Pierre Monnin
email: pierre.monnin@loria.fr

Amedeo Napoli

Adrien Coulet

Combining Concept Annotation and Pattern Structures for Guiding Ontology Mapping

Keywords: Concept Annotation, Pattern Structures, Formal Concept Analysis

Formal Concept Analysis (FCA) is a mathematical framework classifying in formal concepts a set of objects w.r.t. their common attributes. To this aim, FCA relies on a binary incidence relationship indicating whether an object has an attribute. On one hand, in order to consider more complex descriptions for objects (e.g., intervals, graphs), FCA has been extended with Pattern Structures. On the other hand, in a previous work, we introduced the notion of Concept Annotation, adding a third dimension to formal concepts, computed over the extent, without modifying the original classification. In this paper, we combine Concept Annotation with the formalism of Pattern Structures and we consider multiple annotation possibilities, i.e., multiple annotations for one concept and computing the annotation over the intent. We illustrate our approach and its interest with two use cases: (i) suggesting mappings between ontology classes and (ii) finding specific classes frequently associated as domain and range of a predicate.

Introduction

An ontology is a formal representation of a particular domain [START_REF] Gruber | A translation approach to portable ontology specifications[END_REF], consisting of two parts: an assertion component (ABox) and a terminological component (TBox). In the TBox, classes and predicates between classes are defined while in the ABox, individuals instantiate classes and predicates. For example, the drug codein can be considered as an individual instantiating the class Analgesics.

Nevertheless, there is a need for structure between ontologies. Indeed, individuals of an ABox may instantiate classes from several ontologies. For example, in the medical domain, individuals representing diseases may instantiate classes of several ontologies of diseases, such as MeSH (Medical Subject Headings), ICD-9-CM and ICD-10-CM (International Classification of Diseases version 9 and 10, Clinical Modification). Corresponding classes from distinct ontologies may be mapped thanks to equivalence relationships resulting from an alignment process [START_REF] Euzenat | Ontology Matching, Second Edition[END_REF]. This alignment may be either a manual review by an expert or a semiautomatic process.

On the other hand, within an ontology, additional knowledge and structuring may be discovered. Thus, the ABox of an ontology may also contains predicate assertions, i.e., relations between individuals. For example, in DrugBank, a database of drug information, a drug can be associated with a gene, indicating an interaction between them. Thereby, the gene VKORC1 is indicated to be an inhibitor of the drug warfarin. Genes and drugs instantiate classes of ontologies, such as ATC (Anatomical Therapeutic Chemical Classification System) for drugs and GO (Gene Ontology) for genes. Therefore, from the predicate assertions, we could discover classes of drugs and genes frequently associated as domain and range of a predicate. Such domain and range associations could be interesting as they may indicate common properties of the gene class or the drug class.

In this article, we aim at addressing both of the aforementioned use cases, (i) suggesting equivalence relationships between ontology classes and (ii) discovering classes frequently associated in domain and range of a predicate, with Formal Concept Analysis (FCA) [START_REF] Ganter | Formal concept analysis -mathematical foundations[END_REF]. FCA is a well-founded mathematical framework adapted to knowledge engineering purposes [START_REF] Baader | Completing Description Logic Knowledge Bases Using Formal Concept Analysis[END_REF][START_REF] Bendaoud | Formal Concept Analysis: A Unified Framework for Building and Refining Ontologies[END_REF][START_REF] Coulet | Using Pattern Structures for Analyzing Ontology-Based Annotations of Biomedical Data[END_REF] that groups objects w.r.t. their common attributes in formal concepts. The association of an attribute to an object is expressed thanks to a binary relationship. Formal concepts are organized by a partial order in a hierarchical structure called a lattice. In a previous work [START_REF] Monnin | Using Formal Concept Analysis for Checking the Structure of an Ontology in LOD: The Example of DBpedia[END_REF], we introduced an extension of FCA, called Concept Annotation, allowing to compare the hierarchical structure of the lattice with the class hierarchy of an ontology for possible refactoring. The main interest of Concept Annotation resides in adding a third dimension to a previously generated lattice without changing its structure. The main contribution of the present article is to extend the initial definition of Concept Annotation by (i) associating multiple Concept Annotations with formal concepts, and (ii) using a formalism similar to Pattern Structures [START_REF] Ganter | Pattern Structures and Their Projections[END_REF] to express the annotations. Pattern Structures are another extension of FCA dealing with objects having complex (non binary) descriptions (e.g., graphs, numerical values, classes of an ontology). In the following, we assume that the reader is familiar with the basics of FCA and Pattern Structures.

This article is organized as follows. In Section 2, necessary basics about ontologies are recalled. In Section 3, we present the initial definition of Concept Annotation as well as our proposed extension. Then, in Sections 4 and 5, we illustrate the combination of Concept Annotation and Pattern Structures by, respectively, suggesting equivalence relationships between classes and discovering classes frequently associated as domain and range of a predicate. Finally, in Sections 6 and 7, we discuss the results we obtain on the two considered use cases as well as the next challenges to tackle.

Basics about Ontologies

An ontology is a formal representation of a particular domain [START_REF] Gruber | A translation approach to portable ontology specifications[END_REF]. It is composed of two parts, the TBox and the ABox. The TBox defines classes and relationships between them. We denote C(O) the set of classes of the ontology O. Classes of an ontology are instantiated by individuals of the ABox. These individuals can also be involved in instantiation of relationships (such as the drug-gene relationship in DrugBank previously explained).

Classes of an ontology are organized in a partial order by a subsumption relation denoted by ⊑. Considering two classes C and D, C ⊑ D states that every instance of C is also an instance of D. The least common subsumer (sometimes named the lowest common ancestor) of two classes C 1 and C 2 of an ontology is the most specific class subsuming both C 1 and C 2 . It is denoted by lcs(C 1 , C 2 ).

In EL ontologies where no cycle appears, the lcs of two classes always exists [START_REF] Baader | Computing least common subsumers in description logics with existential restrictions[END_REF]. Considering a set of classes

C n = {C 1 , C 2 , . . . , C n }, we define the set of the most specific classes of C n as min C n = {C ∈ C n | ̸ ∃ D ∈ C n , D ⊑ C}.

