F Blanc

F Peters

E Lemaire

Local transient rheological behavior of concentrated suspensions

]. The minimum seems to be

associated with the viscosity of an isotropic suspension, or at least of a suspension whose particles do not interact through non-hydrodynamic forces, while the plateau value would correspond to the viscosity of a suspension structured by the shear where the nonhydrodynamic forces play a crucial role.

I. INTRODUCTION

Concentrated suspensions are very common in several engineering fields such as civil engineering, food or pharmaceutical industry or geophysical situations such as debris flows, sediment transport, and submarine avalanches. In order to understand their flowing behavior, it is desirable to know their response to imposed forces and motions at their boundary. A broad variety of complex rheological behaviors such as shear thinning or shear thickening, yield stress, normal stress differences or shear banding are exhibited. This diversity issues from the diversity of the interactions (Brownian, Van der Waals, steric, electrostatic…) and of the particles properties (size, shape, polydispersity…) that are possibly involved in the flow behavior. Nevertheless even a suspension whose physicochemical characteristics are very simple (non-Brownian spherical particles, negligible colloidal interactions, Newtonian suspending liquid) can exhibit a complex rheological behavior. For instance, non-Brownian concentrated suspensions are usually shear thinning [START_REF] Stickel | Fluid mechanics and rheology of dense suspensions[END_REF]] and, if the particle and fluid density are not carefully matched, a yield stress has been shown to develop [START_REF] Fall | Yield Stress and Shear Banding in Granular Suspensions[END_REF]]. The transient response of such suspensions is also of interest as reported by Gadala-Maria and [START_REF] Gadala-Maria | Shear-induced structure in a concentrated suspension of solid spheres[END_REF], [START_REF] Kolli | Transient normal stress response in a concentrated suspension of spherical particles[END_REF] and [START_REF] Narumi | Transient response of concentrated suspensions after shear reversal[END_REF]. As emphasized by [START_REF] Stickel | Fluid mechanics and rheology of dense suspensions[END_REF] and [START_REF] Morris | A review of microstructure in concentrated suspensions and its implications for rheology and bulk flow[END_REF] in their review articles on the rheology of dense suspensions, such a complexity of the rheological behavior is most likely due to the formation of a microstructure induced by the shear flow together with the non-hydrodynamic forces and the key problem is to understand the relationship between the macroscopic or bulk properties of the medium and its microscopic structure.

The first direct experimental evidence of the shear-induced anisotropic microstructure has been provided by [START_REF] Parsi | Fore-and-aft asymmetry in a concentrated suspension of solid spheres[END_REF] who measured the pair distribution function for a suspension whose particle volume fraction was 0.4. They note a maximum of the pair distribution function in a direction which is roughly that of the compression axis.

Furthermore, they show that when the direction of the shear is reversed, the particles rearrange into the mirror image of the structure. This direct observation of the microstructure confirms the hypothesis advanced by Gadala-Maria and [START_REF] Gadala-Maria | Shear-induced structure in a concentrated suspension of solid spheres[END_REF], a few years before, to explain the transient shear stress response of a suspension when it undergoes a shear reversal. The suspension viscosity was shown to depend only on the strain after the shear reversal. In 2002, these results were supplemented by the measurement of the transient normal stresses [START_REF] Kolli | Transient normal stress response in a concentrated suspension of spherical particles[END_REF], [START_REF] Narumi | Transient response of concentrated suspensions after shear reversal[END_REF]] which present the same kind of behavior, i.e. a collapse of the recovery normal stresses if plotted against the strain, whatever the shear rate step magnitude.

The characterization of the microstructure has also motivated a lot of numerical studies based on various numerical techniques such as Stokesian Dynamics [START_REF] Singh | Normal stresses and microstructure in bounded sheared suspensions via Stokesian dynamics simulations[END_REF], [START_REF] Sierou | Rheology and microstructure in concentrated noncolloidal suspensions[END_REF], [START_REF] Drazer | Microstructure and velocity fluctuations in sheared suspensions[END_REF], Bricker and Butler (2007)] or Force Coupling Method [START_REF] Yeo | Dynamics of concentrated suspensions of non-colloidal suspensions[END_REF], [START_REF] Abbas | Fully coupled simulations of non-colloidal monodisperse sheared suspensions[END_REF]] which all conclude to the formation of an anisotropic shear-induced microstructure.

Achieving accurate rheological measurements with a non colloidal suspension is quite challenging because perturbative effects such as wall slip [START_REF] Jana | Apparent wall slip velocity coefficients in concentrated suspensions of non colloidal particles[END_REF]], shear localization [START_REF] Huang | Flow of Wet Granular Materials[END_REF]] or particle migration [Leighton andAcrivos (1987), Chow et al. (1994)] can arise during the experiment. The local rheological measurements have been shown to be very powerful to measure the true rheological response of a dense suspension. Furthermore, they measured the particle concentration profiles and showed that for such high concentrated suspensions, the migration is almost instantaneous. It should be stressed that all their measurements have been performed after a period of pre-shear.

On the opposite, the present paper is focused on the transient shear rate response of rather concentrated suspensions (0.3<φ<0.5) of non-Brownian spheres (diameter 31µm) when a shear stress is suddenly applied or reversed. Suspensions of various concentrations are sheared between two concentric cylinders so that the shear stress field is controlled in the gap.

The suspensions are designed to be almost transparent allowing to determine the velocity field in the gap using PIV measurements. The local transient response of the suspension is then deduced from the local shear stress and shear rate. The main characteristics of the suspensions are presented in section II. The third section is devoted to the description of the experimental procedure. The results are displayed in section IV where two transient regimes with two different characteristic times are studied. The long time variation of the velocity profile coming from the shear induced particle migration is presented in Sec. IV.A while the rapid changes happening after the shear reversal are described in Sec. IV.B. All these results are discussed in Section V.

II. SUSPENSION CHARACTERISTICS

The suspension is made of monodisperse spherical PMMA particles (Microbeads CA30), 31µm in diameter, dispersed in a Newtonian fluid that has been specially designed for our application (Cargille Laboratories, Immersion Liquid Code 11295101160). The fluid has both the same density (d=1.18) and the same refractive index (n=1.49) as the PMMA particles in order to perform the PIV measurements detailed below. To improve the index matching, the temperature is controlled and the best matching is obtained for T=34°C. At this temperature, the suspending fluid has been found to be fully Newtonian and its viscosity has been measured to be η 0 =1.02 Pa.s. [START_REF] Blanc | Particle Image Velocimetry in concentrated suspensions: application to local rheometry[END_REF]]. The particle volume fraction varies from 0.3 to 0.5. The shear viscosity measurements have been performed for shear rates, γ ɺ , between 10 -2 and 1 s -1 . Therefore, given the above characteristics of the suspension, the Peclet number that measures the ratio of convection to thermal diffusion is very large: and the Reynolds number based on the particle size is very small: 2 9 7 p 0 a Re 10 10

- - ργ = - η ɺ ∼
where ρ is the density of the suspending fluid and a, the radius of the suspended particles.

Thus, our measurements were carried out in the creeping flow regime with negligible Brownian motion.

The viscosity of the suspensions has been studied either in a narrow-or in a wide-gap

Couette geometry mounted on a rheometer Mars II (Haake, Thermofisher). The possible variation of the viscosity of the suspension with the shear rate was first studied in a narrow-gap Couette cell (inner radius, 10 mm, outer radius, 10.85 mm). After the suspension had been poured in the cup and degassed, the shear stress was increased step by step. At each step, the steady value of the shear rate was recorded and the value of the viscosity deduced. The measurements have then been performed upon increasing or decreasing the stress, and gave the same results, suggesting that no migration had occurred. Figure 1 shows the flow curve obtained for a φ=0.47 suspension. The data are well fitted by a power law: 0.88 40 σ = γ ɺ where σ and γ ɺ are measured in Pa and s -1 , respectively. Such a slight shear thinning behavior is often observed in concentrated non-Brownian suspensions [START_REF] Gadala-Maria | Shear-induced structure in a concentrated suspension of solid spheres[END_REF], [START_REF] Zarraga | The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids[END_REF]]. Acrivos and coworkers [START_REF] Acrivos | On the measurement of the relative viscosity of suspensions[END_REF]] have shown that such a shear thinning could occur when the host liquid and the particles do not have the same density, due to the viscous resuspension of the particles. Nevertheless we never noticed any evidence for sedimentation in our experiments and the origin of the low shear thinning observed in our experiments remains unclear.

As it is well known [START_REF] Jana | Apparent wall slip velocity coefficients in concentrated suspensions of non colloidal particles[END_REF]] and shown again in Appendix B, the viscosity measurements are difficult to perform in narrow-gap Couette cell because of the wall slip that significantly changes the value of the apparent viscosity. For that reason and since the observed shear thinning is rather weak (when the shear stress is multiplied by two, the viscosity decreases by only 9%), we decided to perform the further viscosity measurements in the wide-gap Couette cell that will be used for the PIV experiments even though the shear stress is not constant in the gap.

