Tan-Sy Nguyen
email: tansy.nguyen@math.univ-paris13.fr

Marie Luong
email: marie.luong@univ-paris13.fr.

Mounir Kaaniche
email: mounir.kaaniche@univ-paris13.fr.

Long H Ngo
email: hailong.ngo@univ-paris13.fr.

Azeddine Beghdadi
email: azeddine.beghdadi@univ-paris13.fr.

T.-S Nguyen

A novel multi-branch wavelet neural network for sparse representation based object classification

Keywords: Object classification, sparse coding, wavelet transform, neural networks, multi-branch architecture

Recent advances in acquisition and display technologies have led to an enormous amount of visual data, which requires appropriate storage and management tools. One of the fundamental needs is the design of efficient image classification and recognition solutions. In this paper, we propose a wavelet neural network approach for sparse representation-based object classification. The proposed approach aims to exploit the advantages of sparse coding, multi-scale wavelet representation as well as neural networks. More precisely, a wavelet transform is firstly applied to the image datasets. The generated approximation and detail wavelet subbands are then fed into a multi-branch neural network architecture. This architecture produces multiple sparse codes that are efficiently combined during the classification stage. Extensive experiments, carried out on various types of standard object datasets, have shown the efficiency of the proposed method compared to the existing sparse coding and deep learning-based methods.

Introduction

Object classification is one of the most common problems in computer vision and pattern recognition fields. It plays a crucial role in applications such as face recognition, event detection [START_REF] Yang | Shared multi-view data representation for multi-domain event detection[END_REF], visual tracking [START_REF] Li | Visual tracking by dynamic matchingclassification network switching[END_REF], and more. Its main goal consists of finding a compact representation or set of relevant features that capture the intrinsic properties of the image contents. Object/image classification remains a challenging problem, especially in the case of large scale and complex datasets [START_REF] Khan | Residual networks based distortion classification and ranking for laparoscopic image quality assessment[END_REF]. In this respect, many efforts have been made in recent years to design new efficient classification methods. Among these methods, sparse representation-based classification (SRC) has attracted much attention due to its several advantages. Namely, SR ensures high robustness to noise and other kinds of degradation while producing a compact representation of the data through only a small number of meaningful features [START_REF] Elad | Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing[END_REF]. Recently, crucial attention has been paid to deep learning-based techniques due to the benefits of using neural networks for image analysis and processing [START_REF] Lee | Connectivity-based convolutional neural network for classifying point clouds[END_REF]. Motivated by the success of such powerful techniques, this work aims to combine sparse representation with neural networks to further enhance their classification performance.

In the following, we will provide an overview of the state-of-the-art classification techniques and then summarize the contributions of the current work.

Conventional classification techniques

The main idea behind most of the conventional (i.e. non-deep learning) classification techniques relies on the sparse representation tool [START_REF] Wright | Robust face recognition via sparse representation[END_REF][START_REF] Zhang | Image classification using spatial pyramid robust sparse coding[END_REF]. For instance, the original SRC method [START_REF] Wright | Robust face recognition via sparse representation[END_REF] aims to estimate the sparsest representation of a test sample using an over-complete dictionary composed of training samples.

The resulting sparse code is then used as a feature descriptor for classification purposes. The latter is often made according to the minimum reconstruction error criterion. Inspired by this original SRC method, different variants have also been developed. Indeed, a kernel sparse representation for image classification and face recognition has been proposed in [START_REF] Gao | Kernel sparse representation for image classification and face recognition[END_REF]. Moreover, to reduce the computational time of SRC, local and block-based SRC schemes have been designed in [START_REF] Li | Local sparse representation based classification[END_REF] and [START_REF] Wang | Block sparse representation for pattern classification: Theory, extensions and applications[END_REF], respectively. Generally, the performance of these methods is limited in the case of large image datasets due to the fact that the dictionary is formed by all the training samples of each class. To cope with this drawback, several methods based on compact dictionary learning have been developed. In this context, an interesting approach is discriminative dictionary learning (DDL), which allows the generation of a dictionary with small size from a selective dataset. This can be achieved using an objective function with reconstructive and discriminative terms. Moreover, a Fisher Discrimination Dictionary Learning (FDDL) method is proposed in [START_REF] Yang | Fisher discrimination dictionary learning for sparse representation[END_REF]. This method uses a Fisher discrimination criterion to learn a structured dictionary whose sub-dictionaries have specific class labels while producing discriminative sparse coding coefficients. In [START_REF] Jiang | Label consistent K-SVD: Learning a discriminative dictionary for recognition[END_REF], the Label Consistent K-SVD (LC-KSVD) method is proposed. This method consists of using only a single small dictionary for jointly learning a discriminative dictionary as well as a linear classifier. To this end, the authors explicitly incorporated a discriminative sparse-code (also called label consistency) constraint term and a classification error term into the objective function which is solved by applying the K-SVD algorithm [START_REF] Zhang | Discriminative K-SVD for dictionary learning in face recognition[END_REF]. While the above methods as well as most of the existing DDL methods rely on a single layer dictionary learning process, a multi-layer DDL approach has recently been proposed in [START_REF] Song | Multi-layer discriminative dictionary learning with locality constraint for image classification[END_REF]. However, dictionary learning methods are still challenging in the case of large datasets. To tackle this problem, a projection step is often performed to transform high dimensional data into a low dimensional space.

For this reason, a new approach for joint dimension reduction and dictionary learning has been developed in [START_REF] Li | A novel dimension reduction and dictionary learning framework for high-dimensional data classification[END_REF]. Another interesting solution will consist of applying a nonlinear projection using neural networks.

Neural networks-based classification techniques

Recent deep learning-based classification methods can be classified into three main categories: (i) spatial-based neural networks, (ii) wavelet-based neural networks, and (iii) sparse coding neural networks. Methods in the first category employ various Neural Network (NN) models to extract deep features for image/object classification. These models include VGG19 [START_REF] Simonyan | Very deep convolutional networks for largescale image recognition[END_REF], ResNet50 [START_REF] He | Deep residual learning for image recognition[END_REF],

Wide ResNet [START_REF] Zagoruyko | Wide residual networks[END_REF] and auto-encoder [START_REF] Gogna | Discriminative autoencoder for feature extraction: Application to character recognition[END_REF]. To further reduce the computational complexity and exploit other compact representations of the original images, the second category of developed methods have focused on the design of NN operating in the wavelet transform domain [START_REF] Said | Deep wavelet network for image classification[END_REF]. For instance, in [START_REF] Williams | Advanced image classification using wavelets and convolutional neural networks[END_REF], the input data is transformed into wavelet subbands that are fed into a Convolutional Neural Network (CNN). In [START_REF] Chen | Learning a wavelet-like autoencoder to accelerate deep neural networks[END_REF], the authors proposed a wavelet-like auto-encoder model that decomposes an input image into two channels corresponding to low and high frequency information. Then, the low frequency channel is fed into a standard classification network (VGG16) to extract deep features. Finally, these features are fused with those extracted from the high frequency channel using a lightweight network. However, by only considering two channels, the edge information is not efficiently taken into account during the classification stage. Moreover, the output scores of these two channels are combined using an average operation. In [START_REF] Li | Wavelet integrated CNNs for noise-robust image classification[END_REF], the authors propose to enhance CNNs by replacing max-pooling, strided-convolution, and average-pooling with Discrete Wavelet Transform. Similarly, a U-Net architecture is deployed in [START_REF] Liu | Multi-level wavelet convolutional neural networks[END_REF] with the intention of embedding wavelet decomposition into the CNN blocks to reduce the resolution of the feature maps. However, this method is mainly designed for image denoising applications. Finally, methods in the third category aim to combine neural networks with sparse representation/sparse coding models.

More precisely, in [START_REF] Zhang | Constructing deep sparse coding network for image classification[END_REF], a deep sparse coding network is developed to combine the advantages of CNN and sparse coding-based classification techniques. A sparse auto-encoder (AE)-based model using an ℓ 1/2 sparsity regularization as a constraint on the hidden representation is proposed in [START_REF] Feng | Sparse representation learning of data by autoencoders with ℓ 1/2 regularization[END_REF]. In [START_REF] Sun | Supervised deep sparse coding networks for image classification[END_REF], the conventional sparse coding scheme is extended to a multi-layer architecture yielding a deep sparse coding network. Another extension of SRC, referred to as Deep Sparse Representation Classification (DSRC), is developed in [START_REF] Abavisani | Deep sparse representation-based classification[END_REF]. It is based on a convolutional AE to learn sparse representation and find the sparse codes for classification. More precisely, an encoder is firstly used to extract embed-ding features which are fed into the sparse coding layer. Then, the sparse codes of the features are estimated by solving a sparse coding optimization problem.

Finally, the recovered embedding features are fed into the decoder for image reconstruction purposes. The obtained sparse codes are exploited during the classification stage based on the minimum reconstruction residual criterion as performed in the standard SRC approach.

Contributions

While previous deep learning-based classification methods aim to extract deep features from original images, or combine neural networks with sparse coding or embed wavelet transform into neural networks, the main goal of this work is to take advantage of neural networks, sparse coding as well as wavelets.

More precisely, we propose a hybrid approach using a neural network-based sparse representation technique that operates in the wavelet transform domain.

Note that a preliminary version of this work, inspired by the Deep Sparse Representation Classification (DSRC) model [START_REF] Abavisani | Deep sparse representation-based classification[END_REF], has recently been presented in [START_REF] Nguyen | Convolution autoencoder-based sparse representation wavelet for image classification[END_REF].

