Mariam Kallakech

Philippe Biela

Ludovic Macaire

Denis Hamad

Mariam Kalakech

Constraint scores for semi-supervised feature selection: A comparative study

Keywords: Feature selection, Pairwise constraints, Kendall's coefficient, Constraint scores, Laplacian score, Fisher score

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Introduction

In machine learning and pattern recognition applications, the processing of high dimensional data requires large computation time and capacity storage. Though, it leads to poor performances when the dimensionality to sample size ratio is high. To improve performances, the sample dimensionality is reduced thanks to feature extraction or selection schemes [START_REF] Liu | Feature extraction construction and selection a data mining perspective[END_REF]; [START_REF] Yu | Feature selection for high-dimensional data: A fast correlation-based filter solution[END_REF]). Let us notice that feature extraction transforms the original input space into a new low dimensional space by combining the initial features, while feature selection retains the most relevant ones in order to build a low dimensional feature space.

Data samples can be either unlabeled or labeled, leading to the development of unsupervised and supervised feature selection techniques. Unsupervised feature selection measures the feature capacity of keeping the intrinsic data structure in order to evaluate its relevance [START_REF] Dy | Feature selection for unsupervised learning[END_REF]). Supervised feature selection consists in evaluating feature relevance by measuring the correlation between the feature and class labels [START_REF] Yu | Efficient feature selection via analysis of relevance and redundancy[END_REF]).

Supervised feature selection requires sufficient labeled data samples in order to provide a discriminating feature space. However, the sample labeling process by the human user is fastidious and expensive. That is the reason why in many real applications, we have huge unlabeled data and small labeled samples. To deal with this "lack labeled-sample problem", recent semi-supervised feature selection schemes have been developed by Zhao et al. (Zhao and Liu (2007a); [START_REF] Zhao | Semi-supervised feature selection via spectral analysis[END_REF]). They propose a semi-supervised feature relevance criterion which takes into account both unlabeled data and labeled samples. Unfortunately, this score requires to define the classifier step in order to compare the initial labels of the samples and those provided by the classifier [START_REF] Ng | On spectral clustering: Analysis and an algorithm[END_REF]).

Beside class labels, there is another kind of user supervision information called the pairwise constraints [START_REF] Bar-Hillel | Learning a mahalanobis metric from equivalence constraints[END_REF]). It consists to simply specify whether a pair of data samples must be regrouped together (mustlink constraints) or cannot be regrouped together (cannot-link constraints).

Zhang et al. [START_REF] Zhang | Constraint score: A new filter method for feature selection with pairwise constraints[END_REF]) propose to evaluate the feature relevance by scores which only take into account these constraints. Zhao et al. de-fine another score which uses both the pairwise constraints defined by the user and the unlabeled nearest neighbors of the samples [START_REF] Zhao | Locality sensitive semi-supervised feature selection[END_REF]).

However, this score considers the neighbors of each sample without explicitly taking into account its local density property.

These authors have experimentally shown that the features selected thanks to these constraint-based scores, may provide results which are comparable with those given by supervised approaches. Unfortunately, these scores strongly depend on the given constraint subsets built by the user. So, when the user slightly modifies the constraint subset, the feature scores may also change.

In this paper, we first present a review of the feature selection scores. We then propose a semi-supervised score which uses both pairwise constraints and the local properties of the unlabeled data. We experimentally demonstrate that this score, thanks to the contribution of the unlabeled data, is less sensitive to constraint changes than the classical constraint scores, while providing satisfying classification results.

Previous works compare the performances of the feature scores by considering the accuracy rates obtained by a classifier operating in the selected feature space. In order to measure the sensitiveness of the scores to constraint changes, we could estimate the dispersion of accuracy rates with respect to different subsets of constraints. Though, this evaluation depends on the be-havior of the used classifier. So, we propose to only examine the dispersion of the feature ranks provided by the examined scores thanks to the Kendall's coefficient [START_REF] Grzegorzewski | The coefficient of concordance for vague data[END_REF]).

The paper is organized as follows. In section 2, we review different supervised and unsupervised feature selection scores. Then, we introduce the spectral semi-supervised score in section 3. Recent constraint scores are detailed in section 4 and our semi-supervised constraint score is presented in section 5.

In section 6, we study the relationships between the constraints given by the user and the feature ranks obtained by the different scores. Comparative experimental results are provided in section 7 in order to assess the efficiency of our semi-supervised constraint score.

Supervised and unsupervised feature selection

Given a dataset of n samples defined in a d-dimensional feature space, let us denote X = (x ir ) i = 1, . . . ,n; r = 1, . . . ,d; the associated data matrix where x ir is the r th feature value of the i th data. Each of the n rows of the matrix X represents a data sample x i = (x i1 , . . . ,x id ) ∈ R d , while each of the d columns of X defines the feature values f r = (x 1r , . . . ,x nr ) T ∈ R n .

Supervised feature selection

The principle of supervised feature selection consists in examining the correlation between projected data samples and their class labels on each feature axis. It looks for features on which the classes are compact and far from each others. For this purpose, one uses the well known Fisher criterion to evaluate the feature relevance [START_REF] Bishop | Neural Networks for Pattern Recognition[END_REF]).

By considering the sample coordinates on the feature f r , each class ω, ω=1, ..., c, populated with n ω labeled samples is characterized by its mean µ ωr and its variance σ 2 ωr . Moreover, let us denote µ r the mean of all data samples on the feature f r .

The Fisher score F r used to evaluate the relevance of the feature f r is defined by:

F r = c ω=1 n ω (µ ωr -µ r ) 2 c ω=1 n ω σ 2 ωr . (1)
In order to select the most relevant features, they are sorted according to the decreasing order of their Fisher score F r .

Unsupervised feature selection

Unsupervised feature selection consists in evaluating the relevance of each feature by examining the dispersion of the data samples projected on its axis.

A feature is considered as being relevant when the data samples projected on this feature axis are scattered as much as possible.

So, the variance score V r is used to evaluate the relevance of the feature f r :

V r = 1 n n i=1 (x ir -µ r ) 2 . ( 2
)
The features are sorted according to the decreasing order of V r , in order to select the most relevant ones.

Rather than measuring the data dispersion along a feature axis, one examines the local properties of the data. The basic idea is to assume that the input data pairwise distances are preserved in the relevant feature space.

So, similar samples have to be close when they are projected on a relevant feature axis.

According to the spectral graph theory (von Luxburg. ( 2007)), data samples

x i , i = 1, . . . ,n are represented by n nodes in a graph structure. The edge between two connected nodes i and j is weighted by a similarity level defined by:

s ij = exp - x i -x j 2 2t 2 , (3)
where x i -x j 2 represents the squared euclidean distance between x i and x j in the d-dimensional initial feature space (von Luxburg. ( 2007)). The parameter t has to be tuned in order to represent the local dispersion of the data (Zelink-Manor and Perona ( 2005)).

The weights of the graph are represented by a similarity matrix S (n × n).

From S, we calculate the Laplacian matrix defined as L = D -S, where D

(n × n) is the diagonal matrix, D ii = n j=1 s ij .
It is interesting to note that the degree D ii of a node i can be considered as a local density measure at x i .

In an unsupervised context, He et al. assume that the projections (coordinates) of similar data on the examined feature axis have to be as close as possible [START_REF] He | Laplacian score for feature selection[END_REF]). They propose to compute the Laplacian score L r of a feature f r as:

L r = ij (x ir -x jr ) 2 s ij i x ir -f r D ii . ( 4
)
It is easy to demonstrate that:

L r = f T r L f r f T r D f r (5)
where

f r = f r -f r and f r = n i=1 x ir D ii n i=1 D ii .
By considering D ii as a density probability measure, f r is the weighted feature average.

In order to select relevant features, they are sorted according to the ascending order of L r .

He et al. have experimentally demonstrated that the classifier operating in the feature space selected by the Laplacian score L r , outperforms the classifier operating in the feature space selected by the variance score V r . Indeed, L r takes into account the locality structure of the data samples.

Semi-supervised feature selection

In many applications, we have huge unlabeled and a few labeled data samples. Indeed, labeling all the data samples by the user is time consuming and fastidious. In that context, the labeled data subset is usually too small to carry sufficient information for the supervised selection while unsupervised approaches ignore this label information which could be yet interesting for feature selection. (Zhao and Liu (2007a); [START_REF] Zhao | Spectral feature selection for supervised and unsupervised learning[END_REF]). This score compares the initial labels of the data samples with the labels provided by a classifier operating with the examined feature f r . It is defined as:

M r = α ij (x ir -x jr ) 2 s ij i x ir -f r D ii + (1 -α) 1 -N M I f r ,y . (6)
where f r is the cluster indicator of the samples generated by the classifier and y is the initial class label vector (Zhao and Liu (2007a)). The first term of equation ( 6) calculates the Laplacian score of the feature f r , while the second term estimates the corresponding classification error of f r according to the labeled data. The term α is a regularization parameter set to 0.1 by (Zhao and Liu (2007a)), in order to favor the contribution of the labeled data. That is the reason why the selected features by using equation ( 6) mainly depend on the labeling decision achieved by the classifier.

