Xiang Sun
email: sunx@nercita.org.cn

Huarui Wu

Huaji Zhu

Cheng Peng
email: pengc@nercita.org.cn

Jingqiu Gu

Research on the Collaboration Service Mechanism for Pig Diseases Diagnosis Based on Semantic Web

Keywords: semantic web, context awareness, collaborative decision, pig diseases diagnosis, agricultural intelligent system

Distributed knowledge resources for the pig diseases diagnosis have the characteristics of heterogeneous, autonomy and difficult to be shared among different systems. Based on the web-based ontology modeling language, a descriptive model to describe the heterogeneous knowledge resources of the pig diseases diagnosis and a formal model to express the decision services of pig diseases are proposed in this work. Based on the two models, a complex decision task can be automatically divided into many dependent simple decision fragments. The relationships between different fragments, the mapping relationships between each fragment which can be dynamically adjusted according the divided fragments and its distributed knowledge resources can be formally described. The models form a mechanism for cooperative decision for pig diseases diagnosis. The method is proposed by achieving the balanced decomposition of decision task and intelligent schedule of knowledge resources. The method is verified in a pig diseases collaborative diagnosis system. The result shows that the method is superior to the traditional intelligent decisionmaking method.

Introduction

Nowadays, along with the application of intelligent information techniques in agricultural information systems, the amount of knowledge resources of raising pig in the process of monitoring, production, operation and management grows rapidly [START_REF] Wu | Architecture of agricultural data grid system[END_REF]. However, these knowledge resources belong to different owners and they are heterogenous in type and representation. The dispersed diagnostic knowledge which has some incomplete issues inevitably reduces the efficiency of knowledge service. Collaborative decision-making is the decision-making process or ability of coordinating different data resources, knowledge resources, terminal devices, applications [START_REF] Paolucci | Semantic matching of web services capabilities[END_REF], [START_REF] Sycara | Dynamic service matchmaking among agents in open information environments[END_REF]. The purpose of collaborative decision-making service in the field of pig diseases diagnosis which has heterogeneous knowledge is rapidly positioning associated knowledge resources and making decision. As a result, it can achieve the

2

The Semantic Description of Decision-making Services Knowledge Resources Distributed environment is the basis environment of achieving resource sharing and collaborative decision-making. A wide range of agricultural knowledge resources, the representation model, location, underlying communication mechanism are different. To realize collaborative service in distributed agricultural resources, the paper puts forward a semantic description model that can support the multiple type, shield heterogeneity and adapt to dynamic environment. This model uses OWL as the agricultural knowledge ontology expression language [START_REF] Sun | Intelligent Service-the Research of Integrating Web Service and Semantic Web[END_REF], [START_REF] Wang | Web-oriented knowledge description language[END_REF], [START_REF] Yang | The OWL Description of Ontology-Based Agriculture Domain Knowledge[END_REF] and explains the related agricultural sharing concept model. Additionally, it provides basic terms and relationship for agricultural knowledge resource model and makes up the extension rules and complex definitions using those relationship and terms. This mechanism realizes the concept system of classifying and describing the agricultural mass knowledge. It is the basic for collaborative knowledge resource organization and management and information exchange.

Ontology Description of Decision-making Services of Agriculture Knowledge

Semantics heterogeneity brings on differences of understanding the agricultural knowledge resources, described by using same descript mode, between agriculture knowledge resources of providers and users. This will affect the effect of the collaborative decision-making. In order to solve semantic heterogeneity problem, this paper introduces the semantic ontology quaternion group to describe the decisionmaking service of agriculture knowledge:

} constraint owner relation kobject { KR > < > < > < > < = , , , (1)
'kobject' is the agricultural knowledge atom, which is the smallest knowledge resources unit in the entire descriptive model. Expressed as:

} , , i 1 {ra kobject i Ω ∈ Φ ∉ ≤ ≤ = ra ra n (2)
'Ra' is the knowledge atom in the field of agricultural knowledge resources; 'relation' means the collection that has reciprocity and affects among the knowledge atom and knowledge entities. Namely:

} Φ r n, l k, j, i, 1 ) rm , (rm r ) ra (ra {r relation j k ij j i ij ∉ ≤ ≤ ∨ + = (3)
'Rm' means a knowledge entity which is constituted by agriculture knowledge resources.

} ra ra , 1 ra ra { rm i i i i Ω ∈ ⋅ Φ ∉ ⋅ ≤ ≤ ⋅ ∏ ∨ ⋅ = ∑ j j j j ra ra n j i ra ra , ; (4)
'rij (rai+raj)' means the connections among knowledge atoms of agricultural knowledge resource. rij (rmk,rmj) means the connections among knowledge entities. Owner is the identifying information that marked the agriculture knowledge resources owner and application quality. It reflects knowledge resource owner and application effect under the distributed environment. Constraint is resource constraints of agricultural knowledge. It includes some labels of agricultural knowledge entities and related resources, rules or operation collection that is generated by relations among knowledge atoms or entities [START_REF] Gruber | Towards principles for the design of ontologies used for knowledge sharing[END_REF].

The Model of Agricultural Knowledge Resources

According to characteristics of agricultural knowledge resources and core idea of collaborative services in complex system, we made the unified description and expression of the agricultural knowledge resources model and defined the model as the seven BNF sub determinant:knowledge pattern, meta-knowledge, rule set, quantitative model, resource label, Ontology Bridge, external link extension. These basic hierarchic group of sub determinants form more complex models of agricultural knowledge resources. Some BNF are shown in Figure 1. Knowledge pattern is mostly used to determine the type of knowledge expression. In process of collaborative decision-making, meta-knowledge, rule set and quantitative model mostly are used to establish the generalized mapping relationship between items of rule antecedent facts and conclusions facts, such as these four basic types:'part of', 'kind of', 'instance of', 'attribute of'. Otherwise, pattern matching relationship, include certainty, uncertainty, fuzzy matching, or the accessibility relation that based on the generalized mapping relationship also can be used. This two By OWL mapping between ontology and ontology, between ontology and the information sources, between ontology and knowledge representation methods, Ontology Bridge solve the semantic heterogeneous problem of agricultural knowledge resources naming, attribute and extension. For example, OWL: equivalentClass and OWL: equivalentProperty are used to solve the problem of naming heterogeneity that different information sources denote the same concept according to multiple languages. Rdfs:subPropertyOf is used to solve the problem of attribute heterogeneity. Owl:equivalentClass and owl:equivalentProperty are used to solve the problem of equivalence element and owl:intersectionOf is used to solve the problem of crossed element. Rdfs:subClassOf and rdfs:subPropertyOf are used to solve the problem of contain elements in ontology syntax.and owl:unionOf in global ontology are used to solve the problem of disjoint elements. The solutions of these problems achieve the unified description of knowledge representations that have many types such as production rules, examples, and semantic web. These solutions also can convert variety of types of knowledge to the personalized knowledge representation required by collaborative decision-making [START_REF] Sun | Technology of Agricultural Production Knowledge Integration Based on Semantic Web[END_REF].

Resource label and exterior extension are made use of combining the information resources and the knowledge resources. It includes plain text, multi-media, HTML, GIS, remote sensing images, database and other resources marks. Moreover, it realizes the knowledge integration of frame, production rule, example, description and other models of knowledge expression.

According to the composition of agricultural knowledge resources and the definition of semantic description system, knowledge resources of agricultural collaborative decision-making service is divided into three layers, decision-making problem layer, correlative information layer, knowledge model layer. Decisionmaking layer includes definition of atom of agricultural knowledge resources. Such as, decision-making facts, concepts, fact attribute those related to agricultural knowledge rules and the relations among the concepts. Correlative information layer mostly includes related information of different decision-making model, resources label information that related to decision-making project; the efficiency of resource reference, etc. Knowledge layer includes meta-knowledge of agricultural knowledge resource model and related index of model, index of decision-making problem, index of resource correlation. By means of related index, it can accurately locate the related model, decision-making problem, related information resource of each node in a distributed environment [START_REF] Zhao | Development platform for agricultural intelligent system based on technocomponentware model[END_REF].

The Representation Formalism of Decision-making Service Problem

Description of Decision-making Problems List

In the semantic description model of decision-making service knowledge resources, decision-making layer provide users with a description of their required decisionmaking service and the abstract of actual decision-making data item, to realize the analysis of decision-making problem requirement. It means an expression of decisionmaking problem requirement.

) , , , , , ( , RU RF RC RA RO UID R R R R R R d dn d2 1 d s dec = ∧ ⋅ ⋅ ⋅ ∧ ∧ ∧ = (5)
R5 means initial limit of decision-making goals. Rd means the restriction of decision-making premise data items. UID is the unique identifier of the decisionmaking service users. RO is the domain limit of knowledge or resources. RA is the restriction of functional properties of agricultural knowledge. RC means the explanation of the initial value of agricultural knowledge. RF is the restriction of information resources. RU is the memory of decision-making service.

There are a large number of agricultural knowledge resources. These knowledge resources provide decision-making knowledge service for problem of decisionmaking service. Based on the decision-making goal of decision-making user and the model of agricultural knowledge resources, the formalism representations of decisionmaking service problem identify the interaction in each step progressively.

)} Rxt , k ( , ), Rxt , k ( ), Rxt , k ( {Rxt, RU n n 2 2 1 1 UID > < ⋅ ⋅ ⋅ > < > < = (6)
'Ki' means the node identification of decision-making problem flow. 'Rxti' is the node information set of decision-making service problem. When collaborative decision-making occurred, the description of problem and the context information are stored in RU as a unique identification. It provides the problem list that interacts with the user for collaborative decision-making services.

Firstly, according to the domain type, content, owner of knowledge resources, property, states of obtained decision-making services goal,the limited value is initialize and then the decision-making knowledge resources are obtained to calculate the decision-making problem list.

Matching of Decision-making Problem and Knowledge Resources

It is the key that to determine the knowledge resources related to decisionmaking problem in the distributed environment and achieve the representation formalism of decision-making service problem. The semantic matching between knowledge resource and user decision-making problem is determined by the matching degreen of the concepts similarities of description terms and functional property constraint.