However, unlike our previous work using only the low frequency subband, we propose here a more general framework, exploiting both the approximation as well as the detail subbands. To this end, a novel multi-branch wavelet neural network (MB-WNN) architecture is designed in this paper. This architecture produces multiple sparse codes at different orientations and resolution levels.

During the classification stage, and in order to handle multiple sparse codes, we have extended the standard residual-based probabilistic approach [START_REF] Wei | A new sparse representation classifier (SRC) based on probability judgement rule[END_REF] which is developed for single sparse code.

The remainder of this paper is organized as follows. In Section 2, we recall some background on Sparse Representation Wavelet-based Classification (SRWC). The proposed multi-branch wavelet neural network architecture is then described in Section 3. The experimental results are shown in Section 4.

Finally, some conclusions and perspectives are drawn in Section 5.

Background on Sparse Representation Wavelet-based Classification

Before describing the proposed architecture, we first recall the main idea behind Sparse Representation Wavelet-based Classification (SRWC) method [START_REF] Ngo | Sparse representation wavelet based classification[END_REF].

In fact, unlike the classical SRC methods, SRWC operates in the wavelet transform domain and consists of using the approximation subband instead of the entire image.

More precisely, by considering k-classes with their labeled training samples, a wavelet decomposition is first applied to all the samples. Then, the generated approximation wavelet coefficients are processed using the Principal Component Analysis (PCA) technique [START_REF] Mooi | Mooi-Reci, Principal component analysis and factor analysis[END_REF]. The resulting vectors of the different samples are referred to as atoms. The sparse coding step consists of constructing subdictionaries D c tr derived from the training atoms d c i , where i ∈ {1, . . . , n c }, n c is the number of atoms in class c, c ∈ {1, . . . , k} and k is the total number of classes. Thus, the sub-dictionary D c tr for a class c is given by

D c tr = [d c 1 , d c 2 , ..., d c nc ] ∈ R m×nc , ( 1
)
where m is the dimension of each atom.

Then, for k classes with total number of atoms n, with n = ∑ k c=1 n c , a dictionary D tr is constructed from the above sub-dictionaries as follows:

D tr = [D 1 tr , D 2 tr , ..., D k tr ] ∈ R m×n . (2)
Based on the sparse representation model, any new test atom x st ∈ R m from a given class can be approximated by a linear combination of the atoms of the same class, yielding:

x st = s c 1 d c 1 + . . . + s c nc d c nc , (3)
where [s c 1 , s c 2 , ..., s c nc ] ⊺ ∈ R nc is the corresponding sparse coefficients vector associated with class c.

The class of a given test atom x st is deduced from the dictionary D tr by selecting a set of samples of the training atoms which can better approximate x st by the associated sparse code s whose non-zero entries correspond to the c th class.

Thus, the sparse representation vector can be expressed as

s = [0, ...0, s c 1 , ..., s c nc , ..., 0, 0] ⊺ ∈ R n . (4)
Given the dictionary D tr , the approximated sparse vector ŝ of x st is obtained by solving the Lasso optimization problem:

ŝ = argmin s ∥x st -D tr s∥ 2 2 + λ 0 ∥s∥ 1 , (5)
where λ 0 is a parameter that controls the balance between the reconstruction error and sparsity.

Finally, the non-zero coefficients in ŝ allow to identify the class of the unlabeled test atom. However, in practice, the non-zero coefficients may be associated with different classes due to the noise or the correlation that may exist within multiple classes. For this reason, the residual between the original test sample

x st and the estimated one is computed for the possible candidate classes, and the predicted class is chosen according to the minimum residual value:

class(x st ) = arg min c ∥x st -D tr δ c (ŝ)∥ 2 2 , (6)
where ŝ ∈ R n and δ c (ŝ) is the characteristic function that selects elements in ŝ that are only associated with class c.

Proposed multi-branch wavelet neural network architecture and classification scheme

Motivated by the advantages of wavelets in producing multi-scale representation with good space-frequency localization, a wavelet transform is applied to the image dataset. More precisely, the luminance component X is firstly extracted from each input color image. Then, the Haar transform is performed on the resulting images to generate four wavelet subbands: One approximation subband X (LL) and three detail subbands oriented horizontally X (LH) , vertically X (HL) and diagonally X (HH) . Thus, for notation concision, X

X (o) tr ∈ R m×ntr , X (o) v ∈ R m×nv ,
and

X (o)
st ∈ R m×nst , respectively.

Architecture description

Once the wavelet decomposition is performed and the approximation X (LL) as well as the horizontal X (LH) , vertical X (HL) and diagonal X (HH) detail subbands are obtained, the latter are fed into different neural networks. Thus, a multi-branch wavelet neural network architecture is obtained. It can be observed from Fig. 1 that each wavelet subband is processed through the following three main blocks:

• An encoder E (o) that generates the encoding features Y (o) in the latent space. Thus, from the input vector

X (o) inp = [X (o) tr , X (o)
ref ], we obtain:

Y (o) = [E (o) (X (o) tr ), E (o) (X (o) ref )] = [Y (o) tr , Y (o) ref ] (7)
where the index ref refers either to the validation or test data. For instance, when the MB-WNN architecture is used in the training (resp. test) phase, X

ref and

Y (o) ref terms correspond to X (o) v and Y (o) v (resp. X (o)
st and Y (o) st ).

• A sparse coding layer, parameterized by S (o) , which aims to find the sparse

encoding features Ŷ(o) from Y (o) . • A decoder D (o) that consists in reconstructing an output X(o) = D (o) ( Ŷ(o) ) close to the encoder input X (o) inp .
It is important to note that the sparse coding layer plays a crucial role in this architecture. For instance, the estimation of the sparse representation of the encoding features Y (o) is achieved by minimizing the approximation error under the sparsity constraint. This leads to the following Lasso optimization problem:

min S (o) ∑ o∈O (λ (o) 1 ∥Y (o) ref -Y (o) tr S (o) ∥ 2 F + λ (o) 2 ∥S (o) ∥ 1 ) , (8)
where

S (o) ∈ R ntr×n ref is the sparse coefficient matrix, λ (o)
1 are positive constants weighting the different approximation errors across the different subbands, λ tr . Thus, we have:

Ŷ(o) ref = Y (o) tr S (o) . (9)
Knowing that the sparse coder's output is given by Ŷ

(o) = [ Ŷ(o) tr , Ŷ(o) ref ] while Ŷ(o) tr = Y (o)
tr , problem (8) can be rewritten as min

S (o) ∑ o∈O (λ (o) 1 ∥Y (o) -Ŷ(o) ∥ 2 F + λ (o) 2 ∥S (o) ∥ 1 ) . (10)

Learning approach

Once the different blocks are described, we focus on the model learning approach. The proposed MB-WNN model will be trained using an appropriate loss function which aims to achieve the following two objectives for each branch of the architecture:

• Learning a meaningful representation of the input wavelet subbands X

L (o) AE = ∥X (o) inp -X(o) ∥ 2 F . (11)
• Estimating the sparse representation of the encoding features in the latent space by minimizing both the approximation error (i.e. first term in [START_REF] Wang | Block sparse representation for pattern classification: Theory, extensions and applications[END_REF])

and the sparsity penalty constraint (i.e. second term in [START_REF] Wang | Block sparse representation for pattern classification: Theory, extensions and applications[END_REF]). Thus, the loss related to the sparse coding layer, L

SC , is defined by

L (o) SC = λ (o) 1 ∥Y (o) -Ŷ(o) ∥ 2 F + λ (o) 2 ∥S (o) ∥ 1 . (12)
Since we are dealing with a multi-branch architecture, the global loss function is defined by considering the weighted sum of the two loss terms L

L t = ∑ o∈O (L (o) SC + λ (o) 3 L (o) AE ) = ∑ o∈O (λ (o) 1 ∥Y (o) -Ŷ(o) ∥ 2 F + λ (o) 2 ∥S (o) ∥ 1 + λ (o) 3 ∥X (o) inp -X(o) ∥ 2 F ), (13)
where λ have already been defined in ( 8)). Note that the choice of these weights will be discussed in the next section to better analyze the impact of the different subbands on the classification performance.

Finally, the loss function L t is minimized using a standard stochastic gradientbased optimization method to get the optimal parameters of the overall architecture.

Multiple sparse codes-based classification stage

Once the model is trained, the obtained sparse codes are used in the test phase to proceed with the classification stage. More precisely, the class labels of the test samples will be identified using a residual-based probabilistic rule [START_REF] Wei | A new sparse representation classifier (SRC) based on probability judgement rule[END_REF].

In the case of highly inter-correlated data, the probabilistic rule revealed good classification performance compared to the classical approach based on the truncated residual scheme where the samples from different classes may yield the same residual [START_REF] Wright | Robust face recognition via sparse representation[END_REF]. While the residual-based probabilistic approach has been developed for a single sparse code [START_REF] Nguyen | Convolution autoencoder-based sparse representation wavelet for image classification[END_REF][START_REF] Wei | A new sparse representation classifier (SRC) based on probability judgement rule[END_REF], we propose to generalize it to deal with a multiple sparse code matrix S (o) .

In this respect, for each class c ∈ {1, . . . , k} and orientation o ∈ O, the residual term r

(o) c
is firstly computed as follows:

r (o) c (x (o) st ) = ∥y (o) st -Y (o) tr δ c (s (o) )∥ 2 2 ∥δ c (s (o) )∥ 2 2 , (14)
where

x (o)
st is the subband at orientation o of the observed sample in the test data

X (o) st , y (o)
st is its embedding feature vector and s (o) is its corresponding sparse code column in the sparse code matrix S (o) .

