Shahid Rahman

Michel Crubellier

Zoe Mcconaughey

Aristotle's ategorical Syllogisms as Dialectical Games Work in Progress

published or not. The documents may come

Introduction

Since the publications of Kurt Ebbinghaus (1964), John Corcoran (1972, 1974b) and Robin Smith (1984;1986, 1989), we learned that Aristotle's theory of categorical syllogism should be understood as a system of rules by the means of which a system of logic is declined. This new perspective contrastsand rightly so -with the axiomatic approach initiated by the work of Jan Lukasiewicz (1957). Recent work by Mathieu Marion and Helge Rückert (2015) shows that the notion of quantifier assumed in the Analytica has its roots in the Topica and more generally it looks as if the dialectical games of the Topica are intimately related to the notion of quantifiers deployed in the theory of syllogism. It is our main claim, that, though we agree with the insight that the theory of syllogism should be understood as a system of rules, these rules in fact can been as determining some specific dialectic games that extend the games of the Topica with inferential moves. Moreover, it is our view that a dialectical understanding of the theory of syllogism provides a uniform frame that links the Topica and the Analyticadespite Sir David Ross's (1949, p. 59)) stern dismissal of the Topics. In fact, our interpretation follows the path pointed out by Marion/Rückert (2015), that starts with Christian August Brandis (1833), and continues with Ernst Kapp (1942), andKurt von Fritz (1984), where the priority of Topics is reaffirmed, and Prior Analytics is understood in terms of its dialectical setting. In France, Brunschwig (1967) and recent work by Michel Crubellier (2008,2011,2014) and Crubellier and André Laks (2009a, b), also point in the same direction. Moreover, there are many commentators, ranging from, say, Le Blond (1939) to G. E. L. Owen's (1986) and Irwin (1999), that have argued in favour of the methodological importance of dialectic for Aristotle's philosophy as a whole. However, with the sole exception of the work of Marion and alii the precise description of the games have not been yet developed and neither has been yet shown how to develop the theory of syllogism within a dialectical setting. The latter is the main aim of the present study.

As mentioned above we read the methods of proof deployed in the Analytica Priora as extending the use of dialectical games of the Topica-based on the presentation of counterexamples and indirect demonstration -to games that include "inferential moves", that is, moves that force the player who endorses the premises of a perfect syllogism, to bring also forward the suitable conclusion. The precise link between the dialectical games of the Topica and those that structure the indirect and direct proofs of the Analytica Priora is what Mathieu Marion and Helge Rückert call the Socratic-Rule of the Topica (a notion that is very close to the so-called formal rule in contemporary dialogical logic). Besides, we claim that the Socratic-Rule links the Topica and the Analytica not only through the notion of quantifier but also through the qualification of the different roles of the players. Indeed, the distinction of the roles of two players responds two pairs of roles, namely: (i) the identification of the player whose moves are restricted by the Socratic rule and the one whose moves are not, (ii) the further distinction of a player whose thesis (theses) are to be tested (Proponent) and an antagonist who triggers the test (Opponent). The interplay between both of these pairs follows from the challenge of a contradiction. Notice that a purely game theoretic way to characterize a contradiction amount identifying a contradiction as a pair of the posits of, say, player X against which the adversary Y wins even when Y's own moves are restricted by the Socratic Rule: in order to win the adversary needs only to make himself use of the positive member of the pair, posited before by X, and challenge with it the negative member of the contradiction. Now, since in the Topica the main objective is to try to find a contradiction between P's assertions it is the moves of O that will be restricted by the Socratic-Rule. In contrast, the use of indirect proofs in the Analytica , is to allow P to prove that the conclusion at stake follows from some contradiction between O's posits. Thus, in the Analytica, it is P's rather than O's moves that are restricted by the Socratic-Rule. In other words, the identification of the player whose moves are restricted by the Socratic-Rule is dependent upon the identification of the player who challenges contradictions of his adversary under the terms established by this rule and in the context of the general aims of the game.

A not negligible point of our overall strategy is to stress the dialectical assumptions in the rule interpretation of Corcoran and Smith. As a matter of fact, in order to link the natural deduction interpretation with Aristotle's own style of demonstration Corcoran and Smith (following Corcoran) introduce operators such as "acceptance" and "But you have already accepted" . These operators that are alien to the natural deduction framework and that stem from a dialectical terminology express, as we will show in the following section; applications of the Socratic rule. Furthermore our perspective integrates quite naturally the indirect demonstration; the proof by ekthesis and the rejection strategy (that Smith associates with the construction of countermodels). More generally we claim that our approach explains Corcoran's worries about the apparent lack of interest of Aristotle for a type of semantics that we nowadays call, model theoretic. Indeed, the point is that the overall theory of meaning of Aristotle's Logic has deep game theoretical roots.

