Myriam Matray
email: myriam.matray@gmail.com

Jacques Poisat
email: jacques.poisat@univ-st-etienne.fr

Territorial clusters of economic cooperation: a new attempt to build entrepreneurial and institutional partnerships within a social economy?

Keywords: deliberation, entrepreneurship, governance, local authorities, social innovation, TCEC

Coming from the civil society and entrepreneurial processes in a bottom-up strategy, territorial clusters of economic cooperation (inspired by French economic clusters) could not emerge and develop without support from public institutions. Indeed, jointly developed local strategies by groups of citizens and institutions tend to foster the emergence of groups of social actors. The study of TCEC in the Rhone-Alps Region highlights the many practical forms that the involvement of public institutions can take in the development of a social cluster, and shows the strategic importance of reaching an agreement, both from the social economy and local public institutions point of views. However the forms of public-civil society governance differ according to the types of clusters and their evolution.

As the political approach to economic problems shows [START_REF] Ostrom | Governing the Commons: The Evolution of Institutions for Collective Action[END_REF], local development projects are partly based on the ability of social actors to create collective intelligence [START_REF] Heurgon | Le Développement durable, c'est enfin du bonheur ![END_REF] through deliberation in public spaces [START_REF] Habermas | L'Espace public[END_REF], involving all stakeholders, including Universities [START_REF] Goujon | L'université citoyenne et solidaire: une modalité exploratoire de co-construction du développement local dans une région en reconversion, 57 ème congrès de l[END_REF]. However, local democracy cannot be imposed and collective intelligence requires broad mobilization. Consequently, the questions of the actors' coordination and of the emergence of new forms of regulations in territories become strategic, including in the social sector.

Thus, after a long period of social innovations, the organizations in social and solidarity economy (SSE) experiment new forms of coordination between actors, companies and local authorities to pool resources and develop cooperative projects in territories. For example, the concept of "territorial clusters of economic cooperation" (TCEC), which was inspired by the French competitiveness clusters launched in 2005, has recently emerged under the influence of the main networks of SSE. This approach was backed by public authorities.

In July 2013, a first government call for projects ("Support to the dynamics of TCEC"), with a budget of three millions euros, allowed to finance 23 clusters (including 4 in the Rhône-Alps Region) among 180 candidates. A second call for projects was launched on April 21 st , 2015. Above all, the TCEC were defined by the law of July 31 st , 2014 on social and solidarity economy: "territorial clusters for economic cooperation are defined by the association of local social businesses and local companies, in connection with local authorities, associations, research centers, higher education and research institutions, training organizations or any other person or entity, in order to implement a common and continuous strategy based on sharing, cooperation or partnership for innovative economic and social projects for local sustainable development." So, TCEC are part of the territorial strategies supported by the French State. In fact, the evolution of TCEC led to the reconsideration of the issue of the governance, understood as the coordination between actors. The approach by local collective strategies [START_REF] Gundolf | Emergence et pérennité des stratégies collectives territorialisées: le rôle de l'entrepreneuriat collectif[END_REF][START_REF] Poisson | Déterminants et processus d'émergence des systèmes agroalimentaires localisés alternatifs[END_REF] shows that entrepreneurial / citizen and institutional processes combine to bring out collectives. If the dynamics of local actors, indeed, is essential in a bottom-up logic, public institutions play a key role. In other words, coming from the civil society and entrepreneurial dynamics, TCEC could not emerge and develop without public authorities.

The aim of this paper is to analyze the cooperation between the various TCEC and the local public institutions in order to identify the keys elements of effective governance.

Our analysis is based on interviews with TCEC leaders in the Rhone-Alps Region, as well as our participative observation of social innovations in the Roanne area. As researchers, we have been following these initiatives as part of a project, financed by the Rhone-Alps Region, called "Citizen and Solidarity University", and since 2013, as part of the ARC8 academic research community in the Rhône-Alps Region. Based on a new cooperation between stakeholders and researchers, this operational working group has identified several key criteria for collaboration between members of the TCEC: the importance of the place, role of each stakeholder, institutional proximity and promotion of the cluster. Regarding the cooperation between the TCEC and the public institutions, the challenge is to consolidate existing partnerships and to initiate new collaborations. This momentum is based on a constructive partnership with local stakeholders, a complementarity between TCEC and local needs as well as an active business intelligence.

After analyzing the many practical forms that the involvement of public institutions can take in the development of TCEC, we will see that the forms of public-civil society governance differ according to the types of clusters and their evolution.

The general framework of the cooperation between institutions and TCEC

Many institutions are involved in the TCEC:

-the State, -local authorities (regions, "départements", municipalities), -semi-public institutions (association for employment), -industrial and commercial public institutions (Chambers of Commerce), -national and regional chambers in charge of SSE interests, -national federations of SSE, -research centers, higher education and research institutions, training organizations, -the European Union (structural funds) Since the 80s, French public authorities have taken an interest in SSE as a way to formalize social innovations. Today many institutions implement and coordinate the development of the SSE sector.

1.1.

A fundamental question: the sharing of power between institutions and TCEC

With the decentralization policies of the 80s in France, a new form of governance has grown and transformed the public forms of governance into a partnership between public authorities and local networks, including those of the civil society. Institutions play a leading role as coordinators in the territory. In the same way, TCEC are SSE organizations which implement a logic of territory development that come from their promoters as well as from various institutions. However, the modus operandi of institutions and TCEC varies, especially as public interventions are more and more regulated by market mechanisms. Consequently, TCEC gradually have to define their own self-financing model (Matray, Poisat, 2015).

The characteristic of TCEC, and more widely of the SSE sector, lies in the plurality of private and public actors that need to come together to create a collective project. In that regard institutions play a role of facilitator and collaborate as partners within a network of local actors, according to the mode of "partnership governance" [START_REF] Enjolras | Le pôle territorial de coopération économique : Une forme originale de polarisation en économie sociale et solidaire ? XVes Rencontres du RIUESS[END_REF]. However, a recurring question must be considered: do institutions have to be members of the TCEC or can they remain partners? In fact, the institutions' involvements vary according to their competences.

-The Rhone-Alps Region subsidizes the TCEC in order to develop the SSE and social innovations. But the Region may assign missions of economic development to TCEC, with the risk of instrumentalization of these clusters. -Being closer to the project managers of the clusters, institutions take part in the professionalization of actors as well as in the evaluation of their actions. In some TCEC, for example, council representatives are part of the board of directors. It doesn't necessarily mean that those TCEC are being used by the public authorities. Indeed, they play different roles and are quite independent. On the other hand, other TCEC set up technical committees involving civil servants from local councils. In that case, they are often confronted with coordination problems (red tape, subdivision of municipal services, frequent changes of project managers in local councils), and their legitimacy are often questioned when local councils assign public service missions to TCEC. The cooperation between institutions and TCEC is inherent to the clusters' projects and indirectly to the SSE's projects in the territory. However, cooperation is difficult to implement because of the risks of power games between the various public and private actors, even within this non profit framework.

Cooperation practices between institutions and TCEC

In France, 23 TCEC benefitted from the first government call for projects in 2013 and were supported by the State. According to the General Secretary for Modernization of Public Action "local authorities were indeed already heavily involved in the structuring of these clusters, through financial support and / or engineering, and continue to be." Ceteris paribus, the first government call for projects, had a relatively modest public budget (€ 3 millions). At the same time, the majority of TCEC were granted public funding for their local SSE actions from local authorities.

Prospective and networking

For some TCEC (such as Pollens, Loire) the role of local institutions preceded the creation of this TCEC. These have involved a large number of SSE organizations and members of the local economic, social and academic worlds in the emergence of the cluster. In this case, local institutions acted as facilitators as they identified and put together an SSE network, which is an essential step when generating a TCEC. Therefore, local institutions are in charge of identifying emerging needs in the territory.

In effect, local public authorities facilitate the creation of networks since they serve the territory and its inhabitants, like the TCEC. Therefore both cooperate with the same actors, and the involvement of institutions in TCEC projects enable them to be legitimized in the eyes of local actors. Furthermore, as TCEC implement a horizontal approach, they interact differently with the local actors. The TCEC follow a structuring logic of vertical organization (TCEC Ulisse, Isère; TCEC InnoVales, Haute Savoie/Ain) or a more horizontal logic of territorial complementarity (TCEC Smac 07, Ardèche).

In all cases, TCEC play a major role in planning networking and territorial structure, and in particular they reduce the "structural holes" [START_REF] Burt | Structural holes. The social structure of competition[END_REF] in social networks, hence the interest of local public institutions in accompanying the TCEC. The first government call for projects funded the projects which fitted their program on priority areas of social policy. The second government call for projects had the same objective and encouraged TCEC' socially and technologically innovative projects, and especially in areas supported by the public policies of the National Steering Committee (for example : priority areas of the social policy, deindustrialised areas, overseas territories, academic support, guidance and support for young people, short circuits, organic agriculture…).

Integrating TCEC in local public policies

In most cases, public financing enables the emergence and development of TCEC. Municipalities / local authorities generally finance clusters by taking over the coordination of the structure. Such financing agreements are granted on condition that the TCEC develop a strategic plan for sustainable local development. Gradually, a partnership grows between local institutions and the TCEC, which begin to appear in the programs of "départements" and regions, for example in the Regional economic development Schemes or in the Regional Innovation Strategies (Cornu, 2014, pp.28-30).

