Ahmed Drissi

El Maliani

Mohammed El Hassouni
email: mohamed.elhassouni@gmail.com

Yannick Berthoumieu
email: yannick.berthoumieu@ims-bordeaux.fr

Driss Aboutajdine

Lrit

T Author

GENERIC MULTIVARIATE MODEL FOR COLOR TEXTURE CLASSIFICATION IN RGB COLOR SPACE

Keywords: Classification, Texture, Copula, Kullback-Leibler divergence

This paper presents a new method for modeling magnitudes of dual tree complex wavelet coefficients, in the context of color texture classification. Based on the characterization of dependency between RGB color components, Gaussian copula associated with Generalized Gamma marginal function is proposed to design the multivariate generalized Gamma density (MGΓ D) modeling. MGΓ D has the advantages of genericity in terms of fitting over a variety of existing joint models. On the one hand, the generalized Gamma density function offers free-shape parameters to characterize a wide range of heavy-tailed densities, i.e. the genericity. On the other hand, the intercomponent,inter-band dependency is captured by the Gaussian Copula which offers adapted flexibility. Moreover, this model leads to a closed-form for the probabilistic similarity measure in terms of parameters, i.e. Kullback Leibler divergence. By exploiting the separability between the copula and the marginal spaces, the closed-form enables us to minimize the computational time needed to measure the discrepancy between two Multivariate Generalized Gamma densities in comparison to other models which imply of using a Monte-Carlo method characterized by an expensive time-computing. For evaluating the performance of our proposal, a K-Nearest Neighbor (KNN) classifier is then used

Introduction

For various practical applications in computer vision, texture analysis is a useful and key component for solving problems such as pattern recognition, classification or segmentation. Thus, in practical applications, an efficient modeling of the variation of intensity that characterizes the texture in the image is central. The relevance of the model determines to a large extent the effectiveness of the feature extraction and then the performance of the image processing method. However, as relevant as the model is, it is also important to propose an appropriate similarity measure and easy to use. Many recent works devoted to texture stressed that accurate feature extraction can be achieved by statistical modeling of subband coefficients in the transformed domain of wavelets. A variety of wavelet decompositions can be applied, ranging from Discret Wavelet Transform (DWT), Steerable Pyramids, to the recent complex wavelet transforms [START_REF] Kingsbury | The Dual-Tree Complex Wavelet Transform: A new Technique for Shift-Invariance and Directional Filters[END_REF]. Each of these transformations has some particularities, but all share the property sparsity of their subband distributions. Histograms are heavy tailed and pickily pronounced, and then need to a sub-Gaussian modeling. To fulfill this need, a pioneering work where Do and Vetterli proposed the Generalized Gaussian Density (GGD) [START_REF] Do | Wavelet-based texture retrieval using generalized Gaussian density and Kullback-Leibler distance[END_REF] as an alternative to the Gaussian model. Recently, the magnitudes of complex subbands coefficients have been modeled by Weibull and Gamma distributions [START_REF] Mathiassen | Texture Similarity Measure Using Kullback-Leibler Divergence between Gamma Distributions[END_REF] [3], leading to a considerable enhancement of the retrieval performances.

In the same way, we have proposed the Generalized Gamma distribution as a model for the coefficient magnitudes issued from the Dual Tree Complex Wavelet Transform (DTCWT), for grey level texture classification issue [START_REF] El Maliani | Texture classification based on the Generalized Gamma distribution and the Dual Tree Complex Wavelet Transform[END_REF]. A more general work was done inspired by the Generalized Gamma density, but in the context of texture retrieval [START_REF] Choy | Statistical Wavelet Subband Characterization Based on Generalized Gamma Density and Its Application in Texture Retrieval[END_REF].

In these previous works, only marginal distributions of subband histograms are taken into account. This is suitable for grey level images when independence between subbands of the same scale is supposed. However in the case of color textures, the dependence between color components is undeniable and must be taken into account by conceiving multivariate models, otherwise the system will suffer from the lack of this crucial information. The joint statistical modeling is then a welcome advantage for color texture retrieval or classification. Such framework has been considered previously by many authors.

Verdoolaege et al. [START_REF] Verdoolaege | Multiscale colour texture retrieval using the geodesic distance between multivariate Generalized Gaussian models[END_REF], proposed a multivariate Generalized Gaussian distribution (MGGD) for multiscale color texture retrieval. The model was devoted to describe dependence across color components while assuming independence among subbands of a single color component. In [START_REF] Kwitt | A joint model of complex wavelet coefficients for texture retrieval[END_REF] [START_REF] Kwitt | Efficient Texture Image Retrieval Using Copulas in a Bayesian Framework[END_REF], Kwitt et al. treated the problem of joint modeling of complex coefficient magnitudes across subbands of different color components. Based on Gaussian copula and student t copula for modeling the dependence structure, in conjunction with Weibull and Gamma densities as parametric margin models, they achieved significant enhancements in the context of texture retrieval. Another Copula based multivariate modeling was proposed by Sakji-Nsibi et al. [START_REF] Sakji-Nsibi | Fast scalable retrieval of multispectral images with kullback-leibler divergence,textquotedblright in Image Processing[END_REF], for multicomponent image indexing using Gaussian Copula in conjunction with the GGD and Gamma densities. Again substantial improvement of classification results was obtained in comparison with the marginal modeling approach. However, texture databases are extremely increasing in term of diversity and heterogeneity of textures, which limits the use of one model over others. Moreover, the lack of analytical expression for the KL divergence in the case of multivariate models based on copulas presents a shortcoming, since the alternative Monte-Carlo based approach [START_REF] Kwitt | A joint model of complex wavelet coefficients for texture retrieval[END_REF][8] is computationally expensive.