Then, a probability value p

p (o) c = e -r (o) c ∑ k c=1 e -r (o) c , (15)
where k denotes the number of classes.

Finally, based on a linear combination of the probability values p

α (o) p (o) c ), (16)
where p c denotes the probability that the sample x st belongs to class c, and the upper branch of the MB-WNN shown in Fig. 1). While such a single branch model presents the advantage of reducing the complexity of the architecture, it may be less efficient as it does not take into account the detail wavelet coefficients (i.e. edge information) of the input images as addressed in the proposed MB-WNN-based framework.

α (o)

Experimental results

To evaluate the performance of the proposed methods, extensive experiments have been conducted. The first round of experiments is devoted to the analysis of the influence of the different parameters involved in the proposed MB-WNN architecture. The latter is then compared to recent state-of-the-art classification methods.

Experimental settings

As mentioned in the previous section, each branch of the proposed architecture relies on three main blocks referred to as encoder, sparse coding and 265 decoder. It is important to note here that the retained architecture, applied to each branch, is quite similar to that investigated recently in [START_REF] Abavisani | Deep sparse representation-based classification[END_REF]. However, few modifications related to the kernel sizes are made to reduce the amount of the involved parameters. For instance, a description of the employed architecture is provided in Table 1. To train our models, a two-stage approach is adopted in our experiments as investigated in [START_REF] Abavisani | Deep sparse representation-based classification[END_REF]. The first one corresponds to a pretraining stage where the model is trained without considering the sparse coding layer (i.e. like a traditional AE model). This step is achieved using 100 epochs.

Then, in the second stage, we continue the training of the overall model, while including the sparse coding layer, by using 900 epochs. The models are trained using the ADAM optimizer [START_REF] Kingma | A method for stochastic optimization[END_REF] with the learning rate 10 -3 while applying a decay of 0.9. These implementations were carried out using Pytorch 2.0 and NVIDIA Quadro RTX 6000 GPU. Note that the source code of the proposed approach is available on github 3 .

Experimental datasets

To validate the effectiveness of the proposed methods, our simulations have been carried out on different types of standard balanced datasets often used for object classification and recognition purposes. More precisely, we have used two digits datasets (USPS [START_REF] Hull | Database for handwritten text recognition research[END_REF] and SVHN [START_REF] Netzer | Reading digits in natural images with unsupervised feature learning[END_REF]), three face datasets (AR face [START_REF] Martinez | The AR face database[END_REF],

AR gender [START_REF] Martinez | The AR face database[END_REF] and UMDAA-01 [START_REF] Zhang | Domain adaptive sparse representation-based classification[END_REF]), and four natural object dataset (COIL-100 [START_REF] Nene | Columbia Object Image Library[END_REF], ETH-80 [START_REF] Leibe | Analyzing appearance and contour based methods for object categorization[END_REF], ARID [START_REF] Loghmani | Recognizing objects in-the-wild: Where do we stand?[END_REF], Tiny ImageNet [START_REF] Le | Tiny imagenet visual recognition challenge[END_REF]). Samples of some datasets are shown in Fig. 2.

(a) USPS [START_REF] Hull | Database for handwritten text recognition research[END_REF] (b) SVHN [START_REF] Netzer | Reading digits in natural images with unsupervised feature learning[END_REF] (c) AR face [START_REF] Martinez | The AR face database[END_REF] (d) UMDAA-01 [START_REF] Zhang | Domain adaptive sparse representation-based classification[END_REF] (e) COIL-100 [START_REF] Nene | Columbia Object Image Library[END_REF] (f) ARID [START_REF] Loghmani | Recognizing objects in-the-wild: Where do we stand?[END_REF] Figure 2: Samples of some employed datasets.

Note that 80%, 10% and 10% of each dataset are the proportions of samples which have been randomly selected for training, validation and test phases, respectively.

Comparison methods and performance evaluation

The proposed single and multi-branch wavelet neural network architectures for sparse representation classification (SRC) will be designated SB-WNN-SRC and MB-WNN-SRC, respectively. It should be noted here that our methods are tested using three resolution levels of the Haar transform while selecting the wavelet subbands obtained at the third level. Our approaches will be compared to different state-of-the-art methods. They include the following conventional sparse representation-based methods:

• SRC [START_REF] Wright | Robust face recognition via sparse representation[END_REF]: It represents the original sparse representation-based classification method.

• LC-KSVD [START_REF] Jiang | Label consistent K-SVD: Learning a discriminative dictionary for recognition[END_REF]: Unlike the previous method where the dictionary is formed by all training samples of each class, this method aims to learn a dictionary of small size from a selective dataset. The learning is achieved by using an objective function with label consistency constraint. The optimization problem is solved by applying the K-SVD algorithm [START_REF] Zhang | Discriminative K-SVD for dictionary learning in face recognition[END_REF].

• FDDL [START_REF] Yang | Fisher discrimination dictionary learning for sparse representation[END_REF]: It consists of learning a structured dictionary whose subdictionaries have specific class labels. In this respect, a Fisher discrimination criterion is included in the objective function to produce sparse codes.

• SRWC [START_REF] Ngo | Sparse representation wavelet based classification[END_REF]: It is an enhanced version of the original SRC method which is performed in the wavelet transform domain as described in Section 2.

• SCCRC [START_REF] Li | Multiplication fusion of sparse and collaborative-competitive representation for image classification[END_REF]: It is a recent improved version of SRC which consists in combining sparse and collaborative representations.

Moreover, we have considered the following deep learning-based methods:

• Wide ResNet-SRC [START_REF] Zagoruyko | Wide residual networks[END_REF]: It corresponds to sparse representation-based classification method using deep features extracted from a Wide ResNet architecture applied to the original dataset images. Note that other standard deep architectures have also been tested as discussed later.

• WAE-VGG16-SRC [START_REF] Chen | Learning a wavelet-like autoencoder to accelerate deep neural networks[END_REF]: It uses a wavelet-like auto-encoder which firstly consists of applying an encoder to decompose the image in two channels containing low and high frequency information. Then, the low frequency information is fed into a standard network (VGG16) to extract its features. Finally, these features are fused with those extracted from the high frequency information.

• Kernel-SARL [START_REF] Sadeghi | On the global optima of kernelized adversarial representation learning[END_REF]: It is based on a kernel formulation of spectral adversarial representation learning framework.

• DSRC [START_REF] Abavisani | Deep sparse representation-based classification[END_REF]: It consists of using an encoder to extract embedding features which are fed into the sparse coding layer. Then, the sparse codes of the features are estimated and used for classification.

It is worth noting that, in addition to Wide ResNet, other existing architectures, developed for deep feature extraction, have also been tested like VGG19 [START_REF] Simonyan | Very deep convolutional networks for largescale image recognition[END_REF],

ResNet50 [START_REF] He | Deep residual learning for image recognition[END_REF] and AE [START_REF] Gogna | Discriminative autoencoder for feature extraction: Application to character recognition[END_REF] as it will be shown in Fig. 3. It should also be noted that DSRC [START_REF] Abavisani | Deep sparse representation-based classification[END_REF] is the recent state-of-the-art method which is more related to the current work.

The comparison is performed using different criteria. First, the standard classification accuracy criterion is used. Then, the different methods are evaluated in terms of robustness against the proportion of employed training samples. Finally, the size of different neural networks (i.e. the number of parameters) will be provided.

Results and discussion

Before evaluating the different methods on the test datasets, we should first determine the optimal values of the parameters introduced by the proposed MB-WNN-SRC approach. Thus, an analysis of the influence of these parameters on the accuracy performance is conducted.

1 = ηλ (o) 2 , (o)
∀o ∈ {LH, HL, HH}, λ

(o) p = 1 ω λ (LL) p , p ∈ {1, 2, 3}, (18)
where η and ω are positive constants.

While η will range from a small value (set to 0.25) to a high value (set to 10), the tested ω values are chosen by considering three cases:

(i) In the first case, we set ω to 3, which means that the three detail subbands as well as the approximation one have the same level of importance during training, and so they contribute in equal manner to the learned model.

(ii) In the second case, we set ω to 4, which means that less importance is given to the three detail subbands compared to the approximation one.

(iii) In the third case, we set ω to 2, which means that more importance is given to the three detail subbands compared to the approximation one.

Once the training is achieved, the probability-based classification rule, defined in [START_REF] Simonyan | Very deep convolutional networks for largescale image recognition[END_REF], is applied. In this respect, let us define the following weight vector:

α = (α (LL) , α (LH) , α (HL) , α (HH) ) ⊺ . (19)
The classification accuracy results, obtained with different η values, are provided in Tables 2, 3 and 4 for cases (i), (ii) and (iii), respectively. values on the classification accuracy (%), while setting λ

(LL) 2 = 1, λ (LL) 3
= 10, ω = 3 (i.e. case (i)), α = (0.5, 0.2, 0.2, 0.1) ⊺ and using

Eqs. ( 17)- [START_REF] Zagoruyko | Wide residual networks[END_REF]. values on the classification accuracy (%), while setting λ

Datasets η = 10 η = 8 η = 4 η = 2 η = 1 η = 0.5 η = 0.
(LL) 2 = 1, λ (LL) 3
= 10, ω = 4 (i.e. case (ii)), α = (0.5, 0.2, 0.2, 0.

