A Blaise

S André

P Delobelle

Y Meshaka

C Cunat

IDENTIFICATION OF THE TRUE ELASTIC MODULUS OF HIGH DENSITY POLYETHYLENE FROM TENSILE TESTS USING AN APPROPRIATE REDUCED MODEL OF THE ELASTOVISCOPLASTIC BEHAVIOR

Keywords: inverse identification, instantaneous Young modulus, tensile behavior, semicrystalline polymer

The rheological parameters of materials are determined in the industry according to international standards established generally on the basis of widespread techniques and robust methods of estimation. Concerning solid polymers and the determination of Young's modulus in tensile tests, ISO 527-1 or ASTM D638 standards rely on protocols with poor scientific content: the determination of the slope of conventionally defined straight lines fitted to stressstrain curves in a given range of elongations. This paper describes the approach allowing for a correct measurement of the instantaneous elastic modulus of polymers in a tensile test. It is based on the use of an appropriate reduced model to describe the behavior of the material. The model comes a thermodynamical framework and allows to reproduce the behavior of an HDPE Polymer until large strains, covering the elastoviscoplastic and hardening regimes. Well-established principles of parameter estimation in engineering science are used to found the identification procedure. It will be shown that three parameters only are necessary to model experimental tensile signals: the instantaneous ('Young's') modulus, the maximum relaxation time of a linear distribution (described with a universal shape) and a strain hardening modulus to describe the 'relaxed' state. The paper ends with an assessment of the methodology. Our results of instantaneous modulus measurements are compared with those obtained with other physical experiments operating at different temporal and length scales.

Introduction

During the last decades, one can observe a significant gap between recommended practices in the field of rheological parameter determination, relying on international standards, and the efforts made in the scientific community. On one hand, mechanical scientists develop more sophisticated models of behavior laws. On the other hand, experimentalists think about the appropriate metrology that can be associated to constitutive laws in order to precisely identify their parameters. The scientific field of inverse methods precisely deals with the mathematical basis of parameter estimation which can referred to as Model-Based Metrology (MBM). It has growth intensively since the eighties. MBM should nowadays be a very common framework when considering the metrology of physical parameters through pertinent physical modeling of experiments. It is necessary for a real enhancement of the quality in measured properties of materials which will, in turn, allows to investigate more deeply the link that can be made with the microstructural organization of materials. In order to illustrate this point of view, an example is given in this paper regarding the determination of the elastic modulus of polymers using the uniaxial tensile test. With respect to international standards, it rests upon the characterization of a linear elastic regime, i.e. proportionality between stresses and strains, at very small strains (short times). Among them, one can cite the ISO 527-1 and ASTM D 638 which suggest such a method leading to approximately the same results. Although this method is efficient and physically well-founded for metallic materials, it is not so satisfactory for polymers. Indeed, these materials quickly manifest a viscoelastic regime which takes place in the beginning of the tensile test. Therefore the obtained response does not show any true linear part. This clearly shows that an inverse approach of parameter estimation is required. This latter consists in the use of an optimization procedure based on uniaxial tensile test data and relying upon an adapted rheological constitutive model. The inverse problem is merely formulated as a least square optimization problem. It must be verified that the assessed parameters are not correlated and that their sensitivity is high enough in order to ensure a good quality of the identification. The aim of this paper is to show that this approach is not only possible but the only way to access pertinent measurement of the elastic modulus of materials like HDPE (High Density PolyEthylene) among SemiCrystalline Polymers (SCP).

The sketch of the paper is as following: first, examples will be given of the observed mechanical behavior of HDPE in uniaxial tensile tests. True stress -true strain curves at different strain rates will illustrate the fact that it is very difficult and even false to measure correctly the elastic modulus of this polymer according to the recommended standards. In a second step, the rheological constitutive model will be presented in its most reduced version. This model will be shown able to describe different behavioral regimes (viscoelasticity, viscoplasticity, material hardening). In view of a parameter estimation problem, a reduced model means a model which meets the identifiability criteria i.e. a good adequation between the number of identifiable parameters and the relevance of the model with respect to the experiments (parsimony principle). In a third part, the identification procedure and the sensitivity analysis of the estimated parameters will be described. At last, the results of the identification will be discussed and compared to those given by the standards, and those measured by two other scientific techniques (ultrasonic device and nanoindentation tests).

Tensile tests on HDPE : experimental results

Material

The material tested in this work is HDPE (High Density Polyethylene, grade "500 Natural") produced by Röchling Engineering Plastics KG. Two different products (A and B) were manufactured under the same reference in an interval of six years and supplied in sheets (extrusion process). Information from the supplier indicates that the molecular weight and density are respectively of 500,000 g/mol and 0.935 g/cm 3 . Differential scanning calorimetry gave a crystallinity index of 68 wt% for product A and 66 wt% for product B. Bone-shaped samples A were cut from a 6 mm thick sheet of polymer (along the extrusion direction " // A " (samples of reference in this paper) and perpendicularly to it " ⊥ A "). Some other samples were cut along the extrusion direction and machined to 3 mm thick (core samples " c A "). Samples B were cut along the extrusion direction only, from a 4 mm sheet produced six years earlier. Even if specimens A and B are supposed to be the same chemical material, this time interval between the dates of production has resulted in a different behavior. As the experiments performed on sample B at delivery and 6 years later lead to the same results, ageing can not be considered as responsible for this difference. Hence, the explanation probably lies in a different manufacturing process. The microstructure and the mechanical behavior of a polymer are clearly dependent on the cooling process which generates more or less crystalline phase and rules its distribution and organization in the bulk among the amorphous one. This unwanted difference will be seen to turn into real opportunity when considering the scope of the present analysis.

Video-controlled tensile test

All mechanical tests were performed on a servo-hydraulic MTS 810 load frame with Flextest SE electronic controller. A video-extensometer (VidéoTraction®) gives access to the elastoviscoplastic response of polymers under uniaxial tension. Local measurements of true strains are performed at the center of the neck. The measurement of the corresponding force enables us to construct the axial true stress -true strain curve of the sample (G'sell et al., 2002). Another important feature of the system is that it controls the servovalve of the machine in real time (through a feed-back loop) so that any desired input path for the true strain 11 ε can be imposed on the system. Seven dot markers are printed on the front face of the sample prior to deformation (Fig. 1), within a representative elementary frame of about 9 mm². The markers are black, nearly round and with a diameter of about 0.4 mm. Local deformations are measured between the dots according to Hencky's definition of the true strain:

ln 0 l l ε   =     (1)
The longitudinal deformations are interpolated through Lagrangian polynomials in order to get precisely the longitudinal strain value 11 ε in the section where the transversal strains are measured (Fig. 1). Thanks to the special geometry of the specimen, these measurements are performed always where necking develops. A standard deviation of 4 10 -is commonly achieved for the noise that corrupts this strain signal. Recent measurements using Digital Image Correlation (DIC system ARAMIS 3D from GOM Instruments) show that the longitudinal strain as measured by Videotraction lies within an error of less than 4%

(accounting for both the 2D plane measurement bias, and the necking phenomenon which do not conserve the principal axis reference frame).The transverse true strain 22 ε is calculated by averaging the strains determined from the two pairs FC and CG (Fig. 1). The second transverse strain 33 ε in the third direction has been simply taken equal to 22 ε , with the assumption that the strain tensor complies with a transversally isotropic material (assumption checked by viewing both lateral faces of the specimen using our 3D DIC system). True stress takes into account the reduction of the cross-sectional area, 0

S S <

, undergone by the sample while it is stretched:

( ) ( ) exp exp 11 22 33 22 0 0 F F F 2 S S S σ ε ε ε = = -- = - (2)
A CCD camera mounted on a telescopic drive records images during the test and follows the elementary frame during deformation. The applied force F is directly measured by a 5 kN load cell. An image analysis software computes the strains in real time and thus, the true stress can be determined according to eq. ( 2).

Figure 1: Principle of the true strain measurements as used by Videotraction system.

In the case where measurements of the transverse strains are not available, an isovolumic strain hypothesis can be used. This hypothesis is nearly met for our HDPE samples as proved with DIC measurements of the strain field. This approximation will be used later because it is an excellent way to demonstrate what an experimental bias consists in and how it can or cannot (in the present case) interfere with the identification of parameters. On Fig. 2, one can see that even if the transverse strain measurements exhibit some lack of reproducibility and of precision, the curves are nearly linear with approximately a -0.5 slope (Poisson coefficientν ).

Hence, we will consider the case where ν is set to 0.5.

This criterion is summarized as follows:

. 05

υ = . 0 5 ν = (3) * * . 22 33 11 11 0 5 ε ε υ ε ε = = - = - (4)
The stars upperscripts denote then the calculated transverse strains obtained from the unique measurement of 11 ε .

This leads obviously to a null volume strain:

* * v 11 22 33 0 ε ε ε ε = + + = (5)
and to a true stress:

( ) ( ) * exp * * exp 11 22 33 11 0 0 F F S S σ ε ε ε = - - = (6)
Finally the observable * σ is considered as depending only on the measurement of 11 ε .

An example of representation of σ and * σ is given in Fig. 2. Fig. 3 represents true stress * σ -true strain 11 ε curves obtained for three different strain rates 11 εɺ . In addition, at large strains, an almost linear relation is observed (Fig. 4) when the true stress is plotted as a function of the strain deformation variable [START_REF] Haward | Strain Hardening of Thermoplastics[END_REF] defined as:

exp( ) exp( )

2 1 HT 2 ε λ λ ε ε - = - = - - (7)
with λ, the extension ratio defined as: exp( )

0 l l λ ε = = (8)
This strain variable naturally arises from a microscopic modeling of the entropic elasticity of a molecular network [START_REF] Treloar | The Physics of Rubber Elasticity[END_REF]. In this regime the hardening component of the stress behavior can be described with the following relation:

hard HT G σ ε = (9)
where G stands for the rubbery (or hardening or hyperelastic) modulus (in MPa).

