Christophe Dolabdjian
email: christophe.dolabdjian@unicaen.fr

Dufay Basile
email: bdufay.basile@unicaen.fr

Saez Sebastien
email: csebastien.saez@unicaen.fr

Yelon Arthur
email: arthur.yelon@polymtl.ca

Menard David
email: edavid.menard@polymtl.ca

Is low frequency excess noise of GMI induced by magnetization fluctuations?

Keywords: GMI, Noise, Magnetization fluctuations, Magnetometers

We have investigated the possible impact of low frequency magnetization fluctuations on the equivalent magnetic excess noise of GMI sensors, which we have recently shown to exhibit 1/f noise. This noise component is not associated with the detailed measuring setup nor with the conditioning electronic noise sources, suggesting that it is intrinsic to the sensing element. Various intrinsic GMI noise sources might be able to explain this observation; these include magnetic domain wall motion, hysteresis loop losses, etc. Since GMI elements are excited by a high frequency current, it has been assumed that these low frequency (lf) intrinsic noise sources cannot interact with the carrier and the sensed signal. We recall that the GMI effect is based on an impedance variation, which is governed by magnetization angle variations. These are modulated by the sensed signal and the lf magnetization noise, which then appear as sidebands around the carrier frequency. Applying the fluctuation-dissipation theorem to the GMI model, we have related both the signal and noise to the magnetic susceptibility spectrum and thus quantified the equivalent magnetic noise of GMI sensors at lf. We then present a preliminary comparison to our previous experimental results.

Introduction

Giant MagnetoImpedance (GMI) in soft magnetic materials has considerable potential for magnetic field sensing applications [1,2], for which high sensitivity to the magnetic field and low equivalent magnetic noise spectral density are required. In the past few years, a theoretical model of GMI sensor noise, treating its dependence upon material properties has been developed, [3,4,]. It has been applied to the white noise regime [5], for which we found that electronic noise dominates and explains the discrepancies between the expected, intrinsic, GMI noise and experimental noise measurement. In contrast, the lf, 1/f, noise, which has been observed, has not been theoretically investigated. This work focuses on this issue. A first theoretical estimate of the intrinsic magnetic noise at lf is proposed, in order to evaluate the fundamental limitations of GMI sensing elements. Expected wire sensor noise performances are reported and compared with experimental observations.

Noise in GMI sensing elements

It has been recognized that thermal fluctuations of the magnetization set fundamental limits to the signal-to-noise ratio of magnetoresistive sensors [6]. The response of any magnetic field sensor, which depends upon the magnetization direction of its sensing elements, is subject to intrinsic magnetic noise arising from thermal fluctuations of the magnetization. For a soft magnetic wire, driven by a high frequency longitudinal current, such as a GMI sensing element, we have presented a model, discussed above, with which we may evaluate intrinsic GMI sensor white noise [3,4].

Here, we extend it to include the fluctuation-dissipation theorem for the magnetization at lf. Based on a small signal analysis, the voltage appearing across the GMI is usually given by

        n Z V t Z b t z t I t B             , ( 1
)
where Z is the impedance of the wire at the working magnetic field point, and ∂Z/∂B is the impedance sensitivity; with B the magnetic induction; b(t) is the small sensed applied magnetic induction, z n (t), the equivalent impedance noise variation and I(t) the sinusoidal rf current of amplitude and angular frequency, I AC and ω 0.

By using classical demodulation technique and considering chain gains to be unity, the Fourier Transform of the output signal of the lock-in is

            0 0 2 AC n Z I V Z b z B                    , (2)
where ω (= 2πf ) is the angular frequency. At the working angular frequency, ω 0 , it yields sensor field sensitivity, in V/T, of

  0 2 AC Z I S B     (3)
and an equivalent voltage noise, in V/√Hz, of

    2 AC n n I v z      . ( 4
)
We assume that the cylindrical shell volume of the GMI wire, defined by the skin depth, is a single domain with a uniform helical magnetization, subject to a uniform effective internal field dominated by magnetoelastic anisotropy. Then, the impedance of the wire, Z, obtained from the solution of Maxwell's equations and the linearized Landau-Lifshitz equation for magnetization dynamics, can be approximated by [3,4]

2 0 cos Z Z Z     , ( 5
)
where θ is the angle between the shell layer average magnetization and the wire longitudinal axis (see Fig. 1 of [3]). In Eq. ( 5),

  0 M Z Z Z   
, where Z 0 and  

0 M t ZZ 
 are the non-magnetic and magnetic parts of the wire impedance, corresponding to small circumferential driving fields which are parallel and perpendicular to the static magnetization, respectively, and µ t is the relative transverse permeability.

We assume that the driving field, h φ , static applied field, H 0 , and effective anisotropy field, H K , are all small compared to the saturation magnetization, M s .as is usually the case for GMI. We further assume that the impedance noise power spectral density, S hh , depends upon the thermal fluctuations of the free layer magnetization. Assuming equipartition theorem and our previous model, as was done in refs. [3,4], we find

      2 2 2 2 2 2 0 2 0 4 1 t hh B n s s S k T '' Z Z Z z S M M                                                  (6)
In Eq. ( 6) S θθ is the spectral density of the magnetization direction fluctuations, χ t , the magnetic susceptibility,  , the free layer wire volume, µ 0 the permeability of free space, and k B T, the thermal energy. The imaginary part of the susceptibility χ'' is usually thought of as the coefficient that describes the losses driven by applied fields.

After some simplification, based on our previous description [3], the equivalent impedance noise spectral density may be rewritten as

      2 2 2 2 2 0 0 4 4 1 tan tan B B n s int k T k T Z Z c z M H f                                           , ( 7
)
where c is a constant and

  tan '' '       
is the magnetic loss tangent factor (assuming χ' ≈ 1 for ω → 0). Let us assume [4], a wire with circumferential anisotropy such that the magnetization versus field relation is given by

0 cos K HH  
, and thus the internal field is

  2 2 int 0 K K H H H H 
. The sensor will be operated at a few MHz with a DC bias field equal to . We notice that

Z    0 3 4 t Z   . ( 8
)
Also, based on these hypotheses, Eqs. ( 7) and (8) lead to a voltage noise power spectral density, in V 2 /Hz, given by

    2 2 2 0 2 0 3 4 Tan 44 t B n int I kT vZ H             22 0 23 0 Tan 4 2 B K m IZ kT f H f           . (9)
In Eq. ( 9) γ is the gyromagnetic ratio, α, the dimensionless Gilbert damping parameter and f m the ferromagnetic resonance frequency of the wire. Then, the equivalent magnetic noise spectral density, in T/√Hz, in the lf, 1/f , regime is given by

      12 2 23 0 0 4 Tan 2 Bm n K k Tf Z bf Hf Zf B        . (10)
Equation ( 10) is the main result of this section. It shows that the equivalent magnetic noise spectral density is expected to scale directly with the impedance sensitivity ratio, with the square root of the ferromagnetic resonance frequency, the magnetic loss tangent coefficient, and the absolute temperature, and inversely with the square root of the wire volume, the gyromagnetic ratio and the anisotropy field.

Results and discussion

Eq. ( 10) may be compared to the equivalent magnetic spectral density in the white noise regime, obtained earlier 3, [4]: Assuming f m ≈ γ µ 0 M S and a loss factor of 0.001 to 0.01, the rhs of Eq. ( 12) is about ((4 -40) 10 6 )/f. So, it yields to an equivalent magnetic noise spectral density of between 27 and 86 pT/√Hz at 1 Hz and having a 1/f noise behavior. This theoretical value maybe compared to our previous experimental results [5,7], which give noise levels of 30-100 pT/Hz at 1 Hz.

    2 2 2 23 0 4 B n _ w K kT Z b H Z B             . ( 11
) so that       2 2 Tan 2 n m n _ w bf f f bf    . ( 12
)
The result, Eq. (10) must be viewed only as a rough estimate of the equivalent GMI magnetic noise at lf. These preliminary calculations suggest that thermal magnetic noise arising from thermal fluctuations of the magnetization could be a significant contribution to the lf intrinsic noise of the sensing element through impedance fluctuations. Further theoretical and experimental studies are underway to clarify this issue and to fix limits of the lf noise performance of GMI, a key point for the future of GMI sensor development.

  Commercial melt-extracted magnetic amorphous wires (MXT, Inc. of Montreal) with a composition of Co 80.89 Fe 4.38 Si 8.69 B 1.52 Nb 4.52 (wt%), were used as reference sensing elements. The samples considered were two centimeter long of approximately 30 µm in diameter. An average resistivity of 130 µΩcm was inferred from the resistance of the wires. This yields an impedance value of around 50 Ω in MHz frequency range with a highest sensitivity close to 100 MΩ/T. The magnetic noise density, calculated from Eq. (11) with α = 0.01, γ =28 GHz/T and H K = 55 A/m, is about 10 fT/√Hz.


Z Yao

C Charrier

C Rosenberger
email: christophe.rosenberger@ensicaen.fr

Utility Validation of a New Fingerprint Quality Metric

I. INTRODUCTION

Fingerprint somehow can be regarded as a relatively fullfledged application in biometrics. The use of this biometric modality is not limited to traditional public security area, but spread into the daily life, smart phone authentication control and e-payment, for instance. However, quality control of biometric sample is still a necessary task due in order to optimize the operational performance. Research works had shown that biometric systems performance could be greatly depressed for those uncontrolled conditions [START_REF] Alonso-Fernandez | Quality measures in biometric systems[END_REF]. For fingerprint, the quality can be intuitively described as the clarity of its ridge and valley pattern, noise condition, and the feasibility of feature extraction such as minutiae points. However, intuitively bad quality fingerprint sample might generate high matching result for some cases and vice versa [START_REF] Grother | Performance of biometric quality measures[END_REF]. In this case, many previous research works contributed to validate fingerprint quality metrics in terms of the relation between biometric sample quality and system performance. Chen et al. [START_REF] Chen | Fingerprint quality indices for predicting authentication performance[END_REF] proposed quality metrics in both frequency domain and spatial domain, and discussed several criteria to evaluate quality metric, such as quality index predicting matching performance. Fernandez et al. [START_REF] Alonso-Fernandez | A comparative study of fingerprint image-quality estimation methods[END_REF] compared correlations between several previously proposed quality metrics. Grother et al. [START_REF] Grother | Performance of biometric quality measures[END_REF] discussed quality evaluation methods in terms of detection error trade-off characteristics (DET), the effect of low quality samples rejection rates on improving performance, Kolmogorov Smirnov (KS) statistic, and etc. Mohamad et al. [START_REF] El-Abed | Quality assessment of image-based biometric information[END_REF] calculated the KS statistics for image-based quality metric of multimodal biometric data.

The study of this paper makes an evaluation of a proposed quality metric generated by using a utility-based multi-features fusion approach [START_REF] Abed | Fingerprint quality assessment using a no-reference image quality metric[END_REF]. The quality metric involves in several quality criteria, including no-reference image quality metric (NR-IQA) [START_REF] Saad | A dct statistics-based blind image quality index[END_REF], measures derived from texture features, and measure based on the triplet representation of fingerprint minutiae.

The paper is organized as follows. Section 2 presents a state of the art on fingerprint quality. In section 3, we describe the proposed quality metric called Q. Section 4 details experimental results. Conclusion and future works are given in section 5.