Direct and indirect interventions of institutions

Direct interventions of institutions take the form of subsidies and collaborative infrastructures. The survey of the "Network of local authorities for a solidarity economy"1 underlines the strategic importance of the place as a tool for identification and visibility, cooperation/mutualisation area, exchanges facilitator, synergies, and social innovations (Cornu, 2014, pp.34). The collaborative infrastructure projects are supported by local institutions that enroll TCEC in territorial planning policies. For example, the TCEC "Culture and Cooperation" (Loire) is a cultural cluster. It offers to its members shared administrative services (Human Resources mutualisation) and takes part in the emergence of innovative activities. Thus "Culture and Cooperation" initiated the development of Arts / Science / Design coworking premises called "Le Mixeur" (collaborative R&D laboratory, FabLab…), in partnership with local councils. In other cases, local councils lend premises to the TCEC free of charge (Pollens, Loire).

However, real estate investment is not enough to create a dynamic cooperation, which requires financial resources and engineering.

Beyond financial support, local public institutions offer skills in technical and administrative engineering, in order to help the emergence of TCEC, such as : collaboration with a project manager, structuring local SSE initiatives (conventions with Regions and/or the State), professionalization of actors (including training…)… Thus, some TCEC offer training periods and develop partnerships with universities as Master'2 in the TCEC Lussas (Ardèche) which is specialized in documentaries.

The limits of cooperation between institutions and TCEC

The interactions between institutions and TCEC have several limits. On the one hand, local authorities are organized vertically around a function (job service, housing service, health service, etc.) and not around an interdisciplinary problem (social exclusion, energy poverty…) like TCEC. Horizontal links between these functions are low, and greatly complicate the tasks of TCEC. For example, the TCEC Ulisse, which fights against energy poverty, must simultaneously apply to different town departments to implement a project. The same difficulties also exist to obtain public subsidies that depend on several budgets for the same project.

On the other hand, integrating TCEC in local public policies does limit their independence, especially when they have to apply for public contracts. Moreover, the second ministerial call for projects, in 2015, required that beneficiaries include at least one conventional business, which would influence the actions and the managerial choices of these TCEC. There must be a right balance between institutions / businesses/ SSE actors and the civil society. Eventually, TCEC have self-financing objective but are aware that their activities cannot be entirely selffinanced. So, they look for support from foundations or for patronage. Such entrepreneurial practices may gradually influence the TCEC forms of governance, which are based on participative democracy (Matray, Poisat, 2015). In that regard, the TCEC must maintain the original sense of their mission and their independence.

Furthermore, the electoral constraints can disturb and/or interrupt the projects of TCEC; that is why some of them try to sign public agreements to protect themselves. In this case, the cooperation between TCEC and academic researchers, through regional programs (like the ARC8 Rhone-Alps research community), may consolidate partnership with the public institutions.

Besides, institutions are sometimes wary about some TCEC projects because: "Institutions must learn to accept that TCEC projects are uncertain since these are "collective experiences"" (Colin Lemaitre, TCEC Culture and Cooperation).

Basically, the involvement of public institutions varies according to the type of TCEC

Variations of public-civil society governance methods according to the types of TCEC

Today, if public institutions are extensively involved in many ways in the emergence and development of territorial clusters of economic cooperation (TCEC), the importance and forms of collaborations, partnerships, interactions differ according to the types of clusters and their evolution.

From questioning the impact of TCEC on social and solidarity based economy...

In previous papers, we questionned the contribution of the TCEC approach to the evolution of the SSE. Don't the TCEC tend to be instrumentalized, institutionalized when they integrate local public policies and don't they become an instrument of pure economic development [START_REF] Matray | Les pôles de coordination de l'économie sociale et solidaire dans un territoire au service de l'innovation sociale et de l'émergence de nouveaux projets. Deux exemples dans le Roannais[END_REF][START_REF] Cornu | Les rôles des collectivités locales dans les dynamiques de PTCE[END_REF][START_REF] Enjolras | Le pôle territorial de coopération économique : Une forme originale de polarisation en économie sociale et solidaire ? XVes Rencontres du RIUESS[END_REF]? The adoption by some TCEC of technocratic managerial logic, which characterizes economic clusters, does risk obscuring the political dimension of the SSE as an alternative to capitalism. However, case studies have revealed significant differences in the position and evolution between the clusters. So we have proposed, as a hypothesis, to classify TCEC into four "ideal types," according to two axes (Matray, Poisat, 2015).

The first axis involves the activities which are initiated and / or coordinated by the clusters for sustainable territorial development. These activities fall within the commercial sphere, are financed by the market and / or by the redistribution (State, local authorities) and / or the reciprocity (civil society).

The second axis refers to the style of governance, i.e the interactions between civil society and public authorities [START_REF] Enjolras | Le pôle territorial de coopération économique : Une forme originale de polarisation en économie sociale et solidaire ? XVes Rencontres du RIUESS[END_REF] 2 . Do TCEC favor a technocratic method of decisions, which associates only leaders, elected representatives and experts from public or private spheres? On the contrary, do they try to seek a deliberative way to practice economy, to involve all stakeholders, recipients of solidarity actions, citizens and employees in the governance? In fact, many combinations are possible. Our target is to identify the position of each TCEC in what seems to be the heart of the political dimension of solidarity economy: the organization of citizen deliberation, understood as a process to buildcommon standards through the confrontation of different points of view expressed by actors with equal rights (Dacheux, Goujon,2013).

... to the sketch of a typology

We finally identified four major types of TCEC, with regard to the design of the SSE that emerges from their activities and their governance. This classification seems relevant to characterize more precisely the interactions with institutions. Types 1 and 2 clusters develop few activities that can be financed by the market. Their emergence and development therefore depend greatly on their ability to legitimize the projects of their members in the eyes of public institutions.

1) The "social" cluster is essentially part of a citizen response to the social needs of the territory. As it develops few commercial activities, it does not include capitalist firms. Acting more like a developing pole of SSE, it primarily seeks to bring together actors, to support project developers and to promote a solidarity-based economy. The active and financial partnership with local authorities is fundamental, both in the emergence of the cluster, when institutions sometimes play the role of initiator, and during its development. However, the "social" cluster should seek to maintain its operational independence towards institutions and to preserve its freedom of decision through its own deliberative processes. In that regard, it is more in accordance with the alternative political conception of SSE and advocates deliberation as governance. Its aim is not only to involve all stakeholders -including institutions -in democratic modes of decision-making, strategy defining and to develop collective intelligence in the territory, but also to associate, in various forms, the recipients of the services offered and, more broadly, the locals. This type of clusters often corresponds to emerging TCEC, which, if they manage to expand their network of members and increase self-financing, may evolve into an "alternative" cluster.

2) In the "institutionalized"cluster, the greater use of public authorities as partners, because of the low self-financing of their operations, may threaten its independence and sustainability. In some cases, a mix between the ESS policies of the institutional actor and the cluster's poses a risk of instrumentalisation. Unlike the first type, that cluster is generally part of a liberal vision of SSE, where the political project of participative democracy is forgotten in favor of the economic and social utility of solidarity-based initiatives. The ultimate goal is to create wealth and jobs, by looking, through cooperation and sharing, for a more efficient structuration of SSE organizations in order to increase their spillover effects on the territory. Governance remains essentially in the hands of the members of the cluster, which combines, as appropriate, experts, technicians or elected representatives of the territory.

Attracted by the managerial practice of capitalist enterprises and economic clusters, the two other types of TCEC are looking for a viable business model, focusing on trade... which does not exclude the research of partnership with institutions.

3) The "entrepreneurial" cluster looks like a traditional economic cluster. Following the guidance of the Law of 31 July 2014 on social and solidarity economy, it does include not only social businesses but also traditional companies, i.e local small and medium sized businesses or large groups. Of course, unlike any economic cluster, the goal is still to build / expose / develop / transmit local resources for the benefit of the territory and its inhabitants. Therefore, social intervention is not neglected but often takes the form of integration through economic activity, training services to increase the employability of the unemployed or classic social assistance. The effectiveness of these clusters, in terms of jobs creation, does enhance the interest of public authorities involved, at various levels and in various ways, in their emergence and development: extending networks of stakeholders, establishing collaborative infrastructure, mobilizing financial resources and engineering, promotion, prospective ... [START_REF] Cornu | Les rôles des collectivités locales dans les dynamiques de PTCE[END_REF]. However, the significant share of self-financing in the "entrepreneurial" cluster tends to guarantee its independence from the institutions, which are perceived rather as partners or clients who seek the TCEC to meet unmet community needs. Besides, as these clusters are used to sign contracts with the administrations, particularly through tenders or calls for proposals, they are part of the paradigm of competitive governance, which is based on market mechanisms and now tends to become hegemonic (Enjolras, 2000). Like economic clusters, the "entrepreneurial" clusters favor a technocratic mode of governance. The management is marked by a concern for rationality, use traditional tools of management and assessment, although social goals and ethical issues are never ignored. In other words, this type of cluster is in the reformist / entrepreneurial conception of SSE that aims to better regulate capitalism, to introduce an ethical dimension by combining profitability and solidarity, but without offering any real alternative.

