E Bodéré

P Y Clément
email: p_clement@hotmail.com

A Genoud
email: genoud_aude@yahoo.fr

M Le Trocquer

M Moras

L Pochard

V Ribaud
email: ribaud@univ-brest.fr

Ph Saliou
email: saliou@univ-brest.fr

INTEGRATING THE JENA RDF API WITHIN SAKAI : TOWARDS A SEMANTIC COLLABORATING LEARNING ENVIRONMENT

Keywords: RDF, semantic web, CLE, Sakai

In this paper, we describe a semantic extension to the CLE Sakai. Semantic means the ability to create, retrieve, query and navigate with knowledge about the entities managed by Sakai. RDF is the support of this semantic extension, through the integration within Sakai of Jena, an open-source RDF API provided by HP laboratory. New features are briefly presented and some use cases discussed. Open issues concludes this on-going work.

INTRODUCTION

Sakai is an open-source course management system (CMS). Described as the "digital home of students' coursework" [START_REF] Lippincott | Net Generation Students & Libraries[END_REF], the CMS is becoming the primary mechanism for faculty to provide students with scholarly information and resources, and in turn, for students to access such materials.

RDF is a W3C standard intended to manage metadata. RDF models metadata as 3-tuples which assert that some resource (identified by URI) has some property (identified by URI) which has a value identified either by URI or given literally. The most common day-to-day use of metadata is to help us find things but it can change the way to organize, manage and use things Social software supports activities in digital social networks. Knowledge sharing and exchange is an important part of these activities. Because knowledge management and learning are closed, the use of social software can support learning. Hence, we investigated the Sakai tools able to support social activities for knowledge management such as resources sharing, "wikiing", blogging and so on. Coupled with RDF metadata, these tools offer new features: providing additional semantic to social usages add significant value allowing view, navigation and query along semantic rather than simply chronological connections.

COLLABORATION AND LEARNING ENVIRONMENT

Sakai tools

Sakai defines itself as an online Collaboration and Learning Environment. Many users of Sakai deploy it to support teaching and learning, ad hoc group collaboration, support for portfolios and research collaboration.

The Resources Tool is a core feature of Sakai that allows an instructor to assemble a collection of documents or other files supporting a course and organize them into folders and subfolders for access by students. When an instructor creates a new resource item in Sakai, he or she can choose to upload an existing file from his or her local system, author a text document or HTML page directly through the Sakai interface, or link to existing content via a URL. The instructor can then enter Dublin Core metadata for the item (as of Sakai 2.0), along with information on copyright and access permissions. Once a resource item has been created, it can easily be selected and attached to postings within other Sakai tools, including Wiki and Blog tools.

An example of RDF application

RDF extends the linking structure of the Web to use URIs to name the relationship between things as well as the two ends of the link (this is usually referred to as a "triple"). RDF is a model describing qualified (or named) relationships between two (Web) resources, or between a Web resource and a literal. Using this simple model, it allows structured and semi-structured data to be mixed, exposed, and shared across different applications. Let us take an example from a W3C Note Describing and retrieving photos using RDF and HTTP. This note describes a project for describing & retrieving (digitized) photos with (RDF) metadata. The pictures are digitized and stored as JPEG images. Metadata is written into the pictures with the data-entry program (and can also be edited if corrections are necessary). Requests from the Web are served by a server, by sending either the picture or the metadata, depending on the form of the request.

The metadata is separated into three different schemas:

Dublin Core schema 1 . The Dublin Core schema [START_REF]Dublin Core metadata element set[END_REF] is a general schema for identifying original works, typically books and articles, but also films, paintings or photos. It contains such properties as creator, editor, title, date of publishing and publisher. Technical schema 2 . This schema captures technical data about the photo and the camera, such as the type of camera, the type and the date of film, etc.

Content schema. This schema is used to categorize the subject of the photo by means of a controlled vocabulary. This schema allows photos to be retrieved based on such characteristics as portrait, group portrait, landscape, architecture, sport, animals, etc. This is an excerpt of some metadata in RDF <rdf:Description rdf:about=""> <s0:creator>Bert Bos</s0:creator> <s0:format>image/jpeg</s0:format> <s0:date>1999-06-26</s0:date> <s1:camera>Canon 5</s1:camera> <s0:title>Marian with sheep</s0:title> <s0:subject>Landscape, Animal</s0:subject> <s0:description>Marian brings the sheep to the field in the morning. The lamb she carries was born that night.</s0:description> </rdf:Description> This means that the resource about we are talking on, was created by Bert Bos, is a jpeg image ...

Semantic annotation of resources

An annotated system is a system which "knows about" its own content in order that automated tools can process annotations to improve the use of the system. The goal is to create annotations with welldefined semantics. For example, the semantic annotations can identify people, organizations, and projects, as well as including traditional metadata, such as the author's name and date of publication. When these statements are integrated within a RDF repository, we can then support SPARQL queries like "give me all the people which work in projects on the Semantic Web". According to the features of the Semantic Blogger [START_REF] Cayzer | Semantic Blogging: Spreading the Semantic Web Meme[END_REF], the software will exploit the semantic annotations to provide three key behaviours:

1 Referred as s0 in the example above 2 Referred as s1 in the example above Semantic View: context sensitive, schema driven views of the content (over and above fixed templates).

Semantic Navigation: new navigation modalities (over and above unlabelled links).

Semantic Query: richer query and discovery mechanisms (over and above free text search).

Uren and al. [START_REF] Uren | Semantic annotation for knowledge management: Requirements and a survey of the state of the art[END_REF] provide a complete survey of semantic annotation for knowledge management.

A SEMANTIC EXTENSION OF SAKAI

Overview of the project

The team

Thanks to a strong partnership with Thales company, Brest University offers a kind of apprenticeship during the last year of a Master of Software Engineering. Young engineers are immersed in an imitation of the real world. The plan of action is built on a 7-month project, followed by a 5-month internship. Young engineers make up two teams of 6; each team is led by one associate professor acting as project manager. A contract defines the customer-supplier relationship. An adaptation of the Thales ISO 9001 baseline is the project baseline. Each project team has its own office with individual working post and common installations. Each project team uses a different and complete software engineering tools suite (Oracle and Open Source). At the end of the academic year, the team members obtain a Master's degree in Software Engineering. This year, one team's project aims to provide a semantic extension to Sakai, called Sakai Semantic Edition (Sakai SE).

Project objectives and outcomes

Information Management can be defined as the set of activities that people perform in order to acquire, organize, maintain and retrieve information for everyday use. A space of information (SI) for a person or a group includes all the information items that are under that person's or that group's control. A SI also includes applications, tools (such as a presentation tool) and constructs (e.g., associated properties, folders) that support the acquisition, storage, retrieval and use of the information in a SI. Sakai offers several spaces of information: personal (MyWorkspace), shared by a group (Site), shared between two persons (DropBox). The main goal of Sakai SE is to enhance these SI with semantic annotations. Main functions are: a) Indexation allows to describe the information item with a set of metadata (called a notice) in order to exploit the notice without the need to access to the item ; b) Search on metadata c) Semantic hyperlinks between notices and/or items d) Navigation offers knowledge walkthrough along the semantic paths e) RDF modelization defines a common sense f) Automatic extraction provides RDF metadata g) Formalization of knowledge through the use of vocabularies and schemas

Jena : a Semantic Web Toolkit

Jena is a leading Semantic Web programmers' toolkit.

It is an open-source project, implemented in Java, and available for download from SourceForge.

Jena overview

This overview is based on the Jena overview in [START_REF] Wilkinson | Efficient RDF Storage and Retrieval in Jena2. First International Workshop on Semantic Web and Databases[END_REF].

Jena offers a simple abstraction of the RDF graph as its central internal interface. The main contribution of Jena is a rich API for manipulating RDF graphs. RDF graphs in memory or in databases. Jena provides additional functionality to support RDFS and OWL.

The two key architectural goals of Jena are:

• Multiple, flexible presentations of RDF graphs to the application programmer. This allows easy access to, and manipulation of, data in graphs enabling the application programmer to navigate the triple structure.

• A simple minimalist view of the RDF graph to the system programmer wishing to expose data as triples.

An simplified overview of the Jena architecture is shown in Figure 1. Applications typically interact with an abstract Model which translates higher-level operations into low-level operations on triples stored in an RDF Graph.

Figure 1 : Jena2 Architectural Overview

Jena storage

At an abstract level, the Jena storage subsystem need only implement three operations: add statement, to store an RDF statement in a database; delete statement, to remove an RDF statement from the database; and the find operation. The find operation retrieves all statements that match a pattern of the form <S,P,O> where each S, P, O is either a constant or a don't-care. Jena's query language, RDQL, is converted to a set of find operations in which variables are permitted in the find patterns. The variables enable joins across the patterns.

A widely used implementation technique is to store RDF statements in a relational database using a single statement table, often called a "triple-store." This is a table that stores each RDF statement as a row and has columns for the subject, predicate and object.