Concept Annotation

Concept Annotation is an extension of FCA that we introduced in a previous work [START_REF] Monnin | Using Formal Concept Analysis for Checking the Structure of an Ontology in LOD: The Example of DBpedia[END_REF]. In this section, we first recall basics about our initial work and then present the proposed extension.

Basics about Concept Annotation

In Concept Annotation, standard FCA is applied on a formal context to build a concept lattice. Then, for each formal concept, an annotation is computed, adding a third dimension to each concept.

For example, in [START_REF] Monnin | Using Formal Concept Analysis for Checking the Structure of an Ontology in LOD: The Example of DBpedia[END_REF], a lattice is built from a formal context (G, M, I) where G is a set of individuals from the ABox of an ontology O and M is a set of predicates of this ontology. (g, m) ∈ I indicates that the individual g is involved in a relationship whose predicate is m. Then, classes of the TBox instantiated by the individuals in G are added as annotation. The resulting hierarchy formed by the lattice and the annotations is compared to the class hierarchy of the ontology. To compute the annotation, we defined a derivation operator ( ) , where 2 C(O) corresponds to the powerset of the set of classes of the ontology. Considering a formal concept (A, B) where A is a set of individuals, the annotation is defined thanks to the derivation operator as follows:

•) ⋄ : 2 G → 2 C(O
A ⋄ = ∩ g∈A {C | O |= C(g)} 3
. Intuitively, it corresponds to the set of all classes of O instantiated by all individuals in A.

Combining Concept Annotation and Pattern Structures

In this paper, we propose to combine Concept Annotation with Pattern Structures. Indeed, such combination will allow us to only keep as annotation the most specific classes instantiated by all individuals.

To do so, we define a function δ : 2 G → 2 C(O) that associates an individual from G with the set of the most specific classes of the ontology O that this individual instantiates. Formally, given g ∈ G, δ(g

) = min {C ∈ C(O) | O |= C(g)}.
δ(g) is considered as the description of the individual g. Given two individuals, g 1 , g 2 ∈ G, we define a similarity operator ⊓ to compare their two descriptions as follows:

δ(g 1 ) ⊓ δ(g 2 ) = min {lcs(C 1 , C 2 ) | ∀ (C 1 , C 2 ) ∈ δ(g 1 ) × δ(g 2 )}
The use of the least common subsumer allows to compute the most specific classes of the ontology that both g 1 and g 2 instantiates. Finally, given a formal concept (A, B), we annotate it thanks to the new derivation operator (•)

• : 2 G → 2 C(O) defined as A • = g∈A δ(g).
In the following sections, the defined functions and operators will be restricted to specific ontologies. Thus, δ i , ⊓ i and (•) •i will only be applicable to classes of an ontology O i .

Suggesting Mappings between Classes of Ontologies

In this section, we illustrate the combination of Concept Annotation and Pattern Structures with the use case of suggesting equivalence relationships between classes of two ontologies, denoted by O 1 and O 2 . To generate these mappings, we consider individuals that instantiate classes of both O 1 and O 2 . They also instantiate classes of another ontology, denoted by O ref , that is considered in this setting as the reference ontology, i.e. the feature we consider to build the original classification of the set of instances. In order to avoid complexity problems, we keep O ref of a small size.

Annotating the Concept Lattice with O 1 and O 2

In this next step, classes from O 1 and O 2 are considered for annotating the concept lattice. To this aim, we define two annotations (one per ontology) using the formalism defined in Subsection 3.2. Thereby, we use two functions δ 1 and δ 2 to associate an individual g with the set of the most specific classes from O 1 and O 2 that this individual instantiates. ⊓ 1 and ⊓ 2 are used to compute the similarity between descriptions of two individuals w.r.t. the two considered ontologies O 1 and O 2 . Finally, to compute the two annotations for each pattern concept (A, D), O2) are defined as follows:

two derivation operators (•) •1 : 2 G → 2 C(O1) and (•) •2 : 2 G → 2 C(
A •1 = 1 g∈A δ 1 (g) A •2 = 2 g∈A δ 2 (g)
As a result, for each pattern concept (A, D), an annotated pattern concept

(A, D, A •1 , A •2 ) is obtained where A •1 (resp. A •2
) is the set of the most specific classes of O 1 (resp. O 2 ) that all individuals in A instantiate.

Reading Mappings from the Lattice

Let us consider an annotated pattern concept (A, D, A •1 , A •2 ). From the previous definitions, we know that A •1 ⊆ C(O 1 ) contains the set of the most specific classes of O 1 that all individuals in A instantiate. Similarly, A •2 ⊆ C(O 2 ) contains the set of the most specific classes of O 2 that all individuals in A instantiate. Therefore, considering each pair of classes

(C 1 , C 2 ) ∈ A •1 × A •2
, we know that they are instantiated by the same set of individuals, i.e., individuals in A. Therefore, an equivalence relationship is suggested between C 1 and C 2 , based on the instances of the two classes.

Discovering Associated Classes as Domain and Range of a Predicate

In this section, we illustrate how Concept Annotation and Pattern Structures could be used to discover the most specific classes frequently associated as domain and range of a predicate from its instantiations in an ABox. Such domain and range characterization could indeed indicate common behavior at the class level. For example, considering DrugBank, a gene can be involved with a drug in a relationship whose action is specified (e.g., inhibitor, antagonist). Such relationships correspond to assertions of a predicate. As drugs instantiate classes of ATC and genes instantiate classes of GO, we could discover that instances of a GO class, considered as a family of genes, are frequently indicated as inhibitors of instances of an ATC class, considered as a family of drugs.

In the following paragraphs, we consider individuals, instantiating classes of an ontology O 1 , that are involved in relationships with other individuals, instantiating classes of an ontology O 24 .

Classifying Relationships in a Concept Lattice

First, a classification of relationships is established. O2) are defined as follows:

two derivation operators (•) •1 : 2 G → 2 C(O1) and (•) •2 : 2 M → 2 C(
A •1 = 1 g∈A δ 1 (g) B •2 = 2 m∈B δ 2 (m)

It is noteworthy that (•)

•2 is applied on the intent of the formal concept. As a result, each formal concept (A, B) is replaced by the annotated concept

(A, B, A •1 , B •2 ) where A •1 (resp. B •2
) is the set of the most specific classes of O 1 (resp. O 2 ) that all individuals in A (resp. in B) instantiate.