The inner and outer cylinder radii are R in =14 mm and R out =24 mm respectively, their height is 60 mm. The rotor basis is hollowed out and is positioned one millimeter above the cup (Fig. 2). We have chosen to set the rotor close to the cup in order to limit shear-induced migration from or toward the bottom [START_REF] Leighton | The shear-induced migration of particles in concentrated suspensions[END_REF]]. Nevertheless, since the rotor is close to the cup and its immersed height is not very large compared to the gap between the coaxial cylinders, the usual relation between the torque and the shear stress does not hold and a correction has to be introduced:

2 bot 2 r L. (r) Γ = π σ + Γ (1)
Upon varying the height L of the sheared liquid, we measure Γ bot , the torque that results from the presence of the bottom [START_REF] Blanc | Particle Image Velocimetry in concentrated suspensions: application to local rheometry[END_REF]] and we deduce the shear stress field in the gap between the coaxial cylinders:

bot cor 2 2 (r) 2 Lr 2 Lr Γ -Γ Γ σ = = π π
(2)

Neglecting the slight shear thinning behavior of the suspension, the shear rate in the gap is obtained from the spin rate of the inner cylinder, Ω:

2 2 in out 2 2 2 out in R R 1 (r) 2 R R r γ = Ω - ɺ (3)
And the viscosity is given by the ratio of the shear stress to the shear rate:

2 2 cor in out app 2 2 in out R R 4 L R R - Γ η = πΩ (4)
The variation of the apparent viscosity as a function of the particle volume fraction is shown on Fig. 3. These measurements have been performed for an average shear rate, ( )

in out (R ) (R ) / 2 γ = γ + γ ɺ ɺ ɺ , of about 0.02s -1 .
The data represented in Fig. 3 are fit to a Krieger-Dougherty law (solid line) with a maximum packing fraction φ*=0.536 and an exponent n=2:

η = η 0 1 1 - φ φ *       n (5)
This value of φ* is very small compared to the values usually found in the literature [START_REF] Ovarlez | Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging[END_REF], [START_REF] Stickel | Fluid mechanics and rheology of dense suspensions[END_REF]] that are around 0.60-0.63. It should be mentioned that [START_REF] Zarraga | The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids[END_REF] have also found a rather low value for the packing fraction (φ*=0.58). As discussed in Appendix A, we attribute this low value to polymeric forces between particles. Indeed, we noticed a viscoelastic behavior of the suspensions which is not expected for a suspension of truly hard, non-Brownian spherical particles.

III. EXPERIMENTAL PROCEDURE

PIV is a now widespread non-invasive technique that allows for flow profile measurements (see [START_REF] Adrian | Twenty years of particle image velocimetry[END_REF]] for a review). However, even if it has been already used in the field of concentrated suspensions [START_REF] Lenoble | The flow of a very concentrated slurry in a parallel-plate device: Influence of gravity[END_REF]], [START_REF] Wiederseiner | Rheophysics of highly concentrated coarse-particle suspensions in a wide-gap Couette rheometer[END_REF]],

PIV is still a challenging technique in such heterogeneous media. The experimental facility and methods are described in detail elsewhere [START_REF] Blanc | Particle Image Velocimetry in concentrated suspensions: application to local rheometry[END_REF]]. The present section summarizes the main information.

A transparent wide-gap Couette cell is mounted on a controlled-stress rheometer (Haake Mars II, Thermo Scientific). The measurement of the rotor spin rate provides macroscopic rheometric data while the velocity profile in the gap between the cylinders is deduced from PIV analysis. A schematic drawing of the set-up is shown on Fig. 2. The inner rotating cylinder (28 mm in diameter, 60 mm in height) and the outer stationary cup (48 mm in diameter, 60 mm in height) are made of PMMA, in order that their refractive index is approximately matched with the index of the suspension. The front face has been polished in the form of a rectangular window that lets the horizontal laser sheet (Lasiris TEC Laser 635 nm, 35mW, Stockeryale) enter the gap between the cylinders. The inner cylindrical walls have been roughened to minimize the suspension slip. The whole apparatus is placed in a thermostated box, and the temperature is set to T=34°C.

A small quantity of the particles, that amounts to 0.25% of the whole suspension in volume, is colored with a fluorescent dye, the Nile Blue A (excitation peak at 635 nm, broad emission peak at 650 nm). Those particles serve as flow tracers. A CCD USB camera (Pixelink PLB 741 U, 1280x1024 pix 2 ) is placed under the cup and records sequential images of the illuminated part of the suspension through the polished bottom of the cup. The laser sheet and the camera are trigged by an external oscillator. In order to enhance the contrast of the images, the reflected and scattered light is filtered out by an optical high pass filter (λ HP =650nm) placed in front of the camera, so that only the fluorescing light is detected. The laser sheet illuminates the horizontal plane 14 mm above the bottom of the cup, where it has been shown [START_REF] Blanc | Particle Image Velocimetry in concentrated suspensions: application to local rheometry[END_REF]] that the influence of the bottom on the velocity profile could be neglected. Fig. 4 displays a typical image of the illuminated plane obtained with a φ=44.4%

suspension.

The image processing is performed using an open source software DPIV Soft available on the web (https://www.irphe.fr/~meunier/). Each image is divided into overlapping subsets named correlation windows (64x64 pixels). The cross correlation of the corresponding windows from two successive images yields the mean velocity of the particles in the window.

The same procedure performed on all windows gives the velocity field (60x30 2D vectors) in the illuminated plane. The concentration of colored particles has been chosen such that the number of particles in a window is sufficient (approximately 5), while keeping acceptable light absorption. Given the size of the image plane (approximately 1cm x 1.5cm), the spatial resolution is around 200 µm. The camera acquires frames at a maximum rate of 10Hz, resulting in a temporal resolution of 100ms.

The radial and azimuthal components of the velocity are then computed, and averaged over the azimuthal angle θ. The mean radial velocity v r (r,t) is always very close to zero (v r /v θ ∼O(10 -2 )). The local shear rate is extracted from the mean azimuthal velocity v θ (r,t): the best fit of the function f

(r, t) = a(t) r + b(t) r + c(t)
r 2 to the velocity data is sought and the shear rate is deduced from the well-known expression:

ɺ γ = r ∂ ∂r f(r, t) r       (6)
As explained previously, the local shear stress is deduced from the torque on the rotating cylinder (eq.( 2)) that is corrected for the influence of the bottom of the cup [START_REF] Blanc | Particle Image Velocimetry in concentrated suspensions: application to local rheometry[END_REF]] and the viscosity profile is computed:

( , ) ( , ) ( , )
σ η = γ ɺ r t r t r t (7)
The statistical error in the viscosity measurement has been estimated lower than 4% [START_REF] Blanc | Particle Image Velocimetry in concentrated suspensions: application to local rheometry[END_REF]] and the uncertainty in the determination of the bulk volume fraction is less than 0.1%.

IV. TRANSIENT RESPONSE

A. long time migration

The main goal of this paper is the measurement of the local transient response of a concentrated suspension to a shear reversal. One parameter plays a major role in this mechanical response, namely the particle volume fraction. As a consequence, we need to efficiently control this parameter. However, it is well known that the particles in a sheared suspension tend to migrate toward regions where the shear rate is lower [START_REF] Phillips | A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration[END_REF]]. This migration is all the more rapid as the particle concentration is higher and can have significant effects after even a few revolutions of the bob. For example, the velocity profile change obtained for a continuously sheared suspension whose mean volume fraction is φ mean =0.444 is displayed in Fig. 5. This curve has been obtained upon controlling the rotor spin rate at a value 1 rpm which corresponds to an average shear rate of 2s -1 . Figure 5 suggests that wall slip is present near the inner cylinder but, due to the finite size of the correlation windows, we are not able to measure accurately the velocity close to the boundaries. So, we cannot conclude anything about the wall slip velocity. After ten revolutions, the velocity profile has changed significantly due to outward migration. From eq. ( 2), ( 6) and ( 7), we can compute the time evolution of the viscosity profile that is shown in Fig. 6. We note that around a special value of r=r c , the viscosity remains approximately constant. We have chosen r c /R out =0.77, which is the position where the viscosity variation is the lowest at the beginning of the migration, even though in Fig. 6, where very long time variation are displayed, a higher value seems more appropriate. Since the viscosity mostly depends on the particle volume fraction through a monotonic increasing function, the volume fraction around the position r=r c should also be constant. The existence of this radial zone where the particle concentration varies slightly throughout the migration process was observed experimentally by [START_REF] Phillips | A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration[END_REF] and is predicted either by the suspension balance model [START_REF] Morris | Curvilinear flows of noncolloïdal suspensions: The role of normal stresses[END_REF]] or by the diffusive flux model [START_REF] Phillips | A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration[END_REF]]. We have observed that the value of r c depends only very weakly on the mean volume fraction. As a consequence, in the following, all local measurements are performed at this particular radial position in the gap, giving the properties of the suspension at the volume fraction φ mean .

