Tan-Sy Nguyen

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

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

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

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

Convolution Autoencoder-Based Sparse Representation Wavelet for Image Classification

Keywords: Sparse representation classification, wavelet domain, autoencoder, probability residual criterion

In this paper, we propose an effective Convolutional Autoencoder (AE) model for Sparse Representation (SR) in the Wavelet Domain for Classification (SRWC). The proposed approach involves an autoencoder with a sparse latent layer for learning sparse codes of wavelet features. The estimated sparse codes are used for assigning classes to test samples using a residual-based probabilistic criterion. Intensive experiments carried out on various datasets revealed that the proposed method yields better classification accuracy while exhibiting a significant reduction in the number of network parameters, compared to several recent deep learning-based methods.

I. INTRODUCTION

Over the past decade, most research in classification has emphasized on the use of Sparse Representation whose effectiveness in many image analysis and processing problems has been well established [START_REF] Elad | Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing[END_REF]. It should be noted that the development of deep learning has opened up other tremendous opportunities in a variety of research areas. Specifically, SR can provide an efficient representation and a simple interpretation of data using a small number of basis vectors. Furthermore, this representation is correlated with the properties of visual neurons in early visual brain areas [START_REF] Olshausen | Emergence of simple-cell receptive field properties by learning a sparse code for natural images[END_REF]. It was proven that sparsity-based learning methods can improve the classification performance [START_REF] Wright | Robust face recognition via sparse representation[END_REF]- [START_REF] Abavisani | Deep sparse representation-based classification[END_REF].

In the classical Sparse-Representation Classification (SRC) method [START_REF] Wright | Robust face recognition via sparse representation[END_REF], it is assumed that a test sample can be sparsely represented by a linear combination of training samples. Then, sparse representation coefficients of the linear combination model are computed by means of optimization, and the reconstruction residuals of each class are calculated to find the label of the test sample, using a given criterion.

However, classification performance can be compromised in the case of complex datasets that may contain highly correlated samples from different classes. Another limitation is due to the fact that the dictionary contains all training samples from all classes, and especially in the case of large datasets.

Therefore, it is necessary to develop an efficient solution for learning the optimal representations of data allowing to extract relevant features from big and complex datasets for classification purpose. It should be noted here that deep learning methods have been successfully investigated in large domains. In particular, unsupervised neural networks such as auto-encoders have been widely and successfully used in various applications and especially data classification [START_REF] Papyan | Convolutional neural networks analyzed via convolutional sparse coding[END_REF], [START_REF] Ngo | Sparse representation wavelet based classification[END_REF].

In order to further improve the classification performance for large and complex datasets, we propose in this paper, a novel Convolution AE model regularized with sparsity constraint for classification in the wavelet domain. This model can be considered as a hybrid approach of Convolution AE [START_REF] Papyan | Convolutional neural networks analyzed via convolutional sparse coding[END_REF] and Sparse Representation Classification model. Because the wavelets promote sparsity, the input data are image features described by the complementary information from the wavelet coefficients in the low-pass sub-band (LL), representing the most important component of the image. Furthermore, the AE loss function is regularized by a sparsity penalty on the features in the latent space of the AE, allowing the network to better learn the distinction between the different class samples, and so, resulting in a more accurate classification process. This approach leads to better accuracy in classifying data with high variations in its contents (e.g. the SVHN [START_REF] Netzer | Reading digits in natural images with unsupervised feature learning[END_REF] dataset, used in Section V).