The concepts similarity is gained by calculating the semantic distance between problem description and description terms of context knowledge resource. The similarity of functional property is get by calculating the problem description and context knowledge resource property [START_REF] Li | Ontology Construction for Semantic Web: A Role-Based Collaborative Development Method[END_REF][START_REF] Ye | Application-Oriented Web Service Discovery and Matchmaking[END_REF][START_REF] Wang | Reconfigurable Platform for Knowledge Management Based on Ontology[END_REF]. The calculate method of concept similarity is as follows:

∑ ∑ = = × + × = n n 1 j j j 1 i i i Kterm) , (
∑ ∑ = = = + n n 1 j j 1 i i 1 ω ω .
The functional property constrain matching of agricultural knowledge atoms is decided by the domain of knowledge atom and the decision-making service premise data items constrain. The value of constrain is related to the initial value of agricultural knowledge atom. So the constrain value of decision-making problem need to be converted to the interval form. 'C(Qterm)' is the value range of problem description sub-constrain. 'C(Kterm)' is the value range of knowledge description sub-constrain. k μ is the weight coefficient of matching degree of constraint. Then the calculating method of similarity degree of functional property is as follows:

⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ Φ = ∩ Φ ≠ ∩ ∩ = = ∑ = 。 , ; , , ) ( ) ( 0 ) ( ) ( )) ( ( )) ( ) ( ( P ) ( ) ( 1 Fmatch n 1 k k Kterm C Qterm C Kterm C Qterm C Kterm C P Kterm C Qterm C Kterm C Qterm C k μ (9)
'Fmatch' is weighted sum of 'Amatch' and 'Sim_question'.

) ( RA , R Fmtach 2 ) R , on(R Sim_questi 1 Amatch s d s × + × = μ μ (10)
In above formula,

Agricultural Cooperative Decision-making Service Mechanisms

The completion of specific agricultural cooperative decision-making service needs multiple process fragments.In the collaborative decision-making process, it's necessary to achieve task decomposition of distributed decision-making services, matching of knowledge resources and intelligent invocation of agricultural cooperative decision-making tasks based on the user demand for services, the context memory of agricultural decision-making service problem flow and the message of user interaction decision-making premise items.

Fig. 2. The collaborative service model of agricultural knowledge resources

Agricultural cooperative decision-making service module has two main functions: the matching of decision-making problems' knowledge resources and the task decomposition of decision-making services [START_REF] Uschold | Ontologies: Principles, methods and applications[END_REF][START_REF] Holsapple | A Collaborative Approach to Ontology Design[END_REF][START_REF] Mei | Logical Foundation of Semantic Web[END_REF][START_REF] Ma | Production and distribution collaborative decision-making based on supply hub[END_REF]. The steps are as follows:

Step 1. Matching of decision-making problem knowledge resources based on the demands of user's decision-making. Model of knowledge resource ontology description can analyse the agricultural knowledge resource, calculate the matching degree of knowledge resource and create a "decision-making service problem chain".

Decisionmaking service task decomposition service character description table Step 2. According to the content of "decision-making service problem chain", decision-making task decomposition modules analyses the type and class degree of service to determine the character of the agricultural knowledge decision-making service and achieve mapping scheduling of decision-making task. Agricultural knowledge service can be divided into compute-intensive knowledge service, communicate-intensive knowledge service and compute-communicate relative equilibrium knowledge service. By analyzing class degree of knowledge service, the number and character of parallel task is determined and each node's application quality of service resources is obtained. Finally, the analysis results are published regularly.

Task mapping

Decision-making task scheduling

Resource ontology description

Demands of user decision

Matching of knowledge resource

Analysis of character

Step 3. The decision-making task mapping module divides knowledge service into many sub-tasks, using "service character description table". These sub-tasks include some dependent or preferred constrain relationship and are mapped as some task sets that adapt to different machine characters and service characters. The goal is to make sure the sub-task can be carried out at suitable node.

Step 4. Agricultural collaborative decision-making service task scheduling module based on the result of task mapping, send task to specific node service queue and schedule the decision-making of distributed nodes to execute decision-making task.

Step 5. In the end the result returns. The conflict resolution is finished and the results return to user.

Analysis and application

Based on the agricultural coordination decision-making service mechanism, we use J2EE platform to develop pig diseases collaborative diagnosis system. As the result of the great variety diseases of live pig symptom, although knowledge databases of various nodes follow the unified regulations, the knowledge engineers of system establishment are different, Therefore, the attention points on diagnosis knowledge are different and there are differences on symptom antecedent, the consequent as well as authentication. For this reason, the pig sickness main body is contructed, and by modeling the policy-making service knowledge resources' semantic description, the existing knowledge, the data, the multimedia resources are labeled.So the integrated applications of pig sickness diagnosis knowledge rules and related disease symptom, prevention method and so on are realized in Beijing, Tianjin, Hebei, Sichuan and other providence.

The system not only can recommend users more semantically related concepts based on knowledge of swine, but also can complete collaborative coordination of different expertise for the same diagnosis, and provide various results with recommendations for the diagnosis and disposal program. It's difficult for the traditional agricultural expert system to finish. Take 4 swine diseases expert systems as the example; take 20 initial symptoms as the example to deduction, the inference experimental result as shown in Table 1. Comparing collaborative decision-making system based on semantic WEB to traditional expert system, the former has remarkable enhancement on the inference conclusion gain rate, the inference conclusion authentication mean value and the inference efficiency. Such as Rg increases 12.5%, Cn increases 14% and Rc improved 13.3%.

Conclusion

In the application service of distributional knowledge resources, how to realize the knowledge resources coordination decision-making service is important. Based on the semantic WEB technology, the paper, proposed a kind of agriculture decision-making service knowledge resources semantics description model which supports the multitypes, shield isomerism, adaptation dynamically. It defines agricultural knowledge resources semantics description as 7 minor BNF. Based on this foundation, the formalized expression method of policy-making question is studied and the demands of agricultural coordination decision-making service are decomposed into many policy-making parts, according to the agricultural decision-making service question class context memory and user's interactive decision-making premise information. As a result, the match between distributional decision-making service task decomposition and the knowledge resources and the intelligent dispatch service of agricultural coordination decision-making task are realized, enhancing the using efficiency and inference efficiency of agricultural knowledge.

The collaborative service of agricultural knowledge resources is a complex engineering system. The collaborative decision-making service based on semantic WEB proposed in this paper took the semantic main body as the foundation of coordination service. The realtionship of the quality of coordination service and the competition of ontology library is high. The future research will focus on the update of the increment of agricultural knowledge resources ontology and the resolution mechanism of coordination service conflict.

Fig. 1 .

 1 Fig. 1. BNF fragment of agricultural knowledge resources model

  <Agricultural knowledge resources model> ::=< knowledge pattern>|<meta-knowledge>|<rule set>| <quantitative model>|<resource label>|<ontology bridge>|<external link extension> <Knowledge pattern> :=< frame pattern>|<production rule pattern>|<example pattern>| <descriptive pattern> <meta-knowledge>:= <knowledge domain>|<knowledge owner>|<knowledge module>| <decision-making item>|<decision-making data item> <Rules set> ::=< rule name>|<rule definition>|< rule mark> <Quantitative model> ::=< model name>|<model class>|<model expression> < Resources label>::=< resources type>|< resources class>|<resource owner>|<resources ontology mark>;{< resources ontology marking sentence>;} <Ontology bridge> ::=< atom of knowledge resources>|< relation of atoms>|<ontology owner>|< resources ontology marks>;{<resources ontology marking sentences>;} <External link extension> ::=< resources label> ;{< plain text>|<multi-media>| <HTML>|<GIS>| <remote sensing image>|<database> ;} unit mode separately finished the qualitative analysis and model calculations in the decision-making process.

Fig. 3 .

 3 Fig. 3. Collaborative decision-making of swine diseases

Fig. 4 .

 4 Fig. 4. Ontology-based Collaborative decision-making of swine diseases

  Cd_sim(Qterm ,Kterm)' is the distance similarity of semantic concept. 'Ca_sim(Qterm, Kterm)' is the concept property similarity. 'Qterm' represents the problem description. 'Kterm' is the description of knowledge resources. ' is the maximum depth of 'Qterm' and "Kterm" in description model of agricultural knowledge resource ontology. ω is corresponding weight coefficient.

	Sim_questi	on(Qterm,	Kterm)		ω	Cd_sim	(Qterm	,	Kterm)	(7)
		ω	Ca_sim	Qterm
	Additionally, '⎩ ⎨ ⎧ = 0 1 Kterm) , (Qterm , Qterm Qterm Cd_sim i ,	Kterm Kterm 、 、	before before	not -i the i	class class parent parent a by same d accompanie with the	(8)
	Ca_sim j	(Qterm	,	Kterm)	=	⎩ ⎨ ⎧ 0 1 , Qterm Qterm 、 Kterm Kterm 、 ,	before before	not j the j	by same d accompanie with the -	class s parentclas parent a
	'N

Table 1 .

 1 Results of reasoning test All the knowledge regular strip counts Kb; Amount of connection resources R; Integer Rn in view of 20 initial symptom's inference conclusions (confidence level threshold 60%); Inference conclusion gain rate Rg= Rn/20*100; In view of confidence level of 20 inference conclusion mean values Cn; Inference efficiency Rc= (Rg +Cn) /2

	System	Kb	R(Gbit)	Rn	Cn	Rg	Rc
	Beijing	450	1.8	16	87	80	83.5
	Tianjin	347	0.7	14	79	70	74.5
	Hebei	512	0.9	17	81	85	83
	Sichuan	401	1.1	15	82	75	78.5
	Coordination	1710 4.5	19	89	95	92

Acknowledgments. This paper is supported by a grant from Beijing Nova program (Grant No. 2009B26), a grant from Beijing Academy of Agriculture and Forestry Sciences for Young Scholars and the National Natural Science Foundation of China (Grant No. 60871042).


Bastien Durand
email: bastien.durand@ensi-bourges.fr

Franck Delvare

Patrice Bailly

Didier Picart

Friction between steel and a confined inert material representative of explosives under severe loadings

Keywords: friction, confinement, Split Hopkinson Pressure Bars, identification

The ignition of a confined explosive submitted to an impact strongly depends on the friction conditions between the explosive and the confinement material (generally steel). A test has been developed to study the friction between steel and a material mechanically representative of an explosive. The scope of interest is that of high pressures and high relative velocities (respectively 20 MPa and 10 m/s). The friction device consists of making a cylinder, formed of the material, slide through a steel tube. Axial prestress enabling the steel-material contact stress to be generated is performed by means of a screw-nut system. This confinement state avoids any fracture of the material from occurring throughout the test.

Two kinds of tests are carried out: low-velocity (around 1 mm/min) and high-velocity (around 10 m/s). The relative displacement is obtained using a testing machine during the low-velocity tests, and thanks to a Hopkinson bars system during the high-velocity tests. Examination of the measurements obtained during high-velocity tests shows that a workable steady state of equilibrium has been reached. As the interface stresses cannot be measured, the friction coefficient must be determined using indirect data: force measurements obtained from the machine or from the Hopkinson bars and strain measurements made on the exterior of the tube. The procedure to identify the steel-material friction coefficient from these measurements entails analytical modelling and finite element simulations of the mechanical behaviour of the tube-specimen assembly. The friction coefficient identified during the high-velocity tests is far higher than the coefficient identified during the low-velocity tests.