Macroscopic behavior of HDPE in a tensile test

Various domains can be clearly identified on the true stress * σ -true strain 11 ε curves (Fig. 3). For very low strains, the mechanical behavior hardly exhibits linear elastic stage. A viscoelastic phase is observed until the apparition of a yield point for a strain value 11 ε close to 0.1 (referred to as yield ε ). At this point, the force reaches a maximum (first Considere condition) [START_REF] Hiss | Network Stretching, Slip Processes, and Fragmentation of Crystallites during Uniaxial Drawing of Polyethylene and Related Copolymers. A Comparative Study[END_REF] and is characterized by the development of necking. After the yield point, the mechanical response reveals the presence of a pseudo-plateau corresponding to the softening of the polymer followed by a plastic or flow regime. Referring to the recent paper by [START_REF] Haward | Strain hardening of High Density Polyethylene[END_REF], the point where the differential nominal stress becomes positive (in terms of extension ratio), sometimes called the second Considere condition, is obtained for a value of 1.2. It corresponds to the stabilization necking condition and is considered as the transition between the viscoplastic flow and the strain-hardening regimes. The hardening phase is followed in the study up to strains of about ≈ 11 ε 1.9.

Measurements of the elastic modulus using standards

Standards ISO 527-1 and ASTM D 638 have been applied on experimental data (see Fig. 5 for an example of implementation of standard ISO 527-1) and the results for the elastic modulus are reported in table 1.

One can notice that different values are obtained if the nominal strain rate is changed (discrepancy of the order of 23 %). This should not be, because the elastic modulus is supposed to correspond to an instantaneous response with respect to the time constant of the excitation. This result is all the more unsettling for two reasons: firstly, because nowadays, parameter estimation in engineering science can not rely on such a poor procedure (adjusting a slope at the origin or between two values of strain). Secondly, because progress in material sciences will be possible only if physical parameters are identified according to a procedure that agrees with their definition. The elastic modulus is the instantaneous thermodynamic coefficient which links the stress and the strain in the corresponding state law and must be determined in full coherence with this status. The elastic modulus of polymers, as measured by standards, does not comply with an appropriate scientific definition and hence does not give satisfying results. We will demonstrate that using a well-thought strategy of A -MPa ).

Figure 5: Nominal stress-nominal strain curve of HDPE at a strain rate of 6 mm/min (dotted curve) -The linear fit illustrates ISO 527-1 standard for determining elastic moduli.

Behavior model of HDPE and sensitivity analysis

3.1 Constitutive modeling

General formulation

In order to characterize accurately and correctly the mechanical behavior of HDPE, a generalized viscoelastic constitutive model is necessary. The model used in this study is generally referred to as the DNLR approach (Distribution of Non-Linear Relaxations, see [START_REF] Cunat | A thermodynamic theory of relaxation based on a distribution of non-linear processes[END_REF][START_REF] Cunat | Lois constitutives de matériaux complexes stables ou vieillissants, apports de la thermodynamique de la relaxation[END_REF][START_REF] Cunat | The DNLR approach and relaxation phenomena : part I : Historical account and DNLR formalism[END_REF]. It has been used to reproduce many different experiments (various polymers and loading conditions [START_REF] Rahouadj | A thermodynamic approach with internal variables using Lagrange formalism[END_REF][START_REF] Mrabet | An irreversible model for semicristalline polymers submitted to multisequence loading at large strain[END_REF]). This model relies on fundamental axioms of Thermodynamics of Irreversible Processes (T.I.P.) as stated for example by [START_REF] Callen | Thermodynamics and an Introduction to Thermostatics[END_REF] or more recently by [START_REF] Kuiken | Thermodynamics of Irreversible Processes: Applications to Diffusion and Rheology[END_REF]. The state laws of irreversible thermodynamics are classically decomposed into an instantaneous (or unrelaxed)

component and a delayed one. In the following, the indexes u and d will refer respectively to the unrelaxed and the delayed components This decomposition is classical in other thermodynamical approaches that make use of internal variables (Maugin and Muschik 1994).

In a tensile test where the strain is imposed, its dual thermodynamic variable is the stress and this variable constitutes the response of the system. The postponed response is described using a modal approach (lower indices j in the equations denote the mode j ) for the irreversible processes at stake in the microstructure. The existence of a modal basis to describe dissipative mechanisms has been proved in thermodynamics by [START_REF] Meixner | Thermodynamik und Relaxationserscheinungen[END_REF]. It is simply described here by a first order kinetic model where the stress 'fluctuation' or difference between the modal true stress j σ and the modal relaxed stress r j σ regresses according to time j τ . This time represents the relaxation time which governs the kinetics of each dissipation mode j . The overall true stress σ is the sum of all the modal components j σ and the general relation of the DNLR formalism is written:

r N N j j u d u j j j 1 j 1 j E = =   σ -σ σ=σ +σ = σ = ε-     τ   ∑ ∑ ɺ ɺ ɺ ɺ ɺ (10) u j
E corresponds to the modal unrelaxed modulus. u j E and r j σ can be simply defined by :

u j u j E p E 0 = and r j r j p σ σ 0 =
where 0 j p is a coefficient that weights the modal component j with respect to the global quantity. r σ refers to a relaxed state corresponding to the steady-state regime of the internal mechanisms, where the non equilibrium forces do not evolve in time. This thermodynamical state is clearly defined by 0 = A ɺ and named "iso-affin" state according to the recommendation of Prigogine (1946). A is indeed the affinity state variable introduced by De Donder (1936) to take into account chemical reactions. It is used by mechanical scientists in solid rheology to describe internal reorganizations which take place in the matter under deformation processes [START_REF] Kuiken | Thermodynamics of Irreversible Processes: Applications to Diffusion and Rheology[END_REF]. u E stands for the common elastic (Young's) modulus and hence, the modal weights must fulfill the normalized condition:

N 0 j j 1 p 1 = = ∑ (11)
From eq. ( 10), we can easily infer that for a tensile test at very low strain rate, the true stress is expected to be very close to the relaxed stress (the non equilibrium forces are constant). According to this approach, the relaxed state can be seen as a pseudo-equilibrium which is still dynamic. It must not be mistaken for the equilibrium state (rigorously defined in T.I.P. by a zero-affinity state) which corresponds to the new microstructural conformation taken by the matter when the driving external forces are stopped. For example, in a same type of modeling (the V.B.O. approach due to [START_REF] Krempl | Relaxation behaviour and modeling[END_REF], the distinction is clearly made although the same stress notation ( g ) and name ("equilibrium") is used for both: g is considered as the equilibrium stress "when all time rates are zero" (must be equal to 0 in the absence of applied stress) and g is considered as an equilibrium stress that has to evolve under application of an external force and is said to correspond to the stress "that must be overcome to generate inelastic deformation" (it corresponds to the variable r σ in the present model).

The DNLR approach in the form of eq. ( 10), offers then two entry points which can be adapted for different modelings: the spectrum of relaxation times and the relaxed state which will be discussed now.

Spectrum of relaxation times

Following the postulate of an equipartition of the created entropy accompanying regression of the fluctuations about an equilibrium state, [START_REF] Cunat | A thermodynamic theory of relaxation based on a distribution of non-linear processes[END_REF][START_REF] Cunat | Lois constitutives de matériaux complexes stables ou vieillissants, apports de la thermodynamique de la relaxation[END_REF][START_REF] Cunat | The DNLR approach and relaxation phenomena : part I : Historical account and DNLR formalism[END_REF] proposed a universal spectrum that links each dissipative modal weight 0 j p to its corresponding relaxation time j τ (Eq. ( 12)).

j 0 j N j j 1 p = τ = τ ∑ (12)
The whole linear spectrum of relaxation times is generally distributed along a logarithmic scale in an interval [ min τ , max τ ]. Only one parameter is needed to describe this spectrum (in linear viscoelasticity). We generally retain the longest relaxation time max τ because it is more close to the physical perception of an experimenter. It is named

T max τ
where the "T" upperscript denotes the Tensile phase of the test) The number of decades d chosen for the temporal scale is a hyper-parameter which means that it is connected to the mathematical structure of the model: the results must be independent of the decade number once it is set to a given upper bound. A number of decades greater than or equal to 6 is generally required to reach convergence and to cover all possible relaxation times of the material (Eq. ( 13)).

= N

dissipative modes are generally considered in order to build a quasi-continuous spectrum of relaxation times. To summarize, d and N will be considered as fixed (model structure parameters having a null sensitivity on the model once they are set to appropriate physically sounded values) and the linear spectrum of relaxation times j τ is defined once parameter T max τ is known, according to the following equation (Eq. ( 14)):

N j d N 1 j max 10 -   -  -   τ = τ (14)
To account for a non-linear spectrum, a multiplying factor (shift-factor) is classically used and denoted by ,...) , ( t T a

. It shifts the time scale according to some dependent law of other variables like temperature T (Arrhenius or V.F.T. law), deformation rate εɺ (power-law form), stress tensor σ (to recover plastic domains)... In this form, the DNLR model rigorously corresponds to the Biot model for viscoelasticity [START_REF] Biot | Linear thermodynamics and the mechanics of solids[END_REF] or to a generalized Maxwell model [START_REF] Tschoegl | The Phenomenological Theory of Linear Viscoelastic Behaviour, An Introduction[END_REF], except that a recursive-type relation links both the relaxation times and weights [START_REF] André | Rheological constitutive equation of solids: a link between models based on irreversible thermodynamics and on fractional order derivative equations[END_REF].