B. Transient response after shear reversal

Besides this long time evolution of the velocity profiles, we have observed a fast transient response of the suspension when the shear is reversed. Figure 7 shows the evolution of the normalized velocity profiles in a suspension with φ=44.4% when the direction of the torque applied on the inner cylinder is reversed. The measurement has been performed right after that the suspension had been mixed and poured in the Couette cell, so that an approximately homogeneous spatial distribution of the particles was expected at the beginning of the experiment. A negative torque of 250 µNm is applied to the inner cylinder until it has rotated quarter turn which corresponds to a mean deformation of 3.5 in the suspension. At time zero, the magnitude of the torque passes from -250 µNm to 250 µNm which, according to expression (2) corresponds to a shear stress of 2 Pa at the bob. In Fig. 7, the dashed line represents the velocity profile expected for a Newtonian fluid. The solid lines are the measured profiles at various times, t, after the torque reversal. The very first ones are nearly Newtonian, the next profiles deviate from this Newtonian curve before returning to it for longer times. From each curve, following the procedure described in section III, we can deduce the viscosity profiles (η(r,t)) as well as the total strain (γ(r,t)):

t 0 (r, t) (r, t ')dt ' γ = γ ∫ ɺ (8)
In particular, we can measure the viscosity and the strain at the special point r=r c

where we are assured that the particle volume fraction has almost not changed.

If we plot the viscosity η(r c ,t) as a function of the strain, γ(r c ,t), we obtain the curve that is shown on Fig. 8. Just after the shear reversal, the viscosity undergoes a step-like decrease. When the strain increases further, the viscosity first decreases, passes through a minimum and increases again to reach a plateau value.

First, we can note that such a strain-dependent viscosity is consistent with the time evolution of the normalised velocity profile shown in Fig. 7 since, in a wide-gap Couette cell, the shear rate is larger close to the bob and so is the strain. Thus as long as the plateau viscosity has not been reached everywhere in the gap, the viscosity is not uniform and the velocity profile is not Newtonian.

Second, it should be stressed that the characteristic total strain for such an experiment is 3.5 which corresponds to approximately 1/4 revolution of the bob. According to Fig. 6, for such a strain, the effect of the particle migration on the viscosity profile is negligible. All further experiments were conducted for an accumulated strain lower than 50 (corresponding to 4 revolutions of the bob) so that the particle migration was weak. Moreover, to minimize further the effect of migration, all viscosity measurements were performed at the radial position r=r c .

The shear reversal experiment is repeated for other values of the applied torque corresponding to shear stresses of 0.85, 2.12, 4.25 and 5.95 Pa and the results obtained with a φ=0.47 suspension are presented on Fig. 9. The first observation is that the viscosity mainly depends on the strain amplitude even though, in accordance with the slight shear thinning behavior of the suspension (Fig. 1), the value of the plateau decreases as the shear rate increases. On the contrary, the value of the minimum of the viscosity seems not to depend on the value of the applied torque.

The same experiment is carried out for other values of the particle concentration and the results are displayed on Fig. 10. For each particle volume fraction, the value of the applied torque is chosen in order that the angular velocity of the internal cylinder at the plateau is roughly the same (around 0.5 rpm which corresponds to an average shear rate of 0.1s -1 ). As the particle concentration increases, the minimum is more pronounced and the strain necessary to reach the plateau value of the viscosity decreases.

A striking result shown on Fig. 11 is that the minimum of the viscosity scales as

φ m * -φ ( ) -1
with φ m * = 0.537 while the plateau viscosity scales as φ p * -φ ( ) -2

with φ p * = 0.534 .

This result is discussed in the following section.

V. DISCUSSION.

The transient response of a concentrated suspension after shear reversal has already been studied through macroscopic rheological measurements either in narrow-gap Couette [Gadala-Maria and [START_REF] Gadala-Maria | Shear-induced structure in a concentrated suspension of solid spheres[END_REF]] or in parallel plate geometry [START_REF] Kolli | Transient normal stress response in a concentrated suspension of spherical particles[END_REF] [START_REF] Vand | Viscosity of solutions and suspensions. I Theory[END_REF]] or of a slip layer [START_REF] Jana | Apparent wall slip velocity coefficients in concentrated suspensions of non colloidal particles[END_REF]].

A more precise comparison between our results and those obtained by Gadala-Maria and Acrivos reveals a small difference: while all their results collapse into a single curve when the viscosity divided by its plateau value is plotted versus the strain, our results are slightly different. Indeed, Figure 9 shows that, as the applied torque increases, the plateau viscosity decreases while the minimum remains almost unchanged. This effect is rather weak so we do not want to give too much importance to this observation but it would mean that the shear thinning behavior of the suspension is related to a change of its microstructure together with the non-hydrodynamic interactions with the magnitude of the shear rate. Indeed, we speculate that the viscosity passes through the minimum when the microstructure has disappeared and the suspension is almost isotropic before the particles have rearranged into the steady-state structure when the viscosity plateau is reached again. In this latter state, the particles are very close together along the compression axis and it is plausible that their arrangement and the non-hydrodynamic interactions depend slightly on the value of the shear rate that causes the plateau viscosity to vary. On the opposite, if the minimum viscosity is reached when the suspension is unstructured and when non-hydrodynamic forces have relaxed, the value of the minimum should not depend on the shear rate magnitude. The results reported in Fig. 11 are more pronounced and have more implications. The minimum viscosity and the plateau value do not scale in the same way with the particle concentration. Both diverge when the particle concentration approaches approximately the same value, φ*=0.535 but with different power laws of (φ*-φ). The minimum viscosity diverges when φ tends to φ* with an exponent (-1) while the plateau viscosity follows a (-2) power law. Following our speculation that the suspension is almost isotropic when the viscosity is minimum and that an anisotropic structure has formed at the plateau, two different behaviors were indeed expected.

As for the plateau, the (-2) power law is usually observed both in experiments and in numerical simulations. From a theoretical point of view, [START_REF] Mills | Apparent viscosity and particle pressure of a concentrated suspension of non-Brownian hard spheres near the jamming transition[END_REF] have proposed to explain this scaling with a model that accounts for the solid friction between particles in dynamic clusters. In the same paper, they also provide a scaling of the viscosity with the solid volume fraction for an isotropic random suspension where the dissipation is purely hydrodynamic. The viscosity is shown to scale as φ 4 3 .(φ * -φ) -1 . Thus they predict the same divergence of the viscosity with the particle concentration as that we measured although we have not observed the φ 4 3 dependence.

In the particle concentration range that we consider here, our results are in close agreement with Stokesian Dynamics simulation of unstructured non-Brownian suspensions [START_REF] Sierou | Accelerated Stokesian dynamics simulations[END_REF]]. Their results together with our experimental data are displayed in Fig. 12 where the inverse of the minimum reduced viscosity, η 0 /η min , is plotted versus the reduced particle volume fraction, φ/φ* (in our case, φ*= 0.535 and in the case of Stokesian Dynamics results, φ*=0.64). For relatively moderate solid volume fractions (0.55<φ/φ*< 0.93), the linear scaling is satisfactory for both results, even though the simulation results

show a clear curvature. For higher particle concentrations, Sierou and Brady have proposed a different scaling of the viscosity with the reduced volume fraction that fits well to their simulations for this higher range of volume fractions. Unfortunately, we did not perform measurements at so high concentrations.

We note that in the work of [START_REF] Sierou | Accelerated Stokesian dynamics simulations[END_REF], the particles are homogeneously and isotropically distributed, whereas Kolli et al. consider the initial values of the shear and normal stresses after shear reversal, that correspond to a structured state where the nonhydrodynamic interactions have relaxed. As for our study, we consider the minimum of the viscosity that should be obtained for the less structured state of the sheared suspension, and our results compare well to the simulations of [START_REF] Sierou | Accelerated Stokesian dynamics simulations[END_REF]. However, it is not clear to us whether the point is the isotropic structure or the lack of non-hydrodynamic interactions.

