Philippe Roudot

Charles Kervrann

Franc ¸ois Waharte

Jérôme Boulanger

LIFETIME MAP RECONSTRUCTION IN FREQUENCY-DOMAIN FLUORESCENCE LIFETIME IMAGING MICROSCOPY

Keywords: FLIM, frequency-domain, fluorescence, spot tracking, template matching, M-estimation

We propose a robust statistical framework for reconstructing lifetime map corrupted by vesicle motion in frequency domain FLIM imaging. Instrumental noise is taken into account to improve lifetime estimation. Robust M-estimators and MLestimators allow to jointly estimate motion and lifetime. Performances are demonstrated on simulated and real samples.

INTRODUCTION

Fluorescence lifetime imaging microscopy (FLIM) is now a widely spread imaging technique for sensing fluorophore environment in a living biological sample (like pH, ions...). Fluorescence lifetime (i.e. average time of a fluorophore staying in excited state before relaxing to its ground state and possibly emitting a photon) is particularly useful to detect the Förster resonance energy transfer (FRET) which can allow to quantify spatial proximity between neighboring molecules [START_REF] Hink | Imaging proteinprotein interactions in living cells[END_REF].

Compared with the well-established time-correlated singlephoton counting (TCSPC) technique, frequency-domain (FD) methods are known to be faster [START_REF] Suhling | Time-resolved fluorescence microscopy[END_REF]. Nevertheless, in addition to the instrumental noise, vesicles motion may corrupt lifetime results [START_REF] Suhling | Time-resolved fluorescence microscopy[END_REF]. Thereby we propose a framework that exploits the FD FLIM output intensity model to jointly estimate lifetimes and trajectories of vesicles.

To our knowledge, no previous work have been done on this specific topic; Fourier transform or sine fitting is often recommended to estimate lifetime in the case of static molecules [START_REF] Spring | Image analysis for denoising fullfield frequency-domain fluorescence lifetime images[END_REF][START_REF] Lakowicz | Principles of fluorescence spectroscopy[END_REF]. However vesicle motion estimation is a heavily studied topic and most of methods rely on particle linking after spot detection in each frame [START_REF] Meijering | Tracking in molecular bioimaging[END_REF], [START_REF] Chenouard | Tracking algorithms chase down pathogens[END_REF]. Spot detection can be performed using for example multi-scale wavelet analysis, top-hat filtering or local curvature analysis (see [START_REF] Smal | Quantitative comparison of spot detection methods in fluorescence microscopy[END_REF]).

In this paper we propose and extension of our work [START_REF] Roudot | Lifetime estimation of moving vesicles in frequency-domain fluorescence lifetime imaging microscopy[END_REF] which takes the Intensified Charged Couple Device (ICCD) Poisson-Gaussian noise model into account. The new estimation scheme improves the accuracy of both background and vesicle lifetime estimation.

The remainder of this paper is organized as follows: Section 2 presents the physical process related to FD FLIM measurements. Section 3 describes our image and noise models and presents a procedure devoted to the estimation of lifetime and movement. Section 4 presents experimental results.

FLUORESCENCE LIFETIME MEASUREMENT BY HOMODYNE DETECTION

FD FLIM images have a very specific intensity footprint which needs to be described for a better understanding of our approach. This section briefly presents the theoretical basis for lifetime measurement in FD-FLIM by homodyne detection (see [START_REF] Lakowicz | Principles of fluorescence spectroscopy[END_REF] for a more detailed description). Given a fluorescence sample excited with a single laser pulse, the impulse response R(t) can be described by a sum of exponential function parameterized by decay times: these lifetimes represent the average time between its excitation time and return time to the ground state of each molecule present in the sample. For the sake of simplicity, we assume the fluorescence decay to be described by a single exponential decay R(t) = R 0 e -t τ where R 0 is the intensity at t = 0 and τ is the decay time.

Whereas the excitation source in time-domain methods is a laser pulse (like with TCPSC), the FD source is a sinusoidally modulated excitation E(t) defined as:

E(t) = C E + A E sin(ω E t + Φ E ) (1)
where ω E denotes the radial frequency of the excitation signal, C E the offset, A E the amplitude and Φ E is the phase delay. Hence, the emitted fluorescent signal is defined as :

F (t) = E(t) * R(t) = C F + A F sin(ω E t + Φ E -arctan(ω E τ )) (2)
where * denotes the convolution operator. Accordingly, the phase delay Φ E -arctan(ω E τ ) of F (t) allows us to recover τ . In an experimental setup, the frequency (> 1 MHz) is too high to be compatible with CCD sensor frame rate. Therefore the signal F (t) is phase-modulated with K sinusoidal signals M k (t), k ∈ [1, K] at frequency ω E (homodyne detection):

M k (t) = C M + A M sin ω E t + Φ M0 + 2πk K . (3)
Thanks to the low pass effect of the CCD detector, the higher frequencies of the phase modulated-signal F (t)M k (t) are negligible. Therefore we obtain K time-independent signals described as a function of k ∈ [1, K] (see Fig. 1):

S θ (k) = G CCD (F (t)M k (t)) = C S + A S cos 2πk K + Φ τ (4) where C S = C F C M , A S = A F A M and Φ τ is the phase.
In what follows, we denote θ = (C S , A S , Φ τ ) T . The phase Φ τ = (Φ M0 -Φ E + arctan(ω E τ )) can be recovered using sine fitting or Fourier decomposition techniques from the lower frequency of S θ (k). The phase delay (Φ M0 -Φ E ) of the system is calibrated by recording a reference frame sequence of a fluorescent sample with known lifetime. Finally, as we monitor the frequency ω E , τ comes straightforwardly. Figure 2 shows an example of lifetime computation using the Fourier transform and specific locations highlights the problem of moving vesicles which generate unrealistic values.

LIFETIME MAP RECONSTRUCTION METHOD

The method presented above has been successfully applied on immobilized samples [START_REF] Suhling | Time-resolved fluorescence microscopy[END_REF]. However we can expect to improve lifetime estimation by taking instrumental noise into account, furthermore considering moving vesicles is crucial when studying lifetime accuracy. In this section, we propose a statistical framework providing a more precise reconstruction of the lifetime map. First we present the image and noise modeling, secondly we describe the lifetime estimator procedure of both background and subresolution vesicles.

Modeling spatial and temporal variations

Several degradations were taken into account to model the vesicle intensity. A Gaussian function approximates the point spread function (PSF) which corresponds to a Bessel function in theory. As the confocal equipment is not rigorously stigmatic the signal is composed of a background and a pinpoint vesicle before microscope diffraction. The background follows the model (4) due to cytoplasmic fluorescence. Finally the background is assumed to be smooth over the PSF support. More formally, let I(k)(x) be the intensity on frame k at location x ∈ Ω (pixel grid):

I(k)(x) = S θ b (k)(x)+S θ0 (k)G γ (x-x 0 (k))+ε(k)(x) (5)
where S θ0 (k) is the vesicle mean intensity parameterized by θ 0 = (C S0 , A S0 , Φ τ0 ) T as in ( 4), S θ b (k)(x) is the spatially varying background signal parameterized by

θ b (x) = (C S b (x), A S b (x), Φ τ b (x)) T , G γ (x)
is a Gaussian function of variance γ 2 describing the shape of the vesicles and centered on the origin, x 0 (k) is the vesicle location on frame k, ε is the noise introduced by the ICCD and described in the next section.

