Mitigating Gender Bias in Face Recognition Using the von Mises-Fisher Mixture Model Jean-R emy Conti 1 2Nathan Noiry 1Vincent Despiegel2Stephane Gentric2Stephan Cl emenc on1

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Mitigating Gender Bias in Face Recognition

Using the von Mises-Fisher Mixture Model

Jean-R´

emy Conti *12 Nathan Noiry * 1 Vincent Despiegel 2St´

ephane Gentric 2St´

ephan Cl´

emenc¸on 1

Abstract

In spite of the high performance and reliability

of deep learning algorithms in a wide range of

everyday applications, many investigations tend

to show that a lot of models exhibit biases, dis-

criminating against speciﬁc subgroups of the pop-

ulation (e.g. gender, ethnicity). This urges the

practitioner to develop fair systems with a uni-

form/comparable performance across sensitive

groups. In this work, we investigate the gender

bias of deep Face Recognition networks. In or-

der to measure this bias, we introduce two new

metrics,

BFAR

and

BFRR

, that better reﬂect

the inherent deployment needs of Face Recog-

nition systems. Motivated by geometric consid-

erations, we mitigate gender bias through a new

post-processing methodology which transforms

the deep embeddings of a pre-trained model to

give more representation power to discriminated

subgroups. It consists in training a shallow neural

network by minimizing a Fair von Mises-Fisher

loss whose hyperparameters account for the intra-

class variance of each gender. Interestingly, we

empirically observe that these hyperparameters

are correlated with our fairness metrics. In fact,

extensive numerical experiments on a variety of

datasets show that a careful selection signiﬁcantly

reduces gender bias. The code used for the ex-

periments can be found at

https://github.

com/JRConti/EthicalModule_vMF.

1. Introduction

In the past few years, Face Recognition (FR) systems have

reached extremely high levels of performance, paving the

Equal contribution

LTCI, T

ecom Paris, Institut Poly-

technique de Paris

Idemia. Correspondence to: Jean-

emy Conti

jean-remy.conti@telecom-paris.fr

, Nathan Noiry

<nathan.noiry@gmail.com>.

Proceedings of the

39 th

International Conference on Machine

Learning, Baltimore, Maryland, USA, PMLR 162, 2022. Copy-

right 2022 by the author(s).

way to a broader range of applications, where the reliability

levels were previously prohibitive to consider automation.

This is mainly due to the adoption of deep learning tech-

niques in computer vision since the famous breakthrough

of (Krizhevsky et al.,2012). The increasing use of deep FR

systems has however raised concerns as any technological

ﬂaw could have a strong societal impact. Besides recent

punctual events

that received signiﬁcant media coverage,

the academic community has studied the bias of FR systems

for many years (dating back at least to (Phillips et al.,2003)

who investigated the racial bias of non-deep FR algorithms).

In (Abdurrahim et al.,2018) three sources of biases are

identiﬁed: race (understood as biological attributes such as

skin color), age and gender (available gender labels from FR

datasets are males and females). The National Institute of

Standards and Technology (Grother et al.,2019) conducted

a thorough analysis of the performance of several FR al-

gorithms depending on these attributes and revealed high

disparities. For instance, some of the top state-of-the-art

algorithms in absolute performance have more than seven

times more false acceptances for females than for males. In

this paper, we introduce a novel methodology to mitigate

gender bias for FR. Though focusing on a single source of

bias has obvious limitations regarding intersectional effects

(Buolamwini & Gebru,2018), it is a ﬁrst step to gain in-

sights into the mechanisms at work, before turning to more

complex situations. Actually, the method promoted in this

paper, much more general than the application considered

here, could possibly alleviate many other types of bias. This

will be the subject of a future work.