4) Yet rare, the "alternative" cluster looks like the "entrepreneurial" cluster in terms of its business model, its search for profitable activities and balanced partnership with traditional companies and institutions. However, its governance is more democratic, more open to employees, volunteers, citizens -in short civil society -than conventional clusters. These clusters are often involved in the cultural or environmental sectors, which are historically characterized by a culture of exchange, deliberation and cooperation.

Of course, this typology is not static. These four categories refer to major trends, ideal types, and they do not perfectly match TCEC. Above all, the study of the Rhone-Alps TCEC showed that the positions of the different poles have evolved more or less rapidly since their creation, and in various ways. In any case, given the reduction in allocations to local authorities and the ministerial guidelines, the clusters with low self-finance will be encouraged to evolve. The second ministerial call for proposals, in 2015, requires, indeed, that future beneficiaries incorporate at least one conventional business and that "the economic model is designed to move towards reasonable self-financing."

Conclusion

Coming from civil society and entrepreneurial processes in a bottom-up strategy, territorial clusters of economic cooperation (inspired by French economic clusters) could not emerge and develop without the support from public institutions. But such governance is not selfevident and needs to be organized, taking into account the purposes, constraints and resources of each stakeholder, while complying with the necessary but relative autonomy of each of them. The study of TCEC in the Rhone-Alps Region highlights the many practical forms that the involvement of public institutions can take in the development of a social cluster, and shows the strategic importance to reach agreements, both for the social economy and local public institutions.

However, the public-civil society forms of governance differ according to the types of clusters and their evolution. If the TCEC are broadly in line with the principles, the issues, the objectives of social and solidarity economy, they differ by the conception of SSE that emerges from their choice of activities and governance, which led us to outline a typology of these clusters. Overall, it seems that technocratic/market models of capitalist enterprises and classic economic clusters prevail today.

"Réseau des collectivités Territoriales pour une Economie Solidaire RTES": This network brings together local authorities that engage around a charter for the development of social economy.

Bernard Enjolras defines governance as "the set of institutional arrangements governing the interactions between actors whose activities contribute to the achievement of objectives under the general interest."


Beijing Chen

Senior Member, IEEE Gouenou Coatrieux
email: gouenou.coatrieux@telecom-bretagne.eu

Member, IEEE Jiasong Wu
email: jswu@seu.edu.cn

Zhifang Dong
email: dongzhifang@seu.edu.cn

Fellow, IEEE Jean Louis Coatrieux
email: jean-louis.coatrieux@univ-rennes1.fr

Huazhong

Jean Louis

Fast Computation of Sliding Discrete Tchebichef Moments and Its Application in Duplicated Regions Detection

Keywords: Tchebichef moments, sliding transform, fast computation, duplicated signal detection

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

Tchebichef moments or transforms were first introduced by Mukundan et al. [START_REF] Mukundan | Image analysis by Tchebichef moments[END_REF]. Since the Tchebichef polynomials are orthogonal in the discrete domain of coordinate space, their implementation does not involve any numerical approximation. This property makes them superior to the conventional continuous orthogonal moments such as Legendre moments and Zernike moments [START_REF] Wang | Recursive computation of Tchebichef moment and its inverse transform[END_REF][START_REF] Shu | Fast computation of Tchebichef moments for binary and grayscale images[END_REF][START_REF] Asli | The fast recursive computation of Tchebichef moment and its inverse transform based on Z-transform[END_REF][START_REF] Zhang | Symmetric image recognition by Tchebichef moment invariants[END_REF]. Thus, Tchebichef moments have been extensively used in pattern recognition [START_REF] Zhang | Symmetric image recognition by Tchebichef moment invariants[END_REF], texture classification [START_REF] Marcos | Texture classification using discrete Tchebichef moments[END_REF], medical image reconstruction [START_REF] Dai | Reconstruction of tomographic images from limited range projections using discrete Radon transform and Tchebichef moments[END_REF] and forensics [START_REF] Huang | Blind integrity verification of medical images[END_REF]. Among these applications, local features play an important role and allow avoiding a whole segmentation of objects [START_REF] Elshoura | Analysis of noise sensitivity of Tchebichef and Zernike moments with application to image watermarking[END_REF]. Local moments have been shown of interest for the detection of key features [START_REF] Cao | A review of object representation based on local features[END_REF][START_REF] Chen | A Zernike moment phase-based descriptor for local image representation and matching[END_REF][START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF][START_REF] Ryu | Rotation invariant localization of duplicated image regions based on Zernike moments[END_REF]. However, one of the main drawbacks in their use as local features is their computational load inducing the needs for fast computation. In this paper, we investigate the local discrete orthogonal Tchebichef moment properties when used over sliding windows and we propose a fast computation algorithm of sliding discrete Tchebichef moments (SDTMs).

Basically, the sliding orthogonal transform of a signal means that the transform is computed on a fixed-size window of the signal, which is continuously updated with new samples while discarding the older ones [START_REF] Macias | Recursive formulation of short-time discrete trigonometric transforms[END_REF]. Let f(x) be the original signal with length L, and assume that the window at time instant p contains N values f(p), f(p+1), …, f(p+N-1), 0 ! p, p+N-1 < L, then, the sliding orthogonal transform is defined by [START_REF] Kober | Fast algorithms for the computation of sliding discrete sinusoidal transforms[END_REF][START_REF] Kober | Fast algorithms for the computation of sliding discrete Hartley transforms[END_REF]

where w m is a window function, and { ( , )} m s ! is an orthogonal basis set. p s Y represents the orthogonal transform of the windowed signal around time p.

In the past decades, many works have been reported in the literature for the fast computation of sliding transforms. They can be classified into two categories: 1) the structures of radix-2 and radix-4 fast algorithms such as sliding FFT [START_REF] Farhang-Boroujeny | Generalized sliding FFT and its application to implementation of block LMS adaptive filters[END_REF] , Hopping DFT [START_REF] Park | The hopping discrete Fourier transform[END_REF], sliding Walsh Hadamard transform [START_REF] Ouyang | Fast algorithm for Walsh Hadamard transform on sliding windows[END_REF], sliding Haar transform [START_REF] Macias | Efficient computation of the running discrete Haar transform[END_REF], sliding conjugate symmetric sequency-ordered complex Hadamard transform [START_REF] Wu | Sliding conjugate symmetric sequency-ordered complex Hadamard transform: fast algorithm and applications[END_REF]; 2) the first-and second-order shift properties of sliding transforms including sliding DFT [START_REF] Jacobsen | The sliding DFT[END_REF][START_REF] Duda | Accurate, guaranteed stable, sliding discrete Fourier transform[END_REF], sliding DCT [START_REF] Kober | Fast algorithms for the computation of sliding discrete sinusoidal transforms[END_REF], sliding DHT [START_REF] Kober | Fast algorithms for the computation of sliding discrete Hartley transforms[END_REF], sliding discrete fractional Fourier transform [START_REF] Tao | Short-time fractional Fourier transform and its applications[END_REF], sliding Walsh Hadamard transform [START_REF] Mozafari | An efficient recursive algorithm and an explicit formula for calculating update vectors of running Walsh-Hadamard transform[END_REF], sliding Haar transform [START_REF] Ouyang | Fast pattern matching using orthogonal Haar transform[END_REF], modulated sliding discrete Fourier transform [START_REF] Orallo | Harmonics measurement with a modulated sliding discrete Fourier transform algorithm[END_REF], and sliding geometric moment (SGM) [START_REF] Martinez | Efficient computation of local geometric moments[END_REF].

In this paper, we suggest to utilize the first-order shift properties of SDTMs for their fast computation. The rest of this paper is organized as follows: Section II defines the one-dimensional and two-dimensional SDTMs. Some properties of orthogonal

Tchebichef polynomials are reviewed. The proposed fast algorithm to compute SDTM and its computational complexity analysis are presented in Section III. Performance test and applications are reported in Section IV in the framework of digital forensics looking at the detection of duplicated signal segments or image regions. Section V concludes the paper.

II. PRELIMINARIES

A. Sliding discrete Tchebichef moment

According to the definition of the sliding orthogonal transform in [START_REF] Mukundan | Image analysis by Tchebichef moments[END_REF], let us first introduce the 1-D SDTM for the orthogonal Tchebichef polynomials set. For a signal f(x) of length L, x = 0, 1, …, L-1, assume that the window of f(x) at time instant p contains N values f(p), f(p+1), …, f(p+N-1), 0 ! p, p+N-1 < L, then the nth order 1-D SDTM is defined as

1 0 ( ) ( ) N p n n x T f p x t x ! = = + " , n = 0,1,…,N-1, (2)
where t n (x) is the nth order orthonormal Tchebichef polynomial defined by 2 0

(1 ) ( ) ( ) (1 ) ( ) , ( !) (1 ) ( , )

n n l l l n l l N n x n t x l N n N ! = " " " + = " # n, x = 0,1,…,N-1. (3)
Here (a) l is the Pochhammer symbol

( ) ( 1)( 2) ( 1) l a a a a a l = + + + ! ! , l ! 1, and (a) 0 = 1, (4)
and the squared-norm !(n, N) is given by

( )! ( , ) (2 1)( 1)! N n n N n N n ! + = + " " . (5)
For an N"N block [p, p+N-1] " [q, q+N-1] of an image f(x, y) at position (p, q), the (n+m)th order 2-D SDTM is defined as

1 1 , , 0 0 ( , ) ( ) ( )
N N p q n m n m x y T f p x q y t x t y ! ! = = = + + "" , n, m = 0,1,…,N-1. (6)
B. Some properties of orthonormal Tchebichef polynomials [START_REF] Shu | Fast computation of Tchebichef moments for binary and grayscale images[END_REF] In [START_REF] Shu | Fast computation of Tchebichef moments for binary and grayscale images[END_REF], Shu et al. established the following relationship between t n (x) at position x and t n (x+a) at position x+a,

0 0 ( ) ( , )( ) ( ) n n l n r r l l r t x a g n l a t x ! = = + = ! "" , (7)
where

(2 1)( 1 )! ( , ) ( 1)
!( )!( 1 )! ( , ) ( 1) , !( 1 )! ( , ) ( )!( )!( 1)!( )!( 1 )! ! ! " = + " " " + + " " " = " " " + " " + + " " " # s n r n r s l l N l l N s n r s N s r g n l r N n n N r s n r s s l s l N s 0 # l # n. ( 8
)
When a = 1 in (7), we have

1 1 0 ( 1) ( ) ( , ) ( ) n n n l l t x t x g n l t x ! = + = ! " , (9)
where g 1 (n, l) is defined by [START_REF] Huang | Blind integrity verification of medical images[END_REF].