Based on these observations and to remedy the problem of complex nature and huge variety of textures in the databases, we propose a generic multivariate model that takes into account the strong dependency between RGB color components. In order to manage the diversity, the Generalized Gamma density is proposed associated to the Gaussian copula for capturing the dependence accross color component subbands of a fixed and/or different pair of orientation at a fixed scale of decomposition, i.e the MGΓ D model. The choice of the Multivariate Generalized Gamma based on the Gaussian copula is justified by the existence of KL divergence for Gaussian copula and Generalized Gamma margin model, since we present a closed form of the KL divergence based similarity measure.

Then, contribution in this work is threefold:

-First, concerning the dataset we consider an inter-band inter-component dependency which seems to provide a rich information for the characterization process.

-Second, concerning the model we present MGΓ D as a generic multivariate model in order to capture the rich information of dependency between R, G and B components subbands.

-Three, concerning the similarity measurement which is a bit hard to calculate in case of copulas based models. We present a closed form expression of the KL divergence between two MGΓ D densities which gives a big advantage over the Monte-carlo based approach.

Experiments using the K-nearest neighbor classifier show the superiority of the Multivariate Generalized Gamma model over the existing joint models, and highlight also the effectiveness of the proposed metric when compared with the Monte-Carlo based approach. This paper is organized as follows. In the next section, we give an overview of the multivariate statistical modeling in the RGB color space, we present the multivariate proposed model and we derive the Kullback-Leibler divergence.

In section 3, we provide texture classification results. Finally, we conclude in the section 4.

2 Multivariate Statistical Modeling in the RGB color space

Wavelet Domain

For texture analysis, wavelet transform presents advantages in terms of locality, sparsity and spatial-frequency characterization. Thus, this analysis tool received a lot of attention and achieved notoriety in the last decades. In this context, the classical Discret Wavelet Transform (DWT), provides an intuitive description and a non-redundant representation of images. However, the decomposition of an image with a DWT leads to only four directional frequency subbands at each decomposition scale, one is an approximation subband and three are detail subbands corresponding to the vertical, horizontal and diagonal orientations. In addition to this lack of directionality, DWT is not shift invariant, thus each small shifting of the image leads to a significant difference of the wavelet coefficient magnitude for the same subband. The main reason of this problem is the real valued nature of the coefficients of DWT, since that the Fourier transform does not suffer from this one. As an alternative, N. Kingsbury [START_REF] Kingsbury | The Dual-Tree Complex Wavelet Transform: A new Technique for Shift-Invariance and Directional Filters[END_REF] proposed the Dual-Tree Complex Wavelet Transform (DTCWT). The basic idea of this approach consists in using two real wavelets to obtain complex wavelet coefficients, which are shift-invariant. DTCWT provides six detail subbands per scale instead of three subbands in the case of DWT, which presents a rich directional selectivity. In the next subsection, we investigate the existence of dependencies among coefficients of these subbands.

Dependency observations

As already mentioned, many works are based on marginal modeling of textures, without accounting for the information that resides across the color components. This approach has the advantage to be simple and tractable, but leads to a considerable loss of the dependence information. Let us scroll some proofs of the nonzero dependence between components, when color textures are represented in the RGB color space. In the following paragraph, some experiments are conducted for some samples of textures from the Vistex database to exhibit the effective dependency between color components in the RGB color space.

Perceptual report

Before providing objective proofs of dependence such as mutual information and scatter plots, one can make a perceptual report even being a subjective

one.

The dependence across RGB color components can be observed perceptually, from the texture presented in Fig. 1. Clearly observed from Fig. 2, the spatial structures of color components of the Fabric.0015 color texture are extremely close in case of RGB representation (Fig. 2(a)), while we observe that in the case of Yuv representation (Fig. 2(b)), the spatial structure of the Y component is different compared to spatial structures of u and v components.

Mutual information

Mutual information is a measure of the statistical dependence between two variables. Let X and Y be two random variables, f (x) and f (y) the marginal probability distribution functions of X and Y respectively, and f (x, y) the joint probability distribution function of X and Y . The mutual information between X and Y is then:

I(X, Y ) = x,y f (x, y)log f (x, y) f (x)f (y)
X and Y are independent if I(X, Y ) = 0. between R, G and B color components for different color textures from Vistex database [START_REF]MIT vision and modeling group[END_REF]. It can be seen that a considerable dependence exists between those components.

Scatter plot test

Scatter plot is a graphical test for assessing dependence between variables.

More the points cluster in a band from lower left to upper right, higher the degree of dependence between these variables is. In (c) and (d), we measure the degree of dependence, when considering also the inter-band dependence, i.e the dependence between component subbands

in different orientations ({R o vs G o }).
Here, the scatter plots show that the degree of dependence is considerable (r = 0.82 for {R 3 vs G 4 }) even it is less than the inter-component one.

The dataset

In order to improve the characterization, we wish to exploit both inter-component and inter-band dependencies. Thus, the dataset considered for our model is Then, in our case, the dependence structure is represented by one 18-by-18 correlation matrix.