The topic corresponding to the study of different types of

bias and to the elaboration of methods to alleviate them is re-

ferred to as fairness in machine learning, which has received

increasing attention in recent years, see e.g. (Mehrabi et al.,

2019), (Caton & Haas,2020), (Du et al.,2020). Roughly

speaking, achieving fairness means learning a decision rule

that does not mistreat some predeﬁned subgroups, while

still exhibiting a good predictive performance on the overall

population: in general, a trade-off has to be found between

See for instance the study conducted by the American Civil

Liberties Union.

arXiv:2210.13664v3 [cs.CV] 22 Feb 2024

Mitigating Gender Bias in Face Recognition

fair treatment and pure accuracy

. In this regard, one needs

to carefully deﬁne what will be the relevant fairness met-

ric. From a theoretical viewpoint, several ones have been

introduced, see e.g. (Garg et al.,2020) or (Castelnovo et al.,

2021) among others, depending on how the concept of eq-

uity of treatment is understood. In practice, these very re-

ﬁned notions can be inadequate, as they ignore speciﬁc use

case issues, and one thus needs to adapt them carefully. This

is particularly the case in FR, where high security standards

cannot be negotiated. The goal of this article is twofold:

novel fairness metrics, relevant in FR applications in partic-

ular, are introduced at length and empirically shown to have

room for improvement by means of appropriate/ﬂexible

representation models.

Contribution 1. We propose two new metrics,

BFAR

and

BFRR

, that incorporate the needs for both security and

fairness (see section 2.2). More precisely, the

BFAR

(resp.

BFRR)

metric accounts for the disparity between false ac-

ceptance (resp. rejection) rates between subgroups of inter-

est, computed at an operating point such that each subgroup

has a false acceptance rate lower than a false acceptance

level of reference.

It turns out that state-of-the-art FR networks (e.g. ArcFace

(Deng et al.,2019a)) exhibit poor fairness performance w.r.t.

gender, both in terms of

BFAR

and

BFRR

. Different strate-

gies could be considered to alleviate this gender bias: pre-,

in- and post-processsing methods (Caton & Haas,2020),

depending on whether the practitioner “fairness” interven-

tion occurs before, during or after the training phase. The

ﬁrst one, pre-processing, is not well suited for FR purposes

as shown in (Albiero et al.,2020), while the second one,

in-processing, has the major drawback to require a full re-

training of a deep neural network. This encouraged us to

design a post-processing method so as to mitigate gender

bias of pre-trained FR models.

In order to improve

BFAR

and

BFRR

disparities, we cru-

cially rely on the geometric structure of the last layer of

state-of-the-art FR neural networks. The latter is a set of

embeddings lying on a hypersphere. Those embeddings

are obtained through two concurrent mechanisms at work

during the learning process: (i) repel images of different

identities and (ii) bring together images of a same identity.

Contribution 2. We set a von Mises-Fisher statistical mix-

ture model on the last layer representation, which corre-

sponds to a mixture of gaussian random variables condi-

tioned to live on the hypersphere. Based on the maximum

likelihood of this model, we introduce a new loss we call

Fair von Mises-Fisher, that we use to supervise the training

of a shallow neural network we call Ethical Module. Taking

This dichotomy somewhat simpliﬁes the problem since an

increase in accuracy could also lead to a better treatment of each

subgroup of the population.

the variance parameters as hyperparameters that depend on

the gender, this ﬂexible model is able to capture the two

previously mentioned mechanisms of repulsion / attraction,

which we show are at the origin of the biases in FR. Indeed,

our experiments remarkably exhibit a substantial correla-

tion between these hyperparameters and our fairness metrics

BFAR

and

BFRR

, suggesting a hidden regularity captured

by the model proposed. More precisely, we identify some

regions of hyperparameters’ values that (i) signiﬁcantly im-

prove

BFAR

while keeping a reasonable performance but

degrading

BFRR

,(ii) signiﬁcantly improve

BFRR

while

keeping a reasonable performance but degrading

BFAR

and (iii) improve both

BFAR

and

BFRR

at the cost of little

performance degradation. This third case actually achieves

state-of-the-art results in terms of post-processing methods

for gender bias mitigation in FR.

pre-trained model

(frozen)

von Mises-Fisher

Loss

shallow MLP

training set

sensitive

attribute

Ethical Module

MS1MV3

gender

ArcFace

size: (512, 1024, 512)

Fair

Figure 1: Illustration of the Ethical Module methodology.

In gray: our experiment choices.

Besides a simple architecture and a fast training (few hours),

the Ethical Module enjoys several beneﬁts we would like to

highlight.