Reading the Domain and Range of a Relation from the Annotated Lattice

Let us consider an annotated concept

(A, B, A •1 , B •2 ). Every individual in A is in a relationship with every individual in B. Furthermore, A •1 is the set of the most specific classes of O 1 that all individuals in A instantiate. Similarly, B •2
is the set of the most specific classes of O 2 that all individuals in B instantiate. Therefore, from this annotated concept, for the considered predicate, classes from A •1 as domain are associated with classes from B •2 as range.

Along the hierarchy of the lattice, the two annotations A •1 and B •2 behave in the opposite way. Indeed, A •1 , computed on the extent, contains more specific classes when the number of individuals in the extent decreases, i.e., when browsing the lattice top to bottom. On the contrary, B •2 , computed on the intent, contains more specific classes when the number of individuals in the intent decreases, i.e., when browsing the lattice bottom to top. Consequently, in annotated concepts at the top of the lattice, general classes of O 1 will be involved in domains and specific classes of O 2 will be involved in ranges. In annotated concepts at the bottom of the lattice, specific classes of O 1 will act as domain and general classes of O 2 will act as ranges. Therefore, this structure could be of interest in an interactive setting with an expert. Indeed by browsing the lattice depending on her specific constraints on the domains or ranges to discover, the expert could leverage on the lattice order to obtain more general or specific classes involved.

Discussion

Regarding the suggestion of equivalence relationships between classes of two ontologies, the approach needs to be validated on a real dataset where mappings already exist. One main identified drawback is that the current annotation process works under the Closed World Assumption. Thus, mappings are suggested considering that all instantiations are correct and none is missing. As many datasets are under the Open World Assumption, the suggested mappings may only be applicable on the considered set of individuals but may not be applied to other sets of individuals. Therefore, there is a need to validate the suggested mappings with an expert. One next challenge would be to define a new derivation operator to compute an annotation working under the Open World Assumption.

The original lattice is built from the set of individuals and the classes of a reference ontology O ref . However, other features could be considered for this original classification, such as the relationships involving the individuals, similarly to our previous work [START_REF] Monnin | Using Formal Concept Analysis for Checking the Structure of an Ontology in LOD: The Example of DBpedia[END_REF]. The choice of the features to consider are of importance as they impact the original lattice, which is the the "pivot" structure from which equivalence relationships are suggested. One user can then choose the specific features to consider to guide the generated mappings. In the proposed approach, equivalence relationships are suggested by considering annotated concepts separately. Nevertheless, the subsumption relations between concepts could be considered to suggest mappings of the form of subsumption relations between classes of the two considered ontologies. Such setting could therefore be used to align and structure folksonomies of various users in a social network.

Finally, the suggestion of mappings may not be the only use case of interest for the proposed setting. Indeed, by annotating the original lattice with classes of different versions of the same ontology, concept drift could be highlighted. For example, a semantic change in the classes between two versions of an ontology would be indicated by discovering in the annotations that the same set of individuals does not instantiate the same classes between the two versions.

Regarding the use case of discovering classes frequently associated as domain and range of a predicate, we could also benefit from an experiment on a real dataset. In this setting, as previously mentioned, there is an issue in selecting the interesting domains and ranges. Additionally to the interactive discovery process previously mentioned, various metrics could be considered to highlight interesting annotated concepts. For example, notions of support or confidence based on the cardinal number of the extent and / or intent could be of interest here. Similarly to the suggestion of equivalence relationships, this work does not benefit from the subsumption relations between formal concepts. Such relations could be used to build a hierarchical organization of associations of domains and ranges. However, as the two annotations behave in opposite ways, this hierarchy should not be read as an order.

Conclusion

In this paper, we combine Concept Annotation with the formalism of Pattern Structures. To illustrate our approach and its interest, we consider two use cases: suggesting equivalence relationships between classes of two ontologies and discovering classes frequently associated as domain and range of a predicate. The formalism of Pattern Structures is an advantage compared to our original work on Concept Annotation as it enables more complex annotations, using descriptions of objects and similarity operations. Moreover, we illustrate some other annotation possibilities: multiple annotations for one concept and annotations computed on the intent instead of the extent. As a result, starting from a concept lattice considered as a "pivot" structure, it is possible to obtain a complex structure representing or highlighting several relations between its components. The next challenges lies in experimenting our approach on real datasets and formalizing the properties of the annotation.

4. 1

 1 Classifying Individuals w.r.t. O ref in a Concept Lattice The first step is to build a concept lattice classifying the individuals w.r.t. the classes of O ref they instantiate. To this aim, we use the pattern structure (G, (2 C(O ref ) , ⊓ ref ), δ ref ). G is the set of individuals and 2 C(O ref ) is the powerset of the set of classes of O ref . δ ref and ⊓ ref are defined as in Subsection 3.2 w.r.t. the classes of O ref . From this pattern structure definition, we obtain pattern concepts (A, D) organized in a concept lattice, where A ⊆ G is a set of individuals and D ∈ 2 C(O ref ) is a set of the most specific classes of O ref that all individuals in A instantiate.

Annotating the Concept Lattice with O 1 and O 2

  To this aim, we consider the formal context (G, M, I) where G is the set of individuals instantiating classes from O 1 and M is the set of individuals instantiating classes from O 2 . Given g ∈ G, and m ∈ M , (g, m) ∈ I if and only if a relationship between g and m exists. Standard binary FCA is applied on this formal context to generate the associated formal concepts organized in a concept lattice. In the DrugBank example, G is the set of genes and M is the set of drugs. Considering g ∈ G and m ∈ M , we have (g, m) ∈ I if and only if a relationship between the gene g and the drug m exists in DrugBank.Consider a formal concept(A, B) from the concept lattice. A ⊆ G is a set of individuals instantiating classes of O 1 and B ⊆ M is a set of individuals instantiating classes of O 2 .From the definition of the derivation operators in FCA, we know that every individual in A is in relationship with every individual in B. To find the most specific ontology classes involved as domain and range of the predicate, two annotations for formal concepts are defined (one per ontology) using the formalism defined in Subsection 3.2. We use two functions δ 1 and δ 2 to associate an individual with the set of the most specific classes from O 1 and O 2 that this individual instantiates. δ 1 is applied on individuals g ∈ A and δ 2 is applied on individuals m ∈ B. ⊓ 1 and ⊓ 2 are used to compute the similarity between descriptions of two individuals w.r.t. the two considered ontologies O 1 and O 2 . Finally, to compute the two annotations for each formal concept (A, B),

	5.2

C(g) indicates that g is an instance of C in the Description Logics formalism

It is not necessary for O1 and O2 to be distinct.