Let us address now an unavoidable issue that concerns most of the work on history of logic. Does it nowadays make sense to study the work of Aristotle in the context of the level of sophistication reached by the contemporary developments in logic? The answer is double, on one hand contemporary researches stress the dialectical aspects of logic and on the other recent studies of reasoning in natural language come back to the syllogistic form of deduction. Let us discuss briefly both points:

(i) As remarked by Clerbout/Rahman (2015) a brief examination on the most recent literature in logic will make it apparent that a host of research in this area is devoted to the study of the interface between games, logic and epistemology. These studies provide the basis of ongoing enquiries in the history and philosophy of logic, going from the Indian, the Greek, the Arabic, the Obligationes of the Middle Ages to the most contemporary developments in the fields of theoretical computer science, computational linguistics, artificial intelligence, social sciences and legal reasoning. In fact, a dynamic turn, as Johan van Benthem puts it, or more precisely a dialectical one is taking place where the epistemic aspects of inference are linked with game theoretical approaches to meaning where interaction play the central role. 1 In regard to the birth of this contemporary turn, it could be placed around the 1960's when Paul Lorenzen and Kuno Lorenz developed dialogical logic ---inspired by Wittgenstein's language games and mathematical game theory ---and when some time later on Jaakko Hintikka combined game-theoretical semantics with epistemic (modal) logic. If we were to pinpoint a precise date, the very beginnings of the dynamic turn could be situated in 1958, with Lorenzen's talk2 ``Logik und Agon''. More generally the dialectical approach is at the heart of the impressive programme of Pragmatist Inferentialism of Robert Brandom [1994,2000], according to whom it is the chain of commitments and entitlements in a game of giving and asking for reasons that deploys conceptual content by tightening up knowledge and inference:

A community precluded from giving reasons for beliefs cannot so much as have the concept of reliability-nor, accordingly (by anyone's lights), of knowledge. Its members can serve as measuring instruments-that is, reliable indicators-both of perceptible environing states and of one another's responses. But they cannot treat themselves or one another as doing that. For they do not discriminate between reliable indication and unreliable indication. Brandom (2000, p. 107).

Moreover, according to Brandom, games of asking for reasons and giving them constitute the base of any linguistic practice:

Sentences are expressions whose unembedded utterance performs a speech act such as making a claim, asking a question, or giving a command. Without expressions of this category, there can be no speech acts of any kind, and hence no specifically linguistic practice. Brandom (2000, p.125).

Nonetheless, it is evident that the contemporary so-called dialectcal turn has its roots in ancient Greece, where the agora emerged as the first public space for discussion and decision-making on diverse and serious matters, and where after the crucial influence of 1 New results in linear logic by J.-Y. Girard at the interfaces between mathematical game theory and proof theory on the one hand and between argumentation theory and logic on the other resulted in the work of, among others, S. Abramsky, J. van Benthem, A. Blass, H. van Ditmarsch, D. Gabbay, M. Hyland, W. Hodges, G. Japaridze, E. Krabbe, H. Prakken, G. Sandu, D. Walton and J. Woods, who placed game semantics at the center of a new concept of logic in which logic is understood as a dynamic instrument of inference. In this context see also Blass[1992], Abramsky & Mellies[1999], Girard [1999], Lecomte & Quatrini[2010,2011], Lecomte[2011], Lecomte & Tronçon[2011]. . the Sophists, of Plato and of Aristotle dialectical reasoning won a place in our understanding of science and constitution of a society which it ever lost any more.

(ii) In contemporary logic inferences are characterized from two main viewpoints the model-theoretic and the proof-theoretic one. By contrast most approaches to theory of inferences in natural language are of the model-theoretic kind. Koji Mineshima, Mitsuhiro Okada, and Ryo Takemura (2012) worked on a proof-theoretical approach to syllogistic reasoning that should provide an appealing and cognitively speaking efficient system to such kind of deductions carried out in natural language and that are close to those of diagrammatic reasoning.

It is our view, that Aristotle's work represent the interface between these two moments, the dialectical and the natural language one, and thus, beyond the obvious historical interest of studying its writings, we do think that a better understanding of his thought on these issues, despite the fact that it was developed in the gone golden era of Ancient Greece, will help us to render a fresh approach the notion of meaning as interaction, present in so many contemporary philosophical programs.

Published as Lorenzen[1960].


Redouane El Moubtahij
email: redouane.el.moubtahij@univ-poitiers.fr

B Augereau

H Tairi

C Fernandez-Maloigne

A spatial image polynomial decomposition with application to video classification

Keywords: Texture extraction, polynomial decomposition, dynamic texture, video classification

This paper addresses the use of orthogonal Polynomial Basis transform in video classification due to its multiple advantages, especially for multiscale and multiresolution analysis similar to the wavelet transform. In our approach, we benefit from these advantages to reduce the resolution of the video by using a multiscale/multiresolution decomposition, to define a new algorithm which decomposes a color image into geometry and texture component by projecting the image on a bivariate polynomial basis and considering the geometry component as the partial reconstruction and the texture component as the remaining part, and finally to model the features (like motion and texture) extracted from reduced image sequences by projecting them into a bivariate polynomial basis in order to construct a hybrid polynomial motion texture video descriptor. To evaluate our approach, we consider two visual recognition tasks, namely the classification of dynamic textures and recognition of human actions. The experimental section shows that the proposed approach achieves a perfect recognition rate in the Weizmann database and highest accuracy in the Dyntex++ database compared to existing methods.