Constraint scores

The prior knowledge about the data can be represented according to two different ways: class labels and pairwise constraints. Class labels require to have detailed information about the classes and to precisely indicate the label of each data sample. Pairwise constraints simply mention for some pairs of data samples that they are similar, i.e. must be regrouped together (must-link constraints), or that they are dissimilar, i.e. cannot be regrouped together (cannot-link constraints).

The user has to build the subset M of must-link constraints and the subset C of cannot-link constraints defined as:

M = {(x i ,x j )
, such as x i and x j must be linked}. C = {(x i ,x j ), such as x i and x j cannot be linked}.

The cardinals of these subsets are usually much lower than the number ( n 2 ) of all possible pairwise constraints.

These pairwise constraints are easier to be obtained by the user than the class labels. They simply formalize that two data samples belong or not to the same class without detailed information about the classes in presence.

Indeed, labeled data samples can be transformed into must-link and cannotlink constraints but not vice versa. It consists in putting must-link constraint between two data samples which share the same label and cannot-link constraint between two data samples sharing different labels.

Zhang et al. have recently proposed a constraint feature selection scheme which uses only a subset of must-link and cannot-link constraints [START_REF] Zhang | Constraint score: A new filter method for feature selection with pairwise constraints[END_REF]).

In the context of the spectral theory (von Luxburg. ( 2007)), two specific graphs are built:

• The must-link graph G M where a connection is established between two nodes i and j if there is a must-link constraint between their corresponding samples (nodes) x i and x j .

• The cannot-link graph G C where a connection is established between two nodes i and j if there is a cannot-link constraint between their corresponding samples (nodes) x i and x j .

The connection weights between two nodes of the graphs G M and G C are respectively stored by the similarity matrices

S M (n × n) and S C (n × n),
and are given by:

s M ij =          1 if (x i ,x j ) ∈ M or (x j ,x i ) ∈ M 0 otherwise, (7)
s C ij =          1 if (x i ,x j ) ∈ C or (x j ,x i ) ∈ C 0 otherwise. ( 8
)
The matrices S M and S C are used to define the constraint Laplacian

matrices L M = D M -S M and L C = D C -S C
, where D M and D C are the degree matrices defined by

D M ii = n j=1 s M ij and D C ii = n j=1 s C ij .
In order to measure the constraint preserving ability of the feature f r , Zhang et al. define two constraint scores C 1 r and C 2 r :

C 1 r = ij (x ir -x jr ) 2 s M ij ij (x ir -x jr ) 2 s C ij = f T r L M f r f T r L C f r , (9)
C 2 r = ij (x ir -x jr ) 2 s M ij -λ ij (x ir -x jr ) 2 s C ij = f T r L M f r -λf T r L C f r , (10)
where λ is a regularization parameter used to balance the contribution of the two constraints terms of C 2 r . Must-link constraints are favored by setting

0 < λ < 1.
The lower these two scores are, the more relevant the feature is. Zhang et al.

have experimentally shown that the features selected by C 1 and C 2 provide similar performances when λ is well balanced.

Constraint scores for semi-supervised feature selection

The scores presented in section 4 use only the available constraints and do not take into account the unlabeled data contribution. Zhao et al. define another score which uses both unlabeled data and pairwise constraints in order to retrieve both locality properties and discriminating structures in the data samples [START_REF] Zhao | Locality sensitive semi-supervised feature selection[END_REF]). They build a new graph G W which connects samples having high probability of sharing the same label:

• G W is the within-class graph: two nodes i and j are connected if (x i ,x j ) or (x j ,x i ) belongs to M, or if the two samples are unlabeled but they are sufficiently close to each other (by using the k-nearest neighbor graph)

The edges in the graphs G W are weighted by using the similarity matrix S W

(n × n) and are expressed as:

s W ij =                              γ if (x i ,x j ) ∈ M or (x j ,x i ) ∈ M 1 if x i or x j is unlabeled but node i ∈ KN N (j) or node j ∈ KN N (i) 0 otherwise (11)
where γ is a constant parameter which has been empirically set to 100 in [START_REF] Zhao | Locality sensitive semi-supervised feature selection[END_REF]).

Zhao et al. introduce a Laplacian score, called the locality sensitive discriminant analysis score and defined as:

C 3 r = i,j (x ir -x jr ) 2 s W ij i,j (x ir -x jr ) 2 s C ij = f T r L W f r f T r L C f r , (12)
where L W = D W -S W , D W being the degree matrix defined by

D W ii = n j=1 s W ij .
This score implicitly takes into account the unlabeled data but favors pairs of must-link data by assigning them high weights in the matrix S W . Moreover, the similarity matrix S W represents the links between the k-nearest neighbors of the data by binary weighting them. By mainly considering the must-link constraints, C 3 is very close to C 2 , and both neglect the unlabeled data samples.

Though, taking into account the unlabeled data samples should catch the data structure and make less sensitive a feature score against the given constraint subset. That leads us to propose another semi-supervised constraint score which is less sensitive to the constraints chosen by the user. Given the matrices S, S M and S C , the semi-supervised constraint score is defined as :

C 4 r = f T r L fr f T r D fr . f T r L M f r f T r L C f r . ( 13
)
The proposed score C 4 r is the simple product between the Laplacien score L r (see equation ( 4)) processed with the unlabeled data and the constraint score C 1 r (see equation ( 9)) defined by Zhang:

C 4 r = L r .C 1 r , (14)
So, it takes into account both the unlabeled data thanks to L r and the available constraints thanks to C 1 r in order to evaluate the relevance of the feature f r . As for the other scores, the features are ranked in ascending order according to score C 4 in order to select the most relevant ones.

Constraint subset influence in feature selection process

The constraint scores introduced above evaluate the relevance of the features based on the must-link and cannot-link constraints. However, these scores strongly depend on the given constraint subsets M and C. Indeed, changing the subset of available constraints could lead to a large change in the feature ranks and so, in the selected features. To illustrate this problem, we examine the following toy example.

Toy example

Let us consider four 3-dimensional data samples:

A(-3, -1,1); B(-3,1,1); C(-1 -1,1) and D(1, -3, -1).

A, B and C belong to the first class whereas D is assigned to the second class. We can see that the feature f 3 is the single feature which corresponds propose to measure how the feature ranks vary with respect to the subset of constraints.

C/M {(A,B)} {(A,C)} {(B,C)} {(A,D)} C 1 f 3 = f 1 , f 2 f 3 = f 2 , f 1 f 3 , f 1 , f 2 C 2 f 1 , f 3 , f 2 f 3 = f 2 , f 1 f 3 , f 1 , f C 3 f 1 , f 3 , f 2 f 3 , f 2 , f 1 f 3 , f 1 , f C 4 f 3 = f 1 , f 2 f 3 = f 2 , f 1 f 3 , f 1 , f {(B,D)} C 1 f 3 = f 1 , f 2 f 3 = f 2 , f 1 f 3 , f 1 = f 2 C 2 f 1 , f 3 , f 2 f 2 , f 3 , f 1 f 3 , f 1 = f C 3 f 1 , f 3 , f 2 f 2 , f 3 , f 1 f 3 , f 2 , f C 4 f 3 = f 1 , f 2 f 3 = f 2 , f 1 f 3 , f 1 , f {(C,D)} C 1 f 3 = f 1 , f 2 f 3 = f 2 , f 1 f 3 , f 1 = f C 2 f 3 = f 1 , f 2 f 3 = f 2 , f 1 f 3 , f 1 = f C 3 f 3 , f 1 , f 2 f 3 , f 2 , f 1 f 3 , f 2 , f C 4 f 3 = f 1 , f 2 f 3 = f 2 , f 1 f 3 , f 1 , f

Kendall's coefficient

The performances of the examined scores are generally compared by measuring the accuracy rates obtained from well classified data projected on the selected features. To measure the dependence of the scores on the given constraint subsets, we could estimate the disparity of the accuracy rates obtained with different constraint subsets. This evaluation requires to define a labeling decision step, such as k-nearest neighbor classifier, which operates in the selected feature space. Since this decision step influences the quality of classification, the comparison between the score performances may depend on the used classifier.

We prefer to examine only the feature ranks deduced from the scores, so that our study is not corrupted by the decision step. More precisely, we study the concordance between feature ranks when the constraint subset changes.

Given the sample set X = (x 1 ,...,x n ) and the associated class label of each data sample, we randomly pick up a subset S q of pairwise constraints (S q = M q ∪ C q ). Then, we rank the d features f r , according to the different constraint scores C * r detailed in sections 4 and 5 ( * = 1, 2, 3, 4).