⊺ and using

Eqs. ( 17)- [START_REF] Zagoruyko | Wide residual networks[END_REF]. values on the classification accuracy (%), while setting λ

Datasets η = 10 η = 8 η = 4 η = 2 η = 1 η = 0.5 η = 0.
(LL) 2 = 1, λ (LL) 3
= 10, ω = 2 (i.e. case (iii)), α = (0.5, 0.2, 0.2, 0.1) ⊺ and using

Eqs. ( 17)- [START_REF] Zagoruyko | Wide residual networks[END_REF]. 2). It is important to note here that setting ω to 2 (Table 4) leads to similar performance to the case where ω is equal to 3 (Table 2). However, a drop in classification performance occurs where ω was set to 4 (Table 3) (i.e. when less importance is given to the three detail subbands compared to the approximation one). This confirms the interest in exploiting both the approximation and detail subbands in the multi-branch architecture. Based on this study, the next simulations will be performed using ω = 3 and η = 8.

Datasets η = 10 η = 8 η = 4 η = 2 η = 1 η = 0.5 η = 0.
Moreover, regarding the multiple sparse codes-based classification stage, we have also studied the impact of the choice of the weighting terms α (o) associated to the probabilities p (o) as shown in [START_REF] Simonyan | Very deep convolutional networks for largescale image recognition[END_REF]. To this end, three cases are considered.

The first case is α = (0.5, 0.2, 0.2, 0.1)

⊺ which means that the same importance is given to the approximation subband (α (LL) = 0.5) and the three detail subbands (α (HL) +α (LH) +α (HH) = 0.5). The second case is α = (0.6, 0.15, 0.15, 0.1)

⊺ which means that more importance is given to the approximation subband. The third case is α = (0.4, 0.2, 0.2, 0.

⊺ which means that more importance is given to the three detail subbands.

Table 5 provides the accuracy results for the above considered three weight Table 5: Influence of the choice of the weight vector α = (α (LL) , α (LH) , α (HL) , α (HH) )

⊺ on the classification accuracy (%) of MB-WNN-SRC approach, while setting η = 8 and ω = 3. vectors. Thus, similarly to the previous analysis, giving the same level of importance to the approximation and the three detail subbands yields the highest accuracy values. For this reason, the weight vector α will be set to Finally, it should be noted that the performance of the proposed MB-WNN-SRC approach has also been evaluated by considering different neural network structures. This has been achieved by modifying the number of convolution and pooling layers as well as the kernel sizes in each branch. While the kernel size slightly affects the accuracy performance, it has been observed that the number of layers has more impact on the results. For instance, it can be seen from Table 6 that a significant improvement can be achieved by increasing the number of layers up to three. However, adding more layers yields similar performance.

Datasets

For these reasons, we have retained the neural network structure described in Table 1.

Comparison with the state-of-the-art classification methods

Once the proposed MB-WNN-SRC method is analyzed and the optimal parameter values are found (i.e. η = 8, ω = 3 and α = (0.5, 0.2, 0.2, 0.1) ⊺ ), we now focus on the comparison of its performance with the aforementioned state-ofthe-art classification methods. The accuracy results of the different methods are reported in Table 7 where the two best values are highlighted in bold. It should be noted that, in addition to the deep learning based-methods 400 kernel-SARL [START_REF] Sadeghi | On the global optima of kernelized adversarial representation learning[END_REF] and DSRC [START_REF] Abavisani | Deep sparse representation-based classification[END_REF], we only reported in Table 7 the results of Wide ResNet-SRC [START_REF] Zagoruyko | Wide residual networks[END_REF] and WAE-VGG16-SRC [START_REF] Chen | Learning a wavelet-like autoencoder to accelerate deep neural networks[END_REF]. The latter have been selected since they have shown better performance than the remaining neural networksbased feature extraction techniques as illustrated in Fig. 3. Thus, it can be firstly seen from Table 7 that the particular SB-WNN-SRC method outperforms achieved thanks to the extended multi-branch architecture (MB-WNN-SRC).

For instance, the multi-branch architecture achieves higher gain compared to the single branch variant, especially for the SVHN [START_REF] Netzer | Reading digits in natural images with unsupervised feature learning[END_REF], COIL-100 [START_REF] Nene | Columbia Object Image Library[END_REF], ETH-80 [START_REF] Leibe | Analyzing appearance and contour based methods for object categorization[END_REF], ARID [START_REF] Loghmani | Recognizing objects in-the-wild: Where do we stand?[END_REF] and Tiny ImageNet [START_REF] Le | Tiny imagenet visual recognition challenge[END_REF] datasets. This can be explained by the fact that the multi-branch architecture combines various sparse codes produced from both approximation and detail wavelet subbands of the image dataset.

Moreover, and since neural networks are well known to be very efficient when they are trained on large scale datasets, we propose to evaluate the performance of different deep learning-based methods with respect to the number of labeled training samples. For this reason, for each employed dataset, four subsets are created by randomly selecting 20%, 40%, 60% and 80% of the samples of the whole dataset. Then, after separating the training/validation/testing samples in each subset, we tested the different classification methods. Fig. 3 shows the performance of these methods for the different employed datasets while varying their sizes. As previously stated, the plots confirm the high efficiency of the recent DSRC [START_REF] Abavisani | Deep sparse representation-based classification[END_REF] method as well as the WAE-VGG16-SRC [START_REF] Chen | Learning a wavelet-like autoencoder to accelerate deep neural networks[END_REF] one compared to the previous deep learning-based methods (i.e VGG19-SRC [START_REF] Simonyan | Very deep convolutional networks for largescale image recognition[END_REF], ResNet50-SRC [START_REF] He | Deep residual learning for image recognition[END_REF], Wide ResNet-SRC [START_REF] Zagoruyko | Wide residual networks[END_REF] and AE-SRC [START_REF] Gogna | Discriminative autoencoder for feature extraction: Application to character recognition[END_REF]) aiming at extracting deep features from original images. Regarding the effect of the dataset size, it can be observed that the aforementioned neural networks-based methods fail and result in a significant drop of performance when the number of samples is relatively small. However, the recent methods WAE-VGG16-SRC [START_REF] Chen | Learning a wavelet-like autoencoder to accelerate deep neural networks[END_REF] and DSRC [START_REF] Abavisani | Deep sparse representation-based classification[END_REF] as well as the proposed method are less sensitive to the training image dataset size. Most importantly, our MB-WNN-SRC method outperforms the state-ofthe-art approaches and appears more robust to the size of the employed training dataset.

Finally, the different deep learning-based methods have been compared in terms of number of the network parameters and execution time as shown in Table 8 and Table 9, respectively. Indeed, one can see that the proposed multi-branch architecture has more parameters compared to the DSRC and SB-WNN models, since the latter are based on a single branch architecture. However, the number of parameters in our MB-WNN architecture are much smaller than that of the other neural networks (VGG19, ResNet50, etc) based methods. It should be noted here that the reduced number of parameters explains the good behavior of the proposed methods, whatever the size of the training dataset. Another advantage of designing an efficient model with a small size is it allows to achieve a gain in storage memory.

Similarly to the complexity of the different methods in terms of number of parameters, the execution times obtained with ARID image dataset (of size 256× 256) show that our proposed multi-branch architecture requires an additional time of about 0.4 seconds compared to the single branch and DSRC approaches.

However, it is much faster than the remaining neural networks-based methods.

Conclusion and perspective

An object classification method, exploiting simultaneously the advantages of neural networks as well as sparse coding techniques and multi-scale wavelet decompositions, is proposed. More precisely, a set of auto-encoders combined with sparse coding layers are applied to different wavelet subbands yielding a new multi-branch neural network architecture. Unlike existing sparse representation classification methods, the proposed architecture presents two main duce various sparse codes resulting in better discrimination ability. However, the main limitation of the proposed approach is the complexity of the architecture, since the number of involved branches is directly related to the retained wavelet subbands. Overall, the simulations carried out on various types of standard datasets have shown the benefits that can be drawn from the proposed architecture. Despite its good performance, the proposed architecture could be improved with further investigation. It is important to recall that the training of our architecture aims only to find the optimal sparse codes that are then exploited in the test phase for classification purposes. Therefore, a more efficient architecture could be designed by integrating the classification stage and resorting to an end-to-end learning approach. Moreover, as the proposed architecture operates in the wavelet transform domain, it would be interesting to investigate other decompositions like the recent neural network-based multiscale transforms [START_REF] Li | Complex contourlet-CNN for polarimetric SAR image classification[END_REF][START_REF] Dardouri | Dynamic neural network for lossy to lossless image coding[END_REF].

  (o) denotes a given wavelet subband at orientation o ∈ O, where O = {LL, HL, LH, HH} is the set of the generated directional wavelet subbands. By repeating the same decomposition process on the approximation subband, a multi-resolution representation of the input image can be obtained. Before describing the proposed architecture, let us introduce the following notations. For a given dataset, n, n tr , n v , and n st represent the respective numbers of all, training, validation and testing samples. Thus, following the wavelet decomposition stage, the resulting subbands, with orientation o ∈ O, of the training, validation and testing samples are designated by

Figure 1 :

 1 Figure 1: Proposed multi-branch wavelet neural network (MB-WNN) architecture.

2

 2 are positive regularization parameters that control the sparsity penalty and the fidelity between the input and output of the sparse coder, and ∥⋅∥ F is the Frobenius norm.According to[START_REF] Gao | Kernel sparse representation for image classification and face recognition[END_REF], the estimated sparse encoding features Ŷ(o) ref can be seen as the output of a Fully Connected Network (FCN) whose input layer is the encoded feature vector Y (o)

  (o) inp by a nonlinear reduction method, using the encoder block of the AE, instead of the linear PCA as performed in SRWC [31]. The decoder block of the AE ensures that the encoded features allow to recover a reconstructed subband X(o) close to X (o) inp . Hence, the loss related to the reconstruction error of the AE, L (o) AE , is given by

  associated to each branch. Therefore, the retained loss function of our MB-WNN architecture becomes:

(o) 3 are

 3 positive parameters used to weight the reconstruction errors resulting from the auto-encoders applied to the different wavelet subbands (while λ

  subbands at orientation o ∈ O, the class of the test sample x st is identified as follows class (x st ) = arg max c

  is a positive constant (with ∑ o∈O α (o) = 1) representing the weight associated to each probability value p (o) c . In other words, α (o) reflects the importance of subband o in the classification stage. Particular case: single branch architecture A particular case of the proposed MB-WNN architecture consists in considering only one wavelet subband, yielding a single branch architecture. Since wavelet transforms allow to concentrate the main information of a given input image in the low frequency subband, the single branch architecture could be easily deduced by selecting the branch dealing with the approximation subband (i.e.