Introduction

Video actions classification is one of the topics of current research in computer vision and pattern recognition. They can be found in many applications such as human action recognition, [START_REF] Weinland | A survey of vision-based methods for action representation, segmentation and recognition[END_REF] anomaly detection, [START_REF] Bertini | Multi-scale and real-time non-parametric approach for anomaly detection and localization[END_REF] face recognition [START_REF] Lu | Face recognition via weighted sparse representation[END_REF] and dynamic texture. [START_REF] Koleini | A new efficient method to characterize dynamic textures based on a two-phase texture and dynamism analysis[END_REF] Its ultimate objective is to develop a video model that is efficient, fast and simple to implement against complex motions and an amazing increase in database sizes. To achieve these objectives, we propose a new fast video classification tool that models the motion and texture of the video block after reducing image sizes by using polynomial basis transformations. Since this technique offers a compact and hierarchical representation of images, and offers several advantages like multiscale and multiresolution decomposition, it has become among the best known methods in the analysis and modeling of videos compared 1 with histograms of the orientation of gradient HOG, [START_REF] Klser | A spatio-temporal descriptor based on 3d-gradients[END_REF] histograms of the orientations of optical flow HOF 6 or spatio-temporal local binary patterns LBP-TOP. [START_REF] Kellokumpu | Human activity recognition using a dynamic texture based method[END_REF] The remainder of this paper is organized as follows: Section 2, discusses the most recent and popular video classification descriptor in the literature based on modeling of motion and texture features to describe the action in video; Section 3, presents our algorithm based on the polynomial transform and used in a video classification, firstly presenting the Polynomial Transform, then the decomposition step. After that, the use of our image decomposition to construct a video descriptor for human action recognition and dynamic texture classification; and finally Section 4, describes and discusses the proposed approach compared to existing methods in the state of the art.

Related Work

The main idea of this paper is to fuse the motion features and the color texture feature after extracting from video database to perform action classification (like human action and dynamical texture), so we review representative works based on modeling of these two different features respectively.

Motion Modeling Methods

Efros et al [START_REF] Efros | Recognizing action at a distance[END_REF] proposed a descriptor based on blurred measures optical flow to recognize the actions of the players at a football and tennis match : firstly, Lucas and Kanade optical flow algorithm [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF] is used to extract the motion information between two consecutive frames. Then, the half-wave rectification technique is adopted to decompose the vector field U and V to four components. The components are then blurred with a Gaussian smoothing. The vector image with four components forms the motion descriptor. The descriptors obtained in this way are compared by a normalized correlation. Using this measure, a motion similarity matrix is constructed and the classification is performed using a k-nearest neighbour classifier. This method is extended by Danafar and al 10 by using a shape estimation. They used Harris detector to find the area where the silhouette is located.

The background and silhouette are separated by using K-means algorithm. The silhouette is then divided into three horizontal parts: head and shoulders, body and arm then legs. The local optical flow for each part of the silhouette is modeled by the method proposed by Efros et al [START_REF] Efros | Recognizing action at a distance[END_REF] Recently, some frequency transformation approaches, [START_REF] Imtiaz | Action recognition algorithm based on optical flow and ransac in frequency domain[END_REF] wavelet-based approaches, [START_REF] Shao | A wavelet based local descriptor for human action recognition[END_REF][START_REF] Palafox | Human action recognition using wavelet signal analysis as an input in 4w1h[END_REF] and polynomial transformation approaches [START_REF] Druon | Modélisation du mouvement par polynômes orthogonaux : application à l'étude d'écoulements fluides[END_REF][START_REF] Kihl | Local polynomial space-time descriptors for actions classification[END_REF] have been shown to offer good recognition accuracy.

For example, in [START_REF] Shao | A wavelet based local descriptor for human action recognition[END_REF] wavelets are used for proposing local descriptors utilizing the capability in compacting and discriminating data, whereas in [START_REF] Palafox | Human action recognition using wavelet signal analysis as an input in 4w1h[END_REF] wavelet processing techniques are applied to solve the problem of real time processing as well as to filter the original signal in order to achieve better classification. For polynomial-based approaches, the descriptors are based on using the orthogonal polynomial basis to describe the motions, and it has been applied very recently for action modeling. The mathematical background to generate a Bivariate Polynomial Basis as following :

First, a Bivariate Polynomial (BP) of degree d is a function of x = (x 1 , x 2 ) ∈ R 2 given by P (x) = (d 1 ,d 2 )∈[0;d] 2 d 1 +d 2 ≤d a d 1 ,d 2 x d 1 1 x d 2 2
(1)

with any a d 1 ,d 2 ∈ R.
Considering a finite set of pairs

D = {(d 1 , d 2 )} ⊂ N 2 ,
K D = K d 1 ,d 2 (x) = x d 1 1 x d 2 2 (d 1 ,d 2 )∈D (2)
Obviously K D satisfies the linear independence and spanning conditions and so, K D is a basis of E D , the canonical basis. In our context of color image decomposition, we look for bases with more suitable properties such as orthogonality or normality. So, to construct a discrete orthonormal BP finite basis we first have to consider the underlying discrete domain