Introduction

Solid explosives are materials able to quickly release energy under excessive loadings.

Our study focuses more particularly on cases of so-called "low energy" impacts (below the shock-to-detonation threshold, which corresponds to impact velocities of some tens to hundreds of metres per second). Different experimental configurations are able to reproduce this situation [START_REF] Field | Ignition mechanisms of explosives during mechanical deformations[END_REF]. One of the most commonly used is the "Steven test" [START_REF] Chidester | A frictional work predictive method for the initiation of solid high explosives from low pressure impacts[END_REF], [START_REF] Vandersall | Experimental and modelling studies of crush, puncture, and perforation scenarios in the Steven impact test[END_REF]. This consists of launching projectiles of a few hundred grams, at a velocity of some tens to hundreds of metres per second, which impact targets composed of a steel/explosive/steel assembly (Figure 1).

Predicting the ignition of the explosive during this type of impact is based on numerical simulations [START_REF] Kim | The Influence of Temporal Profile on Hyperbolic Heat Conduction in Materials Subjected to Repetitively Pulsed Laser Radiation[END_REF].

However, prerequisite to such predictions is a deep and full awareness of the friction conditions at the explosive-steel interface. Indeed, numerical analysis shows that the time of ignition and the location of the ignition point in the target is highly influenced by the kinetic friction coefficient magnitude [START_REF] Dickson | Frictional Heating and Ignition of Energetic Materials[END_REF], [START_REF] Hoffman | Aspect of the tribology of the plastic bonded explosive LX-04[END_REF], [START_REF] Picart | Ignition of HMX-based PBX submitted to impact: strain localization and boundary conditions[END_REF]. The purpose of our study is thus to develop an experimental procedure enabling the friction coefficient between an explosive and a steel to be identified under loadings generated by "low energy" impacts. This range is that of high contact pressures (several tens of MPa) combined with high sliding velocities (10 m/s). Few set-ups satisfy the pressure and velocity requirements set in this study: tribometer with explosively-driven friction [START_REF] Kim | Nanostructures generated by explosively driven friction: Experiments and molecular dynamics simulations[END_REF], target-projectile assembly with oblique impact [START_REF] Prakash | A pressure-shear plate impact experiment for investigating transient friction[END_REF], [START_REF] Rajagopalan | Novel experimental techniques for investigating time resolved high speed friction[END_REF], Hopkinson torsion bars [START_REF] Rajagopalan | Novel experimental techniques for investigating time resolved high speed friction[END_REF], [START_REF] Espinosa | A Novel Dynamic Friction Experiment Using a modified Kolsky Bar Apparatus[END_REF], [START_REF] Huang | Dynamic Friction of SiC Surfaces: A Torsional Kolsky Bar Tribometer Study[END_REF], [START_REF] Rajagopalan | A modified Kolsky bar for investigating dynamic friction[END_REF], dynamometrical ring with parallelepipedic specimen launched using a gas gun or hydraulic machine [START_REF] Philippon | An experimental study of friction at high sliding velocities[END_REF], [START_REF] Philippon | A Device Enhancement for the Dry Sliding Friction Coefficient Measurement Between Steel 1080 and VascoMax with Respect to Surface Roughness Changes[END_REF] and, as a final possibility, the friction of a pin on a revolving disc [START_REF] Dickson | Frictional Heating and Ignition of Energetic Materials[END_REF].

Moreover, the desired pressures (several tens of MPa) are habitually reserved for metals [START_REF] Prakash | A pressure-shear plate impact experiment for investigating transient friction[END_REF], [START_REF] Rajagopalan | Novel experimental techniques for investigating time resolved high speed friction[END_REF] and ceramics [START_REF] Huang | Dynamic Friction of SiC Surfaces: A Torsional Kolsky Bar Tribometer Study[END_REF] whereas the explosives have little resistance to simple compression. This therefore requires a totally new device to be designed which enables us to confine the explosives. Indeed, a confinement configuration makes it possible to apply high pressures on an explosive sample without fracturing it. Consequently, the decision was made to design a tubular test chamber to act as a confinement for the explosive. For safety reasons, specimens made of an inert material mechanically representative of explosives are used to carry out our tests. A pressure of the order of 10 MPa is firstly imposed at the specimen -tube interface.

Then, the friction characterisation test consists of forcing the specimen to slide in the tube.

The advantage of this device lies in that it may be used with a classical test machine for tests at low sliding velocities (around 1 mm/min) or can be mounted on a dynamic test bench of the Hopkinson bar type to reach high sliding velocities (around 10 m/s).

In section 2, the description of the experimental set-up is followed by a presentation of the low-velocity and high-velocity tests. The raw results are discussed, which demonstrate the interest in performing the study at high sliding velocities. In section 3, the issue of the establishment of a procedure to identify the friction is raised, such procedure being based on analytical and numerical models. The test has to be modelled since there is no way to directly measure the stresses on the sliding interfaces. Two kinds of model are proposed: an analytical one similar to the Janssen's one [START_REF] Janssen | Versuche über Getreiedruch in Silozellen[END_REF] and a numerical one using the finite element method. In section 4, the limits of the experimental device and of the friction identification procedure are discussed.

Experimental configurations

The inert material which is a mechanical equivalent of explosives

An equivalent inert material, denoted I1 and largely described in [START_REF] Bailly | Dynamic behavior of an aggregate material at simultaneous high pressure and strain rate: SHPB triaxial tests[END_REF], was used for safety reasons. Its mechanical properties are known and are relatively similar to those of explosives. The Young's modulus E is 2 GPa and the Poisson's ratio ν is 0.4. The non-elastic behaviour has been studied by carrying out triaxial compression tests. Under compressive loading, the material is able to flow when its plasticity threshold has been attained (here the maximal constraints obtained using triaxial tests are assimilated to a plasticity threshold in order to simplify the behaviour model). The plasticity flow threshold thus identified is defined by a Drucker-Prager criterion [START_REF] Bailly | Dynamic behavior of an aggregate material at simultaneous high pressure and strain rate: SHPB triaxial tests[END_REF]:

(1)

C < αP + σ eq
where P is the hydrostatic pressure and σ eq the Von Mises equivalent stress.

As the identified behaviour is assumed to be perfectly plastic [START_REF] Bailly | Dynamic behavior of an aggregate material at simultaneous high pressure and strain rate: SHPB triaxial tests[END_REF], the plasticity flow is defined by:

(2)

C αP + σ eq =
The model's parameters have been identified: C = 25 MPa and α = 0.63.

According to relation [START_REF] Chidester | A frictional work predictive method for the initiation of solid high explosives from low pressure impacts[END_REF], in the case of a simple compression loading, the maximum axial stress is only 31 MPa. The I1 specimen has therefore to be confined to reach pressures of several tens of MPa without fracturing it (the material cannot flow indefinitely).

Friction test cell enabling confinement

One way of confining a material is to enclose it in a ring [START_REF] Bailly | Dynamic behavior of an aggregate material at simultaneous high pressure and strain rate: SHPB triaxial tests[END_REF], [START_REF] Forquin | Influence of free water on the quasi-static and dynamic of strength of concrete in confined compression tests[END_REF]. The idea retained is to slide a specimen of our material into a steel tube. The tube thus acts both as a sliding surface and a confining ring (Figure 2). The inner wall of the steel tube was reamed and the specimen was turned on a sliding lathe. Both have a weak surface roughness representative of the roughness of pyrotechnic structures such as the Steven-test (Figure 1). The normal pressure at the tube-specimen interface is generated by a screw-nut system (Figure 2). The tightening of this system creates an axial prestress and as the specimen is constrained in the tube, it induces the normal pressure by Poisson effect. The relative displacement of the specimen in the tube, and thus, the tangential stress linked to the friction, is then obtained using a classical test machine for the low-velocity tests (around 1 mm/min) or by using Hopkinson bars (Figure 3) for the high-velocity tests (around 10 m/s). Thus, there are no extra effects due to radial inertia. The stresses and the sliding velocity at the interface are deduced from measurements performed by a circumferential strain gauge bonded to the external face of the tube and from measurements of forces and velocities collected by the machine or the bar system. Bar and striker diameters: 20 mm

Low-velocity tests

A series of low-velocity tests was performed by positioning the test cell onto a classical test machine. The speed of the machine is set at 1 mm/min. Different values for the normal contact pressure were obtained by modifying the prestress due to the tightening of the screw-nut system (Figure 2). Since the device is in equilibrium during these low-velocity tests, the forces F i and F o (Figure 2) are equal and denoted F. They correspond to the friction force.

Figure 4 gives, for two different values of ε θθ (due to the pre-tightening of the screwnut system), the evolution of F as a function of the displacement of the cross-piece ∆. These evolutions can be split into two phases: a first sticking phase during which the friction force increases until reaching a peak; then a second phase corresponding to the sliding during which the friction force remains constant. Relative motion does not occur everywhere along the interface during the first phase, which thus corresponds to the static friction; whereas relative motion occurs everywhere during the second phase, which thus corresponds to the kinetic friction. The curve forms a peak as the static friction coefficient is slightly higher than the kinetic friction coefficient. Only the second phase of the test, during which the sliding and the stress and strain states of the material specimen are assumed to be steady, is exploited.

A mean friction stress τ mean can be calculated during the sliding phase as the length L (L = 40 mm) and external radius R (R = 10 mm) of the specimen are known (Figure 5).

(

) RL F = τ mean 2π 3
An indicative value of the pressure level at the interface p ind is obtained using the strain measurement ε θθ given during the steady phase by the circumferential strain gauge bonded on the tube (Figure 2):

(4)

(

)

θθ e t ind ε R R E = p 2 2 2

2R

where E t is the Young's modulus of the tube (E t = 210 GPa) and R e its external radius (R e = 14 mm) (Figure 2). Indeed, if no friction occurs at the tube-specimen interface, the formula (4) gives the pressure at this interface exactly. L tube =90 mm (length of the tube)

See Figure 2 for the other symbols.