Let us denote R * qr the rank of the feature f r when the score C * r considers the constraint subset S q . In order to evaluate the influence of constraint subset, the feature selection is run p times. For a score C * , the ranks of the d features obtained with the p different subsets S q are represented by the matrix R * (p × d):

R * =             R * 11 R * 12 ... R * 1d R * 21 R * 22 ... R * 2d ... ... ... ... ... R * p1 R * p2 ... R * pd            
The q th row of R * represents the d feature ranks by using the subset S q (q = 1,...,p) while the r th column represents the ranks of the feature f r by using the p different constraint subsets. Therefore, each row of R * is a per-mutation of the d feature ranks which depends on the available constraint subset S q .

We use the Kendall's coefficient to measure the concordance or agreement between the feature ranks with p constraint subsets [START_REF] Grzegorzewski | The coefficient of concordance for vague data[END_REF]).

The Kendall's coefficient K * takes into account the different rows of the matrix R * and is defined as [START_REF] Siegel | Nonparametric statistics for the behavioral sciences, 2nd Edition[END_REF]):

K * = 12∆ * p 2 (d 3 -d) -p.τ * , (15)
where

∆ * = d r=1 R * r -R * 2 , R * r = p q=1 R * qr , R * = 1 d d r=1 R * r and τ * = m v=1 (τ * 3 v -τ * v )
The term τ * v is the number of tied ranks in each of the m groups of ties in R * . The sum τ * is computed over all the groups of ties found in all p rows of the table R * . The Kendall's coefficient which measures the dispersion ∆ * of the feature ranks, ranges from 0 (no agreement) to 1 (complete agreement).

Kendall's coefficient for the toy example

Let us examine the toy example of section 6.1. We propose to compute the Kendall's coefficient from table 1 in order to measure the concordance of the feature ranks obtained by each examined constraint score. The Kendall's coefficients K 1 (see eq. ( 9)), K 2 (see eq. ( 10)), K 3 (see eq. ( 12)) and K 4

(see eq. ( 13)) are respectively 0.4325, 0.2258, 0.3333 and 0.4333.

These low values reflect the dependence of the feature ranks on the available constraints subsets.

This example shows that the Kendall's coefficient K 4 of C 4 is slightly higher than the other ones. The objective of this simple example, with low sample population, is not to compare the Kendall's coefficient of the different scores, but to arise that the selected features depend on the given constraint subset.

This conclusion concurs with that of Sun et al. [START_REF] Suna | Bagging constraint score for feature selection with pairwise constraints[END_REF]).

Comparative experimental results

In this section, we first measure the sensitivity of the scores against the given constraints. So, we compare the Kendall's coefficient obtained by the tested scores (C 1 , C 2 , C 3 , C 4 ).

We also compare the performances obtained by a classifier operating in the feature space selected by C 4 score and those obtained with the classical feature selection methods: Fisher score (supervised), Laplacian score (unsupervised), C 1 , C 2 (constraint scores) and C 3 (semi-supervised constraint score).

Examined databases

For this purpose, our experiments are achieved with six well known and largely used benchmark databases, and more precisely the 'Wine', 'Image segmentation' and 'Vehicle' databases from the UCI repository (Blake et al.

(1998)), the face database 'ORL' [START_REF] Samaria | Parameterisation of a stochastic model for human face identification[END_REF]) and the two gene expression databases, i.e., 'Colon Cancer' [START_REF] Alon | Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays[END_REF]) and 'Leukemia' [START_REF] Golub | Molecular classification of cancer: class discovery and class prediction by gene expression monitoring[END_REF]). These databases have been retained since the features are numeric and since the label information of each data sample is clearly defined.

In our experiments, we first normalize the features between 0 and 1, so that different features have the same scale. For each dataset, we follow an Holdout partition and choose the first half of samples from each class as the training data and the remaining data for testing.

Results on UCI datasets and ORL database

First, we achieve experiments on 3 UCI datasets and the the 'ORL' face database. Here is a brief description of the four considered databases:

• 'ORL' database

The 'ORL' database (Olivetti Research Laboratory) contains a set of face images representing 40 distinct subjects. There are 10 different images per subject, so that the database contains 400 images. For each subject, the images have been acquired according to different conditions: lighting, facial expressions (open / closed eyes, smiling / not smiling) and facial details (glasses / no glasses) (see figure 1).

In our experiments, original images have been normalized (in scale and orientation) so that the two eyes are aligned at the same horizontal position. Then, the facial areas have been cropped in order to build images of size 32 × 32 pixels. Thus, each image can be represented by a 1024-dimensional sample data. The gray level of each pixel is quantified with 256 levels.

We randomly select 5 images from each class (subject) to build the training data. The remaining data are organized as the test data subset.

Kendall's coefficient results

In our experiments, the feature selection is performed on the training data and features are ranked according to the different scores. At each feature selection run q, q=1,..., p, we simulate the generation of pairwise constraints as follow: we randomly select pairs of samples from the training data and create must-link or cannot-link constraints depending on whether the underlying classes of the two samples are the same or different. We iterate this scheme until we obtain card(Sq) 2 must-link constraints and card(Sq)

2 cannot-link constraints.
This operation is repeated over p = 100 runs in order to measure the Kendall's coefficients K 1 , K 2 , K 3 and K 4 .

We achieve experiments for different cardinals of S q ranging from 4 (2 mustlink and 2 cannot-link) to 40 constraints as Zhang et al. do.

Figure 2 shows the Kendall's coefficients K 1 , K 2 , K 3 and K 4 calculated over p = 100 runs for different numbers of constraints. At each run, the same given constraint subset is of course considered by the four tested criteria. The low values of K 1 , K 2 , K 3 and K 4 show clearly that the selected features using C 1 , C 2 , C 3 and C 4 depend on the constraint subsets.

By examining the different curves of figure 2, we see that the different coeffi- cients K 1 , K 2 , K 3 and K 4 increase with the number card(S q ) of constraints.

's coefficient K 1 K 2 K 3 K 4 (d) 'ORL'.
The higher the number card(S q ) of constraints is, the more complete the supervision information is. Ideed, when card(S q ) is high, the semi-supervised learning context tends to become a supervised learning context. That explains why the agreement of feature ranks increases when card(S q ) increases in the different curves of figure 2.

We can also notice that K 4 has the highest values for the different cardinals of constraint subsets in the four databases. This shows that our score C 4 is less sensitive to the must-link and cannot-link constraints built by the user than the classical scores.

We also observe that the values of the Kendall's coefficients on the 'ORL' database are lower than those obtained with the other databases. Indeed, the number of features extracted from the 'ORL' database (1024 features) is bigger than that of the other databases.

Zhao and Zhang have experimentally shown that by picking-up a small number card(S q ) of constraints, their scores allow to select the features which carry a good discriminating power. Figure 2 shows that, between two different runs, the feature ranks change, and so the selected features change when card(S q ) is low. So, the quality of data discrimination would strongly vary with respect to the chosen constraint subsets.

Comparison of the performances

We also compare the performances obtained by the nearest neighbor classifier which operates in the feature space selected thanks to the considered supervised, unsupervised and semi-supervised feature scores.

The classification accuracies of the test data are used to evaluate the performance of each criterion.

As for measuring the Kendall's coefficients, the rates of good classification are averaged over p = 100 runs with different generations of constraints.

Figure 3 shows the plots of accuracy vs. the desired number of selected features on the 'Wine' 3(a), 'Image segmentation' 3(b), 'Vehicle' 3(c) and 'ORL' 3(d) databases, respectively when card(S q ) is set to 10 .

From figure 3 , we can see that the classification rates obtained with constraint scores (C 1 , C 2 , C 3 and C 4 ) range between those obtained with the unsupervised method (Laplacian score) and those obtained with the supervised method (Fisher score). Since card(S q ) is set to 10, these semi-supervised scores use a bit more supervision information than Laplacian score, but much fewer than Fisher Score.

So, these results confirm that a classification scheme reaches higher performance thanks to a supervised learning than thanks to a semi-supervised learning. They also confirm that taking into account a few constraints allows to improve classification results, compared with the unsupervised learning.

However, the curves of C 1 , C 2 , C 3 and C 4 are confused in figure 3. It is also difficult to compare the performances of these scores since they are averaged over 100 runs with different generations of constraints.

That leads us to compare these scores while examining their accuracies at each of the 100 runs. For a fixed number of selected features, in each of card(S q ) / T T of constraints on the 'image' database.

the 100 runs, we propose to rank the 4 criteria in descending order of their accuracy. Let us denote rank * q the rank of the criterion C * at the run q.

This rank takes the values 1, 2, 3 or 4. At each run q, the method having the highest accuracy is ranked as 1 and the method with the lowest accuracy value, is ranked as 4. Methods with the same accuracy have the same rank.

We calculate a rank sum T * for each semi-supervised constraint score as follow:

card(S q ) / T T of constraints on the 'ORL' database.

T * = 100 q=1 rank * q , ( 16
)
where * is 1, 2, 3 or 4 corresponding to the score C 1 , C 2 , C 3 or C 4 respectively.