Components of Sakai SE

Schemas

A schema contains properties. RDF provides no mechanisms for describing these properties, nor does it provide any mechanisms for describing the relationships between these properties and other resources. That is the role of the RDF vocabulary description language, RDF Schema. RDF Schema defines classes and properties that may be used to describe classes, properties and other resources.

The Dublin Core is used for all entities of Sakai SE.

A technical schema groups properties other than Dublin Core properties (see § 2.2). A technical schema may be present in several versions. Versions are managed through a schema group.

SPARQL queries

SPARQL is a query language for getting information from RDF graphs. SPARQL contains capabilities for querying by triple patterns, conjunctions, disjunctions, and optional patterns. Results of SPARQL queries can be ordered and presented in several different forms. Through the integration of the Jena SPARQL engine, Sakai SE offers triple querying and queries management.

Sakai tools

Sakai SE is essentially made of tools supporting the management of spaces of information (cf. § 3.1.2). Some tools were modified in order to incorporate RDF facilities, mainly the Resources tool and the Users tool.

New tools were developed including the Semantic Schema that allows schemas management and the Semantic Search that provides SPARQL queries use.

Application

Model

Other graphs Database Graphs

Specialized Graphs

Property tables

Database-specific Drivers

Sakai provides tools for content authors to represent structures that link together smaller units of content such as resources, pages into larger structures such as portfolios and sites. After an investigation of tools such as OSP portfolio, blogger and wiki, we select wikis because they can be seen as a lightweight platform for the management of these structures. One way to organize the organic growth of Wiki content is to add structure by enriching Wiki pages with RDF metadata. This adds significant value allowing semantic view, navigation and query.

Blogging is also a lightweight Web publishing paradigm that provides a very low barrier to entry but the components of the Blogger tool (blogs and entries) are not managed as Sakai entities and we were unable to bind these components with RDF URI (see § 3.4.1 for details).

Some features of Sakai SE

Technology

Jena has object classes to represent graphs, resources, properties and literals. The interfaces representing resources, properties and literals are called Resource, Property and Literal respectively. Recall that in Jena, a graph is called a model and is represented by the Model interface.

RDF triples can be associated to different Sakai entities : sites, users, groups, content resources, wiki, wiki pages and so on. Each instance of these entities is associated within Sakai with a kind of internal identifier. In order to be able to write or read a triple related to one of theses instances, we have to bind the triple with the right instance of Sakai entities. Hence, for each instance, we build an URI based on the type of the instance (the entity its belongs to) and the proper value of this internal identifier. For example, the file Overview.doc (a ContentResource instance) has as an internal identifier the value c40dd7fc-...508d and its associated URI is sakai:content: c40dd7fc-…508d.

Hence, there is a one-to-one relationship between Sakai objects and RDF URI allowing to bind objects to the related graph (the graph having this URI as root).

Multi-dimensional search and browsing

Web information portals provide a point of access onto an integrated and structured body of information about some domain. Many portals use a hardwired navigation structure based on a single rich classification scheme (e.g. Dublin Core) coupled to hyperlinking of related items and free text search. Yahoo is a canonical example. Portals handling more structured information typically have some more richly structured internal descriptive schema (often directly mirroring the database schema used for storage) and offer a rich search interface which can exploit this schema. This allows search to be tied to specific facets of the descriptive metadata and to exploit controlled vocabulary terms -leading to much more precise searches [START_REF] Reynolds | Semantic Portals -Requirements Specification[END_REF].

In Sakai SE, we describe properties, relationships and classifications of the various information items via the content schemas. This allows the users to choose the structure of the portal and allows searches to exploit the structure of the ontology

Evolution and extension of information structure

Information requirements change over time. This sort of evolutionary change requires us to change the metadata and any associated database schema, not just the descriptive ontology. This can be complex. We need to permit metadata to be added, without invalidating existing metadata. This is greatly facilitated by use of RDF semi-structured data representation. RDF enables incremental additions of properties and relations by virtue of its propertycentric, rather than record-centric, approach to representation [START_REF] Reynolds | Semantic Portals -Requirements Specification[END_REF].

Querying

Creating an intuitive and powerful Web tool for metadata querying is not an easy task, the more so if we want to develop this tool in a generic way, i.e. without 'hard coding' a particular metadata scheme into it. Moreover, the tool should be flexible in the sense that the query functionality itself must be configurable: the user interface should reflect the important metadata fields and provide a user friendly, yet powerful, mechanism to query the metadata instances.

For the moment, querying is done through a direct editing of SPARQL queries but it requires a SPARQL knowledge out of reach from end-users.

Automation

Easing the metadata acquisition bottleneck is the provision of facilities for automatic mark-up of resources. No one wants to spent time to produce metadata, especially if they are still present in some manner with the resources. Creators, dates, description, keywords are provided within most of resources format (word processors, spreadsheet, images …) and metadata are present in the Resources tool. Sakai SE provides an extractor that translates existing metadata in RDF triples.

PERSPECTIVES AND CONCLUSION

A special effort has to be done to provide a quasi natural language querying access to RDF metadata.

An example is Ginseng, a guided input natural language search engine for Querying Ontologies [START_REF] Bernstein | Ginseng A Guided Input Natural Language Search Engine for Querying Ontologies[END_REF].

When the user starts typing, the system predicts the possible completions of what the user enters and presents the user with a choice popup box. Ginseng, thus, guides the user through the set of possible queries while avoiding ungrammatical queries. When a query is completed, Ginseng translates it into SPARQL statements. Ginseng is based on Jena API and it should be possible to incorporate it in Sakai SE.

Social software is used widely in organizational knowledge management and professional learning.

Integrating RDF management into social software should lower the barriers between knowledge and learning management. Semantic blogging is a promised issue but the Blogger tool need to be revisited.

Sakai SE is a first step towards a semantic collaborating and learning environment. The entryticket for Sakai development is now paid and we will build new tools on these semantic foundations.


Miguel Couceiro

Jimmy Devillet

Jean-Luc Marichal

CHARACTERIZATIONS OF IDEMPOTENT DISCRETE UNINORMS

Keywords: Discrete uninorm, idempotency, conservativeness, bisymmetry, contour plot, axiomatization

In this paper we provide an axiomatic characterization of the idempotent discrete uninorms by means of three conditions only: conservativeness, symmetry, and nondecreasing monotonicity. We also provide an alternative characterization involving the bisymmetry property. Finally, we provide a graphical characterization of these operations in terms of their contour plots, and we mention a few open questions for further research.

Introduction

Aggregation functions defined on linguistic scales (i.e., finite chains) have been intensively investigated for about two decades; see, e.g., [3-6, 8-13, 15, 17]. Among these functions, discrete fuzzy connectives (such as discrete uninorms) are binary operations that play an important role in fuzzy logic.

This short paper focuses on characterizations of the class of idempotent discrete uninorms. Recall that a discrete uninorm is a binary operation on a finite chain that is associative, symmetric, nondecreasing (in each variable), and has a neutral element.

A first characterization of the class of idempotent discrete uninorms was given by De Baets et al. [START_REF] Baets | Idempotent uninorms on finite ordinal scales[END_REF]. This characterization reveals that any idempotent discrete uninorm is a combination of the minimum and maximum operations. In particular, such an operation is conservative in the sense that it always outputs one of the input values.

The outline of this paper is as follows. After presenting some preliminary results on conservative operations in Section 2, we show in Section 3 that the idempotent discrete uninorms are exactly those operations that are conservative, symmetric, and nondecreasing (Theorem 12). This new axiomatic characterization is rather surprising since it requires neither associativity nor the existence of a neutral element. We also present a graphical characterization of these operations in terms of their contour plots (Theorem 15). This graphical characterization shows us a very easy way to generate all the possible idempotent discrete uninorms on a given finite chain. In Section 4 we provide an alternative axiomatic characterization of this class in terms of the bisymmetry property. Specifically, we show that the idempotent discrete uninorms are exactly those operations that are idempotent, bisymmetric, nondecreasing, and have neutral elements. More generally, we also show that the whole class of discrete uninorms can also be axiomatized by simply suppressing idempotency in the latter characterization and that this result also holds on arbitrary chains (Theorem 21). Finally, Section 5 is devoted to some concluding remarks and open questions.

Preliminaries

In this section we present some basic definitions and preliminary results. Let X be an arbitrary nonempty set and let ∆

X = {(x, x) | x ∈ X}. Definition 1. An operation F ∶ X 2 → X is said to be • idempotent if F (x, x) = x for all x ∈ X. • conservative (or selective) if F (x, y) ∈ {x, y} for all x, y ∈ X. • associative if F (F (x, y), z) = F (x, F (y, z)) for all x, y, z ∈ X.
Remark 1. Conservative operations were introduced in [START_REF] Pouzet | A projection property[END_REF]. By definition, the output value of such an operation must always be one of the input values. In particular, any conservative operation F ∶ X 2 → X is idempotent. Moreover, such an operation can be "discreticized" in the sense that, for any nonempty discrete subset S of X, its restriction to S 2 ranges in S. More precisely, it can be shown [START_REF] Couceiro | Representations and characterizations of polynomial functions on chains[END_REF] that the following conditions are equivalent:

(i) F is conservative. (ii) For any ∅ ≠ S ⊆ X, we have F (S 2 ) ⊆ S. (iii) For any ∅ ≠ S ⊆ X and any x, y ∈ S, if F (x, y) ∈ S then x ∈ S or y ∈ S.
Definition 2. Let F ∶ X 2 → X be an operation.

