sequences of characters of variable length. The interest of using such decomposition have been highlighted in subsection 3.1.

Here, we focus on the description of the data-driven statistical model that allows to learn a set of sub-lexical units from textual data. Textual data can be either the transcriptions from the training corpus, or any other textual resource of the languages considered. The aim is to obtain a set of sub-lexical units that will serve as the main components of the multi-lingual language model in our recognition system. The problem must be considered as unsupervised, because the sequence of multigrams corresponding to a given observation sequence of character (a word or a sentence) is unknown. To address this problem, we define a generative Hidden Semi-Markovian Model (HSMM, [START_REF] Murphy | Hidden semi-markov models (hsmms)[END_REF]; Yu (2010)) where the k hidden states of the model account for the character length of the multigrams. In other words, the HSMM is trained to model the segmentation of a sequence of characters into multigrams in an unsupervised manner. As in [START_REF] Deligne | Language modeling by variable length sequences: Theoretical formulation and evaluation of multigrams[END_REF], it is assumed that the multigrams are independent from one another. This means that multigrams follow a zero-order Markov process. We have a 0/1 duration probability of the hidden states, and there is a bijective relation between states and durations.

Let O 1:T = O 1 . . . O t . . . O T be an observation sequence of length T (T characters), where O t is the observation (character) at time t. One denotes S = {s 1 , . . . , s M } the set of possible states and Q 1:T a state sequence of length T , where each term

Q [t 1 ,t 2 ]
is an element of S starting at time t 1 and ending at time t 2 with a duration d = t 2t 1 + 1. The joint probability of an observation sequence and a state sequence is expressed as follows:

P(O, Q) = l P(O [t-d l +1,t] | Q [t-d l +1,t] )P(Q [t-d l +1,t] ) = l P(O [t-d l +1,t] |d l ) (3)
Equation 3 presents a way to define a Discrete Hidden Semi-Markov Model of zero order to model multigrams of variable length. The HSMM is trained using the well-known Baum-Welch algorithm, based on the Forward-Backward algorithm. Given a maximal length d max for multigrams, any possible sequence of characters for which the length does not exceed d max is considered in the model.

Once the HSMM has been trained, the decoding step consists in assigning a sequence of multigrams to any sequence of characters of the training corpus. In other words, the goal is to find the most probable sequence of multigrams Q * , given any observation sequence O 1:T of the training corpus, which is defined as follow:

Q * = arg max

q 1:T P(O 1:T , Q 1:T = q 1:T ) (4)
Equation 4 can be solved using the Viterbi algorithm. The optimal path is obtained using a backward pass defined as follow:

d * i = arg max d arg max d δ t-d (d ) b(O [t-d+1:t] ) 1/d (5) t * i = t * i-1 -d * i (6)
where δ t (d)

∆ = max q 1:t-d P(O 1:t , Q [t-d+1
,t] = d, q 1:t-d ) and t * 0 = T . The exponent term 1/d is a penalty term defined to favor longer multigrams at the expense of shorter ones. Indeed, the Viterbi algorithm tends to produce short multigrams because the emission term b(O [t-d+1:t] ) is related to the occurrences of multigrams and the shorter multigrams are often more frequent than the longer ones.

We train one HSMM per language. Each one is optimized to produce the most frequent multigrams which appear in the training texts of a given language. Finally, language specific multigram lexicons are combined. After the tokenization step of the whole training corpus into multigrams, the unified language model is trained as described in section 3.2.2. At test time, the unified language model is applied on a text whatever its language, using the decoding step described in section 3.2.3.

Multilingual system evaluation: English and French case study

Datasets and lexicon units properties

We apply our multilingual system on the English IAM (EN) and the French RIMES (FR) handwriting databases. The RIMES database [START_REF] Gorski | A2ia check reader: A family of bank check recognition systems[END_REF]) contains handwritten mails of different writers. The IAM database [START_REF] Marti | The iam-database: an english sentence database for offline handwriting recognition[END_REF]) is inspired from the LOB corpus [START_REF] Johansson | The lob corpus of british english texts: presentation and comments[END_REF] and it is composed of texts from different writers. These databases are divided into three parts: the training, the validation and the test datasets. Our unified system is trained on the combination of the FR and EN training datasets. The hyperparameters γ and β used for decoding (see section 3.2.3) are optimized using the FR+EN validation dataset. The system is evaluated on the FR and EN specialized test datasets independently. Table 2 shows the number of text line images per dataset. The unified optical model is trained on the joint FR+EN training dataset. The FR dataset is composed of 100 character classes while the EN dataset contains only 79 character classes. 77 of them (including white space) are shared between the two datasets so that the unified dataset sets contains 102 character classes only. The entire EN character set is included in the FR training dataset, except the two characters "#" and "&". Most of the additional character classes in the FR dataset are accented characters.