Relaxed state model adopted for HDPE under tensile tests

The second entry point for the modeling lies in the description chosen for the relaxed stress an elasto-visco-plastic behavior but with only linear hardening in plastic regime. As we saw in section 2, a constitutive law for HDPE must comply with a more complex continuous modeling from the viscoelastic to viscoplastic and hardening behaviors. Hence, a different modeling must be considered for the relaxed state. A SCP in highly deformed state is generally considered as a tridimensional network of rubber type phase with entanglements. A first model for elastomers hardening considered by Wang and Guth (1952) and then modified by [START_REF] Arruda | A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials[END_REF] put forward cells with eight sub-chains by node which is close to a random distribution of the chains in a real material and does not favour any spatial direction.

By considering the works of Haward & Thackray [START_REF] Haward | Strain Hardening of Thermoplastics[END_REF]Treolar (1975), Arruda

and Boyce have shown that during hardening, the "back stress" (or anisotropic resistance to chain alignment) can be described by the following formula: For a given temperature, eq. ( 15) requires the knowledge of the two parameters 0 N and n .

backstress 1 0 B c HT c N k T n L 3 n - λ   σ = ε   λ   (
It can be reduced to the simplified version:

backstress HT G σ ≈ ε ( 16
)
where G is the single parameter, referred to as the hardening modulus.

According to [START_REF] Krempl | Relaxation behaviour and modeling[END_REF], the term "back stress" is often used in the literature but is not relevant for describing the subtle differences existing between equilibrium stress and overstress (relaxed state in our approach) and already pointed out in section 3.1.1. Besides, [START_REF] Negahban | Experimentally Evaluating the Equilibrium Stress in Shear of Glassy Polycarbonate[END_REF] considers the "back stress" as the delayed response (or inelastic response)

and that it can correspond to the "back stress" involved in the Arruda & Boyce models [START_REF] Arruda | A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials[END_REF]. In the framework of Irreversible Thermodynamics based on the concept of Affinity, such considerations are aimless. It makes clear the distinction between actual (or unrelaxed), relaxed, and equilibrium states. Turning now our attention to the experimental results presented in section 2 (Fig. 4), it is clear that the true stress plotted as function of the Haward-Thackray strain variable HT ε exhibits an almost linear relation with constant slope, whatsoever the strain rate. This fact has been already highlighted in other works [START_REF] Van Melick | On the origin of strain hardening in glassy polymers[END_REF]. Therefore in this paper, we consider that the "back stress" in

Arruda & Boyce model could be associated to the relaxed state of our model and thus, eq. ( 19) will be considered. Indeed, the Haward-Thackray variable puts forward the behavior at large strains, when all the relaxation times have been depleted.

r HT G σ = ε (17)
Finally the reduced model that will be considered in this study, contains only three parameters: u E , G and T max τ and is described by eq. ( 20).

N N j HT u j j N j d j 1 j 1 N 1 max G E 10 -   -  = = -     σ -ε   σ= σ = ε-     τ   ∑ ∑ ɺ ɺ ɺ (18)
The relevance of this model will be proven now through the sensitivity analysis and parameter identification procedure (section 4).

Sensitivity analysis fundamentals

Let us consider

( ) t y m
as the measured output (the stress) of a system (our material specimen), and ( ) t y , β the theoretical stress (output) of the behavioral model with parameter vector β of dimension p , representing in our case the p constitutive material parameters.

We can then define the output error ( ) t e (or the residuals) by the following formula:

( ) = - m e( t ) y ( t ) y ,t β (19)
The estimator aims at minimizing this output error. A least square criterion can be used classically which is written as follows:

( ) ( ) ( ) = = - ∑ q 2 m LS i i i 1 E y t y ,t β (20)
where the summation is made over the th i data points corresponding to the successive acquisition times i t ( q stands for the total number of experimental data points).

The minimization of the criterion is done when its derivative with respect to parameters j

β ( p j ... 1 = ) is null: [ , ], ∀ ∈ j 1 p LS j E 0 β ∂ = ∂ → → → → ( ) ( ) ( ) =   ∂   - =       ∂   ∑ q i m i i i 1 j y ,t y t y ,t 0 β β β → → → → ( ) ( ) ( ) ( ) q m j i i i i 1 X ,t y t y ,t 0 β β =   - =   ∑ (21)
From this equation, the sensitivity coefficient vector j X associated to parameter j β can be clearly recognized as:

( )

( , ) , ∂ = ∂ i j i j y t X t β β β (22)
Sensitivity coefficients express how much a model reacts to some small variation of the parameters. Sensitivity coefficients have a fundamental role in the conditioning of the inverse parameter estimation process and, as a consequence, upon the errors made in the estimations (confidence bounds). It is obvious that large sensitivities are sought for when designing an experiment in view of metrological purposes. Note that in almost all cases, sensitivity coefficients are non linear functions of the parameters themselves (when the model y is a non linear function of the j β ).

Switching to a matrix formulation and defining vectors m Y and Y such as:

( ) ( ) ( ) ...       =           m 1 m 2 m m q y t y t Y y t and ( ) ( ) ( ) , , ... ,       =           1 2 q y t y t Y y t β β β (23)
makes the minimization process now expressed as

( ) -= t m X Y Y 0 (24)
with X , the (rectangular) sensitivity matrix whose rows are for each observation time i t and columns are for each parameter j β :

In the case of a linear model with respect to the parameters (linear estimation problem), we have:

= Y X β (25)
where the matrix of the sensitivity coefficients does not depend on the parameters.

The estimated parameter vector, denoted by β ˆ, corresponds to the value reached by β when the criterion is minimized. Therefore by using (25) we can then rewrite the equation ( 24) into:

( )

- = t m X Y X 0 β (26)
Relation ( 26) can be inverted in order to obtain the expression of β ˆ in the case of a linear estimation problem only:

( ) - = 1 t t m ˆX X X Y β (27)
Of course, most parameter estimation problems are not linear and require an iterative linearizing procedure. It is obtained by developing the solution (at rank n+1) in the neighborhood of the solution obtained for the prior iteration (rank n):

( )

+ + = + - ( n 1 ) ( n ) ( n ) ( n 1 ) ( n ) ˆŶ Y X β β (28)
By combining relation (24) written at rank n+1 for parameters estimated at rank n:

( ) 0 ) 1 ( ) ( = -+ n m n t Y Y X
and relation (28), we obtain the following relation of recurrence between estimated parameters at rank n+1 and rank n:

( ) ( ) - + = + - 1 ( n 1 ) ( n ) t ( n ) ( n ) t ( n ) m ( n ) ˆˆX X X Y Y β β (29)
which defines the iterative procedure that can be used for estimating the parameters (Gauss-Newton algorithm).

In the followings, we discuss the statistical properties of the estimator, which depends on the noise ) (t ε of the signal. If the theoretical model is assumed unbiased (perfect) then we have:

( ) ( , ) ( ) m Y t Y t t β ε = + (30)
If classical statistical assumptions are made regarding the experimental noise ) (t ε on the measured signal (the stress) [START_REF] Beck | Parameter Estimation in Engineering and Science[END_REF], it is possible to get an estimation of the errors that can be made in the estimation process for the different parameters. These hypotheses are:

-zero mean value of the signal in the absence of excitation, which corresponds to a zero expectancy for the noise (expected value 0

) ( = ε E ) ;
-constant variance or standard deviation of the noise:

2 0 ) ( σ ε = V .
In the case of a 1 st order linearized estimation, its expected value can be proved to be:

( )

E β β = (31)
This means that there is no error or bias made on the identified parameters.

The variance-covariance matrix ∆ on the estimated parameters (generalization of the scalar-valued variance to higher dimension) naturally involves the sensitivity coefficients. It is calculated as

( ) ( ) ( ) ( ) ( ) ˆˆˆt 1 2 t 0 E E E X X ∆ β β β β σ -   = - - =    
1 1 2 1 p 1 2 2 2 p rs 1 p 2 p p V V V β β β β β β β β β β ∆ β β β β β       =           (32)
A stochastic analysis has been made which consists in calculating theoretically (according to a given noise and given set of parameters) the variance-covariance matrix through eq. ( 32).

This matrix is symmetric squared with dimension equals to the number of parameters. The diagonal terms correspond directly to the variance of each parameter ) ( j V β . They can be used to determine the error made on each parameter. We will present this error (expressed in %) as: ( ) ( )

j j j V Err β β β = ( 33
)
The off-diagonal terms can be used to calculate the correlation coefficients mn ρ which express the degree of correlation of the parameters :

cov( , ) ( ) ( ) r s rs r s V V β β ρ β β = (34)
The values for rs ρ lie between 0 and 1. In the case of a model with strongly correlated parameters, the correlation coefficients are 'close' to 1 which means that two columns of the sensitivity matrix X are nearly proportional. The resulting confidence bounds interval for two correlated parameters are therefore generally very high. This means that a large number of solutions exist for these two parameters to allow for a good fit of the experimental curve. A deterministic algorithm used for the minimization process (like the steepest gradient technique) is consequently very sensitive to the initial guess made for the parameters. A strategy to produce first approximate estimates using physical background is highly recommended. But still, the estimation problem is ill-posed and indicates to the experimentalist that the involved physical description is probably not appropriate and must be changed.