• An element e ∈ X is said to be a neutral element of F (or simply a neutral element) if F (x, e) = F (e, x) = x for all x ∈ X. In this case we easily show by contradiction that such a neutral element is unique. • The points (x, y) and (u, v) of X 2 are said to be connected for F (or simply connected ) if F (x, y) = F (u, v). We observe that "being connected" is an equivalence relation. The point (x, y) of X 2 is said to be isolated for F (or simply isolated ) if it is not connected to another point in X 2 .

Proposition 3. Let F ∶ X 2 → X be an idempotent operation. If the point (x, y) ∈ X 2 is isolated, then it lies on ∆ X , that is, x = y.

Proof. Let (x, y) be isolated. From the identity F (x, y) = F (F (x, y), F (x, y)) we immediately derive (x, y) = (F (x, y), F (x, y)). □ Remark 2. We observe that idempotency is necessary in Proposition 3. Indeed, consider the operation F ∶ X 2 → X, where X = {a, b}, defined as Some conservative operations have neutral elements (e.g., F (x, y) = max{x, y} on X = {a, b, c} has the neutral element e = a) and some have not (e.g., F (x, y) = x). The following lemma provides an easy graphical test for checking whether a conservative operation has a neutral element. Proposition 4. Let F ∶ X 2 → X be a conservative operation and let e ∈ X. Then e is a neutral element if and only if (e, e) is isolated. (Sufficiency) If e is not a neutral element, then there exists u ∈ X ∖ {e} such that F (u, e) = e = F (e, e) or F (e, u) = e = F (e, e). In both cases, (e, e) is not isolated, a contradiction. □ Corollary 5. Any isolated point (x, y) of a conservative operation F ∶ X 2 → X is unique and lies on ∆ X . Moreover, x = y is a neutral element.

F (x, y) = a, if (x, y) = (a
Remark 3. Proposition 4 no longer holds if conservativeness is relaxed into idempotency. Indeed, by simply taking X = {a, b, c} we can easily construct an idempotent operation with an isolated point on ∆ X and no neutral element (see Figure 2). Also, it is easy to construct an idempotent operation with a neutral element and no isolated point (see Figure 3). It is also noteworthy that there are idempotent operations with more than one isolated point (see Figure 4). Proposition 6. An operation F ∶ X 2 → X is conservative if and only if it is idempotent and every point (x, y) ∈ X 2 ∖ ∆ X is connected to either (x, x) or (y, y).

Proof. Clearly, F is conservative if and only if it is idempotent and for every distinct x, y ∈ X we have either F (x, y) = x = F (x, x) or F (x, y) = y = F (y, y). □ Remark 4. Proposition 6 provides an easy graphical test for checking whether an idempotent operation is conservative. For instance, none of the idempotent operations represented in Figures 234is conservative because in each case the point (a, c) is connected to neither (a, a) nor (c, c).

s s s s s s s s s (a, a) (b, a) (c, a) (a, b) (b, b) (c, b) (a, c) (b, c) (c, c) @ @ @ @ @ @ @ Figure 3. An opera- tion with no isolated point s s s s s s s s s (a, a) (b, a) (c, a) (a, b) (b, b) (c, b) (a, c) (b, c) (c, c) Figure 4.
An operation with two isolated points Proposition 7. An operation F ∶ X 2 → X has a neutral element if and only if there are a vertical section and a horizonal section of X 2 that intersect on ∆ X and such that the restriction of F to each of these sections is the identity function.

Proof. The result immediately follows from the definition of a neutral element. □ Remark 5. Proposition 7 provides a graphical test for checking the existence of a neutral element. For instance, we can easily see that the operation represented in Figure 3 has b as the neutral element. Note that when the operation is conservative, by Proposition 4 it suffices to search for an isolated point on ∆ X .

Main results

We now focus on characterizations of the class of idempotent discrete uninorms. These operations are defined on finite chains. Without loss of generality we will only consider the n-element chains L n = {1, . . . , n}, n ≥ 1, endowed with the usual ordering relation ≤.

Recall that an operation F ∶ L 2 n → L n is said to be nondecreasing in each variable (or simply nondecreasing) if F (x, y) ≤ F (x ′ , y ′ ) whenever x ≤ x ′ and y ≤ y ′ . Definition 8 (see, e.g., [START_REF] Baets | Idempotent uninorms on finite ordinal scales[END_REF]). A discrete uninorm on L n is an operation U ∶ L 2 n → L n that is associative, symmetric, nondecreasing, and has a neutral element.

A first characterization of the class of idempotent discrete uninorms was established by De Baets et al. [START_REF] Baets | Idempotent uninorms on finite ordinal scales[END_REF]. We state this result in the following theorem. Although this characterization is somewhat intricate, it shows, together with Lemma 10 below, that any idempotent discrete uninorm is conservative.

Theorem 9 (see [START_REF] Baets | Idempotent uninorms on finite ordinal scales[END_REF]Theorem 3]). An operation F ∶ L 2 n → L n with a neutral element 1 < e < n is an idempotent discrete uninorm if and only if there exists a nonincreasing map g∶ [1, e] → [e, n] (nonincreasing means that g(x) ≥ g(y) whenever x ≤ y), with g(e) = e, such that

F (x, y) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ min{x, y}, if y ≤ g(x) and x ≤ g(1), max{x, y}, otherwise,
where g∶ L n → L n is defined by

g(x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ g(x), if x ≤ e, max{z ∈ [1, e] | g(z) ≥ x}, if e ≤ x ≤ g(1), 1, if x > g(1)
.

Remark 6. The fact that any idempotent discrete uninorm is conservative can also be easily proved by following the first few steps of the proof of [2, Theorem 3].

We now show that the idempotent discrete uninorms are exactly those operations that are conservative, symmetric, and nondecreasing (see Theorem 12).

First consider the following lemma, which actually holds on arbitrary, not necessarily finite, chains (i.e., totally ordered sets). Proof. Let us first prove that F has a neutral element. We proceed by induction on the size n of the chain. There is nothing to prove if n = 1. We can easily see by inspection that there are only two possible operations if n = 2 and four possible operations if n = 3. The contour plot of these operations are given in Figures 5 and6, respectively. Now suppose that the result holds for any (n -1)-element chain and consider an operation F ∶ L 2 n → L n that is conservative, symmetric, and nondecreasing. By conservativeness and symmetry we then have F (1, n) = F (n, 1) ∈ {1, n}. We may suppose that F (1, n) = F (n, 1) = 1; the other case can be dealt with dually. By nondecreasing monotonicity, we also have F (1, x) = F (x, 1) = 1 for all x ∈ L n . Consider the subchain L ′ = L n ∖ {1}. Clearly, the operation F ′ = F | L ′ is conservative, symmetric, and nondecreasing. By the induction hypothesis, F ′ has a neutral element e ∈ L ′ . Let us show that e is also a neutral element of F . Suppose that this is not true. Then, by Proposition 4 the point (e, e) is isolated for F ′ but not for F . This means that there exists x ∈ L n such that 1 = F (1, x) = F (x, 1) = F (e, e) = e, which contradicts the fact that e ∈ L ′ . Now, let F ∶ L 2 n → L n be an operation that is conservative, symmetric, and nondecreasing. We just showed that F must have a neutral element e ∈ L n . To see that F is associative, let x, y, z ∈ L n be arbitrary and let us show that the identity F (F (x, y), z) = F (x, F (y, z)) holds. Assume that x ≤ y ≤ z (the other five permutations can be treated similarly). We have three cases to examine:

Lemma 10. If F ∶ L 2 n → L n is idempotent, nondecreasing, and has a neutral element e ∈ L n , then F | [1,e] 2 = min and F | [e,n] 2 = max. Proof. For any x, y ∈ [1, e] such that x ≤ y, we have x = F (x, x) ≤ F (x, y) ≤ F (x, e) = x and x = F (x, x) ≤ F (y, x) ≤ F (e, x) = x. This shows that F | [1,e] 2 =
• Suppose x ≤ y ≤ z ≤ e or e ≤ x ≤ y ≤ z. Then the result immediately follows from Lemma 10. • Suppose x ≤ y ≤ e ≤ z. By Lemma 10, we have F (x, y) = min{x, y} = x.