Ω = x (u,v) = x 1,(u,v) , x 2,(u,v) (u,v)∈D (3)
where D will now represent the set of pairs associated to Ω. Starting from K D we intend to construct a new orthonormal basis applying the Gram-Schmidt process. That implies that we need some product and norm for functions defined on Ω. Given two bivariate functions, F and G, their discrete extended scalar product is defined by

F |G = (u,v)∈D ω(x (u,v) ) F (x (u,v) ) G(x (u,v) ) (4)
with ω a real positive function over Ω [Legendre, Chebichev, Hermite, ...]. Then, the actual construction process of an orthonormal basis

B D,ω = {B d 1 ,d 2 } (d 1 ,d 2 )∈D (5)
is a recurrence upon (d 1 , d 2 )

T d 1 ,d 2 (x) = K d 1 ,d 2 (x) - (l 1 ,l 2 )≺ 2 (d 1 ,d 2 ) K d 1 ,d 2 |B l 1 ,l 2 ω B l 1 ,l 2 (x) (6) B d 1 ,d 2 (x) = T d 1 ,d 2 (x) |T d 1 ,d 2 | ω (7)
where ≺ 2 is the lexicographical order and | | ω the norm induced by | ω . The resulting set of B polynomials verifies

B d 1 ,d 2 |B l 1 ,l 2 ω =        0 if (d 1 , d 2 ) = (l 1 , l 2 ) 1 if (d 1 , d 2 ) = (l 1 , l 2 ) (8)
and so B D,ω is effectively an orthonormal basis with respect to a weighting function ω. A special case, later used in this paper, is the complete base where D exactly represents the set of pairs associated to Ω, that is

D = [0; N 1 ] × [0; N 2 ] (9)
The space E D being dense in the space of functions over Ω, it allows to well approximate any bivariate function I by an appropriate combination of elements of a B D,ω orthonormal basis

P I (x) = {(d 1 ,d 2 )}⊂D b d 1 ,d 2 B d 1 ,d 2 (x) (10)
where b d 1 ,d 2 is the scalar resulting of the projection b d 1 ,d 2 = I|B d 1 ,d 2 ω . In fact, with a complete orthonormal basis, the polynomial approximation of I is a first order osculatory polynomial interpolation : for all points of the domain we have P I (x) = I(x). An other nice property is that the projection on polynomial B d 1 ,d 2 can be considered as an approximation of the partial derivation

∂ d 1 ∂ d 2 .
Finally and in practice, the discrete projection process supposes that both I and B d 1 ,d 2 can be evaluated on the common domain Ω. So, the set of collocation points can be obtain by uniform or non-uniform discretization of given intervals. For example, with [-1; 1] 2 and referring to equation ( 9), the collocation points obtained by uniform discretization are

x 1,(u,v) = -1 + 2u N 1 x 2,(u,v) = -1 + 2v N 2 (11)
Based on this background, Druon et al [START_REF] Druon | Modélisation du mouvement par polynômes orthogonaux : application à l'étude d'écoulements fluides[END_REF] Their descriptor is based on two polynomial transformations: Spatial transformation is defined by projecting of displacement fields on a bivariate orthogonal polynomial basis and the temporal transformation is defined by projecting the temporal evolution of spatial polynomial coefficients on the univariate polynomial dimension basis.

Texture Modeling Methods

For a dynamic texture (DT) classification, several studies were carried out in the literature in various applications like video indexing, [START_REF] Dubois | Indexation de Textures Dynamiques à l'aide de Décompositions Multi-échelles[END_REF] spatial-temporal segmentation, [START_REF] Chen | Automatic dynamic texture segmentation using local descriptors and optical flow[END_REF] synthesis videos, [START_REF] Doretto | Dynamic textures[END_REF] etc.

They can be defined as a varying spatio-temporal phenomenon and having a spatial and temporal repeatability. Among the recent techniques developed for DT recognition, Xu et al [START_REF] Xu | Dynamic texture classification using dynamic fractal analysis[END_REF] have proposed a model based on a dynamic Fractal analysis to model DT with a multi-slice volumetric and dynamic Fractal spectrum. In, [START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF] Zaho et al. proposed a method for dynamic texture recognition. Their approach is based on modeling of texture with the LBP [START_REF] Otsuka | Feature extraction of temporal texture based on spatiotemporal motion trajectory[END_REF] in space-time volume. LBP method is also used in [START_REF] Ren | Dynamic texture recognition using enhanced lbp features[END_REF] to solve the unreliable information problem of LBP features by applying a Principal Histogram Analysis and a super histogram of each of all LBP histograms patch.

In more recent work, few authors were able to develop a descriptor that can classify the global video actions whatsoever human actions or dynamic textures. In [START_REF] Norouznezhad | Di-rectional space-time oriented gradients for 3d visual pattern analysis[END_REF] Ehsan et al used a directional spatio-temporal oriented gradients over nine symmetric planes for the classification of dynamic textures, human gestures and human actions. They proved the relevance of their descriptor and compared their method with the majority of descriptors existing in the action classification field.