Noise modeling

In this section we characterize the noise induced by the ICCD setup. We present an alternative approach to the derivation described in [START_REF] Spring | Image analysis for denoising fullfield frequency-domain fluorescence lifetime images[END_REF].

In our setup, ICCD is composed of a CCD sensor and a third generation image intensifier which amplifies the photonic signal of the sample. In a nutshell, intensifiers photocathode transforms photons into electrons which are concentrated and accelerated in microchannel plates, the ouput electrons are transformed back into burst of photons using a phosphor screen. As a result, the spatially correlated photons output follows a filtered Poisson process. Considering CCD gain and read out noise, Boddeke showed that the ICCD response I(x) at location x in Ω is [START_REF] Boddeke | Quantitative fluorescence microscopy[END_REF]:

I(x) = g INT g CCD ℵ(x) + ξ (6)
where ℵ(x) is the incident photon number on the ICCD which follows a Poisson law of parameter λ(x), ξ is the CCD read out Gaussian noise such as ξ ∼ N (m ξ , σ 2 ξ ), g CCD is the gain of the CCD sensor and g INT is the gain of the intensifier assumed to be a realization of a random variable. Indeed if CCD gain is often considered as constant, Boddeke experimentally showed that phosphor screen uncontrolled emission impact is non-negligible so that the resulting gain cannot be considered as constant [START_REF] Boddeke | Quantitative fluorescence microscopy[END_REF]. Under the proper assumptions of independence of random variables it comes:

Var[I(x)] =g 2 CCD Var[g INT ℵ(x)] + σ 2 ξ ( 7
)
=g 2 CCD Var[g INT ] Var 2 [ℵ(x)] + g 2 CCD (E 2 [g INT ] + Var[g INT ]) Var[ℵ(x)] + σ 2 ξ
By taking the expectation of ( 6) and knowing that Var[ℵ(x)] = λ(x) = E[ℵ(x)], we have :

Var[I(x)] = Var[g INT ] E 2 [g INT ] (E[I(x)] -m ξ ) 2 (8)
+ g CCD (E[g INT ] + Var[g INT ] E[g INT ] )(E[I(x)] -m ξ ) + σ 2 ξ .
As proposed in [START_REF] Boulanger | Patch-based non-local functional for denoising fluorescence microscopy image sequences[END_REF] where a linear noise model is considered on CCD cameras, we estimate the parameters of the quadratic relation using a robust M-estimator. Each image of the stack is first divided in 7×7 tiles. The values E[I(x)] are approximated by the median of the tile pixels and Var[I(x)] values are approximated using least trimmed square estimation of the pseudo-residual variance. Figure 3 shows an example of robust fitting on a phase calibration stack. In what follows, Var[I(x)] is estimated using the parameter ruling the quadratic relation [START_REF] Roudot | Lifetime estimation of moving vesicles in frequency-domain fluorescence lifetime imaging microscopy[END_REF] and the average local image value for E[I(x)].

Background lifetime estimation

Due to the high number of photon on the ICCD output we can consider the Poisson-Gaussian noise as Gaussian noise with a spatially varying variance Var[I(k)(x)]. This model enables to cope with non-stationary noise and select appropriately the level of confidence. Thus, the background lifetime parameters are estimated by minimizing the following least mean square criterion:

θb = argmin θ b k∈[1,K] I(k)(x) -S θ b (k)(x) 2 Var[I(k)(x)] (9)
where Var[I(k)(x)] is computed by applying the relation [START_REF] Smal | Quantitative comparison of spot detection methods in fluorescence microscopy[END_REF] on pixel value I(k)(x). Figure 4 shows the enhancement provided by taking into account heteroscedasticity of the noise.

Vesicle parameter estimation

This section presents the estimation procedure of the decay time τ 0 for each detected vesicles. In this procedure, the vesicle location x 0 (k) and shape γ and the lifetime parameter θ 0 are jointly estimated. This optimization problem is implemented by successive minimization of criteria with respect to (x 0 (k), γ) and to θ 0 until |

θ n 0 -θ n+1 0 θ n 0 | ≤ δ with n the iteration
number and δ a constant (set to 10 -6 in our experiments).

Iterative minimization procedure

The lifetime parameters θ 0 are estimated by minimizing a weighted least square similar to [START_REF] Boddeke | Quantitative fluorescence microscopy[END_REF]. At iteration n of the minimization algorithm we have :

θn 0 = argmin θ0 (10) k∈[1,K] x∈W (x n 0 (k)) I(k)(x) -S θ0 (k)G γ n (x -xn 0 (k)) 2 Var[I(k)(x)]
where

W (x n 0 (k)) is a 7×7 window centered at xn 0 (k). The motion parameters x n+1 0 (k) are estimated for each frame I(k), k ∈ [1, K] given θn 0 .
Tracking is performed frame-by-frame by minimizing a robust error function, at iteration n + 1 of the algorithm we have :

(x n+1 0 (k), γn+1 ) = argmin (x0(k),γ) (11) x∈W (x0(k)) ρ η I(k)(x) -S θn 0 (k)G γ (x -x 0 (k)) .
The function ρ is an influence function chosen and η = λσ res (k) is a scale parameter used to to weight down outliers during the minimization process. Experiments show that the Leclerc influence function defined as ρ η

(r) = 1 -e -r 2 η 2
gives more stable results even if no significant difference is noticeable with other M-estimators (e.g. Tukey Bi-weighted or German-McClure). The scale parameter σ res (k) is defined for each frame as [START_REF] Rousseeuw | Robust regression and outlier detection[END_REF]:

σ res (k) = 1.4826 median i∈[1,|W (x0(k))|] |r i -median i∈[1,|W (x0(k))|] r i | (12)
where {r i } i∈[1,|W (x0(k)|] is the set of residuals of estimation introduced in (11) when using the L 2 -norm as influence function which is equivalent to a least mean square estimation.

initialization

Motion x 0 0 (0) are initialized by over detecting spots using derivative of Gaussian filtering. Then Gaussian fitting [START_REF] Anderson | Tracking of cell surface receptors by fluorescence digital imaging microscopy using a charge-coupled device camera[END_REF] is applied to initialize x 0 0 (k) and γ 0 . Then, for each iteration n, we set :

γ n+1 = γn , x n+1 0 (k + 1) = xn+1 0 (k -1), for k ∈ [1, K -1]
I(k)(x) G γ n (x-x n 0 (k)) k∈[1,K],x∈W (x n 0 )(k)
.

Φ n τ0 = Φ M0 -Φ E + arctan(ω E τ EGF P ) where τ EGF P is the fluophore lifetime measured without FRET (e.g. 2.5 ns for the EGFP) and (Φ M0 -Φ E ) is obtained by calibration.