High-velocity tests (using a Hopkinson bar apparatus)

The previously described confinement and sliding cell (section 2.2 and Figure 2) is mounted on a system of Hopkinson bars [START_REF] Bailly | Dynamic behavior of an aggregate material at simultaneous high pressure and strain rate: SHPB triaxial tests[END_REF], [START_REF] Forquin | Influence of free water on the quasi-static and dynamic of strength of concrete in confined compression tests[END_REF] (Figure 3). The impact of the striker generates a compressive incident strain wave ε i in the input bar. When this wave reaches the cell, a reflected wave ε r appears in the input bar and a transmitted wave ε t appears in the output bar. Longitudinal gauges are bonded on the two bars enabling the three strains ε i , ε r and ε t to be determined. The bars and the striker are made of a hard stainless steel (a nickel alloy), whose yield strength is 1 GPa. The forces and velocities at the input and output ends of the cell (section 2.2, Figure 2 and Figure 3) can be deduced from the three measured strain waves by using the following relations:

(

( ) correspond here to the strains to be measured at the input bar -cell interface (which corresponds to the left hand end of the cell, see Figure 2 and Figure 3) and ε t corresponds to the strain to be measured at the output bar -cell interface (which corresponds to the right hand end of the cell).

   - = - = t b o i r b i C V C V ε ε ε (6) ( )    - = + - = t b b o r i b b i E S F E S F ε ε ε V i and V o
A phase of quasi-equilibrium of the forces applied to the cell enables a simple exploitation of the test. This quasi-equilibrium state is verified when the forces are identical at the two ends of the cell:

(7) o i F F ≈
When this quasi-equilibrium state becomes steady, the sliding velocity V at the interface between the steel confinement tube and the specimen I1 is estimated by: ( 8)

o i V V V - =
It is this phase which constitutes the useful part of the test.

Figure 6 gives the raw signals recorded by the strain gauges for two pre-tightening values. The first non-zero values (positive) appearing on the signal given by the gauge bonded to the input bar (on the red curve) correspond to ε i and the second (negative) to ε r . The following non-zero values appearing on the signals given by the gauges bonded to the output bar (green curve) and on the tube (blue curve) respectively correspond to ε t and to ε θθ (section 2.2). The appearance of a steady phase can be observed on all the signals. It should be noted that the gain (the raw signal over strain ratio) for the gauges bonded to the bars has been set to a value of 1,040 V whereas that of the gauge bonded to the tube has been set to a value of 6,240 V. Furthermore, the signals of the gauges bonded to the bars have been inverted.

The forces are determined by using the equations [START_REF] Hoffman | Aspect of the tribology of the plastic bonded explosive LX-04[END_REF]. Figure 7 gives the evolution of the different forces for the two pre-tightening values. Naturally, the equilibrium of the forces is not reached immediately. There is, however, a relatively long steady equilibrium state phase (approximately from 150 µs to 550 µs) characterised by the equality given by equation [START_REF] Picart | Ignition of HMX-based PBX submitted to impact: strain localization and boundary conditions[END_REF] and by constant forces and sliding velocity V (Figure 8). This corresponds to the sliding state being established over the full interface. Any mistake in the time-shifting of the waves has an effect on the estimation of the forces and velocities in the transient phase but nevertheless remains without effect upon the analysis of the steady state phase.

For two pre-tightening values, Figure 9 gives the evolution of the normal pressure and of the friction stress over time. A steady state is established after around one hundred microseconds. The signal transmitted by the gauge bonded to the tube, from which p ind is calculated, can be seen to be noisy. The mean friction stress τ mean is calculated from the output force F o rather than from the input force F i since the aim is to sidestep the problem of the timing of the incident and reflected waves: This study shows the feasibility of the high-velocity tests. The sensitivity of the friction to the sliding velocity emphasizes the necessity of reproducing in the test the velocities encountered in so-called "low energy" impacts (whose magnitudes are of the order of 10 m/s). Indeed, it is apparent that for the intended application (simulation of "low energy" impacts) friction conditions deduced from tests conducted at 1 mm/min may not reasonably be taken into consideration.

Identification of the friction coefficient

The interface stresses cannot be directly measured and the test can be exploited (and the friction coefficient identified) only if they can be estimated. One model is proposed based on an approximate solution of the elastic equilibrium of the cylinder of material I1. This simplified approach consists of considering the equilibrium of slices of elementary thickness constituting the cylinder [START_REF] Janssen | Versuche über Getreiedruch in Silozellen[END_REF]. Numerical simulations will then enable the level of approximation of this model to be judged and the assumption of an elastic behaviour will be verified too.

Analytical modelling

The stress and strain states in the I1 specimen have rotational symmetry and the stress tensor σ and the strain tensor ε are as follows:

θ ε ε θθ ε ε ε ε           =
The axis are defined on Figure 5. Therefore, u ri tube is the initial difference in radii between the exterior of the specimen and the interior of the tube. u ri tube is defined such that u ri tube < 0 where there is radial clearance and u ri tube > 0 where there is radial extra thickness. Similarly, u ri screw is the initial difference in radii between the screw and the interior of the specimen. u ri screw is defined such that u ri screw < 0 where there is radial clerance and u ri screw > 0 where there is radial extra thickness (see Figure 11). The tube and the screw are assumed to be perfectly rigid. There is therefore no coupling between the behaviours of the tube, of the screw and of the specimen. If the specimen is slender enough, then ε rr and σ rr can be assumed to be independent of r and ε θθ can be assumed to be linearly dependent of r:

(11) ( ) ( )                - + - - = - + - = r R R u r R R u R R r R R u u i screw ri i tube ri i i screw ri tube ri rr 1 ) ( θθ ε ε
where R i is the inner radius of the specimen (R i = 3 mm), see Figure 2. Using Hooke's law, this gives:

(12) ( ) ( ) ( )( )       + +         - + - - + + - = screw ri tube ri i screw ri i tube ri i rr zz u u R R r u R R r u R R E z z r ν ν ν σ ν ν σ 1 1 ,
E and ν are respectively the Young's modulus and the Poisson's ratio of I1 whose numerical values are given in section 2.1.

If the friction stresses at the interface with the screw (at r = R i , see Figure 2) are put aside, the elastic equilibrium of an infinitesimal slice of the specimen with a thickness dz located at z results in:

(13) ( ) ( ) [ ] ( ) ∫ = + - R i R rz zz zz dz z R R dr dz z r z r r , , ,

σ σ σ

The friction condition is expressed as:

(14) ( ) ( ) ( ) z f z R f z R rr rr rz σ σ σ = = , ,
where f is the friction coefficient at the interface with the tube (at r = R), see Figure 2 and In this case, relation [START_REF] Rajagopalan | A modified Kolsky bar for investigating dynamic friction[END_REF] becomes:

(15) ( ) ( ) z R f dr z r r rr R i R z zz σ σ = -∫ , ,
However, according to equation ( 12):

(16) ( ) ( ) z z r z rr z zz , , 1 , σ ν ν σ - =
Using the equality [START_REF] Philippon | A Device Enhancement for the Dry Sliding Friction Coefficient Measurement Between Steel 1080 and VascoMax with Respect to Surface Roughness Changes[END_REF], the differential equation governing the distribution of the radial stress is obtained:

(17) ( ) ( )( ) ( ) z R R fR z rr i z rr σ ν ν σ 2 2 , 1 2 - - - =
Let p(z) and τ(z) be the pressure and friction stresses at the interface with the tube (at r = R), see Figure 2 and Figure 5:

(18) ( ) ( ) ( ) ( ) ( )    > - = > - = - = 0 , 0 , z R z z z R z p rz rr rr σ τ σ σ
According to relations ( 14) and ( 17), the interface stresses can be expressed as a function of τ mean :

(19) ( ) ( ) ( ) ( ) [ ] ( ) ( )        = - - = f z z p L f z L f L f z mean τ β β τ β τ 1 exp
exp where β being defined as:

(20) ( )( ) 2 2 1 2 i R R R - - = ν ν β
Since the confinement tube is of reduced thickness, the circumferential strain along the external face of the tube ε θθ (z) (at r = R e , see Figure 5) can be estimated. For z ∈ [0,L], the normal stresses p(z) make a direct contribution to ε θθ (z) of the form:

(21) ( ) ( ) z p R R E R e t 2 2 2 2 -
Since the tube is blocked to the right (Figure 2), for z ∈ [0,L], the tangential stresses τ(z) make an indirect contribution by Poisson's effect to ε θθ (z) of the form:

∫ - z e t t d R R R E 0 2 2 2 ζ ζ τ π π ν (22) ( ) ( )
where ν t is the Poisson's ratio of the tube (ν t = 0.33).

For z ∈ [0,L] the following formula can be deduced: (

]

L f R R E z f R RL z e t t t mean β ν β ν β τ ε θθ - - - + - - = exp 1 exp 2 ) ( 2 2
If z g represents the position of the strain gauge glued on the tube (Figure 2), the indicative values of the interface pressures can be expressed as:

(24) ( ) ( ) g e t ind z R R R E p θθ ε 2 2 2 2 - =
According to relation [START_REF] Sheu | Tool Surface Morphologies for Friction and Wear Control in Metalworking Processes[END_REF] this gives:

]

t g t ind mean z f R L L f R p ν β ν β β τ + - - - - = exp exp 1
In relation [START_REF] Qi | Adhesion and Adhesive Transfer at Aluminum/Diamond Interfaces: A First Principles Study[END_REF], all the parameters are known with the exception of f, this relation thus enables the form of the dependence of The strain gauge is glued on the tube (Figure 2). As the specimen moves relatively to the tube, z g (the position of the gauge relatively to the specimen) varies during the tests and initially, z g = 20 mm (Figure 2). The f relative uncertainty f f ∆ due to the z g uncertainty ∆z g can be deduced from relation [START_REF] Qi | Adhesion and Adhesive Transfer at Aluminum/Diamond Interfaces: A First Principles Study[END_REF] which leads to:

( ) [ ]( ) ( ) g g t t g t g z z f R L f L f L z f R z f f - - - - - - + - - ∆ = ∆ β β ν β β ν β ν β exp exp 1 exp exp (26) ( ) ( ) [ ] ( )
In the case of the low-velocity tests, the displacement of the specimen in the tube is less than 1 mm (Figure 4) and f = 0.14, which leads to f f ∆ < 2 %. As a result, the friction coefficient obtained with z g = 20 mm can be considered to be reliable.

In the case of high-velocity tests, the sliding velocity between the specimen and the tube can reach almost 10 m/s during 500 µs (Figure 8). It leads to a relative displacement of 5 mm, which cannot be neglected relatively to the specimen length. Thus, the position of the strain gauge z g varies from 20 mm to 15 mm. Even if the specimen remains in a stationary state (which implies that τ mean remains constant), p ind , which depends on z g , increases but this variation cannot be seen on Figure 9 because of the noise. The uncertainty due the relative displacement can be taken into account by identifying f with z g = 15 mm, which leads to f = 0.45 (whereas z g = 20 mm leads to f = 0.37).