The method with the lowest rank sum is considered as being the score which provides the best results.

For the 'Wine', 'Image segmentation', 'Vehicle' and 'ORL' databases, the accuracy of each of the 4 semi-supervised criteria seems to be stable when the number of desired features is higher than 6, 5, 8 and 300, respectively (see figure 3). So, we propose to calculate the rank total of each of the semi-supervised criteria by considering the 6 first selected features for the 'Wine database', the 5 first selected features for the 'Image segmentation' database, the 8 first selected features for the 'Vehicle' database and the 300 first selected ones for the 'ORL' database .

Tables 2, 3, 4 and 5 show the rank sum T * for different numbers S q of constraints (4, 10 and 40). Each cell indicates the rank sum of the tested score with the considered number of constraints. From these tables, we can clearly see that T 1 , T 2 , T 3 and T 4 are very close. The features selected thanks to C 4 provide accuracy rates comparable with those obtained by the features selected by C 1 , C 2 and C 3 . Indeed, our score provides the lowest rank sum T (indicated in bold) for 6 times over the 12 rows of tables 2, 3, 4 and 5.

Results on gene expression databases

In this subsection, several experiments are carried out on two gene expression databases, i.e., 'Colon Cancer' [START_REF] Alon | Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays[END_REF]) and 'Leukemia' [START_REF] Golub | Molecular classification of cancer: class discovery and class prediction by gene expression monitoring[END_REF]).

• 'Colon Cancer'

This database contains the expression of 2000 genes measured on 62 tissues (40 tumors and 22 normals).We randomly select 20 and 11 data samples from each class to build the training data. The remaining data are organized as the test data subset.

• 'Leukemia'

This database contains information on gene-expression in samples from human acute myeloid (AML) and acute lymphoblastic leukemias (ALL).

From the originally measured 6817 probe sets we removed genes that were not present in at least one sample so a total of 5147 genes are est values for the different cardinals of constraint subsets in these two gene expression databases. This confirm the fact that our score C 4 is less sensitive to the must-link and cannot-link constraints built by the user than the classical scores. Moreover, since the number of features is very high, 2000 for the Colon Cancer database and 5147 for the leukemia one, we can notice that the gap between K 4 on a hand and K 1 , K 2 and K 3 on the other hand is higher on these databases than the previous ones (a voir).

Figure 5 shows the plots for accuracy vs. different numbers of selected features on Colon Cancer and Leukemia databases. card(S q ) here is set to 60 as Sun et al. do [START_REF] Suna | Bagging constraint score for feature selection with pairwise constraints[END_REF]), so 60 pairwise constraints including 30 must-link and 30 cannot-link are used. From figure 5, we can see that the classification rates obtained with constraint scores (C1, C2, of constraints on the 'Colon Cancer' database.

Tables 6 and7 show the rank sum T * for different numbers S q of constraints (4, 10, 40 and 60) on the 'Colon Cancer' and the 'Leukemia' databases respectively. The rank total of each of the semi-supervised criteria is calculated considering the half of the original features of each of the gene expression databases (see [START_REF] Suna | Bagging constraint score for feature selection with pairwise constraints[END_REF])).

From these tables, we can see that, for the 'Colon Cancer' database, our score provides the lowest rank sum T (indicated in bold) for 3 times over the card(S q ) / T T

Conclusion

Constraint scores which use pairwise constraints for semi-supervised feature selection have shown good performances of classification. Unfortunately, these scores depend on the subset of constraints built by the user since they do not take into account the information provided by the unlabeled data. In this paper, we propose a new semi-supervised constraint score that considers both the pairwise constraints and the local properties of the unlabeled data.

Moreover, we study the relationships between the features selected by the constraint scores and the constraints chosen by the user. We measure the sensitiveness of the scores to constraint changes by the Kendall's coefficient.

40

To the best of our knowledge, this is the first work that studies the influence of the constraint subsets change on the features selected by the constraint scores.

Experimental results on three UCI datasets and a face database show that the proposed score is less sensitive to the constraint changes while selecting features that provide satisfying classification results.

  to the class label (1 for the first class, -1 for the second class). So, an efficient feature selection algorithm should identify the feature f 3 as the most relevant one. Let us consider that the user builds one single must-link constraint and one single cannot-link constraint from the labeled samples. The constraint scores introduced in sections 4 and 5 are used to rank the 3 features based on these constraints. The different feature ranks are shown in table 1. Pairs {(A,B)}, {(A,C)} and {(B,C)} are the single must-link constraints that can be built from the data samples. Pairs {(A,D)}, {(B,D)} and {(C,D)} are the single cannot-linkconstraints that can be also built from the data samples. So, there are 9 constraint combinations which correspond to 9 cells in table 1. The first, the second, the third and the fourth row of each cell represent the ranks of the 3 features given by C 1 , C 2 (α is fixed to 0.1 as suggested by the authors), C 3 and C 4 scores, respectively. The sign '=' between two features means that their scores are equal.For example, let us examine the cell which corresponds to the must-link constraint {(A,C)} and the cannot-link constraint {(B,D)}. Scores C 1 and C 4 select the features f 2 and f 3 as the best features. and the feature f 1 as the third feature. Scores C 2 and C 3 sort feature f 2 as the best feature, feature f 3 as the second feature and feature f 1 as the third feature.

Figure 1 :

 1 Figure 1: Sample face images from the ORL database (2 subjects).

Figure 2 :

 2 Figure 2: Kendall's coefficient for different number S q of constraints on the four databases.

Figure 3 :

 3 Figure 3: Accuracy vs. different number of selected features on the four datasets. 10 pairwise constraints including 5 must-link and 5 cannot link are used.

  used in the experiments. Because Leukemia has a predefined partition of the objects into training (27 ALL and 11 AML) and testing (20 ALL and 14 AML) subsets, the ensembles on this dataset are performed on the predefined training and testing sets.

Figure 4

 4 Figure4shows the Kendall's coefficients K 1 , K 2 , K 3 and K 4 calculated over

Figure 4 :

 4 Figure 4: Kendall's coefficient for different number S q of constraints on the 'Colon Cancer' and the 'Leukemia' databases.

Figure 5 :

 5 Figure 5: Accuracy vs. different number of selected features on the gene expressiondatabases. 60 pairwise constraints including 30 must-link and 30 cannot link are used.

  C3 and C4) range between those obtained with the unsupervised method (Laplacian score) and those obtained with the supervised method (Fisher score). These semi-supervised constraint scores even outperforms the supervised Fisher score on the 'Colon Cancer' database for a number of selected features more than 1050.

Figure 6 :

 6 Figure 6: Accuracy vs. different number of pairwise constraints (for C 1 , C 2 , C 3 and C 4 ) on the gene expression databases. The desired number of selected features is half of the number of original features

Table 1 :

 1 Feature ranks by semi-supervised scores using the chosen constraints.First, table1shows that the feature f 3 is always ranked as the first 276 feature by C 1 and C 4 whatever the given constraint subset. C 2 and C 3 do 277 not retain feature f 3 as the first feature three times. This table also shows that 7 different ranks (indicated in bold in table 1) of the 3 features are obtained by comparing each of the 4 examined scores for the 9 constraint combinations. This simple example clearly shows that the features ranks deduced from these scores strongly depend on the chosen constraints. So, we

275

Table 2 :

 2 The rank sums of the different semi-supervised criteria for different number S q of constraints on the 'wine' database.

	card(S q ) / T	T 1 T 2	T 3	T 4
	4 constraints 208 228 168 179
	10 constraints 228 210 154 183
	40 constraints 184 177 153 146

Table 3 :

 3 The rank sums of the different semi-supervised criteria for different number S q

Table 4 :

 4 The rank sums of the different semi-supervised criteria for different number S q of constraints on the 'vehicle' database.

	card(S q ) / T	T 1	T 2 T 3 T 4
	4 constraints 194 212 311 200
	10 constraints 185 189 250 196
	40 constraints 190 202 300 203

Table 5 :

 5 The rank sums of the different semi-supervised criteria for different number S q

Table 6 :

 6 The rank sums of the different semi-supervised criteria for different number S q

Table 7 :

 7 1 T 2 T 3 T 4 The rank sums of the different semi-supervised criteria for different number S q of constraints on the 'Leukemia' database.4 rows of table 6. For the 'Leukemia database', our score provides the lowest rank sum T (indicated in bold) for the different numbers of constraints (4, 10, 40 and 60). These results improve the fact that the features selected thanks to C 4 provide accuracy rates comparable with those obtained by the features selected by C 1 , C 2 and C 3 . Furthermore, figure6shows the plot for accuracy under fixed number of selected features ( half of the number of original features) vs. different numbers of pairwise constraints on the gene expression databases. For almost all the number of constraints, our score C 4 achieves higher accuracy than C 1 , C 2 and C 3 .