RESULTS AND DISCUSSION

This section presents our experimental results conducted on both synthetic and real samples. Synthetic data Our simulations are based on our intensity model presented in Section 3.1 and the noise model presented in Section 3.2. Background is simulated with an experimental vesicle-free sample and cytoplasmic fluorescence is simulated in the background with constant parameter θ b . Vesicle intensities follow [START_REF] Meijering | Tracking in molecular bioimaging[END_REF] and motions are limited to a two a displacement of pixels. The non-stationary Gaussian noise parameters are estimated on the stack and then imposed on synthetic vesicles. Table 1 shows the convergence of lifetime and motion parameter on simulated vesicles.

Real samples Experiments were conducted on living cells using a confocal microscope with spinning disk setup (UMR 144 CNRS-PICT Institut Curie). The intensifier is a third generation from Lambert Instruments (model II18MD). Figure 4 exhibits the accuracy of our lifetime estimation method presented Section 3.3 on a real sample, showing an improvement of the standard deviation from 0.48 to 0.27. This enhancement is consistent with expected results by biologist experts. Figure 4 also highlights our tracking results (70 % of completed track) on the same cell with 30 moving vesicles we previously tracked manually using the MJtrack software [START_REF] Smal | Quantitative comparison of spot detection methods in fluorescence microscopy[END_REF]. We compare our tracker with Gaussian fitting and the particle tracker plugin in ICY and presented in [START_REF] Genovesio | Multiple particle tracking in 3-d+ t microscopy: Method and application to the tracking of endocytosed quantum dots[END_REF] after normalization. Figure 2 shows the reconstruction of the lifetime map. It is built by estimating lifetime on the background (Section 3.3), then by drawing a patch of size γ and value τ0 at location x0 (0) for each vesicle on the background lifetime map.

CONCLUSION

Our method allows to reconstruct lifetime map of sample usually corrupted by vesicle motion in a efficient and fast way in FD FLIM imaging. This study shows the importance of taking the instrumental noise and the intensity model into account while tracking vesicles. Other trackers such as linking method will be compared to assess the differences in performance of our procedure which is much less demanding in computation compared to spot detection at each frame.

Fig. 1 .

 1 Fig. 1. An example of FD FLIM measurement with K = 6 phase-modulation signal.

Fig. 2 .

 2 Fig. 2. Left: Lifetime map using Fourier decomposition. Dark blue and red spots are irrealistic values due to organel movements. Right: lifetime map reconstruction using our method.

Fig. 3 .

 3 Fig. 3. Robust fit between E[I(k)(x)] and Var[I(k)(x)] using a phase calibration stack.

Fig. 4 .

 4 Fig. 4. Results on real sequences : lifetime histograms on the whole image and tracking performance on fluorescently tagged live cell RPE1 (GFPtagged IFNAR1 and mCHerry-tagged Tyk2).

Table 1 .

 1 Convergence of tracking and lifetime errors is reach after 3 iteration on 30 simulated vesicles and 95 % of vesicle tracked.

	Iteration n	0	1	2	3
	Tracking mean error (pixel) 0.240 0.121 0.081 0.080
	Lifetime mean error (ns)	0.387 0.348 0.345 0.343
	lifetime parameters θ n 0 = (C n S0 , A n S0 , Φ n τ0 ) T are initialized as follows : C n S0 and A n S0 are respectively the mean and the max-
	imum value of the set

ACKNOWLEDGEMENT

The authors thanks Cedric Blouin for providing the experimental data.


Denis Fortun

Charles Kervrann

SEMI-LOCAL VARIATIONAL OPTICAL FLOW ESTIMATION

Keywords: Optical Flow, variational methods, minimization domain, motion segmentation, discrete optimization

Global variational methods for optical flow estimation usually suffer from an over-smoothing effect. We propose a semilocal estimation framework designed to integrate and improve any variational method. The idea is to implicitly segment the minimization domain into coherently moving windows. In a first time, local variational estimations are performed in overlapping candidate square regions. Then, a global discrete optimization, non subject to the over-smoothing introduced by variational approaches, selects the optimal window for each pixel. Experimental results show an increasing of the sharpness of discontinuities and a significant improvement of global registration errors compared to the results of the baseline global variational method.

INTRODUCTION

The optical flow approximates the projection of the motion of a 3D scene on the image plane. Any optical flow estimation have thus to be based on a conservation assumption of some optical properties of the image able to capture the real motion (intensity, gradient, image descriptor . . . ). This data conservation constraint provides in general a single equation and is consequently insufficient to recover the two components of the motion field (aperture problem). The typical way to overcome this under-determination is to add to the data conservation constraint a spatial coherency constraint. Existing methods can be classified regarding their local or global strategy to impose such a constraint.

The spatial coherency of local approaches is ensured at a pixel x ∈ Ω ⊂ R 2 by the assumption of common parametric motion (translational in [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF]) in a neighborhood V (x) ⊂ Ω, where Ω is the image domain. The global approach allows to compute a dense motion field and explicitely adds to a data potential ρ data (•) which penalizes deviations from the data conservation constraint, a regularization potential ρ reg (•) which penalizes high values of the norm of the gradient ∇w This work was realized as part of the Quaero Program, funded by OSEO, French State agency for innovation. of the velocity field w : Ω → R 2 . A global energy combining these two potentials is minimized [START_REF] Horn | Determining optical flow[END_REF]:

E global (w) = Ω ρ data (x, I, w) + λ ρ reg (x, ∇w)dx (1)
where I : Ω × [0, T ] → R is an image sequence and λ is a balance parameter between data fitting and regularization.

The best results of the state-of-the-art are achieved by global variational motion estimation methods. However, this kind of methods still suffer from an over-smoothing effect. This phenomenon is particularly visible in the seminal work of [START_REF] Horn | Determining optical flow[END_REF] which uses quadratic penalty function for the regularization potential. This shortcoming has been greatly reduced by the introduction of robust penalty functions [START_REF] Black | The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields[END_REF][START_REF] Mémin | Dense estimation and object-based segmentation of the optical flow with robust techniques[END_REF], adaptation of the regularization along image discontinuities [START_REF] Wedel | Structure-and motionadaptive regularization for high accuracy optic flow[END_REF] or non-local regularization strategies [START_REF] Werlberger | Motion estimation with nonlocal total variation regularization[END_REF], but it still remains. Indeed these methods are limited by the coarse-to-fine scheme [START_REF] Mémin | Dense estimation and object-based segmentation of the optical flow with robust techniques[END_REF][START_REF] Brox | High accuracy optical flow estimation based on a theory for warping[END_REF], necessary to cope with large displacements. This approach avoids most local minima due to the non-convexity induced by the non-linearized data potential, at the price of an over-smoothing of the discontinuities.