4. 4 . 1 . 3 to 10 .

 41310 Influence of the parameters introduced by the proposed MB-WNN-SRC approach With the proposed multi-branch architecture, and according to the loss function defined in[START_REF] Zhang | Discriminative K-SVD for dictionary learning in face recognition[END_REF], the different parameters λ (o) p with p ∈ {1, 2, 3} and o ∈ O should be appropriately selected since their choice may impact the classification performance. On the one hand, for a given subband o, our analysis aims to find the relationship between the different values λ to weight respectively the error due to the sparse coding layer, the sparsity of the estimated codes and the reconstruction error. On the other hand, for a given p value, the contributions of the different wavelet subbands o should be established. Therefore, based on the previous study of the single branch architecture [29], we first propose to set λ Then, different tests are performed to find the best relation between λ , LH, HL, HH}, λ

Wavelet transform FCN Sparse Coding CNN Decoder CNN Encoder FCN Sparse Coding CNN Decoder CNN Encoder FCN Sparse Coding CNN Decoder CNN Encoder FCN Sparse Coding CNN Decoder CNN Encoder Reconstructed LL subband Reconstructed LH subband Reconstructed LL subband Reconstructed LL subband Wavelet LL subband Wavelet LH subband Wavelet HL subband Wavelet HH subband

Table 1 :

 1 Description of the neural network model used in each branch of the overall architecture.

			Feature maps
		Layer			Kernel size # Param
			Number	Size
		Conv2D-1	10	48×48	3×3	208
		Max-pool-1	10	24×24	∅	0
		Conv2D-2	20	24×24	3×3	1168
	Encoder
		Max-pool-2	20	12×12	∅	0
		Conv2D-3	30	12×12	1×1	4640
		Max-pool-3	30	6×6	∅	0
	Sparse	FCN	30	6×6	∅	540225
	Coding
		Conv2D-4	30	6×6	1×1	9248
		Upsampling-1	30	12×12	∅	0
		Conv2D-5	20	12×12	3×3	4640
	Decoder	Upsampling-2	20	24×24	∅	0
		Conv2D-6	10	24×24	3×3	1168
		Upsampling-3	10	48×48	∅	0
		Conv2D-7	1	48×48	3×3	208
	Param	Total parameters: 561,474

Table 2 :

 2 Influence of the relationship between λ

	(o)
	p

Table 3 :

 3 Influence of the relationship between λ

	(o)
	p

Table 4 :

 4 Influence of the relationship between λ

	25

Table 6 :

 6 Influence of the number of layers in the proposed MB-WNN architecture on the classification accuracy.

		Number of convolution and pooling layers
		2	3	4	5
	SVHN [35]	68.2640 69.3525 69.3525	69.3540
	UMDAA-01 [37]	93.2550 95.4489 95.4489	95.4450
	COIL-100 [38]	89.3685 93.2050 93.2050	93.2050
	ARID [40]	80.5500 81.3680 81.3680	81.3680
	Tiny ImageNet [41] 82.3250 84.7750 84.7750	84.5030

Table 7 :

 7 Classification accuracy (%) of the proposed approach as well as the state-of-the-art methods.

										Tiny
	Methods/ Datasets USPS [34]	SVHN	AR face	AR gender	UMDAA-01	COIL-100	ETH-80	ARID	ImageNet
			[35]	[36]	[36]	[37]	[38]	[39]	[40]
										[41]
	SRC [6]	0.8778	0.1571	0.9761	0.9300	0.7900	0.9116	0.9177	0.6980	0.7115
	FDDL [11]	0.9134	0.2254	0.9616	0.9400	0.8122	0.8822	0.9320	0.7115	0.7330
	LC-KSVD [12]	0.8745	0.3531	0.9770	0.8680	0.8482	0.9142	0.9425	0.7320	0.7750
	SRWC [31]	0.9545	0.2821	0.9839	0.9420	0.8529	0.9229	0.9315	0.7535	0.7660
	SCCRC [42]	0.9465	0.6850	0.9363	0.9580	0.9450	0.8910	0.9420	0.7050	0.7840
	Wide ResNet-	0.9523	0.5050	0.9833	0.9580	0.8850	0.9221	0.9267	0.7082	0.7840
	SRC [18]
	WAE-VGG16-	0.9625	0.6850	0.9750	0.9650	0.9320	0.9100	0.9720	0.6520	0.7530
	SRC [22]
	Kernel-	0.9680	0.6820	0.9810	0.9708	0.9482	0.9235	0.9775 0.8220	0.8335
	SARL [43]
	DSRC [28]	0.9625	0.6775	0.9812	0.9648	0.9339	0.9112	0.9573	0.7890	0.8078
	SB-WNN-	0.9682	0.6824 0.9837	0.9654	0.9510	0.9235	0.9625	0.7930	0.8120
	SRC
	MB-WNN-	0.9718	0.6935 0.9837	0.9705	0.9545	0.9320	0.9877 0.8137	0.8478
	SRC

Table 8 :

 8 Evaluation of different deep neural networks in terms of number of parameters.

				Wide	WAE-
		VGG19-	ResNet50-			Kernel-	DSRC	SB-WNN-	MB-WNN-
				ResNet-	VGG16-
		SRC [16]	SRC [17]			SARL [43]	[28]	SRC	SRC
				SRC [18]	SRC [22]
	#param	138M	25.6M	8.8M	57.4M	2.83M	24.5K	12.7K	2.25M

Table 9 :

 9 Execution times (in seconds) for different deep learning-based classification methods.

				Wide	WAE-
		VGG19-	ResNet50-			Kernel-	DSRC	SB-WNN-	MB-WNN-
	Method			ResNet-	VGG16-
		SRC [16]	SRC [17]			SARL [43]	[28]	SRC	SRC
				SRC [18]	SRC [22]
	Time	4.91	2.36	1.36	2.87	1.02	0.19	0.15	0.57

https://github.com/tansyab1/MB-WNN-SRC

Funding

This work has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 722068.

the existing ones for most of the employed datasets. Further improvements are VGG19-SRC [START_REF] Simonyan | Very deep convolutional networks for largescale image recognition[END_REF] ResNet50-SRC [START_REF] He | Deep residual learning for image recognition[END_REF] Wide ResNet-SRC [START_REF] Zagoruyko | Wide residual networks[END_REF] AE-SRC [START_REF] Gogna | Discriminative autoencoder for feature extraction: Application to character recognition[END_REF] WAE-VGG16 [START_REF] Chen | Learning a wavelet-like autoencoder to accelerate deep neural networks[END_REF] DSRC [START_REF] Abavisani | Deep sparse representation-based classification[END_REF] MB-WNN-SRC (a) USPS [START_REF] Hull | Database for handwritten text recognition research[END_REF] VGG19-SRC [START_REF] Simonyan | Very deep convolutional networks for largescale image recognition[END_REF] ResNet50-SRC [START_REF] He | Deep residual learning for image recognition[END_REF] Wide ResNet-SRC [START_REF] Zagoruyko | Wide residual networks[END_REF] AE-SRC [START_REF] Gogna | Discriminative autoencoder for feature extraction: Application to character recognition[END_REF] WAE-VGG16 [START_REF] Chen | Learning a wavelet-like autoencoder to accelerate deep neural networks[END_REF] DSRC [START_REF] Abavisani | Deep sparse representation-based classification[END_REF] MB-WNN-SRC (b) SVHN [START_REF] Netzer | Reading digits in natural images with unsupervised feature learning[END_REF] advantages: First, it exploits both low and high frequency information located in the approximation and detail wavelet subbands. Secondly, it allows to pro-


PhD Marie-Françoise Calmette

Marie-Françoise Mrs

PhD Calmette

Isabelle Péchoux

Are environmental policies counterproductive?

Keywords: environmental policies, agglomeration of activities environmental policies, concentration of activities JEL classification: F12, J61, R38

Are environmental policies counterproductive?