For training the multilingual language model, we used the combined training sets from the French and English languages. By nature, this dataset brings a relatively small vocabulary which we refer to as "small FR+EN". In addition, we also trained the language model on a very large dataset of text samples made of both English and French texts which we later refer to as "large FR+EN". This large dataset was collected as follows. Regarding the FR dataset, we collected 52, 930 paragraphs from French Wikipedia pages. This Wikipedia and the RIMES training datasets are joint and the 29.1k most frequent words are used for training the language model. Regarding the EN dataset, we combined the LOB (excluding IAM validation and test examples), Brown and Wellington datasets to form the large EN lexicon used for training the language model. Figure 2 gives the lexicon sizes when considering lexicons of k-multigrams or words. For example the small FR+EN lexicon of words contains 12.5k words, while the corresponding 2-multigram lexicon is made of 1.3k elements only. In contrast the multilingual "large FR+EN" lexicon contains 117k elements while there is only 3.1k 2-multigrams to describe the same dataset. This highlights a significant reduction of the lexicon size when using k-multigrams sub-lexical units instead of words.

System configurations and Evaluation protocol

The observations given in input of our system are gray-scales images of text-lines. The input images are normalized to a fixed height of 100 pixels, preserving the aspect ratio. The gray scale is normalized to zero mean and unity variance (standardization). The unified optical model is a Recurrent Neural Network (RNN) composed of 4 Bi-directional Long Short Term Memory (BLSTM) layers with 200 cells in each direction. Our network is trained using the Connectionist Temporal Classification (CTC) approach [START_REF] Graves | Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks[END_REF]). The model is implemented using the EESEN toolkit [START_REF] Miao | Eesen: End-to-end speech recognition using deep rnn models and wfst-based decoding[END_REF]) that has been first introduced for speech recognition. We used the curriculum learning algorithm for training the observation sequences from the shorter sequence to the longer one. Training starts with a learning rate of 10 -5 which decreases during training. Training ends after 108 training epochs using an early stopping criterion.

We used 9-gram language models of k-multigrams or words, and 10-gram language models of characters in any of our experiments. We explored k-multigrams models for 2 ≤ k ≤ 5. To ensure that long distance dependencies are estimated in the model, all text lines are concatenated in order to form a single text line. Every language model is estimated using the MIT language modeling toolkit from [START_REF] Hsu | Language modeling for limited-data domains[END_REF], where the backoff coefficients are estimated using the modified Kneser-Ney smoothing method [START_REF] Kneser | Improved backing-off for m-gram language modeling[END_REF]).

Viterbi's two-pass decoding algorithm is applied at the paragraph level to track contextual dependencies as much as possible. The beam search parameter has been fixed to 0.8 in all the experimentations while language model scale and word insertion penalty are optimized for each scenario (see below).

During test, we are interested in quantifying the contribution of each stage (i.e. optical model and language model) to the performance of the unified system. In this respect, we conducted four experiments (scenarios). The first one (UU) consists in having a whole unified system (both the optical model and the language model are unified). The second experiment (US) consists in combining a unified optical model with a specialized language model. The third experiment (SU) combines a specialized optical model with a unified language model. Finally, the last one (SS) is composed of specialized models (both the optical model and the language model). In the following section we report and analyze the performance of these four configuration of the system.

Recognition results

We now evaluate our approach on the small and large EN and FR vocabularies. We first present observations we obtained for the four experiments described above in sections 4.3.1 and 4.3.2 and then discussed the results in section 4.4. Finally, we compare our system with state-of-the-art methods.

Evaluation with small EN and FR lexicons

Figure 3 presents the Word Error Rate (WER) obtained after training the language model on the small FR and EN datasets respectively. These figures present the performance of the systems for 6 differents types of language models: words, characters, and k-multigrams (with 2 ≤ k ≤ 5).

One can first notice that the unified optical model (US and UU) significantly improves the specialized framework whatever the test set (FR and EN) and the configuration of the system (specialized or unified). The optical model takes advantage of language similarities to be more robust and more efficient.

Besides, multigram-based language models often outperform traditional language models based on words or characters. Especially, the 2-multigrams language models are always better than the traditional ones. This confirms our hypothesis that multigrams are a good trade-off between words and characters for a language modeling task. Compared to the specialized frameworks, the unified scheme generally provides similar results except for the FR dataset where a slight improvement is observed when the optical model is unified.

Finally one can notice that the performance obtained on the IAM dataset is rather low for any of the language model type (word, character or k-multigrams), and system unification type (SS, SU, US, UU), when a small vocabulary is used for training the language models. This is a particular difficulty encountered on the IAM dataset because of the low lexicon coverage rate of the training dataset lexicon on the test dataset. This is the reason why most studies have come to use additional linguistic resources to get better performance. To the next, we analyze tour systems using a large vocabulary setting.