In the followings, the identifiability of the model parameters will be analyzed through array ∆ ~ which combines the variance-covariance matrix and the correlation matrix. ∆ ~ puts the main diagonal of ∆ on its main diagonal and the correlation coefficients on the offdiagonal terms.

β ρ ρ ρ β ρ ∆ ρ ρ β       =           ɶ ( 35
)
When there is no model bias (perfect agreement between the conditions of the experiment and the model based on it), the relation between the estimated parameter vector for the non linear estimation problem and its 'exact' value can be given at convergence with the following formula:

( )

ˆ1 t t X X X β β ε - = + (36)
In that case, the residuals ( ) t e (the difference between the model and the data) correspond exactly to the noise. Their standard deviation corresponds then to the experimental standard deviation of the noise and the residuals remain unsigned (no large fluctuation around the zero level)

The matrix ∆ ~ is a good tool to investigate identifiability conditions of a parameter estimation problem. Besides, use can be made of the normalized sensitivity coefficients:

( , )

j j j j y t X β β β ∂ = ∂ ɶ (37)
in order to check graphically the level of sensitivity and possible correlations between parameters (different intervals of the independent variable, the time t here may be advantageously considered). These coefficients will be dimensionally homogeneous to the signal itself, the stress, and will be specified in MPa.

Identification procedure

Adjustment results

The model ) ( * ε σ described previously is now applied to experimental data in order to identify the following parameter

vector: ] G , [ max T u E τ β , =
. Computations are made with Matlab Software and use is made of the Levenberg-Marquardt and/or Simplex algorithms to minimize the least square criterion. Once the convergence is reached and the parameters determined, the sensitivity coefficients at this point of the parameters space will be calculated along with the resulting optimal confidence bounds. The initial values used as starting guesses for the parameter vector are determined according to physical basis (a general rule that should be followed : if the literature uses the wording "knowledge" model in the field of system identification, this means that the researcher should have ideas or at least basic methods to roughly determine its parameters). An initial instantaneous modulus can be derived from the slope at the origin of the stress-strain curve (or using the standards). An initial value for the maximum relaxation time can be simply derived by assuming a pure exponential-type behavior of the rising part of the curve as a result of a step input. For the experiment performed at a strain rate of 3 1 5 10 s --× for example, a decaying strain of the order of 0.04 can be deduced from the rule of tangents (or logarithmic plot) and leads to a typical time of about 8 seconds. Finally, a "visual" linear regression applied on the experimental data of Figure 3 gives a first rough estimate of the order of magnitude of 110 50 2.2 G = = which later will be seen also as a perfect initialization value of the minimization algorithm.

Two different identification intervals can be considered. In case where small strains levels ( . ε . The experiment carried out at the lowest strain rate (Fig. 7) is the closest to this picture. In that case, the parameters are perfectly identified and the confidence interval is mainly due to the contribution of the noise (eq. ( 35)). But still, the residuals show a signed character, which means that some bias exist (inadequation between model and data) that induces an additional error on the estimated parameters (biased estimations). The existence of a bias generally helps the experimentalist to improve the couple model-experiment. If the bias remains limited (of the order of the noise level) then the parameters can be considered as well-identified. Wellidentified means here that according to the sensitivity analysis made in section 3.2, the estimation process produces the unique set of parameters (the optimal one) minimizing the residuals in the least-square sense. This means that in case of no bias (in the data, in the model) but in the presence of the same amount of noise, the variances on the un-biased parameter estimates will be lower-bounded by the estimated variances calculated in a stochastical manner (eq. 37). Here, the bias seems more important when the strain rate increases. For a strain rate of 1 2 10 - -s , the residuals lie in the [-10%, 10%] range of the current stress values. A possible origin of this bias may come, in such mechanical tensile tests, from a non ideal input command. In our case, the input ramp ( )

t t ε ε ɺ =
is very good, due to the special care taken in the selection of the PID parameters used by the control feed-back loop.

Furthermore, the real input signal is used in the calculation of the model response. Here, the bias is more likely due to an approximation introduced in the model for the material behavior.

This approximation comes from the use of * σ instead of σ (see eqs. ( 2) and ( 6)) which avoids introducing the measurements made on the transversal strain to calculate the crosssection variations of the specimen with respect to time. The bias is also due to the approximation used in the relaxed state modeling (Haward-Thackray relation, eq. ( 17)) as evidenced by the curves shown on fig. 4 (the linear behavior is not perfect). Trying to reduce the bias by tracking defects with respect to the idealized experimental conditions is the first thing to do. Then a refinement of the model can be legitimate as a second step. Otherwise, a refinement of the model stimulated by biased data can be dramatic in view of the Parameter Estimation Problem. For example, trying to use here the Arruda & Boyce model of eq. ( 15) will introduce additional unnecessary parameters which may in return distort the parameter estimation problem (poor identifiability conditions). Here, as far as the residuals remain small with respect to the signal to noise ratio and because the aim of the paper is mainly devoted to show how this model can lead to better estimations of the instantaneous Young modulus, these attempts are not presented here. It is interesting also to discuss the results of the identification procedure when the tensile test is followed by a relaxation (initiated around a strain of 1.9). The reduced model can be used here too but with some modifications. Indeed, when relaxation starts, the material is highly deformed. This new excitation applies on a material which is different and implies to "reset" the knowledge we have on the microstructure. The relaxed state and the times spectrum must be described differently. For the relaxed state a simple constant stress value is assumed to be reached at long times and noted ∞ = σ σ r ("equilibrium" state at rest). The spectrum conserves its properties (linearity, number of decades and number of modes) but it is simply shifted in time according to a new maximum relaxation time noted R max τ , which can be considered as a new characteristic of the material (in fibrillar state for such a high deformation when relaxation is initiated). Note that in view of our constitutive model, the instantaneous (or unrelaxed) modulus has not to be considered different. For the tensilerelaxation test, the parameter vector that is considered

is now ] , , G , [ max max R T u E τ σ τ β ∞ , = .
The results of the identification with this modified (but still reduced) model are shown on Fig. 10 for specimen B. It can be seen that the agreement is very good both in the tensile and relaxation stages, with the same u E value. Physically, this means that elastic (instantaneous)

properties of the SCP are the same in the initial bi-phasic microstructure and in the fibrillar state (when unloading is triggered). It is emphasized again that this reduced approach is made for a pure metrological purpose. This model would have to be made more complex to account for more sophisticated loading paths such as multiple loading-unloading stages.

Sensitivity Analysis

Fig. 11 plots the sensitivity coefficients (Eq. ( 24)) of the model parameters and table 2 presents the results of the stochastic analysis.

(a) (b) 2a and2b) as a result of the high degree of correlation between both parameters for most part of the curves (especially during the plastic regime).

Interval I E u τ max T E u 2.
Nevertheless, the identification is possible as confirmed by the relative error calculated on both u E and T max τ parameters. This is due to the sensitivities behavior around . ). The robustness of the identification has also been checked by changing the initial values of the optimization algorithm. If two parameters were correlated, then the estimated values would have changed for each different run. Converging to the identified parameter vector whatever its initialization, is a good verification that the estimation is well-posed.

Regarding the use of interval I to perform the identification on parameters u E and T max τ only, the sensitivity analysis helps to its definition. For strains below 0.13, the sensitivity to G is only 5% of the maximum sensitivities of u E and T max τ and therefore, this parameter can be omitted. It is interesting to note by comparing table 2a and 2b that the variances on the parameters u E and T max τ is lowered by a factor of 2 when considering interval II instead of interval I. Therefore it is clear that using the whole curve is preferable.

Estimated parameters and discussion

Table 3 gives the values estimated for specimen A on both interval I ( u E ,

T max τ ) and interval II ( u E , T max τ , G ). Table 3. Identified values of parameters u E , T max τ (Identification interval I) and u E , T max τ ,G (Identification interval II) -Specimen // A .

Identification of the elastic modulus E u

As can be seen from Table 3, the elastic moduli identified for sample A, specimen // A , lie in the range from 2700 to 2950 MPa for all experiments (including those at different strain rates). This represents a variation of % 4 ± around the central value which corresponds to the order of magnitude of the estimated variances yielded by the stochastic analysis (Table 2b).

The uncertainty on these measurements has been (over)estimated by changing the value of the identified u E within the 4 and will be commented later.

In order to check the validity of the results obtained by this MBM approach, two independent direct measurements of the elastic modulus of the material have been carried out.

Ultrasound measurements:

The first one rests upon a pulse-echo technique and an ultrasonic device (Optel OPBOX system). Direct measurements are made of the velocity of a sonic wave in the longitudinal and transverse directions of the sample. Two transducers enable sonic excitation (range 0.5 to 10 MHz) thanks to piezoelectric components which apply shear and compression stresses on a ceramic membrane. The transducers collect echoes of the sonic wave after reflection on the backside of the sample. Then, the measured signal enables calculating the average period T between two successive echoes. For a semicrystalline polymer, three echoes can usually be obtained (millimetric thicknesses).

Longitudinal and transverse speeds of the wave are derived using the following formula:

T e V 2 = (38)
where e 2 is twice the thickness of the sample.

Once the density ρ of the material, the transverse and the longitudinal velocities

( T V , L V )
are known, the elastic modulus can be calculated as follows:

( )

2 2 2 T L T u PE 2 2 L T V 3 V 4 V E V V ρ - = - (39)
The OPBOX system offers different ways to calculate the velocities.

• A single identified echo can be located by the experimentalist. The system, through identification of the envelope curve of the oscillating signal, determines the timeof-flight (maximum of the echo) and calculate the velocity according to (38).

• Multiple echoes can be used (successive or not). The time interval separating two echoes is determined thanks to idealized envelope curves accounting for attenuation of the signal.