Acknowledgments

This work is supported by the PractiKPharma project, founded by the French National Research Agency (ANR) under Grant No.: ANR-15-CE23-0028, and by the Snowball Inria Associate Team.


Raphael Dumas

Laurent Frossard

Pierre-Marc Beaulieu

Rickard Branemark

Hip power analysis in individuals with transfemoral amputation: a different strategy different from stabilisation during gait stance

Introduction and Objectives

Joint moments and joint powers during gait are widely used to determine the effects of rehabilitation programs as well as prosthetic fitting. Following the definition of power (dot product of joint moment and joint angular velocity) it has been previously proposed to analyse the 3D angle between both vectors, αMw. [START_REF] Dumas | Hip and knee joints are more stabilized than driven during the stance phase of gait: An analysis of the 3D angle between joint moment and joint angular velocity[END_REF] Basically, joint power is maximised when both vectors are parallel and cancelled when both vectors are orthogonal.

In other words, αMw < 60° reveals a propulsion configuration (more than 50% of the moment contribute to positive power) while αMw > 120° reveals a resistance configuration (more than 50% of the moment contribute to negative power).

A stabilisation configuration (less than 50% of the moment contribute to power) corresponds to 60° < αMw < 120°.

Previous studies demonstrated that hip joints of able-bodied adults (AB) are mainly in a stabilisation configuration (αMw about 90°) during the stance phase of gait. [START_REF] Dumas | Hip and knee joints are more stabilized than driven during the stance phase of gait: An analysis of the 3D angle between joint moment and joint angular velocity[END_REF][START_REF] Samson | 3D joint dynamics analysis of healthy children's gait[END_REF] Individuals with transfemoral amputation (TFA) need to maximise joint power at the hip while controlling the prosthetic knee during stance.

Therefore, we tested the hypothesis that TFAs should adopt a strategy that is different from a continuous stabilisation.

The objective of this study was to compute joint power and αMw for TFA and to compare them with AB.

Methods

Three trials of walking at selfselected speed were analysed for 8 TFAs (7 males and 1 female, 46±10 years old, 1.78±0.08 m 82±13 kg) and 8 ABs (males, 25±3 years old, 1.75±0.04, m 67±6 kg). The joint moments are computed from a motion analysis system (Qualisys, Goteborg, Sweden) and a multi-axial transducer (JR3, Woodland, USA) mounted above the prosthetic knee [START_REF] Vertriest | Static load bearing exercises of individuals with transfemoral amputation fitted with an osseointegrated implant: reliability of kinetic data[END_REF][START_REF] Frossard | Development and preliminary testing of a device for the direct measurement of forces and moments in the prosthetic limb of transfemoral amputees during activities of daily living[END_REF][START_REF] Lee | Magnitude and variability of loading on the osseointegrated implant of transfemoral amputees during walking[END_REF][START_REF] Frossard | Fall of a transfemoral amputee fitted with osseointegrated fixation: loading impact on residuum[END_REF][START_REF] Dumas | Load during prosthetic gait: Is direct measurement better than inverse dynamics?[END_REF][START_REF] Frossard | Load applied on a bone-anchored transfemoral prosthesis: characterisation of prosthetic components -A case study[END_REF][START_REF] Frossard | Categorization of Activities of Daily Living of Lower Limb Amputees During Short-Term Use of a Portable Kinetic Recording System: A Preliminary Study[END_REF][START_REF] Frossard | Load on osseointegrated fixation of a transfemoral amputee during a fall: Determination of the time and duration of descent[END_REF][START_REF] Frossard | Daily activities of a transfemoral amputee fitted with osseointegrated fixation: continuous recording of the loading for an evidence-based practice[END_REF][START_REF] Frossard | Load on osseointegrated fixation of a transfemoral amputee during a fall: loading, descent, impact and recovery analysis[END_REF][START_REF] Frossard | Apparatus for monitoring load bearing rehabilitation exercises of a transfemoral amputee fitted with an osseointegrated fixation: a proof-of[END_REF][14][15][16][17] for TFAs and from a motion analysis system (Motion Analysis, Santa Rosa, USA) and force plates (Bertec, Columbus, USA) for ABs. The TFAs were fitted with an OPRA (Integrum, AB, Gothengurg, Sweden) osseointegrated implant system and their prosthetic designs include pneumatic, hydraulic and microprocessor knees. [START_REF] Frossard | Load applied on a bone-anchored transfemoral prosthesis: characterisation of prosthetic components -A case study[END_REF]14,[18][19][20][21][22][23] Previous studies showed that the inverse dynamics computed from the multi-axial transducer is the proper method considering the absorption at the foot and resistance at the knee. [START_REF] Dumas | Load during prosthetic gait: Is direct measurement better than inverse dynamics?[END_REF]

Results

The peak of positive power at loading response (H1) was earlier and lower for TFA compared to AB. Although the joint power is lower, the 3D angle between joint moment and joint angular velocity, αMw, reveals an obvious propulsion configuration (mean αMw about 20°) for TFA compared to a stabilisation configuration (mean αMw about 70°) for AB.

Figure 1: Dimensionless hip joint power (m is the body mass, g is 9.81 m/s2 and L is length length) and 3D angle between joint moment and joint angular velocity: Red and blue lines and areas are means standard deviations for individuals with transfemoral amputation (TFA) and ablebodied adults (AB), respectively.

The peaks of negative power at midstance (H2) and of positive power at preswing / initial swing (H3) occurred later, lower and longer for TFA compared to AB. Again, the joint powers are lower for TFA but, in this case, αMw is almost comparable (with a time lag), demonstrating a stabilisation (almost a resistance for TFA, mean αMw about 120°) and a propulsion configuration, respectively. The swing phase is not analysed in the present study.