Taking advantage of foundation models. In the recent

survey (Bommasani et al.,2021), the authors judiciously

point out a change of paradigm in deep learning: very ef-

ﬁcient pre-trained models with billions of parameters they

call foundation models are at our disposal such as BERT

(Devlin et al.,2018) in NLP or ArcFace (Deng et al.,2019a)

in FR. Many works rely on these powerful models and ﬁne

tune them, inheriting from both their strengths and weak-

nesses such as their biases. Hence the need to focus on

methods to improve the fairness of foundation models: our

method is in line with this approach.

No sensitive attribute used during deployment. Though

the Ethical Module requires access to the sensitive label

during its training phase, this label (e.g. gender) is not

needed anymore, once the training is completed. This is

compliant with the EU jurisdiction that forbids the use of

protected attributes for prediction rules.

Organization of the paper. Section 2.1 presents the widely

spread usage of FR and its main challenges. It is followed

Mitigating Gender Bias in Face Recognition

by section 2.2 where we discuss different fairness metrics

that arise in FR and introduce two new ones we think are

more relevant with regards to operational use cases. In sec-

tion 3, we present the von Mises-Fisher loss that is used for

the training of the Ethical Module and discuss its beneﬁts.

Finally, in section 4, we present at length our numerical

experiments, which partly consist in learning an Ethical

Module on the ArcFace model, pre-trained on the MS1MV3

dataset (Deng et al.,2019b). Our results show that, remark-

ably, some speciﬁc choices of hyperparameters provide high

performance and low fairness metrics both at the same time.

Related works. The correction of bias in FR has been

the subject of several recent papers. (Liu et al.,2019) and

(Wang & Deng,2020) use reinforcement learning to learn

fair decision rules but despite their mathematical relevance,

such methods are computationally prohibitive. Another line

of research followed by (Yin et al.,2019), (Wang et al.,

2019a) and (Huang et al.,2019) assumes that bias comes

from the unbalanced nature of FR datasets and builds on

imbalanced and transfer learning methods. Unfortunately,

these methods do dot completely remove bias and it has

been recently pointed out that balanced dataset are actually

not enough to mitigate bias, as illustrated by (Albiero et al.,

2020) for gender bias, (Gwilliam et al.,2021) for racial

bias and (Wang et al.,2019b) for gender bias in face detec-

tion. (Gong et al.,2019), (Alasadi et al.,2019) and (Dhar

et al.,2021) rely on adversarial methods that can reduce

bias but are also known to be unstable and computationally

expensive. All of the previously mentioned methods try to

learn fair representations. In contrast, some other works

do not affect the latent space but modify the decision rule

instead: (Terh

orst et al.,2020) act on the score function

whereas (Salvador et al.,2021) rely on calibration methods.

Despite encouraging results, these approaches do not solve

the source of the problem which is the bias incurred by the

embeddings used.

2. Fairness in Face Recognition

In this section, we ﬁrst brieﬂy recall the main principles of

deep Face Recognition and introduce some notations. The

interested reader may consult (Masi et al.,2018) or (Wang

& Deng,2018) for a detailed exposition. Then, we present

the fairness metrics we adopt and argue of their relevance

in our framework.

2.1. Overview of Face Recognition

Framework. A typical FR dataset consists of face im-

ages of individuals from which we wish to predict the

identities. Assuming that the images are of size

h×w

and that there are

identities among the images, this

can be modeled by i.i.d. realizations of a random variable

(X, y)∈Rh×w×c× {1, . . . , K}

, where

corresponds to

the color channel dimension. In the following, we denote

by Pthe corresponding probability law.

Objective. The usual goal of FR is to learn an encoder

function

fθ:Rh×w×c→Rd

that embeds the images in a

way to bring same identities closer together. The resulting

latent representation

Z:= fθ(X)

is the face embedding of

. Since the advent of deep learning, the encoder is a deep

Convolutional Neural Network (CNN) whose parameters

are learned on a huge FR dataset

(xi, yi)1≤i≤N

made of

i.i.d. realizations of the random variables

(X, y)

. There are

generally two FR use cases: identiﬁcation, which consists

in ﬁnding the speciﬁc identity of a probe face among several

previously enrolled identities, and veriﬁcation (which we

focus on throughout this paper), which aims at deciding

whether two face images correspond to the same identity

or not. To do so, the closeness between two embeddings

is usually quantiﬁed with the cosine similarity measure

s(zi,zj) := z⊺

izj/(||zi|| · ||zj||)

, where

|| · ||

stands for

the usual Euclidean norm (the Euclidean metric

||zi−zj||

is also used in some early works e.g. (Schroff et al.,2015)).