II. STATE OF THE ART ON FINGERPRINT QUALITY

Quality assessment of fingerprint samples is mostly studied considering feature extraction. Earlier studies considered feature extraction in both at a local and global level [START_REF] Bolle | System and method for determining the quality of fingerprint images[END_REF], [START_REF] Ratha | Fingerprint image quality estimation[END_REF], [START_REF] Bazen | Segmentation of fingerprint images[END_REF], [START_REF] Shen | Quality measures of fingerprint images[END_REF], [START_REF] Lim | Fingerprint quality and validity analysis[END_REF], [START_REF] Chen | Fingerprint image quality analysis[END_REF]. Most of these studies qualifying fingerprint quality in terms of ridge-valley structure clarity, orientation variations, and so on. Studies in that time considered fingerprint quality from image features and gray level contrast, loosely speaking. Details of fingerprint such as minutiae characteristics and singularity points are rarely considered due to the development of this issue in that time. In addition, performance evaluation was taken into account in terms of the comparison between quality metric and manual quality assessment result. Later, NBIS quality algorithm [START_REF] Tabassi | Nist fingerprint image quality[END_REF] had been proposed, in which fingerprint quality has been defined as a predictor of the further matching performance. The evaluation of fingerprint quality metric had been considered by the interaction between sample quality and biometric system performance [START_REF] Lee | A novel measure of fingerprint image quality using the Fourier spectrum[END_REF], [START_REF] Chen | Fingerprint quality indices for predicting authentication performance[END_REF]. ISO/INCITS then established a standard for biometric quality, among which the quality of a biometric sample is considered in 3 different aspects [START_REF] Alonso-Fernandez | A comparative study of fingerprint image-quality estimation methods[END_REF]:

(1) Character, referring to the quality of subject physical features; (2) Fidelity, which means that how similar is a biometric sample to its reference sample; (3) Utility, indicating the impact of biometric sample quality on the overall performance of a biometric system.

Mohamad et al. [START_REF] Abed | Fingerprint quality assessment using a no-reference image quality metric[END_REF] proposed an utility-based quality assessment approach, in which several criteria of quality were linear combined by using an optimization method [START_REF] Houck | A genetic algorithm for function optimization: a matlab implementation[END_REF], genetic algorithm (GA). The quality metric proposed in that study shows a relatively ideal result for fingerprint samples with several kinds of alterations. The quality metric carried out in this study is an extension of the one that had just mentioned, and mainly focuses on the original database of fingerprint, and the purpose of evaluating its performance was achieved.

III. PROPOSED FINGERPRINT QUALITY METRIC

The quality metric defined in [START_REF] Abed | Fingerprint quality assessment using a no-reference image quality metric[END_REF] is in the form of

Q = 1 A N i=1 α i C i , (1)
where N is the number of quality criteria C i (i = 1, , N ), α i is the linear coefficients, and A is a normalization constant. . In this study, the feature involved in generating quality metric includes 3 classes, one is features proposed in [START_REF] Abed | Fingerprint quality assessment using a no-reference image quality metric[END_REF], the second class is consisted of several texture features extracted directly from fingerprint image, and the third type of feature is derived from the triplet representation of minutiae-based template, m i = {x, y, θ}, (x, y) is the location of minutia point and θ is the orientation of the minutia point. The framework of the quality metric in this study can be illustrated by figure 1.

A. Features in Previous Works

In the previous work, there were 11 features contributing to obtain the quality metric, including one derived from a NR-IQA algorithm and several image-based features.

1) NR-IQA [START_REF] Saad | A dct statistics-based blind image quality index[END_REF] is an algorithm to evaluate the quality of an image, and it is classified into non-distortion specific approaches. This quality metric generally involves Then, a global quality score was calculated by using a multi-scale approach as:

BLIIN DS = L i v α i 1 1 v α i 2 2 v α i 3 3 v α i 4 4 , (2)
where α i j are calculated by using the correlation of v i with the subjective notes given by human observers.

2) Salient features based on features extracted by using

Scale Invariant Feature Transform (SIFT) operator, which include the number N B of detected keypoints in the image, DC coefficient of the descriptor matrix, mean and standard deviation of the scales related to the keypoints. 3) Patched features, in addition to the blocks number of the image, were computed from the root mean square (RMS) values of each block of the image, including mean value of blocks RMS values, standard deviation of RMSs, median value of RMSs, skewness of RMSs and kurtosis value of them.

B. Texture Features

Texture features are widely used for image classification and retrieval applications. But it is not sure that whether some of them are able to contribute distinctive result for quality assessment of fingerprint image. In this study, 4 classes, 11 texture features were selected as the components for generating quality metric, given in table I. 1) The first class is local binary pattern (LBP) feature and its extensions or transforms. The LBP feature is proposed by Ojala et al [START_REF] Ojala | Multiresolution gray-scale and rotation invariant texture classification with local binary patterns[END_REF], using for image classification. This feature is simple yet efficient so that it is widely used for texture analysis. The idea of LBP operator was that the two-dimensional surface textures can be described by two complementary measures: local spatial pattern and gray scale contrast [START_REF] Pietikäinen | Computer vision using local binary patterns[END_REF]. Basic LBP operator generates a binary string by thresholding each 3-by-3 neighborhood of every pixel of the image. Figure 2 illustrates the procedure of extracting LBP histogram from fingerprint image. The basic operator also extended by many improvements such as uniformed LBP. The transforms of LBP involved in this study include four-patch LBP (FLBP), completed LBP (CLBP), LBP histogram Fourier transform (LBPHFT [START_REF] Nanni | Survey on lbp based texture descriptors for image classification[END_REF], and median LBP (MLBP) [START_REF] Hafiane | Median binary pattern for textures classification[END_REF]. The original LBP is a gray-scale invariant operator and its general form (with radius) is rotation invariant. Thus, the general form of basic LBP is robust to changes of both luminance and rotation. Because of this, the general form LBP has been widely adopted in the applications of face modality. CLBP modeling the general form of LBP operator and splits it into 3 components, in which sign components is said to be most informative, and they proposed two forms of combination of the components to obtain more effective performance. 2) The second class is Haralick features or gray level cooccurrence matrix (GLCM) [START_REF] Haralick | Textural features for image classification[END_REF]. Figure 3 presents the calculation of several statistic measures generated from the GLCM matrix which involves in 4 directions combination of neighbor pixels. 3) The 2D Gabor function is a sinusoidal function modulated by a Gaussian window. In this case, the basis of Gabor function is complete but not orthogonal. In the last few decades, it has been widely applied to fingerprint image and other biometric data, such as classification and segmentation tasks. Shen et al. [START_REF] Shen | Quality measures of fingerprint images[END_REF] proposed using Gabor response to qualify fingerprint image, in which it is said that one or several Gabor features of 8-direction Gabor response are larger than that of the others. Olsen et al. [START_REF] Olsen | Gabor filters as candidate quality measure for nfiq 2.0[END_REF] proposed a quality index based on 4-direction Gabor response and it is said that 4-direction is sufficient to qualify fingerprint. However, in this study, it is observed that 2-scale 4-direction Gabor filters do not bring out distinctive regularity for a specified database.

4) The last one is local relational string (LRS) [START_REF] Hafiane | Local relational string for textures classification[END_REF]

C. Measures based Minutiae Template

It can be observed that minutiae are greatly associated with minutiae distributions, locations, and their orientation through the past studies of fingerprint minutiae [START_REF] Chen | The statistical modelling of fingerprint minutiae distribution with implications for fingerprint individuality studies[END_REF]. In this study, several simple measures based on minutiae locations and orientations were calculated to generate quality metric. Some of these measures were inspired by features for matching operation, and some others were derived from studies of classification. This is due to the consideration of utility of image quality, i.e. features carried out for matching and classification somehow might be associated with fingerprint image quality and one or several distinctive properties of the image. In addition, the qualities of fingerprint images decided by human vision do not fully accord with the practical matching performance. The first kind of measures are based on minutiae numbers and DFT of three components of minutia template. In addition to minutiae number-based features, several statistic-based rotation and translation invariant measures based on minutiae orientations and neighboring minutiae are also calculated [START_REF] Ross | Toward reconstructing fingerprints from minutiae points[END_REF]. These features were proposed to distinguish fingerprint images of different classes in terms of the relations of both minutiae orientations and locations.

IV. EXPERIMENTAL RESULTS

In order to validate the behavior of the quality metric of this study, a comparative study have been made between the proposed quality metric and a representative quality assessment tool, NFIQ [START_REF] Tabassi | Nist fingerprint image quality[END_REF].

A. Protocol and Databases

In this study, three FVC databases [START_REF] Maio | Fvc2004: Third fingerprint verification competition[END_REF] have been used for experiments. Details on databases are given in table II. Each of these 3 databases involves 100 fingertips, and 8 samples for each fingertip. In this case, the matching scores involved in the experiment have been calculated by using NBIS tool [START_REF] Watson | User's guide to nist biometric image software[END_REF], Bozorth3. The intra-class scores contain 1×7×100 = 100 genuine scores, and the inter-class scores are consisted of 1×7×99×100 = 69300 impostor scores for the whole database. Minutiae template used in the experiment was also extracted by using NBIS tool, MINDTCT. This software generates a quadruple representation of minutia point, m i = {x, y, o, q}, where (x, y) is the location of minutia point, o indicates orientation, and q is a quality score of minutia point. In the experiment, just the first 3 components have been used for calculating minutiae-based measures.

B. Evaluation

In the experiments, the EER values of the 3 databases have been calculated using the FVC principles [START_REF] Maio | Fvc2004: Third fingerprint verification competition[END_REF], i.e. the first sample of each fingerprint is used as the reference while other seven samples are used as the authentication samples. The results are shown by ROCs in figure 4, in which the EER values of FVC2002DB3A, FVC2004DB1A and FVC2004DB3A are 11.6 %, 12.9%, and 8.5%, respectively. Second, we computed the EER values of 3 databases by choosing the best samples as the reference (considering the resulting EER values). This is due to the consideration that good quality references should decrease matching error rate. In this case, two kinds of EER values are computed based on NFIQ and proposed approach, respectively. Here, the graphic results of FVC2004DB1A are presented only because the Pearson correlation value of this database is the lowest one among the 3 databases, illustrated by figure 5. The EER value based on NFIQ is 13.2%, and 10.6% corresponds to the proposed approach. For FVC2004DB1A and FVC2004DB3A, the EER values are 14.7% (NFIQ), 14% (proposed), 8.3% (NFIQ), and 6.7% (proposed). Third, the correlation between genuine matching score and the corresponding quality index of each database was calculated, as given in table III. It can be observed that NFIQ does not show high numbers for the correlation values. This fact have been mentioned in [START_REF] Alonso-Fernandez | A comparative study of fingerprint image-quality estimation methods[END_REF]. In addition, this experiment shows a similar result given by [START_REF] Alonso-Fernandez | A comparative study of fingerprint image-quality estimation methods[END_REF], i.e. it is relatively difficult to qualify fingerprint images captured by optical sensors. The proposed metric give much better results (but not perfect).