We mention two non-variational approaches related to our method that have been investigated to reduce the oversmoothing effect of global variational methods: (i) parametric motion estimation based on motion field segmentation; (ii) discrete optimization of the energy (1). In the first case, a parametric model of the flow field is estimated inside coherently moving regions. The estimation of the discontinuities is thus transfered to the segmentation step [START_REF] Sun | Layered image motion with explicit occlusions, temporal consistency, and depth ordering[END_REF]. In the second case, discrete optimization of the energy (1) is able to find strong minima for non-convex functionals without coarse-tofine schemes, but is limited by the quantization of the flow field range [START_REF] Boykov | Fast approximate energy minimization via graph cuts[END_REF].

In this paper, we present a method combining local estimations and discrete optimization to sharpen the discontinuities of a global variational method by implicitly segmenting the flow field. It is composed of two stages: first, local variational estimations are performed on a regular grid of overlapping windows; second, the resulting local motion vectors are used as candidates for a global discrete optimization. In this scheme, the discrete optimization module selects of the optimal spatial minimization domains, yielding an implicit segmentation of the flow field. It is worth noting that our framework is general and can be used to improve any baseline variational method. The results with the popular and representative method [START_REF] Brox | High accuracy optical flow estimation based on a theory for warping[END_REF] show significant improvements over the global variational approach when applied on several sequences of the Middlebury database [START_REF] Baker | A database and evaluation methodology for optical flow[END_REF].

VARIATIONAL OPTICAL FLOW ESTIMATION

Global variational approach

All global variational optical flow estimation methods are based on the minimization of the energy (1). The method described in [START_REF] Brox | High accuracy optical flow estimation based on a theory for warping[END_REF] contains most of the basis concepts still used in the most recent methods. Therefore we used this method to evaluate our semi-local framework, which can integrate any other variational optical flow estimation methods. The main features of the method [START_REF] Brox | High accuracy optical flow estimation based on a theory for warping[END_REF] are described in this section.

Global energy functional The data potential penalizes deviations from intensity and gradient conservation constraints with a L 1 penalty function:

ρ data (x, I, w) = φ |I(x + w(x), t + 1) -I(x, t)| 2
(2)

+ γφ ∇I(x + w(x), t + 1) -∇I(x, t) 2
where γ > 0 is a balance parameter and φ z 2 = √ z 2 + ǫ 2 is a regularized form of the L 1 norm with ǫ a small constant.

The regularization potential penalizes high gradients with the same convex and discontinuity-preserving L 1 norm:

ρ reg (x, ∇w) = φ ∇u(x) 2 + ∇v(x) 2 (3)
Energy minimization The minimization of ( 1) is performed by solving the Euler-Lagrange equations. To make these equations tractable, the data potential (2) must be linearized, which limits the estimation to small displacements. Therefore all recent variational methods adopted a coarse-to-fine scheme to handle large displacements [START_REF] Mémin | Dense estimation and object-based segmentation of the optical flow with robust techniques[END_REF][START_REF] Brox | High accuracy optical flow estimation based on a theory for warping[END_REF]. The coarseto-fine levels are interpreted in [START_REF] Brox | High accuracy optical flow estimation based on a theory for warping[END_REF] as fixed point iterations enabling the minimization of the initial non-linear energy.

At each level, a second fixed point allows to cope with the remaining non-linearity of the Euler-Lagrange equations due to the L 1 norm. The resulting linear system is then solved with Successive Over Relaxation (SOR).

Restriction to local domains

The minimum reached by global variational methods is usually suboptimal. Indeed, variational optimization is proved to find the global minimum only for convex energies, which is not the case of (1) with the non-linearized data potential [START_REF] Horn | Determining optical flow[END_REF]. Actually, the coarse-to-fine scheme transforms the problem into successive minimizations of convex approximations of (1) which tend to smooth the discontinuities of the flow field.

To reduce this over-smoothing effect we propose to minimize several energies of the form (1) over local regions, inspired by the localization of Total Variation for denoising [START_REF] Louchet | Total variation as a local filter[END_REF]. The local motion field w

V : V ⊂ Ω → R 2 is the minimizer of E(w V ) = V ρ data (x, I, w V ) + λ ρ reg (x, ∇w V )dx (4)
obtained with the method described in the previous section. If V is a coherently moving region without strong discontinuities, the over-smoothing does not occur. We emphasize that contrary to other approaches computing optical flow in local regions [START_REF] Sun | Layered image motion with explicit occlusions, temporal consistency, and depth ordering[END_REF][START_REF] Cw Zitnick | Consistent segmentation for optical flow estimation[END_REF], where the motion is restricted to a parametric model, we prefer to compute a regularized flow field. Thus we overcome the difficulty of local parametric approaches to handle complex motion fields, where the region V must be small enough to ensure the validity of the parametric assumption, and large enough to avoid the aperture problem. Figure 1 shows how the localization of variational methods can improve the accuracy of the global approach when the region is suitably chosen: as the window size increases, the details of the flow tends to disappear. Our strategy to select the optimal region of each pixel is presented in the next section.

SEMI-LOCAL FRAMEWORK

Our aim is to estimate the motion at each pixel with a local variational model (4) in coherently moving regions. Existing similar approaches are restricted to parametric motion and are based either on an image gradient segmentation [START_REF] Cw Zitnick | Consistent segmentation for optical flow estimation[END_REF], subject to over-segmentation of the flow field, or on global variational minimization of non-convex functionals [START_REF] Sun | Layered image motion with explicit occlusions, temporal consistency, and depth ordering[END_REF] for joint estimation of regions and motion, suffering from the drawbacks described in the previous section. Our semi-local framework performs an implicit segmentation of the flow field, without global variational estimation or image segmentation.

Local estimations

Local variational estimations are performed on overlapping square windows of different sizes.

For a fixed size s, let V s,α be a set of regularly spaced windows covering the whole image, with an amount of overlap α between neighbors (see Fig. 2). For a set of varying sizes S = {s 0 , . . . , s n }, we define V S,α = s∈S V s,α . For each window V ∈ V S,α , a local motion field w V is estimated by minimization of (4).

One pixel is contained in several overlapping windows with different locations and sizes. We denote N V (x) the set of windows containing the pixel x (see Fig. 2, N V (x) = V 1 , .., V 4 }). The computed flow fields over these windows provide a set {w V (x)} V ∈NV (x) of candidate motion vectors at each pixel x.

Global aggregation

The aggregation step aims at combining the locally estimated flow fields to compute an optimal global flow field. The goal is to select at each pixel the most appropriate window. To this end, we consider the aggregation as a multi-label assignment problem, where local candidates {w V (x)} V ∈NV (x) constitute the discrete label space at pixel x. The global flow field w Ω resulting from the aggregation is then the minimizer of a global objective energy:

w Ω = arg min w E Ω (w) s.t. w(x) ∈ {w V (x)} V ∈NV (x) . (5)
Thus, the solution is found by selecting the best motion vector among the small set of candidates {w(x)} V ∈NV (x) . We define E Ω as:

E Ω (w) = x∈Ω ρ data (x, I, w) + λψ reg (x, w) (6)
where ψ reg (•) is a Markov Random Field prior:

ψ reg (x, w) = y∈∆(x) φ |u(x) -u(y)| 2 + |v(x) -v(y)| 2
where ∆(x) is a 4 or 8 pixel neighborhood. The energy E Ω is actually a discrete version of the energy (1). The difference with global variational minmization of ( 1) is that we impose the solution to belong to the set of local variational estimations {w V (x)} V ∈NV (x) . Our discrete optimization scheme does not suffer from the over-smoothing effect of the global variational approach. Consequently, it selects the candidates coming from coherently moving regions, which are not affected by this over-smoothing (see the smallest window in Fig. 1). This selection of the optimal window at each pixel can be seen as an implicit segmentation of the spatial minimization domain used for the baseline variational method.