The proposed method is inspired by two recent previous works on Deep Sparse Representation Classification (DSRC) [START_REF] Abavisani | Deep sparse representation-based classification[END_REF] and Sparse Representation Wavelet Classification (SRWC) [START_REF] Ngo | Sparse representation wavelet based classification[END_REF]. However, unlike the DSRC method [START_REF] Abavisani | Deep sparse representation-based classification[END_REF], the proposed model is performed in the wavelet domain in order to take advantage of the sparsity to improve the classification. An important contribution is a deep convolution autoencoder architecture (see Table I) which reduces the number of network parameters significantly, and improves classification accuracy. Furthermore, the classification criterion from the residuals is not based on minimum residuals criterion [START_REF] Abavisani | Deep sparse representation-based classification[END_REF], but rather on a residual-based probabilistic rule [START_REF] Wei | A new sparse representation classifier (SRC) based on probability judgement rule[END_REF]. This work is a first step of a study dedicated to video distortions classification in the context of video-assisted laparoscopic surgery [START_REF] Khan | Towards a video quality assessment based framework for enhancement of laparoscopic videos[END_REF], [START_REF] Khan | Residual network based distortion classification and ranking for laparoscopic image quality assessment[END_REF].

The remainder of this paper is organized as follows. Section II will present related works. Then, we provide some backgrounds on the SRWC method in Section III. Afterward, the methodology of the proposed classification scheme is presented in Section IV. Subsequently, in Section V, experimental results are discussed before a conclusion in Section VI.

II. RELATED WORK

It is important to note that the performance of the classification is strongly conditioned by the way the data is represented. Among the representations of the data, SR has proven to be very effective in several applications, particularly in image analysis and processing [START_REF] Elad | Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing[END_REF], [START_REF] Lu | Group sparse reconstruction for image segmentation[END_REF].

Since the publication of the well-known SRC [START_REF] Wright | Robust face recognition via sparse representation[END_REF], many other SR-based classification methods have been proposed [START_REF] Jiang | Label consistent k-svd: Learning a discriminative dictionary for recognition[END_REF], [START_REF] Yang | Fisher discrimination dictionary learning for sparse representation[END_REF], [START_REF] Ngo | Sparse representation wavelet based classification[END_REF]. In SRC [START_REF] Wright | Robust face recognition via sparse representation[END_REF], a new sample is assumed to be a sparse linear combination of the labeled training samples. The sparse representation coefficients are then computed via a sparse optimization problem over the dictionary formed by all the training samples. Using the estimated sparse code, the test sample can then be assigned to a class that corresponds to a minimum reconstruction residual. Other strategies based on the construction of selective dictionaries by learning from a group of training samples instead of using all of them as a dictionary, make it possible to improve the classification performance [START_REF] Jiang | Label consistent k-svd: Learning a discriminative dictionary for recognition[END_REF], [START_REF] Yang | Fisher discrimination dictionary learning for sparse representation[END_REF]. For instance, the label consistent K-SVD (LC K-SVD) approach [START_REF] Jiang | Label consistent k-svd: Learning a discriminative dictionary for recognition[END_REF] relies on dictionary learning, using the objective function defined by the reconstruction and classification error terms and more particularly by a discriminant sparse code error to impose discrimination in the sparse coding. In another approach, namely Fisher discrimination dictionary learning (FDDL) [START_REF] Yang | Fisher discrimination dictionary learning for sparse representation[END_REF], the objective function is constructed with a discrimination constraint based on a Fisher discrimination criterion to learn a structured dictionary.

A key success in the classification of any complex system is to have a good representation of features. Owing to surge of interests in Machine Learning, deep learning models including VGG19 [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF], ResNet50 [START_REF] He | Deep residual learning for image recognition[END_REF], Inception [START_REF] Szegedy | Rethinking the inception architecture for computer vision[END_REF], WideResNet [START_REF] Zagoruyko | Wide residual networks[END_REF], are developed and have been considered as powerful models for learning feature representation of data. More particularly, learning models that use sparse coding to extract sparse features from complex data have applications in classification. Sparse representation is used to eliminate the useless features and extract the latent structure of data, improving the interpretation of the input data and the generalization of the model.