VI. CONCLUSION

We have been able to characterize the bulk transient response of concentrated non-Brownian suspensions of various particle volume fractions after shear reversal, using a PIV technique. Our measurements confirm those previously obtained using classical viscosimetry [START_REF] Gadala-Maria | Shear-induced structure in a concentrated suspension of solid spheres[END_REF], [START_REF] Kolli | Transient normal stress response in a concentrated suspension of spherical particles[END_REF], [START_REF] Narumi | Transient response of concentrated suspensions after shear reversal[END_REF]]: the transient viscosity mostly depends on the strain. After shear reversal, the viscosity undergoes a steplike reduction, decreases slower, passes through a minimum and increases again to reach a plateau value. The plateau viscosity obeys the classical Krieger-Dougherty law with an exponent n=2 and a packing concentration φ*=0.535 whose low value could be explained by interparticle forces. As for the minimum viscosity, its values compare well with the results of numerical simulation performed on random suspensions [START_REF] Sierou | Accelerated Stokesian dynamics simulations[END_REF]] and is consistent with the predictions of [START_REF] Mills | Apparent viscosity and particle pressure of a concentrated suspension of non-Brownian hard spheres near the jamming transition[END_REF]: in the range of particle volume mention that we have attempted to wash the particles with deionized water and with ethanol without noting any change on the suspension rheology. According to the constructor (Microbeads AS), it is plausible that the particles are covered by polymer chains (cellulose) that served as surfactant during the manufacturing of the particles.

We can estimate the characteristic interaction length, δ, by considering the excluded volume arising from the repulsion between particles:

a + δ a       3 ≈ φ nc φ * = 0.61 0.535 (A 1)
where a is the particle radius. Eq.(A1) leads to δ≈1µm.

Moreover, the presence of repulsive forces between particles is corroborated by the observation of an elastic behavior of the suspension which is illustrated in The protocol is represented in Fig. 14.a and the bob angular velocity response is plotted in Fig. 14.b. The longer is the rest period, the higher is the angular velocity overshoot (or lower the viscosity). This would mean that, during shear periods, the particles are forced to approach each other in the compression direction and, when the shear is stopped, the particles move away one from each other under the repulsive interactions. Thus after a sufficiently long rest time, the shear-induced structure of the suspension tends to disappear and its viscosity approaches the minimum viscosity recorded during shear reversal. Again, this

s u η = γ ɺ r a q (B.1)
where a is the radius of the particles and η r , the relative viscosity of the suspension. q is a constant that is determined experimentally: in [START_REF] Jana | Apparent wall slip velocity coefficients in concentrated suspensions of non colloidal particles[END_REF]], q=8.

In our work, the total strain during pre-shear is small, so that we can neglect migration.

For comparison, Gadala-Maria and [START_REF] Gadala-Maria | Shear-induced structure in a concentrated suspension of solid spheres[END_REF], in similar conditions, measured a characteristic strain for migration of around 4.32.10 4 . Following [START_REF] Jana | Apparent wall slip velocity coefficients in concentrated suspensions of non colloidal particles[END_REF], we introduce a slip velocity at the bob u si and at the cup u so . In the case of a purely viscous homogeneous liquid, the velocity profile is given by:

u r = R o 2 -r 2 R o 2 -R i 2 R i r       2 Ω - u si R i - u so R o             + u so R o (B.2)
and the shear rate:

ɺ γ = r d dr u r       = 2 R o 2 R o 2 -R i 2 R i r       2       Ω - u si R i - u so R o             (B.3)
where Ω is the bob angular velocity.

From (B.1), (B.2) and (B.3), the slip velocity is found at the boundaries, and the velocity profile can be computed.

2 2 2 2 2 2 2 2 2 2 2 2 ( ) 2 2 Ω η γ = -   + +   η -   η = γ   + +   η -   ɺ ɺ o r i o i o i r o i i o r app o i r o i i o q R a R R R R R q a R R R R q a R R q a R R R R (B.4)
with γ ɺ app the apparent shear rate, at the bob.

Using eq. (B.4) and setting that the stress is controlled:

app app r r σ = η γ = η γ ɺ ɺ (B.5)
we obtain the expressions of the bulk shear rate, γ ɺ , and relative viscosity, η r , versus their apparent values app γ ɺ and app r η :

2 2 2 2 2 2 2 2 2 1 2 1     η γ = γ - +       -     η η =   η - +   -   ɺ ɺ app app o i r o i i o app r r app o i r o i i o R R a q R R R R R R a q R R R R (B.6)
as well as the expression of the bulk strain: At last, we would like to mention a striking result shown on Fig. 17 where the slip correction has been calculated upon replacing η r (γ) with its plateau value (η r (γ->∞)), in eq.

2 2 2 2 0 2 ( ) 1 γ     η γ = - +       -     ∫ app app o i r o i i o R R u a du q R R R R (B.7)
(B.1). Although we do not have any explanation for it, it appears clearly that the agreement is far better when eq. (B.1) is solved with η r (γ->∞) rather than with η r (γ).

FIGURE CAPTIONS

Figure 1: Flow curve of a φ=0.47 suspension obtained in a narrow-gap Couette rheometer.

Inset: power-law fit: ln(σ(Pa))=0.88.ln( γ ɺ (s -1 ))+3.7. ) )

PIV φ * =0.53 SD φ * =0.64 [START_REF] Sierou | Accelerated Stokesian dynamics simulations[END_REF] Figure

  For instance, in 2002 Shapley et al. use Laser Doppler Velocimetry to measure particle velocity fluctuations and velocity profiles in concentrated suspensions sheared in a narrowgap Couette cell. Another powerful technique to visualize both the velocity and the concentration profiles in suspensions is the Magnetic Resonance Imaging. In 2006, Ovarlez and his collaborators used such MRI techniques to perform local measurements on very concentrated suspensions sheared in a wide-gap Couette viscosimeter. They have been able to measure both the velocity and the particle concentration profiles in the suspension sheared between two concentric cylinders. They have obtained very impressive results on the stationary behavior of suspensions with particle concentration as high as 60%. One of their main results is that contrarily to what is observed with macroscopic rheological measurements, i.e. a pseudo yield stress, suspensions have a purely viscous behavior.

  [START_REF] Narumi | Transient response of concentrated suspensions after shear reversal[END_REF]] . In particular, it has been shown that the transient viscosity is a function of the strain only. Gadala-Maria and[START_REF] Gadala-Maria | Shear-induced structure in a concentrated suspension of solid spheres[END_REF] proposed to explain the transient response by the development of a shear-induced microstructure. This microstructure has been observed by[START_REF] Parsi | Fore-and-aft asymmetry in a concentrated suspension of solid spheres[END_REF] and is characterized by the asymmetry of the pair distribution function with an excess of particles in the compressional quadrant. When the direction of the shear is reversed, the microstructure is broken and the particles rearrange into the mirror image of this microstructure. In 2002, Kolli et al. studied the transient shear and normal force response to a shear reversal in non-Brownian suspensions in an annular plate-plate geometry. They observed that the normal force exhibited a transient behavior too. It abruptly went to a negative small value before increasing again to its positive value. They connected the initial value of both shear and normal stresses with the hydrodynamic contribution (in a structured state) and the plateau value with a mixed contribution of hydrodynamic and contact forces. They noted that the hydrodynamic contribution compared to the plateau value decreased as the concentration increased. Haan and Steif (1998) found previously the same type of behavior for the particle pressure in numerical simulations of 2D suspensions of cylinders. Thus both shear-induced microstructure and relative contributions of the hydrodynamic and non-hydrodynamic effects are presumably involved in the transient rheological response of non-Brownian suspensions. Basically, the local response of the suspension presented in the current paper displays the same features as those obtained through conventional viscosimetric measurements. The main contribution of our work is to confirm that the transient response is actually a bulk property of the flowing suspension and cannot originate from a boundary problem such as the development of a Vand zone [

  Fig. 13. This recovery test has been carried out on a φ=0.444 suspension. The recovery curve (solid line) is almost well fitted by an exponential function (dashed line) with a Maxwell time, τ, of about 2s which corresponds to an elastic modulus of approximately η/τ≈70 Pa. Furthermore, we speculate that the repulsive forces between particles are responsible for a perturbation of the shear-induced microstructure. This has been observed during transient tests where shear stress steps have been applied always in the same direction after rest periods of different durations.

Figure ( 16

 16 Figure (16) displays the PIV measurements together with the viscosity measured in the

Figure 2 :

 2 Figure 2: Experimental set-up.

Figure 3 :

 3 Figure 3: Steady relative viscosity of the suspension versus the particle volume fraction.

Figure 4 :

 4 Figure 4: Typical image of a φ=0.444 suspension recorded 14 mm above the cup bottom.

Figure 5 :

 5 Figure 5: Long time evolution of the velocity profile in a φ=0.444 suspension sheared in a

Figure 6 :

 6 Figure 6: Long time evolution of the normalized viscosity profile for the φ=0.444 suspension.

Figure 7 :

 7 Figure 7: Transient velocity profiles after shear-reversal (at t=0 the applied torque passes from

Figure 8 :

 8 Figure 8: Transient reduced viscosity after shear reversal versus local strain measured at r R out = 0.77 . φ=0.444

Figure 9 :

 9 Figure 9: Transient reduced viscosity after shear reversal versus local strain. The experiment

Figure 10 :

 10 Figure 10: Transient viscosity after shear reversal for different particle volume fractions vs local strain. ɺ γ plateau ≈ 0.1 s -1 . The applied torque values have been chosen in order the bob

Figure 11 :

 11 Figure 11: Scaling law for the plateau viscosity and the minimum viscosity. The applied

Figure 12 :

 12 Figure 12: Inverse of the relative viscosity η 0 η versus φ/φ*: ( ) minimum viscosity from

Figure 13 :

 13 Figure 13: Recovery test after shear stress cancellation. φ=0.444.

Figure 14 :

 14 Figure 14: Sequence of shearing and shear stress break with increasing of the break duration.

Figure 15 :

 15 Figure 15: Viscosity response to a shear stress reversal for a φ=0.47 suspension. Wide-gap + PIV measurements: σ=4.25 Pa ( ), σ=5.95 Pa ( ). Narrow-gap measurements: σ=4.25 Pa ( ____ ), σ=5.95 Pa (----).