We used another validation approach defined in [START_REF] Chen | Fingerprint quality indices for predicting authentication performance[END_REF] to test the proposed quality metric. In order to have a comparative study with the reference algorithm, the discrete quality values of each database have been divided into 5 isometric bins which correspond to the 5 quality labels of NFIQ. Then, the EER values of the divided bins have been calculated. The result is given in figure 6. For NFIQ-based quality values, it is easier to calculate the EER values of the 5 label bins, as depicted in figure 7.

We are able to observe that the matching performances on FVC2002DB3A and FVC2004DB3A were monotonically increased by pruning bad quality samples gradually. NFIQ generated quality levels from 1 to 4 for FVC2002DB3A, and no samples of level 5 were figured out for this database. This might be due to the minutiae points detected on the images of this database, because NFIQ algorithm involves in minutia quality of the fingerprint image. This situation was observed when calculated the correlation between 14 minutiae measure and genuine matching scores in the experiment of this study. It shows a relatively higher correlation on FVC2002DB3A, while the values of two other databases are relatively lower. For FVC2004DB1, both the proposed quality metric and the reference algorithm showed certain difficulties. This is consistent with the past result in [START_REF] Alonso-Fernandez | A comparative study of fingerprint image-quality estimation methods[END_REF] and the previous result in this study. Here, just graphic results on FVC2002DB3A are presented only, while the 5 bins' EER values based on proposed approach and NFIQ of FVC2004DB1A and FVC2004DB3A are given in table IV.

V. CONCLUSION

This study evaluated the performance of a multi-feature fusion-based quality metric for fingerprint samples. In the study, the proposed quality metric was evaluated on 3 different FVC databases, FVC2002 DB3 A, FVC2004 DB1 A, and FVC2004 DB3 A. Among the evaluation result, it can be observed that database consisted of images of capacitive sensor is relatively easier to be qualified. This is due to several factors impacted on image quality and matching performance. In addition to external factors such as sensor type and environment, it might be involved in image factors, such as contrast, image size, pixel density, foreground and background area; and correspondingly the factors caused by minutiae template, such as minutiae location, minutiae reliability, and other minutiae properties if they are considered.

Future works of this study will focus on improving the current quality metric and feature processing for the quality metric.

Fig. 1 .

 1 Fig. 1. Illustration of the calculation of quality metric.

  3 kinds of DCT-based feature, DCT-based contrast feature v 1 , DCT-based structure feature v 2 , and DCT-based anisotropy orientation features v 3 and v 4 ; v 1 is an average value of the contrast of k th DCT path of an image, v 2 is the a global image kurtosis based on the kurtosis value of each DCT patch, v 3 and v 4 are variance and max value of the mean value of each DCT patchs Renyi entropy.

Fig. 2 .

 2 Fig. 2. Illustration of general LBP operator.

  FLBP is a patch-based descriptor based on central symmetric patches in different scales around a pixel. This operator considers the similarities between neighboring patches of pixels, and is a complementary of pixel based descriptors. LBPHFT is a rotation invariant image descriptor based on the Fourier transform of ULBP histogram of the original image. This operator considers complex conjugate of the ULBP histogram Fourier transform to deal with the cyclic shift in ULBP histogram caused by image rotation. MLBP is and transform of LBP operator and invariant to monotonic change of gray-scale. Instead of using central pixel, it uses the median value of the local block as the threshold to obtain LBP string.

Fig. 3 .

 3 Fig. 3. Illustration of Haralick features.

Fig. 4 .

 4 Fig. 4. ROCs of 3 databases.

Fig. 6 .

 6 Fig. 6. Monotonic increasing matching perfromance validation of FVC2002DB3A for proposed quality metric, calculated by dividing quality values into 5 isometric bins.

Fig. 7 .

 7 Fig. 7.Monotonic increasing matching perfromance validation of FVC2002DB3A for NFIQ, calculated by dividing quality values into 5 isometric bins.

  which is an illumination invariant operator, and it reflects variation of local gray level of the image. The operator is based on the local pixels relation in a specified scale, and it uses 3 relations to generate local relation motif histogram for measuring local spatial variations of the image.

TABLE II DETAILS

 II OF 3 EXPERIMENT DATABASES.

	DB	Sensor Type	Resolution	Image Dim	DB Size
	2002DB3A	Capacitive	500dpi	296×560	100×8
	2004DB1A	Optical	500dpi	480×640	100×8
	2004DB3A	Thermal Sweeping	512dpi	300×480	100×8

TABLE III CORRELATION

 III ANALYSIS BETWEEN SCORES AND QUALITY METRICS

	Quality Metric	Database	02DB3A	04DB1A	04DB3A
		NFIQ	-0.2690	-0.2067	-0.2458
		Q	0.4600	0.3164	0.5351

Fig. 5. ROCs and EERs of FVC2004DB1A, calculated by using the best samples as the reference.

TABLE IV 5

 IV BINS EER VALUES BASED ON PROPOSED APPROACH AND NFIQ OF FVC2004DB3A.

	Bin No.	B1	B2	B3	B4	B5
	Q (04DB1)	22.2%	16.6%	17.2%	17.8%	13.3%
	NFIQ (04DB1)	15.8%	18.1%	17.7%	23.2%	26.5%
	Q (04DB3)	14.2%	8.9%	7.4%	5.8%	4.2%
	NFIQ (04DB3)	7.5%	8.1%	13.4%	12.9%	29.8%


A Ghintran

E González-Arangüena

C Manuel

A probabilistic position value

Keywords: Game Theory, TU Games, Graph-restricted Games, Position Value. 1991 MSC: 91A06, 91A12, 91A43

). We provide two characterizations of this new allocation rule. Following in Slikker's (2005a) footsteps, we characterize the probabilistic position value using probabilistic versions of component efficiency and balanced link contributions. Then we generalize the notion of link potential, defined by Slikker (2005b) for the class of deterministic communication situations, to the class of generalized probabilistic communication situations, and use it to characterize our allocation rule. Finally, we show that these two characterizations are logically equivalent.

Introduction

Various economic or social situations in which a group of agents cooperate to achieve a common goal can be appropriately formalized via cooperative games with transferable utility, or TU games. In such games, agents are referred as players. A TU game summarizes all the necessary information concerning the worth produced by each coalition of players when they agree to cooperate. It is assumed that any coalition can form. On the other hand, in many situations the collection of possible coalitions is restricted by some social, hierarchical, economical, communicational or technical structures. In this article, we restrict ourselves to the special case of communication situations, introduced by [START_REF] Myerson | Graphs and cooperation in games[END_REF], in which the cooperation among players is limited because only some undirected and bilateral relations are possible. Then, a communication situation consists of a TU game and a network of possible relations modelled via a graph. The vertices in the graph correspond to the players and the edges correspond to bilateral communication links. In order to measure the impact of restrictions on communication on the gains from cooperation, [START_REF] Myerson | Graphs and cooperation in games[END_REF] associates to each communication situation a new TU game, the so-called graph-restricted game. The Myerson value of a communication situation is the Shapley value of its graph-restricted game. [START_REF] Myerson | Graphs and cooperation in games[END_REF][START_REF] Myerson | Conference structures and fair allocation rules[END_REF], [START_REF] Borm | On the position value for communication situations[END_REF] and [START_REF] Slikker | Social and economic networks in cooperative games[END_REF] provide various characterizations of the Myerson value. [START_REF] Meessen | Communication games[END_REF] suggests to associate to each communication situation an alternative TU game that focuses on the role of links: the so-called link game, in which the set of players is the set of links. This link game associates to every set of links the worth produced by the grand coalition of players when the links in this set are the only ones available. The position value, defined by [START_REF] Meessen | Communication games[END_REF], shares the Shapley value of each link in the link game equally between its two incident players. The position value of a player is the sum of the gains he collects in this way. [START_REF] Borm | On the position value for communication situations[END_REF] provide a characterizations of the position value that is valid on the class of communication situations in which the graph is cycle-free, whereas Slikker (2005a,b) obtains two new characterizations without the restrictions on the graph existing in the previous one.

In this article, we are interested in situations where the network structure is not given and fixed, i.e. several alternative networks can form and thus players are able to form occasional alliances as well as long term relationships. In this setting, the Myerson approach can be think of as a particular case in which only one network is possible. There are two different ways to model such communication networks.

The first one [START_REF] Calvo | Values of games with probabilistic graphs[END_REF]) consists of considering the communication between two players as a Bernoulli trial. A probabilistic graph maps to each link a probability of realization. These probabilities are considered as independent. A probabilistic communication situation is made up of a TU game and a probabilistic graph on the same set of players. Because of the independence assumption, this approach fails to take into account those situations in which the probability of realization of a set of links is correlated with the realization of another one. In the second approach [START_REF] Gómez | A value for generalized probabilistic communication situations[END_REF]), we dropped the independence assumption and considered the so-called generalized probabilistic graphs, that assign a probability, measuring the likelihood of each one of the potential networks. Restricting the cooperation in a TU game with a generalized probabilistic graph we obtained a generalized probabilistic communication situation. This approach is a generalization of the previous one and extends the range of situations to which can be applied.

The aim of this article is to carry on the work of extension of allocation rules to (generalized) probabilistic communication situations initiated by [START_REF] Calvo | Values of games with probabilistic graphs[END_REF] and [START_REF] Gómez | A value for generalized probabilistic communication situations[END_REF]. [START_REF] Calvo | Values of games with probabilistic graphs[END_REF] define and characterize a natural extension of the Myerson value in terms of component efficiency, fairness and balanced contributions on the class of probabilistic communication situations. [START_REF] Gómez | A value for generalized probabilistic communication situations[END_REF] extend these results to the class of generalized probabilistic communication situations, and describe some properties of the graph-restricted game.

In this article, we extend the definition of the position value to the class of generalized probabilistic communication situations and provide two characterizations of this new allocation rule. Following in Slikker's (2005a) footsteps, we characterize the probabilistic position value using probabilistic versions of component efficiency and balanced link contributions. Slikker (2005b) also characterizes the position value using a link potential for communication situations that extends the potential for TU games of [START_REF] Hart | Potential, value, and consistency[END_REF] in a natural way. This link potential focuses on the marginal contributions of a player's link. We will extend too this characterization to the class of generalized probabilistic communication situations.

As [START_REF] Thomson | On the axiomatic method and its recent applications to game theory and resource allocation[END_REF] argues, analysing the logical relations between the axioms permits to highlight their relative power. We show that the probabilistic link potential is equivalent to the probabilistic balanced link contributions property. Ortmann (1998) obtains a similar result for TU games. This article is organized as follows. In Section 2, we introduce notations and preliminaries. Section 3 is devoted to the study of the generalized probabilistic graphs and its subgraphs. The definition of the probabilistic position value can be found in Section 4. In section 5, we present two characterizations of the defined value extending the ones existing for the deterministic case. And finally, the logical relation between these two characterizations is given in Section 6.