Conclusion

The analysis of hip joint power may indicate that TFAs demonstrated less propulsion and resistance than ABs during the stance phase of gait. This is true from a quantitative point of view. On the contrary, the 3D angle between joint moment and joint angular velocity, αMw, reveals that TFAs have a remarkable propulsion strategy at loading response and almost a resistance strategy at midstance while ABs adopted a stabilisation strategy.

The propulsion configuration, with αMw close to 0°, seems to aim at maximising the positive joint power. The configuration close to resistance, with αMw far from 180°, might aim at unlocking the prosthetic knee before swing while minimising the negative power.

This analysis of both joint power and 3D angle between the joint moment and the joint angular velocity provides complementary insights into the gait strategies of TFA that can be used to support evidence-based rehabilitation and fitting of prosthetic components.

Disclosure of

  Interest  R. Dumas: None Declared,  L. Frossard: None Declared,  C. Robert-Leblanc: None Declared,  P.-M. Beaulieu: None Declared,  R. Branemark Conflict with: stock in Integrum. The authors believe that this not of nature to influence their interpretation of the results.


Rémi Colom

Ross Mcphedran

Brian Stout

Nicolas Bonod

Modal Expansion of the Scattered Field: Causality, Non-Divergence and Non-Resonant Contribution

Modal analysis based on the quasi-normal modes (QNM), also called resonant states, has emerged as a promising way for modeling the resonant interaction of light with open optical cavities. However, the fields associated with QNM in open photonic cavities diverge far away from the scatterer and the possibility of expanding the scattered field with resonant contributions only has not been established. Here, we address these two issues while restricting our study to the case of a dispersionless spherical scatterer. First, we derive the rigorous pole expansion of the T -matrix coefficients that link the scattered to the incident fields associated with an optical resonator. This expansion evinces the existence of a non-resonant term. Second, in the time domain, the causality principle allows us to solve the problem of divergence and to derive a modal expansion of the scattered field that does not diverge far from the scatterer.

INTRODUCTION

Light can resonantly interact with dielectric and metallic particles that behave as 3D open cavities with radiative and possibly absorption losses. The analysis of the resonance is of uppermost importance in understanding and optimizing the process. Among the several theories developed for such an analysis, the Quasi Normal Mode (QNM) theory, also called resonant state expansion, has attracted much attention since it allows for an interpretation of the electromagnetic response of optical resonators with respect to their eigenmodes associated with complex eigenfrequencies ω p,n [START_REF] Leung | [END_REF][2][3][4][5][6][7][8][9][10][11][12]. Even though the use of QNM is quite recent in nanophotonics, it has a long history in quantum scattering theory, electromagnetism and nuclear physics [13][14][START_REF] García-Calderón | Selected Scientific Papers Of Sir Rudolf Peierls[END_REF][START_REF] Zel'dovich | [END_REF][START_REF] Kapur | Proc. R. Soc. Lond. A[END_REF] starting from the Gamow states [START_REF] Gamow | [END_REF] and the singularity expansion method [START_REF] Baum | On the singularity expansion method for the solution of electromagnetic interaction problems[END_REF][START_REF] Vincent | [END_REF].

The imaginary part of the eigenfrequencies ω p,n reflects the energy decay experienced by the modes of open and passive systems. That means that under an exp(-iωt) time dependence, the imaginary part of ω p,n must be negative. This negative imaginary part of the eigenfrequencies in turn causes a divergence of the associated eigenmodes at large distances from the scatterer r → ∞ as they verify outgoing boundary conditions meaning that their radial dependence asymptotically tends towards e iknr /r, k n = ωp,n c [21]. It is interesting to note that this problem was raised in a different context by Lamb as early as 1900 [22][23][START_REF] Nussenzveig | Causality and dispersion relations[END_REF] who introduced the terminology "exponential catastrophe" following the pioneering work of J.J. Thomson [23][START_REF] Nussenzveig | Causality and dispersion relations[END_REF][START_REF] Thomson | [END_REF]. Because of the divergence of these resonant states, their normalization has to be redefined [START_REF] Leung | [END_REF]3,4,7,[START_REF] García-Calderón | Selected Scientific Papers Of Sir Rudolf Peierls[END_REF][START_REF] Zel'dovich | [END_REF][26][27][28][29][30]. QNM analysis of the scattering problem of optical resonators has been carried out by expanding the internal field on the QNM basis in order to study the scattering problem [4]. The scattered field was then computed by means of the Green function [4,5,31]. QNM expansions was recently used to investigate the temporal dynamics of photonic resonators [32,33]. However, two major problems still remain and prevent us from having a clear understanding of the QNM analysis of the scattering problem: 1. A fundamental problem remains regarding the divergence of the QNM fields: how is it possible to express the scattered field as a sum of QNM whereas QNM diverge when r → ∞? We will show that in the time-domain, the causality principle solves the problem of divergence faced by time harmonic QNM fields. The modal expansion in the time domain of the outgoing and scattered fields reveals that a part of the scattered field keeps the same temporal dependence as the incident field (mere reflection) while the second part involves the eigenmodes of the scatterer and modifies the temporal dependence of the incident field.

2. The other problem we want to treat is related to the expansion of the scattered field: can the scattered field of a resonator be expanded as a sum of resonant contributions only or should a non-resonant contribution also be taken into account? This question remains, although it has long been established that the internal field existing inside a spherical resonator could be expanded as a sum of QNMs only [START_REF] Leung | [END_REF]34]. This result is only valid for spherically symmetric scatterers but has been assumed in most of the QNM analysis of arbitrary shaped scatterers [4,35] for which the internal field was recast as a superposition of QNMs. Here, we derive an expression of the scattered field possessing a non-resonant part in addition to resonant parts associated with the QNM fields. The non-resonant contribution is needed to derive convergent expansion of the scattered field. It is different from the incident field and is not required for expanding the internal field.