Figure 16 :

 16 Figure 16: Viscosity response to a shear stress reversal for a φ=0.47 suspension. Wide-gap + PIV measurements: σ=4.25 Pa ( ), σ=5.95 Pa ( ). Narrow-gap measurements corrected for the wall slip: σ=4.25 Pa ( ____ ), σ=5.95 Pa (----).

Figure 17 :

 17 Figure 17: Viscosity response to a shear stress reversal for a φ=0.47 suspension. Wide-gap + PIV measurements: σ=4.25 Pa ( ), σ=5.95 Pa ( ). Narrow-gap measurements corrected for the wall slip: σ=4.25 Pa ( ____ ), σ=5.95 Pa (----). The wall slip correction is calculated using

  Figure 7

  Figure10

Figure 14

 14 Figure 14

  Figure 15

fractions that we have studied (0.55<φ/φ*< 0.93), the minimum viscosity diverges as (φ*-φ) -1 , suggesting that the suspension passes through an almost isotropic structure where the nonhydrodynamic forces have relaxed.

ACKNOWLEDGMENTS

This work was supported by the ANR (program ANR-08-BLAN-0048 -CSD 2). We are grateful to L. Lobry and P. Mills for fruitful discussions.

elastic behavior together with the small value of φ* suggest that some polymer molecules at the surface of the particles give rise to soft steric repulsion.

APPENDIX A: A POSSIBLE EXPLANATION FOR THE LOW VALUE OF THE PACKING FRACTION.

Upon fitting the data of Fig. 3 with the Krieger-Dougherty law, the particle packing fraction, φ*, is found to be around 0.536. This value is much smaller than the usual values obtained for non-colloidal suspensions, φ nc ≈0.61. We speculate that even though the particles are as large as 31µm in diameter, they undergo soft interactions. One can wonder about the nature of the inferred interactions. We can rule out the electrostatic interactions since we have measured the same value of φ* when the particles were dispersed in an aqueous solution whose polarity is of course very different from that of the mineral oil used in the present experiment. The Van der Waals forces should also be discarded as their interaction range is too short for explaining the low value of the packing volume fraction. Furthermore, they should be weak since the refractive indices of the particles and of the liquid are matched.

Finally, polymeric forces remain the only possible interaction between particles. We have to

APPENDIX B: COMPARISON WITH MACROSCOPIC NARROW-GAP

MEASUREMENTS

In this section, we perform classical rheometric measurements in a narrow-gap Couette geometry Z40Ti mounted on the rheometer Mars II (Haake, Thermofisher). The bob diameter is 2R i =41.42 mm, the gap is g=R o -R i =0.99 mm, its height is L=55mm.The transient viscosity of the φ= 47% suspension is measured. After it had been poured in the cup, the suspension was shortly pre-sheared (total strain around 365) in order to obtain a well defined structure in the suspension. Then the shear reversal measurements were performed.

The raw viscosity measured for two values of the imposed shear stress (4.25 and 5.95 Pa) is displayed on Fig. 15, together with the corresponding wide-gap local measurements. The apparent viscosity is significantly lower in the narrow-gap cell, suggesting that wall slip occurs at the boundaries. [START_REF] Jana | Apparent wall slip velocity coefficients in concentrated suspensions of non colloidal particles[END_REF] performed local measurements in a narrow-gap geometry to determine the slip length at the wall. They used a Laser Doppler Velocimetry technique to measure the velocity profile across the gap. In this paper, they sheared the suspensions for 2 days before they acquired the velocity profile, so that they had to take the radial migration into account. The wall slip was characterized by an apparent slip velocity, i.e. the difference between the velocity of the walls and the velocity of the suspension at the same position.

They showed that the slip velocity u s was related to the local shear rate γ ɺ and the local volume fraction φ by:


Achraf Ghorbel
email: achraf.ghorbel@irisa.fr

Sébastien Macé
email: sebastien.mace@irisa.fr

Aurélie Lemaitre
email: aurelie.lemaitre@irisa.fr

Eric Anquetil
email: eric.anquetil@irisa.fr

Interactive competitive breadth-first exploration for sketch interpretation

Keywords: sketch recognition, breadth-first exploration, interactive recognition, 2D architectural floor plans

In this paper, we present a new generic method for an interactive interpretation of sketches. This method is based on a competitive breadth-first exploration of the analysis tree. As opposed to well known structural approaches, this method allows to evaluate simultaneously several possible hypotheses of recognition in a dynamic local context of document. At each step of the analysis, the decision process selects the best hypotheses. If it detects an ambiguity, it will solicit the user to select the right hypothesis. In fact, the user participation has a great impact to avoid error accumulation during the analysis step and overcomes the combinatory due to the sketch complexity. This paper demonstrates this interactive method on 2D architectural floor plans.

I. INTRODUCTION

Digital documents are becoming more and more omnipresent in our life. Many reasons such as the flexibility offered by digital processing have led to transform handwritten documents to digital ones. In this context, we are working on mapping technical paper documents, like architectural floor plans, to numerical ones. We aim at offering a complete and homogeneous solution to unify paper document recognition and pen-based sketch interpretation (for instance: with Tablet PC).

In the literature, two major approaches to interpret documents are perceived: statistical and syntactic. Choosing one of these two approaches depends on the document to analyze: the syntactic approach is the best approach to interpret a well structured document, and the statistical approach is the best adopted when dealing with a big homogeneous and labeled base allowing the learning. A major advantage of methods based on statistical approach [START_REF] Lemaitre | Preliminary experiments in layout analysis of handwritten letters based on textural and spatial information and a 2d markovian approach[END_REF] [START_REF] Montreuil | Unconstrained handwritten document layout extraction using 2d conditional random fields[END_REF] is the incorporation of uncertainty. However, these methods usually lack the ability to convey the hierarchical structure of the document and need a wide learning on a homogeneous and labeled base. Syntactic methods [START_REF] Chan | An efficient syntactic approach to structural analysis of on-line handwritten mathematical expressions[END_REF] [4] [START_REF] Mao | Document structure analysis algorithms: a literature survey[END_REF] [START_REF] Coüasnon | Dmos, a generic document recognition method: Application to table structure analysis in a general and in a specific way[END_REF] segment the image in primitives and are based on rules or graphs to describe how to compose these primitives. However, syntactic methods have difficulties to incorporate the uncertainty.

In this work, we focus on structured documents. We present a new syntactic approach based on an interactive lazy interpretation of the document. For modeling this approach, we adopt the bidimensional grammars. Unlike the classical syntactic and the static approaches, the suggested method does not always select the first or the best found hypothesis. The associated analysis process is able to take into account the uncertainty so that the user can be solicited by the analyzer to choose between two or more possible hypotheses. This intervention of the user avoids a false decision and thus the propagation of errors during the analysis. This interactive recognition strategy allows lazy interpretation of complex structured documents. It can be applied to off-line documents (image), as illustrated in this paper (Figure 8(a)), as well as on-line or vectored documents. One of the main originalities is to allow a breadth-first exploration of the analysis tree driven by a contextual focus on a delimited zone of the document. Each branch is characterized by a score so that the decision process is able to take the right decision or to ask the user in case of ambiguities.

The remaining of the paper is organized as follows. In the section II, we introduce our analysis method. The implementation of our interactive analyzer is described in section III. Experimental results are reported in section IV and finally, section V concludes the paper.

II. INTERACTIVE BREADTH-FIRST EXPLORATION

In this section, we present the analyzer by first describing its main characteristics. Then, we detail its internal design.

A. Analyzer characteristics

In structured documents, the primitives must not be interpreted in an isolated manner. Indeed, the interpretation of each primitive has an impact on the interpretation of its neighbors. For that, to interpret a primitive, it is necessary to take into account the neighbor primitives. However, we work in a bidimensional context (images), which induces a large combinatory. To overcome this complexity problem, we propose to guide our breadth-first exploration using a spatial contextual focus.