Preliminaries

A game in characteristic function form (a TU game or a coalitional game) is a pair (N, v), where N = {1, 2, . . . , n} is a finite set of players and v (the characteristic function) is a real map defined on 2 N , the set of all subsets (coalitions) of N, satisfying v(∅) = 0. For each S ⊂ N, v(S ) represents the value produced by S when its players agree to cooperate. When there is no ambiguity with respect to the set of players N, we identify the game (N, v) with its characteristic function v. For each S ⊂ N, the cardinal of S is denoted by s. The 2 n -1 dimensional vector space of all games with players set N is denoted by G N . A game v in G N is zero-normalized if v({i}) = 0 for all i ∈ N. [START_REF] Shapley | A value for n-person games[END_REF] introduces a point solution for the class of cooperative games, widely used in the literature. For each v ∈ G N , the Shapley value of player i ∈ N is a convex linear combination of his marginal contributions:

S h i (N, v) = S ⊂N, i S s!(n -1 -s)! n! [v(S ∪ {i}) -v(S )] .
This solution is a linear map from G N on R n , and thus it can be computed using the so-called unanimity games. Given a coalition S ⊂ N, S ∅, the unanimity game u S is defined by u S (T ) = 1 if S ⊂ T and u S (T ) = 0 otherwise. The collection of games {u S } ∅ S ⊂N is a basis of G N , and then, for v ∈ G N , it holds that:

v = ∅ S ⊂N ∆ v (S )u S .
For each ∅ S ⊂ N, ∆ v (S ) is known as the Harsanyi dividend [START_REF] Harsanyi | A bargaining model for the n-person game[END_REF]) of the coalition S and can be obtained from:

∆ v (S ) = T ⊂S (-1) s-t v(T ).
It is easy to see that S h i (u S ) = 1/s, if i ∈ S and 0 otherwise. As a consequence, for all v ∈ G N ,

S h i (N, v) = S ⊂N, i∈S ∆ v (S )
s .

To formalize social or economic network, we use a graph (N, γ), where N = {1, 2, . . . , n} is the finite set of nodes (actors) and γ a collection of links (edges or ties). A link is an unordered pair {i, j}, such that i j and i, j ∈ N. A graph (N, δ) is a subgraph of (N, γ) if δ ⊂ γ. When there is no ambiguity with respect to N, we refer to (N, γ) as γ. The complete graph with nodes set N is given by K N = {{i, j} | i j and i, j ∈ N}. Denote by 2 K N the class of all networks with nodes set N. Each graph (N, γ) is both an element of 2 K N and a subgraph of K N .

Two nodes i and j are directly connected in (N, γ) if i = j or {i, j} ∈ γ. If i and j are not directly connected in (N, γ) but there exists a sequence (i 1 = i, i 2 , . . . , i k = j) such that {i h , i h+1 } ∈ γ for h = 1, . . . , k -1, then i and j are connected in (N, γ). A graph (N, γ) is connected if any two nodes i, j ∈ N are connected. A subset S of N is connected in (N, γ) if the partial graph (S , γ | S ) is connected, where γ | S is the set of links of γ of which both incident nodes are in S .

Given a graph (N, γ), the notion of connectivity induces a partition of N in connected components. A connected component is a maximal connected subset. Two distinct nodes i and j are in the same connected component if and only if they are connected. Let N/γ denote the set of all connected components of N in γ and, more generally, for each S ⊂ N, S /γ is the set of all connected components in the partial graph (S , γ | S ). If the connected component to which node i belongs is a singleton, we say that i is an isolated node. Obviously, γ is connected if and only if |N/γ| = 1. Finally, L i (N, γ) = {l ∈ γ|i ∈ l} denotes the set of links in γ incident on i and γ -i = γ \ L i (N, γ) is the subgraph of γ obtained by removing links in L i (N, γ). When there is no ambiguity with respect to N we write L i (γ) instead of L i (N, γ).

A communication situation is a triplet (N, v, γ), where v is a game in G N and γ a graph in 2 K N . For convenience we restrict ourselves to communication situations with a zero-normalized underlying TU-game. In a communication situation, the nodes of the graph are the players of N. The only feasible coalitions are formed by players connected in the graph, i.e. the graph introduces the available channels of communication among players.

An allocation rule for communication situations is a function Ψ : C N → R n , where C N is the set of communication situations with players set N. The real number Ψ i (N, v, γ) is the payoff of player i in the game v when the communication possibilities are restricted by the graph γ. Two well known allocation rules for communication situations are the Myerson value and the position value. [START_REF] Myerson | Graphs and cooperation in games[END_REF] introduces the graph-restricted game (N, v γ ), of which characteristic function v γ is defined by:

Given (N, v, γ) ∈ C N ,
v γ (S ) = T ∈S /γ v(T ), S ∈ 2 N ,
v γ (S ) representing the value1 produced by S when the communication is restricted by γ. The Myerson value, µ, is the Shapley value of this graph-restricted game. [START_REF] Myerson | Graphs and cooperation in games[END_REF] characterizes this allocation rule in terms of two appealing properties: component efficiency and fairness. Component efficiency states that the payoffs of the players of a component add up to the worth of this component. The fairness property establishes that the deletion of the link {i, j} changes the payoffs of i and j by the same amount. Then [START_REF] Myerson | Conference structures and fair allocation rules[END_REF] provides an alternative characterization using component efficiency and the balanced contributions property. This property states that the payoff difference player i experiences if all the links incident to player j are deleted is equal to the payoff difference player j experiences if all the links incident to player i are deleted. [START_REF] Meessen | Communication games[END_REF] and [START_REF] Borm | On the position value for communication situations[END_REF] introduce an alternative associated game for communication situations. Given a communication situation (N, v, γ) ∈ C N , the link game (γ, r v γ ) is defined by:

r v γ (ξ) = T ∈N/ξ v(T ), ξ ⊂ γ.
Then, they define the position value, π, as the allocation rule on C N that equally allocates the Shapley value of each link in the previous game between its two incident nodes2 , i.e.: [START_REF] Borm | On the position value for communication situations[END_REF] provide a characterization of the position value that is valid on the class of communication situations with a cycle-free graph. Slikker (2005a) characterizes this allocation rule using component efficiency and the balanced link contributions property. In the same spirit as Myerson's balanced contributions, the balanced link contributions property states that the payoff difference player i experiences when player j sequentially delete all his links is equal to the payoff difference player j experiences when player i sequentially delete all his links. Slikker (2005b) provides characterizations of the Myeron value and the position value on the class of reward games using potentials. These characterizations are in the same spirit as the corresponding one of [START_REF] Hart | Potential, value, and consistency[END_REF] for the Shapley value.

π i (N, v, γ) = 1 2 l∈L i (N, γ) S h l (γ, r v γ ).
Hamiache (1999) introduces another allocation rule that satisfies five specific properties. This allocation rule is not related with our one. [START_REF] Calvo | Values of games with probabilistic graphs[END_REF] extend the model of Myerson by restricting the cooperation in a probabilistic way. They define a probabilistic graph as a pair (N, p), where p is a function that assigns to each link {i, j} ∈ K N its probability of realization. They assumed the independence of these probabilities and characterized the Myerson value for probabilistic communication situations (N, v, p) using the extension of component efficiency, fairness and balanced contributions properties to this new setting. [START_REF] Gómez | A value for generalized probabilistic communication situations[END_REF] give another step in this direction considering a probability distribution defined on the set of all possible communication graphs 2 K N . For each γ ∈ 2 K N , p(γ) is the probability of realization of γ. It satisfies p(γ) ≥ 0 for all γ ∈ 2 K N and γ∈2 K N p(γ) = 1. The pair (N, p) is referred as a generalized probabilistic graph. A generalized probabilistic communication situation is a triplet (N, v, p) where v is a cooperative game and (N, p) is a generalized probabilistic graph. The class of probabilistic communication situations with player set N is denoted by [START_REF] Gómez | A value for generalized probabilistic communication situations[END_REF] define and study the properties of the induced or restricted game (N, v p ), where, for each S ⊂ N,

G N . Given (N, v, p) ∈ G N ,
v p (S ) = γ∈2 K N p(γ)v γ (S )
is the expected value of coalition S in this probabilistic framework. They define the generalized probabilistic Myerson value, µ(N, v, p), as the Shapley value of the game v p , and they characterize it using probabilistic extensions of component efficiency, fairness and balanced contributions.

Generalized probabilistic graphs

Denote by S p the support of graph (N, p), that is, the set of graphs γ ∈ 2 K N such that p(γ) > 0, and let γ p = ∪ γ∈S p γ. The set of all generalized probabilistic graphs with nodes set N is denoted by P N . When there is no ambiguity with respect to the set N, we identify (N, p) with the probability function p.

The notion of connectivity can be extended to generalized probabilistic graphs in the following way: two nodes i and j in N are directly connected in (N, p) when there exists γ ∈ S p such that {i, j} ∈ γ. Two nodes i and j are connected in (N, p) if there is a sequence of nodes (i 1 = i, i 2 , . . . , i k = j) such that i h and i h+1 are directly connected in (N, p) for all h = 1, 2, . . . , k -1. This notion of connectivity in (N, p) induces a partition of N in probabilistic connected components. A probabilistic connected component is a maximal connected set. Denote by N/p the set of all connected components in (N, p). Note that N/p = N/γ p . Let us observe that two nodes i and j can be in the same connected component of N/p even if there is no γ ∈ S p such that i and j are connected in γ.

Each generalized probabilistic graph (N, p) induces 2 ( n 2 ) generalized probabilistic subgraphs (N, p ξ ) ∈ P N , one for each ξ ∈ 2 K N , where:

p ξ (γ) =              δ⊂K N \ξ p(γ ∪ δ) = δ⊂γ p \ξ p(γ ∪ δ) if γ ⊂ ξ 0 otherwise.
If (N, p ξ ) is a generalized probabilistic subgraph of (N, p), we note (N, p ξ ) ⊂(N, p). The interpretation of (N, p ξ ) is the following: only deterministic subgraphs γ of ξ have a strictly positive probability of realization. The probability p ξ (γ) is the sum of the probabilities that p gives to those graphs in 2 K N having in common with ξ the links in γ, i.e.:

p ξ (γ) =              η∈2 K N , η∩ξ=γ p(η) if γ ⊂ ξ 0, otherwise.
The following proposition provides the condition under which two generalized probabilistic subgraphs of (N, p) ∈ P N coincide. This allows to identify the subgraphs that are different.

Proposition 3.1 For each (N, p) ∈ P N and each pair (N, p ξ ), (N,

p ξ ′ ) ⊂(N, p), it holds that p ξ = p ξ ′ if and only if ξ ∩ γ p = ξ ′ ∩ γ p .
Proof : Suppose that p ξ = p ξ ′ and ξ ∩ γ p ξ ′ ∩ γ p . Then there exists l ∈ ξ ∩ γ p such that l ξ ′ , or l ∈ ξ ′ ∩ γ p such that l ξ. Without loss of generality, let us consider the former possibility. We have

0 = γ⊂ξ\{l} p ξ ′ (γ ∪ {l}) = γ⊂ξ\{l} p ξ (γ ∪ {l}),
where the first equality follows since, for any γ ⊂ ξ \ {l}, ξ ′ γ ∪ {l} and the second one since p ξ = p ξ ′ . As

γ⊂ξ\{l} p ξ (γ ∪ {l}) = γ⊂ξ\{l} δ⊂γ p \ξ p(γ ∪ {l} ∪ δ) = η⊂γ p \{l} p(η ∪ {l}),
we can conclude that p(η ∪ {l}) = 0 for all η ⊂ γ p \ {l}. Then l γ p , which is a contradiction.