Evaluation with large EN and FR lexicons

We now report experimentation results on the large EN and FR datasets described in section 4.1. Figure 4 presents the Word Error Rate (WER) obtained on the FR and EN test datasets respectively. Similarly to small lexicons, we observe that unifying the optical models (US and UU) improves the recognition rate. Unifying the language models does not impact the recognition performance, even if the unified lexicons are highly imbalanced (the unified word lexicon has 116k words with only 29k from the French language). Moreover, dealing with multigrams language models generally reduces the WER, compared to systems based on traditional language models of words or characters.

Discussion

We have seen that unifying English and French optical models allows to significantly improve the recognition performance of the system. This may be explained by two facts: on the one hand, there is a lot of shared characters between the two languages, which increases the number of training samples for theses character classes; on the other hand, the shared characters appear in a different context (i.e. into different words), which gives more variability in the data as well. The model can benefit of it for increasing its generalization capacities. The unified optical model achieves a Character Error Rate (CER) of 10.0% alone (i.e. without language model) on the FR test set, while the specialized model has a CER of 12.3%. Similarly, we obtain a CER of 15.3% on the EN test set with the unified optical model while the CER of the specialized model is 18.9%. Compared to the specialized language model, the contribution of a unified language model appears rather moderate, as results are generally equivalent whatever the lexicon size. In fact, there is no reason to believe that some improvements should be gain by combining the English and French languages. Indeed, the language modeling task should become more complex due to some language similarities. The key point is that unifying both languages into the language model does not affect the system performance, which allows to design a system with only one language model whatever the languages.

There is only one case where unifying the language model improves the performance of the specialized system. This is for the UU setting on the FR test dataset with a small lexicon. In this case, the system benefits from both a better CER with a unified optical model and a better coverage of the unified language model. This improvement may be explained by the fact that the OOV rate, computed at a word-level, is reduced in the unified cases (compared to the specialized ones) and that the effective coverage rate, computed at a sub-lexical-level, is high. For instance, the OOV rate on the FR set for the 3-multigrams is of 1.4% in the specialized case and of 0.7% in the unified one, while the effective coverage rate for the unified dataset is equal to 98.5%. In contrast, the OOV rates on the EN set are very small (0.06% and 0.03% for the specialized and unified case respectively) and the effective coverage rate stay low enough even if the lexicon has been unified (89.9%). Besides, when lexicons are large, unifying the languages does not significantly reduce the OOV rate, which does not impact the recognition performance. For example, the unification of languages for the 3multigrams only reduce the OOV rate from 0.5% to 0.4% on the FR dataset and from 2.2% to 1.6% on the EN dataset.

Moreover, languages models based on multigrams often outperform traditional language models based on words or characters and there is always a system based on multigrams which is better than systems based on traditional models. Another advantage of dealing with multigrams is related to the lexicon size, which is highly reduced compared to word lexicons. For instance, there is 116k words in the large unified EN + FR lexicon while there is 71.6k 5-multigrams, 16.9k 3-multigrams and only 3.1k 2-multigrams.

Finally, we analyze the system complexity according to the search graph size (number of states and transitions in the FST automaton S defined in equation 1), the search graph volume on disk and the decoding time. Besides, the memory and the processing time during the decoding process depend on the search graph size [START_REF] Mohri | Weighted finite-state transducers in speech recognition[END_REF]). Note that complexities are computed from the unified FR+EN lexicon and that the character language model is a 9-gram model (in contrast to previous results based on a 10-gram model) to provide comparable complexities. As illustrated in Table 3, the number of states and can perform sometimes poorly (as shown on the EN dataset in Figure 4). Table 5 shows the decoding time on the FR valida- tion set, which relates to a measure computed on 1.7 millions of frames. The decoding time is reduced using multigram language models compared to traditional language models based on words. For instance, using a 5-multigram language model reduces the time by 18% and by 24% using 2-multigrams on the small lexicon while there is a reduction of 22% using 2multigrams on the large lexicon. To conclude, we compare our multilingual recognition system with state-of-the art systems (Table 6). In our system, the optical model has a simpler architecture compared to stateof-the-art systems while these systems rest on language models based on words or hybrid models of words and characters. Multilingual systems based on multigrams reach performance closed to most of the state-of-the-art systems, while the system complexity is reduced. The specialized system presented in Table 6 refer to a selective approach where the right specialized model is always selected (an ideal case). We show that, whatever the test set (FR or EN), our unified system makes profit of the combination of the languages to outperform the selective approach.

Conclusion

We presented an end-to-end unified multilingual system for handwriting recognition where both the optical model and the language model are trained on datasets composed of examples from several languages. Our proposal allows, on the one hand, to optimize a unique system, whatever the languages that are in training and, on the other hand, to avoid the use of a decision process for selecting one specialized system trained on a specific language.