• The experimentalist can measure directly the time interval using cursors located on well-identified extrema of repeated pattern (echoes). This method was found less accurate.

These three possibilities allow us to check the confidence that can be placed in the measurements (through reproducibility and consistence) and to calculate a measurement uncertainty.

For specimens // [START_REF] Piché | Ultrasonic velocity measurement for the determination of density in polyethylene[END_REF] and both longitudinal and transversal velocities reported by [START_REF] Legros | Ultrasonic evaluation and application of oriented polymer rods[END_REF].

Nanoindentation and CSM method: second direct measurement is based on nanoindentation tests where the Young's modulus (and hardness) are deduced using the continuous stiffness method (CSM) (Olivier et al., 1992;[START_REF] Le Rouzic | Comparaison of three different scales techniques for the dynamic mechanical characterization of two polymers (PDMS and SU8)[END_REF]. Experiments The true modulus of the specimen can be derived from the apparent measurement by taking into account the stiffness of the indenter through relation:

2 2 i i 1 1 1 E E E ν ν - - = + ′ (42)
where (ν , E ) and ( i ν , i E ) stand for the Poisson ratio and elastic modulus respectively for the material and the indenter. But for polymer application,

E E i >> 1040 ( ) i E GPa =
and eq. ( 42)

is reduced to:

2 E E 1 ν ′ = - (43)
Combining equations ( 40), ( 41) and ( 43) leads to:

. 4).

The technique also offers the possibility to measure the stress at 3.3% of true strain through In view of the uncertainty intervals given above (not reported in Table 4), the following comments can be made. For all specimens, the three techniques give very close results. The agreement is very good between MBM and PE values but local differences can occur with the Nanoindentation test. These differences must be analyzed carefully. For specimen //

Comparison between MBM, PE and Nanonindentation CSM methods:

A , the value given by the Nano technique is 16% lower than for the MBM and PE techniques. This can be explained as the Nanotechnique probes a small volume just behind the surface.

Therefore the identified value may be sensitive to some slight skin effect which may be present on the 6 mm extruded sample ( // A ). On the contrary, both the MBM and PE methods probe the entire volume. As specimen B was manufactured directly with a 4 mm thickness, this effect is not visible (within the confidence intervals of both three measurements). Now looking at specimen c A (3mm core of specimen // A ), this effect should disappear. Higher Despite the measurement uncertainty, it seems that a general trend can be observed: for all three techniques, the measured elastic modulus is higher for the core specimen. This may prove that (i) core samples are slightly more rigid (small differences in microstructure) and

(ii), the MBM technique is sensitive to such small variation.

Regarding now the incidence of these results on the material property, it is clear that also this commercial HDPE is produced under the same reference, it can exhibit large changes in its elastic behaviour (around 600 MPa ). The increase in elasticity for samples A , thanks to X-ray microtomography [START_REF] Blaise | Investigation of the mesostructure of a mechanically deformed HDPE by synchrotron microtomography[END_REF].

This result is then conform to those given in [START_REF] Legros | Ultrasonic evaluation and application of oriented polymer rods[END_REF] where un-oriented and oriented HDPE are investigated through Pulse-Echo technique. Oriented HDPE is shown to produce higher longitudinal and shear wave signal and hence higher elastic modulus. where τ max , the maximum relaxation time of the spectrum, is used for the material characteristic time and 1 ε ɺ ɺ ɺ ɺ for the experimental characteristic time of excitation. It can be pointed out that the Deborah numbers found for the range of investigated strain rates remain very close. They are of the order of 0.03 which means that the time response of the material is two decades below the time scales imposed by the experiment. This argues in favor of a similar scenario for the succession of internal mechanisms taking place at the microstructure levels. A slight increase of this value is maybe observed when strain rate increases. The difference between the highest and smallest values is about 15% for specimen A which is a little bit higher than the expected variance on this parameter. The maximal relaxation time identified by the model seems then to express a slight sensitivity of the material with respect to the applied strain rate (as confirmed by the tensile curves of Fig. 3). This may confirm the results of an in-situ microstructural investigation based on light scattering and infrared imaging during cold drawing [START_REF] Baravian | Optical techniques for in situ dynamical investigation of plastic damage[END_REF], where kinetic effects appear starting from a strain rate of 2 1 10 s --. The increase in Deborah number agrees well with the observation that plasticity of the material is enhanced when the strain rates are lower. Finally it is interesting to recall that a simple linear viscoelastic assumption was made for the spectrum specification. The relatively constant Deborah numbers found whatever the applied strain rates (and for a same quality of the identification process, with same behavior of the residuals) validates this hypothesis.

Identification of the maximal relaxation time τ max T

Identification of the hardening modulus G

Concerning the value of the hardening modulus G describing the behavior of HDPE at high strains, the identified values for all strain rates are also very close (discrepancies are less than 5% as shown by Table 5). Those identified values for G are close to the ones found in the experimental results of uniaxial tensile tests reported by Haward on HDPE [START_REF] Haward | Strain Hardening of Thermoplastics[END_REF][START_REF] Haward | Strain hardening of High Density Polyethylene[END_REF] and the value found by [START_REF] Bartczak | Influence of molecular parameters on high-strain deformation of polyethylene in the plane-strain compression, Part I. Stress-strain behaviour[END_REF] for linear polyethylene in plane-strain compression. The estimated hardening modulus appears to be the same for both specimens A and B. A slight increase of the modulus is observed for the highest strain rate (10 -2 s -1 ) but this value might also be slightly biased as shown by the quite important signed character of the residuals. This may indicate that at large strain rates, the Haward-Thackray approach used as relaxed state in the model may become inappropriate.

Conclusion about the identification of the parameters

Parameters u E , T max τ and G of the reduced model presented in section 3 have been shown to be easily identifiable thanks to an appropriate sensitivity analysis. For each tested strain rate, the same elastic modulus has been identified as expected by the instantaneous character of this physical parameter. The hardening modulus and Deborah number based on the maximal relaxation time of the spectrum also are found to be nearly constant. A slight effect due to strain rate is obvious, but remains limited. It is accompanied by more pronounced wavy residuals, indicating that some bias may exist between the idealized model and its experimental realization when the strain rate increases. Thanks to the application of this physical model, one can retrieve much precise information regarding the behavior of the polymer. Finally, the results found by this model for the Young modulus u E have been corroborated by two other independent and direct physical measurements probing the material at high frequencies (MHz) or at small spatial scales (µm). This is a clear evidence of the consistence of the approach with the underlying physics because this result is the corollary of an indentified maximum relaxation time of the order of a few seconds. As a spectrum of 6 decades is considered in our approach to get the convergence of the model, this means that relaxation times as small as 1 µs are necessary to allows for a good description with the model and a determination of a 'truly' instantaneous modulus. If one considered that the pulse-echo technique relies on an excitation at 5 MHz, this means that the time scale required by the model is in perfect agreement with this technique which in turn give a support to our identification of the elastic modulus. Note that for this technique, the spatial size of the representative elementary volume is of the order of a few micrometers, which means also that small time scale events are probed.

Conclusion

An inverse approach has been presented for the estimation of the parameters of a rheological law of SCP's in the case of a tensile test. Both ideas of using an appropriate

Figure 2 :

 2 Figure 2: True stress σ or * σ (left axis) and transverse true strain 22 ε (right axis) versus longitudinal true strain 11 ε . Repeated experiments on specimen // A at

Figure 3 :

 3 Figure 3: True stress * σ versus true strain 11 ε curves. (Specimen // A -Strain rates of

Figure

  Figure 4: True stress * σ versus strain variable HT ε

rσ.

  The most basic form consists in assuming the linear form ε σ

  N stands for the density of chains per unit volume, n is the number of segments per chain, which controls the behavior at large strains until the extreme extensibility of the network, B k is the Boltzmann constant and T , the temperature. 1 -L represents the inverse Langevin function.

  Fig.6, the experimental data are plotted along with the model for the optimal identified values (insert of the figure). Leaving aside the first points of the curve, highly dependent on the feedback loop control efficiency, the identification residuals lie in the [-1, 1] MPa range,

Figure 6 :

 6 Figure 6: Experimental data, fitted curve and residuals obtained when identifying parameters u E and T max τ

Figure 7 :

 7 Figure 7: Experimental data, fitted curve and residuals obtained when identifying parameters u E , G and T max τ

Figure 8 :

 8 Figure 8: Experimental data, fitted curve and residuals obtained when identifying parameters u E , G and T max τ

Figure 9 :

 9 Figure 9: Experimental data, fitted curve and residuals obtained when identifying parameters u E , G and T max τ

Figure 10 :

 10 Figure 10: Tensile test followed by a relaxation. Experimental data, fitted curve and residuals as a function of time: identified parameters∞

  , if the identification was carried out only on the plateau, it would be completely impossible to get simultaneous information about u E and T max τ as both elastic and viscoelastic regimes are over. Fig.12represents the normalized sensitivity to the elastic modulus u E versus the sensitivity to T max τ in order to check the relation between both parameters all along the test. One can see that at small strains also strongly correlated as the curve is not so far from the straight line with a zero intercept. The correlation coefficient between u E and

  marks the transition between the different regimes (insert of figure12).One can also notice that parameter G can be estimated with a very low confidence bound (error of 0.06 %) as a consequence of its very weak (inexistent) correlation with the

Figure

  Figure 11: Sensitivities to parameters u E , G and T max τ

Figure

  Figure 12: Sensitivity ) ( u E X as a function of ) ( max T X τ

  that the direct model produces theoretical curves that flank the experimental ones. This allows to take into account the bias effects evidenced earlier by the residuals plotting. For specimen // A , at a strain rate of

E

  room temperature (Table4). (Of course, there is no possibility of measuring a Young modulus in a specific direction, hence u must be considered as the modulus of the 6mm thick material). Likewise, the measurements on specimen B gave a the longitudinal sound velocity very conform to those correlated to density in

h∆

  have been carried out on our samples with CSM at L.M.A. Femto-ST (Laboratoire de Mécanique Appliquée, Besançon-France) on a Nanonindenter II S using a pyramidal Berkovich probe tip. In this technique, the indenter vibrates at a frequency of 45Hz for an amplitude 0 of 1-2 nm during the indenter penetration over a depth h varying from 0.5 to 6 m µ . This process generates elastic and plastic deformations which results in a print conforming to the shape of the indenter. In the CSM method, the small harmonic load oscillation modulus can be identified, the real component being the apparent instantaneous (storage) modulus E′ .It is measured through direct calculations according to Le Rouzic et al.φ is the phase lag due to viscous dissipation, , and A , the projected area of the elastic contact. For a Berkovich indenter, constant . 40) and A is given by the following formula .