Reciprocally, to prove that ξ ∩ γ p = ξ ′ ∩ γ p implies p ξ = p ξ ′ , let us show that for each ξ ⊂ K N , p ξ = p ξ∩γ p . For any γ ⊂ ξ ∩ γ p ,

p ξ∩γ p (γ) = δ⊂K N \(ξ∩γ p ) p(γ ∪ δ) = δ⊂(K N \ξ)∪(K N \γ p ) p(γ ∪ δ) = δ⊂K N \ξ p(γ ∪ δ) = p ξ (γ)
where the third equality follows since, for each

δ such that δ ∩ (K N \ γ p ) ∅, p(γ ∪ δ) = 0. Moreover, if γ ξ ∩ γ p , then p ξ∩γ p (γ) = p ξ (γ) = 0, which completes the proof.
As a direct consequence of previous proposition, we can restrict the family of generalized probabilistic subgraphs of (N, p) ∈ P N to the set {(N, p ξ )} ξ⊂γ p , that we denote by 2 (N, p) . In particular, note that (N, p K N ) coincides with (N, p) and (N, p γ p ). Let us extend the standard notation for the deterministic case and denote by (N, p -ξ ) (or p -ξ when there is no ambiguity on the nodes set N) the probabilistic subgraph (N, p γ p \ξ ), ξ ⊆ γ p . In the special case ξ = {l}, we will simplify the notation using p -l instead of p -{l} . Moreover, for all C ⊂ N, we will note p C = p γ p | C . Proposition 3.2 For each (N, p) ∈ P N and ξ, ξ ′ ⊂ γ p such that ξ ⊂ ξ ′ , it holds that (N, p ξ ) ⊂(N, p ξ ′ ).

Proof : If γ ⊂ ξ, p ξ (γ) = δ⊂γ p \ξ p(γ ∪ δ) = α⊂ξ ′ \ξ η⊂γ p \ξ ′ p(γ ∪ α ∪ η) = α⊂ξ ′ \ξ p ξ ′ (γ ∪ α) = (p ξ ′ ) ξ (γ) (1)
where the fourth equality in (1) follows since γ p ξ ′ = ξ ′ .

Thus we can define an inclusion relation, noted ⊂, in 2 (N, p) , in the following way:

for ξ, ξ ′ ⊂ γ p , (N, p ξ ) ⊂(N, p ξ ′ ) if and only if ξ ⊂ ξ ′ .

The probabilistic position value

For each (N, v, p) ∈ G N , we can define the probabilistic link game (p, r v p ) as follows: p) , where (N, v p ξ ) is the probabilistic Myerson game associated to the probabilistic communication situation (N, v, p ξ ). The following proposition states that the probabilistic link game is component additive.

r v p (p ξ ) = v p ξ (N), p ξ ∈ 2 (N,
Proposition 4.1 Given (N, v, p) ∈ G N and p ξ ⊂p, we have

r v p (p ξ ) = C∈N/ξ r v p (p ξ | C ).
Proof : For each p ξ ⊂p, we have:

r v p (p ξ ) = γ⊂ξ p ξ (γ)r v ξ (γ) = γ⊂ξ p ξ (γ)v γ (N) = γ⊂ξ p ξ (γ) C∈N/ξ v γ | C (C) = C∈N/ξ γ⊂ξ p ξ (γ)v γ | C (C) = C∈N/ξ          η⊂ξ | C δ⊂(ξ\ξ | C ) p ξ (η ∪ δ)          v η (C) = C∈N/ξ η⊂ξ | C p ξ |C (η)v η (C) = C∈N/ξ r v p ξ |C (p ξ |C ) = C∈N/ξ r v p (p ξ | C ).
The next result states that the defined probabilistic link game can be identified with a convex linear combination of the associated link games to certain particular (deterministic) communication situations: the corresponding ones to the graphs in the support of the probabilistic graph (N, p). This identification is based on the fact that, for each (N, p) ∈ P N , the linear maps3 g p : G N → G p and g p :) → G γ p , respectively defined by g p (v) = r v p and g p (v) = r v p , induce, in a natural manner, a linear isomorphism between the respective image sets.

Proposition 4.2 For all (N, v, p) ∈ G N , it holds that:

r v p = γ∈S p p(γ) r v γ p | γ .
Proof : For each ξ ⊂ γ p , we have:

γ∈S p p(γ) r v γ p | γ (ξ) = γ⊂γ p p(γ)r v γ p (ξ ∩ γ) = δ⊂ξ          γ⊂γ p , ξ∩γ=δ p(γ)          r v γ p (δ) = δ⊂ξ p ξ (δ)r v γ p (δ) = δ⊂ξ p ξ (δ)r v ξ (δ) = r v p (ξ)
which completes the proof.

Next, using the introduced probabilistic link game, we can define, in a parallel way to the deterministic case, an alternative allocation rule for generalized probabilistic communication situations, that we will call the probabilistic position value.

Definition 4.1 Given (N, v, p) ∈ G N , the probabilistic position value for player i is defined by:

π i (N, v, p) = p l ∈L i (p) 1 2 S h p l (p, r v p ) L i (p) being the set of probabilistic links {p l ⊂p | l ∈ L i (γ p )}.
The following proposition states that the probabilistic position value can be obtained as a convex linear combination of the (deterministic) position values of the appropriated (deterministic) communication situations.

Proposition 4.3 For each (N, v, p) ∈ G N , it holds that π(N, v, p) = γ∈S p p(γ)π(N, v, γ).
Proof : As the aforementioned isomorphism permits to identify games r v p and r v p , we have, for each l ∈ γ p :

S h p l (p, r v p ) = S h l (γ p , r v p )
and thus, for each i ∈ N:

π i (N, v, p) = p l ∈L i (p) 1 2 S h p l (p, r v p ) = l∈L i (γ p ) 1 2 S h l (γ p , r v p ) = l∈L i (γ p ) 1 2 γ∈S p p(γ) S h l (γ p , r v γ p | γ ) = l∈L i (γ) 1 2 γ∈S p p(γ) S h l (γ, r v γ ) = γ∈S p p(γ) l∈L i (γ) 1 2 S h l (γ, r v γ ) = γ∈S p p(γ)π i (N, v, γ),
where the third equality follows by the linearity of the Shapley value and Proposition 4.2, and the fourth since, for all l ∈ γ p \ γ, l is a dummy player in the game (γ p , r v γ p | γ ), and, for each l ∈ γ, S h l (γ p , r v

γ p | γ ) = S h l (γ, r v γ ).

Characterizations of the probabilistic position value

In this section we present two characterizations of the probabilistic position value which extend the ones existing for the deterministic case. The deterministic position value π : C N → R n is characterized by Slikker (2005a) as the unique allocation rule on C N satisfying: i) component efficiency, i.e: for each C ∈ N/γ, i∈C π i (N v, γ) = v(C), and ii) balanced link contributions, i.e.: for i, j ∈ N,

l∈L j (γ) [π i (N, v, γ) -π i (N, v, γ \ {l})] = l∈L i (γ) [π j (N, v, γ) -π j (N, v, γ \ {l})].
The following definitions extend these two properties to the probabilistic case.

Definition 5.1 An allocation rule Ψ :

G N → R n satisfies component efficiency if, for each (N, v, p) ∈ G N and each C ∈ N/p, i∈C Ψ i (N, v, p) = v p (C) holds.
Definition 5.2 An allocation rule Ψ : G N → R n satisfies the balanced probabilistic link contributions property if, for each (N, v, p) ∈ G N and each i, j ∈ N,

p l ∈L i (p) π j (N, v, p) -π j (N, v, p -l ) = p l ∈L j (p) π i (N, v, p) -π i (N, v, p -l ) .
In the next three propositions, it is proved that the defined probabilistic position value is characterized by these two properties.

Proposition 5.1 The probabilistic position value satisfies component efficiency.

Proof : Let (N, v, p) ∈ G N and C ∈ N/p. Then:

i∈C π i (N, v, p) = i∈C p l ∈L i (p) 1 2 S h p l (p, r v p ) = i∈C l∈L i (γ p ) 1 2 S h l (γ p , r v p ). ( 2
)
The last term in (2) coincides with:

l∈γ p | C S h l (γ p | C , r v p C ) = r v p C (γ p | C ) = v p C (C) = v p (C).
Proposition 5.2 The probabilistic position value satisfies the balanced probabilistic link contributions property.

Proof :

Given (N, v, p) ∈ G N and i, j ∈ N, p l ∈L i (p) π j (N, v, p) -π j (N, v, p -l ) = l∈L i (γ p )          γ∈S p p(γ)π j (N, v, γ) - γ∈S p -l p -l (γ)π j (N, v, γ)          = l∈L i (γ p )          γ⊂γ p p(γ)π j (N, v, γ) - γ⊂γ p \{l} [p(γ ∪ {l}) + p(γ)]π j (N, v, γ)          = l∈L i (γ p )          δ⊂γ p \{l} p(δ ∪ {l})π j (N, v, δ ∪ {l}) - δ⊂γ p \{l} p(δ ∪ {l})π j (N, v, δ)          = l∈L i (γ p ) γ⊂γ p p(γ) π j (N, v, γ) -π j (N, v, γ \ {l}) = γ⊂γ p p(γ) l∈L i (γ p ) π j (N, v, γ) -π j (N, v, γ \ {l}) = γ⊂γ p p(γ) l∈L j (γ p ) π i (N, v, γ) -π i (N, v, γ \ {l})
where the last equality follows since the deterministic position value satisfies balanced link contributions. This last term coincides with

p l ∈L j (p) π i (N, v, p) -π i (N, v, p -l )
by the same arguments as for the previous equalities.

The proof of the following theorem mimics the corresponding one in Slikker (2005a) and then it is omitted.

Theorem 5.1 The probabilistic position value is the unique allocation rule on G N that satisfies component efficiency and balanced probabilistic link contributions. Slikker (2005b) provides another characterization of the deterministic position value using a natural extension of the potential function defined by [START_REF] Hart | Potential, value, and consistency[END_REF] to characterize the Shapley value.

Let P be a real function defined on C = n∈N C N . 4 The marginal contribution of player i to a communication situation can be defined as the sum of the marginal contributions of each of his incident links:

D i P(N, v, γ) = l∈L i (γ) [P(N, v, γ) -P(N, v, γ \ {l})]
for each (N, v, γ) ∈ C N ⊂ C and each i ∈ N.

A function P : C → R is a link potential function if P(N, v, ∅) = 0 for each (N, v, ∅) ∈ C
and the sum of the marginal contributions of players with respect to D equals the value produced by the connected components, i.e.

i∈N D i P(N, v, γ) = v γ (N)
for each (N, v, γ) ∈ C such that γ ∅.

Slikker (2005b) obtains the following result.

Theorem 5.2 There exists a unique link potential function P :

C → R. Moreover, for each (N, v, γ) ∈ C and each i ∈ N, D i P(N, v, γ) = π i (N, v, γ) holds.
Let us extend this result to the class of generalized probabilistic communication situations. Let G = n∈N G N and Q a real function defined on G . The expected marginal contribution of a player to a generalized probabilistic communication situation can be defined as the sum of the expected marginal contributions of each of his incident probabilistic links:

M i Q(N, v, p) = p l ∈L i (p) [Q(N, v, p) -Q(N, v, p -l )] (3)
for each (N, v, p) ∈ G such that γ p ∅ (p p ∅ ) and each i ∈ N.