F (x 1 , ..., x d ) = P (X 1 ≤ x 1 , ..., X d ≤ x d )) (1)
Sklar's theorem [START_REF] Sklar | Fonctions de répartition à n dimensions et leurs marges[END_REF] shows that there exist a d-dimensional copula C such that:

F (x 1 , ..., x d ) = C(F 1 (x 1 ), ..., F d (x d )) (2)
Further, if C is continuous and differentiable, the copula density is given by:

c(u 1 , ..., u d ) = ∂ d C(u 1 , ..., u d ) ∂u 1 ...∂u d (3)
The joint pdf is then deduced uniquely from the margins and the copula density as follows:

f (x 1 , ..., x d ) = c(F 1 (x 1 ), ..., F d (x d )) d i=1 f i (x i ) (4)
where f i , i = 1, ..., d, represent the marginal densities.

In order to represent structure of dependence, one has in hand different families of copulas such as Archimedian (Fig. 4), Gaussian (Fig. 5) and t-Student (Fig. 6) copulas.

It appears that the Gaussian copula is suitable to model linear dependence which is the most popular in texture modeling and which is the case of the inter-band inter-component we study in this paper. Fig. 7 shows that dependence structure for subband coefficients from different components has an elliptical behavior that is well fitted by the Gaussian copula (Fig. 5).

The Gaussian copula density is given by:

c(u, Σ) = 1 |Σ| 1/2 exp[- 1 2 ϑ T (Σ -1 -I)ϑ] (5)
with

ϑ i = φ -1 (F i (x i )) ,

Generalized Gamma distribution

In [START_REF] El Maliani | Texture classification based on the Generalized Gamma distribution and the Dual Tree Complex Wavelet Transform[END_REF] we used the univariate Generalized Gamma density (GΓ ) in order to fit marginal distributions of the dual tree complex wavelet coefficient magnitudes.

Generalized Gamma was first introduced by Stacy [START_REF] Stacy | A Generalization of the Gamma Distribution[END_REF] as a generic distribution for modeling duration. In contrast to the Generalized Gaussian Density (GGD), Generalized Gamma has an additional shape parameter, which allows more flexibility in fitting larger classes of subbands histograms. Moreover, the Generalized Gamma provides more genericity since it covers GGD, Weibull, Gamma and a variety of well-known models as special cases. The probability density function of GΓ is defined as:

f (y; w) = τ λ ατ Γ (α) y ατ -1 exp -( y λ ) τ , y ≥ 0, α, τ, λ > 0 ( 6
)
where w = (α, τ, λ) denotes the GΓ parameters, α and τ are shape parameters, λ is the scale parameter, and Γ (.) is the Gamma function.

From Fig. 8, we can visually conclude that Generalized Gamma model gives a better marginal fitting in comparison with its special cases, i.e Gamma and

Weibull densities, considering one subband from a given texture. This behavior remains the same for the whole set of textures within the Vistex database as it is shown in Table 2, where we compute the KL divergence between marginal distributions of textures from the Vistex database and the parametric marginal fitting of our model and its special cases, such as:

KL(f m ||f ) = i f m (i)log f m (i) f (i) (7)
where f m represents the empirical marginal probability density function, and f denotes the parametric marginal probability density function (Weibull, Gamma or GΓ ). Results show that GΓ model offers the best marginal fitting since it minimizes the KL divergence. is based on a Gaussian copula in conjunction with the GΓ density. Thus, from equations ( 4) and ( 6), we derive the (MGΓ D) density as:

f (x, θ) = 1 |Σ| 1/2 exp[- 1 2 ϑ T (Σ -1 -I)ϑ]× ( τ λ ατ Γ (α) ) d exp[- d i=1 ( x i λ ) τ ] d i=1 x ατ -1 i ( 8
)
where θ = (α, τ, λ, Σ) denotes the hyperparameters of the joint model.

It is worth recalling here that the joint modeling presents big advantages over the marginal modeling in terms of consideration of the dependence information that exists between subband coefficients. Thus, it is natural that MGΓ D model is gainful compared to univariate GGD or GΓ models for example.

Parameter estimation 281

For estimating parameters of MGΓ D, we use the IFM (Inference From Mar-282 gins) method [START_REF] Joe | Multivariate Models and Dependence Concepts[END_REF]. Firstly, this consists in estimating the parameters of the 283 marginals using the Maximum Likelihood procedure. Let w i = (α i , τ i , λ i ) be 284 the parameters of the marginal f i . ML estimators ŵi are deduced as shown in

In the case of Gaussian Copula, Σ can be estimated by the following matrix:

289 Σ = 1 M M i=1 ϑ i ϑ i T ( 10
)
with ϑ i = (φ -1 (F 1 (x 1 ).. 295 Do and Vetterli [START_REF] Do | Wavelet-based texture retrieval using generalized Gaussian density and Kullback-Leibler distance[END_REF] proposed the KL divergence as a similarity measure between 296 parametric representations, when independence is supposed among transform 297 coefficients. However, deriving the KL divergence in the case of copula based 298 multivariate models is a challenging task. In [START_REF] Kwitt | A joint model of complex wavelet coefficients for texture retrieval[END_REF] and [START_REF] Sakji-Nsibi | Fast scalable retrieval of multispectral images with kullback-leibler divergence,textquotedblright in Image Processing[END_REF], authors proposed a Monte-carlo approximation of the KL divergence. But, this approach is computationally expensive and is not deterministic, since the KL divergence differs depending on the random number generation. An alternative approach was proposed in [START_REF] Kwitt | Efficient Texture Image Retrieval Using Copulas in a Bayesian Framework[END_REF], by employing the bayesian CBIR maximum likelihood selection rule as a similarity measure. This significantly reduced the execution time in comparison with the Monte-carlo based similarity measurement. However, authors stressed that, even if the joint copula based approach leads to better retrieval rates using the ML selection rule, the marginal approach excels in terms of the computational time due to the simple closed form expressions of the KL divergence when the marginal-only approach is considered. Hence, merging the joint approach with a closed form expression of the KL divergence as a similarity measure will be a major advantage.