MODAL EXPANSION OF THE SCATTERING OPERATORS

When considering light-scattering, an excitation field E exc (kr) illuminates a scatterer and gives rise to a scattered field E scat (kr) outside the scatterer and an internal field E int (kr). The total field outside the scatterer is the superposition of the excitation and scattered field: E tot (kr) = E exc (kr) + E scat (kr) and can be also decomposed as a sum of incoming E in (kr) and outgoing E out (kr) fields. The incoming and outgoing fields E in (kr) and E out (kr) are expanded on a set of incoming (N

(-) n,m (kr), M (-)
n,m (kr)) and outgoing (N (+) (kr), M (+) (kr)) Vector Partial Waves (VPWs). Given the definitions of the incoming and outgoing VPWs (see Supplemental Material), E in (kr) and E out (kr) can be expressed as follows:

E out (kr) = E 0 ∞ n,m s (h,+) n,m (ω)M (+)
n,m (kr) + s (e,+) n,m (ω)N (+) n,m (kr)

E in (kr) = E 0 n,m s (h,-) n,m (ω)M (-) n,m (kr) + s (e,-) n,m (ω)N (-) n,m (kr) (1)
where E 0 is the amplitude of the field, s

(i,-) n,m (resp. s (i,+)
n,m ) are the coefficients of expansion of the incoming (resp. outgoing) field on the incoming (resp. outgoing) VPWs, with i = (e, h), e and h denoting electric and magnetic modes, n and m denote the multipolar order (see Supplemental Material). The S-matrix operator provides the outgoing field with respect to the incoming field: E out (ω) = S(ω)E in (ω). For sphericallysymmetric scatterers, it takes the form of a diagonal matrix in a multipolar representation with coefficients defined as

S (i) n = s (i,+) n,m /s (i,-)
n,m . The analytic properties of the S (i) n coefficients are linked with the energy conservation and causality. In particular, for passive media, the norm of all the S-matrix elements are bounded |S The causality principle also has important implications on the analytic properties of the S-matrix coefficients [START_REF] Nussenzveig | Causality and dispersion relations[END_REF]30,36]. Causality notably implies that the coefficients S (e,h) n (ω) have to be regular, i.e. do not admit poles, in the upper part of the complex plane. Demonstrations of this analytic property of the S-matrix coefficients can be found in [START_REF] Nussenzveig | Causality and dispersion relations[END_REF]36]. Poles satisfy the condition S (ω) should also admit zeros in the upper part of the complex plane that are mirror images of the poles of S (e,h) n (ω) with respect to the real axis for a lossless scatterer [START_REF] Nussenzveig | Causality and dispersion relations[END_REF]. These zeros z (e,h) n,α satisfy the condition S (e,h) n (z (e,h) n,α ) = 0 and are associated with modes verifying incoming boundary conditions [37,38]. The reality of the field and the time reversal symmetry impose that the mirror images of ω p,n,α and z n,α with respect of the imaginary axis: -ω . The S-matrix coefficients can be cast as a product between a holomorphic function S (e,h) R,n (ω) and an exponential phase factor e -2i ω c R :

S (e,h) n (ω) = e -2i ω c R S (e,h)
R,n (ω). The prefactor exp(-2ikR) expresses the causality requirement, and arises from the ratio of outgoing and incoming zero-order Hankel functions at the sphere surface. This term will be necessary to guarantee the convergence of the sum form in Eq.( 3) when M → ∞. These considerations lead to the following infinite-product form of the S-matrix coefficients [START_REF] Nussenzveig | Causality and dispersion relations[END_REF]30,36,37,39](see also Supplemental Material):

S (e) n (ω) = (-1) n+1 e -2ikR ∞ α=-∞ ω -z (e) n,α ω -ω (e) p,n,α S (h) n (ω) = (-1) n e -2ikR ∞ α=-∞ ω -z (h) n,α ω -ω (h) p,n,α (2)
where k = ω c . This infinite product converges because of the link between the positions of zeros and poles. The poles and zeros with a positive index α have a positive real part while those with a negative index are their mirror images with respect to the real axis (a pole on imaginary axis can be designated α = 0). If we expand S (e,h) R,n (ω) into partial fractions, it is possible to transform the product in Eq. ( 2) into an infinite sum [37,39]. In practice, one has to truncate this sum to a finite number M of terms. This leads to the following expression of S (e,h) n (ω) (see Supplemental Material):

S (i) n (ω) e -2ikR S (i) nr,n + M α=-M r (i) n,α ω -ω (i) p,n,α (3)
where i = (e, h) and r

(i) n,α is the residue of S (i) R,n at the poles ω (i)
p,n,α and the non-resonant term

S (i) nr,n = 1 + M α=-M r (i) n,α ω (i) p,n,α .
So far, the use of the S-matrix formalism has proved very useful for deducing the analytic properties of the scattering matrix coefficients from general properties like causality and energy conservation. However, the Tmatrix formalism will be preferred in the following as it provides a more intuive description of the scattering problem by linking the excitation field to the scattered field. The excitation field E exc will be expanded on the set of regular VPWs (N n,m (ω)), while coefficients of expansion of the scattered field E scat on the outgoing VPWs (N (+) (kr),M (+) (kr)) will be denoted (f

(e) n,m (ω),f (h) n,m (ω)) (see Supplemental Material).
The T-matrix is also diagonal for a spherically symmetric scatterer and its coefficients are defined as follows:

T (e,h) n (ω) = f (e,h) n,m (ω)/e (e,h)
n,m (ω). A pole expansion of the T-matrix coefficient can also be derived from Eq. ( 3) by means of the following relation T = S-I 2 [40], where I is the identity matrix, leading to:

T (i) n (ω) A (i) n (ω) + B (i) n (ω) M α=-M r (i) n,α ω -ω (i) p,n,α (4)
where i = (e, h), A

(i) n (ω) = e -2ikR S (i) nr,n -1 2 and B (i) n (ω) = e -2ikR 2
. This pole expansion of the T-matrix coefficients allows to determine the spectral response of a scatterer over a broad range of frequencies from a discrete set of QNMs. This is illustrated in Fig. 1a where the accuracy of this pole expansion and the importance of these non-resonant terms are assessed. We compare in particular the dipolar electric partial scattering efficiency:

Q (e) 1 = 6 z 2 |T (e)
1 | 2 obtained by exact calculation on one side and using Eq. ( 4) with M = 100 for a spherical scatterer of dielectric permittivity ε = 16 in air as it is close to the permittivity of silicon in the visible spectrum [29]. Poles were found by using their asympotic values [30] along with pole finding methods (ε = 16 in the whole complex ω-plane). A good agreement between the exact calculations and Eq. ( 4) is found. Fig. 1a) also illustrates the importance of the non-resonant term in Eq. ( 4) since a poor prediction of the electric dipolar partial scattering efficiency is obtained when neglecting it even when a large number of poles is taken into account. Similar results can be obtained for other multipolar orders. The link between the pole expansion of the T-matrix coefficients and the QNM expansion of the scattered field will be clarified in the second part of this study.