  Figure 13 gives the results in terms of modulus u Nano E E = as a function of the penetration depth. The measurements are repeated over several indents (between N=15 and N=20 for each sample) with a 50 m µ distance between them. Only measurements obtained for penetration depths ranging from 3 6 m µ and above have been considered in order to avoid uncertainties in the detection of the surface sample and specimen roughness effects (Qasmi and Delobelle,2006). Although, the samples were polished using a 2 m µ alumina based polishing paste.Compared to measurements made on the initial rough surface, this treatment has been shownto have a great impact on the reproducibility and the confidence bounds, which have been reduced by a factor of nearly 20. The measurements made on polished specimens are routinely within a 50MPa ± confidence interval. According to the imposed displacement path of the indenter, the experiments correspond to a strain rate of about

  very close to the one that can be read on Fig 6.

Figure 13 :

 13 Figure 13: Modulus identified from Nanoindentation tests versus penetration depth (all specimen, N repeated experiments).

  and both three techniques give more convergent results.

  reduced model in relationship to sensitivity analysis principles are shown to be the key for pertinent measurements through fitting procedures. Illustration is given for the characterization of different HDPE specimens through a tensile test. The identified instantaneous Young modulus is shown to be twice the value determined by contemporary recommended standards thus showing that progress has to be made in this metrological field to improve the characterization of polymers which exhibit elastoviscoplastic behavior. It is unsensitive to the strain rates, as expected from such a property. Finally it is shown to be very close to the determinations made according to two different physical techniques probing the sample at very low time scales (Pulse-echo technique) or very small length scales (Modulated Nanoindentation test). Other conclusions regarding the viscoelastic and hardening behaviors also suggest the relevancy of the approach.

Table 2 .

 2 Variance-covariance matrix of parameters u E and

			Interval II	E u	G	τ max T
			E u	1.37 %	-0.18	-0.9988
		15 % -0.9667
			G	-0.18	0.06 %	0.147
	τ max T	-0.9667 3.61 %	τ max T	-0.9988 0.147	1.38 %

Table 4

 4

	( // A , ⊥ A and c A ) and B by the three metrological techniques (indices MBM, PE and Nano
	for respectively the

gathers the values of the elastic modulus obtained for all HDPE Natural 500 specimens A

Model Based Metrology applied to tensile curves, the Pulse-Echo and Nanonindentation technique

  ). The data obtained by applying standards on tensile curves are also given. It is evident from these latter values that standards are very far from producing (i) reliable data and (ii) physically founded estimations. The discrepancy can be as large as 300% and the values depend on the applied displacement rate (Table1)! Nevertheless, using tensile curves to identify precisely the elastic modulus of polymers is still possible on the condition that a correct MBM is used. This can be asserted from the comparison made in Table4with the values yielded by the two other physical techniques based on direct measurements.

Table 4 .

 4 Comparison between elastic modulus values given by standards (from tensile curve), the present model-based parameter estimation, the pulse-echo technique and the nanoindentation technique, for all samples A ( //

	A , ⊥ A , c A ) and B (in MPa). Discrepancy

Table 5

 5

	gathers the estimated maximal relaxation times max T τ	for all specimens A and B and
	the three different strain rates. Also shown is the corresponding Deborah number (ratio
	between the time constant mat t characterizing intrinsic 'fluidity' of the material, and the time
	scale of the experiment exp t	or of the observer). The fluidity of the material is inversely
	proportional to the Deborah number. In the present case, its maximal value can be calculated
	by the following formula:
		max De	=	exp mat t t	=	max τ ε ɺ	(45)

Table 5 .

 5 Average estimated maximal relaxation timeDe and hardening modulus G for specimens //A and B and different strain rates.

	τ	max	T	, corresponding Deborah number

max


N Lermé

F Malgouyres

Simultaneous Segmentation and Filtering via Reduced Graph Cuts

Keywords: graph cuts, reduction, image segmentation, filtering

Recently, optimization with graph cuts became very attractive but generally remains limited to small-scale problems due to the large memory requirement of graphs, even when restricted to binary variables. Unlike previous heuristics which generally fail to fully capture details, [8] proposes another band-based method for reducing these graphs in image segmentation. This method provides small graphs while preserving thin structures but do not offer low memory usage when the amount of regularization is large. This is typically the case when images are corrupted by an impulsive noise. In this paper, we overcome this situation by embedding a new parameter in this method to both further reducing graphs and filtering the segmentation. This parameter avoids any post-processing steps, appears to be generally less sensitive to noise variations and offers a good robustness against noise. We also provide an empirical way to automatically tune this parameter and illustrate its behavior for segmenting grayscale and color images.

Motivation and scope

Graph cuts have become increasingly popular due to their ability to efficiently compute the Maximum A Posteriori of Markov Random Fields (MRF). This popularity is notably driven by the introduction of a fast maximum-flow (max-flow) algorithm [START_REF] Boykov | An experimental comparison of min-cut/maxflow algorithms for energy minimization in vision[END_REF] making near real-time performance possible for solving numerous labeling problems such as denoising, image segmentation, stereovision, etc.

In parallel, technological advances in image acquisition have both increased the amount and the diversity of data to process. As an illustration, in the satellite SPOT-5 launched by Arianespace in 2002, the embedded high resolution sensors can capture multispectral and panchromatic images of about 1GB.

Processing this type of data amounts to solve large scale optimization problems. In the image segmentation context, almost all graph cuts-based methods are impractical to solve such problems due to the memory requirements for storing the underlying graphs. To overcome this situation, some amount of work has been done in this direction and a number of heuristics [START_REF] Li | Lazy Snapping[END_REF][START_REF] Lombaert | A multilevel banded graph cuts method for fast image segmentation[END_REF][START_REF] Sinop | Accurate banded graph cut segmentation of thin structures using laplacian pyramids[END_REF][START_REF] Kohli | Uncertainty driven multi-scale energy optimization[END_REF] and exact methods [START_REF] Lempitsky | Global optimization for shape fitting[END_REF][START_REF] Delong | A scalable graph-cut algorithm for N-D grids[END_REF][START_REF] Strandmark | Parallel and distributed graph cuts by dual decomposition[END_REF] have been proposed in recent years.

To our best knowledge, this problem seems to be first addressed in [START_REF] Li | Lazy Snapping[END_REF] where the underlying graph is built upon a pre-segmentation. Although this approach greatly reduce the computational burden of graph cuts, the results strongly depend on the algorithm used for computing the pre-segmentation. Also, better results are obtained when over-segmentation occurs, losing in this way the main benefit of such a reduction.

Others have also reported band-based methods [START_REF] Lombaert | A multilevel banded graph cuts method for fast image segmentation[END_REF][START_REF] Sinop | Accurate banded graph cut segmentation of thin structures using laplacian pyramids[END_REF][START_REF] Kohli | Uncertainty driven multi-scale energy optimization[END_REF]. A low-resolution of the image is first segmented and the solution is propagated to the finer level by only building the graph in a narrow band surrounding the interpolated foreground/background interface at that resolution. While such an approach drastically reduce time and memory consumption, it is limited to segment roundish objects. This problem is notably reduced in [START_REF] Sinop | Accurate banded graph cut segmentation of thin structures using laplacian pyramids[END_REF] but still present for low-contrasted details. In [START_REF] Kohli | Uncertainty driven multi-scale energy optimization[END_REF], finer bands are obtained using an uncertainty measure associated to each pixel.

Exact methods have been also investigated [START_REF] Lempitsky | Global optimization for shape fitting[END_REF][START_REF] Delong | A scalable graph-cut algorithm for N-D grids[END_REF][START_REF] Strandmark | Parallel and distributed graph cuts by dual decomposition[END_REF]. In [START_REF] Lempitsky | Global optimization for shape fitting[END_REF], binary energy functions are minimized for the shape fitting problem with graph cuts in a narrow band while ensuring the optimality on the solution. One makes a band evolve around the object to delineate by expanding it when the minimum-cut (min-cut) touches its boundary. This process is iterated until the band no longer evolves. Although the algorithm generally converges in few iterations toward the optimal solution, an initialization is required and no bound on the band size is given.

A parallel max-flow algorithm yielding a near-linear speedup with the number of processors is described in [START_REF] Delong | A scalable graph-cut algorithm for N-D grids[END_REF]. Nevertheless, the algorithm is relatively sensitive to the available amount of physical memory and remains less efficient on small graphs.