This idea is close to the well known LL(k) analysis, where reading the next k tokens enables to choose without ambiguities which rule must be applied. However, in our bidimensional analysis, we have to limit the number of token k to explore, i.e. the depth of the analysis. Consequently, contrary to the LL(k) analysis, the exploration of the following tokens does not allow to take a unique decision on the rule to apply, because we voluntarily limitate the value of k. Moreover, sometimes, the grammar is not LL(k) for any k and the analyser meets ambiguities. In those two cases, the process can not take the right decision, and may hesitate between several hypotheses. In order to validate the right decision, we propose an analysis process that can, through its decision process, take the right decision or ask the user in case of ambiguities. For this purpose, the analyzer is based on the following characteristics:

• a rule based analysis,

• a bidimensional descending breadth first analysis,

• the attribution of scores to each hypothesis, • a spatial contextual focus of the exploration.

B. Analyzer design

The analyzer explores the matching between the documents structure defined by the production rules and the set of primitives contained in the document. The interactive breadth-first analysis process consists of three stages: 1) defining the local context, 2) building the analysis tree and 3) making the decision . Figure 1 shows the complete process of analysis and the relationship between the three parts of the analyzer.

User intervention

Defining the local context

Building the analysis tree

Making the decision

Starting analysis : document to analyze

End of analysis : document analyzed 2) Building the analysis tree: each primitive can be interpreted in several ways. The number of analysis trees corresponds to the number of possible interpretations for the current primitive. Each root is the production rule which consumes this primitive. Each node or leaf is the application of a production rule deduced from the previous node. Figure 4 illustrates an example of the analysis for a given bidimensional local context. The system tries to explore all the possible productions in the local context.

Every leaf or node of the tree has a calculated score from both its local score and the score obtained from the preceding nodes. Every score determines the adequate degree to validate a production. It is calculated from each rule.

3) Making the decision: after building all the analysis trees, the decision process determines the right root to validate. Two cases are presented: case of a single root and case of several roots.

• Case of a single root: this root is validated.

• Case of several roots: the branch having the highest leaf score in each root is considered. These branches are sorted by the score of each leaf. Then we compare the obtained branches. Two cases may appear:

-Implicit validation: when the analyzer is confident enough to choose the correct root without integrating the user. The analyzer implicitly validates the root which has the branch having the highest score. -Explicit validation: when the decision process requires the user to validate the right decision. In practice, if the difference between the branch with the highest score and another branch is below a threshold, called threshold of confidence and these two branches are contradictory (at least one joint primitive is not consumed by the same rule production), the user intervention is required. When the correct root is validated, other roots are put on hold and the new roots are either the sons of this root if exists, or the waiting roots otherwise. We take the first step (cf section II-B1). The analysis is complete when no production rule is applicable.

III. IMPLEMENTATION OF THE INTERACTIVE ANALYZER

In this section, we describe the implementation of our interactive analysis method and illustrate it on 2D architectural plans. For this, we consider a set of production rules modeling 2D architectural plans. These rules are illustrated in Figure 2. We detail, during this section, the building of the analysis tree for the interpretation of a primitive, the impact of moving the local context on the analysis tree and the decision process in a case of ambiguity.

A. Interpretation of a primitive

In order to simplify the explanation of this part, we rely on a simple example of 2D architectural plan. We consider that each primitive can be interpreted as a wall or as a part of a door. A door is composed of a sequence of primitives between a wall and a segment that are collinear. A wall is composed of a single primitive. This is described by the production rules P1, P2 and P4 presented in Figure 2.

Figure 3 shows the image during the interpretation. An element is already interpreted (a wall), and other primitives

have not yet been interpreted. We now present how we build the analysis tree for the next primitive to interpret (here "prim2"). Building the analysis tree of a primitive "prim2" begins with the productions which consumes the primitive "prim2" and describes all possible productions in a local context (Figure 4). The number of analysis trees corresponds to the number of possible productions to interpret "prim2". "prim2" can be interpreted in two different manners (rules P1 and P2). P1 allows the transformation of "prim2" to a wall and P2 transforms "prim2" in a sequence (one of the components of a door). Then, we continue the analysis while the following primitives belong to the local context. Here, we study the different interpretations of "prim3" and "prim4". Each branch of the tree has a score deduced from the set of productions applied in this branch. The decision process will validate the right decision based on the criteria described in section II-B3.

B. Transition between two successive analysis trees

After building the analysis trees and validating the right root, the analyzer examines the rest of the document. This processing consist in building new analysis trees. The roots of these trees are the sons of the last validated production (last root validated) if they exist, or roots placed on hold, otherwise. The transition of an analysis tree to the next analysis tree involves a shift of the local context. This movement allows applying the other productions. In fact, the building of the new analysis trees is not the rebuilding of the whole branch but only the new found productions by shifting the context.

In order to better assimilate this transition, we describe an example of interpretation of the 2D architectural plan. This example is composed of seven primitives; three of them build up a door. Figure 5 a single tree appears, the decision process validates its root. Otherwise it takes the best hypothesis or can solicit the user in case of ambiguity.

C. Ambiguity cases

In this section, we focus on a particular case of the analysis process: the case of user intervention. Figure 7(a) shows a part of architectural floor plan to interpret. We launch the analyzer on the same principle described in Section II-B. When arriving the stage illustrated in Figure 7(b), two contradictory hypotheses are considered: with almost the same score (see section II-B3). The decision process evaluates two production rules: apply the first hypothesis, which corresponds to the branch described in Figure 7(c), i.e. the interpreted document will consist of two doors, or apply the second hypothesis, which corresponds to the branch described in Figure 7(d), i.e. a single window is interpreted.

The decision process decides to report the decision on the user because the two branches in competition are contradictory and the difference between the scores is lower than the threshold of confidence. The decision process launches an interaction with the user, and proposes to validate the first hypothesis (Figure 7(c)). If the user validates this hypothesis, the corresponding root is validated. If the user declines the first hypothesis, the system implicitly validates the second hypothesis, and then we will have the result shown in Figure 7(d).

IV. EXPERIMENTAL RESULTS

In this section we illustrate the presented interactive breadth-first exploration on handwritten 2D architectural floor plans (cf Figure 8(a)). We present the formal rule description of the document, and how we estimate the size of the local context before presenting preliminary results.

A. Document description

In order to describe the structure of the document, we use the context-driven constraint multiset grammars (CD-CMG), developed in our previous work [START_REF] Macé | Eager interpretation of on-line handdrawn structured documents: The dali methodology[END_REF]. The CD-CMG allows the use of statistical pattern recognizers. The analysis process is driven by the document structure that involves a significant decrease in the complexity of the analysis process. We have adapted this eager analysis process to address a lazy (a posteriori) interpretation process based on the interactive breadth-first exploration. The score calculated by each production is due to preconditions and constraints of the rule production [START_REF] Macé | Eager interpretation of on-line handdrawn structured documents: The dali methodology[END_REF]. This grammar has been used for the interpretation of on-line hand-drawn structured documents. In this work, we adopt this grammar for a off-line interpretation of structured document.

In the application context of handwritten 2D architectural floor plan recognition, we focus on floor plans composed of walls, doors and windows. The set of primitives is composed of segments extracted from the image using Kalman filtering [START_REF] Leplumey | Kalman filter contributions towards document segmentation[END_REF]. Each primitive can be a component of an opening (door or window), or a wall. An opening element is a set of segments localized between two collinear walls, or between a collinear wall and segment. Figure 2 shows some production rules used for the description of architectural plans.

B. Determination of the local context size

An important question is the definition of the size of the 2D local context. Increasing the size of the 2D local context can increase the rate of document recognition, but also creates more combinatory and ambiguity problems, and therefore requires more user intervention. The idea is to minimize the size of the 2D local context while conserving enough contextual information to take the right local interpretation of the primitives according to the productions rules. In our case, this size is calibrated according to the maximum length of an opening.

C. Preliminaries results

We tested this interactive sketch recognition approach on 26 images of handwritten 2D architectural floor plans (some examples are reported in Figure 8(a) and Figure 9) drawn by five persons. Each image consists of walls, doors and windows. These images contain 128 doors and 133 windows. During the interpretation of these images, the decision process solicits the user on average of one time by image. About 60% of these images do not require a user intervention. In 70% of the cases of ambiguity, the user validates the first hypothesis. In 30% cases, the user does not validate the first hypothesis, and so he does not validate the branch that has the best score. This demonstrate the importance of the user intervention to avoid interpretation errors. The Figure 8 In this paper, we have presented an original interactive analysis process to interpret structured documents such as 2D architectural floor plans. This analyzer is based on a competitive breadth-first exploration of the analysis tree according to a dynamical local context of the document. The decision process is able to solicit the user in the case of strong ambiguity. The first tests of this interactive analyzer have been made on the 2D architectural floor plans. Integrating the user in the analysis process is in our view a key point to adress complex off-line sketch reognition. Future work will focus on applying this new analysis strategy in different types of structured documents and on extending the experimental results on large image databases.