Let us now extend this work to the internal field, the field inside the resonator, by introducing the Ξ matrix that relates the internal field to the incoming field. Let (u

(e) n,m (ω),u (h)
n,m (ω)) be the coefficients of the internal field on the regular VPWs (N

(1) n,m (k s r),M (1)
n,m (k s r))

where k s = √ ε s ω c . The coefficients are then simply defined as: Ξ

(i) n (ω) = u (i) n,m (ω) s (i,-) n,m (ω)
. Similarly, it is useful to introduce the Ω matrix that relates the excitation field to the internal field and whose coefficients are defined as:

Ω (i) n (ω) = u (i) n,m (ω) e (i) n,m (ω) , with Ω (i) n (ω) = Ξ (i) n (ω) 2
. A pole expansion can also be derived for Ξ these coefficients do not possess zeros that are symmetric to their poles with respect to the real axis. One can show that Ξ (i) n (ω) and Ω (i) n (ω) admit the following pole expansion (see Supplemental Material):

Ξ (i) n (ω) = ∞ α=-∞ r (i) Ξ,n,α ω -ω (i) p,n,α ≡ 2 Ω (e,h) n (ω) (5)
where i = e or h. Unlike the pole expansions of S (e,h) n

(ω) and T (e,h) n (ω), there is no non-resonant term in the pole expansion of Ξ (e,h) n (ω) and Ω (e,h) n (ω). To study the accuracy of this pole expansion, we calculate the averaged square internal field of a spherical resonator excited by a plane wave [30] (see Supplemental Material):

|Eint| 2 |E0| 2 = ∞ n=1 |E (e) i,n | 2 |E0| 2 + |E (h) i,n | 2 |E0| 2 .
This expression is plotted in Fig. 1b with ε s = 16 while considering the electric dipole contribution only, i.e.

|E (e)

i,1 | 2 |E0| 2 . Fig. 1b shows a good agreement between exact calcu-lations and the predictions obtained while using Eq. 5 with 20 poles (10 poles with a positive real part along with their symmetric with respect to the imaginary axis).

CAUSALITY AND TIME DOMAIN

Let us now derive the expression of the outgoing and scattered fields in terms of the QNM fields. We will in particular show how the divergence of these QNM fields can be dealt with. That is why we will here focus on the far-field region where this divergence occurs. The modes of spherically symmetric scatterers have to belong to one type of VPWs. Let us denote the electric-type (resp. magnetic-type) modes of the n -th multipolar order E

E (e),FF n,m,α (r) ∝ e i ω ( e) p,n,α c r Z n,m (θ, φ) E (h),FF n,m,α (r) ∝ e i ω (h) p,n,α c r X n,m (θ, φ) . (6)
where X n,m (θ, φ) and Z n,m (θ, φ) are vector spherical harmonics (see Appendix A of [41]). Since e i ω (i) p,n,α c r = e i ω (i) p,n,α r c e -ω (i) p,n,α r c and as, due to causality, ω

p,n,α < 0, E

n,m,α (r) is exponentially diverging as r → ∞ (i = e, h). One may believe that this divergence could hinder the derivation of an expression of the scattered field in terms of (E

(e) n,m,α ,E (h)
n,m,α ). However, this divergence can be understood by noting that, due to causality, considering a scattered field at r → ∞ amounts to assuming that the system was excited at t → -∞ [22,[START_REF] Nussenzveig | Causality and dispersion relations[END_REF].

In what follows, we will show that a divergence free expansion of the scattered and outgoing fields of 3D open optical cavities in the time domain can be obtained. It will require the use of a causal incoming and excitation fields, i.e. fields with a sharp cut-off in the time domain [START_REF] Nussenzveig | Causality and dispersion relations[END_REF], together with rigorous calculations based on the theorem of residues. Causality can be better understood in the framework of the S matrix while considering incoming and outgoing waves in the far-field region. The incoming field defined in Eq. ( 1) takes the following asymptotic expression in the far-field region that is denoted E FF in :

E FF in (r, ω) = E 0 ∞ n=1 n m=-n s (h,-) n,m (ω)i n+1 e -ikr kr X n,m (θ, φ) -s (e,-) n,m (ω)i n+2 e -ikr kr Z n,m (θ, φ) (7)
where s

(i,-) n,m (ω) = s (i,-) n,m g(ω), s (i,-)
n,m being the coefficient of expansion of the incoming field on the VPWs and g(ω) being the Fourier transform of the temporal dependence of the incoming field g(t). Let us take the Fourier transform of one of the terms in this sum:

E (e),FF in,n,m (r, t) = - i n+2 E 0 2π +∞ -∞ s (e,-) n,m (ω)
e -ikr kr ×

Z n,m (θ, φ)e -iωt dω .

(

) 8
As shown in the Supplemental Material, we choose the time dependence g(t) = H(t)e -iω0t for the incoming field since it corresponds to an incoming causal wavefront. This causal wavefront can be built up by choosing the following frequency dependence for s n,m a constant) as it is the Fourier transform of g(t). In a sake of generality, an additional phase factor e iωts is introduced to yield an additional time delay t s . This definition of s (e,-) n,m (ω) can be used in the expression of E (e),FF in,n,m (r, t) (see Supplemental Material):

E (e),FF in,n,m (r 0 , t) = -H t -t s + r 0 c i n+2 E 0 s (e,-) n,m × e -ik0r0 k 0 r 0 Z n,m (θ, φ)e -iω0(t-ts) ,( 9
)
where H is the Heaviside step distribution. The time-dependent outgoing field in the far-field region can be calculated with respect to the S-matrix coefficients of the incoming field: (t-τ ) (10) with