In [START_REF] Feng | Sparse representation learning of data by autoencoders with L 1/2 regularization[END_REF], a sparse AE-based model is proposed for improving the classification capability, by learning sparse representation using 1/2 regularization method as a sparsity constraint on the average hidden activation. Another approach consists of using a deep learning model in the process of feature extraction for SRC. In [START_REF] Abavisani | Deep sparse representation-based classification[END_REF], the main idea of the DSRC is to use an AE in learning sparse representation for classification. The network consists of a sparse coding layer located between the encoder and the decoder. The role of this layer is to estimate the sparse representation of features from the encoder. The learning objective includes the training of the encoderdecoder and sparse coding. The estimated spare codes are then used to determine the class labels of testing samples, using the minimum reconstruction residual as in SRC approach [START_REF] Wright | Robust face recognition via sparse representation[END_REF]. The drawback of this method relies on the conventional residual criterion [START_REF] Wright | Robust face recognition via sparse representation[END_REF], which is not discriminant when samples of different classes may have the same residual. These are the main reasons that led us through this study to turn to SRWC.

In the next section, we will briefly present the SRWC method [START_REF] Ngo | Sparse representation wavelet based classification[END_REF] based on SRC approach operating in the wavelet domain. This is one of the main ingredients of the proposed method.

III. BACKGROUND ON SRWC

The SRWC is applied in the wavelet domain to construct a dictionary built from the training samples. Ngo et al. [START_REF] Ngo | Sparse representation wavelet based classification[END_REF] showed that using only features described by wavelet coefficients in low-pass subband (LL) in SRC can significantly improve the image classification accuracy. Therefore, given a set of labeled training samples from k classes, the Haar wavelet transform is applied to the original images and the training samples, to generate the low-pass subband coefficients (LL) corresponding to the rough description of the image signal, and the objective is to classify test samples in the wavelet domain.

D train = [D 1 train , D 2 train , ..., D k train ] ∈ R m×n , (1)
Under sparse representation assumption, any new test atom x test ∈ R m from the same class c, will be a sparse linear combination of the atoms associated with the same class, as below:

x test = s c 1 d c 1 + . . . + s c nc d c nc , (2)
where [s c 1 , s c 2 ...s c nc ] ∈ R nc is the sparse coefficients vector corresponding to class c. It is ideally possible to detect the class of a given unlabeled data x test by finding a set of samples in the training atoms that can better approximate x test . Its sparse representation can be determined by finding a set of sparse approximation vector s whose nonzero entries are those associated with the c th class, s = [0, ...0, s c 1 , ..., s c nc , ..., 0, 0] ∈ R n , in solving the following Lasso optimization problem:

ŝ = min s x test -D train s 2 2 + λ 0 s 1 (3)
Then, we can determine the class to which x test belongs based on the non-zero coefficients in ŝ. The minimum difference between the sparse sample and the original one is finally calculated to deal with this problem:

class pred (x test ) = argmin c x test -D train δ c (ŝ) 2 (4)
where ŝ ∈ R n , δ c is the characteristic function, and δ c (ŝ) selects elements in ŝ that are associated with class c.

IV. PROPOSED METHOD (CAESRWC)

In this paper, we propose a novel method, namely Convolution Autoencoder based Sparse Representation Wavelet Classification (CAESRWC) to solve the large dataset classification problem while achieving space efficiency with reduced number of parameters and better performance, compared to other deep learning methods [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF]- [START_REF] Zagoruyko | Wide residual networks[END_REF].

The proposed method takes advantages of learning both from a deep convolution AE, providing the latent space with features that capture the useful structure of the input wavelet coefficients in the low-pass subband (LL), and from the sparse coding of the latent variables, providing sparse code for the classification of the unlabelled test sample. Another advantage is the fact that the method is performed in the wavelet domain, promoting sparsity and enhancing the classification reliability. As mentioned in [START_REF] Zou | Image classification using wavelet coefficients in low-pass bands[END_REF], the extracted low-pass subband coefficients which lead to better discrimination (see Section IV-A) will enhance the classification accuracy. The proposed classification scheme consists of four main modules as illustrated in Fig. 1 and described below: The loss function is derived for training the autoencoder by minimizing both the reconstruction error and sparse coding regularization term. The resulting sparse code of the encoding features can be used to derive the class labels using a residualbased probabilistic model. Let X train ∈ R m×ntr , X val ∈ R m×n val , and X test ∈ R m×nte be the given vectorized LL wavelet coefficients of the training, validation, and testing data, respectively. Likewise, Y train ∈ R my×ntr , Y val ∈ R my×n val and Y test ∈ R my×nte are their corresponding encoding features. In our experiments (section V-B), X train , X val , and X test are three sub-packages extracted from the observed dataset for training, validation, and testing in the proportion of 0.8, 0.1, and 0.1, respectively (n tr = 0.8 * n, n val = n te = 0.1 * n). The input of the encoder is defined as