A function Q : G → R is a probabilistic link potential function if Q(N, v, p ∅ ) = 0 for each (N, v, p ∅ ) ∈ G
, and the sum of the marginal contributions of players with respect to M equals the expected value produced by the grand coalition, i.e.

i∈N M i Q(N, v, p) = v p (N) (4)
for each (N, v, p) ∈ G such that p p ∅ .

In the following theorem, we show that the probabilistic link potential function is uniquely defined. Moreover, the expected marginal contributions that correspond to this potential coincide with the probabilistic position value.

Theorem 5.3 There exists a unique probabilistic link potential function

Q : G → R. Moreover, for each (N, v, p) ∈ G and each i ∈ N, M i Q(N, v, p) = π i (N, v, p) holds.
Proof : Firstly, we show that a probabilistic link potential function exists. Consider (N, v, p) ∈ G , and define:

Q(N, v, p) = ξ⊂γ p ∆ r v p (p ξ ) 2|ξ| (5)
Obviously, for each (N, v, p ∅ ) ∈ G we have Q(N, v, p ∅ ) = 0. Let us prove that Q, as defined in (5), satisfies (4) for each (N, v, p) ∈ G such that p p ∅ . It is easy to see that, in this case, for p ξ ⊂p -l , it holds that r v p -l (p ξ ) = r v p (p ξ ). Thus, for each p ξ ⊂p -l , ∆ r v p -l (p ξ ) = ∆ r v p (p ξ ). Then, from (3) and (5), and using the component efficiency property of the probabilistic position value, we obtain:

i∈N M i Q(N, v, p) = i∈N p l ∈L i (p)          ξ⊂γ p ∆ r v p (p ξ ) 2|ξ| - ξ⊂γ p \{l} ∆ r v p -l (p ξ ) 2|ξ|          = i∈N p l ∈L i (p) ξ⊂γ p ,l∈ξ ∆ r v p (p ξ ) 2|ξ| = i∈N 1 2 p l ∈L i (p) S h l (p, r v p ) = i∈N π i (N, v, p) = C∈N/γ p v p (C) = v p (N).
Therefore, Q(N, v, p), as defined in (5), is a probabilistic link potential function. Moreover, by the same arguments used to obtain the first four equalities, it holds that

M i Q(N, v, p) = π i (N, v, p) for each (N, v, p) ∈ G and each i ∈ N.
Secondly, we show that the probabilistic link potential is unique. The combination of (3) and (4) gives: , v, p) for each (N, v, p) ∈ G in a unique way.

Q(N, v, p) = 1 2|γ p |          v p (N) + 2 l∈γ p Q(N, v, p -l )          . Starting with Q(N, v, p ∅ ) = 0 for each (N, v, p ∅ ) ∈ G , one can recursively define Q(N
In next proposition we prove that the probabilistic link potential function admits an expression in terms of the deterministic link potential function.

Proposition 5.3 The unique probabilistic link potential function Q : G → R sat- isfies Q(N, v, p) = γ∈S p P(N, v, γ)p(γ)
, where P : C → R is the unique link potential function.

Proof : To prove this, let us show that Q ′ (N, v, p) = γ∈S p P(N, v, γ)p(γ) is a probabilistic link potential function. By the uniqueness of Q, the result holds. Note that Q ′ (N, v, p ∅ ) = 0 trivially holds for each (N, v, p ∅ ) ∈ G . If (N, v, p) is such that p p ∅ , we have:

i∈N M i Q ′ (N, v, p) = i∈N p l ∈L i (p) Q ′ (N, v, p) -Q ′ (N, v, p -l ) = i∈N p l ∈L i (p)          γ∈S p P(N, v, γ)p(γ) - γ∈S p -l P(N, v, γ)p -l (γ)          = i∈N p l ∈L i (p)          γ⊂γ p p(γ)P(N, v, γ) - γ⊂γ p \{l} [p(γ ∪ {l}) + p(γ)]P(N, v, γ)          = i∈N p l ∈L i (p)         
= γ⊂γ p p(γ)v γ (N) = v P (N),
where the seventh equality follows since P is the unique potential link function for the deterministic case.

6 On the relation between the potential and the balanced probabilistic link contributions property

In this section, we prove the equivalence between the two previous characterizations. Before providing the main result of this section, we introduce two more definitions and highlight a property of the probabilistic link potential function.

In a communication situation, players who are not in the same connected component are not able to coordinate their actions. Then, the connected components operate independently: there is no externalities between them. A function Q : G → R satisfies component additivity if it captures this idea.

Definition 6.1 A function Q : G → R is component additive if for each (N, v, p) ∈ G , Q(N, v, p) = T ∈N/γ p Q(T, v |T , p T ).
In the next proposition, we state that the probabilistic link potential satisfies this property.

Proposition 6.1 The probabilistic link potential is component additive.

Proof : Let us recall that, for a deterministic communication situation (N, v, γ) ∈ C , the dividends ∆ r v γ (ξ) are equal to zero if ξ is not connected (see, for example, [START_REF] Gómez | A unified approach to the Myerson value and the position value[END_REF]). Given a graph (N, γ) and S ⊆ N, let us note S (γ) = {i ∈ S such that ∃ l ∈ γ with i ∈ l} the set of not isolated players of S in the graph (N, γ). Thus, if P : C → R is the deterministic link potential function, we have:

P(N, v, γ) = ξ⊂γ ∆ r v γ (ξ) 2|ξ| = T ∈N/γ ξ⊂γ | T |N(ξ)/ξ|=1 ∆ r v γ (ξ) 2|ξ| = T ∈N/γ ξ⊂γ | T |N(ξ)/ξ|=1 ∆ r v |T γ | T (ξ) 2|ξ| = T ∈N/γ P(T, v | T , γ | T ).
Then P is component additive. For the probabilistic link potential function Q, we have,

Q(N, v, p) = γ∈S p p(γ)P(N, v, γ) = γ∈S p p(γ) T ∈N/γ P(T, v | T , γ | T ) = C∈N/γ p γ∈S p p(γ) T ∈N/γ | C P(T, v | T , γ | T ) = C∈N/γ p γ⊂γ p p(γ)P(C, v | C , γ | C ) = C∈N/γ p γ=δ∪ξ δ⊂γ | C ξ⊂γp\γ |C p(γ)P(C, v | C , γ | C ) = C∈N/γ p δ⊂γ | C p C (δ)P(C, v | C , γ | C ) = C∈N/γ p Q(C, v | C , γ | C ).
An allocation rule is component decomposable if the payoff of a player is not af-fected by the values created by players who do not belong to his component.

Definition 6.2 An allocation rule

Ψ : G → R is component decomposable if, for each (N, v, p) ∈ G and each i ∈ N, Ψ i (N, v, p) = Ψ i (T i , v |T i , p T i ),
where T i is the connected component of N/p to which i belongs.

In the following theorem we prove that the balanced link contributions property and the variations used to define the probabilistic link potential are equivalent. , v, p) being defined as in (3).

: G 0 → R such that Q(N, v, p ∅ ) = 0 for each (N, v, p ∅ ) ∈ G and M i Q(N, v, p) = Ψ i (N, v, p) for each (N, v, p) ∈ G with p p ∅ and each i ∈ N, M i Q(N
Proof : Suppose that a component additive function

Q : G → R satisfying Q(N, v, p ∅ ) = 0 for each (N, v, p ∅ ) ∈ G and M i Q(N, v, p) = Ψ i (N, v, p) for each (N, v, p) ∈ G with p p ∅ and each i ∈ N exists.
Let us prove that Ψ verifies balanced probabilistic link contributions. We have:

p l ∈L j (p) [Ψ i (N, v, p) -Ψ i (N, v, p -l )] = p l ∈L j (p) [M i Q(N, v, p) -M i Q(N, v, p -l )] = p l ∈L j (p)         p k ∈L i (p) [Q(N, v, p) -Q(N, v, p -k )] - p k ∈L i (p) [Q(N, v, p -l ) -Q(N, v, p -{k, l} )]         = p l ∈L j (p) p k ∈L i (p) [Q(N, v, p) -Q(N, v, p -k )] - p l ∈L j (p) p k ∈L i (p) [Q(N, v, p -l ) -Q(N, v, p -{k, l} )] = p k ∈L i (p) [Ψ j (N, v, p) -Ψ j (N, v, p -k )].
Note that the expression after the third equality sign is symmetric in i and j. The last equality follows by the same arguments as for the first three ones.

Conversely, let Ψ be a component decomposable allocation rule on G verifying balanced probabilistic link contributions and such that Ψ i (N, v, p ∅ ) = 0 for each (N, v, p ∅ ) ∈ G and each i ∈ N. We have to show that we can define a component additive function , v, p) for each (N, v, p) ∈ G with p p ∅ and each i ∈ N. We proceed by induction on |γ p |. Firstly, note that this assertion is trivially verified for |γ p | = 0. Then, by induction hypothesis, let us assume that there exists a component additive function

Q : G → R, such that Q(N, v, p ∅ ) = 0 for each (N, v, p ∅ ) ∈ G and M i (N, v, p) = Ψ i (N
Q such that Q(N, v, p ∅ ) = 0 for each (N, v, p ∅ ) ∈ G and M i Q(N, v, p) = Ψ i (N, v, p) for each (N, v, p) ∈ G with p p ∅ , |γ p | < k and each i ∈ N. Consider (N, v, p) ∈ G such that |γ p | = k. Remark that if T ∈ N/γ p , T = {i}, Q({i}, v |{i} , p {i} ) = 0. Consider now T ∈ N/γ p such that |T | > 1. Note that, for each i ∈ T , |L i (p)| 0 and L i (p T ) = L i (p). Let us define: Q(T, v |T , p T ) = Ψ i (T, v |T , p T ) + p l i ∈L i (p) Q(T, v |T , (p T ) -l i ) |L i (p)|
where i is whatever element in T . We relegate to the Appendix the proof that Q(T, v |T , p T ) is well defined in the sense that it does not depends on i ∈ T . Then, we define

Q(N, v, p) = T ∈N/p Q(T, v |T , p T ).
Consider i ∈ N. By the induction hypothesis, we have:

M i Q(N, v, p) = p l i ∈L i (p) Q(N, v, p) -Q(N, v, p -l i ) = p l i ∈L i (p)          T ∈N/p Q(T, v |T , p T ) - T ∈N/p -l i Q(T, v |T , (p T ) -l i )          .
For each T ∈ N/p such that i T , Q(T, v |T , p T ) = Q(T, v |T , (p T ) -l i ). Denote by T i the connected component of N/p containing i. It holds that:

M i Q(N, v, p) = p l i ∈L i (p) [Q(T i , v |T i , p T i ) - S ∈T i /(p T i ) -l i Q(S , v |S , (p S ) -l i )] = p l i ∈L i (p) Q(T i , v |T i , p T i ) -Q(T i , v |T i , (p T i ) -l i ) .
By (6) and by component decomposability of Ψ, we obtain: , v, p).