E (e),FF out,n,m (r, t) = (-i) n+2 E0 2π +∞ -∞ S (
C n,m (θ, φ) = (-i) n+2 E 0 s (e,-)
n,m Z n,m (θ, φ); g(t) = H(t)e -iω0t , and τ = t s + r c -2R c . Two contributions can be identified in Eq. ( 10): (i) The non-resonant contribution equal to the convolution between the temporal dependence of the incoming field g(t) and the Dirac distribution δ t -t s -r c + 2R c . This contribution corresponds to a mere reflection of the incoming field by the surface of the scatterer. Importantly, it keeps the same temporal dependence as the incoming field which means that it does not distort the incoming field that is simply translated in time by a factor τ ; (ii) The resonant contributions that correspond to the convolution between g(t) and a sum of exponentially decreasing terms that are characteristic of the response of each mode. This resonant response strongly distorts the temporal dependence of the incoming field which was expected since it results from a resonant interaction between light and eigen-modes of the cavity. These two contributions are sketched in Fig. 2. The scattered field can finally be obtained by removing the outgoing part of the excitation field from E (e),FF out,n,m (r, t) in Eq. ( 10):

E (e),FF scat,n,m (r, t) = D n,m (θ, φ) 2k 0 r g(t) * S (e) nr,n δ (t -τ ) -δ t -t s - r c -iH (t -τ ) M α=-M
r (e) n,α e -iω (e) p,n,α (t-τ ) (11) with

D n,m (θ, φ) = (-i) n+2 E 0 e (e)
n,m Z n,m (θ, φ). The scattered field in the time domain results from a convolution between the excitation field g(t) and the function characterizing the response of the scatterer, described by two terms. The first term, S (e) nr,n δ (t -τ )δ t -t s -r c , is associated with the non-resonant part S (e) nr,n of the S-matrix coefficients from which is substracted the outgoing part of the excitation field. This evinces the fact that the non-resonant part does not include the excitation field. The second term includes the far-field limit of the QNM fields E The Heaviside distribution results from causality and simply means that, given the initial conditions imposed on the incoming field in Eq. ( 9), the outgoing field has only been able to propagate up to a distance r = c (t -t s )+2R. Due to the Heaviside distribution, the outgoing field is different from zero only when t-t s -r c + 2R c ≥ 0. Therefore the term e ω (e) p,n,α (t-ts-r c + 2R c ) < 1 since one has both ω (i) p,n,α < 0 and t -t s -r c + 2R c ≥ 0. and consequently this term does not diverge.

CONCLUSION

To conclude, in addition to the resonant contributions due to the excitation of QNM of the scatterer by the excitation field, a non-resonant contribution must be taken into account in the expansion of the scattered field. This result has been demonstrated by deriving pole expansions of the S and T matrix coefficients of a dispersionless 3D scatterer. We benefited from the symmetry of the spherical scatterers to provide the explicit expressions of both resonant and non-resonant contributions of the scattered field expansion. Then, the expansion of the scattered field in the time domain was computed by means of an inverse Fourier transform of the frequency-dependent scattered field obtained from the pole expansion of the T-matrix coefficients. Rigorous calculations based on the theorem of residues together with causality principle lead to QNM expansion terms that do not diverge far from the scatterer. This result obtained in the framework of the multipolar theory can, on physical grounds, be generalized to arbitrarily shaped scatterers. Moreover, QNM theory is applicable to an extremely broad range of physical studies far beyond optics.

A*MIDEX project (no. ANR-11-IDEX-0001-02) funded by the Investissements d'Avenir French Government program, managed by the French National Research Agency (ANR). The authors acknowledge Emmanuel Lassalle for fruitful discussions.

  (e,h) n | ≤ 1 (see Supplemental Material) and |S (e,h) n | = 1 for lossless scatterers (where | | corresponds to the norm).

  can be identified as the QNM eigenfrequencies of the scatterers. The poles of the S-matrix, ω (e,h) p,n,α = ω (e,h) p,n,α + iω (e,h) p,n,α , consequently have a negative imaginary part: ω (e,h) p,n,α . Moreover, energy conservation and time-reversal symmetry yield for a lossless scatterer: S (e,h) n (ω) = 1/S (e,h) * n (ω * ) [24, 37]. As a consequence, S (e,h) n

  (kr)) with the corresponding coefficients of expansion (e (e) n,m (ω),e (h)

n

  (ω) and Ω (i) n (ω). Energy conservation does not require the coefficients Ξ (i) n (ω) to be unitary. Consequently, and in contrast to S (i) n (ω),

Figure 1 : 1 |

 11 Figure 1: (a) Dipolar electric partial scattering efficiency Q (e) 1 = 6 z 2 |T (e) 1 | 2 and (b) averaged square modulus of the internal electric field with |E 0 | = 1 for an ε = 16 spherical scatterer in air: exact calculations (red solid curve), Eq. (4) with M = 100 poles for (a) and Eq. (5) with M = 10 poles for (b) (gray dashed line), Eq. (4) with M = 100 for (a) while assuming S (e,h) nr,n = 0 (blue dashed curve).

  ,α (r) (resp. E (h) n,m,α (r)). Outside the scatterer, i.e. for r > R:E (e) n,m,α (r) ∝ N n,m,α (r) ∝ M (+) n,m ω (h)p,n,α c r . Moreover, in the far-field region (i.e. when r → ∞), they possess the following asymptotic expressions E (e),FF n,m,α (r) and E (h),FF n,m,α (r) (see Supplementary Material):

  (ω) e ikr kr Z n,m (θ, φ)e -iωt dω. The easiest way to calculate E (e),FF out,n,m (r, t) is to use the convolution theorem associated with the theorem of residues:E (e),FF out,n,m (r, t) = C n,m (θ, φ) k 0 r g(t) * S (e) nr,n δ (t -τ ) -iH (t -τ ) M α=-Mr (e) n,α e -i ω (e) p,n,α c

Figure 2 :

 2 Figure 2: Schematic representation of non-resonant and resonant terms in Eq.10

  (e),FF n,m,α (r) ∝ e i ω (e) p,n,α c r Z n,m (θ, φ). The Heaviside step function in front of the sum prevents the field from diverging when r → ∞.

ACKNOWLEDGMENTS

Research conducted within the context of the International Associated Laboratory ALPhFA: Associated Laboratory for Photonics between France and Australia. This work has been carried out thanks to the support of the