Therefore, an operating point

t∈[−1,1]

(threshold of ac-

ceptance) has to be chosen to classify a pair

(zi,zj)

genuine (same identity) if

s≥t

and impostor (distinct

identities) otherwise.

Training. For the training phase only, a fully-connected

layer is added on top of the deep embeddings so that the

output is a

-dimensional vector, predicting the identity of

each image within the training set. The full model (CNN +

fully-connected layer) is trained as an identity classiﬁcation

task. Until 2018, most of the popular FR loss functions were

of the form:

L=−1

i=1

log eκµ⊺

yizi

k=1 eκµ⊺

kzi!,(1)

where the

µk

’s are the fully-connected layer’s parameters,

κ > 0

is the inverse temperature of the softmax function

used in brackets and

is the batch size. Early works

(Taigman et al.,2014;Sun et al.,2014) took

κ= 1

and

used a bias term in the fully-connected layer but (Wang

et al.,2017) showed that the bias term degrades the per-

formance of the model. It was thus quickly discarded in

later works. Since the canonical similarity measure at the

test stage is the cosine similarity, the decision rule only

depends on the angle between two embeddings, whereas

it could depend on the norms of

µk

and

during train-

ing. This has led (Wang et al.,2017) and (Hasnat et al.,

2017) to add a normalization step during training and take

µk,zi∈Sd−1:= {z∈Rd:||z|| = 1}

as well as introduc-

ing the re-scaling parameter

in Eq. 1: these ideas signiﬁ-

cantly improved upon former models and are now widely

adopted. The hypersphere

Sd−1

to which the embeddings

belong is commonly called face hypersphere. Denoting by

θi

the angle between

µyi

and

, the major advance over

Mitigating Gender Bias in Face Recognition

the loss of Eq. 1(with normalization of

µk,zi

) in recent

years was to consider large-margin losses which replace

µ⊺

yizi= cos(θi)

by a function that reduces intra-class an-

gle variations, such as the

cos(mθi)

of (Liu et al.,2017) or

the

cos(θi)−m

of (Wang et al.,2018). The most efﬁcient

choice is

cos(θi+m)

and is due to (Deng et al.,2019a) who

called their model ArcFace, on which we build our method-

ology. A ﬁne training should result in the alignment of each

embedding

with the vector

µyi

. The aim is to bring to-

gether embeddings with the same identity. Indeed, during

the test phase, the learned algorithm will have to decide

whether two face images are related to the same, potentially

unseen, individual (one refers to an open set framework).

Evaluation metrics. Let

(X1, y1)

and

(X2, y2)

be two

independent random variables with law

. We distinguish

between the False Acceptance and False Rejection Rates,

respectively deﬁned by

FAR(t) := P(s(Z1, Z2)≥t|y1̸=y2)

FRR(t) := P(s(Z1, Z2)< t |y1=y2)

These quantities are crucial to evaluate a given algorithm

in our context: Face Recognition is intrinsically linked to

biometric applications, where the usual accuracy evaluation

metric is not sufﬁcient to assess the quality of a learned

decision rule. For instance, security automation in an airport

requires a very low FAR while keeping a reasonable FRR

to ensure a pleasant user experience. As a result, the most

widely used metric consists in ﬁrst ﬁxing a threshold

that the

FAR

is equal to a pre-deﬁned value

α∈[0,1]

and then computing the

FRR

at this threshold. We use the

canonical FR notation to denote the resulting quantity:

FRR@(FAR = α) := FRR(t)with FAR(t) = α.

The

FAR

level

determines the operational point of the FR

system and corresponds to the security risk one is ready to

take. According to the use case, it is typically set to

10−i

with i∈ {1,...,6}.

2.2. Incorporating Fairness

While the

FRR@FAR

metric is the standard choice for

measuring the performance of a FR algorithm, it does not

take into account its variability among different subgroups

of the population. In order to assess and correct for potential

discriminatory biases, the practitioner must rely on suitable

fairness metrics.