Another method was based on non-linear Stationary Subspace Analysis to separate a stationary part of the video from its non-stationary parts as proposed by Mahsa et al in. [START_REF] Baktashmotlagh | Discriminative non-linear stationary subspace analysis for video classification[END_REF] Their technique is used especially for signals and has adapted to the video action classification. [START_REF] Baktashmotlagh | Discriminative non-linear stationary subspace analysis for video classification[END_REF]

Texture Extraction Models

Yves Meyer [START_REF] Meyer | Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures[END_REF] has proposed a model of image decomposition using the algorithm of Rudin-Osher-Fatemi. [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF] According to this model, an image is split in two parts, one containing the structure u, the other one containing the texture u. The result is provided by the minimization of the functional

F(u, v) = f F + λ g G ( 12
)
where f ∈ F , g ∈ G and λ is the parameter of the model. More precisely, F is the space of functions with bounded variations and G the space of oscillating functions with the property that more a function is oscillating, more its standard norm g G will be low. This model can be solved numerically due to the formulation proposed in J-F.Aujol, [START_REF] Aujol | Image decomposition: application to textured images and SAR images[END_REF][START_REF] Aujol | Structure-texture image decompositionmodeling, algorithms, and parameter selection[END_REF] by the introduction of an additional parameter µ corresponding to the maximum norm of textures in the space G. The use of non-linear projectors defined by A.Chambolle [START_REF] Chambolle | An algorithm for total variation minimization and applications[END_REF] provides the decomposition of the image by an iterative algorithm (see [START_REF] Aujol | Image decomposition: application to textured images and SAR images[END_REF][START_REF] Aujol | Structure-texture image decompositionmodeling, algorithms, and parameter selection[END_REF] for more details).

A. Buades [START_REF] Buades | Cartoon/Texture Image Decomposition[END_REF] has created a method that, as we know, is the fastest and most efficient implementation of the theory given Yves Meyer. [START_REF] Meyer | Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures[END_REF] It is a fast approximate solution to the original variational problem obtained by applying non-linear filtering to the image. For each image pixel, a decision is made whether it belongs to the geometric part or to the texture part. This decision is made by computing a local total variation of the image around the point, and comparing it to the local total variation after a low pass filter has been applied. In fact, edge points in an image tend to have a slowly varying local total variation when the image is convoluted by a low pass filter while textural points instead show a strong decay of their local total variation. After the selection of the points belonging to the geometrical part, the texture part is considered as the difference between the original image and the geometrical part. (See Figure 1 for image decomposition with Buades 31 method).

In fact, there is no unique decomposition and the algorithm relies on an important parameter, the scale parameter which is directly related to the granularity of textures distinguished.

Action Recognition Method

In this section, we describe our algorithm based on the polynomial transform and used for classification of videos. Firstly, we define the multiscale/multiresolution decomposition used to reduce This method provides some flexibility especially in the choice of the resolution factors, depending of the sub-domains size and of their offset, the transform can be perform with juxtaposed or overlapped sub-domains. Moreover, the choice of the weighting function ω allows, at the same time, to perform a multi-scale and a multiresolution transformation. As an illustration, Figure 2 shows an example of a first level transformation using a 4 × 4 Hermite complete basis.

Image Decomposition by Partial Reconstruction

A Polynomial Transform performed with a complete basis is perfectly reversible. However, it is possible to obtain many kind of approximations by selecting the coefficients during the reconstruction phase, i.e. partial reconstruction. This choice of coefficients may follow various strategies, among them we have : (a) brutal restriction to a given subset, for example the polynomials of degree less than a threshold; (b) restriction based on energies, for example by using the normality of the basis to assimilate the absolute value of its coefficients to a part of the energy of a sub-domain, then sort the coefficients and finally retain a fixed number of these coefficients or those satisfying a certain condition (cf. Principle Component Analysis (PCA)).

In our case, to decompose the image into geometric and texture component, we assume that the geometrical part is given by a partial reconstruction Ĩ of the original image I in an overlapped Polynomial Transform context. As seen before, this transform is very flexible, so there are many conceivable solutions. In order to get a compromise between quality and computation time, we choose to use an Hermite basis, to set the sub-domain size to 3 × 3, with an offset of 2, and to select the three dominant coefficients for each sub-domain. The partial reconstruction of a color image I = (I j ) j=1•••3 , i.e. the construction of the geometrical part, can then be summarize by

Ĩj (x) = 1 c(x) {Ω M x}   Ψ(Ω M ) ω(x M ) (d 1 ,d 2 )∈P j,M b j,d 1 ,d 2 (I j,M ) B d 1 ,d 2 (x M )   ( 13
)
where x is a point referring to the global image domain Ω, x M is the same point referring to a given sub-domain Ω M , I j,M the restriction of I j to sub-domain Ω M , P j,M the set of selected polynomials for I j,M approximation, ω the weighting function of the scalar product and b j,d 1 ,d 2 the coefficient of the projection of I j,M on the basis polynomial