X = [X train , X ref ] ∈ R m×(ntr+n ref )
where "ref " refers to "validation" or "test".

A. Wavelet transform block

This pre-processing step allows extracting LL wavelet coefficients of input image data to be fed into the autoencoder. Each RGB image is converted into gray-scale. Then a 5level Haar wavelet decomposition is performed on each image. A discriminant analysis can be conducted to show the discrimination ability of these coefficients and to analyze which wavelet decomposition level gives the best discrimination. For that, two classes are randomly selected from the observed data for the computation of the discrimination index using five LL subbands corresponding to five levels of wavelet transform. The fisher ratio defined below could be used as a discrimination index.

v = (µ 1 -µ 2 ) 2 σ 2 1 + σ 2 2 , (5)
where µ 1 , µ 2 , σ 1 , σ 2 are mean and deviation vectors of normalized LL coefficients of the two classes. In Fig. 2, we can clearly see that in almost datasets, using 3 levels of wavelet decomposition is enough to obtain good discrimination.

B. Sparse coding block

The function of the sparse coder in the latent space is to estimate the SR of the encoding features

Y = [Y train , Y ref ] ∈ R my×(ntr+n ref )
, by solving the Lasso minimization problem, which is similar to [START_REF] Wright | Robust face recognition via sparse representation[END_REF].

min S Y ref -Y train S 2 F + λ 1 S 1 , (6)
where S ∈ R ntr×n ref is the sparse coefficients matrix and λ 1 is a positive regularization parameter that controls the sparsity penalty and the fidelity between the input Y and the output (estimated sparse encoding features) Ŷ of the sparse coder. From (6), the estimated sparse encoding features, Ŷref is obtained by

Ŷref = Y train S ∈ R my×n ref . If the sparse coder's output is Ŷ = [ Ŷtrain , Ŷref ], (6)
can be reformulated as:

min S Y -Ŷ 2 F + λ 1 S 1 , (7)

C. Loss function block

The objective of the proposed AE-based sparse representation classification in the wavelet domain is to :

• Learn efficient representation of wavelet coefficients X of an original data I, through extraction of reduced encoding wavelet features Y from original wavelet features X, with a convolution AE, instead of using PCA as in SRWC [START_REF] Ngo | Sparse representation wavelet based classification[END_REF]. The inherent advantages of AEs over PCA is that they are capable of learning, hence modelling complex non-linear data, and to reconstruct back the input wavelet features, X = decoding( Ŷ) = decoding(sparse coder(Y)). The loss for the reconstruction error of the AE between the input X and reconstructed input X using sparse encoding wavelet features Ŷ is:

L AE = X -X 2 F ( 8
)
• Estimate the sparse representation of encoding wavelet features in the latent space to be used for classification based on SR. This is accomplished by adding a penalty on the sparsity of encoding wavelet features (second term in ( 7)) while minimizing the reconstruction error of these features (first term in ( 7)). The regularization loss for sparse coding layer is:

L SR = min S Y -Ŷ 2 F + λ 1 S 1 (9)
Note that the loss function to be minimized when training the AE-based model is composed of a global reconstruction term (L AE ), making the encoding-decoding scheme efficient, and a sparsity regularization term, to learn sparse representation in latent space (L SR ). The total loss of the proposed architecture is L t = L SR + L AE :