-If F (x, z) = x, then F (F (x, y), z) = F (x, z) = x and by conservativeness we also have -

F (x, F (y, z)) ∈ {F (x, y), F (x, z)} = {x}. s s s s (1, 1) (2, 1) (1, 2) (2, 2) s s s s (1, 1) (2, 1) (1, 2) (2, 2)
(1, 1) (2, 1) (3, 1) (1, 2) (2, 2) (3, 2) (1, 3) (2, 3) (3, 3) s s s s s s s s s (1, 1) (2, 1) (3, 1) (1, 2) (2, 2) (3, 2) (1, 3) (2, 3) (3, 3) s s s s s s s s s (1, 1) (2, 1) (3, 1) (1, 2) (2, 2) (3, 2) (1, 3) (2, 3) (3, 3)
s s s s s s s s s (1, 1) (2, 1) (3, 1) (1, 2) (2, 2) (3, 2) (1, 3) (2, 3) (3, 3)
If F (x, y) = x, then we have F (F (x, y), z) = F (x, z) = F (x, F (y, z)).
-If F (x, y) = y, then by conservativeness and nondecreasing monotonicity we have F (x, z) = z and hence

F (F (x, y), z) = F (y, z) = z = F (x, z) = F (x, F (y, z)
). This completes the proof of the proposition. □ Remark 7.

(a) The existence of a neutral element in Proposition 11 is no longer guaranteed if the chain is not finite. For instance, the real operation

F ∶ [0, 1] 2 → [0, 1] defined by F (x, y) = min{x, y}, if x, y ∈ [0, 1
2 ) 2 , and F (x, y) = max{x, y}, otherwise, is conservative, symmetric, and nondecreasing, but it does not have a neutral element. Here we have provided a simpler proof based on the existence of a neutral element. (c) We observe that conservativeness cannot be relaxed into idempotency in Proposition 11. For instance the operation F ∶ L 2 3 → L 3 whose contour plot is depicted in Figure 2 is idempotent, symmetric, and nondecreasing, but one can show that it is not associative and it has no neutral element. (d) We also observe that each of the conditions of Proposition 11 is necessary.

Indeed, we give in Figure 7 an operation that is conservative and symmetric but that is not nondecreasing. We also give in Figure 8 an operation that is conservative and nondecreasing but not symmetric. Finally, we give in Figure 9 an operation that is symmetric and nondecreasing but not conservative. None of these three operations is associative and none has a neutral element. (Sufficiency) Trivial. □

Theorem 12. An operation

F ∶ L 2 n → L n is conservative, symmetric,
The following result gives the exact number of idempotent discrete uninorms on L n . A proof that essentially relies on Theorem 9 can be found in [START_REF] Baets | Idempotent uninorms on finite ordinal scales[END_REF]Theorem 4].

Here we provide an alternative proof based on Theorem 12. Theorem 14. There are exactly 2 n-1 idempotent discrete uninorms on L n . Proof. By Theorem 12 it is enough to count the number c n of operations on L n that are conservative, symmetric, and nondecreasing. As already observed in the proof of Proposition 11 we can see by inspection that c 1 = 1, c 2 = 2, and c 3 = 4 (see Figures 5 and6). Suppose now that c n-1 = 2 n-2 for some integer n ≥ 3 and let us prove that c n = 2 n-1 . Let F ∶ L 2 n → L n be an arbitrary idempotent discrete uninorm. By conservativeness and symmetry we then have F (1, n) = F (n, 1) ∈ {1, n}. Suppose first that F (1, n) = F (n, 1) = 1. By nondecreasing monotonicity, we also have

F (1, x) = F (x, 1) = 1 for all x ∈ L n . Consider the subchain L ′ = L n ∖ {1}. Clearly, the operation F ′ = F | L ′ is
conservative, symmetric, and nondecreasing and there are c n-1 = 2 n-2 possible such operations (see Figure 10, on left). We arrive at the same conclusion if F (1, n) = F (n, 1) = n (see Figure 10, on right). In total, we then have

c n = 2 n-2 + 2 n-2 = 2 n-1 .
□ Theorems 12 and 14 together enable us to provide the following graphical characterization of the idempotent discrete uninorms in terms of their contour plots.

Theorem 15 (Graphical characterization). All the idempotent discrete uninorms

F ∶ L 2 n → L n can

be constructed recursively in terms of their contour plots by the following algorithm:

Step 1. Choose the neutral element e ∈ L n and set C 1 = {e}. The point (e, e) is necessarily isolated with value e Step 2. ). We clearly have c 1 (1) = 1. It is also easy to see that c 2 (1) = c 2 (2) = 1 (see Figure 5). Suppose now that c n-1 (e) = ( n-2 e-1 ) for some integer n ≥ 3 and let us compute c n (e) for any e ∈ L n . We clearly have c n (1) = c n (n) = 1 so we can assume that 1 < e < n. It is then easy to see that if a 1 = e -1 (resp. a 1 = e + 1), then the number of possible

For k = 1, . . . , n -1 1. Pick a closest element a k to C k in L n ∖ C k 2. Set C k+1 = {a k } ∪ C k 3. Connect all the points in C 2 k+1 ∖ C 2 k with common value a k
operations constructed in Step 2 is c n-1 (e -1) = ( n-2 e-2 ) (resp. c n-1 (e) = ( n-2 e-1 )). In total we obtain c n (e) = ( n-2 e-2 ) + ( n-2 e-1 ) = ( n-1 e-1
). Let us now show that the algorithm enables us to generate all the idempotent discrete uninorms on L n . On the one hand, we clearly see that the algorithm enables us to construct ∑ n e=1 c n (e) = 2 n-1 possible operations. On the other hand, all these operations are clearly conservative, symmetric, and nondecreasing. We then conclude the proof by Theorems 12 and 14. □ Figure 11 gives two possible idempotent discrete uninorms on L 4 . All the possible idempotent discrete uninorms on L 2 and L 3 are given in Figures 5 and6, respectively.

We end this section by giving a graphical test for checking whether a conservative operation is associative. Proposition 16. Let F ∶ X 2 → X be a conservative operation. Then the following assertions are equivalent.

(i) F is not associative.

(ii) There exist a, b, c ∈ X pairwise distinct such that F (a, b), F (a, c), F (b, c) are pairwise distinct. (iii) There exists a rectangle such that one of its vertices is on ∆ X and the three remaining vertices are in X 2 ∖ ∆ X and pairwise disconnected.

Bisymmetric operations

In this section we provide a characterization of the class of discrete uninorms in terms of the bisymmetry (or mediality) property. From this result we immediately derive a new characterization of the class of idempotent discrete uninorms. Definition 19. An operation F ∶ X 2 → X is said to be bisymmetric if

F (F (x, y), F (u, v)) = F (F (x, u), F (y, v))
for all x, y, u, v ∈ X.

Lemma 20. Let F ∶ X 2 → X be an operation. Then the following assertions hold:

(x, z) ∈ {x, z}. If F (x, z) = x, then F (F (x, y), z) = F (F (x, y), F (z, z)) = F (F (x, z), F (y, z)) = F (x, F (y, z)). If F (x, z) = z, we have F (F (x, y), z) = F (F (x, y), F (x, z)) = F (F (x, x), F (y, z)) = F (x, F (y, z)).
This shows that F is associative. □ Remark 8.

(a) Using Lemma 20 we immediately see that if an operation is conservative and symmetric, then it is associative if and only if it is bisymmetric. Combining this observation with Proposition 16 provides a test for bisymmetry under conservativeness and symmetry. (b) We observe that the conjunction of bisymmetry and symmetry implies neither associativity nor the existence of a neutral element (take for instance the arithmetic mean over the reals). (c) Also, the conjunction of conservativeness and associativity does not imply bisymmetry, even in the presence of a neutral element. We give in Figure 13 an operation that is conservative, associative, and has 1 as the neutral element. However it is not bisymmetric since

F (F (1, 2), F (3, 2)) ≠ F (F (1, 3), F (2, 2)).
Using Lemma 20, we immediately derive the following two characterizations. The first one was already established in [START_REF] Su | On the discrete bisymmetry[END_REF]Corollary 1] in the special setting of aggregation operators (i.e., operations F ∶ L 2 n → L n that are nondecreasing and that satisfy F (1, 1) = 1 and Since Theorem 21 and Corollary 22 are derived from Lemma 20 only, we immediately see that these results still hold on any chain, or even on any ordered set, provided the discrete uninorm is replaced with a uninorm (i.e., a binary operation that is associative, symmetric, nondecreasing, and has a neutral element). Figure 14. An operation that is neither associative nor symmetric Remark 9. We observe that bisymmetry is necessary in Corollary 22. For instance the operation F ∶ L 2 4 → L 4 whose contour plot is depicted in Figure 14 is conservative, nondecreasing, and has a neutral element. However, it is neither associative nor symmetric.

F (n, n) = n). Theorem 21. An operation F ∶ L 2 n → L n is bisymmetric,

Concluding remarks

In this paper we established three main characterizations of the class of idempotent discrete uninorms on finite chains. Two of them are of axiomatic nature (Theorem 12 and Corollary 22) while the third one is of graphical nature (Theorem 15). These axiomatic characterizations are essentially based on conservativeness, which is a rather strong property that can be easily justified in some contexts (see Remark 1). The graphical characterization is a rather surprising result that shows that the idempotent discrete uninorms on a given finite chain can be very easily generated from a graphical viewpoint. This result contrasts with the rather intricate descriptive characterization given in Theorem 9.