III. FAST COMPUTATION OF SLIDING DISCRETE TCHEBICHEF MOMENTS AND ITS COMPUTATIONAL COMPLEXITY

In this section, we propose an efficient algorithm for the fast computation of 1-D SDTMs defined by (2) and 2-D SDTMs by [START_REF] Marcos | Texture classification using discrete Tchebichef moments[END_REF]. We also analyze its computational complexity.

A. Fast algorithm of 1-D sliding discrete Tchebichef moments

According to 1-D SDTM defined in (2), the 1-D SDTM at time instant p+1 is given by

1 1 0 ( 1 ) ( ) N p n n x T f p x t x ! + = = + + " , n = 0, 1, …, N-1. ( 10
)
Equation ( 10) can be rewritten as

2 1 0 ( 1) ( ) ( 1 ) ( ) N p n n n x T t N f p N f p x t x ! + = = ! + + + + " . ( 11
)
Similarly, (2) can also be rewritten as

2 0 (0) ( ) ( 1 ) ( 1) N p n n n x T t f p f p x t x ! = = + + + + " . (12)
Using ( 9), [START_REF] Chen | A Zernike moment phase-based descriptor for local image representation and matching[END_REF] and the symmetry property

( 1 ) ( 1) ( ), n n n t N x t x ! ! = ! (13)
as well as making the notation g 1 (n, n) = -1, the second term in (12) becomes

2 0 2 1 2 1 0 0 0 1 1 1 1 0 1 1 0 ( 1 ) ( 1) ( 1 ) ( ) ( , ) ( 1 ) ( )
( 1) ( ) ( , ) ( 1) ( )
( , ) ( 1)
(0) ( ) . N n x N n N n l x l x n p p n n l l l n p l l l l f p x t x f p x t x g n l f p x t x T t N f p N g n l T t N f p N g n l T t f p N ! = ! ! ! = = = ! + + = + = + + + = + + ! + + " # = ! ! + ! ! ! + $ % " # = ! ! ! + $ % (14)
By introducing ( 14) into [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF], for all the moments up to order M, M < N, [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF] can be expressed in a matrix form as

1 0 0 1 1 1 1 ( ) ( ) ( ) ( ) .
( 1) ( ) ( )

+ + + ! " + ! " # $ # $ ! " % & ' ( % & ' ( % & ) + % & ' ( % & ' ( % & = ) ) + % & ' ( % & ' ( % & % & ' ( % & ' ( % & ' ( ) + % & % & * + , - , - * + , - ! ! ! ! p p p p M M M M p p M M f p N f p T T f p N f p T T A B B f p N f p T T ( 15
)
where

1 1 1 1 1 1 1 1 1 1 (0, 0) 0 0 0 (1, 0) (1,1) 0 0 (2, 0) (2,1) (2, 2) 0 ( ,0) ( ,1) ( ,2) ( , ) M g g g A g g g g M g M g M g M M ! " # $ # $ # $ = # $ # $ # $ % & ! ! ! " " " " " ! , (16)
( )

0 1 2 (0), (0), (0), , (0)
M M B diag t t t t = ! . ( 17
)
Since the lower triangular matrix A M is non-singular, (15) can be written as

1 0 0 1 1 1 1 1 ( ) ( ) ( ) ( ) .
( ) ( 1) ( )

+ + ! + " # + " # $ % $ % " # & ' ( ) & ' ( ) & ' ! + & ' ( ) & ' ( ) & ' = ! ! + & ' ( ) & ' ( ) & ' & ' ( ) & ' ( ) & ' ( ) ! + & ' & ' * + , - , - * + , - ! ! ! ! p p p p M M M M p p M M f p f p N T T f p f p N T T A B B f p f p N T T (18)

B. Generalization of the proposed fast algorithm to 2-D

The algorithm derivation presented in the previous section can be easily generalized to 2-D SDTMs.

The 2-D SDTM defined in (6) of the next sliding N " N block [p+1, p+N] " [q, q+N-1] in the horizontal direction at position (p+1, q) is given by

1 1 1, , 0 0 ( 1 ,
) ( ) ( )

N N p q n m n m x y T f p x q y t x t y ! ! + = = = + + + "" , n, m = 0, 1, …, N-1. ( 19
)
Following the same procedure as for 1-D SDTM, the compution of the moments 1, , p q n m T + of order up to (M, M), i.e., 0 # n, m # M, M < N, can be obtained as follows:

1, 1, 1, 0,0 0,1 0, 1, 1, 1, 1,0 1, 1 1, 1, 1, 1, ,0 ,1 , , , , , 0,0 0,1 0, 0 , , , 1,0 1,1 1, 1 , , , ,0 ,1 ,
+ + + + + + + + + ! " # $ % $ % $ % $ % $ % & ' ( ) * + * + = ! ! * + * + * + , - ! ! " " " " ! ! ! " " " " ! p q p q p q M p q p q p q M p q p q p q M M M M p q p q p q p M p q p q p q M M M p q p q p q M M M M T T T T T T T T T T T T TC T T T A B T T T , , 1 , , , 0 1 , , , 0 1 , , , 0 1 , , , 0 1 , , , 0 1 ( 1) ( 1) ( 1)
+ + + + + + + + + " # ( ) $ % * + $ % * + $ % * + $ % * + * + $ % , - & ' " # $ % ! ! ! $ % + $ % $ % ! ! ! $ & ' ! ! " " " " ! ! ! " " " "
! q p q p q M p q p q p q M p q p q p q M p N q p N q p N q M p N q p N q p N q M M M p N q M p N q M p N q M TC TC

where A M and B M are given in ( 16) and [START_REF] Farhang-Boroujeny | Generalized sliding FFT and its application to implementation of block LMS adaptive filters[END_REF], respectively;

, p q m TC and , p N q m TC + are the mth order 1-D SDTMs of an outgoing column f(p, q+y), 0 # y # N-1, and an incoming one f(p+N, q+y) , 0 # y # N-1, respectively, given by

1 , 0 ( , ) ( ) N p q m m y TC f p q y t y ! = = + " , (21) 1 , 0 ( , ) ( )
N p N q m m y TC f p N q y t y ! + = = + + " . ( 22
)
They can be calculated by the fast algorithm for 1-D SDTMs described in previous section.

Similarly, for the vertical SDTM of the sliding block [p, p+N-1] " [q+1, q+N] at position (p, q+1) defined as

1 1 , 1 , 0 0 ( , 1 ) ( ) ( ) N N p q n m n m x y T f p x q y t x t y ! ! + = = = + + + "" , n, m = 0, 1, …, N-1, ( 23
)
we can obtain the following relationship , 1

, 1 , 1 0,0 0,1 0, ,

,0 0,1 0, 0 0 , , 1 , 1 , 1 1,0 1,1 1, , 1 , 1 , 1 ,0 ,1 , , , , , , 0
+ + + + + + + + + ! " # $ # $ # $ # $ # $ % & ' ( ) * ) * = + + ) * ) * ) * , - ! ! " " " " ! ! ! " , 1,0 1,1 1, , , , ,0 ,1 ,
" " " ! p q p q p q M p q p q p q M p q p q p q M M M M p q p q p q p q p M p q p q p q M p q p q p q M

M M M T T T T T T T T T T T T TR TR T T T T T T ( ) , 0 , , , 1 1 1 1 , , , , , , 0 0 0 , , , 1 1 1 , , , ( 1) ( 1)
( 1)

+ + + + + + + + + + ! " ' ( # $ ) * # $ ) * # $ ) * # $ ) * ) * # $ , - % & ! " + + # $ + + # $ + # $ # $ + + # $ % & ! ! " " " " ! ! ! " " " " ! q p q p q p q p q T M M p q p q p q M M M p q N p q N M p q N p q N p q N M p q N p q N p q N M p q N M M M TR TR TR TR B A TR TR TR TR TR TR TR TR TR B TR TR TR , M (24)
where the superscript T is the matrix transpose operation, , p q n TR and , p q N n TR + are the nth order 1-D SDTMs of an outgoing row f(p+x, q), 0 # x # N-1, and an incoming one f(p+x, q+N) , 0 # x # N-1, respectively, given by

1 , 0 ( , ) ( ) N p q n n x TR f p x q t x ! = = + " , (25) 1 , 0 ( , ) ( )
N p q N n n x TR f p x q N t x ! + = = + + " . ( 26
)
They can also be computed using the proposed fast algorithm for 1-D SDTMs.