Finite element simulations

Numerical simulations based on the finite element method are performed using ABAQUS CAE / Standard. The computations are made in two-dimensional axisymmetric configuration with quadrangular elements having quadratic interpolation. Only the I1 specimen and the steel tube are modelled and their mechanical behaviours are assumed to be perfectly elastic (which is checked in section 4.1). They are discretized by 0.2 mm x 0.2 mm size square elements. The normal contact between both parts is defined by a direct hard contact (no interpenetration) and separation is enabled (no adhesion). The tangential contact obeys a Coulomb's law imposed by Lagrange multipliers. The test is simulated with friction coefficients f varying from 0.10 to 0.45 by step of 0.05. The presence of the screw is modelled by an imposed displacement boundary condition. The radial displacement is assumed equal to u ri screw on the interface (Figure 13). The influence of the friction between screw and specimen is assumed to be negligible (no tangential stresses). The tightening of the screw-nut assembly is modelled by axial displacements imposed at the two ends of the specimen. Moreover, the specimen being pushed in the tube, a displacement is imposed on the right face of the tube (Figure 13). According to relation [START_REF] Durand | Numerical solution of Cauchy problems in linear elasticity in axisymmetric situations[END_REF] this gives:

(27) ( ) 1 exp 1 exp ) ( -               - = L f L z L f L f z mean β β β τ τ
It is thus possible for the stress profiles provided by the analytical model and by the numerical simulations to be compared. Figure 15 shows that formula [START_REF] Durand | Numerical solution of Cauchy problems in linear elasticity in axisymmetric situations[END_REF] gives results that are similar to those given by the numerical simulations although it does not allow any edge effects to be taken into account. The error produced by the analytical model is essentially due to the approximation of a reduced thickness for the confinement tube used to establish the formula [START_REF] Sheu | Tool Surface Morphologies for Friction and Wear Control in Metalworking Processes[END_REF]. The term σ rz 2 is the highest at the contact with the tube (at r = R), the criterion defined by inequality (1) can therefore be reformulated by using the notations defined in relation ( 18):

(29) C p f <         + - +       - ν ν α ν ν 3 1 3 2 1 2 2
The condition (29) is verified whatever the p magnitude if:

(30) ( ) ( ) 3 3 2 1 9 1 2 2 2 ν ν ν α - - + < f
In this particular case, this gives f < 0.31. The condition (29) is thus verified for the low-velocity tests for which f = 0.15.

For higher friction coefficients, the elasticity of the material is preserved if the pressure p remains under a certain value:

(31) ( ) ν ν α ν ν 3 1 3 2 1 2 2 + - +       - < f C p
For the high-velocity tests, it has been established that f < 0.56. f = 0,56 leads to the minimal value of ( )

ν ν α ν ν 3 1 3 2 1 2 2 + - +       - f C
and the corresponding maximal pressure p is 70 MPa.

According to relation [START_REF] Zhang | Origin of Static Friction and its Relationship to Adhesion at the Atomic Scale[END_REF], the maximal value of the stresses is 3.4 times higher than the value of the mean stress, which limits the mean pressure p mean to around 20 MPa. Given the values reached by the mean pressures p mean (Figure 14), the elasticity condition is respected.

Limits of the experimental device

Due to the fracture of the screw (which suddenly breaks in torsion) during its tightening, the pressure value of the device is limited to roughly 20 MPa and the compressed air gun used during the Hopkinson bar tests limits the possible range of relative velocity to roughly 5-20 m/s.

The proposed configuration merely gives macroscopic information on the interface conditions. It does not, therefore, enable less global information to be supplied on the sliding and namely on the transition between the sticking phase and the sliding phase.

As only the measurements of the friction force and of the external strain of the tube are available, a model is needed to estimate the friction coefficient and there are no overabundant measurements to confirm its adequacy.

Conclusion

The purpose of the study was to design an experimental configuration enabling the friction measurement between an inert material, mechanically representative of an explosive, and steel. The desired sliding velocities were of the order of 10 m/s and the desired pressures These results are useful since there is otherwise very little data. Future researches should focus on:

-Understanding local behaviour (namely the onset of sliding), and in this objective there are ongoing developments to improve the metrology (measurement of fields) and analysis (inverse method) to obtain more localised data on the interface [START_REF] Durand | Numerical solution of Cauchy problems in linear elasticity in axisymmetric situations[END_REF].

-Designing devices enabling the extension of the pressure range to 100 MPa and over. It has been done in [START_REF] Durand | Identification of the friction under high pressure between an aggregate material and steel: experimental and modelling aspects[END_REF] with a low-velocity tribometer which need to be adapted to high velocities.

-The achievement of higher velocities (100 m/s), for which another dynamic testing device would have to be designed.

-Modelling the friction process to understand the influence of the roughness [START_REF] Hector | Focused Energy Beam Work Roll Surface Texturing Science and Technology[END_REF], [START_REF] Hector | Analysis of Engineering Surfaces[END_REF], [START_REF] Sheu | Tool Surface Morphologies for Friction and Wear Control in Metalworking Processes[END_REF] and the influence of the chemical interaction [START_REF] Adams | Adhesion, Lubrication and Wear at the Atomic Scale[END_REF], [START_REF] Qi | Adhesion and Adhesive Transfer at Aluminum/Diamond Interfaces: A First Principles Study[END_REF], [START_REF] Zhang | Atomic Simulation of Kinetic Friction at Its Velocity Dependence at Al/Al and α-Al 2 O 3 /α-Al 2 O 3 Interfaces[END_REF], [START_REF] Zhang | Origin of Static Friction and its Relationship to Adhesion at the Atomic Scale[END_REF].

Figure 1 :

 1 Figure 1: Diagram of the impact of a projectile on a target composed of a thin front plate

Figure 2 :

 2 Figure 2: Diagram of the test cell constituted by a steel confinement and a pre-stressed

R e = 14

 14 mm (external radius of the confining tube) R = 10 mm (internal radius of the confining tube and external radius of the specimen) R i = 3 mm (internal radius of the specimen and radius of the screw) L = 40 mm (length of the specimen)

F

  i : force applied to the input end of the cell

FFigure 3 :

 3 Figure 3: Hopkinson bar device with confinement and sliding cell.

Figure 4 :

 4 Figure 4: Evolutions (for the smallest and the largest values of ε θθ (µ) due to the pre-

Figure 5 :

 5 Figure 5: Definition of the sliding direction, of the axis r u , θ u and z u , and of the radial r and

d = 15

 15 mm (initial position of the specimen relatively to the tube)

  respectively designate the velocities at the input and output ends of the cell. S b and E b represent respectively the cross-section of the bars and the Young's modulus of the bars (E b = 140 GPa), and C b the celerity of the longitudinal waves through the bars. ε i and ε r

Figure 6 :

 6 Figure 6: Raw signals from the strain gauges.

  (a): smallest pre-tightening value (b): largest pre-tightening value

Figure 7 :

 7 Figure 7: Forces F i and F o respectively applied to the input and output ends of the device.

Figure 8 :

 8 Figure 8: V i , V o and V, respectively, velocity of the input end of the cell, velocity of the output

  (a): smallest pre-tightening value (b): largest pre-tightening value

Figure 9 :

 9 Figure 9: Levels of normal pressure (p ind ) and shear stress (τ mean ) at the interface during a test.

Figure 10 :

 10 Figure 10: Shear stress τ mean as a function of the normal stress p ind . Results obtained during

Figure 11 :

 11 Figure 11: Definition of u ri tube : initial difference in radii between the exterior of the specimen and the interior of the tube and of u ri screw : initial difference in radii between the screw and the

Figure 5 .

 5 Figure 5.

.

  be estimated.Using a first order Taylor expansion of the function exp, the formula (25) leads to an Figure12shows the validity of this approximation. This approximation enables f to be identified from the values of Figure10). This gives, with z g = 20 mm, f = 0.14 for the low-velocity tests and f = 0.37 for the high-velocity tests.

Figure 12 :

 12 Figure 12: Numerical and analytical (from relation (25)) evolutions of f 1 as a function of

Figure 13 :

 13 Figure 13: Axial and radial displacements (u z and u r ) imposed on the tube-specimen

Figure 15 :

 15 Figure 15: Numerical and analytical axial profiles of the friction stress for f = 0.15 and for

  were several tens of MPa. A new confinement set-up was designed because of the low mechanical resistance of the material when submitted to the classical compression of commercial tribometers. The stresses and the friction coefficient between steel and the inert material have been identified from indirect measurements, from an analytical model and from numerical simulations of the mechanical response of the set-up.The mean pressures reached almost 20 MPa whereas the classical pin-on-disk tribometer limits this pressure to 5 MPa[START_REF] Dickson | Frictional Heating and Ignition of Energetic Materials[END_REF]. The sliding velocities reached around 1 mm/min during low-velocity tests and around 10 m/s during high-velocity tests. An influence of the sliding velocity on the friction has been clearly demonstrated since the kinetic friction coefficient identified is around 0.15 for the low-velocity tests and of the order of 0.4-0.5 for the high-velocity tests.

Acknowledgements

The authors would like to thank the reviewers for their valuable comments.

For the high-velocity tests, the uncertainty due the relative displacement is taken into account by applying a 5 mm axial displacement on the tube (Figure 13) in additional simulations (instead of 0.1 mm). f = 0.56 is identified from these simulations.

The friction coefficient thus being calculated for both types of test (low and high sliding velocities), it is possible, for each test, for the mean pressure p mean at the interface to be estimated from the values of τ mean obtained by measurement (Figure 10). Thus, we can check if the desired pressures have been reached. The results obtained with the friction coefficient estimated from the numerical simulations are presented in Figure 14.


Thanh-Trung Vo

Patrick Mutabaruka

Jean-Yves Delenne

Saeid Nezamabadi

Farhang Radjai

Strength of wet agglomerates of spherical particles: effects of friction and size distribution

We investigate the mechanical behavior of wet granular agglomerates composed of spherical particles by means of molecular dynamics simulations. The capillary cohesion force is modeled as an attraction force at the contact between two particles and expressed as an explicit function of the gap and volume of the liquid bridge. We are interested in the effect of the friction coefficient between primary particles. The agglomerates are subjected to diametrical compression tests. We find that the deformation is ductile involving particle rearrangements. However, a well-defined stress peak is observed and the peak stress is used as a measure of the compressive strength of the agglomerate. The strength increases with friction coefficient but levels off at friction coefficients above 0.4. Furthermore, the compressive strength is an increasing function of particle size span.

Introduction

Agglomeration of fine particles is used in many sectors including powder metallurgy [1], chemical engineering [2], iron-making industry [START_REF]Best available techniques (BAT) reference document for iron and steel production[END_REF] and pharmaceutical industry [START_REF] Wu | [END_REF]. Fine granular materials are prepared and mixed in required proportions, compacted into a green granule or tablet and finally sintered to reach sufficient mechanical strength and toughness required to resist subsequent loading. The strength is generally measured by performing diametrical compression on the granules or tablets [5,6]. It is crucially dependent on the agglomeration process but also on the nature of binding mechanism.