In this work we put a particular emphasis on the graphical properties of operations by looking into their contour plots. In particular, we have presented graphical tests to verify whether an operation:

• is conservative (Proposition 6),

• has a neutral element (Proposition 7),

• is an idempotent discrete uninorm (Theorem 15), and whether a conservative operation:

• has a neutral element (Proposition 4),

• is associative (Proposition 16). In view of these results, some questions emerge naturally, and we end this paper by listing a few below.

(a) Enumerate and/or generate all the conservative (resp. conservative and associative, etc.) operations on a finite chain. (b) Provide a graphical test for checking whether a conservative operation is bisymmetric. Using Remark 8(a) we already have such a test for symmetric operations.

(c) Knowing that F ∶ L 2 n → L n is bisymmetric and symmetric, does it imply that it is associative and that it has a neutral element? We know from Remark 8(b) that this is not true on an arbitrary chain.

  , b), and F (x, y) = b, otherwise. Then (a, b) is isolated and a ≠ b. The contour plot of F is represented in Figure 1. Here and throughout, connected points are joined by edges. To keep the figures simple we sometimes omit the edges obtained by transitivity.

Figure 1 .Figure 2 .

 12 Figure 1. A non-idempotent operation

Figure 5 .

 5 Figure 5. Possible operations when n = 2

Figure 6 . 3 -

 63 Figure 6. Possible operations when n = 3

Figure 7 .Figure 8 . 3 Figure 9 .

 7839 Figure 7. An operation that is not nondecreasing

Figure 11 .

 11 Figure 11. Two possible idempotent discrete uninorms

Proof.Example 17 .Figure 12 .Proposition 18 .

 171218 Figure 12. An associative operation (left) and the six rectangles to be checked (right)

( a )Figure 13 .

 a13 Figure 13. An associative operation that is not bisymmetric

  min. The other identity can be proved similarly.

□ Proposition 11. If F ∶ L 2

n → L n is conservative, symmetric, and nondecreasing, then it is associative and it has a neutral element.

  and nondecreasing if and only if it is an idempotent discrete uninorm. Figure 10. Illustration of the proof of Theorem 14 Corollary 13. Let F ∶ L 2 n → L n be an operation that is associative, symmetric, nondecreasing, and has a neutral element. Then F is idempotent if and only if it is conservative. Proof. (Necessity) By definition, F is an idempotent discrete uninorm, and hence it is conservative.

	s		s	s	s
					s
		F | L ′
	s		F | L ′
	s	s	s		s
	Proof. (Necessity) The result immediately follows from Proposition 11.
	(Sufficiency) By definition, any idempotent discrete uninorm is symmetric and
	nondecreasing. It is also conservative. Indeed, this follows from Theorem 9 if
	1 < e < n and from Lemma 10 if e = 1 or e = n (for an alternative proof see
	Remark 6).				□

  nondecreasing, and has a neutral element if and only if it is a discrete uninorm.

	Corollary 22. An operation F ∶ L 2 n → L n is idempotent (or conservative), bisym-
	metric, nondecreasing, and has a neutral element if and only if it is an idempotent
	discrete uninorm.

Acknowledgments

The authors would like to thank Gergely Kiss for fruitful discussions and valuable remarks. This research is partly supported by the internal research project R-AGR-0500-MRO3 of the University of Luxembourg.


Franc Eric Marchand

¸ois Chaumette

Visual Servoing through mirror reflection

Apart the use of catadioptric cameras, only few visual servoing works exploit the use of mirror. Such a configuration is however interesting since it allows to overpass the limited camera field of view. Based on the known projection equations involved in such a system, this paper introduces the theoretical background that allows the use of planar mirror for visual servoing in different configurations. Limitations intrinsic to such systems, such as the number of d.o.f actually controllable, is then discussed. Experiments using a mirror mounted on the end-effector of a 6 d.o.f robot validate the proposed approach.

I. INTRODUCTION

Visual servoing [START_REF] Chaumette | Visual servoing and visual tracking[END_REF] has proved to be a very efficient way to control robot using information provided by one or multiple cameras. Vision sensors have the main advantage to get a rich information of the robot environment. However standard perspective cameras have a limited field of view. Considering mirrors may extend the camera field of view. This is usually achieved with catadioptric camera [START_REF] Mariottini | Image-based visual servoing with central catadioptric cameras[END_REF] or with controlled mirror such as in the recent Saccade robot [START_REF] Iida | Saccade mirror 3: High-speed gaze controller with ultra wide gaze control range using triple rotational mirrors[END_REF]. This paper deals with the combination of visual servoing and a planar mirror. Although coupling visual servoing and mirror within catadioptric system (omnidirectional camera) has been seen as a powerful option, few research papers consider the use of mirrors in visual servoing when the relative position between the camera and mirror varies.

Most of the literature in this area is related to the use of catadioptric camera e.g. [START_REF] Barreto | Visual servoing/tracking using central catadioptric images[END_REF][13] [START_REF] Hadj Abdelkader | Imagebased control of mobile robot with central catadioptric cameras[END_REF]. In such systems, the camera and the mirror are rigidly attached and the catadioptric system is mounted on the robot. The main difference between such system and a classical visual servoing system is the projection model (central projection model) and consequently the interaction matrix. The resulting control has also proved to feature good decoupling properties for some particular visual features [START_REF] Tahri | Robust imagebased visual servoing using invariant visual information[END_REF].

The use of planar mirrors has been considered to simulate multi-camera system. Indeed, with a mirror, it is possible to obtain a stereo view using a single camera (planar catadioptric stereo [START_REF] Mariottini | Planar catadioptric stereo: single and multi-view geometry for calibration and localization[END_REF]). Although such a system may allow to provide useful information for robot control, the mirrors positions are not controlled. Multiple stationary planar mirrors systems for micro alignment of optical fiber using a visual servoing techniques was also reported in [START_REF] Mikawa | Visual servoing for micro mass axis alignment device[END_REF]. The control of the alignment process is achieved in closed loop using the projection and reflection of the optical fiber on both mirrors. Eric Marchand is with Université de Rennes 1, IRISA, Inria Rennes, Rennes, France

Eric.Marchand@irisa.fr.

Franc ¸ois Chaumette is with Inria, IRISA, Rennes, France

Francois.Chaumette@inria.fr Such visual servoing approach avoid precise calibration of the mirror-based stereo system. A more classical visual servoing system is reported in [START_REF] Kulpate | Precise 3d positioning of a robotic arm using a single camera and a flat mirror[END_REF] [START_REF] Kulpate | An eye-to-hand visual servoing structure for 3d positioning of a robotic arm using one camera and a flat mirror[END_REF] where the motion of the robot end-effector is controlled through its reflection in a mirror. Let us note that in this work, the mirror and the camera are stationary. We will see that such a system is in fact very similar to a classical visual servoing structure.

In all previous works, the mirrors are motionless and the relative pose between the camera and the mirror(s) is fixed. Few works considered mobile mirrors or a varying mirror/camera position. Laser steering application for microsurgery is reported in [START_REF] Tamadazte | Weakly calibrated stereoscopic visual servoing for laser steering: Application to phonomicrosurgery[END_REF], [START_REF] Andreff | Laser steering using virtual trifocal visual servoing[END_REF]. A laser spot controlled thanks to the motion of a 2 d.o.f mirror is achieved using a stereo visual servoing approach. In the context of high speed visual servoing, some systems that consider mirrors, have been proposed to achieve high-speed gaze control: Saccade mirror [START_REF] Okumura | High-speed gaze controller for millisecond-order pan/tilt camera[END_REF], [START_REF] Iida | Saccade mirror 3: High-speed gaze controller with ultra wide gaze control range using triple rotational mirrors[END_REF]. These systems consider two [START_REF] Okumura | High-speed gaze controller for millisecond-order pan/tilt camera[END_REF] or three [START_REF] Iida | Saccade mirror 3: High-speed gaze controller with ultra wide gaze control range using triple rotational mirrors[END_REF] one d.o.f mirror to provide fast tracking capabilities (1000Hz) of moving target. The proposed Saccade Mirror system allows to simulate a pan/tilt camera while allowing very fast saccade and increasing the field of view of the camera. Open loop control law is considered for the tracking process.

In this paper we provide the theoretical background required for the development of closed-loop visual servoing. We consider the three possible configurations which may allow to either control a moving camera, a moving target attached to the robot or a moving mirror. In all these cases the target is observed by the camera through its reflection in the mirror. After recalling the reflection and the resulting projection equations, we will see that the two former configurations can be easily formulated using classical visual servoing control law. The last case, that is controlling the mirror, is the key contribution of this paper. We will exhibit the interaction matrix that links the motion of a point in the image to the motion of the mirror. We will also demonstrate that in practice only 3 mirror d.o.f. are actually controllable. We will thus propose to use the remaining d.o.f. to cope with the issue induced by the fact that the mirror has not an infinite size. Experiments using a mirror mounted on the end-effector of a 6 d.o.f robot validates the proposed approaches.