Discrete optimization

In this subsection we detail our approach for the discrete optimization problem [START_REF] Wedel | Structure-and motionadaptive regularization for high accuracy optic flow[END_REF]. The formulation of (5) differs from the classical multi-label assignment scheme because the regularization is not applied to the discrete labels but to the continuous-valued motion vectors assigned to the labels. This problem has been addressed in the context of optical flow estimation in [START_REF] Lempitsky | Fusion moves for markov random field optimization[END_REF] with candidates estimated by several global methods. The authors achieved the multi-label optimization with the fusion-move algorithm, which operates by successive fusions of proposal labellings (see [START_REF] Lempitsky | Fusion moves for markov random field optimization[END_REF] for more details about fusion-move and its applications). We apply this technique for solving [START_REF] Wedel | Structure-and motionadaptive regularization for high accuracy optic flow[END_REF].

The fusion-move algorithm requires a set of global flow fields {w Ω1 , . . . , w ΩN } to be fused. Let us consider that the set of candidates at each pixel {w V (x)} V ∈NV (x) is composed by a maximum number of N candidates. Then for i ∈ {1 . . . N }, we assign to w Ωi (x) one arbitrarily chosen candidate in {w V (x)} V ∈NV (x) , so that each candidate is assigned to at least one global flow field w Ωi (x). Our experiments showed that the assignment strategy of the local candidates in the global flow fields has negligible influence on the final result.

RESULTS AND DISCUSSION

We use the Average Angular Error (AAE) to evaluate the performance of our method on sequences of the Middlebury benchmark [START_REF] Baker | A database and evaluation methodology for optical flow[END_REF]. Two aggregation procedures are considered: SL-fusion performs the discrete optimization described in Section 3.2 and 3.3; SL-Mean performs a weighted mean of the local candidates, favoring central pixels with a gaussian filter centered on each window. In all our experiments we fix the parameters of [START_REF] Brox | High accuracy optical flow estimation based on a theory for warping[END_REF] λ = 40 and γ = 5.

Table 1 compares the AAE of the baseline variational method [START_REF] Brox | High accuracy optical flow estimation based on a theory for warping[END_REF] with SL-fusion and SL-Mean, for several sets of window sizes S. The superior performance of SL-fusion over SL-Mean highlights that the selection of the best window is crucial to prevent the global flow field from being influenced by outliers coming from inappropriate regions. For SL-fusion, the errors obtained with single sizes are always higher than those obtained with their combination. This result shows that a single window size is not able to capture all types of coherently moving regions and that the aggregation procedure successfully combines the advantages of each size by selecting the best region. The results of SL-fusion with S = {15, 49, 129} are significantly better than those of the baseline variational method for the sequences Grove3, Rubberwhale and Grove2. This is due to the better preservation of the discontinuities illustrated in Fig. 3. We mention that even for very smooth sequences like Dimetrodon, large window sizes (here 129) ensure that the result cannot be worse than the global variational method, and is even slightly improved. The influence of the amount of overlap α is shown in Fig. 4. As it can be intuitively expected, the error decreases when the overlap increase. Indeed, the overlap determines the number of candidate regions, and thus the probability that the windows fall in appropriate regions.

CONCLUSION AND FUTURE WORK

We proposed a new approach for optical flow estimation, combining global methods with local minimization regions. Our experiments showed that our semi-local framework improves the estimation accuracy of a baseline variational method along discontinuities of the motion field, by performing an implicit segmentation of the spatial minimization domain. In the future we plan to extend this general framework to non-variational baseline methods and adaptive region shapes.

Fig. 1 .

 1 Fig. 1. Influence of the spatial minimization domain. 1 st row: two frames and ground truth of the motion field with a zoom on a region of interest (green square). 2 nd and 3 d rows: variational estimations over windows of different sizes centered on the same green square.

Fig. 2 .

 2 Fig. 2. Illustration of the patches distribution for a given size s and overlapping ratio α = 0.3. The pixel x is contained in four patches V1, . . . , V4 providing four candidates for x: wV 1 (x), . . . , wV 4 (x).

Fig. 3 .Fig. 4 .

 34 Fig. 3. Visual comparison between global variational estimation (1 st column) and its integration in SL-fusion with S = {15, 49, 129} and α = 0.75 (2 nd column) on Grove3 and Rubberwhale.

Table 1 .

 1 Comparison of the results (AAE) obtained with our implementation of[START_REF] Brox | High accuracy optical flow estimation based on a theory for warping[END_REF], SL-Mean and SL-fusion for α = 0.75.

		Grove2 Grove3 Rubber-Dime-
				whale	trodon
	Variational [7]	2.38	5.97	3.92	1.83
	SL-Mean
	S = {15} S = {49} S = {129} S = {15, 49, 129} SL-fusion	4.91 2.43 2.38 4.21	17.7 6.11 6.01 16.2	5.42 4.04 3.98 5.05	4.40 1.91 1.83 3.54
	S = {15} S = {49} S = {129} S = {15, 49, 129}	2.95 2.27 2.30 2.10	12.8 5.85 5.83 5.60	4.47 3.69 3.71 3.34	3.30 1.87 1.81 1.79


H Bannour
email: hajer.bannour@irstea.fr

C Barral

N Touze-Foltz

FLOW RATE IN COMPOSITE LINERS INCLUDING GCLS AND BITUMINOUS GEOMEMBRANES

Keywords: Geosynthetics, composite liners, geosynthetic clay liner, bituminous geomembrane, flow rate

Some studies were performed in the past years regarding the behavior of geosynthetic clay liners (GCLs) as part of a composite liner, focused on the situation where a GCL is located under a hole in a high density polyethylene (HDPE) geomembrane (GM). In this case, the contact between the geomembrane and the GCL was quantified in terms of flow rate through the composite liner and in terms of interface transmissivity. Various situations were tested in the past in order to evaluate the situation where a smooth GM is used in contact with the GCL. The situation where a textured geomembrane is used in contact with the GCL was also evaluated. More recently, the effect of the nature of the bentonite in the GCL, sodium or calcium bentonite, leading to different flow rates in the GCL was evaluated. In all cases an HDPE GM was used. This choice is linked to the fact that it is the most commonly GM used in composite liners including GCLs for chemical compatibility reasons. However one could imagine that the use of other GMs, like bituminous geomembranes (B-GMs) associated to a GCL, could be adapted for hydraulic applications. A quantification of flow rates was thus performed in laboratory tests at the decimetric scale for the case of a damaged B-GM located on top of a GCL. This corresponds to an alternative design for a canal projected in France at the moment. The effect of the side of the GM (polymeric film or sand layer for the bituminous geomembrane) in contact with a GCL containing calcium bentonite was evaluated. Results obtained are presented in this paper and compared depending on the GM side in contact with the GCL.