Framework. In order to incorporate fairness with respect

to a given discrete sensitive attribute that can take

A > 1

different values, we enrich our previous model and consider

a random variable

(X, y, a)

where

a∈ {0,1, . . . , A −1}

With a slight abuse of notations, we still denote by

the

corresponding probability law and, for every ﬁxed value

we can further deﬁne

FARa(t) := P(s(Z1, Z2)≥t|y1̸=y2, a1=a2=a)

FRRa(t) := P(s(Z1, Z2)< t |y1=y2, a1=a2=a).

In our case, we focus on gender bias so we take

A= 2

with

the convention that

a= 0

stands for male,

a= 1

for female.

Existing fairness metrics. Before specifying our choice for

the fairness metric used here, let us review some existing

ones (Mehrabi et al.,2019) that derive from fairness in the

context of binary classiﬁcation (in FR, one classiﬁes pairs

in two groups: genuines or impostors). The Demographic

Parity criterion requires the prediction to be independent

of the sensitive attribute, which amounts to equalizing the

likelihood of being genuine conditional to

a= 0

and

a= 1

Besides heavily depending on the number and quality of

impostors and genuines pairs among subgroups, this crite-

rion does not take into account the

FAR

s and

FRR

s, which

are instrumental in FR as previously mentioned. An at-

tempt to incorporate those criteria could be to compare

the intra-group performances:

FRR0@(FAR0=α)

v.s.

FRR1@(FAR1=α)

. However, the operational points

and

satisfying

FAR0(t0) = α

and

FAR1(t1) = α

generically differ as pointed out by (Krishnapriya et al.,

2020). To fairly assess the equity of an algorithm, one

needs to compare intra-groups

FAR

s and

FRR

s at the same

threshold. Two such criteria exist in the fairness literature:

the Equal Opportunity fairness criterion which requires

FRR0(t) = FRR1(t)

and the Equalized Odds criterion

which additionally requires

FAR0(t) = FAR1(t)

. Never-

theless, working at an arbitrary threshold

does not really

make sense since, as previously mentioned, FR systems

typically choose an operational point achieving a predeﬁned

FAR

level so as to limit security breaches. This is why most

current papers consider a ﬁxed operational point

such that

the global population False Acceptance Rate equals a ﬁxed

value α. For instance, (Dhar et al.,2021) computes

|FRR1(t)−FRR0(t)|with FAR(t) = α. (2)

However, we think that the choice of a threshold achieving

a global

FAR

is not entirely relevant for it depends on the

relative proportions of females and males of the considered

dataset together with the relative proportion of intra-group

impostors. For instance, at ﬁxed images quality, if females

represent a small proportion of the evaluation dataset, the

threshold

of Eq. 2is close to the male threshold

satis-

fying

FAR0(t0) = α

and away from the female threshold

satisfying

FAR1(t1) = α

. Such a variability among

datasets could lead to incorrect conclusions.

New fairness metrics. In this paper, we go one step further

and work at a threshold achieving

maxaFARa=α

instead

FAR = α

. This alleviates the previous proportion de-

pendence. Besides, this allows to monitor the risk one is

willing to take among each subgroup: for a pre-deﬁnite rate

Mitigating Gender Bias in Face Recognition

deemed acceptable, one typically would like to compare

the performance among subgroups for a threshold where

each subgroup satisﬁes

FARa≤α

. Our two resulting

metrics are thus:

BFRR(α) := maxa∈{0,1}FRRa(t)

mina∈{0,1}FRRa(t)(3)

and

BFAR(α) := maxa∈{0,1}FARa(t)

mina∈{0,1}FARa(t),(4)

where tis taken such that maxa∈{0,1}FARa(t) = α.

One can read the above acronyms “Bias in FRR/FAR”. In

addition to being more security demanding than previous

metrics,

BFRR

and

BFAR

are more amenable to interpreta-

tion: the ratios of

FRR

s or

FAR

s correspond to the number

of times the algorithm makes more mistakes on the discrim-

inated subgroup. Those metrics generalize well for more

than 2distinct values of the sensitive attribute.

3. Geometric Mitigation of Biases

Contrary to a common thinking about the origin of bias,

training a FR model on a balanced training set (i.e. with as

much female identities/images than male identities/images)

is not enough to mitigate gender bias in FR (Albiero et al.,

2020). It is therefore necessary to intervene by designing a

model to counteract the gender bias.