In this work, we aim to exploit the copulas properties to come up with a closed form of the KL divergence between two MGΓ D models.

As it is known, the most attractive feature of the Copula approach is the separability between the marginal space and the dependence structure. From Sklar's theorem (subsection 3.1), we see that for continuous multivariate distribution functions, the margins and the dependence structure can be separated [START_REF] Bohdalova | A note to Copula Functions[END_REF]. That is, we can analyze the dependence structure of multivariate distributions without studying the marginal distribution. Moreover, it was proved in [START_REF] Embrechts | Correlation And Dependence In Risk Management: Properties And Pitfalls, RISK MANAGEMENT: VALUE AT RISK AND BEYOND[END_REF], that given a Copula C and under increasing and continuous functions of the marginals, C remains invariant (see Appendix A). So, the independency between the marginals space and the Copula space allows us to use the Kullback Leibler divergence for measuring similarity between two Gaussian copula based models. If we consider f (x; θ 1 ) and g(x; θ 2 ) two joint pdfs that respectively model two datasets T 1 and T 2 :

f (x; θ 1 ) = c(F 1 (x 1 ), ..., F d (x d )) d i=1 f i (x i ) (11)
and

g(x; θ 2 ) = c(F 1 (x 1 ), ..., F d (x d )) d i=1 g i (x i ) ( 12
)
as the sum of the KL divergences between the marginals and the Gaussian dependence structure.

Where

θ 1 = {(w (1)
1 , ..., w

d ), Σ 1 } and θ 2 = {(w (2) 1 , ..., w (1)
d ), Σ 2 } are the hyperparameters of f and g respectively.

KL(f (x; θ 1 ), g(x; θ 2 )) = KL margins (f (x; (w (1) 1 , ..., w (1)
d )), g(x; (w

(2) 1 , ..., w (2)
d ))+ KL Gaussian (f (x; Σ 1 ), g(x; Σ 2 )) (13)
This can also be proved mathematically, as shown in the recent work of Lasmar & al. [START_REF] Lasmar | Gaussian Copula Multivariate Modeling for Image Texture Retrieval Using Wavelet Transforms[END_REF]. That is,

KL(f (x; θ 1 ), g(x; θ 2 )) = d i=1 KL(f i (x i ; w (1) i ), g i (x i ; w (2) i )) +0.5(tr(Σ -1 2 Σ 1 )) + log |Σ 2 | |Σ 1 | -d) (14)
So, using the KL between two univariate GΓ pdfs [START_REF] El Maliani | Texture classification based on the Generalized Gamma distribution and the Dual Tree Complex Wavelet Transform[END_REF][10] we deduce a closed form of KL divergence between two MGΓ D models as:

KL(f (x; θ 1 ), g(x; θ 2 )) = d(log( τ 1 λ α2τ2 2 Γ (α 2 ) τ 2 λ α2τ2 1 Γ (α 1 ) )+ +( λ 1 λ 2 ) τ1 Γ (α 1 + τ2 τ1 ) Γ (α 1 ) + ψ(α 1 ) τ 1 (α 1 τ 1 -α 2 τ 2 ) -α 1 ) +0.5(tr(Σ -1 2 Σ 1 )) + log |Σ 2 | |Σ 1 | -d) (15)
This presents a huge advantage in terms of lower computational complexity when compared with the Monte-Carlo based approach proposed in [START_REF] Lasmar | Gaussian Copula Multivariate Modeling for Image Texture Retrieval Using Wavelet Transforms[END_REF].

Experimental Results

To evaluate the effectiveness of the proposed model we adopt color texture classification as an application. Note that the proposed model can be used for segmentation issue also. Our purpose is not to evaluate the performance of classifiers, we aim at quantifying the performance of the pair model/similarity, i.e. MGΓ D/KLD. Thus, we consider only one classifier. The K-nearest neighbor (KNN) was chosen among a variety of classifiers, since it is straightforward, widely used and well referenced in the literature [START_REF] Cover | Nearest neighbor pattern classification[END_REF]. KNN is a kind of instance-based classifier, where the main idea is that decision is achieved from K-nearest neighbors and letting to the majority vote decide the outcome of the class labeling. The decision is defined from a given similarity measure. For all of our experiments, we consider the DTCWT [START_REF] Kingsbury | The Dual-Tree Complex Wavelet Transform: A new Technique for Shift-Invariance and Directional Filters[END_REF] which is an oriented complex decomposition. In addition to its directional analysis, shift invariance and low redundancy properties, the DTCWT was chosen for its reduced computational time. The two-scale DTCWT with a Q-shift [START_REF] Song | Globally Convergent Algorithms for Estimating Generalized Gamma Distributions in Fast Signal and Image Processing[END_REF][START_REF] Song | Globally Convergent Algorithms for Estimating Generalized Gamma Distributions in Fast Signal and Image Processing[END_REF]-tap filter which is used.