Figure 1 . Analysis process 1 )

 11 Figure 1. Analysis process 1) Defining the local context: the bidimensional local context is defined for an analysis tree as the maximum distance between the elements of the root and the elements of any leaves. The size of the local context is a parameter to define according to each type of document. Figures 5(a) and 5(b) show the evolution of the local context (the frame which surrounds primitive to interpret) during the analysis process.2) Building the analysis tree: each primitive can be interpreted in several ways. The number of analysis trees corresponds to the number of possible interpretations for the current primitive. Each root is the production rule which consumes this primitive. Each node or leaf is the application of a production rule deduced from the previous node. Figure4illustrates an example of the analysis for a given bidimensional local context. The system tries to explore all the possible productions in the local context.Every leaf or node of the tree has a calculated score from both its local score and the score obtained from the preceding nodes. Every score determines the adequate degree to validate a production. It is calculated from each rule.3) Making the decision: after building all the analysis trees, the decision process determines the right root to validate. Two cases are presented: case of a single root and case of several roots.

Figure 2 .

 2 Figure 2. Example of production rules used for the architectural plans

Figure 3 .

 3 Figure 3.Document being analyzed and position of the local context during the analysis. "prim2" is the next primitive to interpret. The box describes the local context and includes "prim2", "prim3" and "prim4"

Figure 4 .

 4 Figure 4. Illustration of the analysis tree for the interpretation of "prim2": two roots are considered. (-) and (+) designate respectively the consumed elements and the created elements.

  (a) and 5(b) show the state of the analyzer in two consecutive stages. Figures 6(a) and 6(b) show the built analysis trees from two consecutive primitives.

Figure 5 .

 5 Figure 5. Document being analyzed and position of the local context (box) during the analysis

Figure 6 .

 6 Figure 6.The analysis trees. (-) and (+) designate respectively the consumed elements and the created elements.

Figure 7 .

 7 Figure 7. Example of ambiguity between two hypotheses

  (a) focuses on the interpretation of one of these floor plan images. During the interpretation of this

  (a) Original image (b) Document misinterpreted. User validates the first hypothesis. Transformation of two doors in a window (cf. dotted line).(c) Document correctly interpreted. User invalidates the 1st hypothesis and the decision process implicitly validates the 2nd hypothesis (two doors).

Figure 8 .

 8 Figure 8. Interpretation's result for the second example

  Figure 8(b) shows the result if the user validates the first hypothesis (wrong interpretation) and Figure8(c)shows the obtained result otherwise (if the user rejects the first hypothesis, the analysis process implicitly validates the second hypothesis).

Figure 9 .

 9 Figure 9. Examples of 2D architectural floor plans V. CONCLUSION

1520-5363/11 $26.00 © 2011 IEEE DOI 10.1109/ICDAR.2011.241

ACKNOWLEDGMENT

The authors would like to thank all the people who took part in the experiments. This work partially benefits from the financial support of the ANR Project Mobisketch.


Achraf Ghorbel
email: achraf.ghorbel@irisa.fr

Abdullah Almaksour
email: abdullah.almaksour@irisa.fr

Aurélie Lemaitre
email: aurelie.lemaitre@irisa.fr

Eric Anquetil
email: eric.anquetil@irisa.fr

Incremental learning for interactive sketch recognition

Keywords: sketch recognition, incremental learning, interactive recognition, 2D architectural floor plans

In this paper, we present the integration of a classifier, based on an incremental learning method, in an interactive sketch analyzer. The classifier recognizes the symbol with a degree of confidence. Sometimes the analyzer considers that the response is insufficient to make the right decision. The decision process then solicits the user to explicitly validate the right decision. The user associates the symbol to an existing class, to a newly created class or ignores this recognition. The classifier learns during the interpretation phase. We can thus have a method for auto-evolutionary interpretation of sketches. In fact, the user participation has a great impact to avoid error accumulation during the analysis. This paper demonstrates this integration in an interactive method based on a competitive breadth-first exploration of the analysis tree for interpreting the 2D architectural floor plans.

I. INTRODUCTION

In this paper, we are working on mapping technical paper documents, like architectural floor plans, to numerical ones. We aim at offering a complete, interactive and auto-evolving solution to unify paper document recognition and pen-based sketch interpretation (for instance: with Tablet PC).

At present, structured documents can be very complex. Faced with this complexity, the various existing methods [START_REF] Chan | An efficient syntactic approach to structural analysis of on-line handwritten mathematical expressions[END_REF] [2] [3] [START_REF] Coüasnon | Dmos, a generic document recognition method: Application to table structure analysis in a general and in a specific way[END_REF] keep a margin of error. Therefore, very often, an a posteriori verification phase will be necessary to ensure there is no recognition error. In this phase the user browses the document to correct the errors due to the interpretation.

To avoid the verification phase on the one hand, and avoid error accumulation during the analysis step on the other hand, we proposed an interactive method of analysis of offline structured document where the decision process solicits the user if necessary. In our previous work [START_REF] Ghorbel | Interactive competitive breadth-first exploration for sketch interpretation[END_REF] the role of user was limited to validate the right hypothesis and then unlocks a situation where the decision process is not sure to make the right decision. In summary, the process can solicit the user to be sure to make the correct decision. Now, we want to exploit the solicitation of the user during the analysis not only to unlock a situation but also to learn the process of analysis. In this context, we focus in this paper on improving the capacity of symbol recognition. In the sketch interpretation method, the classifier is responsible for symbol recognition.

The classification systems can be generally categorized into two types: static and evolving systems. Static systems are trained in batch mode using a predefined learning dataset, while incremental learning algorithms are used to train evolving classifiers, like for our symbol recognition system. In incremental learning algorithms, new instances from existing classes can be progressively introduced to the system to improve its performance. Moreover, new unseen classes can be added to the system at any time by the incoming data.

In this work, we present the advantage of soliciting the user to improve the recognition capacity of the classifier by incremental learning able to dynamically add new classes. The remaining of the paper is organized as follows. In the section II, we introduce our existing interactive analysis method. Section III describes principles of the incremental classifier. The coupling of this incremental classifier with our interactive analysis of sketches is described in section IV. Experimental results are reported in section V and finally, section VI concludes the paper.

II. INTERACTIVE BREADTH-FIRST EXPLORATION

In this section, we summarize our interactive method of structured document interpretation [START_REF] Ghorbel | Interactive competitive breadth-first exploration for sketch interpretation[END_REF] in which we propose to integrate our incremental classifier.

This analyzer is based on the following characteristics:

• a rule based analysis,

• a bidimensional descending breadth first analysis.

• the attribution of scores to each hypothesis,

• a spatial contextual focus of the exploration to limit the combinatory.

User intervention

Defining the local context

Building the analysis tree

Making the decision

Starting analysis : document to analyze

End of analysis : document analyzed

Figure 1. Analysis process

These characteristics were chosen in order to ensure the best interactivity with the analysis system. Figure 1 shows the complete process of analysis and the relationship between the three parts of the analyzer. The analyzer begins by defining a spatial contextual focus. Once the context is well defined, the analyzer goes to the second stage. In this stage, the analyzer explores all possible hypotheses of interpretation in the spatial context using a set of bidimensional rules that describe the structure of the document. The next stage is the decision making. The role of the decision process is to validate the right hypothesis among a set of competing hypotheses generated with a descending breadth first analysis. Sometimes the decision process is not sure to make the right decision. In this case, it solicits the user. In practice, if the difference of scores between the top two branches is below a threshold of confidence and if these two branches are contradictory (at least one joint primitive is not consumed by the same rule production), the user intervention is required. When the correct hypothesis is validated, we return to the first step. The analysis is complete when no more production rule is applicable. This intervention of the user avoids a false decision and thus the propagation of errors during the analysis. This interactive recognition strategy allows lazy interpretation of complex structured documents.

In the current state, the information provided by the user is only used to unlock situations. In this paper, we want to exploit more widely this information by learning continuously during the analysis. In this context, we propose to integrate an incremental classifier which uses information supplied by the user to improve its capabilities during the analysis.