L t = min S Y -Ŷ 2 F + λ 1 S 1 + λ 2 X -X 2 F (10)

D. Classification block

Assigning the class labels to the test samples can be computed using the sparse codes matrix S and a residualbased probabilistic rule [START_REF] Wei | A new sparse representation classifier (SRC) based on probability judgement rule[END_REF] that is proven as a better metric than truncation of residuals. The probability-based residual is obtained as follows. First, the residuals for the c th class is computed:

r c (x test ) = y test -Y train δ c (s) 2 2 δ c (s) 2 2 , ( 11
)
where x test is the observed sample in X test with its embedding feature vector y test and its corresponding sparse code column s in the sparse code matrix S.

Then, the label of the test sample x test is identified using a probability value defined in [START_REF] Khan | Towards a video quality assessment based framework for enhancement of laparoscopic videos[END_REF], as: where p c denotes the probability that the x test sample belongs to class c, while k denotes the number of the classes. From ( 12), we can see that 0 ≤ p c ≤ 1 and k c=1 p c = 1. Furthermore, the smaller r c (x test ) is, the bigger p c is. The test sample x test belongs to class c if the probability is higher than a threshold (0.99 in our experiments). Otherwise, the class prediction is determined by the minimum residual from [START_REF] Wei | A new sparse representation classifier (SRC) based on probability judgement rule[END_REF]. Comparing to the traditional SRC residual criterion [START_REF] Wright | Robust face recognition via sparse representation[END_REF], this probability judgement rule depends on two parts (see Eq. ( 11)): the numerator which is the basic residual term often used in SRC method [START_REF] Wright | Robust face recognition via sparse representation[END_REF] and the denominator which represents the 2 -norm of the sparse code.

class (x test ) = argmax c (p c ) = argmax c e -rc k c=1 e -rc , (12)

V. EXPERIMENTAL RESULTS

The following provides some pertinent details on how to conduct the experimental study to evaluate the performance of the proposed classification scheme.

A. Experimental settings

The model is trained using Tensorflow 2.0, and NVIDIA Tesla T4 GPU. In this task, the momentum Adam optimizer is used with the learning rate 1e -3, and an epoch decay factor of 0.9. Besides that, there are two sub-steps in the training process. In the first step considered as a pre-training stage, our model is launched in 100 epochs without sparse coding layer. In this time, the proposed architecture is the same as normal autoencoder for the purpose of initializing the resultcloser parameters based on extracted abstract features.

After that, in the second step, the sparse coding layer is added to the pre-trained architecture. The summary of the endto-end model is shown in Table . I. We continue training the end-to-end network in 900 epochs. The first layer is also the input layer, which has the number of neurons corresponding to the dimension of wavelet coefficients vector. The kernel size is 3×3 for the first and second convolution layer and 1×1 for the third convolution layer. To tackle overfitting, dropout and random permutation cross-validation are used during training. In all experiments, the parameters λ 1 and λ 2 in (10) are set to 10 and 1, respectively.

B. Datasets

In this paper, we evaluate our method against state-of-the art SRC methods (SRWC [START_REF] Ngo | Sparse representation wavelet based classification[END_REF], FDDL [START_REF] Yang | Fisher discrimination dictionary learning for sparse representation[END_REF], LC-KSVD2 [START_REF] Jiang | Label consistent k-svd: Learning a discriminative dictionary for recognition[END_REF] and DSRC [START_REF] Abavisani | Deep sparse representation-based classification[END_REF]). 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], YaleB [START_REF] Georghiades | From few to many: Illumination cone models for face recognition under variable lighting and pose[END_REF] and UMDAA-01 [START_REF] Zhang | Domain adaptive sparse representation-based classification[END_REF]), one object recognition dataset COIL-100 [START_REF] Nene | Columbia object image library (coil-20)[END_REF], and AR gender binary dataset [START_REF] Martinez | The ar face database[END_REF] are used in our experiments. Fig. 3 shows sample images from these datasets. We randomly select a smaller subset of the used datasets and perform all the experiments on the selected subset.