For all color texture samples, every color band of each subimage was normalized by subtracting its mean and dividing by its standard deviation. We recall that the similarity measure is given by:

D(T 1 , T 2 ) = Ns s=1 KL(f (x T1 s , θ s 1 ), f (x T2 s , θ s 2 )) (16)

Experiments versus data diversity

In order to conduct representative experiments, we use different databases and various configurations of the experimental protocol leading to a large view of the proposed model performance in comparison with other ones from the state of the art. Firstly, the conventional Vistex databases [START_REF] Do | Wavelet-based texture retrieval using generalized Gaussian density and Kullback-Leibler distance[END_REF][START_REF] Kwitt | Image similarity measurement by Kullback-Leibler divergences between complex wavelet subband statistics for texture retrieval[END_REF][START_REF] Verdoolaege | Multiscale colour texture retrieval using the geodesic distance between multivariate Generalized Gaussian models[END_REF][START_REF] Kwitt | A joint model of complex wavelet coefficients for texture retrieval[END_REF][START_REF] Lasmar | Gaussian Copula Multivariate Modeling for Image Texture Retrieval Using Wavelet Transforms[END_REF][START_REF] Stitou | Copulas based multivariate gamma modeling for texture classification[END_REF] are used and also the Outex database which is a more challenging color texture database, since the color and texture information are not easily distinguishable.

Secondly, the experiments were conducted on two sizes of sample respectively 32 × 32 and 128 × 128. We consider the following scenarios:

-First scenario, i.e. DB1: 32 × 32 Vistex, addressing a set of 24 textured images of size 512 × 512 from the Vistex database [START_REF]MIT vision and modeling group[END_REF], shown in Fig. 11.

The protocol follows the work of [START_REF] Permuter | A study of gaussian mixture models of color and texture features for image classification and segmentation[END_REF] and [START_REF] Qazi | Color spectral analysis for spatial structure characterization of textures in IHLS color space[END_REF]. Each image was divided into subimages of size 32 × 32 pixels.We consider 96 from the resulting 256 subimages as the training set, while the remaining 160 subimages are considered as the test set. This dataset is used in order to evaluate robustness of our model even if very local spatial structures are considered.

-Second scenario, i.

Quantitative evaluation of performance

For evaluating performance, we repose on two criteria namely the percentage of classification and the precision (or predictive positivity) [START_REF] Dorizzi | ECG signal analysis through hidden Markov models[END_REF] to quantify informative measures respectively the true positive rate and false positive rate.

For each class of color textures, let TP be the number of true positives, i.e.

subimages correctly classified. We consider the two measures:

-The percentage of classification is the proportion of textures which were well labeled by the classifier over the number of false negatives, noted FN.

The false negative is the number of subimages being wrongly considered as not class members given a specific class of textures. The Percentage classification is given as follows: P recision = T P T P + F P .

P ercentage = T P T P + F N × 100%. (17)
For a given class, the system can achieve a percentage classification with a score of 100%, and a lower precision. This means that the classifier is good at labeling subimages of this class, but attributes other elements to this class while, in fact, they are not members of this latter. proposed by Verdoolaege et al. [START_REF] Verdoolaege | Multiscale colour texture retrieval using the geodesic distance between multivariate Generalized Gaussian models[END_REF]. We also compare the proposed approach with the marginal modeling approach proposed by Do et al. [START_REF] Do | Wavelet-based texture retrieval using generalized Gaussian density and Kullback-Leibler distance[END_REF] which assume independence between the color components. Let us give descriptions of these approaches:

Results in

-Stitou et al. : In this approach [START_REF] Stitou | Copulas based multivariate gamma modeling for texture classification[END_REF], authors proposed a Gamma based marginal modeling in conjunction with a Gaussian copula in order to characterize the local structure of wavelet subbands. We call this model, the Multivariate Gamma density (MGam). L1 distance was used as similarity measure for this model, however, we will use the KL divergence as a similarity measure between MGam features in order to have a comparative look of the models (MGΓ D vs MGam). For this, we use the expression of KL divergence between Gamma densities [START_REF] Kwitt | Image similarity measurement by Kullback-Leibler divergences between complex wavelet subband statistics for texture retrieval[END_REF] and the KL divergence of the Gaussian copula (Equation ( 13)).

- the case of grey level texture retrieval. For color textures characterization, the marginal approach consists in modeling each of the color components independently using GGD model and then concatenate the features in one global vector.

Table 4 and Table 5 show respectively the percentage classification and the precision of the KNN classifier for all 24 textures of DB1. We firstly remark the large gain of performances of the multivariate approach over the marginal GGD based approach. This approves our dependence observations in the RGB color space, and shows the huge loss of information when the correlation between color components is omitted.

The second main observation that can be deduced from Table 4 and Table 5, is the flexibility of the MGΓ D over the MWbl and MGam models which can be considered as special cases of MGΓ D. Our model gives better performances in term percentage and precision of the classification. This behavior remains the same for most of the 24 color textures of DB1, and is due to the genericity of the characterization that offers MGΓ D reposing on its additional shape parameter.

In Table 6 and Table 7, we show the average percentage classification and the average precision for DB1, DB2 and DB3. For all these databases, our Table 9 Execution time (in minutes) for calculating similarity measure matrix between the learning and testing set for DB1, DB2 and DB3 using MGΓ D model culty of distinguishing color and texture information in this database. We, also, clearly observe that our approach achieves higher rates in comparison with the MGGD presented in [START_REF] Verdoolaege | Multiscale colour texture retrieval using the geodesic distance between multivariate Generalized Gaussian models[END_REF]. This approves our consideration of non-Zero dependence across same color component subbands (see Fig. 3) since we consider inter-component and inter-band dependencies.