M i Q(N, v, p) = Ψ i (T i , v |T i , p T i ) = Ψ i (N
Remark 6.1 Note that the definition of Slikker's (2005b) deterministic player potential can be generalized to the probabilistic case in the following way. Denote Substituting this previous expression in the right-hand term of (6) we obtain: (9) Suppose that i and j have no probabilistic link in common. In this case, (9) can be written as:

1 |L i (p)|                  
Ψ i (T, v |T , (p T ) -l j ) = |L i (p)|Q(T, v |T , (p T ) -l j ) - p l i ∈L i (p)
Q(T, v |T , (p T ) -{l i , l j } ). (10) Note that ( 9) and (10) hold for an arbitrary l j . Using (10) we have that (8) is equal to:

Ψ j (T, v |T , p T ) |L j (p)| + 1 |L i (p)|                  
Q(T, v |T , (p T ) -{l i } )                         .
And after some straightforward calculations we obtain that previous expression, and then (8), equals to:

  δ⊂γ p \{l} p(δ ∪ {l})P(N, v, δ ∪ {l}) -δ⊂γ p \{l} p(δ ∪ {l})P(N, v, δ)          = i∈N p l ∈L i (p) γ⊂γ p p(γ) P(N, v, γ) -P(N, v, γ \ {l}) = γ⊂γ p p(γ) i∈N l∈L i (γ p ) P(N, v, γ) -P(N, v, γ \ {l})

  p l j ∈L j (p) Ψ i (T, v |T , (p T ) -l j ) |L j (p)| + |L i (p)|Ψ j (T, v |T , p T ) |L j (p)| -p l i ∈L i (p) Ψ j (T, v |T , (p T ) -l i ) |L j (p)| + p l i ∈L i (p) Q(T, v |T , (p T ) -l i) the induction hypothesis in the proof of Theorem 6.1., we know that:Ψ i (T, v |T , (p T ) -l j ) = |L i (p)\{p l j }|Q(T, v |T , (p T ) -l j )p l i ∈L i (p)\{p l j } Q(T, v |T , (p T ) -{l i , l j } ).

  p l j ∈L j (p) |L i (p)|Q(T, v |T , (p T ) -l j ) |L j (p)| -p l j ∈L j (p) p l i ∈L i (p) Q(T, v |T , (p T ) -{l i , l j } ) |L j (p)| -p l i ∈L i (p) |L j (p)|Q(T, v |T , (p T ) -l i ) |L j (p)| + p l i ∈L i (p) p l j ∈L j (p) Q(T, v |T , (p T ) -{l i , l j } ) |L j (p)| + p l i ∈L i (p) Q(T, v |T , (p T ) -l i )                   = Ψ j (T, v |T , p T ) + p l j ∈L j (p) Q(T, v |T , (p T ) -l j ) |L j (p)| .Finally, as the right-hand side in (6) coincides with (8) and thus with the secondΨ j (T, v |T , p T ) |L j (p)| + 1 |L i (p)|                         p l j ∈L j (p) l j {i, j} |L i (p)|Q(T, v |T , (p T ) -l j ) |L j (p)| -p l j ∈L j (p) l j {i, j} p l i ∈L i (p) Q(T, v |T , (p T ) -{l i , l j } ) |L j (p)| + (|L i (p)| -1)Q(T, v |T , (p T ) -{i, j} ) |L j (p)| -p l i ∈L i (p) l i {i, j} Q(T, v |T , (p T ) -{{i, j}, l i } ) |L j (p)| -p l i ∈L i (p) l i {i, j} |L j (p)|Q(T, v |T , (p T ) -l i ) |L j (p)| + p l i ∈L i (p) l i {i, j} p l j ∈L j (p) Q(T, v |T , (p T ) -{l i , l j } ) |L j (p)| -(|L j (p)| -1)Q(T, v |T , (p T ) -{i,j} ) |L j (p)| + p l j ∈L j (p) l j {i, j} Q(T, v |T , (p T ) -{l j , {i, j}} ) |L j (p)| + p l i ∈L i (p)

  Theorem 6.1 Let Ψ be a component decomposable allocation rule on G verifying Ψ i (N, v, p ∅ ) = 0 for each (N, v, p ∅ ) ∈ G and each i ∈ N. This allocation rule satisfies balanced probabilistic link contributions on G 0 if and only if there exists a component additive function Q

Observe that v γ (∅) = 0 because of the zero-normalization of (N, v).

For communication situations where (N, v) is not zero-normalized, the position value of a player is defined as the sum of his individual value and his position value in the communication situation(N, w, L), where (N, w) is the zero-normalization of (N, v), i.e. w(S ) = v(S ) -i∈S v({i}) for each S ⊂ N. Our results hold for this more general setting, but to keep the notation simple, we restrict ourselves to zero-normalized games.

In a natural way, we denote by G p and G γ p the vector linear spaces of games with players sets p and γ p respectively.

Recall that N = {1, . . . , n}.

⋆ This research has been partially supported by the Plan Nacional de I+D+i of the Spanish Government, under the project MTM2008-06778-C02-02/MTM. Authors would like to thank Sylvain Béal, Marc Fleurbaey, Guillaume Haeringer, Agnieszka Rusinowska and Philippe Solal for helpful comments and suggestions.

by Q : G → R the function that assigns to every generalized probabilistic communication situation in G a real number. The expected marginal contribution of a player to a generalized probabilistic communication situation can be defined as the expected marginal contribution of all his links:

for each (N, v, p) ∈ G such that p p ∅ . Results in Theorems 5.3 and 6.1 can be provided for the probabilistic Myerson value of [START_REF] Gómez | A value for generalized probabilistic communication situations[END_REF] and the probabilistic player potential.

Appendix

Suppose Ψ is a component decomposable allocation rule defined on G satisfying balanced probabilistic link contributions and such that 6) is well defined, i.e., it does not depend on i ∈ T .

Proof : Consider i, j ∈ T . By balanced probabilistic link contributions and component decomposability of Ψ,

term in previous equality, we have:

and so the result is proved for the case in which i, j ∈ T have no probabilistic link in common. Now, suppose that i and j have a probabilistic link in common. Thus, if l j {i, j}, (9) can be written as:

and if l j = {i, j},

(12) Then, substituting (11) and ( 12) in (8) we have that ( 8) is equal to:

Note that in previous expression the sum of the second to sixth terms inside the brackets vanishes and so, the result is proved.


B Vibert

J Lebouteiller

F Keita

C Rosenberger
email: christophe.rosenberger@ensicaen.fr

Biometric Sensor and Match-On-Card Evaluation platform

I. INTRODUCTION

Biometric systems are increasingly used to check or determine the identity of an individual. IT Industry is very interesting to this authentication solution in order to embed it in daily life products. The need of the evaluation of biometric sensors and Match-On-Card (MOC) algorithms is more more important to help them to choose the best system for a specific product. Among the different biometric systems, which one provides the best False Rejection Rate (FRR) given the False Acceptance Rate (FAR) set to a specific value ? How much time is needed to achieve a biometric verification on a smartcard ? Researchers are also interested to improve their MOC algorithm an also to compare it with the ones in the state of the art. These aspects become to be crucial for many applications like e-payment, physical access control... The purpose of this paper is to propose an evaluation platform on biometric sensors and MOC for testing their performance and security. This platform allows to perform tests given scenarios and benchmarks for comparing MOCs, and permit to test fake biometric data. We illustrate the usefulness of this platform on a commercial MOC, four commercial sensors and attacks on fingerprint (fake and dead fingers).

The paper is organized as follows. Section 2 is devoted to the state of the art on the evaluating platform. Section 3 describes the proposed platform and its different modules. In Section 4, we describe the sensor acquisition platform and attacks on sensor. In section 5, we illustrate results on a commercial sensor and MOC. We conclude and give some perspectives on this work in Section 6.

II. STATE OF THE ART

In this section, we first give some generalities of biometric systems. We also present the different evaluation methods of a biometric system: quality of biometric data, performance, security and usability. We describe the different existing benchmark databases that can be used for the evaluation task of biometric systems. Finally, we present the existing platforms for testing and characterizing MOC based biometric systems.

In a complete biometric system, we can evaluate each part to quantify its impact on the final result. As for example, a poor fingerprint quality can affect the performance of the system. The evaluation of a complete biometric system is based on several criteria.

The quality of the captured biometric data: In the literature, we find many elements that address the quality of fingerprints [START_REF] Abed | Evaluation de la qualit de donnes biomtriques[END_REF]. As for example, Alonso-Fernandez and al. [START_REF] Alonso-Fernandez | A comparative study of fingerprint image-quality estimation methods[END_REF] presented an overview of existing methods to quantify the quality of fingerprints. The authors show the impact of poor image quality on the overall performance of biometric systems. Other methods for measuring the quality of the fingerprints are given in [START_REF] Grother | Performance of biometric quality measures[END_REF], [START_REF] Lee | Model-based quality estimation of fingerprint images[END_REF]. These methods have proved effective in predicting the quality of fingerprint images. NFIQ metric proposed by NIST is now the reference for this task and is part of all industrial sensors SDK fingerprint [START_REF] Tabassi | A novel approach to fingerprint image quality[END_REF]. Another metric is used in research it is the metric Q [START_REF] Abed | Evaluation de la qualit de donnes biomtriques[END_REF] and this method has a better distribution of quality than NFIQ. The metric Q used multi-criteria to determine the quality of the data.

Performance: We intend here to measure the efficiency of a biometric system in terms of recognition errors in a given context of use. It is quantified by statistical measures (error rate, processing time, etc.). The measures proposed by the International Organization for Standardization ISO/IEC 19795-1 [START_REF]Information technology -biometric performance testing and reporting -part 1 : Principles and framework[END_REF] to evaluate and compare the performance of biometric systems are effective and comprehensive.

Security: With regard to security, a biometric MOC have vulnerabilities, Ratha and al. [START_REF] Ratha | Enhancing security and privacy in biometrics-based authentication systems[END_REF] have combined attacks of a generic biometric system in 8 classes (falsified biometric data, interception of biometric data during its transmission, attack on the extraction module parameters, altered extracted parameters, matching module replaced by a malicious software, alteration of the database, man in the middle attack between the database and the matching module and alteration of the verification decision). For each point, there are different types of attacks. Figure 1 illustrates the possible locations of the attacks in a generic biometric system. Fig. 1. Vulnerabilities locations of a biometric system (defined from [START_REF] Ratha | Enhancing security and privacy in biometrics-based authentication systems[END_REF])

Usability: This evaluation aspect is to analyze the user perception of the system and to quantify its satisfaction and acceptability. The work presented by El-Abed et al. [START_REF] El-Abed | A study of users' acceptance and satisfaction of biometric systems[END_REF], Jain and al. [START_REF] Jain | Biometrics : A grand challenge[END_REF], Kukula and Proctor [START_REF] Kukula | Human-biometric sensor interaction : Impact of training on biometric system and user performance[END_REF] and Kubula and al. [START_REF] Kukula | Effect of human-biometric sensor interaction on fingerprint matching performance, image quality and minutiae count[END_REF] show the importance of this evaluation in the design and comparison of biometric systems [START_REF] Kukula | Human-biometric sensor interaction : Impact of training on biometric system and user performance[END_REF]. An effective system in terms of performance but not acceptable, is not considered interesting (as in the case of DNA verification systems for physical access control).