(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) UMDAA-01 [START_REF] Zhang | Domain adaptive sparse representation-based classification[END_REF] (d) YaleB [START_REF] Georghiades | From few to many: Illumination cone models for face recognition under variable lighting and pose[END_REF] (e) AR [START_REF] Martinez | The ar face database[END_REF] (f) Coil-100 [START_REF] Nene | Columbia object image library (coil-20)[END_REF] Fig. 3: Some data samples from the six employed datasets.

1) USPS [START_REF] Hull | Database for handwritten text recognition research[END_REF]: This dataset contains 7291 training grayscale images of ten digits (0-9). We experiment on a subset of 2000 samples. We randomly select 160, 20, and 20 samples per digit from the training, validating, and testing, respectively.

2) Street view house numbers (SVHN) [START_REF] Netzer | Reading digits in natural images with unsupervised feature learning[END_REF]: This dataset contains over 600,000 labeled real-world images of house numbers. We randomly select 160, 20, and 20 samples per digit for the training, validating, and testing, respectively.

3) AR face [START_REF] Martinez | The ar face database[END_REF]: It contains over 4,000 color images corresponding to 126 people's faces. We experiment on a subset of 2000 samples including 1600, 200, and 200 samples for the training, validating, and testing, respectively.

4) Yale B face [START_REF] Georghiades | From few to many: Illumination cone models for face recognition under variable lighting and pose[END_REF]: The database contains 5760 single light source images of 10 subjects. We experiment on a subset of 2000 samples including 160, 20, and 20 samples per person for the training, validating, and testing, respectively.

5) UMD mobile face [START_REF] Zhang | Domain adaptive sparse representation-based classification[END_REF]: This dataset contains 750 frontfacing camera videos of 50 users captured by smartphones. We randomly select 40, 5, and 5 facial samples from each subject for training, validating, and testing, respectively.

6) COIL-100 [START_REF] Nene | Columbia object image library (coil-20)[END_REF]: This dataset contains color images of 100 objects. We perform the experiments on a subset with 72 samples per object. We randomly select 48, 12, and 12 samples per object for the training, validating, and testing, respectively.

7) AR gender [START_REF] Martinez | The ar face database[END_REF]: This is the last considered dataset which contains 50 male and 50 female images. In each one, their face is captured 26 viewing conditions. We select 20, 3, and 3 images for training, validating, and testing, respectively.

C. Performance and Comparison

The overall recognition rates for all datasets are presented in method offers more accuracy than both traditional SRC [START_REF] Wright | Robust face recognition via sparse representation[END_REF] and state-of-the-art methods [START_REF] Jiang | Label consistent k-svd: Learning a discriminative dictionary for recognition[END_REF], [START_REF] Yang | Fisher discrimination dictionary learning for sparse representation[END_REF], [START_REF] Ngo | Sparse representation wavelet based classification[END_REF]. In the case of the AR database [START_REF] Martinez | The ar face database[END_REF], our method is ranked second and compared to the SRWC method [START_REF] Khan | Towards a video quality assessment based framework for enhancement of laparoscopic videos[END_REF] the difference is somewhat insignificant.

In a second round of experiments, in order to study the effect of the number of training samples, we analyzed the proposed method in a random subset of data with some different sizes using 20%, 40%, 60%, 80%, and 100% of the observed datasets. These subsets are then subdivided into training/ validation/ testing subsets. Comparisons and quality assessment are conducted between the proposed method (CAESRWC) and some modern deep learning architectures. These deep models are used to extract the feature and conduct sparse coding in the next step. The obtained results are shown in Fig. 4 for the different datasets. We can obviously see that the modern Deep Learning models depend greatly on the amount of training data. This is an inevitable part of using training models. Specifically, when the training data is small, deep learning models like Inception Network or ResNet would not achieve good accuracy. Through the results obtained, it is clear that our method remains superior to the others even with limited amount of training data.