The second issue to be discussed is the similarity measure. We compare the proposed KL divergence with the Monte-Carlo based approach [7][8]. In [START_REF] Sakji-Nsibi | Fast scalable retrieval of multispectral images with kullback-leibler divergence,textquotedblright in Image Processing[END_REF],

Sakji-Nsibi et al. used the Monte-carlo based KL divergence, as a similarity measure between generalized Gaussian copula based GGD (CopGGD) feature representations. Table 8, presents average percentage classification using both similarity measure approaches for the MGΓ D, MWbl and CopGGD models, i.e. parametric closed form and Monte-Carlo method. The proposed similarity measure slightly outperforms the Monte-Carlo based approach in term of percentage classification. However, the improvement is more significative in term of execution time as can be clearly seen from Table 9. The computational time can be estimated as 12 times less when using our approach for measuring similarity between copula-based joint models. We note that the experiments were done using Matlab environment on an HP Compaq dc 5800SFF, equipped with an Intel Core 2 Duo CPU at 3GHZ and 1GB of RAM, with a 32-bit Windows vista operating system.

Conclusion

We have proposed a joint generic model for characterizing DTCWT coefficient magnitudes of color textures. MGΓ D presents a pertinent color texture description in comparison with the marginal approach that assumes independence among color component subbands. MGΓ D, also presents a flexible modeling when compared with variety of joint models. We, further, proposed a simple, faster and closed form expression similarity measure using the independence between the marginals space and the dependence structure. The genericity of the proposed model and the pertinence of the similarity measure allowed us to achieve good improvement in terms of classification rate and computational time.

In future works we would like to use copulas for characterizing color textures in luminance-chrominance color spaces such as L*a*b* or HSV. In such color spaces, the separation between luminance and chrominance information need to a multi-model characterization in order to improve classification rates.

  Since we are dealing with color textures in the RGB color space, from each image from the database, color components R, G and B are decomposed via the DTCWT. Let r os = |R s,o |, g os = |G s,o |, b os = |B s,o |, be the subbands representing magnitudes of the coefficients at a fixed scale s and an orientation o.

Fig. 1

 1 Fig. 1 Color texture Fabric.0015.

Fig. 2

 2 Fig. 2 Spatial structure of color components for texture Fabric.0015 on RGB and Yuv color spaces. (a) R, G and B spatial structures. (b) Y, u and v spatial structures.

Fig. 3 6 vs G 6 }

 366 Fig. 3 shows the scatter plots between color component detail subbands for second scale decomposition. The coefficients presented on Fig. 3 are the Pearson (r), Kendall (ρ) and Spearman (τ ) correlation coefficients. Three kinds of dependence between R, G and B component subbands are presented, namely the inter-component only dependence ((a) and (b)), the {inter-component, inter-band} dependence ((c)), and the inter-band only dependence ((d)). In (a) and (b), we observe a huge dependence between R o , G o and bB o subbands as illustrated by the Pearson coefficient (r = 0.98 for {R 6 vs G 6 } and r = 0.95 for {R 1 vs B 1 }).

  represented by the vector x s = [r os g os b os ], where o = {1, ..., N }, and N represents the number of orientations. Since we use the DTCWT as a wavelet decomposition (with 6 orientations per scale), we may have a dataset with 3 × 6 columns such as x s = [r 1s ... r 6s g 1s ... g 6s b 1s ... b 6s ].

Fig. 3

 3 Fig. 3 Scatter plots for different combinations of subbands. (a) and (b) show scatter plots of different color component subbands in a same orientation o (inter-component only dependence). (c) shows scatter plot of different color component subbands in different orientations (inter-component,inter-band dependence). (d) shows scatter plot of different subbands of the same color component (inter-band only dependence).

  and φ represents the standard normal cumulative distribution function. Σ denotes the correlation matrix, and I denotes the d-dimensional identity matrix.

Fig. 4

 4 Fig. 4 Left: the Archimedian copula pdf. Right: the structure of dependence generated from an Archimedian copula (with correlation parameter ρ = 0.1).

Fig. 5

 5 Fig. 5 Left: the Gaussian copula pdf. Right: the structure of dependence generated from a Gaussian copula (with correlation parameter ρ = 0.1).

Fig. 6

 6 Fig. 6 Left: the t-Student copula pdf. Right: the structure of dependence generated from a t-Student copula (with correlation parameter ρ = 0.1).

Fig. 7

 7 Fig. 7 Structure of inter-component,inter-band dependence between different subbands

3. 3 Fig. 8

 38 Fig.8Marginal fitting for texture 'Fabric.0015' from Vistex database, using different univariate models.

Fig. 10

 10 Fig.10shows scatter plots of original color subband coefficient magnitudes against samples from a MGΓ D modeling after estimating subbands combination between color channels (column 1), along with the joint fitting of the MGΓ D model on the empirical joint distributions of subband coefficients (column 2). We observe that the model fits very well the combination of subbands, either when these latter are extremely correlated (first line with correlation coefficient rho = 0.94) or when they exhibit a small correlation (second line with correlation coefficient rho = 0.44).