The approach used in [START_REF] Strandmark | Parallel and distributed graph cuts by dual decomposition[END_REF] is different: instead of reducing the graphs, the problem is decomposed into optimizable sub-problems, solved independently and updated according to the results of the adjacent problems. This process is iterated until convergence and optimality is guaranteed by Lagrangian decomposition.

Finally, another band-based method called Reduced Graph Cuts (RGC) was proposed for reducing graphs in binary image segmentation [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF]. The graph is progressively built by only adding nodes which locally satisfy a condition. In the manner of [START_REF] Sinop | Accurate banded graph cut segmentation of thin structures using laplacian pyramids[END_REF][START_REF] Kohli | Uncertainty driven multi-scale energy optimization[END_REF], the graph nodes are typically located in a narrow band surrounding the object edges to segment. Empirically, the authors show in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] that the solutions obtained with and without reduction are identical and the time for reducing the graph is even compensated by the time for computing the min-cut in the reduced graph when the regularization is of moderate level.

The rest of this paper is organized as follows. We first briefly review the graph cuts framework in Section 2 as well as the band-based strategy of [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] in Section 3 for reducing graphs. Afterwards, a new parameter is introduced in Section 4 for further reducing the graphs and removing small undesired segments in the segmentation due to noise. The sensitivity of this parameter as well as its robustness against noise are also evaluated through experiments for segmenting grayscale and color images.

Preliminaries

Consider a multi-channels image I : P ⊂ Z d → [0, 1] c (c > 0) as a function, mapping each pixel p ∈ P to a vector I p ∈ [0, 1] c 1 . We define a binary segmentation as an application u affecting to each pixel p ∈ P either 0 (background) or 1 (object) and we write u ∈ {0, 1} P . A popular strategy to segment I is to minimize a MRF of the form [START_REF] Boykov | Interactive graph cuts for optimal boundary and region segmentation of objects in N-D images[END_REF]:

E(u) = β p∈P E p (u p ) + (p,q)∈N E p,q (u p , u q ), (1)
among u ∈ {0, 1} P and for a fixed parameter β ∈ R + . The neighborhood system N ⊂ (P × P) is a subset of all pixel pairs. In the sequel, "connectivity 0" will denote 4 and 6 neighbors in 2D and 3D images while "connectivity 1" will denote 8 and 26 neighbors for the same images. In [START_REF] Bouman | Cluster: An unsupervised algorithm for modeling Gaussian mixtures[END_REF], the data term E p (.) is defined as the negative log-likelihood of a label being assigned to pixel p and is computed from its color and the appearance models of the object and background seeds 2 :

E p (1) = -log P(I p |p ∈ O) E p (0) = -log P(I p |p ∈ B) (2)
For any pixels pair (p, q) ∈ N , the corresponding smoothness term in ( 1) is defined as a contrast-sensitive Ising model:

E p,q (u p , u q ) = 0 if u p = u q , 1 p-q 2 exp - Ip-Iq 2 2 2σ 2 otherwise, (3)
where . 2 is the Euclidean norm (either in R d or R c ) and σ is a free parameter generally related to noise acquisition. As an illustration, when pixels p and q belong to a uniform area, we have I p -I q 2 < σ. This implies a large cost in the exponential and discourages any cut between pixels p and q. Conversely, when these pixels are on both sides of a contour, we have I p -I q 2 > σ. This encourages any cut between pixels p and q due to low value of the exponential.

When the smoothness terms are submodular [START_REF] Kolmogorov | What energy functions can be minimized via graph cuts[END_REF], the minimizer of (1) can be efficiently obtained by computing a min-cut in a weighted digraph G = (V, E) with a set of nodes V = P ∪ {s, t}, a set of edges E ⊂ (V × V) and edge capacities c : (V × V) → R + . The terminal nodes s and t are called the source and the sink, respectively. The set of edges E is split into two disjoint sets E n and E t denoting respectively n-links (edges linking two nodes of P) and t-links (edges linking a node of P to the terminal s or t). Once the min-cut is computed in G, we set u p = 1 if a node p is connected to the source s and u p = 0 if p is connected to the sink t.

1 Usually, P corresponds to a rectangle. 2 In this setting, the distributions are estimated using a Gaussian Mixtures Model.

The number of Gaussians is automatically computed using a statistical criterion [START_REF] Bouman | Cluster: An unsupervised algorithm for modeling Gaussian mixtures[END_REF].

Reduction

As explained before, the memory consumption of graph cuts for segmenting highresolution data can be very large. As an illustration, the max-flow algorithm of [START_REF] Boykov | An experimental comparison of min-cut/maxflow algorithms for energy minimization in vision[END_REF] v3.01 used in the experiments of Section 4, allocates 25 P + 16 E n bytes3 . For a fixed amount of RAM, one clearly see that the maximum image size quickly decreases as the dimensionality d of P increases. As shown in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF], most of the nodes in the graph are however useless during the max-flow computation since they are not traversed by any flow. Ideally, one would like to extract the smallest possible graph G = (V , E ) from G = (V, E) while keeping the max-flow value f * in G identical or very close to the max-flow value f * in G. This corresponds to an ideal optimization problem which we will not try to solve since the method for determining G also needs to be (very) fast.

Let us first introduce some terminology before reviewing the method of [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] for building G . For the sake of clarity, the same notations are used as in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF]. In accordance with the construction given in [START_REF] Kolmogorov | What energy functions can be minimized via graph cuts[END_REF], we consider (without loss of generality) that a node is connected to at most one terminal

(s, p) ∈ E t ⇒ (p, t) ∈ E t , ∀p ∈ P. (4)
We also summarize t-links capacities by

c(p) = c(s, p) -c(p, t), ∀p ∈ P. (5)
For any B ⊂ Z d4 and a node p ∈ P, we denote by B p the set translation of B at p

B p = {q + p | q ∈ (B ∩ P)}. (6)
For Z ⊂ P and B ⊂ Z d , we denote by Z B the dilation of Z by B as

Z B = {p + q | q ∈ (B ∩ P), p ∈ Z} = p∈Z B p . (7)
From here, the idea developed in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] for building G is to remove from the nodes of G any Z ⊂ P where all nodes are linked to s (resp. to t) and such that all the flow that might get in (resp. out of) the region Z B does so by traversing its boundary and can be absorbed (resp. provided) by the band Z B \ Z. Building such Z is done by testing each individual pixel p ∈ P. In the manner of [START_REF] Sinop | Accurate banded graph cut segmentation of thin structures using laplacian pyramids[END_REF][START_REF] Kohli | Uncertainty driven multi-scale energy optimization[END_REF], the remaining nodes are therefore located in a narrow band surrounding the object edges to segment. In practice, the authors of [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] use a more conservative test by testing each node p ∈ Z in a square window B of size (2r + 1) d centered in p:

   either ∀q ∈ B p , c(q) ≥ +δ r , or ∀q ∈ B p , c(q) ≤ -δ r . (8)
where δ r = P (B) (2r+1) d -1 and P (B) is defined as P (B) = max( {(p, q) : p ∈ B, q ∈ B and (p, q) ∈ N }, {(q, p) : p ∈ B, q ∈ B and (q, p) ∈ N }).

In words, for any node p ∈ Z satisfying the first (resp. second) condition of ( 8), all its neighbors q ∈ B q are only linked to s (resp. t) and the flow that might get in (resp. out) through t-links in B p \ {p} suffices to saturate the n-links going out (resp. in) B p . Thus, p becomes useless and need not to be added to G . An algorithm of complexity O(1) (i.e. independent of r) is also mentioned in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] for computing [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF]. Additionally, the extra memory storage required by this algorithm is a table of dimensionality (d -1) and is therefore negligible over the image and the graph size. A key point of [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] is that the pixel error between segmentations obtained with and without reduction remains extremely low, hence preserving thin structures which are ubiquitous in some applications.

The experiments presented in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] confirm the intuitive dependence between the size of the reduced graph G and the model parameters. Indeed, when minimizing (1) by graph cuts, the t-links capacities are all multiplied by β. It is therefore straightforward to observe that the test ( 8) is harder to satisfy as β decreases. In such a situation, we need a larger window radius for decreasing δ r in order to reduce the size of the reduced graph G . This results in wider bands around the object contours. Conversely, when β is large, we can afford a large δ r and therefore a small window radius to decrease the size of the reduced graph G . An ideal situation therefore consists of large area of nodes connected to the same terminals separated by rough borders. As opposite, a less favorable situation occurs when these area consist of nodes connected to different terminals. This is typically the case when dealing with noisy images. In the next section, we embed a new parameter in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] to both further reducing G and filtering the segmentation while keeping β large.

Simultaneous segmentation and filtering

A naive approach to filter the segmentation is to apply morphological operators (e.g. opening or closing). Nonetheless, when the amont of noise is large, such an approach fail during the reduction since a lot of nodes would be added to the reduced graph. Another approach would consist in denoising the image first and then applying [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF]. This would lead to unsatisfactory results in the case for instance of echographic images since contours of objects would be over-smoothed.

Another way to filter the segmentation is to relax ( 8) is to allow some nodes in B p to fail complying the test. The proportion of nodes satisfying (8) can be controlled by a parameter η ∈ [0, 1]. As η decreases, the test (8) becomes easier to satisfy since a larger proportion of nodes can be connected to opposite terminals. Embedding η in (8) leads to

   either {q ∈ B p | c(q) ≥ +δ r } ≥ η B p , or {q ∈ B p | c(q) ≤ -δ r } ≥ η B p . (10)

Further reducing graphs

The parameter η can be used for decreasing the memory consumption of graph cuts. The Figure 3 illustrates how far the test (8) can be relaxed for further reducing graphs while getting nearly the same segmentation. In this experiment, the segmentation as well as the reduced graph are shown for segmenting a 2D noisy image. Since the test ( 10) is easier to satisfy as η decreases, the reduced graph G becomes thicker around the object contours.