The efficiency of the new method is also due to the considerably smaller number of network parameters, compared to the other methods, as it can be seen in Table III. Obviously, the reduced number of parameters makes the proposed method perform better with limited amount of training data.

VGG [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF] ResNet [START_REF] He | Deep residual learning for image recognition[END_REF] Inception [START_REF] Szegedy | Rethinking the inception architecture for computer vision[END_REF] Wide

VI. CONCLUSION

In this paper, we have presented a new sparse representation classification method operating in the wavelet transform domain and using deep learning model. More precisely, the (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) UMDAA-01 [START_REF] Zhang | Domain adaptive sparse representation-based classification[END_REF] (d) YaleB [START_REF] Georghiades | From few to many: Illumination cone models for face recognition under variable lighting and pose[END_REF] (e) AR [START_REF] Martinez | The ar face database[END_REF] (f) Coil-100 [START_REF] Nene | Columbia object image library (coil-20)[END_REF] Fig. 4: Effects of the size of the employed subsets of data on the accuracy for the different deep learning methods.

low-pass subbands are fed into an Autoencoder architecture to learn the sparse codes and extract more discriminant features.

Intensive experiments are performed on different datasets and show the benefits of the proposed approach compared to recent state-of-the-art deep learning based approaches. However, the proposed method has just focused on the low frequency information of image. Therefore, some important features of image can be ignored during the wavelet decomposition process. In the future, and instead of focusing only on the approximation subband, the exploitation of high frequency subbands (i.e. the detail wavelet subbands) will be investigated and extended to distortion classification in the context of video guided laparoscopic surgery.

  Given a set of labeled training samples from k classes, the samples are first processed via wavelet transform. Then, Principal Component Analysis (PCA) is used to reduce the dimension of each vectorized sample, which is then called atom. All the vectorized training atoms d i (i = 1...n c ) of the class c are used to create the sub-dictionary matrix D c train = [d c 1 , d c 2 , ..., d c nc ] ∈ R m×nc where m is the dimension of each vectorized training atom, n c is the number of atoms in class c. With n (n = k c=1 n c ) atoms from k categories, the training matrix can be constructed as:

Fig. 1 :

 1 Fig. 1: End-to-end architecture of the proposed CAESRWC.

•

  The Wavelet Transform: extracts the LL wavelet coefficients of the original image data I (including test (or) validation samples and training samples). • The encoder extracts encoding features Y in the latent space, from the LL wavelet coefficients input X. • The Sparse Coder finds the sparse code of the encoding features that are received by the encoder, yielding the recovered encoding features Ŷ (concatenating the recovered encoding features and the training encoding features) to be fed into the decoder. • The decoder computes from the sparse encoding Ŷ, a reconstruction X close to the original input X.

Fig. 2 :

 2 Fig. 2: Discriminant analysis based on different decomposition levels for various standard datasets.

TABLE I :

 I Description of the proposed model's parameters

  Table. II. It can be noticed that the proposed method outperforms the traditional SRC method on most datasets. Furthermore, our

				Method
	Acc. (%)	SRC [3]	FDDL [5]	SRWC [9]	LC-KSVD2 [4]	DSRC [7]	CAE SRWC
	USPS [20]	87.78	91.34	95.45	87.45	96.25	96.82
	SVHN [10]	15.71	22.54	28.21	35.31	67.75	68.24
	ARface [21]	97.61	96.16	98.39	97.70	98.12	98.37
	ARgender [21]	93.0	94.0	94.2	86.8	96.48	96.54
	YaleB [22]	97.54	97.52	98.06	97.80	97.20	98.35
	UMDAA-01 [23]	79.00	81.22	85.29	84.82	93.39	95.10
	COIL 100 [24]	91.16	88.22	92.29	91.42	91.12	92.35

TABLE II :

 II Classification accuracy (%).

TABLE III :

 III Comparison of the number of network parameters between different deep learning methods.

ACKNOWLEDGMENT

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