Benchmark: To evaluate biometric systems, it is necessary to have a database containing biometric data. This database ensures that the systems are tested under the same conditions and allows for reproducible results to compare biometric MOC. Examples of biometric databases from research competitions are FVC2002 or FVC2004 [START_REF] Maio | Second Fingerprint Verification Competition International Conference on Pattern Recognition (ICPR)[END_REF], [START_REF] Maio | Third Fingerprint Verification Competition[END_REF]. Moreover, it is also interesting to perform tests of the same MOC when we use multiple databases acquired under different conditions (biometric sensor, population, environment, etc.). It is also necessary to define test scenarios (number of biometric data for enrollment, number of data for testing . . . ).

Platforms: With regard to platforms, there are quite a few in the literature. We can already cite the NIST platform [START_REF] Grother | MINEX II "Performance of Fingerprint Match-on-Card Algorithms[END_REF], which is used in their annual research competitions. It allows manufacturers to test their MOC or minutiae extractors, in terms of interoperability. In the NIST report, information on FAR (False Acceptance Rate) and FRR (False Rejected Rate) rates for every MOC and different extractors are disseminated. We can also mention the online FVC-Ongoing platform [3] dedicated to algorithms for fingerprint verification (evolution of the FCV competitions). The platform offers multiple databases grouped into two parts. The first one (Fingerprint Verification) quantifies both enrollment and verification modules, while the second one (ISO Fingerprint Matching) quantifies only the verification module on ISO Templates [START_REF]Information technology -biometric data interchange format -part 2 : Finger minutiae data[END_REF] based on minutiae. Performance metrics are: the failure to acquire rate (FTA) and the failure to enroll rate (FTE), the false non match rate (FNMR) for a defined false match rate and vice versa, the average enrollment and verification times, the maximum size required to store the biometric template on the SE, the distribution of legitimate and impostors users scores and the ROC curve with the associated equal error rate (EER). The main drawback of this platform is that it is necessary to submit the executable or source code of the MOC to the online platform which can cause confidentiality issues.

The evaluation of biometric verification algorithms is most of time used during the algorithm prototyping by researchers and relatively little by the industrial world. The evaluation methods and platforms fail to satisfy all of our needs. We intend to realize security analysis of a research or a commercial biometric MOC, to measure performance in terms of errors or verification average time.

III. MATCH ON CARD EVALUATING PLATFORM

The proposed platform architecture is presented in figure 2 and is composed of modules. The platform is made up of different modules allowing to make specific treatments, such as the interface to connect biometric databases. The central element is the Core, all other modules have no knowledge on others. This allows to modify a module without changing the overall operation of the platform. All modules are independent, we can change one and then see the effect of this change on the results. This will allow us, for example, to quantify the impact of a biometric database on the results or if an algorithm is better than another. The proposed platform uses active mechanisms of communication by event allowing multiple modules simultaneously access data exchanged between the client application and the MOC, thus offering the possibility of analyzing "on the fly" results.

Core: The Core is the main module that interfaces and manages all modules. It orchestrates the interaction with the different modules. It only knows the type of data as input of the MOC and the type of data returned by the MOC. As for example, to communicate with the Secure Element, the Core transparently manages the connection and communication with the MOC, it is realized by Personal Computer/Smart Card (PCSC) communication or Java Card OpenPlatform (JCOP) simulator with the software library developed through WSCT in [START_REF] Vernois | WSCT a software for smartcard transactions[END_REF].

Database Interface: The module Interface manages all biometric databases. The Core requests to the interface the next biometric data for processing and delegates to the interface the connection and management of all biometric databases. This allows to abstract the storage format of biometric data for example.

Scenario:

The module Scenario permits to create or use an evaluation scenario. It defines the biometric database to query, the number of biometric data to be used for enrollment or the number of users to consider. This allows us to make reproducible testing only by setting these elements. The module Performance quantifies the impact of these changes.

Performance: This module allows to evaluate the performance of the MOC with different metrics: FAR, FRR, EER, NIFQ value of each capture, ROC curve, enrollment and verification time. It also allows us to save the results in a database to compare several MOC based on the same test scenario.

Security: This module contains various attacks on the MOC. It is possible to use fuzzing approaches [START_REF] Lancia | Un framework de fuzzing pour cartes puce : application aux protocoles emv[END_REF] consisting in injecting fault data to the biometric MOC. It can be a biometric template respecting the ISO format but containing random biometric data (brute force attack). It is also possible to test the interoperability of the MOC by providing biometric templates ISO in which faults have been injected.

GUI Interface:

The proposed platform has a main graphical interface that allows to choose the test scenario and evaluation metrics. From the main interface, you have the option of using "plugins" that allow us to get information about one or more elements (eg minimum, average and maximum time for enrollment and verification). As mentioned earlier, the proposed platform uses active communication by event mechanisms, which provides access to information that you want in just developing a plugin that allows to visualize evaluation results as for example.

A. Evaluation metrics

As a first step, we use classical performance metrics commonly used in the literature and more specific ones:

• False Acceptance Rate (FAR): it measures how many times the biometric data of a user provides positive verifications with biometric data of another user.

• False Rejection Rate (FRR): it measures how many times the biometric data of a user gives a negative verification of biometric data with the same user,

• Success rate of attack: it measures the ratio of successful attacks (number of positive result over a number of transactions).

• Measuring interoperability: it quantifies the ratio of successful tests when providing an ISO template to the MOC.

• ROC curve: It describes the behavior of the biometric MOC for each value of the decision threshold (from which a test is positive). This implies that it is possible to obtain the comparison score from the MOC or to set decision threshold. For industrial MOCs, this is rarely the case but for research ones, this information is always available.

• Verification Time: we measure the time required to achieve a MOC enrollment or to obtain a verification result (after sending the ADPU (Application Data Protocol Unit defined in [19]) to the SE. It is also possible to generate several statistics on computation times such as histogram verification time, average, minimum or maximum time.

• Correlation between verification time and score : In general, a positive verification is slower than a negative one. This information can be exploited by an attacker as it can analyze the response time for the MOC to identify the extent to where the transmitted data is near the biometric reference stored on the SE (approach called Hill Climbing attack in the literature [START_REF] Martinez-Diaz | Hillclimbing and brute-force attacks on biometric systems: A case study in match-on-card fingerprint verification[END_REF]). In order to quantify if a MOC could be attacked by the Hill climbing attack, we measure the Pearson correlation factor between the verification time and score returned by the MOC (when known). A strong correlation highlights a flaw in the biometric MOC, as indication the template is similar to the reference if the time decrease.

IV. SENSOR ACQUISITION PLATFORM

We have built another platform which permits to acquire biometric templates and raw data and save it on databases. During an acquisition campaign, we have used several sensors and the proposed platform can detect which sensor the finger has been placed. This platforms offer to us the possibility to acquire fingerprint on several sensors during a campaign. Each acquisition generates fingerprint images and minutiae ISO templates. These data are saved with an identification number, the hand, the finger and user's profile information. The sensor acquisition platform is shown in figure 3, we can see the different sensors and the display screen which permits to indicate to the user the hand and finger to used for the capture.

With regard to security, a biometric sensor has vulnerabilities. Ratha and al. [START_REF] Ratha | Enhancing security and privacy in biometrics-based authentication systems[END_REF] have combined attacks of a generic biometric system in 8 classes (falsified biometric data, interception of biometric data during its transmission, attack on the extraction module parameters, altered extracted parameters, matching module replaced by a malicious software, alteration of the database, man in the middle attack between the database and the matching module and alteration of the verification decision). For each point, there are different types of attacks. Figure 1 illustrates the possible locations of the attacks in a generic biometric system. We have tested 2 attacks on sensor, one with fake fingerprint data taken on a Fake fingerprint: We have created a fake fingerprint database with real fingers and fingerprints (see figure 4) and we tested them on sensors face to real ones. To build this database, we have used wax and gelatin because these materials are not thick. With this test, we calculate the FTAR and if the sensors output a negative verification or if they are able to detect fake fingerprints or a spoofing attack (see figure 4).

V. ILLUSTRATIONS

To illustrate how the platform works, we have evaluated a commercial MOC on a fingerprint database (home made). We obtained the performance of the MOC, the operating point figure 5 with False Acceptance Rate (FAR) and False Rejection Rate (FRR) , the distribution of FAR FRR vs the Q metric (fingerprint quality metric developed in the GREYC research lab) on figure 7. We also have additional informations such as Q distribution on the database, time distribution figure 6. Industry wants to know this kind of information because it is important to choose the best MOC for their applications. We are able to compare MOC because we use the same testing scenario for each MOC. When we have a high FRR value, a good user is rejected frequently and it has a high impact on acceptability. If you have a high FAR, an impostor user is accepted by the system and it is a vulnerability. The time is another important information because it has an impact on user acceptance. Considering the evolution of the FRR value face to the quality of samples is important to consider especially during the enrollment to guarantee a good operational performance.

For fake fingerprints, we got a FTAR equal to 100% for sensors 1, 3 and 4. Consequently, for these sensors, the tested spoofing attacks are not working. For the sensor 2, the FTAR is 0% meaning that there is no problem to acquire the biometric data. 96 tests have been performed, 65% led to a negative verification and 35% to a positive one.

For the dead fingers, e have used the Q quality metric, because it has been shown in [START_REF] Abed | Evaluation de la qualit de donnes biomtriques[END_REF] that it provides a better quality assessment than NFIQ in Table 1. Metric We note for sensors 2, 3 and 4 that the dead fingers have a lower fingerprint quality (the higher value is Q, the better is the quality). For the sensor 1 (the only swipe sensor), it is not the case and the quality is largely worst than the three others. We think, this is due to its use, the operator for the dead fingers used correctly the sensor (see figure 8) contrary to real users. In this paper, we have presented the proposed platforms which permit the evaluation of biometric sensors (FTAR, quality...) and Match-On-Card algorithms. For the sensors evalution, we have illustrated two attacks on sensors: fake biometric data and fingerprint of dead people. For the dead fingerprint, we have generally observed a lower quality for the data. We have illustrated the possibility of the MOC evaluation platform on commercial MOC. We explained the importance of providing these results for industry in order to help tem to choose a MOC or for research to test if a new development is better than the previous.

In perspectives, we want to improve the attack module by providing new attack scenarios on MOC and to improve the Q metric for fingerprint quality assessment.

Fig. 2 .

 2 Fig. 2. general schema of the platform

Fig. 3 .

 3 Fig. 3. Sensor acquisition platform in working

Fig. 4 .

 4 Fig. 4. Example of fake fingerprint Dead fingerprint: We have created a fingerprint database with dead fingers. We went on a mortuary at the Caen Hospital and we acquired fingerprints of 4 dead people. The protocol is the following : • 3 sensors • 4 fingers (except the thumb) • 2 hands (left and right) • 6 captures per individual per finger per sensor We have 144 (6*2*4*3) data (fingerprint images and ISO template) per individual for all the sensors. For the 4 individuals, we have in total 576 (144*4) data. We have calculated the Failure To Acquire Rate (FTAR) and equals 36.11% (1 -(368/576).

Fig. 5 .

 5 Fig. 5. Operating point for Q metric

Fig. 8 .

 8 Fig. 8. Acquisition in Mortuary

TABLE I .

 I AVERAGE Q METRIC VALUE FOR FINGERPRINT COMING FROM A SENIOR DATABASE THE DEADE FINGERS ONE.