Introduction

A large literature has studied the interactions between trade and environment. In this framework, a lot of papers have analyzed the effects of freer trade on environment with local [START_REF] Barrett | Strategic environmental policy and international trade[END_REF][START_REF] Antweiler | Is free trade good for the environment?[END_REF] or transboundary [START_REF] Copeland | Trade and transboundary pollution[END_REF] pollution. In contrast, in our paper, we focus on the interaction between environmental policy and agglomeration of environmentally sensitive industries. Many papers have discussed the spatial dimension of environmental policies. Most of them consider an oligopoly market producing a homogeneous good and analyze the problem of plant relocation decisions in response to environmental restrictions [START_REF] Markusen | Competition in regional environmental policies when plant locations are endogenous[END_REF][START_REF] Hoel | Environmental policy with endogenous plant locations[END_REF][START_REF] Petrakis | Location decisions of a polluting firm and the time consistency of environmental policy[END_REF]. Other papers study the trade off between the negative effects of deterioration in the environmental quality and positive agglomeration externalities in a spatial equilibrium of cities [START_REF] Papageorgiou | Externalities, indivisibility, nonreplicability, and agglomeration[END_REF][START_REF] Tabuchi | Urban agglomeration and dispersion: a synthesis of Alonso and Krugman[END_REF]. However, there are still two related issues where there is a lack of theoretical research in the spatial dimension of environmental problems. One is the differentiated-goods models and the other is the effect of environmental policies on the agglomeration of activities. Most of the time, the pollution problems are aggravated by agglomeration. Traffic-jams, hurtful emissions by clustered polluting industries, or pollution of soil and rivers by concentrated agricultural activities are some examples. However empirical evidence shows that such agglomerations exist. This paper discusses the impact of environmental policies on the agglomeration of people and activities. We argue that environmental policies may be subject to a kind of moral hazard problem: because each firm (or agent) is less polluting when a regulator enforces an emission tax or quota, the global disutility of pollution decreases with environmental policies and this strengthens the incentive for people and activities to agglomerate. The final result may be more pollution because of the scale effect, even if each firm is cleaner. [START_REF] Copeland | Trade, spatial separation, and the environment[END_REF], have also shown, in a model where pollution policy is ruled out, that, under some conditions, feedback effects can occur between trade and environment. What is new in our results is that the environmental degradation is brought on by the domestic environmental policy itself. We adapt the analytical framework from the economic geography model that has highlighted the conditions under which geographic concentration of activities occurs (Krugman, 1991a(Krugman, , 1991b)). In that basic model without congestion costs, the market size effect always leads to agglomeration of industries in the larger region unless the transport cost is prohibitive. In this paper we assume that firms are polluting and we introduce a disutility of pollution in the agents' welfare. We show that this disutility undermines the incentive to agglomerate in the large region. Finally, we introduce an environmental regulation which decreases at first the global pollution. It follows that the agents' welfare, net of the disutility of pollution, is now higher and that the larger region becomes, once again, more attractive.

The model

At first, we follow Krugman's model closely. In two regions, i=1,2, we consider two categories of consumers, mobile workers and immobile farmers. The regions differ only in one respect: the number L i of workers. The overall number of mobile workers is µ , and L1=f µ and L2 = (1-f)µ, with 0≤ f ≤1.

In each region there are (1-µ)/ 2 farmers. Farmers and workers have identical preferences. Their utility (equivalent in this particular model to their real wage) is described by:

µ µ - = 1 a m C C U (1)
where C a is consumption of the homogeneous agricultural good (used as numeraire) and C m is the [START_REF] Dixit | Monopolistic competition and optimum product diversity[END_REF], subutility function:

1 1 ] [ 1 - - ∑ = = σ σ σ σ n k k m C C (2)
with σ >1, the elasticity of substitution among the n manufactured goods (and the perceived price elasticity of demand). The amount of labor l k required to produce a quantity

x k of good k (k=1…n) is k k x l β α + = (3)
where α is fixed cost and β constant marginal cost. Increasing returns to scale imply that each firm in each region produces a single product. The firms are Chamberlinian monopolistic competitors and, in equilibrium, set a price P i that is uniform within each region:

i i W P β σ σ 1 - = (i=1,2) (4)
with Wi the wage rate of workers in region i.

Free entry and exit conditions imply

) 1 ( - = * σ β α x (5)
where x * , common to every active firm, is the equilibrium quantity produced by each firm, whatever the region. The full employment condition determines the number of firms in each region:

ασ i i L n = 2 , 1 = ∀i . ( 6
)
Krugman assumes that the agricultural good can be freely transported from one region to the other. Transportation costs for manufactured goods take Samuelson's iceberg form: 1 0 ≤ ≤r , an inverse index of transportation cost, is the fraction of one unit of manufactured good that, shipped from one region, reaches the other. Demand for each industrial good is obtained by maximizing consumers' utility (given by ( 1) and ( 2)), subject to their budget constraint.

Equating supply (5) and demand for each good in both regions, together with the full employment condition (6) determines the equilibrium prices and wages for a given allocation of workers, L i

1 2 1 2 1 2 2 1 2 1 2 1 1 1 1 2 1 2 1 ) 1 ( - -         + - + +         + - + = - σ σ µ µ µ σ β α r W W n n L W W r W n n L W P (7) 1 1 2 1 2 2 2 1 1 2 1 2 1 1 2 2 1 2 1 ) 1 ( - -         + - + +         + - + = - σ σ µ µ µ σ β α W r W n n L W r W W n n L W P . ( 8
)
3

Using the results of maximizing agents' utility and relation (1), the workers' real wages in each region are

      - - -       - - -                           + +         + - = 1 ) 1 ( 1 2 1 1 2 1 2 2 1 1 1 2 2 1 1 1 ) 1 ( 1 1 1 1 ) ) 1 (( 1 σ µσ σ σ σ σ σ σ µ µ µ µ r W W n n r W P n r W W n n W P n W U (9)       - - -       - - -                           + +         + - = 1 ) 1 ( 1 2 1 1 2 2 2 2 1 1 1 2 2 1 2 1 1 ) 1 ( 2 2 1 1 ) ) 1 (( σ µσ σ σ σ σ σ σ µ µ µ µ r W W n n W P n r W W n n r W P n W U .(10)
Assuming that workers move, from the low to the high real wage region, Krugman shows that when transportation cost falls below some critical level (r*), the larger region has the higher real wage because of the larger size of the local market: the result is a tendency to concentration of all manufacturing in this region. We assume now that firms are polluting. Pollution, purely local, is a by-product of the manufactured good production process. As assumed most of the time in the environmental literature, we consider that the relation between emission (e) and production (x) by each firm is linear and given by e= θx, where θ>0 represents the firms' environmental performance (a lower θ means less polluting emissions). Assume that workers are harmed by the global pollution (n i e) and take into account this disutility when they decide to migrate or not. They move now from the low to the high welfare region: in region "i" the workers' welfare net of the global disutility of pollution is:

S i = U i -D(n i θx) (11)
where U i is the real wage in region i (given by ( 1)) and D(n i θx) measures the global negative impact of pollution on health or utility. Following most of the models in the related literature, we assume damages are separable. As a result, in our model (as in [START_REF] Chichilnisky | North-South trade and the global environment[END_REF]Copeland and[START_REF] Copeland | Trade and transboundary pollution[END_REF] there are no substitution effects between environment and marketed goods. Smith and Espinoza, 1996 have considered in a computable general equilibrium model, the jointness of marketed and environmental goods. They show that the non-separability assumption has an impact on the evaluation on trade policy and demonstrate that feedback effects may produce counterintuitive results. Because we focus on the linkage between agglomeration and environmental policy, we assume here that pollution affects only the level of utility: this simple framework allows to isolate the centripetal and centrifugal effects which are central arguments in this kind of model. Global disutility is increasing in the global emission, and thus in the number of firms and the output per firm. In the remainder of the article, it will be useful to work with a linear function D(n i θx)≡ δ n i θx with 0<δ <1.

Clearly, the workers' welfare has now two components working in opposite ways: if region 1 is the larger region (f > 0.5), on the one hand, as in the basic model, it is more attractive to workers than the other region because the real wage is higher, U1> U2 (the market size effect) but, on the other hand, it is less attractive because the disutility of pollution increases in the region where agglomeration occurs and plays the role of a centrifugal force. The result of the tension between centripetal and centrifugal forces depends on the relative values of the parameters. For given values of α, β, µ and σ, the critical level of the transportation cost (leading to a concentrated sustainable equilibrium because S1 >S2) decreases when the marginal damage δ increases (r*'(δ)>0). It follows that a concentration of all activities in a single region is now less likely to occur.

Obviously, for given values of µ, σ, and r, there exists a limit value of δ such that workers are so deterred by pollution that agglomeration of firms never occurs.

The effect of environmental policy on agglomeration equilibrium

It is well known that polluting agents need to be induced to internalize the social cost of pollution damage, otherwise they would engage in excessive levels of polluting emission. We assume that a national environmental regulator enforces in both regions a quota of emission. It follows that firms must undertake a costly abatement effort in order to reduce their emission level, by reducing θ. . This effort results in an increase in marginal cost β( θ) with β'( θ) <0. This is the only way to introduce a pollution control in this kind of partial equilibrium and particular economic geography framework (in CGE analyses, like in Smith and Espinoza, 1996, a more complex pollution control can be incorporated in order to show how emission control will both affect and be affected by final good consumption).

Let us underline that we focus entirely on the costs imposed by the regulator: we are neither concerned, here, by the choice among policy instruments, nor by the optimal quota level (for a survey, see [START_REF] Cropper | Environmental economics : A survey[END_REF]. We simply assume that the firms' behavior is Paretoefficient: given the announced quota, the polluting agent undertakes an effort increasing its costs such that the marginal benefit from reduced pollution equals the marginal abatement cost. What is the impact of such an environmental policy on the agents' welfare, then on their location decision? We have to consider the two components of S i . First, let us consider the centripetal force. It depends on the difference between the real wages (U 1- U 2 ). Several results are apparent from equations ( 4), ( 5) and ( 6). Faced with a higher marginal cost, firms lower their production level (5) and raise their price (4).