For a binding liquid, the important parameters are the amount, surface tension and viscosity of the liquid [7][8][9]. However, there are other important parameters that influence the strength of a green granule. The size distribution of primary particles controls the structure of the granule and the capillary force depends on particle size. Furthermore, although capillary attraction force is the main source of cohesion, the friction force between primary particles should affect the behavior of granules under loading. None of these parameters has been investigated in a systematic way in the past since particle size distribution and homogeneous blending of particles with the binding liquid are difficult to control in experiments. Numerically, broad size distributions require large numbers of particles in 3D.

In this paper, we present a numerical investigation of the effects of particle size distribution and friction on the strength of granules of spherical shape composed of primary spherical particles. For the simulations, we use the molecular dynamics (MD) method [START_REF] Radjai | Discrete-element modeling of granular materials[END_REF] and the capillary cohesion is modeled as an explicite function of the the distance between primary particles, liquid volume, surface tension and particle diameters [START_REF] Richefeu | [END_REF][START_REF] Radjai | Mathematical Modeling and Physical Instances of granular Flows[END_REF][START_REF] Richefeu | [END_REF][14][15][16]. We also assume that the liquid is homogeneously distributed between all eligible particle pairs. In the following, we first introduce numerical procedures. Then we present the a few results and conclude with a brief discussion.

Numerical model

The primary particles interact via normal repulsion force f e n , normal damping force f d n , Coulomb friction force f t and capillary force f c n . The total normal force is given by

f n = f e n + f d n + f c n . (1)
The normal repulsion force is modeled as a linear spring: f e n = k n δ n , where δ n is the normal elastic deflection approximated by the overlap between two particles. The normal damping force is a viscous contribution proportional to the relative normal displacement: f d n = γ n δn , where γ n is the damping coefficient.

The capillary force is due to the liquid bond and it depends on the gap δ n , water volume V b , surface tension γ s , and the particle-liquid-gas contact angle θ. The capillary cohesion force can be determined by integrating Laplace-Young equations [17], but we use the following explicite expression [START_REF] Richefeu | [END_REF]:

f c n = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -κ R, for δ n < 0, -κ R e -δ n /λ , for 0 ≤ δ n ≤ δ max n , 0, for δ n > δ max n , (2)
where R = R i R j is the geometrical mean of particle radii, κ = 2πγ s cos θ, and δ max n is the debonding distance, distance beyond which the bond breaks. It is related to liquid volume by [17]

δ max n = 1 + θ 2 V 1/3 b . ( 3
)
The length λ is is a function liquid volume V b , the harmonic mean (R = 2R i R j /(R i + R j )) and the size ratio

r = max{R i /R j ; R j /R i }: λ = α h(r) V b R 1/2 ( 4
)
where α is a constant prefactor. This form fits the capillary force obtained from direct integration of the Laplace-Young equation by setting h(r) = r -1/2 and α 0.9 [18] The total amount of liquid is evenly distributed among all pairs of particles that have a gap below δ max n . When a bond breaks, the liquid volume is redistributed among the contacts belonging to the same particles in proportion to particle sizes [START_REF] Richefeu | [END_REF]. The largest value of the capillary force occurs when two particles are in contact (δ n ≤ 0) [18]. This is the case of most liquid bonds, and the capillary force at those contacts is independent of liquid volume.

For the tangential force f t , we use a combination of elastic force and a frictional threshold [START_REF] Richefeu | [END_REF][START_REF] Luding | Collisions and Contacts between two particles[END_REF]:

f t = -min (k t δ t + γ t δt ), μ( f n -f c n ) (5)
where k t is tangential stiffness, γ t is the tangential damping parameter, δ t is tangential displacement, μ is friction coefficient, δt is tangential velocity.

To create the granules, we first prepare a large sample under isotropic compaction in a box. The distribution of particle diameters is either monodisperse or uniform in particle volume fractions between d min and d max . This distribution generally leads to high packing fraction. The size ratio α = d max /d min was 1 (monodisperse), 2 or 5. The largest particle size was kept to a constant value and the size of the smallest size was decreased. We extract spherical agglomerates from this sample comprising nearly 5000 particles. The cohesion forces are activated when the granule is subjected to diametrical compression between two platens. The friction coefficient was varied from 0 to 1. For each value of μ, several independent samples are simulated. Table 1 displays the values of all parameters used in the simulations. slow constant downward velocity, we measure the vertical force F between the top mobile wall and the granule. By dividing this force by the sectional area of the granule, we get the mean vertical stress inside the granule, which can also be calculated by using the values of forces

Compression of a granule

σ zz = 4F πD 2 ≈ 1 V N c k=1 f k z k z ( 6
)
where D is granule diameter, V is the total volume of the granule, N c is the number of contacts, f k z and k z are the zcomponents of the contact force vector and branch vector, respectively, at contact k. This stress can be normalized by the reference stress Figure 3 shows the vertical stress as a function of ε for 11 different values of μ and α = 1. We see that the stress increases to a peak value with strain all the more rapidly that the friction coefficient is higher. After the peak, the stress declines quite smoothly with cumulative strain towards zero. This ductile behavior is a consequence of particle rearrangements and dissipation due to friction and debonding. The debonding events between primary particles do not lead to spontaneous formation of a fracture surface. The particles are well-connected and the loss of one or two bonds of a particle does not lead to macroscopic rupture. Furthermore, the cohesion is reversible as the liquid is redistributed at each step homogeneously between particles within the debonding distance. The peak stress σ p increases with friction coefficient. Fig. 4 shows σ p /σ c as a function of friction coefficient for three different values of size ratio. It is a rapidly increasing function of μ but levels off beyond μ = 0.4. This means that the granule strength for most geomaterials, which generally have friction coefficients above 0.4, has practically no effect. But lubrication forces reducing friction coefficient may substantially reduce the granule strength. For example, during agglomeration in a rotating drum, the liquid lubrication forces activated by collisions between primary particles may affect the process [7,8].

σ c = γ s R ( 7
We also observe that the peak strength at high friction coefficient increases with size ratio for all values of friction coefficient. As shown in Fig. 5(a), it is nearly a linear function that may be a consequence of the uniform distribution by particle volume fractions. Its value is nearly doubled for a size ratio α = 5 as compared to the monodisperse case. Both the packing fraction Φ and wet coordination number Z (including all capillary bonds) increase with size ratio α.

Conclusions

In this paper, we used molecular dynamics simulations to investigate the effects of friction coefficient and size polydispersity on the mechanical strength of agglomerates (granules) composed of spherical particles interacting via capillary cohesion and Coulomb friction. The stressstrain behavior under diametrical compression is ductile and shows no abrupt rupture of the granule. The peak strength is of the order of the liquid-gas surface tension divided by the mean particle size, but increases considerably with both friction coefficient and size ratio. For a uniform size distribution in particle volume fractions, this increase of the mechanical strength as a function of size ratio can be attributed to the larger connectivity of the particles, which increases nearly in the same proportion. A more detailed analysis of the microstructure should later allow us to better understand the effect of different particle size classes inside the granule.

Figure 2 Figure 1 .

 21 Figure2shows three snapshots of a granule during diametrical compression. During diametrical compression at

Figure 2 .Figure 3 .

 23 Figure 2. Three snapshots of a granule under diametrical compression. The lines represent force chains.

Figure 4 .

 4 Figure 4. Normalized peak strength as a function of friction coefficient between primary particles for three different values of size ratio α.

Figure 5 .

 5 Figure 5. Normalized compressive strength (a), packing fraction Φ (b) and coordination number Z (c) of the granules as a function of size ratio α.

Table 1 .

 1 Parameters used in these simulations

	Parameter	Symbol Value
	Maximum diameter	d max	10.0 μm
	Size ratios	α	[1; 2; 5]
	No. of primary particles	N p	5000
	Friction coefficient	μ	[0,1]
	Normal stiffness	k n	1000 N/m
	Tangential stiffness Normal damping Tangential damping	k t γ n γ t	800 N/m 5.10 -5 Ns/m 5.10 -5 Ns/m
	Capillary force prefactor	κ	0.45 N/m
	Surface tension of water Density of primary particles η γ s	0.072 N/m 2600 kg.m -3
	Time step	dt	10 -9 s

e-mail: thanh-trung.vo@umontpellier.fr e-mail: pmutabar@mit.edu e-mail: jean-yves.delenne@inra.fr e-mail: saeid.nezamabadi@umontpellier.fr † e-mail: franck.radjai@umontpellier.fr

Powders & Grains 2017


Maksym Fedorchuk
email: m.fedorchuk-2017@kpi.ua

Igor Sikorsky

Bart Lamiroy
email: bart.lamiroy@loria.fr

Statistic Metrics for Evaluation of Binary Classifiers without Ground-Truth

Keywords: statistical evaluation, performance evaluation, image binarization, segmentation, document image analysis

In this paper, are presented a number of statistically grounded performance evaluation metrics capable of evaluating binary classifiers in absence of annotated Ground Truth. These metrics are generic and can be applied to any type of classifier but are experimentally validated on binarization algorithms. The statistically grounded metrics were applied and compared with metrics based on annotated data. This approach has statistically significant better than random results in classifiers selection, and our evaluation metrics requiring no Ground Truth have high correlation with traditional metrics. The experiments were conducted on the images from the DIBCO binarization contests between 2009 and 2013.

I. INTRODUCTION

In machine learning and statistics, classification is the problem of identifying to which of a set of categories a new element belongs. In many cases, this is a fully automated task and there are many different classification systems, developed for specific tasks and data types. Usually, there exist a huge number of possible classifiers for a given task, and it is important to be able to determine the ones most suitable for the problem at hand. In this paper, we will try to determine if standard performance metrics can be extended to situations where no reference benchmark data (Ground Truth) are available.

Indeed, the traditional approach to assessing the validity and performance of classifiers generally consists of 3 phases:

 assemble a representative collection of reference data;

 use human annotators to create a set of reference interpretations for the data collection (Ground Truth);

 run the classifiers to be evaluated on the reference data and measuring their discrepancy with the expected Ground Truth.

While this paradigm is well understood, and has been largely adopted to assess and measure the quality of stateof-the-art classifiers, it has a number of limitations and drawbacks [START_REF] Smith | Effect of 'Ground Truth' on Image Binarization[END_REF], one of which is that it assumes Ground Truth is void of errors.