II. REFLECTION AND PROJECTION MODELING

Let us define the mirror frame such that x and y axes are in the mirror plane Π and z axis is perpendicular to the mirror (see Figure 1). The equation of the mirror in this frame is given by:

m n m X = 0 (1)
where m n = (0 0 1) is the normal to the mirror Π expressed in the mirror frame. Let m X R be the symmetric of the point m X wrt the mirror. m X R coordinates are given by (eg, [START_REF] Mariottini | Planar catadioptric stereo: single and multi-view geometry for calibration and localization[END_REF]):

m X R = (I -2 m n m n ) m X (2)
Note that this equation is true whatever the choice of m n (subject that m n = 1) if the origin of F m belongs to plane Π. Let c X be the coordinates of a point in the camera frame F c . Using classical notations for rotation and translation,we have

c X = c R m m X + c t m . (3)
Let

n c X = d (4)
be the equation of the mirror plane in the camera frame where n is the unit vector that represents the normal to the plane in the camera frame and d is the orthogonal distance between the camera and the plane (see Figure 1). We thus have n = c R m m n and c t m = dn. The coordinates of the reflection c X R of c X wrt. the mirror is given by [START_REF] Mariottini | Planar catadioptric stereo: single and multi-view geometry for calibration and localization[END_REF] (see proof in Appendix A):

c X R = (I -2nn ) c X + 2dn (5)
Finally, the projection x of c X R in the image plane is given by:

x = c X R / c Z R ( 6
)
where c Z R is the third component of c X R .

III. VISUAL SERVOING THROUGH MIRROR

Let us consider a basic visual servoing that is specified by observing a set of points x i at given positions in the image x * i . Thus we want to minimize the norm of the error e defined by:

e =     . . . x i -x * i . . .     (7)
From this quite general formulation of the problem one has to consider three different configurations of the triplet camera, mirror and target: A the mirror is mounted on the robot end-effector (see Figure 2). B the camera is mounted on the robot-end-effector (see Figure 4). C the target is mounted on the robot-end-effector (see Figure 5). In the three cases, one wants to observe a target at a given position in the image through its reflection in the mirror. In any case, to design the control law one has to determine the Jacobian J x that links the motion of each point in the image to the motion of the end effector in the joint space, that is

ẋ = J x q (8)
where q is the robot joint velocity. In that case, considering that we want an exponential decrease of e, the control law is given by:

q = -λJ + x e (9)
To determine J x , one has to derive the relation (the interaction matrix [START_REF] Chaumette | Visual servoing and visual tracking[END_REF]) that links the motion ẋ of a point x to the motion of the mirror (A), the camera (B), or the target (C). This is the purpose of the two next sections.

IV. CONTROLLING THE MIRROR POSITION

This case is actually the most interesting one. In that case the mirror is mounted on the end effector of the robot while the camera is motionless in the scene (see Figure 2).

A. Modeling issue and control law

As already said, one has to define the relation (the interaction matrix) that links the motion ẋ of an image point x to the motion of the mirror. From equation ( 6), deriving the normalized coordinates x, we can write:

ẋ = c ẊR c Z R - c X R c Z 2 R ċZ R = c ẊR c Z R - x c Z R ċZ R (10)
where ċZ R is the third component of c ẊR . As the main result of this paper, the interaction matrix that links the motion c ẊR of c X R to the mirror motion c v m expressed in the camera frame ( c v m = ( c v m , c ω m ) where c v m is the mirror translational velocity and c ω m is the mirror rotational velocity) is given by:

c ẊR = L v L ω c v m c ω m (11)
with

L v = -2nn (12)
and

L ω = -2 d[n] × + [ c X R ] × [n] 2 × ( 13
)
where [a] × denotes the skew symmetric matrix associated to the vector a.

Proof. Only mirror is moving, thus, from [START_REF] Hadj Abdelkader | Imagebased control of mobile robot with central catadioptric cameras[END_REF], one can see that

c ẊR = 2n ḋ + 2 ṅd -2( ṅn + n ṅ ) c X ( 14
)
where ṅ and ḋ are the motion of the mirror parameters expressed in the camera frame. Since c X is unknown, it is more convenient to express c ẊR wrt. c X R ( c X R is a priori also unknown, but its projection can be observed and measured in the image, thus only its depth c Z R is in fact unknown). From (5), since c X R and c X are symmetric with respect to the mirror plane, we have:

c X = (I -2nn ) c X R + 2dn (15)
Injecting ( 15) in ( 14), it can be demonstrated that (see proof in appendix B):

c ẊR = 2n ḋ -2 ṅd + 2( ṅn -n ṅ ) c X R ( 16
)
The interaction matrix related to a plane given by ( 4) is given by [START_REF] Urban | Real-time visual servoing using controlled illumination[END_REF][16]:

ṅḋ = 0 3×3 [n] × -n 0 1×3 c v m c ω m (17)
According to equation [START_REF] Remazeilles | Qualitative visual servoing: Application to the visibility constraint[END_REF] we have

2n ḋ = 2n(-n 0 1×3 ) c v m c ω m = -2nn c v m (18)
and

2 ṅd = 2 d (0 3×3 [n] × ) c v m c ω m = 2 d [n] × c ω m (19)
and finally, ṅnn ṅ being an anti-symmetric matrix, it can be demonstrated that

ṅn -n ṅ = [n] × ṅ × = [n] × [n] × c ω m × = [n] 2 × c ω m × ( 20
)
leading to

( ṅn -n ṅ ) c X R = [n] 2 × c ω m × c X R = -[ c X R ] × [n] 2 × c ω m (21)
Considering ( 18), ( 19) and ( 21) in ( 16) leads to the result given in ( 12) and [START_REF] Mariottini | Image-based visual servoing with central catadioptric cameras[END_REF].

Injecting [START_REF] Marchand | ViSP for visual servoing: a generic software platform with a wide class of robot control skills[END_REF] in [START_REF] Marchand | Addendum to "visual servoing through mirror reflection[END_REF] leads to the interaction matrix L x that is defined by:

ẋ = L x c v m c ω m (22)
For this configuration, it is convenient to control the motion of the mirror in the mirror frame F m , we thus have:

ẋ = L x c V m m v m m ω m (23)
with c V m the spatial motion transform matrix from the camera frame to the mirror frame 1 . For an eye-to-hand system J x , introduced in (8), can be expressed as [START_REF] Chaumette | Visual servoing and visual tracking[END_REF]:

J x = -L x c V m m V e e J(q) ( 25
)
where the classical robot Jacobian e J(q) is expressed in the end effector frame, m V e is the spatial motion transform matrix from the mirror frame to the end effector frame. It is a constant matrix as soon as the mirror is rigidly attached to the end effector. Note that, to use (25), the pose of the mirror in the camera frame has to be estimated at each iteration of the control law. Alternatively, one can consider the robot base frame F ∅ and the Jacobian given by [START_REF] Chaumette | Visual servoing and visual tracking[END_REF]:

J x = -L x c V ∅ ∅ V e e J(q) ( 26
)
where c V ∅ can be computed off-line in a calibration step and ∅ V e is obtained thanks to the robot odometry. This latter formulation will be considered in the reported experiments.

B. Actual number of controllable d.o.f

As expected from such a configuration, motions in the mirror plane (ie, along x and y axes and around z axis of F m ) do not modify the position of x in the image (see Figure 3). When the system is perfectly calibrated first, second and sixth column of L x c V m are equal to zero. That is: Proof. From (10), rank

rank L x c V m = 3. (27)
L x c V m = rank L v L ω c V m . L v L ω c R m d[n] × c R m 0 3×3 c R m = -2 L A L B
1 Spatial transform matrix a V b allows transforming velocities expressed in frame F b to velocities expressed in Fa. It is given by:

a V b = a R b [ a t b ] × a R b 0 3×3 a R b . (24)
with, on one hand,

L A = nn c R m = n m n
Since by definition m n = (0 0 1) and denoting n = (n 1 n 2 n 3 )

L A =   0 0 n 1 0 0 n 2 0 0 n 3   . ( 28
)
On the other hand, we have:

L B = dnn [n] × c R m + d[n] × c R m + [ c X R ] × [n] 2 × c R m = (dI 3 + [ c X R ] × [n] × )[n] × c R m since n [n] × = 0 = (dI 3 + [ c X R ] × [n] × ) c R m [ m n] × (29) since [n] × c R m = c R m [ c R m n] × = c R m [ m R c n] × = c R m [ m n] × . Then, from (29) L B = (d c R m + [ c X R ] × c R m [ m n] × )[ m n] × (30)
Expending [ m n] × with m n = (0 0 1) it is immediate to see that the third column of L B is (0 0 0) .

Thus in practice, only 3 d.o.f. of the mirror produce to motion in the image (see Figure 3). Nevertheless, the other d.o.f of the mirror can be considered to achieve other tasks such as maintaining the target visible in the mirror.