Based on the above theoretical results, the 2-D SDTMs up to order (M, M) of all overlapping blocks with size N " N for the image f(x, y) can be computed as follows:

Algorithm for computing the 2-D SDTMs up to order (M, M)

Step 1) Compute all the Tchebichef polynomials displayed in (3) and the coefficient matrix given in [START_REF] Kober | Fast algorithms for the computation of sliding discrete Hartley transforms[END_REF] up to order M.

Step 2) Compute the moments for all N-length row vectors. Firstly, consider the first row vector of each row y of the image f(x, y) and calculate the moments of this vector using the direct algorithm. After that, consider other remaining row vectors of each row using the proposed fast algorithm for 1-D SDTMs.

Step 3) Calculate the moments for the N-length column vectors of the first row using the direct algorithm.

Step 4) Calculate the moments for the first block of the image f(x, y), and then consider all the blocks of the first row using the proposed fast algorithm (20) for 2-D SDTMs based on the results obtained in Step 3).

Step 5) Process the remaining blocks using the proposed fast algorithm [START_REF] Tao | Short-time fractional Fourier transform and its applications[END_REF] based on the results obtained in Step 2) and Step 4).

C. Computational complexity analysis

Since the elements of the matrices A M , B M and the polynomial values t n (x), 0 # n # M, 0 # x # N-1, are independent of the signal, they can be pre-computed and stored in a look-up table . Using this effect, we analyze the computational complexity of the proposed fast sliding algorithm based on (18) for 1-D signal and (20) or (24) for 2-D one as well as that of the conventional direct algorithm treating each window as a separate sub-signal.

(

1) Computation of 1-D SDTMs

For simplicity, we consider here the computational complexity of moments up to the maximum order M for a sliding N-length window of 1-D signal.

For the conventional direct algorithm, we see from [START_REF] Cao | A review of object representation based on local features[END_REF] that the computational complexity is N(M+1) multiplications and

(N-1)(M+1) additions. However, by using the symmetry property of t n (x) mentioned in [START_REF] Ryu | Rotation invariant localization of duplicated image regions based on Zernike moments[END_REF], we can improve the direct algorithm [START_REF] Cao | A review of object representation based on local features[END_REF] as follows

/ 2 1 1 0 ( ) ( 1 ) ( 1) ( ) N p n n n x T t x f p x f p N x ! + = " # = + + + ! + ! $ % & , n = 0, 1, …, M. ( 27
)
This improved algorithm is called compact direct algorithm [START_REF] Wang | Recursive computation of Tchebichef moment and its inverse transform[END_REF]. It is also easy to observe that the compact direct algorithm based on [START_REF] Orallo | Harmonics measurement with a modulated sliding discrete Fourier transform algorithm[END_REF] requires N(M+1)/2 multiplications and (N-1)(M+1) additions.

For the proposed fast algorithm, it can be seen from ( 18) that: (i) the computation of the matrix between the bracket requires (M+1) multiplications and (M+1) additions; (ii) the multiplication of the column vector obtained in the bracket by the lower triangular matrix

1 M
A ! needs M(M+1)/2 additions and M(M+1)/2 multiplications since all the diagonal elements g 1 (n, n) of

1 M
A ! are equal to -1; (iii) other operations in (18) also require (M+1) multiplications and (M+1) additions. Thus, the total computational complexity is (M+1)(M+4)/2 multiplications and (M+1)(M+4)/2 additions.

In [START_REF] Marcos | Texture classification using discrete Tchebichef moments[END_REF], Marcos and Cristóbal proposed an algorithm to compute the Tchebichef moments in the frequency domain. Their algorithm uses the DFT of the input signal and the Tchebichef polynomials. This algorithm is also considered for comparison purpose. Note that: (i) the computation of the Tchebichef polynomials and their DFT are not taken into account due to their independence to the signal; (ii) only half of DFT coefficients are considered for the computation owing to the conjugate symmetry of DFT coefficients when the input signal is real; (iii) the computation of the DFT of the input signal is based on radix-2 FFT algorithm.

The computational complexities of the four algorithms are depicted in Table 1. It can be observed from this table that: (i) The proposed algorithm is independent of the window length N while this is not the case for the other three algorithms; (ii) the proposed algorithm requires fewer arithmetic operations than the other three algorithms when M is not greater than (3N/2-5).

Note that the order M must be smaller than the window length N according to the definition of Tchebichef moments in [START_REF] Wang | Recursive computation of Tchebichef moment and its inverse transform[END_REF]. Now, if we consider the worst case where M = N-1, it appears that (i) the proposed algorithm is superior to the other three algorithms when N > 8 and (ii) the proposed algorithm has the same number of arithmetic operations to the compact direct algorithm when

N = 8.
N(M+1) (N-1)(M+1) (2N-1)(M+1) Compact direct algorithm[2] N(M+1)/2 (N-1)(M+1) (3N/2-1)(M+1)
Frequency algorithm [START_REF] Marcos | Texture classification using discrete Tchebichef moments[END_REF]

Nlog2N+(N/2+2)(M+1) 3Nlog2N/2-2N+N(M+1)/2 5Nlog2N/2-2N+(N+2)(M+1) Proposed fast algorithm (M+1)(M+4)/2 (M+1)(M+4)/2 (M+1)(M+4) (2) Computation of 2-D SDTMs
In this section, we turn to the analysis of 2-D SDTMs and we provide the computational complexity of moments up to order (M, M) for a sliding block of size N " N.

For the conventional direct algorithm, (10) can be rewritten as

1 1 1, , 0 0 ( ) ( 1 , ) ( ) N N p q n m n m x y T t x f p x q y t y ! ! + = = = + + + " " , 0 ! n, m ! M. ( 28
)
Then, the computational complexity is N(N+1)(M+1) 2 multiplications and (N 2 -1)(M+1) 2 additions, while it is decreased to N(N+2)(M+1) 2 /4 multiplications and (N 2 -1)(M+1) 2 additions for the compact direct algorithm given by

/2 1 /2 1 1, , 0 0 ( 1 , ) ( 1) ( 1 , 1 ) ( ) ( ) ( 1) ( , ) ( 1) ( , 1
)

m N N p q n m n m n m n x y f p x q y f p x q N y T t x t y f p N x q y f p N x q N y ! ! + + = = " # + + + + ! + + + ! ! = $ % + ! + ! + + ! + ! + ! ! $ % & ' ( ( , 0 ! n, m ! M. ( 29
)
Now, let us analyze the proposed fast algorithm of 2-D SDTMs using the same approach. It can be seen from ( 20) or ( 24) that:

(i) the computation of the matrix between the bracket requires (M+1) 2 multiplications and (M+1) 2 additions; (ii) the multiplication of the upper anti-triangular matrix gotten in the bracket by the lower triangular matrix

1 M A ! needs M(M+1) 2 /2
multiplications and M(M+1) 2 /2 additions; (iii) the remaining operations in [START_REF] Macias | Efficient computation of the running discrete Haar transform[END_REF] or [START_REF] Tao | Short-time fractional Fourier transform and its applications[END_REF] still require (M+1) 2 multiplications and (M+1) 2 additions; (iv) the computation of 1-D horizontal SDTMs in [START_REF] Macias | Efficient computation of the running discrete Haar transform[END_REF] or 1-D vertical SDTMs in (24) using the proposed fast algorithm for 1-D SDTMs needs (M+1)(M+4)/2 multiplications and (M+1)(M+4)/2 additions. Thus, the total computational complexity is (M+1)(M+2)(M+4)/2 multiplications and (M+1)(M+2)(M+4)/2 additions.

Table 2 summarizes the computational complexities of the four algorithms. We can draw similar conclusions as 1-D SDTMs from Table 2. Moreover, when compared to 1-D SDTMs, the proposed algorithm for 2-D SDTMs is more efficient than the other three algorithms, even for the worst case where M = N-1 as far as N " 3. In fact, there are few cases in real applications with N < 4. Since the proposed algorithm is independent of the block size, it is very suitable for applications which need to traverse the whole image using a block-based method, such as image forensics, pattern matching and texture analysis and so on.

N(N+1)(M+1) 2 (N 2 -1)(M+1) 2 (2N-1)(N+1)(M+1) 2 Compact direct algorithm[2] N(N+2)(M+1) 2 /4 (N 2 -1)(M+1) 2 (5N 2 +2N-4)(M+1) 2 /4
Frequency algorithm [START_REF] Marcos | Texture classification using discrete Tchebichef moments[END_REF] 2N 2 log2N

+(N 2 -2N+8)(M+1) 2 3N 2 log2N-4N 2 +(N 2 -2N+3)(M+1) 2 5N 2 log2N-4N 2 +(2N 2 -4N+11)(M+1) 2 Proposed fast algorithm (M+1)(M+2)(M+4)/2 (M+1)(M+2)(M+4)/2 (M+1)(M+2)(M+4)

IV. PERFORMANCE TEST AND APPLICATION

In this section, we first evaluate the performance of the proposed fast algorithm for SDTMs through a comparison of their computational time and we then consider its application in tampering detection. These tests were implemented in VS2010 on a ThinkPad notebook E420 with 2.40 GHz CPU and 4GB RAM.