3.1. A Geometrical Embedding View on Fairness

In fact, impostor scores (cosine similarities of impostor

pairs) are higher for females than for males while genuine

scores are lower for females than for males (Grother et al.,

2019;Robinson et al.,2020). This puts females at a disad-

vantage compared to males in terms of both

FAR

and

FRR

Typically, this is due to (i) a smaller repulsion between

female identities and/or (ii) a greater intra-class variance

(spread of embeddings of each identity) for female identities,

as illustrated in Figure 2. Thus, we present in the follow-

ing a statistical model which enables to set the intra-class

variance for each identity on the face hypersphere.

3.2. von-Mises Fisher Mixture Model

In order to mitigate the gender bias of deep FR systems, we

set a statistical model on the latent representations of images.

Recall that we assumed that each individual of a FR dataset

is an i.i.d. realization of a random variable

(X, y, a)

, where

is the image,

the identity and

the gender attribute.

Also, recall that, both at the training and the testing stages,

the embeddings are normalized on the hypersphere, meaning

that

Z=fθ(X)∈Sd−1

. As previously mentioned, a ﬁne

learning should result in an alignment of the embeddings

{zi}

of a same identity

around their associated centroid

Figure 2: Illustration of the geometric nature of bias. Each

point is the embedding of an image. In green: two male

identities. In red: two female identities. The overlapping

region between two identities is higher for females than for

males. The grey circles are the acceptance zones, centered

around an embedding of reference, associated to a constant

threshold tof acceptance.

µyi∈Sd−1

. It is therefore reasonable to assume that the

embeddings of a same identity are i.i.d. realizations of

a radial distribution of gaussian-type on the hypersphere,

centered at

µyi

. A natural choice is thus to take the so-called

von-Mises Fisher (vMF) distribution which is nothing but

the law of a gaussian conditioned to live in the hypersphere.

Before turning to the formal deﬁnition of the statistical

model we put on the hypersphere, let us give the deﬁnition

of this vMF distribution.

The von Mises-Fisher distribution. The vMF distribution

in dimension

with mean direction

µ∈Sd−1

and concen-

tration parameter

κ > 0

is a probability measure deﬁned on

the hypersphere Sd−1by the following density:

Vd(z;µ, κ) := Cd(κ)eκµ⊺z,

with

Cd(κ) = κd

2−1/((2π)d

2Id

2−1(κ))

Iν

stands for the

modiﬁed Bessel function of the ﬁrst kind at order

, whose

logarithm can be computed with high precision (see supple-

mentary material A.1). The vMF distribution corresponds

to a gaussian distribution in dimension

with mean

and

covariance matrix

(1/κ)Id

, conditioned to live on

Sd−1

Figure 3 illustrates the inﬂuence of the concentration param-

eter κon the vMF distribution.

Mixture model. Since the vMF distribution seems to reﬂect

well the distribution of the embeddings of

identity around

their centroid, we extend the model to include all the

identities from the training set by considering a mixture

model where each component

(

1≤k≤K

) is equiproba-

ble and follows a vMF distribution

Vd(z;µk, κk)

.Figure 4

provides an illustration of the mixture model.

Maximum likelihood. Let

N≥1

and

(xi, yi, ai)1≤i≤N

be i.i.d. realizations of

(X, y, a)

. Under the previous vMF

mixture model assumption, the probability

pij

that a face

embedding zi=fθ(xi)belongs to identity jis given by

pij =Vd(zi|µj, κj)

k=1 Vd(zi|µk, κk)=Cd(κj)eκjµ⊺

jzi

k=1 Cd(κk)eκkµ⊺

kzi

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MitigatingGenderBiasinFaceRecognitionUsingthevonMises-FisherMixtureModelJean-R´emyConti*12NathanNoiry*1VincentDespiegel2St´ephaneGentric2St´ephanCl´emenc¸on1AbstractInspiteofthehighperformanceandreliabilityofdeeplearningalgorithmsinawiderangeofeverydayapplications,manyinvestigationstendtoshowthatalo...

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Mitigating Gender Bias in Face Recognition Using the von Mises-Fisher Mixture Model Jean-R emy Conti 1 2Nathan Noiry 1Vincent Despiegel2Stephane Gentric2Stephan Cl emenc on1

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