Automatic tuning of η

Lower bound For a fixed window radius, notice first that the value of η must be large enough to not increase the number of components in the reduced graph G (see Figure 2). Indeed, below some value (denoted by η min ), the reduced graph G is split into multiple pieces in areas with high-curvature and the min-cut is no longer ensured of being fully embedded into G . This implies that some voxels could be wrongly labeled in the segmentation.

The Figure 1 illustrates a situation where η min can be easily computed with an image consisting of two highly-contrasted areas. Using [START_REF] Lombaert | A multilevel banded graph cuts method for fast image segmentation[END_REF] with a square window of radius r and η = 1, the reduced graph G corresponds to a thin band of size 2r. An easy under-estimation of η min is obtained by imposing that η min permits to segment these two contrasted areas. In order to do so, we want the test [START_REF] Lombaert | A multilevel banded graph cuts method for fast image segmentation[END_REF] to be false for any pixel p at the boundary between these areas. For such a pixel, we have (assuming e.g. that c(p) ≥ +δ r )

{q ∈ B p | c(q) ≥ +δ r } = (r + 1)(2r + 1) d-1 . ( 11
)
As a consequence, if

η ≤ (r + 1)(2r + 1) d-1 (2r + 1) d , ( 12
)
the node p does not belong to the reduced graph G . Since we want to avoid the situation, we must therefore have

η > (r+1)(2r+1) d-1 (2r+1) d = 1 -r 2r+1 = η min . (13)
Remark that [START_REF] Strandmark | Parallel and distributed graph cuts by dual decomposition[END_REF] does not depend on the dimensionality d of P. By observing [START_REF] Strandmark | Parallel and distributed graph cuts by dual decomposition[END_REF], it is straightforward to see that, as the window radius r tends to infinity, the proportion of nodes allowed to be connected to opposite terminals tends to 1 2 . In practice, we also observed that ( 13) is less accurate in connectivity 0 than in connectivity 1 (see Figure 2).

Upper bound For a fixed window radius r and a positive amount of noise ξ, one can observe in Figure 3 that there exists a value of the parameter η for which most of the nodes in noisy regions are removed from the graph G, leading to a diminution of the size of the reduced graph G .

The purpose of this paragraph is to identify, from a statistical point of view, a reliable value of the parameter η for which all nodes of P are very likely to be removed from G. For a fixed amount of noise ξ in the image I, we therefore want to find an upper bound on η by finding the maximum value of η in such a way that we control the proportion of nodes corresponding to noisy pixels in homogeneous areas.

Consider a noisy constant image I with a noise generated by a Bernoulli distribution of parameter ξ ∈]0, 1[, corresponding to the amount of noise in I5 . The two cases where ξ = 0 and ξ = 1 are trivial and are not considered in our analysis. Assume now that the graph G is defined as in Section 3 where the nodes corresponding to noise-free pixels are connected to the sink t with a capacity c(q) ≤ -δ r and the nodes corresponding to noisy pixels have a capacity c(q) > -δ r .

First, let X be a discrete random variable counting degraded pixels in a square window B of size n = (2r + 1) d in the image I. Then, the probability that at least k pixels are corrupted in B is given by

P(X > k) = n i=k+1 n i ξ i (1 -ξ) n-i , (14)
where

n i = n! i!(n-i)! .
For a fixed window radius r, it is straightforward to see that ( 14) is decreasing in k and tends to ξ n if we impose that ξ ∈ ]0, 1[. According to the test [START_REF] Lombaert | A multilevel banded graph cuts method for fast image segmentation[END_REF] and the hypothesis on G: a node p ∈ P can be removed from G if and only if

{q ∈ B p | c(q) ≤ -δ r } = {q ∈ B p | q is noise-free} ≥ ηn (15)
Moreover, we assumed

{q ∈ B p | q is noise-free} ∼ (n -X). (16)
Therefore, we have

P(p is not removed) = P((n -X) < ηn) = P(X > (1 -η)n). (17)
Fixing a proportion ε 0 of wrongly constructed nodes, we choose

η + = max {η ∈ [0, 1] | P(X > (1 -η)n) ≥ ε}, (18)
Considering the lower bound η min defined in (13), we set

η max = max {η min , η + }. (19)
Combining the definitions of the lower and upper bounds (see ( 13) and ( 19)), it now becomes easy to get an estimation of the parameter η * for a fixed window radius by setting

η * = η min + η max 2 . ( 20
)
Let us now analyze the joint behavior of the lower and the upper bounds. When the amount of noise ξ is fixed, one can easily observe that the gap

∆η = (η max -η min ), (21)
grows as the window radius r increases. Indeed, we have previously seen that the lower bound η min tends to 1 2 as the window radius r increases (see [START_REF] Strandmark | Parallel and distributed graph cuts by dual decomposition[END_REF]). The previous observation is also due to the fact that the upper bound η max grows as the window radius r increases.

Similarly, when the window radius r is fixed, remark that (21) decreases when the amount of noise ξ increases. This situation is consistent because η min remains the same but η max tends to 1 2 since it is more likely that the number of degraded pixels increase in the window B. Increasing the window radius r can compensate the augmentation of the amount of noise only up to ξ = 0.5. Finally, we empirically found that setting ε = 0.05 gives best estimate of η max in (18).

Filtering

The parameter η can also serves to filter the segmentation. This behavior is illustrated in Figure 4 for segmenting a 3D noisy image acquired from a confocal microscope. White spots correspond to cell nuclei in a mouse cerebellum. Observe how far the filtering acts for small values of η: small regions in the reduced graph G as well as in the segmentation are progressively removed as η decreases.

The robustness (see Figure 6 and 5) and sensitivity to noise of the parameter η are now (see Figure 7) are now analyzed. Let us describe the experimental setup. The experiment consists in segmenting four grayscale and five 2D color images in connectivity 1 with an increasing noise level ranging from 4 to 48%. For each image, we compute a reference segmentation on the noise-free image by placing the seeds by hand. We set β = +∞ and automatically estimate the σ parameter as explained in [START_REF] Rother | GrabCut": Interactive foreground extraction using iterated graph cuts[END_REF]. Then, for each impulsive noise level, we select the segmentation maximizing the Dice Similarity Coefficient (DSC) between the reference image and all segmentations obtained through a fixed range of window radii and η values. We choose window radii r from 1 to 12 and eight linearly spaced values of η from η min (w.r.t. r) to 1. Then, each segmentation is computed using the same seeds as those used for the computing the reference segmentation. Again, the σ parameter is automatically estimated as in [START_REF] Rother | GrabCut": Interactive foreground extraction using iterated graph cuts[END_REF].

As shown in Figure 6, for an impulsive noise level up to 45%, the parameter η appears to be reasonably robust with a DSC always greater than 94% for all images, except for the image "rice". However, such high and stable noise robustness can only be reached by increasing the amount of seeds (see Figure 6). The reason why the algorithm behaves poorly on the "rice" image is the following. As said earlier, r must be large enough when ξ increases for removing a maximum number of segments due to noise. This implies wider bands in G around the object contours. However, the object contours further oscillate as ξ increases. Another reason is due to the proximity of the objects to segment. As an illustration, consider two circles over a uniform background, separated by a distance d 0 > 0. We clearly see that the test [START_REF] Lombaert | A multilevel banded graph cuts method for fast image segmentation[END_REF] becomes more and more difficult to satisfy when the window radius r increases. When (2r + 1) ≥ d 0 , the reduced graphs of both circles fuse into one component. This is typically the case for the image "rice" because it consists of small assembled rice grains near from each other. Finally, the Figure 7 also illustrates that the parameter η is not very sensitive to the variations of r and η. The DSC does not vary much with respect to noise, except for the image "rice". This exception can be explained for the same reasons as before.

Conclusion

In this paper, we have presented a new parameter to embed in [START_REF] Lermé | Reducing graphs in graph cut segmentation[END_REF] for simultaneously reduce graphs and filter segmentations with the same computational complexity. We have also described an original manner to automatically tune it with the window radius parameter. In the proposed experiments, this new parameter generally appears to be less sensitive to noise but only for a limited amount, typically less than 50%. To overcome this situation, we could for instance inspect larger neighborhoords to further discriminate signal from noise. Fig. 7: Sensitivity of η for segmenting the images in Figure 6 with an impulsive noise level of 36%. The seeds and model parameters are the same than those used in Figure 6.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Toy example for computing the lower bound ηmin.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Memory gain when segmenting a 2D synthetic image corrupted by 10% of impulsive noise (left). Top row shows the nodes of the reduced graph in light gray while bottom row shows the corresponding segmentation. In this experiment, we set r = 3 and use connectivity 1.

Fig. 5 :

 5 Fig.5: Qualitative analysis of the robustness to noise for segmenting the images "f117" (left-most column), "pyramid" (left column), "sunflower" (right column) and "flamingo" (right-most column) in with a fixed impulsive noise level of 36%. Seeds and model parameters are the same than those used in Figure6(top row).

Fig. 6 :

 6 Fig. 6: Quantitative analysis of the robustness to noise for segmenting in connectivity 1 four 2D grayscale images (top-most curves in the list) and five 2D color images with an impulsive noise level ranging from 4 to 48%.

The operator ' ' stands for the cardinality of a set.

In practice, B is a square centered at the origin.

Histogram-based techniques can be for instance used to estimate ξ.