A. Test of the fast SDTMs in terms of computational time In the first experiment, the 1-D SDTMs of the ten signals were respectively computed using the aforementioned three algorithms. Considering our next targeted applications of 1-D SDTMs in audio signal, the window length N and the maximum order M were respectively set to 16 and 5. Table 3 presents their respective computational time. It can be observed from this table that: (i) our proposed algorithm is faster than the three others; (ii) the computation time remains stable whatever the signal used. The objective of the second experiment is to evaluate the influence of the window size and of the moment order for the four algorithms. We used the same ten signals. In order to test the influence of the window size, two cases were considered: a) the window size N was varied from 6 to 44 with an increment of 2 and the corresponding order was set to M = N-1, which is the worst case of the proposed algorithm; b) the window size N was varied from 8 to 44 with an increment of 4 using a fixed order set to M = 5, which is the order used in the two applications that will be described later. In order to test the influence of moment order, the maximum order M was varied from 5 to 15 and the window size N is 16. Fig. 1 shows the average computational time for the ten signals. Note that Fig. 1 (a) and (b) only shows the results of the frequency algorithm for the window sizes equalling to the power of 2 due to the radix-2 FFT algorithm. For other window sizes, zero padding can be applied but at the cost of an increased complexity: let N be the length of the original signal. If 2 k < N < 2 k+1 , where k is a positive integer, the complexity for the signal after zero padding will be equal to that for the signal with the length 2 k+1 . It can be observed from Fig. 1 that: (i) regarding to the window size, the proposed algorithm, although considered in the worst case (M = N-1), leads to a better performance; (ii) when a fixed order M = 5 is considered, the computational time of the proposed algorithm is independent of the window size. However, it is not the case for the other three algorithms, whose computational load increases at a fast rate. It is in agreement with the computational complexity shown in Table 1; (iii) although the computational time of the proposed algorithm increases with the moment orders, the advantage of the proposed algorithm over the direct algorithms and frequency algorithm is evident. This is also consistent with the computational complexity analysis. Conventional direct algorithm Compact direct algorithm [START_REF] Wang | Recursive computation of Tchebichef moment and its inverse transform[END_REF] Frequency algorithm [START_REF] Marcos | Texture classification using discrete Tchebichef moments[END_REF] Proposed fast algorithm from the public image manipulation dataset [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF]. The sizes of the six images are 504"760. In [START_REF] Yap | Image focus measure based on Chebyshev moments[END_REF], Yap and Raveendran pointed out that the matrix representation and its operators are very effective in some software packages, especially in Matlab, and they proposed a matrix-based implementation method to compute the Tchebichef moments.

So, in this subsection, the proposed fast algorithm is compared with the two direct algorithms presented in Section III.C implemented in VS2010 as well as their matrix-based implementation in MATLAB R2006b. Note that the matrix-based implementation does not change the computational complexity.

In the first example, the SDTMs of the above six images were respectively calculated by using these different implementations.

The block size was set to 16"16 and the maximum order of moments to 5, which is the same one used in [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF] for Zernike moments. The resulting computational time is shown in Table 4. Although the matrix-based scheme leads to a significant benefit, it is far from the performance of our proposed algorithm. In the second experiment, we evaluated the influence of block size, moment order, and image size. The six images used in the previous experiment were considered again. For the first test (i.e. influence of block size), two case were examined: (i) the block size N#N was varied from 6#6 to 44#44 with an increment of 2 and the maximum order (M, M) was set to (N-1, N-1) in all cases; (ii) the block size was varied from 8#8 to 80#80 with an increment of 8 and the order was set to [START_REF] Zhang | Symmetric image recognition by Tchebichef moment invariants[END_REF][START_REF] Zhang | Symmetric image recognition by Tchebichef moment invariants[END_REF]. For the second test, the block size was set to 16"16 and the maximum order of moments (M, M) was varied from [START_REF] Zhang | Symmetric image recognition by Tchebichef moment invariants[END_REF][START_REF] Zhang | Symmetric image recognition by Tchebichef moment invariants[END_REF] to [START_REF] Kober | Fast algorithms for the computation of sliding discrete sinusoidal transforms[END_REF][START_REF] Kober | Fast algorithms for the computation of sliding discrete sinusoidal transforms[END_REF]. For the third test, the block size was still fixed at 16"16, the order of moments was chosen equal to [START_REF] Zhang | Symmetric image recognition by Tchebichef moment invariants[END_REF][START_REF] Zhang | Symmetric image recognition by Tchebichef moment invariants[END_REF], and then the six images were rescaled with a factor ranging from 0.5 to 2.1 with step 0.2. The results of the average computational times of these tests are respectively given in Fig. 3. Note that, the same as 1-D SDTMs, Fig. 3 Conventional direct algorithm Compact direct algorithm [START_REF] Wang | Recursive computation of Tchebichef moment and its inverse transform[END_REF] Matrix-based conventional direct algorithm [START_REF] Yap | Image focus measure based on Chebyshev moments[END_REF] Matrix-based compact direct algorithm [START_REF] Yap | Image focus measure based on Chebyshev moments[END_REF] Frequency algorithm [START_REF] Marcos | Texture classification using discrete Tchebichef moments[END_REF] Proposed fast algorithm Conventional direct algorithm Compact direct algorithm [START_REF] Wang | Recursive computation of Tchebichef moment and its inverse transform[END_REF] Matrix-based conventional direct algorithm [START_REF] Yap | Image focus measure based on Chebyshev moments[END_REF] Matrix-based compact direct algorithm [START_REF] Yap | Image focus measure based on Chebyshev moments[END_REF] Frequency algorithm [START_REF] Marcos | Texture classification using discrete Tchebichef moments[END_REF] Proposed fast algorithm Conventional direct algorithm Compact direct algorithm [START_REF] Wang | Recursive computation of Tchebichef moment and its inverse transform[END_REF] Matrix-based conventional direct algorithm [START_REF] Yap | Image focus measure based on Chebyshev moments[END_REF] Matrix-based compact direct algorithm [START_REF] Yap | Image focus measure based on Chebyshev moments[END_REF] Frequency algorithm [START_REF] Marcos | Texture classification using discrete Tchebichef moments[END_REF] Proposed fast algorithm 0.5 0.7 0.9 Conventional direct algorithm Compact direct algorithm [START_REF] Wang | Recursive computation of Tchebichef moment and its inverse transform[END_REF] Matrix-based conventional direct algorithm [START_REF] Yap | Image focus measure based on Chebyshev moments[END_REF] Matrix-based compact direct algorithm [START_REF] Yap | Image focus measure based on Chebyshev moments[END_REF] Frequency algorithm [START_REF] Marcos | Texture classification using discrete Tchebichef moments[END_REF] Proposed fast algorithm Although the proposed algorithm is computationally efficient, it suffers from the problem of accumulated errors, which also exists for the fast computation of other sliding transforms, for example the sliding DFT [START_REF] Duda | Accurate, guaranteed stable, sliding discrete Fourier transform[END_REF]. For the SDTMs, the accumulated errors are introduced by the iterative updated equations ( 18), ( 20) and ( 24) on each sliding window and two matrices A M and B M in these equations. These two matrices must be calculated recursively, involving matrix multiplication and truncation and so leading to several sources of numerical approximation. In fact, if A M and B M are calculated in double precision, the accumulated errors remain small as it will be shown below. If one wants to further reduce the accumulated errors, the application of the proposed fast algorithm can be slightly modified by reinitializing the process at some regular interval K (the procedure is restarted from the step 3) depicted in Section III.B). The following experiment was carried out to better quantify these accumulated errors.

For the six tampered images of size 504"760 considered in the previous subsection, we computed the SDTMs on 364305 (=(504-15)"(760-15)) blocks of size 16"16 for each image by using different values of the interval K. The definition of the relative errors between SDTMs can be found in [START_REF] Chen | Combined invariants to similarity transformation and to blur using orthogonal Zernike moments[END_REF]. The experimental results are provided in Table 5. They show that: (i) the modified algorithm allows reducing the accumulated errors from e-7 to e-14; (ii) however, the initial proposed algorithm with no interval leads also to small errors (lower than 2.56e-7).

B. Application of SDTMs to duplicated signal segments and image regions detection

As a kind of sliding orthogonal transform, SDTMs can be used to the same applications in signal processing as other sliding transforms, such as signal filtering, spectrum analysis, matching, and tampering detection. Here, tampering detection, more precisely the detection of duplicated signal segments in an audio wave or regions in an image, is considered. The general pipeline for copy-move forgery detection (CMFD) provided in [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF] will be closely followed in our application examples.

With the rapid development of low cost and sophisticated image processing software tools, digital images can be tampered easily with no obvious visual traces. Copy-move manipulations, a common form of local processing, copy parts of an image and paste them somewhere else in the same image [START_REF] Fridrich | Detection of copy-move forgery in digital images[END_REF]. Therefore, automatic detection of duplicated regions is becoming one of the most important and popular digital forensic techniques currently [START_REF] Li | Segmentation-based image copy-move forgery detection scheme[END_REF]. In [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF], Christlein et al. provided a general pipeline for blind passive CMFD without using source images shown in Fig. 4.