INTRODUCTION

Bituminous geomembranes (B-GMs) are used in many geotechnical and environmental protection applications and specifically for hydraulic, infrastructure and transportation fields such as water storage and transportation where B-GMs were used in lining drinking and navigation water canals [START_REF] Domange | The use of bituminous geomembranes in canals reclaiming[END_REF], Duquennoi et al. 1995[START_REF] Etienne | The use of bituminous geomembranes for rehabilitating the navigation canal watertightness[END_REF][START_REF] Breul | Bitumen geomembranes in irrigation-case histories from a range of climates[END_REF][START_REF] Potié | Bitumen geomembranes in canals[END_REF][START_REF] Fagon | Record of 10 years of use of GEOSYNTHETICS ASIA 2012 5 th Asian Regional Conference on Geosynthetics 10 to 15[END_REF], Flaquet-Lacoux et al. 2005), lining, reinforcement and protection of dams and reservoirs [START_REF] Tisserand | Reservoir of Guazza. a large reservoir of unprotected membrane[END_REF][START_REF] Alonso | Results of seventeen years of using geomembranes in dams and basins[END_REF][START_REF] Girard | Waterproofing and rehabilitation of dams by geomembranes in France[END_REF], Poulain el al. 1997, Breul and Herment 1997[START_REF] Breul | Bitumen geomembranes in irrigation-case histories from a range of climates[END_REF][START_REF] Breul | Two case histories of dams waterproofing in South America with BGMs[END_REF] and also for retention ponds [START_REF] Breul | Bituminous geomembrane in extremely cold conditions[END_REF]) and aquifer protection from environmental impact of roads [START_REF] Coppinger | Proc. International Conference of Geosynthetics[END_REF]. Apart from environmental applications, B-GM were also used as components of the liner in cover systems in landfills [START_REF] Ossena | Design and construction of impervious cap to low level nuclear waste repository. Cover system by geosynthetic clay liner in Hautes Gayeulles municipal waste landfill[END_REF], Potié et al. 1997[START_REF] Faure | Water Proofing by using bituminous geomembrane on a maturation clinkers area[END_REF][START_REF] Peggs | Prefabricated bituminous geomembrane: a candidate for exposed geomembrane caps fro landfill closures[END_REF][START_REF] Marchiol | Examination of a bituminous geomembrane after 12 years of use as a cover for a waste storage area[END_REF]) mining [START_REF] Breul | Use of bituminous geomembrane (BGM) liner for agnico eagle mine in Kittila (Finland)[END_REF], tunnels [START_REF] Benchet | Partitioned waterproofing of a double archway work made by bituminous geosynthetic barrier[END_REF], ditches [START_REF] Imbert | Ditch-Coating bu bituminous geomembranes-two pilot sites[END_REF], railways (Imbert et al. 1997), and road foundations (Breul and[START_REF] Breul | BGMs road foundation soils[END_REF] As presented by [START_REF] Breul | Use of bituminous geomembrane (BGM) liner for agnico eagle mine in Kittila (Finland)[END_REF], the structure of a B-GM is generally composed by (Fig. 1): (i) a non-woven polyester geotextile whose mass per unit area is 200 to 400 g/m², (ii) a glass fleece reinforcement which provides stability during fabrication and contributes to the strength of the GM, (iii) a bituminous mastic consisting of a blown 100/40 pen bitumen, and filler. This mastic impregnates the whole structure and gives the waterproofing of the product and ensures the longevity and the high resistance of the product, (iv) a Terphane film bonded to the underside when the membrane is hot, which prevents penetration of the geomembrane by plant roots, and (v) a coating of fine sand on the upper surface to provide a greater traction on slopes, giving greater operator safety and security, and to give protection from the degrading effects of UV radiation [START_REF] Breul | Use of bituminous geomembrane (BGM) liner for agnico eagle mine in Kittila (Finland)[END_REF]. On another hand, design engineers working on projects requiring the use of a GM in a composite liner with a geosynthetic clay liner (GCL) often limit their consideration of options to a high density polyethylene (HDPE) GM, perhaps on the basis of their experience with landfill design. Research performed on HDPE GMs and GCLs as parts of a composite liner focused on the situation where the GM is presenting a hole, aims to quantify the flow rate and the interface transmissivity between the GM and the upper geotextile of the GCL. The objective of this paper is to provide design engineers with elements to consider a broader range of GMs in their designs and to use a rational approach for GM selection for hydraulic applications.

In this paper, first, previous results on interface transmissivity measurements in composite liners composed by a HDPE GM and a GCL are presented. Second, materials tested and interface transmissivity measurement procedure is presented. Finally flow rates and interface transmissivity test results obtained on a composite liner made with a B-GM-GCL containing calcium bentonite are presented and discussed.

FLOW RATES AND INTERFACE TRANSMISSIVITY IN COMPOSITE LINERS WITH A GCL, INCLUDING A DAMAGED GM

The work performed in the past years regarding the behaviour of a composite liner containing a GCL and a punctured HDPE GM was focused on obtaining flow rates and interface transmissivities. The flow through a defect in the GM depends, as indicated by [START_REF] Brown | Quantification of Leakage Rates through Holes in Landfill Liners[END_REF], on the contact between the GM and the underlying soil liner. According to these authors, if the contact is not perfect, once fluid has migrated through the defect, it spreads laterally through the gap existing between the GM and the underlying soil, called interface. This interface flow covers an area called wetted area. Finally, the liquid migrates into and through the soil liner. The contact between the GM and the GCL was quantified in terms of flow rate through the composite liner and in terms of interface transmissivity.

Analytical Solution for Calculating Interface Transmissivity in a Composite Liner

To calculate the GM-GCL interface transmissivity θ, an analytical solution developed by [START_REF] Touze-Foltz | Liquid flow through composite liners due to geomembrane defects: Analytical solutions for axi-symmetric and two-dimensional problems[END_REF] for the case of a circular defect in the GM was used. This analytical solution assumes that: (i) the interface transmissivity is uniform; (ii) the liquid flow in the transmissive layer is radial; (iii) the flow occurs under steady-state conditions; (iv) the compacted clay liner (CCL), the GCL and the GM-GCL interface are saturated; and (v) the additional flow through the passive barrier (CCL + GCL) is one-dimensional and vertical. The final flow rates (steady-state conditions) measured in the transmissivity tests were used in the calculations. It should be pointed out that the interface transmissivity calculated based on the analytical solution described above should be viewed as an apparent transmissivity, due to the fact that preferential flow paths occurred in the tests, as discussed above, which were not considered in the development of the analytical solution employed.