III. INCREMENTAL LEARNING OF A FUZZY INFERENCE

SYSTEM

Our classification system is based on first-order Takagi-Sugeno (TS) fuzzy inference system [START_REF] Almaksour | Improving premise structure in evolving takagi-sugeno neuro-fuzzy classifiers[END_REF]. It consists of a set of fuzzy rules of the following form:

Rule i : IF x is close to P i THEN y 1 i = l 1 i ( x), ..., y k i = l k i ( x) (1
) where l m i ( x) is the linear consequent function of the rule i for the class m:

l m i ( x) = π m i x = a m i0 + a m i1 x 1 + a m i2 x 2 + ... + a m in xn ( 2
)
where n is the size of the input vector. The Prototype P is defined by a center and a fuzzy zone of influence. To find the class of x, its membership degree β i ( x) to each fuzzy prototype is first computed. After normalizing these membership degrees, the sum-product inference is used to compute the system output for each class:

y m ( x) = r i=1 βi ( x) l m i ( x) (3)
where r is the number of fuzzy rules in the system. The score of each class y m is between 0 and 1. The higher is the score, the higher is the degree of confidence in that class to be associated to the given input. The winner class is that with the maximum score. The membership degree is computed by the prototype center µ i and its variance-covariance matrix A i using the multivariate Cauchy probability distribution.:

β i ( x) = 1 2π |A i | 1 + ( x -µ i ) t A -1 i ( x -µ i ) -n+1 2 (4)
The incremental learning algorithm of our model consists of three different tasks: the creation of new rules, the adaptation of the existing rule's premises, and the tuning of the linear consequent parameters. These three tasks must be done in an online incremental mode and all the needed calculation must be completely recursive.

The importance of a given new sample in an incremental clustering process can be evaluated by its potential value. The potential of a sample is defined as inverse of the sum of distances between a data sample and all the other data samples. A recursive method for the calculation of the potential of a new sample has been introduced in [START_REF] Angelov | An approach to online identification of takagi-sugeno fuzzy models[END_REF]. The recursive formula avoids memorizing the whole previous data but keeps -using few variables -the density distribution in the feature space based on the previous data (see [START_REF] Almaksour | Improving premise structure in evolving takagi-sugeno neuro-fuzzy classifiers[END_REF] for more details). A Premise adaptation process allows to incrementally update the prototype centers coordinates according to each new available learning data, and to recursively compute the prototype covariance matrices in order to give them the rotated hyper-elliptical form. For each new sample, the center and the covariance matrix of the prototype that has the highest activation degree are updated recursive manner [START_REF] Almaksour | Improving premise structure in evolving takagi-sugeno neuro-fuzzy classifiers[END_REF]. The tuning of the linear consequent parameters in a firstorder TS model can be done by the weighted Recursive Least Square method (wRLS). More details are available in [START_REF] Almaksour | Improving premise structure in evolving takagi-sugeno neuro-fuzzy classifiers[END_REF].

The incremental learning algorithm is supposed to be supervised. The recognition of each data sample must be followed by a validation or a correction action in order to learn it. If the system answer is validated, the data sample will reinforce the system knowledge associated to its class. If an external correction signal is sent, the confusion between the (wrong) winner class and the true class is solved by the incremental learning algorithm. A third scenario may take place when the input data sample is declared as the first sample from a new unseen class.

IV. USER INTERVENTION IN THE INTERACTIVE ANALYSIS PROCESS

In this section, we present the possibilities offered by the introduction of a classifier based on incremental learning in our interactive sketch recognizer. In particular we detail when and how the user can interact with the incremental classifier. During the analysis, each time the classifier is sought to identify a symbol, the decision process uses the confident degree given by the classifier to make its decision. If the decision process considers that the confidence degree is sufficiently high to make the right decision, it validates the recognition. Otherwise, The decision process will solicit the user. The user is then in front of four possibilities (cf. No action is done by the classifier. With this interaction process, the classifier continuously learns to improve its interpretations. The more the analysis is going on, the more the classifier is accurate, the less the user is solicited. This incremental learning is able to deal with the recognition of new classes of symbols. It is a key point to absorb the great variability of symbols that can occur in a sketch.

Example

To illustrate the interaction process, we develop the analysis on 2D handwritten architectural floor plans. We focus the demonstration on cases where the decision process invokes user interaction. Figure 3 illustrates a handwritten architectural plan to interpret. The incremental classifier initially contains two classes: one kind of door and one kind of window.

Figure 4 shows a case where the solicitation of the user is judged necessary. In this intervention the user is in front of the four possible actions described in section IV. The system presents an interface that contains the hypothesis given by the classifier, the other available classes of the classifier and a field where the user can add a new class. Figure 4(a) shows a case in which the user indicates that the symbol to recognize is a classical window.

V. EXPERIMENTAL RESULTS

In this section we analyze the performance and the behavior of our model of incremental learning on the two types of experiements. The first experiment is conducted on an incremental learning problem related to our specific application context, the recognition of the openings (windows and doors) in architectural plans. The first experiment evaluates the performance of the model of incremental learning on a set of openings belonging in two classes (window and door).

For this evaluation, we choose:

• a classifier which contains two classes (door and window). This classifier is initially learned using 3 samples per class. • an incremental learning database containing 10 doors and 10 windows. • a test database containing 99 windows and 118 doors. All symbols used in this experiment are extracted from 26 images of handwritten 2D architectural floor plans. The main idea of this experiment is to demonstrate the capability of incremental learning to improve the recognition rate of opening belonging in existing classes. For this, we use the test database after each incremental learning step. A learning step is a phase in which the classifier learns using one door and one window of the incremental learning database. Figure 6 illustrates the evolution of the error rate of opening (Door and window) during the incremental learning of a set of samples. Thanks to the adaptation of the classifier, the error rate found on the test database decreases from 9.7% to 2.8%. The most of misrecognized symbols are badly drawn.

Learning phase: a door and a window Evolution of performance during the incremental learning process

The second experiment describes the behavior of incremental classifier and its ability to add new classes during the learning phase. We start this experiment by:

• classifier which contains two classes (door and window). This classifier is the same used in the first experiment. It is initially similar to the classifier used in the first experiment (the same initial learning phase). • incremental learning database containing 10 doors, 10 windows and 10 sliding windows. • test database containing 99 windows, 118 doors and 14 sliding windows. All symbols used in this experiment are extracted from 29 images of handwritten 2D architectural floor plans. This experiment has two phases of adaptation. The first phase is an existing class adaptation. This phase is similar to the first experiment. We use the test database after each learning step. Like the first experiment, an incremental learning step is composed of one door and one window. In the second phase, we introduce a new class (sliding window). Indeed, the classifier is incrementaly learned using at each step one door, one window and two sliding windows.

Figure 6 describes the evaluation of the classifier during this experimentation. The classifier begins to learn the symbol of existing classes. During this phase, the error rate decreases from 15.1% to 8.7%. During the second phase, the classifier needs to learn symbol of a new class (the pic in the curve of Figure 6). After a few samples of adaptation to this new class, the classifier finds its stability and absorbs this class. During this phase, the error rate decreases from 18.2% to 3.9%

We have shown in these first results the performance of a classifier based on incremental learning and its ability to integrate new classes during the analysis phase of structured documents. Our next work is to interpret more complex structured documents containing several classes of symbols.

We are rather confident in this new step because the incremental classifier have been already successfully tested on other database composed of several classes in others domains [START_REF] Almaksour | Improving premise structure in evolving takagi-sugeno neuro-fuzzy classifiers[END_REF].

VI. CONCLUSION In this paper, we have presented the integration of a classifier based on an incremental learning method, in an interactive method for interpreting the 2D architectural floor plans. The role of classifier is to recognize the symbol with a degree of confidence. If this degree of confidence is considered insufficient by the decision process to take the right decision, the analyzer solicits the user to validate the right hypothesis. The user is then in front of four possibilities. He can either confirm the recognition proposed by the classifier, or associates the symbol to an existing class, a new class, or ignores this recognition. The classifier is incrementally learned during the analysis phase. This strategy offers an auto-evolutionary method for sketch interpretation.

Figure 2 Figure 2 .

 22 Figure 2. Interaction scheme of symbol recognition • The user validates the hypothesis proposed by the classifier in spite of the low degree of confidence given by the classifier. The classifier enhances the model of this class. • The user associates the symbol to recognize to other existing class in the classifier. The classifier reduces the confusion between two classes. • The user associates the symbol to a new class: the user considers that the symbol does not belong to an existing class. With this new information, the classifier starts to learn a new class of symbols. • The user ignores the symbol to recognize: the rejection case. The user considers that the recognized symbol is an outlier (noise in the image).No action is done by the classifier. With this interaction process, the classifier continuously learns to improve its interpretations. The more the analysis is going on, the more the classifier is accurate, the less the user is solicited. This incremental learning is able to deal with the recognition of new classes of symbols. It is a key point to absorb the great variability of symbols that can occur in a sketch.

  Figure 4(b) shows a case in which the user associates the symbol to a new class of windows (a sliding windows).

Figure 3 .

 3 Figure 3. Example of an architectural plan to interpret

Figure 4 .

 4 Figure 4. User interventions. Four possibilities exist. The user associates the set of primitives located in the bounding box 'O' to a right class.

Figure 5 .

 5 Figure 5.Evolution of performance during the incremental learning process

  Learning a new classLearning phase: a door and a window Learning phase: a door a window and two sliding windows Learning phase: a door and a window Learning phase: a door, a window and two sliding windows

Figure 6 .

 6 Figure 6. Evolution of performance during the incremental learning (Integration of a new class)