Preprocessing

Feature Extraction

Matching Filtering Postprocessing Keypoint Detection Block Tiling Fig. 4 General processing pipeline for the region duplication forgery detection using the keypoint-based features or the block-based ones [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF].

The detailed steps for block-based methods are as follows:

1) Convert the color image to gray image;

2) Subdivide the test image into overlapping blocks of size N"N;

3) Compute a feature vector for every block; 4) Match each feature vector by searching its nearest one with the minimum Euclidean distance in the feature domain using kd-tree matching; 5) Remove the matched block pairs that are spatially close to each other, i.e.

1 2 ij v ! < !! "
, where ij v !! " is the translation difference ("shift vector") between the matched blocks i and j, and 2 ! is the Euclidean norm; 6) Cluster the remaining pairs that adhere to a joint pattern. Let P(A) be the number of pairs satisfying the same affine transformation T. Remove all pairs that belong to a small number P(A), i.e. P(A) < " 2 ; 7) Preserve the connected regions with the number of pixels more than " 3 and highlight them with bright color as tampered regions.

Christlein et al. [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF] have used two objective criteria Recall (or true positive rate) and Precision to evaluate the efficiency of the algorithms for duplicated region detection. Recall represents the probability that a forgery is detected, while Precision shows the probability that a detected forgery is truly forged. These two objective criteria are also used in the following experiments.

(1) Audio signal duplicated segments detection using 1-D SDTMs

Audio forgery techniques could be used to falsify court evidence, conduct piracy over the Internet, or modify security device recordings or recordings of events taking place in different parts of the world. There are many ways to tamper digital audio. The duplicated audio inserting is a common one. For example, suppose that we have several audio wave files corresponding to some sentences. One can identify a segment of audio wave including the word "not" and insert this segment into any sentences to change their meaning [START_REF] Xiao | Audio authenticity: Duplicated audio segment detection in waveform audio file[END_REF].

So, in the present experiment, 1-D Tchebichef moments were considered as the feature set for detecting the duplicated segments in the signal. The procedure is similar to the one used in image and described by the steps 1 to 7 above. The differences are as follows: 1) the first step is replaced by a normalization of the signal values between 0 and 1 to deal with the different ranges of audio signal values; 2) the second step consists to subdivide the signal into overlapping N-length segments; 3) our fast method based on 1-D Tchebichef moments is applied in the third step. Note that the detection is also a blind passive one (i.e. the source signal is not used). The parameters N, " 1 , " 2 and " 3 have been respectively set as: N = 16, " 1 = 50, " 3 = " 2 = 300. The maximum order M of 1-D SDTMs has been set to 5.

In order to evaluate the detection performance of the proposed algorithm based on 1-D SDTMs, 100 tampered audio signals were considered with 80 signals randomly selected from the public TIMIT dataset [START_REF] Garofalo | The DARPA TIMIT Acoustic-Phonetic Continuous Speech Corpus CDROM, Linguistic Data Consortium[END_REF] and 20 signals created by us. One or two segments of each source file were copied and pasted somewhere else in the same signal to get the tampered signal. Furthermore, to these 100 tampered signals, Gaussian white noise with different standard deviations (STDs) were added and converted into MP3 files with different compression ratios to test the robustness of the proposed algorithm.

We also compared the proposed algorithm with the recent work reported by Xiao et al [START_REF] Xiao | Audio authenticity: Duplicated audio segment detection in waveform audio file[END_REF]. In [START_REF] Xiao | Audio authenticity: Duplicated audio segment detection in waveform audio file[END_REF], Xiao et al. first defined the similarity between two segments and then used a fast convolution algorithm to calculate the similarity. The same two objective criteria (recall and precision) used in image tampering detection experiments were considered for comparison. The results are provided in Table 6 and Table 7. Two examples of detection are given in Fig. 5 and Fig. 6 for illustration. These two examples correspond to two worst cases of noise attack and MP3 compression attack. The detected duplicated segments are highlighted in black. It can be observed from these tables and figures that: 1) the two algorithms well detect duplicated segments in the attack-free scenario; 2) the proposed algorithm has a better performance than Xiao's algorithm [START_REF] Xiao | Audio authenticity: Duplicated audio segment detection in waveform audio file[END_REF] in case of the attack. principal component analysis (PCA), kernel-PCA (KPCA), Hu's moment invariants, blur invariant moments and Zernike moments. Experimental results in [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF] demonstrate that Zernike moments have on the overall the best performance. So, in this paper, we compared the feature sets based on Tchebichef and Zernike moments under the same CMFD pipeline as [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF] given at the beginning of this section IV.B to evaluate the performance of the fast algorithm for 2-D SDTMs.

a) Experimental dataset

To do so, the public image manipulation dataset used in [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF] was considered. This dataset contains 48 source images and many snippets manually extracted from these source images. The tampered images are obtained by copying and pasting these snippets into the source images. They cover different levels of sophistication with duplicated regions of varying numbers and sizes. In order to test the robustness, these snippets are also scaled, rotated, noised, or JPEG compressed before being copied and pasted. Some representative tampered images and their corresponding source images are shown in Fig. 2. The tampered regions are marked with red line. Note that the detection of duplicated regions discussed in this section is a blind one (no use of the source image). Here, the source images are only displayed for illustration.

The sizes of the images in this dataset vary from 800"533 to 2613"3900. In our experiments, the images larger than 1440"1440 were down-sampled with a scaling factor equal to 0.25 and the rest with a scale factor of 0.5. The reasons of this pre-processing are the following: (i) we want to accelerate the process of duplicated regions detection according to our PC configuration since a large image leads to a great number of feature vectors to be matched; (ii) the down-sampling does not affect the comparison of the three algorithms here used as shown in the experiments conducted on images of various sizes; (iii) experimental results reported in [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF] have shown that the down-sampling leads to a decrease of the overall detection performance.

So, here, we have tested attacks combined with down-sampling. These attacks are more complex than those carried out in [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF]. Note that due to the down-sampling, the parameters " 3 and " 2 were set to 300 instead of 800 for the Zernike moments in [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF]. The other parameters N, " 1 , and M remained identical as in [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF]: N = 16, " 1 = 50 and M = 5.

b) Experimental results

We compared the proposed algorithm using 2-D SDTMs with two other algorithms. The first one, which is named "Algorithm 1" hereafter, is the algorithm reported in [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF], combining the Zernike moment features with a kd-tree matching. The second one, called "Algorithm 2", also based on Tchebichef moments, was proposed by Li et al. [START_REF] Li | Passive digital image authentication algorithm based on Tchebichef moment invariants[END_REF] where the moments of every block are computed using the direct algorithm coupled with a lexicographic sorting for matching similar feature vectors.

The results are shown in Table 8 and Fig. 7-Fig. 10. They show that:

1) The proposed algorithm, which combines the Tchebichef moments with kd-tree matching, is superior to the other two algorithms in terms of both Recall and Precision, no matter the manipulations done, except for the rotation operations for Algorithm 1 [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF]. This is mainly due to the following facts: (i) Tchebichef moment features are superior to Zernike moment features [START_REF] Mukundan | Image analysis by Tchebichef moments[END_REF][START_REF] Wang | Recursive computation of Tchebichef moment and its inverse transform[END_REF][START_REF] Shu | Fast computation of Tchebichef moments for binary and grayscale images[END_REF][START_REF] Asli | The fast recursive computation of Tchebichef moment and its inverse transform based on Z-transform[END_REF][START_REF] Zhang | Symmetric image recognition by Tchebichef moment invariants[END_REF]; (ii) the kd-tree matching can achieve better results than the lexicographic sorting [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF]; (iii) the features used in Algorithm 1 are the magnitudes of the Zernike moments, invariant to rotations;

2) Both criteria decrease when increasing the level of attacks (higher noise, larger compression, etc.) except for few cases;

3) Due to the image down-sampling, the overall performances of all compared algorithms are slightly inferior to the results obtained in [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF] when the original full size images are used: both of the Recall and Precision for the plain region duplication forgery is lower than 100%. This behavior has also been observed and pointed out by Christlein et al. in [12](Algorithm 1).

( 1 )

 1 1-D SDTMsIn order to evaluate the performance of the proposed fast algorithm for 1-D SDTMs, ten signals (each with 100000 samples) were randomly generated. The values of these ten signals were normalized to the range [-1, 1].

16 Fig. 1

 161 Fig. 1 Average computational time (millisecond) for the different algorithms with various window sizes N and maximum moment orders M (2) 2-D SDTMs In order to test the efficiency of the proposed fast algorithm for 2-D SDTMs, considering the following application of 2-D SDTMs in image tampering detection, a set of six tampered images shown in Fig. 2 (a), (c), (e), (g), (i) and (k) has been chosen

Fig. 2

 2 Fig. 2 Eight tampered images (first and third columns) and their corresponding source images (second and fourth columns) from the public image manipulation dataset. (a) and (k) are plain copy-move forgery. (c), (e), (g), (i), (m) and (o) are the forgery combined with adding Gaussian noise (standard deviation is 0.1), JPEG compression (quality factor is 50), rotation (angle is 10 degree), scaling (factor is 109%), rotation (angle is 6 degree), and scaling (factor is 91%), respectively.