In a composite liner, a great fraction of the liquid that passes through the GM puncture flows along the GM-GCL interface, moving laterally to a certain distance from the GM puncture before infiltrating into the GCL and underlying layers. The contour of the region reached by the fluid defines the wetted area. Under the conditions of the tests performed in this work, the radius of the wetted area is the internal radius of the cell. Equations 1 to 5 below apply to the boundary condition where the hydraulic head is equal to zero at a certain radius in the specimen, which in the present case is the cell radius:

(1) Where: Q is the flow rate under steady-state conditions; r 0 is the circular defect radius; k s is the hydraulic conductivity of the liner GCL + (CCL); h w is the hydraulic head; d s is the thickness of the liner (GCL + CCL); H s is the thickness of the soil component of the composite liner (GCL + CCL); θ is the interface transmissivity; I 1 and K 1 are modified Bessel functions of the first order; and α, A and B are parameters given by Equations 2 to 5, as follows (2)

(3) Sanding Bitumen impregnation Non woven polyester geotextile

Glass fleece reinforcement

Anti-root film

( ) ( ) [ ] 0 1 0 1 0 s s w s 2 0 2 r BK r AI r H d h k r Q α α α θ π π - - + = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 0 r I R K R I r K r K R K H R K h A s w α α α α α α α - - + - = s s d k θ α =
GEOSYNTHETICS ASIA 2012 5 th Asian Regional Conference on Geosynthetics 10 to 15 December 2012 | Bangkok, Thailand 3 (4)

(5)

Where K 0 and I 0 are modified Bessel functions of zero order and R is the radius of the wetted area.

It should be pointed out that the interface transmissivity calculated based on the analytical solution should be viewed as an apparent transmissivity, due to the fact that preferential flow paths can occur in the tests due to the application of the load through a granular layer having points of contact with the GM. The type of contact between the GM and the GCL, were not considered in the development of the analytical solution employed.

Previous Interface Transmissivity Studies Carried Out on Composite Liners Made with GCLs

Various situations were tested in the past in order to evaluate the effect of a smooth GM in contact with the GCL [START_REF] Harpur | Evaluation of the contact between geosynthetic clay liners and geomembranes in terms of transmissivity[END_REF][START_REF] Barroso | Laboratory investigation of flow rate through composite liners involving GCLs[END_REF][START_REF] Mendes | Influence of structural and material properties of GCLs on interface flow in composite liners due to geomembrane defects[END_REF]. [START_REF] Harpur | Evaluation of the contact between geosynthetic clay liners and geomembranes in terms of transmissivity[END_REF] verified that under steady-state conditions, the most significant fraction of the flow takes place along the interface between the GM and the cover geotextile of the GCL, through the cover geotextile, and along gaps between the cover geotextile of the GCL and the bentonite. A less important amount of fluid percolates through the bentonite and below the GCL. [START_REF] Barroso | Laboratory investigation of flow rate through composite liners involving GCLs[END_REF]Barroso et al. ( , 2010) ) examined the influence of the hydraulic head, pre-hydration of the GCL and confining stress on the GM-GCL interface transmissivity. The results obtained by those authors showed that it is difficult to establish general trends expressing the influence of pre-hydration, confining stress and hydraulic head on the interface transmissivity. Nevertheless, it seems that, regarding the flow rate, it is important to take into account both the initial water content of the specimen and the confining stress [START_REF] Barroso | Laboratory investigation of flow rate through composite liners involving GCLs[END_REF]). The confining stress affects differently the flow rate, depending on the initial water content of the specimen. In fact, the flow rate in pre-hydrated GCLs is about one order of magnitude larger in tests under a confining stress of 50 kPa than in tests under 200 kPa. On the other hand, for non-pre-hydrated specimens, the flow rates are similar for the two confining stresses under steady-state flow conditions [START_REF] Barroso | Laboratory investigation of flow rate through composite liners involving GCLs[END_REF].

The situation where a textured HDPE GM was used in contact with the GCL was also evaluated [START_REF] Barroso | Influence of the textured structure of geomembrane on the flow rate through geomembrane GCL composite liners[END_REF]. Results showed that tests were reproducible and that the texture had a small impact on flow rates obtained at steady-state, although, at the beginning of the tests, larger flow rates were obtained with smooth GMs than with textured ones. This suggests that, at the early phases of the tests, the water flows more easily at the interface when smooth GMs are used. The texture seems to reduce the space available at the interface for the water flow. However, with time, the sodium bentonite in the GCL swelled resulting in a better contact between the GM and the GCL.

More recently, the effect of the nature of the bentonite in the GCL, sodium or calcium bentonite, leading to different flow rates in the GCL was evaluated [START_REF] Mendes | Influence of structural and material properties of GCLs on interface flow in composite liners due to geomembrane defects[END_REF]. Those authors concluded that the nature of the bentonite and the manufacturing process of the GCLs studied did not affect the GM-GCL interface transmissivity when steady-state flow conditions were reached. They did also notice that for hole diameters in the range 4 to 10 mm the diameter of the hole in the GM did not significantly influence the flow rate through the GM-GCL composite liner: the expansion of the sodium bentonite was effective in blocking the puncture in the GM, yielding to a significant reduction on the flow rate. The results suggest that GCLs containing sodium bentonite, whose hydraulic conductivity increases due to cation exchange, can still maintain a good performance in a composite liner in terms of GM-GCL interface transmissivity and flow rate through the composite liner.