The work presented in this paper revisits the concepts presented in [START_REF] Smith | Effect of 'Ground Truth' on Image Binarization[END_REF] by investigating the idea of comparing and evaluating classifiers if no Ground Truth is available and focus on binary classifiers (i.e. classifiers that arrange elements in only two categories). We will be reviewing some of the commonly used metrics for measuring discrepancy: F-Measure, Peak Signal-to-Noise Ratio, Normalized Cross Correlation and Negative Rate Metric, and establish whether they can be approximated by statistical counterparts. We experimentally validate our approach by applying these proposed metrics to a specific type of binary classifier: image binarization. The reader should be aware that findings of this paper do generalize to any kind of binary classifier, however.

Image binarization is a preprocessing step, often necessary for document analysis, medical image analysis, pattern recognition, computer vision and any other, contentfrom-image extraction systems. It converts the image in a bilevel form for further image processing. Depending on the type of images or the type of subsequent needs down the image treatment pipelining it can be a daunting task to decide which binarization method to apply, even using adaptive systems with multiple algorithms of binarization. Unsuitable binarization can be in many cases the reason to fail forthcoming processing steps or reduce their performance.

For the experiments, we will be using state-of-the-art binarization algorithms in the light of the previously described evaluation protocol. The simplest way of estimating which one is more is suitable for a given type of image is to take an image with known Ground Truth (GT) and evaluate the quality by standard comparison metrics. However, in cases where we have no access to suitable images or we have not precise enough Ground Truth [START_REF] Smith | Effect of 'Ground Truth' on Image Binarization[END_REF], [START_REF] Kesiman | An Analysis of Ground Truth Binarized Image Variability of Palm Leaf Manuscripts[END_REF], [START_REF] Smith | An analysis of binarization ground truthing[END_REF] it is necessary to use alternative tools. This paper develops the idea of using statistical tools for evaluation by calculating precision and recall [START_REF] Lamiroy | Computing precision and recall with missing or uncertain ground truth[END_REF], [START_REF] Lamiroy | Statistical Performance Metrics for Use with Imprecise Ground Truth[END_REF] without GT and using the consensus coming from multiple classification systems as a reference [START_REF] Raj | A paired test for recognizer selection with untranscribed data[END_REF].

The rest of this paper is organized as follows: first, we provide an overview of the experimental framework, and the collection of binarization algorithms that were used, followed by the analysis of standard existing performance metrics for which we suggest statistical counterparts in Section IV. Section V provides experimental validation and a conclusion is given in Section VI.

II. TESTED BINARIZATION ALGORITHMS

It is common to categorize binarization algorithms in global and local methods. The general approach for every binarization system is similar: if a pixel (m, n) in the input image has a higher gray level value than a given threshold, then this pixel labeled as background, otherwise, it is labeled as foreground. Individual binarization approaches differ in how the threshold is computed: global algorithms calculate one threshold for the entire image, while local algorithms calculate different threshold values for each pixel of the image, depending on their surrounding region. Global binarization is faster and simpler to implement than local algorithms.

Multiple global binarization algorithms exist in the literature, based on various classification procedures: histogram operations; clustering; entropy analysis and Gaussian distributions. But according to the results of their performance in DIBCO images dataset we decided to retain only one: global Otsu`s binarization method.

Locally adaptive binarization methods are able to compute a threshold for each separate pixel using the information contained in neighborhood pixels and therefore usually show better performance than global ones.

This paper is based one global and nine locally adaptive algorithms:

1) Otsu's global binarization method [START_REF] Otsu | A threshold selection method from gray-level histograms[END_REF];

2) Local Otsu's method;

3) Brensen's method [START_REF] Bernsen | Dynamic Thresholding of Grey-level Images[END_REF]; 4) Niblack's method [START_REF] Niblack | An Introduction to Digital Image Processing[END_REF]; 5) Breadly's method [START_REF] Bradley | Adaptive thresholding using the integral image[END_REF]; 6) Method of local medium value; 7) Modified Gato's method; 8) Wolf's method [START_REF] Wolf | Extraction and Recognition of Artificial Text in Multimedia Documents[END_REF]; 9) Kittler's method [START_REF] Kittler | Minimum Error Thresholding[END_REF]; 10) Sauvola's method [START_REF] Sauvola | Adaptive Document Image Binarization[END_REF];

The selected algorithms have significant reported performance differences. All of them can show high enough quality for some types of images according to the DIBCO evaluation campaigns. Local Otsu's method and Gato's method were slightly modified with respect to their published versions, as will be explained below. Local Otsu's method is based on a global threshold binarization method described by N. Otsu. The idea of selecting an automatic threshold level according to this method was based on the analysis of the image graylevel histogram. Traditionally, the threshold is selected by maximizing the measure of separability between the classes in graylevel. Here, a modified version of the method was used: a local threshold level was chosen for every pixel by applying Otsu's method over a local square window. The optimal size found to be good a choice was 100 by 100 pixels.

Gato's method [START_REF] Gatos | Adaptive degraded document image binarization[END_REF] includes several distinct steps. It is a pre-processing procedure using a low-pass Wiener filter; estimation of foreground regions; background surface calculation by interpolating neighboring background intensities and thresholding by combining the calculated background surface with the original image while incorporating image up-sampling.

Gato's method was modified by adding histogram equalization and median filtering after using a low-pass Wiener filter. Optimal size for low-pass and Wiener filter was 5 by 5 and 17 by 17 pixels.

All the chosen algorithms depend on input parameters and their performance is sometimes sensitive to subtle changes. We applied consistent and near-optimal parameters for all experiments, either by applying the recommended published parameters, either experimentally determined parameters. It should be clear to the reader that the scope of this paper concerns Ground Truth-less performance metrics, and not binarization. Therefore, whether the choice of parameters is actually optimal or not is of no incidence to the conclusions drawn from our study on the various metrics developed in Section IV.

III. INPUT DATA

In the experiments reported in Section V, we use all 56 images from the Digital Image Binarization Contest (DIBCO) editions between 2009 and 2013. DIBCO is organized in the context of the International Conference of Document Analysis and Recognition (ICDAR). The general objective of the contest is to identify advances in document image binarization by applying evaluation performance measures. All editions focus on the evaluation of document image binarization methods using a variety of scanned machineprinted and handwritten documents for which the organizers created the binary Ground Truth images using a semiautomatic procedure [START_REF] Gatos | Proc. International Conference on Document Analysis and Recognition[END_REF]. Binarization methods competing in DIBCO are compared to each other in function of 4 metrics. These metrics are the F-Measure, PSNR, NCC and NRM, and consist of different means of measuring the discrepancy of various binarization outputs (resulting from the competing methods) with the established Ground Truth.

One of the motivations of this paper is that this approach has been challenged with as main argument that the Ground Truth cannot be considered unique and therefore that subsequent performance evaluation is biased [START_REF] Smith | Effect of 'Ground Truth' on Image Binarization[END_REF], [START_REF] Smith | An analysis of binarization ground truthing[END_REF] .

IV. EVALUATION METRICS

Given the legitimate objections to Ground Truth-based evaluation expressed in [START_REF] Smith | Effect of 'Ground Truth' on Image Binarization[END_REF], [START_REF] Kesiman | An Analysis of Ground Truth Binarized Image Variability of Palm Leaf Manuscripts[END_REF], [START_REF] Smith | An analysis of binarization ground truthing[END_REF] we explore the idea of using performance metrics that can be used in absence of Ground Truth as has already been experimented in [START_REF] Lamiroy | Computing precision and recall with missing or uncertain ground truth[END_REF]. The main idea behind the approach is to replace the standard Ground Truth with a consensus metric resulting from the collection of compared methods. This section reviews the conventional performance metrics and reformulates them in the context of this idea of consensus metric. In Section V will then measure and experimentally establish their validity.

A. F-Measure

B. PSNR

In image analysis, Peak Signal-to-Noise Ratio (PSNR) is the maximum value between the power of a signal and corrupting noise. In our case, this ratio measures how close an estimator image is to the estimated image. It is expressed in terms of the logarithmic decibel scale:

𝑃𝑆𝑁𝑅 = 10𝑙𝑜𝑔( 𝐶 2 𝑀𝑆𝐸 ) (4)
Where C 2 is a maximum possible value of separate pixels (difference between foreground and background), and the mean squared error (MSE) described by following equation:

𝑀𝑆𝐸 = ∑ ∑ (𝐴(𝑥,𝑦) -𝐵(𝑥,𝑦)) 2 ) 𝑁 𝑦=1 𝑀 𝑥=1 𝑀𝑁 (5)
The higher are value of PSNR, the higher is the similarity of the two images.

C. NCC

Normalized Cross Correlation is often used for comparing multidimensional arrays and is defined by the following equation: Where 𝐴 ̅ is the mean value of one array and 𝐵 ̅ is the mean of another, and M and N are the dimensions of the arrays.

A higher NCC indicates better matching of arrays.

D. NRM

The Negative Rate Metric is a numerical equivalent of the relation between misclassified elements and all other elements in the class. It is the average value of two negative rates: false negative rate NRFN and false positive rate NRFP:

𝑁𝑅𝑀 = 𝑁𝑅 𝐹𝑁 + 𝑁𝑅 𝐹𝑃 2 (7) 𝑁𝑅 𝐹𝑁 = 𝐹𝑁 𝑇𝑃+ 𝐹𝑁 (8) 𝑁𝑅 𝐹𝑃 = 𝐹𝑃 𝑇𝑁+𝐹𝑃 (9)
A higher NRM indicates a worse mismatch between two classifiers. Section V provides the experimental validation of their relevance.

E. Pseudo F-Measure

We are using the concept of Pseudo F-Measure introduced in [START_REF] Smith | Effect of 'Ground Truth' on Image Binarization[END_REF]. The idea developed in this paper is that the statistical equivalent of ground truth is an array of probabilities that the documents (separate pixels in image binarization) δi belongs to the foreground cluster Δ + . Under this hypothesis these probabilities are:

𝑃(δ i ) = 1 𝑠 ∑ 𝑆 𝑘 (δ i ) 𝑘=1..𝑠 (10)
Where Sk(δi) represents the classification result of document δi by classifier Sn, and s the number of classifiers.

Given the hypothesis of equivalent distribution of all documents δi in their set ∆*, the authors state that Precision "is the probability that a random document retrieved by a query is relevant", and define Pr(Sk) as:

𝑃𝑟(𝑆 𝑘 ) = ∑ 𝑃(δ i ) 𝑆 𝑘 (δ i ) 𝑖=1..𝑑 ∑ 𝑆 𝑘 (δ i ) 𝑖=1..𝑑 (11)
Where d is the total number of elements (in our case, pixels in the image) classified by classifier Sk as belonging to one of classes (foreground or background).