	4 constraints 184 240 194 182
	10 constraints 176 276 150 134
	40 constraints 170 196 148 118
	60 constraints 186 198 168 126


F.-X Chevillotte

Laurent Bécot

A Gautron

Duval

Camille Perrot
email: camille.perrot@univ-paris-est.fr

Guy Bonnet

Minh Tan Hoang

Fabien Chevillotte

François-Xavier Bécot

Laurent Gautron

Arnaud Duval

MICROSTRUCTURE, TRANSPORT, AND ACOUSTIC PROPERTIES OF REAL FOAM SAMPLES

Keywords: Microstructure, transport, and acoustic properties

Introduction

The determination from local scale geometry of the acoustical properties which characterize the macro-behavior of porous media is a long-standing problem of great interest [START_REF] Allard | Propagation of Sound in Porous Media: modeling sound absorbing materials[END_REF] , for instance for the oil, automotive, and aeronautic industries. Recently, there has been a great interest in understanding the low Reynolds viscous flow, electrical, and diffusive properties of fluids in the pore structure of real porous media on the basis of microstructural parameters [START_REF] Adler | Porous Media: Geometry and Transports[END_REF] , as these transport phe-nomena control their long-wavelength frequency-dependent properties [START_REF] Lafarge | The equivalent fluid model (Chapter 6, Part II) in Materials and Acoustics Handbook[END_REF][START_REF] Perrot | Bottom-up approach for microstructure optimization of sound absorbing materials[END_REF] . Our aim in this paper is to get insight into the microstructure of real porous media and to understand how it collectively dictates their macro-scale acoustic properties, from the implementation of first-principles calculations on a three-dimensional idealized periodic unit-cell.

Direct static characterization of foam samples

Three real and commercially available polymeric foam samples have been studied. They are denoted R 1 , R 2 , and R 3 . Real foam samples are disordered and possess a complex internal structure which is difficult to grasp quantitatively. However, our objective is to be able to quantify the local geometry of such foams by an idealized packing of polyhedral Periodic Unit Cells (PUC). Apart from the intrinsic need for characterizing the cell morphology itself, insight into the morphology of an idealized PUC is helpful for understanding the microphysical basis behind transport phenomena. Fig. 1 shows typical micrographs of these real polyurethane foam samples (based on a polyester or polyether polyol), taken with the help of a binocular.

Foam samples were cut perpendicularly to the plane of the sheet. To get an idea of the cellular shape of these samples, the number of edges per face n was measured from 30 different locations for each material. From these measurements follows an average number of edges per face for each foam sample: R 1 , n 1 = 5.10 ± 0.82; R 2 , n 2 = 5.04 ± 0.68; R 3 , n 3 = 5.03 ± 0.71. Next, ligaments lengths were measured on optical micrographs of the foam samples. Since the surface contains exposed cells whose ligament lengths are to be measured on micrographs obtained by light microscopy, great care was taken during measurements to select only ligaments lying in the plane of observation. Ligament length measurements were performed on three perpendicular cross-sections of each sample. Transverse isotropy of the foam samples cellularity was assumed. Here, only results of ligament length measurement distributions on the horizontal surface were plotted in Fig. 2.

Ligament thicknesses constitute also an important geometrical parameter. However, they were difficult to measure. Therefore, the ligament thicknesses were not used. It was observed that computing the permeability of the unit-cell from the averaged ligaments length m L and open porosity φ , produces a significant overestimate of the permeability which results from the exaggerated con- tribution of the very large cells with atypically high permeability. More precisely, the permeability is related primarily to critical sections and not to mean sections. This argument is supported by critical path considerations. As a consequence, the parameters characterizing the geometry of the PUC are determined in this work from two routinely available measured macroscopic parameters: the porosity , and the static permeability k 0 , which are measured using direct and standard methods. Results are presented in Tab. 1. 3. Prediction of transport properties from a three-dimensional periodic unit-cell

The local geometry

As observed from the micrographs, the network of ligaments appears to be similar to a lattice within which the ligaments delimit a set of polyhedra. In this work, it is therefore considered that a representation of the microstructure which can be deduced from this observation is a packing of identical polyhedra. More precisely, truncated octahedra with ligaments of circular cross section shapes and a spherical node at their intersections were considered. It has been shown in a recent paper by using 2D computations [START_REF] Perrot | Bottom-up approach for microstructure optimization of sound absorbing materials[END_REF] that the FEM results are not significantly affected by this approximation, even if the real cross-section of ligaments can be rather different.

A regular truncated octahedron is a 14-sided polyhedron (tetrakaidecahedron) having six squares faces and eight hexagonal faces, with ligament lengths L and thicknesses 2r. The average number of edges per face, another polyhedron shape indicator, is equal to (6×4 + 8×6) / 14 ≈ 5.14 and close to the experimental data presented in Section 2. The cells have a characteristic size D equal to (2√2)L, between two parallel squared faces. An example of regular truncated octahedron for such packings is given in Fig. 3.

The simplest macroscopic parameter characterizing a porous solid is its open porosity, defined as the fraction of the interconnected pore fluid volume to the total bulk volume of the porous aggregate, . The porosity of such a packed polyhedron sample might be expressed as a function of the aspect ratio L/2r, ( ) ( ) ( ) ( )

23 32 2 11 1-3 2 /16 2 / -2 /16 2 / , 2 ( 1)
1 rL C rL C f f f φπ π == -+ - -, ( 1 )
and f is a node size parameter related to the spherical radius R by R = f × r, with f ≥ √2. This last constraint on the node parameter ensures that the node volume is larger than the volume of the connecting ligaments at the node.

Tab. 1. Foam samples macroscopic parameters. The second parameter which is widely used to characterize the macroscopic geometry of porous media, and thus polyhedron packing, is the specific surface area S p , defined as the total solid surface area per unit volume. The hydraulic radius is defined as twice the ratio of the total pore volume to its surface area. This characteristic length may also be referred to as the "thermal characteristic length" ' in the context of sound absorbing materials 1 , so that ' = 2 /S p . As for the porosity, the "thermal characteristic length" might be expressed in terms of the microstructural parameters by ( )

Foams Method (-) φ ' (µm) Λ () 0 k 2 m (-) 0 α (µm) Λ (-) ∞ α () 0 k' 2 m ' α 0 R 1
3 22 12 2 22 2 ' 16 2 / 6 / 2 3 2 / , 2 1 1 rr r CC r C f f f LL L ππ π ⎧⎫ ⎡⎤ ⎡⎤ ⎪⎪ ⎛⎞ ⎛⎞ ⎛ ⎞ Λ= - - + × = -+ - - ⎢⎥ ⎨⎬ ⎜⎟ ⎜⎟ ⎜ ⎟ ⎢⎥ ⎝⎠ ⎝⎠ ⎝ ⎠ ⎣⎦ ⎢⎥ ⎪⎪ ⎣⎦ ⎩⎭ . (2)

Determination of the unit cell aspect ratio from porosity

When a laboratory measurement of porosity is available, the unit-cell aspect ratio / L 2r can be identified through Eq. (1). For a given value of the spherical node size parameter f, the unit-cell aspect ratio 2r/L is given by the solution of a cubic equation that has only one acceptable solution. Once 2r/L is obtained, Eq. ( 2) gives r if a laboratory measurement of S p is available. Then, the idealized geometry of the foam could be completely defined. The main problem in this method is that the specific surface area evaluation from non-acoustical measurements, such as the standard Brunauer, Emmett, and Teller method (BET) based on surface chemistry principles, is not routinely available. In fact, the most widely measured parameter after the porosity to characterize the physical macro-behavior of real porous media is unarguably the static viscous (or hydraulic) permeability k 0 , a quantity having units of a surface (squared length).

Therefore, obtaining the local characteristic sizes of the PUC will be performed thereafter in four steps. Step 1 consists in acquiring the aspect ratio L/2r from the porosity measurements as explained before. For a given spherical node size parameter, this produces all characteristic length ratios of the cell. At this stage, the ligament length of the cell is still unknown, but a nondimensional PUC can be built. Step 2 is to characterize the permeability of the foam from routine measurements. Step 3 is to get the permeability of the set of non-dimensional periodic cells from first principle calculations. As explained before, the non-dimensional cell has a unit side of square faces. The Finite Element computation described thereafter implemented on the non-dimensional cell produces the non-dimensional permeability k d . Let D be the side of square faces of homothetic periodic cells producing the static permeability k 0. . Then, a simple computation shows that k 0 = D 2 k d . Finally, comparing the non-dimensional permeability to the true permeability produces in step 4 the size of the PUC. All other parameters are obtained from the non-dimensional results through a similar scaling.