The number of firms in each region is unchanged (6). It follows that pollution, as a by-product of production, falls since θ is lower and because global output (ni x*) decreases in both regions. This is the positive impact of the environmental policy.

Workers' wages are unchanged when β increases: using (4), it is easy to see in ( 7) and ( 8) that nominal wages are not modified because supply and global demand fall in the same proportion. Real wages are lowered, through higher prices. In equations ( 9) and ( 10) we can check that workers' utilities (U 1 , U 2 ) decrease in the same proportion in both regions when β increases. It follows that the difference (U 1-U2) falls also in the same proportion, and obviously reaches zero (U 1 =U 2 ) for the same critical level of r (r*) whatever the value of β. The interesting result is that the break point, r*, where the real wage becomes higher in the large region (U 1 >U 2 ), is the same whatever the marginal cost level. The result is that the centripetal force is unchanged when β increases.

Let us consider now the centrifugal force. It depends on the difference between the regions' global disutility of pollution. We have underlined that pollution decreases in both region with a higher marginal cost. The important result here is that, because n i is proportional to L i (6), the decrease in the global disutility of pollution (δn i θx) is higher in the larger region. As a consequence, the centrifugal force is softened by the environmental policy. Because the centripetal force is unchanged, the striking result of such an environmental policy is to worsen the tendency for firms to agglomerate: compared to the case without environmental policies, concentration occurs now for a higher transportation cost (r*'(θ )>0).

Conclusions

The empirical evidence indicates that agglomeration of activities occurs, even if it generates excessive pollution. Focusing on the consequences of an environmental policy in a model of economic geography, we show the basis of a moral hazard problem: by reducing the effect of centrifugal force and enlarging the range of transportation costs for which the market size effect dominates, environmental policies increasing the firms' marginal cost may stimulate agglomeration, then global pollution.

*Corresponding author. Tel:33(0)561128533; fax:33(0)561128538.E-mailaddress: calmette@univ-tlse1.fr.Manuscript


Said Jebors

Jeremie Ciccione

Soultan Al-Halifa

Benjamin Nottelet

Christine Enjalbal

Cøline M'kadmi

Muriel Amblard

Ahmad Mehdi

Jean Martinez

Gilles Subra

A New Way to Silicone-Based Peptide Polymers**

Keywords: biopolymers, comb polymers, dimers, hybrid peptides, siloxane bonds

We describe a new class of silicone-containing peptide polymers obtained by a straightforward polymerization in water using tailored chlorodimethylsilyl peptide blocks as monomeric units. This general strategy is applicable to any type of peptide sequences, yielding linear or branched polymer chains composed of well-defined peptide sequences.

Hybrid copolymers composed of biomimetic and synthetic elements gather considerable attention as they open new and exciting perspectives both in the biomaterial and biomedical fields. Peptide-based hybrid copolymers are a class of biomimetic materials that significantly become important because of their specific properties such as biocompatibility, biodegradability, and bioactivity. [START_REF] Deming | [END_REF][2][3] Recent studies focused on the preparation of well-defined homopolypeptides, [4] block copolypeptides [5][6][7][8]9] or more controlled architectures such as comb-like peptide polymers prepared by polymerization of original N-carboxyanhydride bearing a peptide sequence. [10] A second trend has emerged to yield hybrids of peptide and synthetic polymer, mainly with poly(ethylene glycol) and either (meth)acrylic chains [11] or with silica matrixes. [12][13][14] On the other hand, silicones, also referred as polysiloxanes, have a long track record as industrial biocompatible, bioinert, and biodurable materials. [15] Silicones have been widely used since the 60s in medicine as orthopedics, catheters, drains, shunts, extra corporeal equipments, implants, and as injectable fluids for soft-tissue augmentation. [16] However, these last properties are not sufficient for the more demanding tissueengineering applications where additional bioactivity is required. This can be achieved by using organic-inorganic hybrids. For example, sugar-silicone or poly(l-lactic acid) (PLLA)-silicone hybrids have been reported for cell recognition and bone tissue regeneration. [17][18][19] The association of silicone and peptides has also been proposed for improved cell-material interactions purposes but it has been mostly limited to the functionalization of silicone surfaces with peptides. [20][21][22] Herein we report on the synthesis of a new class of silicone biopolymers composed of tailored peptide sequences assembled through a one-step polymerization process initiated in water at neutral pH value. Noteworthy, this straightforward method does not require any post-grafting modification to obtain a functional polymer (Figure 1). The key point of the methodology was the synthesis of pseudopeptide monomers selectively functionalized by chlorodimethylsilyl (-SiMe 2 Cl) moieties. Solution-and solid-phase synthesis strategies were successfully implemented for the preparation of such peptide building blocks. Interestingly, this methodology can be generalized to any peptide sequence.

The chlorodimethylsilyl group (-SiMe 2 Cl) was chosen for assembly of peptide-based polymers. Its reactivity has been extensively studied in the preparation of silicones. [23,24] Hydrolysis of chlorosilane (SiCl) to hydroxysilane (SiOH) groups followed by condensation yields siloxane bonds. The condensation occurs quickly in slightly basic media, at room temperature and in a wide range of solvents, advantageously in water. Interestingly, both reactions are completely chemoselective and take place in the presence of unprotected peptide side-chains (amines, carboxylic acids, alcohols, guanidine etc.). At last, a commercially available reagent (3isocyanatopropyl chlorodimethylsilane) is used for the introduction of the chlorodimethylsilyl functionality onto peptides using their free amino groups.

For this study, three types of hybrid peptide building blocks were prepared (Scheme 1): 1) monofunctional hybrid peptides (type a) derivatized on their N-terminus and used for straightforward homodimerization (noted d); 2) bifunctional hybrid peptides (type b) functionalized both at the N-and Cterminus using a diamino derivative to yield linear polymers (noted B). Finally, Lys-bifunctional hybrid peptides (type c) gave a comb-like polymer (noted C), the peptide sequence of which is pendant on the main polymer chain (Figure 1). Five different sequences (1 to 5) were selected to cover most of the functional groups of unprotected amino acid side chains [i.e. aliphatic (Leu, Ala, Ile, Gly), aromatic (Trp, Phe), carboxylic acid (Asp and Glu), primary amine (Lys) and guaninidine (Arg)]. We also selected the protected sequence 5 bearing bulky and hydrophobic arylsulfonic derivative [Arg(Pbf)]. The combination of the five sequences with the three different types of blocks yields 15 hybrid dimethylhydroxysilane derivatives (Table 1).

Homodimerization of receptor ligands is a well-known strategy to improve binding affinity by increasing the local concentration of binders at the neighborhood of binding site. Enhancement of the biological effects was reported using dimeric analogues of numerous hormones (i.e. enkephalins, [25] neurokinins, [26] bradykinin, [27] serotonin [28] ), of pharmaceutical drugs [29] and was also reported to inhibit protein-protein interaction. [30] The reported strategies mostly rely on the linking of a spacer arm on the solid support and the simultaneous synthesis of the two "arms" from the C to the N-terminus, on the coupling of protected bearing a single free reactive function on the spacer or on chemoselective ligation. [31] In contrast, monofunctional hybrid peptide blocks a were dimerized chemoselectively in phosphate buffer (pH 7.4) at room temperature within two hours with good recovery yield (> 70 %; d1-5 in Table 1). Advantageously, no protection was required on the side chains, and the chlorodimethylsilane function could be introduced easily at the N-terminus (or at any point of the sequence through a lysine side chain). 29 Si NMR spectroscopy demonstrated that all silicon atoms are engaged in siloxane bonds. On the contrary, LC-MS analyses showed the presence of monomers and dimers because of hydrolysis of dimers in the acidic eluents. Siloxane bonds are stable in neutral conditions, enabling the dimerized compounds to be used in bioassays. As a proof of concept, the binding affinity of dimer d4 that displayed two copies of the cholecystokinin (CCK) binding minimal sequence Trp-Nle-Asp-Phe was determined on CCK B receptors on Jurkat cell line. [32] With the C-terminus amide being particularly important for receptor binding, dimerization was achieved at the N-terminus. Displacement experiments of the reference radiolabelled ligand 3-[ 125 I]iodo-4-hydroxyphenyl-propionyl-CCK (K d = 0.5 nm) were performed. Dimer d4 has a K i of 4.9 nm versus 18.9 nm for the monomeric sequence H-bAla-Trp-Nle-Asp-Phe-NH 2 and 0.26 nm for the CCK-8 reference ligand H-Asp-Tyr(SO 3 H)-Met-Gly-Trp-Met-Asp-Phe-NH 2 [33] (Figure 2). Hybrid blocks b1-5 generated linear polymers B1-5 through condensation between two SiOH groups. Hybrid blocks were dissolved in a 1/1000 v/v trifluoroacetic acid (TFA)/water solution (0.5 g mL À1 ). Phosphate buffer (pH 7.4) was added dropwise under stirring to slowly increase the pH of the reaction mixture above 7. Condensation occurred during buffer addition and induced precipitation of the polymer. Comb polymers C1-5 were obtained from hybrid blocks c1-5 with the same procedure. Only comb polymer C1 remained soluble upon buffer addition. Occurring in water, polymerization stops quickly when the polymer starts to precipitate. Thus the degree of polymerization is limited by the solubility of the growing polymer in buffer, thus depending on the nature of the peptide sequence. Longer chains could possibly be obtained using solvent mixtures such as dimethyl formamide (DMF)/water. In any case, the medium was freeze-dried and polymers were obtained as a powder or a sticky gel. The polymers were insoluble in most solvents (i.e. DCM = dichloromethane, MeOH, THF = tetrahydrofuran, EtOH, chloroform) while B4, C4, B5, and C5 were soluble in dimethyl sulfoxide (DMSO) and DMF. Thermal stability of monomers and their corresponding polymers was studied by thermogravimetric analysis (TGA) from room temperature to 800 8C, under air with 5 8C min À1 heating rate. In all case, the inception of the thermal decomposition was detected from about 200 8C until complete oxidation into silica at about 800 8C (Supporting Information). From thermolysis curves, we could perceive, as expected, that polymers were more stable than their corresponding monomers, while no significant difference appeared between monomers and dimers. For the polymers, the general trend involved a two-step thermolysis. An initial 60-70 % weight loss was observed between 200 8C and 500 8C, followed by a 15-25 % weight loss between 500 and 800 8C. All compounds presented residual weight due to the inorganic matrices (SiO 2 ) roughly in agreement with the theoretical calculated values. Comb polymers C1-5 had improved thermal stability compared to their linear counterparts B1-5. Again this might be attributed to improved interactions taking place between the pending peptides of polymers C1-5 as a consequence of their higher spatial proximity in comb polymers compared to the linear ones. [34] Glass-transition temperatures (T g ) were determined by differential scanning calorimetry (DSC) analyses, which were performed on monomers a, b and c, on dimers d and on polymers B and C from À60 8C to 150 8C (Table 2). All T g s, when detectable, were in the range 25 to 50 8C, much higher than for classical silicones as an expected consequence of the hybrid peptide structure of the polymers. No clear influence of the polymer type was observed.