Similarly, Recall was defined as "the probability for a random relevant document to be retrieved by the query" and described by the formula:

𝑅𝑐(𝑆 𝑘 ) = ∑ 𝑃(δ i ) 𝑆 𝑘 (δ i ) 𝑖=1..𝑑 ∑ 𝑃(δ i ) 𝑖=1..𝑑 (12)
In these cases, relevancy has been replaced by P(δi).

These two described statistically-grounded metrics can be combined into Pseud F-Measure in a similar way as for the ordinary F-measure by computing the harmonic mean between Precision and Recall [START_REF] Smith | Effect of 'Ground Truth' on Image Binarization[END_REF].

Besides this extension of [START_REF] Lamiroy | Computing precision and recall with missing or uncertain ground truth[END_REF], we have added three more statistical metrics for classifier evaluation: Pseudo Negative Rate Metric (Pseudo NRM), Pseudo Normalized Cross Correlation (Pseudo NCC) and Pseudo Peak Signal-Noise Rate (Pseudo PSNR).

F. Pseudo NRM

The statistically grounded alternative to Negative Rate Metric should be defined in function of the statistical equivalents of False Negative, False Positive, True Positive and True Negative values. In this case, relevancy also was replaced by P(δi) which had been described above [START_REF] Niblack | An Introduction to Digital Image Processing[END_REF].

According to our assumptions, and in accordance with of [START_REF] Smith | An analysis of binarization ground truthing[END_REF], the value of the Pseudo True Positive for a given classifier Sk can be expressed as the dot product of the array of P(δi) and the array given by the classification results Sk(δi). In the same way, the value of the Pseudo False Negative determined elements is the result of the dot product of P(δi) with the array of inverted binary result given by the classifiers S k .

These two values allow for computing False Negative rates:

𝑃𝑠𝑒𝑢𝑑𝑜 𝑁𝑅 𝐹𝑁 = ∑ 𝑃(δ i ) 𝑆 𝑘 (δ i ) ̅̅̅̅̅̅̅̅ 𝑖=1..𝑑 ∑ 𝑃(δ i ) 𝑆 𝑘 (δ i ) ̅̅̅̅̅̅̅̅ 𝑖=1..𝑑 + ∑ 𝑃(δ i ) 𝑆 𝑘 (δ i ) 𝑖=1..𝑑 = ∑ 𝑃(δ i ) 𝑆 𝑘 (δ i ) ̅̅̅̅̅̅̅̅ 𝑖=1..𝑑 ∑ 𝑃(δ i ) 𝑖=1..𝑑 = 1 - ∑ 𝑃(δ i ) 𝑆 𝑘 (δ i ) 𝑖=1..𝑑 ∑ 𝑃(δ i ) 𝑖=1..𝑑 (13)
One can observe that NRFN actually is (1-Recall) and that this translates into Pseudo NRNF being (1-𝑅𝑐).

We can express the Pseudo Negative Rate of False Positive elements similarly:

𝑃𝑠𝑒𝑢𝑑𝑜 𝑁𝑅 𝐹𝑃 = ∑ (1 -𝑃(δ i )) 𝑆 𝑘 (δ i ) 𝑖=1..𝑑 ∑ (1-𝑃(δ i ))𝑆 𝑘 (δ i ) 𝑖=1..𝑑 + ∑ (1-𝑃(δ i )) 𝑆 𝑘 (δ i ) ̅̅̅̅̅̅̅̅̅ 𝑖=1..𝑑 (14)
We can simplify this equation to

𝑃𝑠𝑒𝑢𝑑𝑜 𝑁𝑅 𝐹𝑃 = ∑ (1 -𝑃(δ i )) 𝑆 𝑘 (δ i ) 𝑖=1..𝑑 𝑑-∑ 𝑃(δ i ) 𝑖=1..𝑑 (15)
The final equation for Pseudo NRM is the same as for ordinary NRM [START_REF] Gatos | Proc. International Conference on Document Analysis and Recognition[END_REF] and is described as the average of the negative rate of false positive and false negative values:

𝑃𝑠𝑒𝑢𝑑𝑜 𝑁𝑅𝑀 = 𝑃𝑠_𝑁𝑅 𝐹𝑁 + 𝑃𝑠_𝑁𝑅 𝐹𝑃 2 (16)
In contrast to F-Measure and PSNR, the lower the value for this metric, the better the classifier.

G. Pseudo NCC

Pseudo Normalized Cross Correlation expresses the level of normalized correlation between the probability that the elements δi belongs to the foreground cluster Δ + given the majority voting P(δ) and the result given by classifier Sk.. Its expression is:

𝑃𝑠𝑒𝑢𝑑𝑜 𝑁𝐶𝐶 = ∑ ∑ (𝑆 𝑘 (𝑚,𝑛) -𝑆 𝑘 ̅̅̅̅ )(𝑃 δ (𝑚,𝑛) -𝑃 δ ̅̅̅̅ ) 𝑁 𝑛=1 𝑀 𝑚=1 √∑ ∑ (𝑆 𝑘 (𝑚,𝑛) -𝑆 𝑘 ̅̅̅̅ ) 2 𝑁 𝑛=1 𝑀 𝑚=1 ∑ ∑ (𝑃 δ (𝑚,𝑛) -𝑃 δ ̅̅̅ ) 2 𝑁 𝑛=1 𝑀 𝑚=1 (16)
The higher this value, the better both arrays correlate with each other.

H. Pseudo PSNR

PSNR is a measure expressing how close image is to another, Pseudo PSNR measures how close the result of a classifier is with respect to an array of probabilities, based on majority voting P(δ) described above [START_REF] Niblack | An Introduction to Digital Image Processing[END_REF]. It is defined by the following equation:

𝑃𝑠𝑒𝑢𝑑𝑜 𝑃𝑆𝑁𝑅 = 10𝑙𝑜𝑔( 𝐶 2 𝑀𝑆𝐸 ) (17)
Where MSE is the mean squared error given by the average of the squares between evaluated array Sk and P(δ) :

𝑀𝑆𝐸 = ∑ ∑ (𝑆 𝑘 (𝑚,𝑛) -𝑃 δ (𝑚,𝑛)) 2 ) 𝑁 𝑦=1 𝑀 𝑥=1 𝑀𝑁 (18)
The higher the value of Pseudo PSNR, the higher the similarity of the two arrays.

The assumption is that the performance of all of these described pseudo-metrics will give acceptable results for evaluation of classification systems without using ground truth or any annotated data at all. The next section describes the experimental protocol and conducted experiments that establish this.

V. EXPERIMENTS

In order to compare above described evaluation metrics we use the correlation coefficient r between each metric and its statistically-grounded pseudo-metric for each image in every dataset. We computed their correlation coefficient using the equation of normalized cross correlation, described in [START_REF] Raj | A paired test for recognizer selection with untranscribed data[END_REF]. For every picture in every dataset, we applied all the binarization systems, mentioned in paragraph II and applied each of the eight metrics described in paragraph IV to all obtained results. After that, was computed the normalized cross-correlation on each couple of metric and its corresponding pseudo-metric. The resulting correlation values for each dataset are shown in Fig. 1. The results obtained from all the datasets are also shown in Table 1. Rows represent datasets of images from DIBCO competition editions of 2009, 2011, 2012 and 2013. All the data consist of four hand-written (HW) and three printed (Pr.) sets of images. The highest average correlation was obtained for Pseudo PSNR, and Pseudo F-Measure. Pseudo F-Measure has a more stable correlation, as shown by its standard deviation. In the all datasets Pseudo NRM has the lowest average correlation.

According to the obtained results, correlation between NRM and Pseudo NRM are too low for further use and these metrics should be discarded. But all the other proposed statistically-grounded metrics can be tested and with some changes and different input parameters for algorithms.

Besides the correlation test we also evaluated how the number of classifiers influences overall correlation. Since every additional classifier has an influence on the statistically grounded metrics, we investigated what exactly happened with the correlation between the metrics when the number of classifiers is increased progressively. Fig. 2 shows how correlation between the F-Measure and Pseudo F-Measure evolves with the number of algorithms. For every quantity of classifiers, all possible combinations of mentioned binarization systems were tested. As an illustration, the data shown in Fig. 2 represents the obtained correlation evaluation for each image in the DIBCO 2011 handwritten dataset. According to the obtained results, the more classification algorithms, the better the correlation coefficient.

VI. CONCLUSIONS AND FUTURE WORK

Aiming to overcome some deficiencies of classifier performance evaluation using annotated data, this paper presents statistically grounded evaluation metrics not requiring Ground Truth. We have presented basic metrics for the evaluation of binary classifiers that can be computed using only statistical tools. These metrics were applied to image analysis and have been tested on DIBCO 2009 -2013 image datasets. Their performance was compared with those of the more traditional performance metrics reported in DIBCO competitions.

The comparison between Ground Truth based and statistically grounded metrics shows high correlation for Pseudo F-Measure, Pseudo PSNR and Pseudo NCC. This means that the approach and metrics described in this paper can be used to find the best classification methods for more than half of analyzed images. In another words, we obtain better-than-random results in the selection of classifiers. This research still needs to be completed, and there are number of questions to resolve. One of them is how to improve the statistically grounded classification metrics such that the correlation with Ground Truth-based classification metrics can be increased and whether it is possible to select the best classifier using statistical tools for a given dataset. Or, if not, how to find criteria to identify those configurations where the approach fails. Moreover, future investigation should extend this work to non-binary classifiers and be applied to approaches in practical cases with post-classification procedures like content extraction.

F

  -measure combines precision and recall by calculating their harmonic mean: of Precision and Recall is based on the relation between true and false determined elements. Precision is the value of retrieved elements that are relevant and Recall is the value of relevant elements that are retrieved. , FN denote the True Positive, False Positive and False Negative values, respectively.

Fig. 1 .

 1 Fig. 1. Correlation between GT-based and statistical metrics for the all DIBCO datasets

Fig. 2 .

 2 Fig. 2.Changes of average correlation with the amount of binarization algorithms (each curve represents a specific test image from DIBCO)

TABLE I .

 I AVERAGE CORRELATION COEFFICIENT OF EVERY METRIC AND

	09'HW	0.76	0.71	0.22	0.16
	Average	0.845	0.856	0.783	0.373
	St. deviation	0.051	0.060	0.234	0.163
						PSEUDO-METRIC
						Average correlation coefficient
						FM &	PSNR &	CrossCorr &	NRM &
					DIBCO	Pseudo	Pseudo	Pseudo	Pseudo
						FM	PSNR	CrossCorr	NRM
					09'Pr.	0.93	0.88	0.93	0.56
					11'HW.	0.85	0.86	0.93	0.56
					11'Pr.	0.85	0.83	0.72	0.21
					12'HW	0.85	0.92	0.90	0.44
					13'HW	0.79	0.87	0.70	0.15
					13'Pr.	0.89	0.83	0.84	0.34