Fig. 9

 9 Fig. 9 Empirical joint distributions considering combinations of subbands of different color components.

Fig. 10

 10 Fig. 10 Column of the left: fitting of samples generated from MGΓ D model (red points) on original subband coefficients (blue points). Column of the right: fitting of the MGΓ D density (red line) on coefficients joint empirical density (blue line)

3 . 5

 35 .φ -1 (F d (x d )) T . M represents the number of observa-290 tions associated to multivariate vectors x d . 291 Similarity measurement 292 As in the context of texture retrieval, texture classification reposes on a perti-293 nent similarity measurement step, especially when opting for an instance-based 294 type of classifiers.

-

  e. DB2: 128 × 128 Vistex, addressing a set of 54 textured images of size 512 × 512 from the Vistex database, shown in Fig. 12. The protocol follows the work of [1]. Each image was divided into subimages of size 128 × 128 pixels. This dataset is available on the Outex web site [23] as test suit Contrib TC 00006. For each texture, subimages are considered to form a checkerboard. The white half of subimages is then considered as the training set and the black half is used as the testing set. Hence, the training procedure will account for non uniformity of the original images. Third scenario, i.e. DB3: 128 × 128 Outex, addressing a set of 68 textured images of size 746 × 538 from the Outex database, shown in Fig. 13. Each image was divided into 20 subimages of size 128 × 128 pixels. Training and testing sets are obtained as with DB2.

Fig. 11

 11 Fig. 11 24 texture classes from Vistex database

4. 3

 3 Average classification rates for the MGΓ D model over other existing models The first issue discussed in this experiment is to provide an experimental evidence of the flexibility and genericity of the proposed model over existing joint parametric models including the copula based Multivariate Gamma density proposed by Stitou et al. [25], the copula based Multivariate Weibull proposed by Kwitt et al. [7], and the Multivariate Generalized Gaussian density

  Kwitt et al.: In[START_REF] Kwitt | A joint model of complex wavelet coefficients for texture retrieval[END_REF], Kwitt et al. proposed a student t copula based multivariate Weibull (MWbl) model for characterizing magnitudes of complex subbands coefficients, besides cross color components dependency. As we already mentioned (subsection 3.5), a computationally expensive Monte-Carlo based approach was adopted to overcome the lack of closed-form of the KL divergence. In[START_REF] Kwitt | Efficient Texture Image Retrieval Using Copulas in a Bayesian Framework[END_REF], authors employed the ML selection rule of the Content Based Image Retrieval (CBIR) framework as an alternative of the Monte-carlo based approach, but without providing a closed-form expression of the KL divergence. Here we use our approach for measuring KL divergence between MWbl models (Equation (13)), in order to compare performances of the MGΓ D and MWbl models. -Verdoolaege et al.: In [5], Verdoolaege et al. proposed the Multivariate Generalized Gaussian (MGGD) model for color texture retrieval. Authors overcome the lack of closed-form expression of the KL divergence by proposing a closed-form expression of the geodesic distance. Verdoolaege et al. modeled dependency across subbands of the R,G and B color components while assuming independence between subbands of the same color component. We use the same parametrization in our experiment, in order to compare our approach to Verdoolaege et al. approach, in terms of the model (MGΓ D vs MGGD), the similarity measure (KL divergence vs Geodesic distance) and in term of the modeled dataset since we consider non-Zero dependence across subbands of a given color component. -Do et al.: The marginal modeling approach proposed by Do & Vetterli [1], has shown efficiency of the Generalized Gaussian density (GGD) in

Fig. 12

 12 Fig. 12 54 texture classes from Vistex database.

Fig. 13

 13 Fig. 13 68 texture classes from Outex database.

Table 1

 1

			shows the mutual information
		{R} vs {G}	{R} vs {B}	{G} vs {B}
	Bark.0000	7.72	7.57	7.59
	Brick.0000	6.13	5.88	5.82
	Fabric.0000	6.19	5.72	5.85
	Clouds.0000	6.14	5.62	5.46

Table 1

 1 Mutual information between R, G and B color components.

Table 2

 2 KLD between the empirical marginal pdf and the parametric ones for several textures.

		Bark Brick Buildings Flowers
	Weibull	0.23	0.22	0.85	0.33
	Gamma	0.12	0.08	0.43	0.19
	GΓ	0.08	0.06	0.35	0.16

Table 3

 3 Average percentage classification of the KNN classifier for different values of k on DB1, BD2 and DB3 using MGΓ D.

		k=1	k=2	k=3	k=4	k=5	k=6
	DB1	90.76 91.37 92.45 93.64	91.24 90.37
	DB2	94.94 95.47 96.72 97.22	96.59 95.36
	DB3	75.25 75.50 77.85 82.64	77.05 76.50

Table 4

 4 Table 3 indicate that better average percentage classification rates are achieved for k=4. Hence, for all the next results we fixe the value of k at 4 neighbors. Percentage classification of the KNN classifier for the MGΓ D, MWbl, MGam and MGGD models for 24 color textures (DB1).