B d 1 ,d 2 . A degree of anisotropy Ψ(Ω M ) is assigned to each sub-domain Ω M and c(x) is the sum of x contributions, c(x) = {Ω M x} Ψ(Ω M ) ω(x M ).
The degree of anisotropy is evaluated according to

Ψ(Ω M ) = 1 1 + λ r ( 14
)
where λ is the largest eigenvalue of a color structure tensor composed with the approximations of partial derivatives, projections on the basis polynomials of degree one

S =     j (b j,1,0 ) 2 j b j,1,0 b j,0,1 j b j,0,1 b j,1,0 j (b j,0,1 ) 2     (15)
The balance between isotropic and anisotropic reconstruction is adjust by the parameter r that controls the degree of anisotropy in a range of 0.25 for isotropic (gaussian) to 2 for highly isotropic.

By doing that, we assure a real color process and avoid marginal treatment deficiencies. Finally, the texture component I T is simply deduce from the partial reconstruction by considering that it is the residual part of the image

I T = I -Ĩ (16)
The results of image decomposition into geometry and texture components by partial reconstruction after an Hermite Polynomial Transform as defined in equation( 13), with the parameter r of equation ( 14) set to 0.75, are shown in Figure 3.

Construction of Video Descriptor

Our video descriptor is based on the modeling of motion and texture information that define the dynamic of the objects from video scene. The motion feature has been chosen because it defines a very important source of visual information. It provides information about the three-dimensional structure of the scene, the trajectory of objects and the activity occurring in the scene. Estimation of all this information requires a precise and relevant measurement of motion in image sequences.

To do this, we start by extracting the motion, notably optical flow, with a method based on color structure tensor, [START_REF] Augereau | Vectorial Computation of the Optical Flow in Color Image Sequences[END_REF] that describes the average local information of orientations and preserves the structure of the motion. A structure tensor corresponds to the product of the gradient vector by its transposed vector. Also, if one is located in a three-dimensional space, the corresponding structure tensor is a real symmetric matrix 3 × 3. The color structure tensor can be constructed by a global displacement tensor from the individual component tensors [START_REF] Benois-Pineau | Motion Estimation in Color Image Sequences[END_REF] (e.g. displacement tensor for each color component). This method allows to evaluate the co-variance between successive frames by describing the spatio-temporal structure of a given neighborhood. In addition, it provides information relative to the local velocity.

Depending on the good results obtained from the use of Polynomial transform for image de- in 3 dimensions. That which allowed us to reduce the video resolution and therefore reduce the computing time to build the descriptive vector so it can be used within a system of dynamic texture recognition. According to these transformations, we obtain an image representation with different scales as shown in Figure 2 for the two dimensional case. In our case, we arrived to represent a video block of 288 × 352 × 120 to images sequences fairly small of size 71 × 96. This can be achieved by projecting the video block into a trivariate polynomial basis with patch 5 × 3 × 8 as shown in Figure 4, which is why the concept of reducing the image size has been introduced.

For the primitive extraction step, we applied the image decomposition method defined in the subsection (3.2) on the reduced image sequences to extract the texture primitive. The motion is extracted directly on the original reduced image sequences by applying the color structure tensor method. Finally, the extracted texture component as well as the motion fields (u, v) would be projected thereafter on a two-dimensional Hermite basis and concatenated into descriptive vector for each video sequence as shown in Figure 5.

Experimental Results

To evaluate the effectiveness and robustness of our approach, we consider two difficult visual tasks of action recognition, namely the classification of dynamic textures and recognition of human actions represented respectively by two databases Dyntex++ [START_REF] Pteri | Dyntex: A comprehensive database of dynamic textures[END_REF] and Weizmann 36 (see Figure 6). A support vector machine with Libsvm library and a radial basis function (RBF) kernel is trained for classification. [START_REF] Chang | Libsvm: A library for support vector machines[END_REF]

Human Action Recognition

In order to classify the human actions, we use the Weizmann database outcome of the work of Gorelick and al, [START_REF] Blank | Actions as space-time shapes[END_REF] which is constituted of videos of a single person, consisting of 10 classes of human actions and performed by 9 different persons. This database is widely used by many authors for the validation of their results. We have chosen this database because it is relatively simple to study as the background of the video is relatively constant on all the videos. In addition, the actions are carried with a static camera (e.g. there is no camera motion or change of illumination).

There are two basic versions of Weizmann databases in the literature: A standard database of basic 10 -classes and a smaller version that does not include skipping class (Weizmann database with 9 classes). In our approach, we use these two databases and we apply the procedure shown in Figure 5 for each video sequence. Similar to, [START_REF] Brahnam | High performance set of features for human action classification[END_REF] all experience is performed using leave-one-out strategy. Figure 7 shows the average recognition rate relative to the size of the patch S 1 , S 2 and S 3

for multiresolution decomposition step (where S 1 = S 2 = S 3 = 3 in this case). The results of the classification are shown in Table 1.