C. Using remaining d.o.f to maintain target in the mirror

Let us consider here that we control the robot in the mirror frame. The Jacobian is then given by (23): J x = -L x c V m and the control law is given by:

v m = -λJ + x e (31)
As stated above, rank J x = 3, three dof remain available to achieve other tasks, which can be done using the redundancy framework [START_REF] Chaumette | Visual servoing and visual tracking[END_REF], [START_REF] Chaumette | A new redundancy-based iterative scheme for avoiding joint limits: application to visual servoing[END_REF].

v m = -λJ + x e + P x J + e2 e 2 (32)
with P x = I -J + x J x . Our goal is to maintain the target in the image. Indeed, the mirror is not of infinite size and some motion may imply that the target reflection moves out the limit of the mirror. To maintain the target reflection in the image, a simple way is to control the remaining degrees of freedom of the mirror to maintain the target reflection in the center of the mirror. Let xM be the center of gravity of the mirror and x the center of gravity of the target in the image. The cost function e 2 is then defined as

e 2 = β(x M -x) ( 33
)
where β is a positive scalar. Note that x, being the reflected point, is invariant to the motion of the mirror involved by this secondary task. Indeed P x belong to the kernel Ker J x . Thus, as demonstrated in the previous paragraph, the motion generated J + e2 e 2 has no effect on the projection of the point in the image. x being invariant, the Jacobian J e2 only links the motion of xM to the motion of the mirror. It is given by the classical interaction matrix related to the point xM [START_REF] Chaumette | Visual servoing and visual tracking[END_REF].

With the proposed scheme, target reflection is maintained in the center of the mirror. It is however possible to relax this constraint, for example by allowing the center of the target reflection to be inside an allowed range inside the mirror following the same strategy as in [START_REF] Remazeilles | Qualitative visual servoing: Application to the visibility constraint[END_REF].

V. SIMPLE CASES OF A MOTIONLESS MIRROR

A. Case of a controlled camera

In that case the camera is mounted on the end effector of the robot. As can be seen on Figure 4, this is a classical visual servoing configuration [START_REF] Chaumette | Visual servoing and visual tracking[END_REF] with respect to c X R . The corresponding interaction matrix L x is thus classically defined by ẋ = L x c v c where c v c = ( c v c , c ω c ) are the translational and rotational velocity of the camera expressed in the camera frame F c . L x is then given by:

L x = -1/Z R 0 x/Z R xy -(1 + x 2 ) y 0 -1/Z R y/Z R 1 + y 2 -xy -x ( 34
)
The proof is given in [START_REF] Marchand | Addendum to "visual servoing through mirror reflection[END_REF]. It has however to be noted that, here, the visual feature is related to a virtual point X R . Indeed X R is not a physical point but the symmetric of X with respect to the mirror. Although the coordinates x = (x y) can be directly measured in the image, the depth Z R is obviously unknown but, it can be noted that this is also usually the case for a classical visual servoing process. In that case the Jacobian J x defined in ( 9) is given by [START_REF] Chaumette | Visual servoing and visual tracking[END_REF]:

J x = L x c V e e J(q) ( 35
)
where e J(q) is the classical robot Jacobian expressed in the end effector frame and c V e is the spatial motion transform matrix from the camera frame to the end effector frame. It is a constant matrix as soon as the camera is rigidly attached to the end effector.

B. Case of a controlled target

In this last case the mirror and camera are motionless and the target is mounted on the end effector of the robot (see Figure 5). This case was reported in [START_REF] Kulpate | An eye-to-hand visual servoing structure for 3d positioning of a robotic arm using one camera and a flat mirror[END_REF][9] [START_REF] Mikawa | Visual servoing for micro mass axis alignment device[END_REF]. This case defines in fact a classical eye-to-hand configuration for which the Jacobian J x can be expressed as [START_REF] Chaumette | Visual servoing and visual tracking[END_REF]:

J x = -L x o V e e J(q) ( 36
)
where L x has been defined in (34) and o V e is the spatial motion transform matrix from the target frame to the end effector frame. It is a constant matrix as soon as the target is rigidly attached to the end effector.

As in the previous case the visual feature is related to a virtual point c X R rather directly c X. Let us note again that the depth Z R is obviously unknown.

VI. EXPERIMENTAL RESULTS

In this section, we consider only the case of a moving mirror. In Section V we demonstrated that the cases of motionless mirrors are equivalent to a classical visual servoing scheme. Therefore, as this is a well known approach, we do not show results for these cases and we focus on the original problem of a controlled mirror.

A. Experiments with mirror control: experimental setup

Figure 6 shows the experimental setup. The mirror is mounted on the end-effector of a 6 d.o.f gantry robot. The camera (which is fully calibrated) is static and looks at the mirror and (thanks to the mirror reflection) to the target. The implementation has be done thanks to the ViSP software [START_REF] Marchand | ViSP for visual servoing: a generic software platform with a wide class of robot control skills[END_REF]. The relative position between the camera and the robot base frame is computed thanks to robot odometry and the localization of the mirror wrt the camera using a nonlinear pose estimation method [START_REF] Marchand | Pose estimation for augmented reality: a hands-on survey[END_REF] using four points initially located on the mirror. This is done in an off-line calibration step to compute the constant matrix c V ∅ (see (26)). The dots (see Figure 6 (left)) are then removed. Note that the measure of the mirror parameters (reported, eg, in Figure 9.c) are computed thanks to robot odometry and the knowledge of c T ∅ obtained during the off-line calibration step. These values are indeed involved in ( 12), [START_REF] Mariottini | Image-based visual servoing with central catadioptric cameras[END_REF] and thus in (26).

B. Servo on one point

This first experiment depicts a basic visual servoing task. The reflection of the target has to be seen in the middle of the image (see Figure 7.a) using the control law given by (26). As expected, 3 d.o.f are actually involved to achieve the task (see Figure 7.c). In this experiment mainly the orientation of the mirror is involved but a small motion along the Z axis can be also observed (see Figure 7.c). The depth c Z R involved in [START_REF] Marchand | Addendum to "visual servoing through mirror reflection[END_REF] has been set to 2.5m, which roughly correspond to the double of the distance between the camera and the mirror. The errors feature an exponential decay (see

C. Servo on four points

In this case we consider a target made of four points. The initial and final (desired) image are shown on Figure 9. As stated in Section IV-B, 3 d.o.f are controlled. Here not only the orientation of the mirror is involved but its depth (in the camera frame) as well. This can be clearly seen on Figure 9, which depicts mirror orientation and depth. The mirror converges toward a desired pose (up to a translation in the mirror planeand a rotation around its normal). The depth c Z R involved in [START_REF] Marchand | Addendum to "visual servoing through mirror reflection[END_REF] is coarsely set to 4m. It can be seen on Figure 9.d than the camera moves backward 50cm to reach a desired position. As stated the measure of the mirror parameters (Figure 9.d) are computed thanks to robot odometry and the knowledge of c T ∅ obtained during the offline calibration step.

Again due to the fact that rank J x = 3 the target may leave the mirror (which is not of infinite size). This is easy to see on Figure 10 that if the mirror does not move down, the target will no longer be on the mirror at convergence. Here a secondary task is thus considered to maintain the center of gravity (COG) of the target at the COG of the mirror (see Section IV-C). In that case all the mirror d.o.f are considered to achieve the global task (mainly a translation along x and y in F m -see Figure 10c).

VII. CONCLUSION

In this paper we investigated visual servoing through mirror reflection and proposed the theoretical background necessary to derive the resulting control law. We demonstrated that when the mirror is static, the problem is very similar to the classical one (with no mirror). In the case where the mirror is moving, we exhibit the analytical form of the interaction matrix and the resulting control law. We demonstrated that, in that case, only

c X R = c R m m X R + dn = c R m (I -2 m n m n ) m X + dn = c R m ( m R c c R m -2 m R c nn m R c ) m X + dn = ( c R m -2nn c R m )( m

Fig. 1 .

 1 Fig. 1. Reflection and projection with a planar mirror

Fig. 2 .

 2 Fig. 2. Case of a controlled mirror

Fig. 3 .

 3 Fig. 3. Invariance of object reflection to in-plane mirror motions

Fig. 4 .

 4 Fig. 4. Case of a controlled camera.

Fig. 5 .

 5 Fig. 5. Case of controlled target.

Fig. 6 .

 6 Fig. 6. Experimental setup: the mirror (red arrows) is mounted on the robot end effector; the camera (blue circles) observes the target (green (green) as its reflection in the mirror. The calibration dots are used to determine the position of the camera in the robot reference frame in an off-line calibration step. They are removed for the visual servoing experiments as can be shown in, eg, Figure 7.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Servo on one point: (a) initial position (left) and desired (and final) position (right). Video is available through youtube here and in the accompanying video ; (b) error xx * , (c) mirror velocity in mirror frame ( m vm, m ωm) in m/s and rad/s. Note that (vx, vy and ωz are always equal to 0 as expected

Fig. 9 .