		MGΓ D	MWbl	MGam MGGD	GGD
	Tex1	56.25	61.25	55.75	53.75	51.25
	Tex2	84.37	85.50	86.12	83.12	48.75
	Tex3	96.87	87.34	77.34	85.62	76.87
	Tex4	91.25	84.37	88.42	86.25	91.25
	Tex5	88.75	88.25	87.37	74.37	78.12
	Tex6	85.00	85.37	89.50	83.12	71.87
	Tex7	93.75	89.37	89.62	81.87	54.37
	Tex8	91.87	94.62	89.12	70.00	90.62
	Tex9	100.00	87.12	97.34	70.00	58.75
	Tex10	100.00	83.00	89.75	83.75	92.50
	Tex11	95.62	89.12	97.50	66.25	89.37
	Tex12	99.37	90.75	93.20	67.50	98.12
	Tex13	95.62	94	96.26	93.75	76.25
	Tex14	99.37	97.37	97.75	88.75	62.50
	Tex15	77.50	87.50	88.12	61.25	70.00
	Tex16	100.00	95.37	87.24	73.12	100.00
	Tex17	99.37	89.75	96.28	88.12	97.50
	Tex18	99.37	87.75	89.75	85.00	93.12
	Tex19	99.37	94.00	95.85	95.00	93.75
	Tex20	98.75	100.00	100.00	100.00	98.12
	Tex21	100.00	96.87	90.87	86.87	69.37
	Tex22	100.00	98.24	99.56	98.12	89.37
	Tex23	100.00	98.37	99.37	99.37	91.25
	Tex24	95	95.86	95.85	95.00	61.25
	Average	93.64	90.04	90.73	82.08	79.34

Table 5

 5 Precision of the KNN classifier for the MGΓ D, Mwbl, Mgam and MGGD models for 24 color textures (DB1).

		MGΓ D MWbl MGam MGGD GGD
	Tex1	1.00	0.97	0.95	0.94	0.98
	Tex2	0.83	0.73	0.81	0.80	0.39
	Tex3	1.00	1.00	1.00	0.89	1.00
	Tex4	0.99	0.95	0.80	0.97	0.95
	Tex5	1.00	0.98	0.83	0.82	0.87
	Tex6	0.94	0.83	0.93	0.90	0.76
	Tex7	0.81	0.78	0.82	0.90	0.54
	Tex8	0.98	0.99	0.98	0.67	0.93
	Tex9	0.96	0.98	0.96	0.81	0.68
	Tex10	0.98	1.00	0.98	0.89	0.81
	Tex11	0.88	0.83	0.86	0.81	0.61
	Tex12	1.00	0.98	0.94	0.83	0.98
	Tex13	0.95	0.88	0.90	0.64	0.80
	Tex14	0.85	0.84	0.81	0.89	0.71
	Tex15	0.97	0.73	0.95	0.58	0.78
	Tex16	1.00	1.00	1.00	0.45	0.93
	Tex17	0.75	0.80	0.76	0.62	0.70
	Tex18	0.89	0.95	0.90	0.74	0.90
	Tex19	0.92	0.98	0.91	0.97	0.89
	Tex20	0.90	0.92	0.90	0.95	0.76
	Tex21	0.88	0.86	0.87	0.90	0.72
	Tex22	1.00	1.00	1.00	0.86	0.71
	Tex23	0.97	0.93	0.98	0.93	0.84
	Tex24	0.91	0.75	0.88	0.99	0.79
	Average	0.93	0.90	0.90	0.82	0.79
		MGΓ D MWbl MGam MGGD GGD
	DB1	93.64	90.04	90.73	82.08	79.34
	DB2	97.22	95.45	95.80	89.75	84.12
	DB3	82.64	77.3	78.61	73.64	70.37

Table 6

 6 Average classification rate for three scenarios.

		MGΓ D MWbl MGam MGGD GGD
	DB1	0.93	0.90	0.90	0.82	0.79
	DB2	0.98	0.95	0.96	0.90	0.87
	DB3	0.85	0.80	0.82	0.80	0.78

Table 7

 7 Average precision for three scenarios.

		MGΓ D	MWbl CopGGD
	Proposed KLD	97.22	95.45	94.50
	Monte-Carlo KLD	96.45	94.70	94.37

Table 8

 8 Average classification rate for different models using Monte-Carlo based KLD and the proposed KLD.

		Proposed approach Monte-Carlo
	DB1	94.95	1.21×10 3
	DB2	4.9	60
	DB3	2.03	25.63

Appendix A.

286

Secondly, the log-likelihood function for the joint distribution is minimized A Supposing y = (y 1 , y 2 , ..., y M ), a set of M independent coefficients, The maximum likelihood function of the sample is defined as:

Thus, the parameters are deduced by solving a system of three equations:

where ψ denotes the digamma function. In [START_REF] El Maliani | Texture classification based on the Generalized Gamma distribution and the Dual Tree Complex Wavelet Transform[END_REF], we tackled the high nonlinearity of the ML equations by using a numerical approximation based on the algorithm of Cohen et al. [START_REF] Cohen | Parameter Estimation in Reliability and Life Span Models[END_REF]. However, a faster algorithm was proposed in [START_REF] Choy | Statistical Wavelet Subband Characterization Based on Generalized Gamma Density and Its Application in Texture Retrieval[END_REF], in which a Scale-Independent Shape Estimation (SISE) method is used to find roots of the ML equations.

B

The proposition presented in [START_REF] Embrechts | Correlation And Dependence In Risk Management: Properties And Pitfalls, RISK MANAGEMENT: VALUE AT RISK AND BEYOND[END_REF], shows one attractive feature of the Copula representation of dependence, namely that the dependence structure when modeled by a Copula is invariant under increasing and continuous transformations of the marginals.

If (x 1 , ..., xn) t has copula C and T 1 , ..., Tn are increasing continuous functions, then (T 1 (x 1 ), ..., Tn(xn)) t also has copula C.