Since we need polynomials coefficients b 0,0,0 , one can use a trivariate basis of degree d B = 1 in order to reduce the resolution size of the images and finally to have the most minimal execution time in multiresolution step. In order to get a compromise between quality and computation time [START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF] 95, 6 SIFT-3D (Scale Invariant Feature Transform) [START_REF] Scovanner | A 3-dimensional sift descriptor and its application to action recognition[END_REF] 82, 6 HOG-3D (Histogram of Oriented Gradients 3D) [START_REF] Klser | A spatio-temporal descriptor based on 3d-gradients[END_REF] 84, 3 OF+CTM (Optical Flow and Correlated Topic Model ) [START_REF] Tu | The complex action recognition via the correlated topic model[END_REF] 89.20 HOG-NSP (Histogram of Oriented Gradients with Nine Symmetry Planes) [START_REF] Norouznezhad | Di-rectional space-time oriented gradients for 3d visual pattern analysis[END_REF]

95, 9

Our approach 100

Recognition of Dynamic Textures

To illustrate the efficiency of our method, we will use it in a dynamic texture classification scheme is among challenging research themes in the image analysis field. The database used in our application is Dyntex ++ database [START_REF] Pteri | Dyntex: A comprehensive database of dynamic textures[END_REF] which is an improved version of the Dyntex database. [START_REF] Péteri | DynTex : a Comprehensive Database of Dynamic Textures[END_REF] Because it is composed of images that represent the textures in motion, it seems appropriate for the present study. Since the videos are not evenly distributed among the 36 classes in this database, we used the same experimental setting as that in [START_REF] Ghanem | Maximum margin distance learning for dynamic texture recognition[END_REF] in the evaluation (e.g without converting the video sequences in grey level) to have the same sequence number of each class. We finally have 3600 dynamic textures, grouped into 36 classes and each sequence have a size of 50 × 50 × 50 pixels.

Thereafter, the data were randomly split into two equal size training and test sets. The random split was repeated 10 times and the average classification accuracy is reported in Table (2).

In • S 1 = height of image and S 2 = width of image.

• d B U,V = 2 and d B T = 2.
The average recognition rate relative to overlap image number and the size of the patch for multiresolution step are shown respectively in Figure 8 and Figure 9. NLSSA (Non-Linear Stationary Subspace Analysis) [START_REF] Baktashmotlagh | Discriminative non-linear stationary subspace analysis for video classification[END_REF] 92.4%

Our method [START_REF] Moubtahij | A polynomial texture extraction with application in dynamic texture classification[END_REF] 93, 13%

From Table 1, we can see that our descriptor is much better than the methods very well known in the field of recognition of human action. We can also see in Table 2 that the descriptor is robustness with the database Dyntex++ where the recognition rate can be ranked among the best rates found in recent methods dealing with the classification of dynamic textures. As a final evaluation, we compare the computational time of our motion texture descriptor against other state-of-the-art video descriptors such as HOG-NSP, 24 3D-SIFT [START_REF] Scovanner | A 3-dimensional sift descriptor and its application to action recognition[END_REF] and LBP-TOP [START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF][START_REF] Ren | Dynamic texture recognition using enhanced lbp features[END_REF] on a set of videos with resolution 160 × 120. For our descriptor, experiments were developed in C++/Matlab environment and performed on an Intel 2.40GHz, whereas for all other descriptors, their experiences are developed and mentioned in. [START_REF] Ren | Dynamic texture recognition using enhanced lbp features[END_REF] Table 3 presents the average run-time for all descriptors. The proposed descriptor is more than 5, 5s times faster than LBP-TOP, HOG-NSP, 3D-SIFT and Morphological Table 3 Average run-time (sec) for computing of our descriptor, LBP-TOP [START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF][START_REF] Ren | Dynamic texture recognition using enhanced lbp features[END_REF] , HOG-NSP [START_REF] Norouznezhad | Di-rectional space-time oriented gradients for 3d visual pattern analysis[END_REF] and 3D-SIFT [START_REF] Scovanner | A 3-dimensional sift descriptor and its application to action recognition[END_REF] descriptors on a video with 160 × 120.

Descriptor Our Descriptor LBP-TOP HOG-NSP 3D-SIFT MCA Run-time (s) 5, 5 7, 4 8, 6 74, 7 7200

Component Analysis MCA [START_REF] Dubois | Analyse de Textures Dynamiques par décompositions spatio-temporelles: application à l'estimation du mouvement global[END_REF] because all main computations can be realized through convolutions.

Besides that, the computational time with MCA is around 2 hours, it serves to decompose an video of the Dyntex database [START_REF] Péteri | DynTex : a Comprehensive Database of Dynamic Textures[END_REF] into geometric and texture component with size 648 × 540 on a classical computer, which is equal to 3, 65s if we use our polynomial decomposition method.