Finally, for B2 and B3 a second T g was seen at À40 8C. As a consequence of the insolubility these polymers only hypotheses can be drawn, however this low T g may be attributed to the presence of cyclic species containing only SiOSi bonds. Soluble polymers were analyzed by steric exclusion chromatography (SEC; Figure 3) and 29 Si NMR spectroscopy. SEC chromatograms clearly showed that type B linear polymers polymers had significantly higher molecular weight than their type C counterparts. As an example M n (B4) is about 25 000 and M n (C4) is about 7600 g mol À1 (Table 2; M n = number average molar mass). This corresponded to an average polymerization degree (DP) of about 19 versus about 6 for B4 and C4, respectively. In opposition to B linear polymers, the polymerization degree of type C comb polymers seemed strongly impacted by the peptide sequence probably because of steric hindrance of pendant chains upon polymerization. 29 Si NMR spectra (Supporting Information) of polymers B4, B5, C1, C4, and C5, which were soluble in DMSO were recorded. Integration of the signal at 7.9 ppm assigned to M 1 substructures, which corresponded to silicon atoms involved in SiOSi bonds, as well as integration of the signal at 11.5 ppm (M 0 substructures) related to silanol end groups (SiOH), were used to calculate the SiOSi/SiOH ratio, which corresponded to half of the polydispersity index estimated by NMR spectroscopy (DP NMR ). Ratios were in good agreement with SEC analyses. As example, for B5 DP NMR was 18 (19 as determined by SEC), whereas for C5 DP NMR was 16 (13 determined by SEC). For polymer B4, the resolution of the NMR spectra did not permit to detect and quantify precisely the SiOH peak, suggesting that the ratio was about 15/1 corresponding to DP > 30. However, for this sample the NMR integration had to be commented carefully as the presence of cyclic species, containing only SiOSi bonds could not be excluded.

For the first time, we propose a bottom-up approach leading to an oligomerization of peptide sequences by straightforward hydrolytic condensation process. Occurring in water, at room temperature and requiring neither catalyst nor chemical reagent, this type of polymerization could be scaled up easily. The diversity of structures and functions of peptides associated with the possibility to mix different hybrid blocks at different ratios, open the door toward multifunctional biopolymers that should find numerous applications, for example as systems for drug or gene vectorization and delivery, scaffolds for tissue engineering or antibacterial films.

Experimental Section

Hybrid peptide synthesis: All peptides were synthesized by the Fmoc/ tBu strategy using DMF as solvent. [30] After selective removal of protecting groups from amino group(s), peptides are derivatized with 3-isocyanatopropyl chlorodimethylsilane in the presence of N,Ndiisopropylethylamine (DIEA) in DMF. Introduction of the dimethyl chlorosilane moiety could be achieved either in solution (Supporting Information) or on the peptide still anchored on the solid support. In this case, the excess of silylating agent was removed by filtration and washing steps. The choice of the solvent was guided by the good swelling of polystyrene (PS) resins and the good solubility of protected amino acids. Special attention was paid on the dryness of DMF used during isocyanate reaction since premature hydrolysis could decrease the reaction yield.

Monofunctional hybrid peptides a1-4 were obtained after Nterminus silylation and cleavage from Rink amide resin (Scheme 1 A). The same approach was used for the synthesis of Lysbifunctional hybrid peptides (c1-3, Scheme 1 C) for which an extra Lysine residue was added to the N-terminus of the sequence to be "comb"-polymerized, the two a and e amino functions being simul- [a] T g s were determined by DSC with a temperature ranging from À60 to + 150 8C. taneously deprotected and silylated with ICPCDS on solid support. N-and C-ter bifunctional hybrid peptide blocks (b1-5) were either prepared on a 2-chlorochloro-trityl PS resin, cleaved and silylated in solution, or obtained by a solid support strategy using the backbone amide linker [5-(4-formyl-3,5-dimethoxyphenoxy)pentanoyl] resin (Scheme 1 B). After final deprotection/cleavage, blocks obtained as hydroxydimethylsilyl species were analyzed by LC-MS and purified by RP HPLC. Hybrid blocks were soluble in water/acetonitrile (1/1 v/ v) 1/1000 TFA. These acidic condition solution (pH 2.5) strongly prevents the oligomerization. Indeed, only very low amounts of intramolecular condensation or dimers (in the case of type a compounds) are detected (ca. 5 %). However, even if they were easily separated by preparative HPLC at different retention times, the collected fractions always contained free (Si-OH) and condensed blocks (Si-O-Si) in equilibrium. This behavior had no significant impact on the further polymerization or dimerization process. After preparative HPLC, hybrid blocks were dehydrated by freeze-drying and can be stored at 4 8C for months.

Figure 1 .

 1 Figure 1. Silicone-containing peptide dimer d and polymers (B and C) obtained respectively from hybrid building blocks a, b, and c.

Scheme 1 .

 1 Scheme 1. Synthesis of dimethylhydroxysilane hybrid peptides. Way a) monofunctional hybrid peptide a4 for dimerization; way b) linear bifunctional hybrid peptide b2 for linear polymerization synthesis; way c) lysine bifunctional hybrid peptide c1 for comb-like polymer synthesis. Fmoc = 9-fluorenylmethoxycarbonyl, SPPS = solid-phase peptide synthesis, and TIS = triisopropylsilane.

Table 1 :

 1 Hybrid peptides dimethylhydroxysilane blocks type a, b or c and dimer d obtained from block a. Hybrids blocks are noted Xy, with X being the type of block that is, a, b or c, and y being the sequence number, that is, 1 to 5Intermolecular condensation lead to dehydrated forms and can be observed during ESI + LC-MS analysis. Retention times (Rt) and m/z values of both species are presented: m/z [M+H] + /m/z [MÀOH] + . MW= molecular weight.

Figure 2 .

 2 Figure 2. CCK B Receptor binding of H-bAla-Trp-Nle-Asp-Phe-NH 2 (!); d4 (*) and H-Asp-Tyr(SO 3 H)-Met-Gly-Trp-Met-Asp-Phe-NH 2 (&).

Figure 3 .

 3 Figure 3. SEC chromatograms (DMF, LiBr 1 %) of polymer C1 (plain line) and monomer c1 (dotted line; UV detection at 254 nm).

  [b] M n (g mol À1 ), polydispersity index and DP were estimated by SEC (in DMF) using PS as standard. [c] DPs were estimated by 29 Si NMR spectroscopy by comparison of the integrals of SiOSi (7.9 ppm) and SiOH (11.5 ppm) signals. [d] Not determined, insoluble polymer.

Table 2 :

 2 Analyses of silicone-containing peptide polymers B1-B5 and C1-C5.

	polymer T g [8C] [a]	M n [b]	[b]	DP SEC	[b]	DP NMR	[c]
	B1	+ 30	insoluble [d] insoluble insoluble insoluble
	B2	+ 24; À41 insoluble	insoluble insoluble insoluble
	B3	+ 25; À39 insoluble	insoluble insoluble insoluble
	B4	no T g	19 100	3.9	19		30
	B5	no T g	23 300	1.27	20		18
	C1	+ 49	16 900	1.24	21		28
	C2	+ 26	insoluble	insoluble insoluble insoluble
	C3	no T g	insoluble	insoluble insoluble insoluble
	C4	no T g	7050	1.44	6		6
	C5	+ 24	16 700	1.07	13		16

Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2015, 54, 3778 -3782

Angew. Chem. Int. Ed. 2015, 54, 3778 -3782 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.angewandte.org 15213773, 2015, 12, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/anie.201411065 by Biu Montpellier, Wiley Online Library on [23/01/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

www.angewandte.org 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2015, 54, 3778 -3782 15213773, 2015, 12, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/anie.201411065 by Biu Montpellier, Wiley Online Library on [23/01/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

www.angewandte.org 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim Angew. Chem. Int. Ed. 2015, 54, 3778 -3782 15213773, 2015, 12, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/anie.201411065 by Biu Montpellier, Wiley Online Library on [23/01/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License