Our unified optical model is optimized to recognize a unified character set. In case of languages of the same origin, unifying the character set reduces the system complexity and increases the number of training examples per character classes that are shared between the languages. While traditional language models are based on words, which can become intractable in case of unified lexicons, we proposed to build a language model based on sub-lexical units, called multigrams. Dealing with multigrams has many advantages: the multigrams are obtained using a data-driven process without the need of linguistic expertise; it reduces the model complexity compared to words; finally, it allows a better modeling of long dependencies than with characters.

Our experiments on English and French languages with small or large lexicons, highlighted that the optical model benefits from the language unification and provides significant improvements compared to specialized systems. A major contribution is to show that unifying languages that have some similarities does not affect the language models which provide similar results than the specialized language models. In addition, dealing with multigrams allows to improve the traditional language models based on words or characters. Finally, our system reaches state-of-the art performance with a unique and less complex system.

Fig. 1 .

 1 Fig.1. Our unified multilingual recognition system composed of a unified optical model dealing with text images of any language and a multilingual language model based on a unified multigram lexicon. At test time, the decoding process consists in combining both models to get a transcription.

Fig. 2 .

 2 Fig. 2. Sizes of the small and large lexicons according to the lexical (words) or sub-lexical (multigrams) decomposition, on the French (FR), English (EN) and unified (FR+EN) training sets.

Fig. 3 .

 3 Fig.3. WER (%) on the FR (left) and EN (right) test datasets for the four experimental setups where language models are based on small vocabularies.

Fig. 4 .

 4 Fig. 4. WER (%) on the FR (left) and EN (right) test datasets for the four experimental setups where language models are based on large vocabularies.

Table 1 .

 1 Illustration

	FR Merci	Me r ci	Mer ci	Merc i	Merci
	EN darling	d ar li ng	dar l ing	dar ling	dar ling

of a French (FR) and English (EN) word decomposition into k-multigrams with 2 k 5 words 2-multigram 3-multigram 4-multigram 5-multigram

Table 2 .

 2 Numbers of text line images in the FR and EN datasets used for training, validation and test.

		Number of text line images
	Databases		per dataset
		Training Validation Test
	FR	9947	1333	778
	EN	6482	976	2915
	FR+EN	16429	2309	--

Table 3 .

 3 Size of decoding search graph (composed automaton S defined in equation 1) for different language model with a small lexicon.

	Language models vs.	number of states	number of arcs	Volume on
	complexity indicators	(×10 6 )	(×10 6 )	disk (MB)
	Words	1.37	3.46	71.9
	m5gram	1.25	3.15	65.5
	m4gram	1.20	3.04	63.1
	m3gram	1.11	2.81	58.2
	m2gram	0.93	2.40	49.5
	characters	0.54	1.48	30.2
	transitions is highly reduced using multigram language models
	compared to the word-based language models. For instance,
	the 2-multigrams model reduces the sum of states and transi-
	tions of the search graph by 31%. The volume on the disk is
	also divided by 1.45. Similar comments can be made on a large

lexicon with more significant results (Table

4

). Dealing with 2multigrams allows to reduce the sum of states and transitions of the search graph by 63% compared to word model. The volume on disk is also reduced by a factor of 2.7. Whatever the lexicon size, the character-based model is lighter than the other models but it never produces the best results in our experiments and it

Table 4 .

 4 Size of decoding search graph (composed automaton S defined in equation 1) for different language model with a large lexicon.

	Language models vs.	number of states	number of arcs	Volume on
	complexity indicators	(×10 6 )	(×10 6 )	disk (GB)
	Words	178	447	9.3
	m5gram	126	317	6.6
	m4gram	108	269	5.6
	m3gram	72	186	4.1
	m2gram	65	166	3.4
	characters	20	50	1

Table 5 .

 5 Decoding time on the RIMES validation set.

	Language models (LMs)	Decoding time with small LMs (Minutes:Seconds)	Decoding time with large LMs (Minutes:Seconds)
	Words	04:28	07:59
	m5gram	03:39	07:32
	m4gram	03:38	07:55
	m3gram	03:34	07:14
	m2gram	03:23	06:14
	characters	03:14	08:18

Table 6 .

 6 Performance comparison of the proposed system with results reported by other studies on the RIMES & IAM test datasets.

	System	WER (%) on RIMES WER (%) on IAM
	Our unified system	9.8 (2-multigrams)	11.2 (5-multigrams)
	Our specialized system	10.8 (2-multigrams)	13.5 (5-multigrams)
	Voigtlaender et al. (2016)	9.6	9.3
	Bluche (2015)	11.8	11.9
	Bluche and Messina (2017) 7.9	10.5