First principles calculations of transport properties

Previous studies [START_REF] Lafarge | The equivalent fluid model (Chapter 6, Part II) in Materials and Acoustics Handbook[END_REF] have shown how the long-wavelengths acoustic properties of rigid-frame porous media can be numerically determined by solving the local equations governing the asymptotic frequency-dependent visco-thermal dissipation phenomena in a periodic unit cell with the adequate boundary conditions. In the following, it is assumed that >> D, where is the wavelength of an incident acoustic plane wave. This means that for characteristic lengths on the order of D ~ 0.5 mm, this assumption is valid for frequencies reaching up to a few tens of kHz. The asymptotic macroscopic properties of sound absorbing materials are computed from the numerical solutions of: (1) the low Reynolds number viscous flow equations (the static viscous permeability k 0 , and the static viscous tortuosity 0 ); (2) the non-viscous flow or inertial equations (the high-frequency tortuosity ∞ , and Johnson's velocity weighted length's parameter ); (3) the equations for thermal conduction (the static thermal permeability k 0 ', and the static thermal tortuosity 0 ').

As previously mentioned, all this procedure assumes that the spherical node size parameter f is known. In our computations, f was set to 1.5. This value respects the constraint f ≥ √2 and is in a rather good agreement with microstructural observations, considering the absence of lump at the intersection between ligaments (see Fig. 1). Application of the above procedure yields the local characteristic sizes of a unit cell ligament for each foam sample: R 1 , L 1 ≈ 123.

Results on asymptotic transport properties obtained from FE modeling

An example of calculated viscous flow velocity, inertial flow velocity, and scaled concentration fields is shown in Fig. 4 for foam sample R 1 . The number of elements and their distribution in the fluid phase regions of the PUC were varied, with attention paid especially to the throat and the near-wall areas, to examine the accuracy and convergence of the field solutions. The symmetry properties of the permeability/tortuosity tensors were also checked, as a supplementary test on convergence achievement. As previously noticed by several authors, due to the slip condition, the fluid flow paths are more homogeneous for the electric-current paths than for the viscous fluid flow. Direct numerical computations of the complete set of macroscopic parameters were performed in reconstructed unit cells from adequate asymptotic field averaging. Results are reported in Tab. 1. Some values are compared to estimations obtained from impedance tube measurements.

Estimate of the frequency dependent responses by a hybrid numerical approach

The acoustic response of foams depends on dynamic viscous permeability and "dynamic thermal permeability". Both of these parameters could be obtained from dynamic FEM computations. The approach presented here relies on the fact that the finite element computations presented previously are easy to implement and provide the asymptotic behavior for both dynamic "permea- bilities". This asymptotic behavior constitutes the input data for the models which are used for predicting the full frequency range of the dynamic "permeabilities". Therefore the hybrid approach employed in our study makes use of the asymptotic parameters of the porous medium obtained by finite elements. Then, it is possible to provide the dynamic permeabilities and to compare these values to experimental ones. As explicated, the comparison between non-dimensional permeability obtained from finite element results and the measured permeability provides the thermal characteristic length ' ; and five remaining input parameters for the models, 0 , ∞ , , k 0 ', and 0 ' are obtained by means of first-principles calculations by appropriate field-averaging in the PUC [START_REF] Lafarge | The equivalent fluid model (Chapter 6, Part II) in Materials and Acoustics Handbook[END_REF] . Finally, we considered the predictions of the three models for the effective dynamic permeabilities. In summary, the Johnson-Champoux-Allard" [JCA] model which uses the 5 parameters ( , k 0 , ∞ , , ') , Johnson-Champoux-Allard-Lafarge" model [JCAL] which uses in addition k 0 ', and Johnson-Champoux-Allard-Pride-Lafarge" [JCAPL] model which uses the full set of parameters ( , k 0 , k 0 ', ∞ , , ', 0 , and 0 ').

Assessment of the methodology through experimental results

Experimental results and comparison with numerical results

One main objective of this section is to produce a comparison between sound absorption coefficients coming from experimental results using the impedance tube technique by Utsuno et al. [START_REF] Utsuno | Transfer function method for measuring characteristic impedance and propagation constant of porous materials[END_REF] , and from numerical computations. In this context, intermediate results were obtained for the acoustic parameters of the JCAL model through the characterization method described in Refs. 6 for viscous and thermal dissipations. These results will be thereafter referenced to on the Figs. and Tab. as obtained from "characterization". Fig. 5 produces the sound absorption coefficient obtained from measurements, from characterization and from numerical computations. Despite the simplicity of the local geometry model used to study the multi-scale acoustic properties of real foam samples predominantly open-cell, there is a relatively good agreement between computed (present microstructural method), measured (impedance tube measurements), and characterized quantities. The general trend given in term of normal incidence sound absorption coefficient A n ( ) by our microstructural approach appears as being particularly relevant, if we notice that it requires only and k 0 , as input parameters, and proceeds without any adjustable parameter.

Discrepancies between measured and computed sound absorption coefficient at normal incidence can be primarily explained from the comparison of a set of parameters obtained from numerical results and from the characterization method reported in Tab. 1; namely ', , ∞ , and k 0 '. From that comparison, it can be seen that the most significant difference is the large overestimation provided for Λ. This means that, at high frequencies, the windows size of the local geometry model, which respectively plays the role of weighting the velocity field for Λ and rapid section changing for ∞ by their small openings (the squares in the case of a truncated octahedron unit-cell), is presumably overestimated by a monodisperse, isotropic, and membrane-free local geometry model.

Consequently, an improvement of the local geometry model would result in the introduction of a second set of characteristic sizes. The purpose of the following is to examine more thoroughly the microstructure in order to provide some means aimed at improving the methodology.

Keys for further improvements of the methodology

A supplementary visual cell inspection is given by electron micrographs at very low magnification, as presented in Fig. 6. The characteristic ligament length L c obtained for the periodic cell is reported on the micrographs, which allows a first visual comparison between observed and computed cell size. Another element of discussion is provided in Fig. 2 where the distribution of the measured ligament lengths is reported (together with its mean value L m ), simultaneously with the length L c obtained from the numerical results and from the calibration coming from (k 0 , ). The characteristic ligament length L c of the local geometry model provides a basis for understanding the influence of certain local geometry features, such as membrane effects and cell anisotropy, on the static viscous permeability of the real foam samples -in connection with ligaments length distribution.

More precisely, if the distribution of the ligament lengths is sharply peaked, one would expect the overall system behavior to be similar to that of the individual elements. This is a configuration close to the one observed for foam sample R 3 , where only isolated residual membranes (thermal reticulation process) and no specific cell elongation were observed, as illustrated on the electron micrograph in Fig. 6(c); and for which the distribution of the ligaments length is relatively sharp. As a result, the ligaments length of the local geometry model for foam sample R 3 is relatively close to the averaged value measured on the micrographs, especially for the horizontal surface through which permeability measurements were performed (L c ≈ 158 m, L m3H ≈ 167 m, and L m3H / L c ≈ 1.06).

On the other hand, if the distribution is broader as shown for foam sample R 2 in Fig. 2, because of cell elongation as it can be seen on Fig. 6 (b), the critical path -made by the small windows at the openings of the cells -is expected to dominate (in Fig. 2, for the horizontal surface L m2H ≈ 227 m, whereas L c ≈ 141 m, and L m2H / L c is now equal to 1.61).

Similarly, as observed for foam sample R 1 in Fig. 6(a), the presence of membranes occludes or significantly reduces the size of some windows which might belong to unit-cells in the class of local permeability sites k ij (in the sense of critical path considerations) much greater, or of the order of k c . This has in addition the effect of disconnecting some critical subnetworks. In this later case, the unit-cells which were belonging to the permeability sites with k ij ≤ k c , may now significantly contribute by participating in a new critical subnetwork, lowering drastically k c (in Fig. 2, for the horizontal surface, L m1H ≈ 193 m, whereas L c = 123 m, and L m1H / L c gives 1.57).

As explained before, reporting the value of L c on the electron micrograph of Fig. 6 can illustrate what is the typical size of a critical path opening. It is also worth mentioning that L c and D c = (2√2)L c provide a rather reliable rough estimate of the characterized values for and ' respectively. This tends to confirm the customarily assumed idea that the small openings (windows) and the pore itself (cell) are respectively associated to viscous and thermal dissipation effects.

Conclusion

A three-dimensional idealized periodic unit-cell (PUC) based method to obtain the acoustic properties of three predominantly open-cell foam samples was described. The first step was to provide the local characteristic lengths of the representative unit cell. For isotropic open cell foams, two input parameters were required, the porosity and the static viscous (hydraulic) permeability. Long-wavelengths acoustic properties were derived from the three-dimensional reconstructed PUC by solving the boundary value problems governing the micro-scale propagation and visco-thermal dissipation phenomena with adequate periodic boundary conditions, and further field phase averaging. The computed acoustic properties of the foams were found to be in relatively good agreement with standing wave tube measurements. A close examination of the real foam sample ligament length distribution as observed from micrographs, and its comparison with the characteristic size of the local geometry model, showed evidences of membrane and cellular anisotropy effects discussed by means of critical path considerations. In summary, we have developed a microcellular approach in which the local characteristic length L c governing the static viscous permeability of a real foam sample can be identified; and from which rough estimates of the viscous Λ and thermal lengths ' Λ may follow (small openings and pore size itself). The overall picture that emerges from that work is that the acoustical response of these materials is governed by their three-dimensional micro-cellular morphology, for which an idealized unit-cell based method is a convenient framework of multiscale analysis displaying the microgeometry features having a significant impact on the overall response function of the porous media.