Conclusion

In this paper, we have proposed a new approach for texture extraction from color image sequence, by using Polynomial Transformations. Partial reconstruction and global approximation are used to build descriptors used in a classification process of dynamic textures. In addition to the simplicity of implementation, we provide a computing time which is especially fast compared to most of methods based on the theory of Yves Meyer due to the cost of the minimization of the total variation. The experimental results show that the proposed approach achieves a very good recognition rate for the Dyntex++ database. This shows the relevance of our texture extraction method in the context of classification of dynamic textures. In some future, we will continue to improve our image decomposition method in order to extract the noise coefficients ignored in the partial reconstruction of the image. We will also investigate the abilities of a derived method which only relies on three dimensional transformations in our classification process of dynamic textures.

  to recognize simple behaviors. Mikolajczyk and Uemura 11 use optical flow information to form a vocabulary forest of local motion-appearance features to recognize actions.

  in 2009 are used the orthogonal polynomial basis for human action recognition. They introduced a polynomial modeling method based on projection into a bivariate polynomial basis and used the polynomial coefficients to analyze the displacement fields from experimental fluid mechanics. Encouraged by their positive results, O. Khil et al 16 in 2013 are based on this formalism to create a spatio-temporal descriptor for human action recognition.

Fig 1 Fig 2

 12 Fig 1 A. Buades method with scale parameter set to 3 : (a and d) original images, (b and e) images of geometry, (c and f) images of texture.

Fig 3

 3 Fig 3 Our decomposition method with r = 0.75 : (a and d) original images, (b and e) images of geometry, (c and f) images of texture.

Fig 4

 4 Fig 4 A three-dimensional multiresolution decomposition : (a) A 3D volume with size 288 × 352 × 8, (b) Reduced image after a multiresolution decomposition by projecting (a) into a three-dimensional Hermite basis with degree d B = 1 and support 3 × 5 × 8 (a patch of color blue).

Fig 5 Fig 6

 56 Fig 5 Spatial Polynomial Motion texture descriptor(SPMT)

Fig 7

 7 Fig 7 The recognition rate versus the support size of 3D Hermite basis for a multiresolution step in Weizmann dataset.

  in the texture extracting step, we choose to use a Hermite basis, to set the sub-domain size to 5 × 5, with an offset of 2. Subsequently, we model the U and V of motion and T of texture by projecting them into a two-dimensional basis with the following settings: • S 1 = (height of image)/3 and S 2 = (width of image)/3. • d B U,V = 6 and d B T = 4. where d B U,V and d B T are the degree of the two-dimensional basis to model U ,V and T respectively.

Fig 8

 8 Fig 8 The recognition rate versus the image overlap in multiresolution decomposition step applied to Dyntex++ database.

  the step of multiresolution decomposition, we project each sequence on a trivariate polynomial basis of degree d B = 1 with a patch of size 3 × 3 × 8 with overlapping temporal of 2 images to reduce the images sizes 16 × 16 instead of 50 × 50. Then, in the texture extracting step, the best results are obtained by projecting the reduced images into a Hermite bivariate basis of degree d B = 2 and a patch S 1 = 3, S 2 = 3. Finally, the features extracted such as motion (U and V ) and texture (T ) are modeled by projecting them into a Hermite bivariate basis with the following parameters:

Fig 9

 9 Fig 9 The recognition rate versus the support size of 3D Hermite basis for multiresolution step Dyntex++ database.

  we represent by E D the space of all BP such as a d 1 ,d 2 ≡ 0 if (d 1 , d 2 ) / ∈ D and by K D the subset of monomials

Table 1

 1 Classification results of the Weizmann database

	Class	Method	Accuracy
	number		(%)
		Fusion (Gabor filter + LBP + HOG) 38	97, 8
		LBP-Top (Local Binary Patterns on Three Orthogonal Planes)
	10 classes

Table 2

 2 Classification results for Dyntex++ database.

VLBP (Volume Local Binary Patterns)

[START_REF] Zhao | Dynamic texture recognition using volume local binary patterns[END_REF]

61.1%

SIFT-3D (3 Dimensional Scale Invariant Feature Transform)

[START_REF] Scovanner | A 3-dimensional sift descriptor and its application to action recognition[END_REF]

63.7%

LBP-TOP(Local Binary Patterns on Three Orthogonal Planes)

[START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF][START_REF] Norouznezhad | Di-rectional space-time oriented gradients for 3d visual pattern analysis[END_REF]

71.2%

DFS (Dynamic Fractal Spectrum)

[START_REF] Xu | Dynamic texture classification using dynamic fractal analysis[END_REF]

89, 9%

HOG-NSP (Histogram of Oriented Gradients with Nine Symmetry Planes)

[START_REF] Norouznezhad | Di-rectional space-time oriented gradients for 3d visual pattern analysis[END_REF]

90, 1%

Biographies and photographs of the other authors are not available. 3 Average run-time (sec) for computing of our descriptor, LBP-TOP [START_REF] Zhao | Dynamic texture recognition using local binary patterns with an application to facial expressions[END_REF][START_REF] Ren | Dynamic texture recognition using enhanced lbp features[END_REF] , HOG-NSP [START_REF] Norouznezhad | Di-rectional space-time oriented gradients for 3d visual pattern analysis[END_REF] and 3D-SIFT [START_REF] Scovanner | A 3-dimensional sift descriptor and its application to action recognition[END_REF] descriptors on a video with 160 × 120.

List of Figures