EXPERIMENTS Materials Tested

A stitch bonded GCL containing powdered calcium bentonite is tested in this study in contact with a B-GM. Both geotextile on the two faces of the GCL were woven. The mass per unit are of dry calcium bentonite in the GCL is 8.10 kg/m². The B-GM, according to the fabric presentation made in Table 1 exhibit two sides with different aspects: a polymeric film is located on one surface whereas a sand layer is encountered at the surface of the second side. These layers are different in terms of their roughness as the polymeric film face (Fig. 2) is smoother than the other one made with a sand impregnation (Fig. 3). 2002), [START_REF] Cartaud | Influence of geotextiles at the interface of landfill bottom composite liners[END_REF] and [START_REF] Barroso | Laboratory investigation of flow rate through composite liners involving GCLs[END_REF], [START_REF] Barroso | Influence of the textured structure of geomembrane on the flow rate through geomembrane GCL composite liners[END_REF], Barroso et al. (2010), [START_REF] Mendes | Influence of structural and material properties of GCLs on interface flow in composite liners due to geomembrane defects[END_REF] it consists of a Plexiglas cell basically composed by four parts: (i) a bottom plate which supports the soil and applies the confining stress; (ii) a 200 mm inside diameter base cylinder, 80 mm high, to accommodate the CCL and the GCL specimen; (iii) a top coarse granular drainage layer; and (iv) an upper cylinder that accommodates the granular layer.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 0 r I R K R I r K r I R I H R K h B s w α α α α α α α - - + = ( ) ( ) 0 1 1 = - + s H R BK R AI α α GEOSYNTHETICS ASIA
In the experimental setup, the B-GM, exhibiting a circular hole, is located on the GCL. To assemble the test, initially the base soil, simulating a CCL, is compacted in the bottom cylinder of the equipment with the upper side contacting a rigid metallic plate in order to ensure a smooth CCL surface underneath the GCL. The final thickness of the CCL is about 6 cm. The internal walls of the bottom cylinder are lubricated before soil compaction, in order to minimise friction between the CCL and these walls during the tests. Once the test cell is closed, the B-GM is ensuring the watertightness of the upper part of the cell. When the flow rate is important, it is noticed at the downstream side of the cell but when flow rate is not visible to the trained eye, readings are made at the upstream side of the cell. In order to get comparable results between the various testing configurations presented in the literature and tests performed in this study, a 4mm diameter hole in the geomembrane is used, and a 50kPa load and a 0.3m hydraulic head applied. The CCL layer used is similar to the one from previous studied performed by [START_REF] Barroso | Laboratory investigation of flow rate through composite liners involving GCLs[END_REF][START_REF] Barroso | Influence of the textured structure of geomembrane on the flow rate through geomembrane GCL composite liners[END_REF] and [START_REF] Mendes | Influence of structural and material properties of GCLs on interface flow in composite liners due to geomembrane defects[END_REF] (see Table . 2). As can be noted, flow rate along the B-GM-GCL interface decreases gradually versus time during 300 hours for the two cases (rough face and smooth face in contact with the GCL) (Fig. 5). Afterwards the flow rate does no longer evolve. This corresponds to steady state. Table 3 shows flow rates, hydraulic conductivities and interface transmissivities results obtained at steady-state. Hydraulic conductivities are obtained using results of previous tests carried out in oedopermeameter cells using NF P 84-705. Apparent interface transmissivities are calculated using the analytical solution proposed by [START_REF] Touze-Foltz | Liquid flow through composite liners due to geomembrane defects: Analytical solutions for axi-symmetric and two-dimensional problems[END_REF] (Equation 1). We conduct a comparison of flow rates obtained in composite liners according to the side in contact with the GCL. Results show that the smoother side in contact with the GCL gives rise to lower flow rates than for the rougher side in transient state (Fig. 5). This could be explained by the fact that the rough side allows the storage of a larger quantity of water at the interface with the GCL, than the smooth side of the B-GM. B-GM-GCL contact may also present interface irregularities so water could have preferential flow paths between the B-GM and the GCL. Once pores are filled and the bentonite is swollen, results were not affected. So in the end of the test, flow rates are similar for both configurations at steady state. As shown, in Table 2, flow rate obtained are equal to 3.89 × 10 -11 m 3 / s for the case of the rough side of the B-GM (sand impregnation) and 3.94 × 10 -11 m 3 / s for the case of the smooth side of the B-GM (film side). It is thought that these minor differences can be associated to the difference on the contact quality between the two configurations at steady state, as water flows more easily at the interface when smoother GMs are used. This steady state results agrees with the one of tests carried out by Barroso et al. (2010) on smooth and textured GMs when flow rates values were about 1.1 × 10 -11 m 3 / s when a smooth GM is used and between 7.6 × 10 -12 m 3 / s and about 1.9 × 10 -11 m 3 / s with a textured GM at steady state. Fig. 6 gives a synthesis of the various interface transmissivity data obtained from the literature and from this study. All data corresponding to the two configurations (rough side or smooth side of the B-GM in contact with the GCL) are located under the GM-GCL contact condition defined by [START_REF] Barroso | Laboratory investigation of flow rate through composite liners involving GCLs[END_REF] linking the interface transmissivity θ to the hydraulic conductivity k GCL of the GCL according to Equation 6:

GCL k . . θ log 7155 0 2322 2 log + - = (6)
This result shows the efficiency of this B-GM-GCL composite liner to reduce interface transmissivity and flow rates along the interface at level that are comparable to the ones obtained in the case of the use of a HDPE GM in composite liner.

CONCLUSIONS

The work performed in this study aims to quantify flow rate measurements throw composite liners defined by a defective B-GM presenting two sides (a rough side defined by a sanding and a smooth side made with a film) and a GCL. Tests are performed under a 50 kPa confining pressure and a 0.3m constant hydraulic head in an interface transmissivity test cell. The B-GM-GCL composite 10 -11 10 -10 10 -9 10 -8 10 -7

GEOSYNTHETICS ASIA 2012 5 th Asian Regional Conference on Geosynthetics 10 to 15 December 2012 | Bangkok, Thailand 6 liner has shown a good performance in term of flow rates and interface transmissivity comparable to the one obtained for HDPE GM-GCL composite liners.

For the two cases, data are located under the GM-GCL contact condition. This provides information on the possibility from a hydraulic point of view to use B-GM in association with a GCL for hydraulic applications, where there is no question regarding chemical compatibility between the GM and the liquid to contain.

Fig. 1

 1 Fig. 1 Typical cross-section of a BGM (adapted from Breul et al. 2008)

Fig. 2

 2 Fig. 2 Polymeric film side of B-GM

Fig. 4

 4 Fig. 4 Interface transmissivity test apparatus (from Mendes et al., 2010)

Fig. 5

 5 Fig. 5 Comparison between interface transmissivity results of the two faces of the B-GM

Fig. 6

 6 Fig. 6 Synthesis of transmissivity data obtained in Fig.6 Synthesis of transmissivity data obtained in the literature for GCLs in contact with HDPE GMs with introduction of results obtained with B-GM

Table 1

 1

		Properties of B-GM
	Composition
	Designation	Functionality	Unit	Values
	Glass-mat	Reinforcement	g/m 2	50
	Non-woven	Reinforcement	g/m 2	400
	geotextile
	Oxidized	Binder	g/m 2	7310
	bitumen
	Sand	Surface finish	g/m 2	217
	Polyester	Under surface	g/m 2	15
	antiriot film	finish
	Thickness (on finished product)	mm	5.6
	EN 1849-1

Table 2

 2 Properties of the base soil used (adapted from[START_REF] Barroso | Laboratory investigation of flow rate through composite liners involving GCLs[END_REF]

	Percent	Percent	Atterberg		Proctor modified	KCCL
	fines	clay	limits		ASTM D 1557	(m/s)
	(%)	(%)	ASTM
			D 4318
			ωL	ωP	ωOPT	γdmax
			(%)	(%)	(%)	(KN/m3)	8 x
	73.6	40.5	54.2	23.7	13.6	19.1	10 -11
	KCCL, hydraulic conductivity of the soil composing the CCL;
	PI, plasticity index; ωL, liquid limit; ωOPT, optimum moisture
	content; ωP; plastic limit; γdmax, maximum dry density.

Table 3

 3 Values of final interface transmissivity calculated by the analytical solution given by Equation1

	Test	Q (m 3 /s)	KGCL (m/s)	R	θ (m 2 /s)
				(m)
	Rough side	3.89 x 10 -11	6.9 x 10 -10	0.1	7.92 x 10 -11
	(sand side)
	Smooth	3.94 x 10 -11	6.9 x 10 -10	0.1	8.02 x 10 -11
	side (film
	side)
	Q, flow rate; Ks, hydraulic conductivity at steady-state; Rc, radius
	of the wetted area; θ, interface transmissivity calculated by the
	analytical solution.

to 15 December 2012 | Bangkok, Thailand