Fig. 1 .

 1 Fig. 1. Typical micrographs of real foam samples: (a) R 1 , (b) R 2 , and (c) R 3 .

Fig. 2 .

 2 Fig. 2. Ligament length distributions for real foam samples R 1 , R 2 , and R 3 .

Fig. 3 .

 3 Fig. 3. Basic 3D periodic foam model geometry

Fig. 4 .

 4 Fig. 4. Asymptotic scaled fields for 1/4 th of the reconstructed foam sample period R 1 : (a) Stokes velocity field, (b) potential velocity field, and (c) diffusion-controlled concentration field.

  3 m, 2r 1 ≈ 19.4 m; R 2 , L 2 ≈ 141.1 m, 2r 2 ≈ 27.3 m; R 3 , L 3 ≈ 157.7 m, 2r 3 ≈ 24.8 m.

Fig. 5 .

 5 Fig. 5. Normal incidence sound absorption coefficient for foam sample R 1 (left), R 2 (centre), and R 3 (right).

Fig. 6 .

 6 Fig. 6. Typical SEM images of real foam samples. (a) R 1 , showing a relatively great number of membranes (indicated by arrows). (b) R 2 , having a degree of anisotropy equal to 1.75 (superimposed ellipse). (c) R 3 , exhibits only few isolated membranes with rather spherical pore shapes.

Acknowledgments

This work was part of a project supported by ANRT and Faurecia Acoustics and Soft Trim Division under convention CIFRE No. 748/2009.


Camille Perrot
email: camille.perrot@univ-paris-est.fr

M T Hoang

Guy Bonnet

F Chevillotte

F.-X Bécot

L Jaouen

Laurent Gautron

R Combes

A Duval

J.-F Rondeau

Microstructure -acoustic properties relationships: Application to membrane and bimodal pore-size distribution effects

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The local scale geometry determination of the acoustical properties which characterize the macro behavior of porous media is a long-standing problem of great interest [START_REF] Rayleigh | The theory of sound, 2 nd Ed[END_REF]- [START_REF] Allard | Propagation of Sound in Porous Media: modeling sound absorbing materials[END_REF], for instance for the oil, automotive, and aeronautic industries. Various methods have been proposed to solve it on a rigorous basis.

The first problem consists in the idealization at the local scale of real media. For example open-celled foams can be idealized as regular arrays of polyhedrons. A presentation of various idealized shapes is given by Gibson and Ashby [START_REF] Gibson | Cellular solids: Structure and Properties[END_REF] for cellular solids, and more specifically by Weaire and Hutzler [START_REF] Weaire | The Physics of Foams[END_REF] for foams.

The second problem consists in the determination of the macroscopic and frequency-dependant transport properties, such as the dynamic viscous permeability [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluidsaturated porous media[END_REF]. The number of media which can be analytically addressed is deceptively small [START_REF] Stinson | The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape[END_REF], and many techniques have been developed in the literature, such as the self consistent method providing mean estimates (see, for instance, the recent work of Boutin and Geindreau and references therein [START_REF] Boutin | Estimates and bounds of dynamic permeability of granular media[END_REF]).

The purpose of this paper is to present a technique based on first-principles calculations of transport parameters [START_REF] Zhou | First-principles calculations of dynamic permeability in porous media[END_REF] in reconstructed porous media [START_REF] Adler | Porous Media: Geometry and Transports[END_REF] which can be applied to model the acoustic properties of real foam samples predominately open-celled; and to compare its predictions to multi-scale experimental data. The main difficulty in modeling the frequency-dependent viscous and thermal dissipation functions through open-celled foams lies in accurately determining micro-structural characteristics, and how they collectively dictate the acoustical macro-behavior. Since the variability in the foam microstructures makes it very difficult to establish and apply local geometry models to study the acoustics of these foams, an inverse model is proposed to quantitatively grasp the complex internal structure of predominately open-celled foam samples. This is achieved from the standpoint of an idealized periodic unit cell (PUC), whose characteristic lengths are directly deduced with routinely available porosity and hydraulic permeability measurements.

The paper is divided into three sections. The method for linking idealized three-dimensional microstructures and macro-acoustic properties of open-cell foams is dealt with in Section 2. A generation of three-dimensional idealized periodic unit cells of real foam samples predominately open-celled is presented, together with a computational technique in reconstructed unit-cells of the macroscopic properties of sound absorbing materials. In Section 3, results at micro-and macro-scales are compared with experimental data obtained from optical microscopy and standing wave tube measurements and discussed. The fourth Section is devoted to some concluding remarks which summarize the progress recently made.

2 Micro-structure and macro-acoustic properties of predominately open-celled foamsOnly packing of identical isotropic polyhedrons was investigated in the current work. Tetrakaidecahedrons with ligaments of circular cross section shapes with a spherical node at their intersections were considered [START_REF] Schmierer | Effective Thermal Conductivity of Fully Saturated High Porosity Foam[END_REF]. They are usually defined from the basis of truncated octahedrons, with ligament lengths L and thicknesses 2r. A tetrakaidecahedron is a 14-sided polyhedron, with six squares faces and eight hexagonal faces. The average number of edges per face, another polyhedron shape indicator, is equal to (6×4 + 8×6) / 14 ≈ 5.14. The cells have characteristic size D equal to (2√2)L, between two parallel square faces, of one unit. All subsequent calculations are performed in a cubic sample of volume D 3 . The ligaments length L is generally taken much larger than the ligaments thickness 2r. An example of tetrakaidecahedron for such packings is given in Figure 1.

The simplest macroscopic parameter characterizing the packing geometry is its porosity, or air volume fraction . The porosities of these packed polyhedron samples might be expressed as a function of the inverse of the ligaments elongation L/2r, . This last constraint on the node parameter ensures that the exposed surface area at the node is positive and the node volume is larger than the volume of the connecting ligaments at the node, assuming a lump of material at ligament intersection. The second parameter which is widely used to characterize the macroscopic geometry of porous media, and thus polyhedron packing, is the specific surface area S p , defined as the total solid surface area per unit volume. The hydraulic radius is defined as twice the ratio of the total pore volume to its surface area. This quantity is generally referred to as the thermal characteristic length ' in the context of sound absorbing materials [START_REF] Champoux | Dynamic tortuosity and bulk modulus in air-saturated porous media[END_REF], such that ' = 2 /S p . As for the porosity, the thermal characteristic length might be expressed in terms of the microstructural parameters, When laboratory measurements of porosity and specific surface area (or thermal characteristic length) are available, average ligaments length and thickness can be identified through equations ( 1) and ( 2). For instance, we seek for the solution of a cubic equation ( 1) in / rL that has only one acceptable solution. Once / rL is obtained by assuming a value for f , Eq. ( 2) gives r . With this relations and laboratory measurements of and S p , the idealized geometry of the foam can be completely defined. The main problem in this analysis is that the specific surface area evaluation from non-acoustical measurements, such as the standard Brunauer, Emmett, and Teller method (BET) [START_REF] Brunauer | Adsorption of Gases in Multimolecular Layers[END_REF] [14] based on surface chemistry principles, is not routinely available. In fact, the most widely measured parameter after the porosity to characterize the physical macro-behavior of real porous media is unarguably the hydraulic permeability, a quantity having units of length squared and obtainable from the Stokes equations. Making use of this property, it is therefore possible to determine from one numerical test with constant porosity, local characteristic lengths of a tetrakaidecahedron unit cell, as an attempt to grasp quantitatively the complex internal structure of real foam samples. Indeed, the permeability obtained from a computational implementation of the incompressible fluid equations can be displayed as a function of the thermal length squared for a given porosity, showing a linear relation. The quantity that is calculated upon numerical calibration is the slope S to be fitted with the k 0 = ' 2 linear relation. The geometrical property that is extracted is the thermal length 2' corresponding to the measured static viscous permeability k 0 . Making use of Eqs. ( 1) and ( 2), local characteristic lengths follows. Hence, there are a priori two routinely available independent measurements to be carried out in order to define the foam geometry: the porosity and the static viscous permeability k 0 . This method for periodic unit-cell reconstruction circumvents the necessary measure of the specific surface area.