  (a) and (b) only shows the results of the frequency algorithm for the block size equalling to the power of 2 owing to the radix-2 FFT algorithm used. It can be seen from these figures that: (i) conclusions similar to those obtained through the previous experiment of 1-D SDTMs can be drawn; (ii) moreover, when compared to 1-D SDTMs, a greater advantage of the proposed fast algorithm over the other algorithms for 2-D SDTMs can be noticed.

27 M

 27 (a) Various block sizes with M = N-1 (b) Various block sizes with M = 5 Average computational time(s)

5 Fig. 3

 53 Fig. 3 Average computational time (second) for the different algorithms with various block sizes N!N and maximum moment orders (M, M)

  Fig.5Example of detection in audio signal with duplicated segments. Original waveform: "He is the murder. I am not the murder". Tampered waveform: "He is not the murder. I am not the murder".

Fig. 6 ( 2 )

 62 Fig. 6 Example of detection for an audio signal with duplicated segments. Original waveform: "One two three four five six seven eight nine ten". Tampered waveform: "One two three four five two seven eight five ten". (2) Image duplicated regions detection using 2-D SDTMs In [12], Christlein et al. compared the performance of 15 most prominent feature sets including the keypoint-based features SIFT and SURF, as well as the block-based ones using discrete cosine transform (DCT), discrete wavelet transform (DWT),

  Original waveform "One two three four five six seven eight nine ten." Tempered waveform "One two three four five two seven eight five ten." Tempered waveform under Gaussian noise attack (STD=0.01) and detected results Tempered waveform under MP3 compression attack (Ratio=16Kbps) and detected results

Fig. 7

 7 Fig. 7 Recall and precision for additive Gaussian white noise

Table 1

 1 Computational complexity for 1-D SDTMs order up to order M

	Algorithms	Multiplication	Addition	Arithmetic operations
	Conventional direct
	algorithm

Table 2

 2 Computational complexity for 2-D SDTMs up to order (M, M)

	Algorithms	Multiplication	Addition	Arithmetic operations
	Conventional direct algorithm

Table 3

 3 Computational time (millisecond) of different algorithms for the ten randomly generated signals (No.1-No. 10)

	Algorithms	1	2	3	4	5	6	7	8	9	10
	Conventional direct algorithm	32	32	31	32	32	31	32	32	32	33
	Compact direct algorithm[2]	26	26	25	25	26	26	26	27	26	27
	Frequency algorithm[6]	35	37	35	36	35	35	36	37	37	36
	Proposed fast algorithm	15	16	14	15	15	15	15	16	16	15

Table 4

 4 Computational time (second) of different algorithms for images shown in Fig.2

	Algorithms	(a)	(c)	(e)	(g)	(i)	(k)
	Conventional direct algorithm	3.949	3.995	3.931	3.998	3.898	3.994
	Compact direct algorithm[2]	2.676	2.686	2.672	2.684	2.671	2.686
	Matrix-based conventional direct algorithm[29]	2.660	2.668	2.661	2.670	2.661	2.667
	Matrix-based compact direct algorithm[29]	2.365	2.370	2.365	2.371	2.363	2.367
	Frequency algorithm[6]	4.521	4.451	4.478	4.464	4.465	4.521
	Proposed fast algorithm	0.083	0.088	0.082	0.089	0.082	0.088

Table 5

 5 Maximum relative error over 364305 blocks for different images using various K

	K	Fig.2(a)	Fig.2(c)	Fig.2(e)	Fig.2(g)	Fig.2(i)	Fig.2(k)
	50	5.43e-14	1.04e-13	1.49e-13	4.03e-14	9.96e-14	1.36e-13
	100	1.83e-13	1.07e-13	3.07e-13	6.87e-14	1.14e-13	1.83e-13
	150	3.25e-13	1.87e-13	4.03e-13	1.19e-13	3.20e-13	4.50e-13
	200	1.93e-12	6.41e-13	8.28e-13	8.93e-13	2.65e-12	1.31e-12
	250	1.80e-11	4.85e-12	3.60e-11	7.89e-12	2.03e-11	8.47e-12
	300	5.32e-11	2.52e-11	3.69e-10	5.05e-11	5.23e-11	6.12e-11
	350	3.32e-10	1.50e-10	1.36e-9	3.59e-10	3.72e-10	4.60e-10
	400	5.23e-10	6.43e-10	2.41e-9	1.00e-9	3.05e-9	2.35e-9
	450	1.51e-9	2.95e-9	2.39e-9	1.68e-9	7.73e-9	1.32e-8
	500	4.37e-9	6.63e-9	3.55e-9	7.40e-9	2.34e-8	3.61e-8
	No interval 1.13e-7	7.11e-8	2.45e-7	6.10e-8	3.79e-8	2.56e-7

Table 6

 6 Recall and precision for additive Gaussian white noise with different STDs

	STD of Gaussian		0	0.001	0.002	0.003	0.004	0.005	0.006	0.007	0.008	0.009	0.01
		Xiao's algorithm[33]	100%	100%	100%	95%	90%	86%	81%	74%	68%	63%	58%
	Recall	Proposed	algorithm
				100%	100%	100%	100%	100%	96%	92%	87%	83%	80%	76%
		using 1-D SDTMs
		Xiao's algorithm[33]	100%	100%	99%	93%	87%	82%	77%	70%	63%	56%	51%
	Precision	Proposed	algorithm
				100%	100%	100%	100%	98%	93%	89%	85%	82%	77%	71%
		using 1-D SDTMs
					Table 7 Recall and precision for MP3 compression
			Compression ratio (Kbps)			16	24	32	40	48
				Xiao's algorithm[33]	74%	83%	87%	93%	100%
			Recall	Proposed	algorithm
								84%	91%	100%	100%	100%
				using 1-D SDTMs
				Xiao's algorithm[33]	69%	79%	85%	90%	100%
			Precision	Proposed	algorithm
								81%	90%	98%	100%	100%
				using 1-D SDTMs

Table 8

 8 Results for plain region duplication forgery (Algorithm 1 is based on Zernike moments and kd-tree matching; Algorithm 2 combines Tchebichef moments and lexicographic sorting; the proposed algorithm uses the fast computation of Tchebichef moments and kd-tree matching)

	Algorithms	Recall	Precision
	Algorithm 1[12]	89.6	87.5
	Algorithm 2[35]	87.5	85.4
	Proposed algorithm
		93.8	87.5
	using 2-D SDTMs

& & & & & &

ACKNOWLEDGEMENTS

The authors thank V. Christlein and C. Riess [12] for sharing their code and dataset as well as their helpful suggestions.

This work was supported by the National Basic Research Program of China under Grant 2011CB707904, the NSFC under Grants 61271312, 61103141, 61232016, 61173141, and 11301074, the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant 13KJB520015, a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, Open Fund of Jiangsu Engineering Center of Network Monitoring (KJR1404), Jiangsu Province under Grants DZXX-031, and BY2014127-11, and the Qing Lan Project. Beijing

Algorithm 1 [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF] Algorithm 2 [START_REF] Li | Passive digital image authentication algorithm based on Tchebichef moment invariants[END_REF] Proposed algorithm using 2-D SDTMs Algorithm 1 [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF] Algorithm 2 [START_REF] Li | Passive digital image authentication algorithm based on Tchebichef moment invariants[END_REF] Proposed algorithm using 2-D SDTMs Fig. 9 Recall and precision when the images are scaled In order to better apprehend these results, visual examples are provided in Fig. 11. These examples correspond to some of those shown in Fig. 2 in the case of a plain copy-move forgery and four different types of additional manipulations. Detected regions are highlighted by colored areas. It can be seen from this figure that the proposed algorithm can correctly detect duplicated regions though the snippets are processed by some manipulations before being pasted. However, this is not the case for the other algorithms. Indeed, sometimes Algorithm1 and Algorithm 2 fail to detect or localize properly duplicated regions with many false positives detection (see for instance, the sky regions of Fig. 11(g) and (h)). 2). Columns are successively the detected results using Algorithm 1 [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF],

Algorithm 2 [START_REF] Li | Passive digital image authentication algorithm based on Tchebichef moment invariants[END_REF] as well as our algorithm.

V. CONCLUSIONS

In this paper, by establishing the link between the Tchebichef moments of two overlapping windows based on some properties of the Tchebichef polynomials, we have presented a fast computation algorithm for 1-D and 2-D sliding discrete Tchebichef moments. The computational complexity analysis and the experimental results demonstrate that our algorithm is more efficient than the conventional direct ones. A notable difference with the direct algorithm is that the proposed algorithm is independent of the block size and is particularly suitable for applications which need to slide a window over the whole signal or image.

Two pplication examples aimed at the detection of duplicated signal segments and image regions has been developed in order to show the efficiency of our approach. On one hand, we have shown that our 1-D SDTMs algorithm leads to a better performance in audio signal than the algorithm reported by Xiao et al [START_REF] Xiao | Audio authenticity: Duplicated audio segment detection in waveform audio file[END_REF]. On the other hand, for image duplicated regions detection, our solution based on 2-D SDTMs is superior to the algorithm based on Zernike moment features and kd-tree technique reported in [START_REF] Christlein | An evaluation of popular copy-move forgery detection approaches[END_REF] and the algorithm proposed by Li et al. [START_REF] Li | Passive digital image authentication algorithm based on Tchebichef moment invariants[END_REF] where the Tchebichef moments of every block are used in combination with a lexicographic technique.