Foams Method (-) Φ ' (µm) Λ 0 k ( ) 2 m 0 α (µm) Λ ∞ α 0 k' ( ) 2 m ' α 0 R 1 Computations 0.
It was then shown how the long-wavelengths acoustic properties of rigid porous media can be numerically determined by solving the local equations governing the asymptotic frequency-dependent dissipation phenomena in a periodic unit cell with the adequate boundary conditions. A detailed presentation of this type of derivation can be found in Ref. [START_REF] Perrot | Bottom-up approach for microstructure optimization of sound absorbing materials[END_REF]. It is assumed that >> D, where is the wavelength of an incident acoustic plane wave. This means that for characteristic lengths on the order of D ~ 0.5 mm, this assumption is valid for frequencies reaching up to 6000 -7000 Hz. The asymptotic macroscopic properties of sound absorbing materials are computed from the numerical solution of the incompressible fluid equations (the static viscous permeability k 0 , and the static viscous tortuosity 0 ), from Laplace equation (the high-frequency tortuosity ∞ , and Johnson's velocity weighted length's parameter ), and from the heat equation (the static thermal permeability k 0 , and the static thermal tortuosity 0 '). The dynamic viscous (respectively thermal) permeability v ( ) (respectively t ( )) is then derived from successive improvements (see Ref. [START_REF] Pride | Drag forces of porous media acoustics[END_REF], and [START_REF] Lafarge | Propagation du son dans les matériaux poreux à structure rigide saturés par un fluide viscothermique[END_REF]- [START_REF] Lafarge | Dynamic compressibility of air in porous structures at audible frequencies[END_REF] respectively) of the model of Johnson et al. [START_REF] Rayleigh | The theory of sound, 2 nd Ed[END_REF] (respectively Champoux and Allard [START_REF] Champoux | Dynamic tortuosity and bulk modulus in air-saturated porous media[END_REF], [START_REF] Allard | Propagation of Sound in Porous Media: modeling sound absorbing materials[END_REF]) from which all the pertaining acoustic indicators such as the normal sound absorption coefficient ( ) can be deduced, and compared with standing wave tube measurements using an adapted experimental setup [START_REF] Utsuno | Transfer function method for measuring characteristic impedance and propagation constant of porous materials[END_REF]. In the following, the 5 parameters ( , k 0 , ∞ , , ') model is referred to Johnson-Champoux-Allard (JCA) simplified model. When k 0 is explicitly taken into account, this is the 6 parameters Johnson-Champoux-Allard-Lafarge (JCAL) model. The complete model relies on 8 parameters ( , k 0 , k 0 , ∞ , , ', 0 , and 0 ') and is referred to the refined Johnson-Champoux-Allard-Pride-Lafarge (JCAPL) model.

Multi-scale results for generated unit-cells and comparison with real foam sample measurements

In order to evaluate our multi-scale methodology, experimental data from microscopy and standing wave tube measurements were used for three real foam samples, R 1 , R 23 , and R 4 .

The porosity was non-destructively measured from the perfect gas law properties using the method described by Beranek [START_REF] Beranek | Acoustic impedance of porous materials[END_REF]. The experimental value of the static permeability k 0 was obtained by means of accurate measurements of differential pressures across serial mounted calibrated and unknown flow resistances, with a controlled steady and non-pulsating laminar volumetric air flow as described by Stinson and Daigle [START_REF] Stinson | Electronic system for the measurement of flow resistance[END_REF]. Starting from and k 0 measurements, f was set to 1.5 and average ligaments length L and thickness 2r where estimated from the original procedure described in Sec. 2. Direct numerical computations of the complete set of macroscopic parameters were performed in reconstructed unit cells (Tab. 1) from adequate asymptotic field averaging (shown in Fig. 2 for foam sample R 1 ). v ( ), t ( ), and ( ) are then analytically derived for JCA, JCAL, and JCAPL models as described in Sec. 2. and compared with standing wave tube measurements, see Fig. 3. Experimental data were provided using the impedance tube technique by Utsuno et al. [START_REF] Utsuno | Transfer function method for measuring characteristic impedance and propagation constant of porous materials[END_REF], in which the complex and frequency-dependant characteristic impedance Z c ( ) and propagation constant q( ) of each material were measured, and the equivalent dynamic viscous permeability v ( ), the equivalent dynamic thermal permeability t ( ), and the sound absorption coefficient at normal incidence ( ), derived from Z c ( ) and q( ). In parallel, the material acoustic parameters of the JCAL model were characterized using the method described in Ref. [START_REF] Panneton | Acoustical determination of the parameters governing viscous dissipation in porous media[END_REF] for viscous dissipation and Ref. [START_REF] Olny | Acoustical determination of the parameters governing thermal dissipation in porous media[END_REF] for thermal dissipation. The computed and characterized viscous (respectively thermal) transition frequencies between viscous and inertial regimes (respectively isothermal and adiabatic), f v = / 2 k 0 ∞ where is the air cinematic viscosity (f t = ' / 2 k 0 ∞ where ' is the air thermal diffusivity [m 2 .s -1 ]) -are also given.

Despite the simplicity of the local geometry model used to study the multi-scale acoustic properties of real foam samples predominately open-celled, there is a relatively good agreement between computed (present microstructural method), measured (impedance tube measurements), and characterized dynamic quantities: v ( ), t ( ), and ( ) (see Fig. 3). Furthermore, the general trend given in term of normal incidence sound absorption coefficient ( ) by our microstructural approach appears as being particularly relevant, if we notice that it requires only and k 0 as input parameters, and proceed without any adjustable parameter. Discrepancies between measured and computed ( ) can be primarily explained from the examination of and ∞ results reported in Tab. 1. Simulations systematically overestimate with a factor ranging between 1.7 and 2.4, and underestimate ∞ . This means that the windows size of the local geometry model, which respectively plays the role of weighting the velocity field for and rapid section changing for ∞ by their small openings (the squares and hexagons in the case of a tetrakaidecahedron unit cell), is presumably overestimated by a monodisperse and membrane-free local geometry model.

A further examination of the limits of the micro-cellular model is provided through micrographs. Ligaments lengths were measured on optical micrographs on the real foam samples R 1 , R 23 , and R 4 . Foam samples were cut perpendicular to the plane of the sheet. The surface contains exposed cells whose ligament lengths are to be measured on micrographs obtained by light microscopy, Fig 4 . Great care was taken during measurements to select only ligaments lying in the plane of observation. Assuming transverse isotropy of the real foam samples, measurements were taken on three-perpendicular planes, and their distributions plotted in Fig. 5.

• For foam sample R 1 , membranes are observed on the micrographs. To reach the permeability of the real foam sample exhibiting membranes, the characteristic size of the membrane-free local geometry model must be considerably smaller than the real foam sample, as observed on the ligaments length distribution. • For foam sample R 23 , no membranes were observed on the micrographs. However, the characteristic size of the local geometry model corresponds to the left-peak of a bimodal distribution, which governs the overall permeability of the real foam sample (this is the criticalpath argument; see Ref.

Conclusions

A three-dimensional idealized periodic unit-cell (PUC) based method to obtain the acoustic properties of three predominately open-celled foam samples was described. The first step was to provide the local characteristic lengths of the representative unit cell. For isotropic open cell foams, two input parameters were required, the porosity and the hydraulic permeability. Long wavelengths acoustic properties were derived from the three-dimensional reconstructed PUC by solving the boundary value problems governing the micro-scale propagation and visco-thermal dissipation phenomena with adequate periodic boundary conditions, and further field phase averaging. The computed acoustic properties of the foams were found to be in relatively good agreement with standing wave tube measurements. A close examination of the real foam samples ligaments length distribution as observed from micrographs, and its comparison with the characteristic size of the local geometry model, showed evidences of membrane and bimodal pore-size distribution effects. The overall picture that emerges from that work is that the acoustical response of these materials is governed by their three-dimensional micro-cellular morphology, for which an idealized unitcell based method is a convenient framework of multi-scale analysis displaying the microgeometry features having a significant impact on the overall response function of the porous media.

Figure 1 :

 1 Figure 1: The (a) tetrakaidecahedron with ligaments of (b) circular cross-section shape and spherical nodes at their intersections, is proposed as an example of idealized periodic unit-cell for identification of open-cell foams average characteristic lengths

  equations are linking micro to macro geometric parameters assuming a tetrakaidecahedron unit-cell with circular cross-section shapes and spherical nodes at their intersections.

Figure 2 :

 2 Figure 2: Asymptotic fields for 1/4 th of the reconstructed foam sample period R 4 : (a) low-frequency scaled velocity field [× 10 -9 m 2 ], (b) high-frequency scaled velocity field [× 10 -11 m 2 ], and (c) lowfrequency acoustic temperature field [° K].

[ 9 ]Figure 3 (Figure 4 (Figure 5 (Figure 4 :

 93454 Figure 3(A): (a) Normal incidence sound absorption coefficient g( ), (b) dynamic viscous permeability v ( ), and (c) dynamic thermal permeability t ( ) for foam sample R 1 : Comparison between measurements, characterization, and computations.

Figure 5 :

 5 Figure 5: Ligament length distributions for real foam samples (left) R 1 , (center) R 23 , and (right) R 4 .In the legend, Measurements (p) accounts for ligaments lengths characterizations through micrographs, whereas (8) Computation stands for ligaments length of the tetrakaidecahedron unitcell used for numerical simulations.

Table 1 : Macroscopic geometric and transport parameters of real foam samples.

 1

	L
	D=2√2L
	2r
	2R