 9 Fig. 9. Servo on 4 points. Camera view: initial position (left) and desired (and final) position on the (right). Videos are available through youtube: external view and camera view and in the accompanying video ; (b) error x i -x * i , i = 1..4, (c) mirror velocity in mirror frame ( m vm, m ωm), (d) mirror equation (n, d) with n = (n 1 , n 2 , n 3 ). Note that only 3 d.o.f are involved to achieve the task (see Section IV-B).

  efficiently considered and propose solutions to use efficiently the remaining dof. Experiments has been achieved with a Gantry robot to show the validity of our approach in the case of controlled mirror.APPENDIXA. Proof of equation[START_REF] Iida | Saccade mirror 3: High-speed gaze controller with ultra wide gaze control range using triple rotational mirrors[END_REF] Let us prove thatc X R = (I -2nn ) c X + 2dnProof. Considering (3) and[START_REF] Barreto | Visual servoing/tracking using central catadioptric images[END_REF]

Fig. 10 .

 10 Fig. 10. Servo on 4 points with large displacement with the use of a secondary task as proposed in section IV-C. This requires the use of the secondary task proposed in Section IV-C: (a) initial position (left) and desired (and final) position (right). Video is available through youtube here and in the accompanying video. Servo on 4 points : (b) error x i -x * i , i = 1..4, (c) mirror velocities ( m vm, m ωm) . Note that in that case all the d.o.f. are used to maintain the target in the mirror.


Rémy Abergel

Saïd Ladjal

Florence Tupin

Jean-Marie Nicolas

A COMPLEX SPECTRUM BASED SAR IMAGE RESAMPLING METHOD WITH RESTRICTED TARGET SIDELOBES AND STATISTICS PRESERVATION

Keywords: Shannon interpolation, complex spectrum, total variation, targets, subpixellic image processing

The aim of this work is to present a resampling scheme for SAR images that preserves spatial resolution and produces statistically accurate images at the same time. Indeed, SAR images are, for reasons due to their acquisition process, well sampled signals according to the Shannon sampling theory. In the presence of strong responses, that we will refer to as targets, a sinc-like function centered at the target is smeared over the entire image and is particularly visible in the range of tens of pixels surrounding the target. To mitigate this phenomenon, the usual solution is to apply an apodization window in the Fourier domain so as to change the cardinal sine impulse response into a much rapidly decaying one. This approach has two major drawbacks. It reduces the resolution of the image and introduces inaccurate statistical dependency between pixels. We propose to resample the image in an adaptive and robust way so that the target smear is canceled and the new sampled image is completely faithful to the underlying signal.

INTRODUCTION

SAR images are provided by complex signal processing being at the heart of the SAR technique (range compression, SAR synthesis). The raw data received by the antenna before these operations are usually not provided by space agencies. The provided Single Look Complex data (SLC) are affected by two important factors that can be seen in the complex Fourier spectrum of the image: over-sampling and weighting of the azimuth and range spectrum [START_REF] Massonnet | Imaging with Synthetic Aperture Radar[END_REF]. These factors can change depending on the data provider even for similar resolutions of the SLC images. For instance, the weighting functions of TerraSAR-X and CSK images are different. These processing have a strong impact on the appearance of the images (spreading of the strong targets) and induce a correlation between neighboring pixels, which can affect further processing like physical parameter estimation [START_REF] Ch | NL-SAR: a unified Non-Local framework for resolutionpreserving (Pol)(In)SAR denoising[END_REF]. In this paper we investigate how the complex spectrum information can be taken into account to improve SAR images by unweighting images while limiting sidelobes of the impulse response of strong targets, and we show how to produce an image suitable both for visualization and further processing. The paper is organized as follows. Section 2 introduces the notations and gives a method to cancel apodization when the weighting function is unknown. Section 3 presents the proposed resampling scheme and is concluded by a proof of statistical accuracy of the result under a reasonable model.

∀(α, β) ∈ Z 2 , u(α, β) = (k,l)∈Ω u(k, l) e 2iπ( αk M + βl N ) . (1)
We note

IK = -K 2 , K 2
∩ Z, then we set Ω = IM × IN , which represents the cannonical frequency domain associated to Ω and is also a period of u.

We display in Fig. 1 the modulus of a SLC TerraSAR-X image u and the modulus of u. We see on that particular example that u is non-zero on a rectangular sub-frequency domain ω = Im × In ⊂ Ω (for some given integers m and n such as m ≤ M , n ≤ N ), and zero outside, showing that u was oversampled. Besides, it happens that the non-zero part of the Fourier spectrum is in fact apodized, which means that it resulted from a multiplication in the Fourier domain by a frequency attenuating function. This function results from the weighting affecting the antenna pattern and the weighting applied to the data [START_REF] Massonnet | Imaging with Synthetic Aperture Radar[END_REF] which depends on the data provider.

In the following, we assume that for any (α, β) ∈ Ω, we have

u(α, β) = u0(α, β) • γ(α, β) if(α, β) ∈ ω 0 otherwise, (2)
where ω = Im× In ⊂ Ω, γ : ω → R++ is a known frequency attenuating function and u0 : ω → C is called the pseudo-raw spectrum.

We call pseudo-raw image the complex-valued image u0 : ω → C with spatial domain ω = Im × In obtained by taking the inverse DFT of u0. Since u0 and u have different resolutions, the pseudoraw image will be compared to the image uω : ω → C obtained by removing the zero-padding from u, that is, the image defined in the Fourier domain by uω(α, β) = u(α, β) for all (α, β) ∈ ω.

Practical estimation of the pseudo-raw spectrum

Now, let us focus on the inversion of (2), that is, on the computation of the pseudo-raw spectrum u0. When the subfrequency domain ω and the frequency attenuating function γ are known (for instance provided by the spatial agency who generated the image) the relation (2) can be easily inverted and we get

∀(α, β) ∈ ω, u0(α, β) = u(α, β) γ(α, β) . (3)
When those pieces of information are unavailable or lost, as we assume in all the experiments that we propose in this document, they must be estimated. We make the reasonable assumption that the subfrequency domain ω can be easily retrieved by looking at the Fourier spectrum u, as it is the case in Fig. 1. In the case γ is not explicitly known, we propose to consider the following separable estimate given by

∀(α, β) ∈ ω, γ(α, β) = aγ • γ1(α) • γ2(β), (4)
where aγ ∈ R++ is a normalization factor whose setting is discussed below, and γ1, γ2 are the signals obtained by averaging | uω| in both directions, that is, by setting for all (α, β) ∈ ω,

γ1(α) = 1 n β∈ In | uω(α, β)| , γ2(β) = 1 m α∈ Im | uω(α, β)| ,
which yields two positively valued functions γ1 : Im → R++ and γ2 : In → R++ as soon as | uω| does not have a column or line being identically-zero. The normalization factor aγ is computed in order that |u0| and |uω| assume the same maximum over ω, which yields

aγ = max (k,l)∈ω |u 0 (k, l)| max (k,l)∈ω |uω(k, l)| , (5)
where

u 0 is such that u 0 (α, β) = uω(α, β)/(γ1(α) • γ2(β)) for all (α, β) ∈ ω.
Finally, u0 can by computed using (3), ( 4) and ( 5). In Fig. 2, we display a cropping of the amplitude images |uω| and |u0| computed from the image u of Fig. 1. As expected the unweighted image is better localized (finer localization of the information) at the price of a large spreading of strong targets in azimuth and range directions.

ADAPTATIVE RESAMPLING SCHEME FOR THE PSEUDO-RAW IMAGE

Model

We illustrate in Fig. 3 that the bright targets observed on the pseudoraw image can be very well approached by a two-dimensional cardinal sine function defined by (as given by the SAR processing [START_REF] Massonnet | Imaging with Synthetic Aperture Radar[END_REF]):

∀(x, y) ∈ R 2 , sinc(x, y) = sin(πx) πx • sin(πy) πy , (6)
modulus of uω modulus of u0 (pseudo-raw) Fig. 2. Comparison between apodized and pseudo-raw images. Removing the apodization from uω yields the pseudo-raw image u0, we display here the modulus of a subpart of those two images. We can see that the pseudo-raw image u0 shows a better level of details than uω, as well as a more precise localization of the information. However, in the presence of strong targets, the signal is polluted by horizontal and vertical patterns.

with the continuity-preserving condition sin(0) 0 = 1. An interpretation of this phenomenon is that the target is sufficiently narrow to be transformed, by the acquisition process, to the impulse response, yielding the cardinal sine function. When the position of the target does not coincide with the sampling grid, an oscillatory pattern peculiar to the sinc function pollutes the values of the pixels in the vicinity of the target, yielding (we drop the double indexes for the spatial coordinates in this section) an observed signal of the type

∀k ∈ ω, u0(k) = A sinc(k -(k0 + δ)) + u * 0 (k) , (7)
where A ∈ C denotes the target amplitude, k0

+ δ (with k0 ∈ ω, δ ∈ [-1 2 , 1 2 ]
2 ) the subpixellic position of the target, and u * 0 the signal that would be acquired in the absence of the target.

The obvious solution to this problem is to resample the image on a grid such that the coordinates of the target are integers, thus suppressing the side lobes contributions. In order to do that, one has to estimate the subpixellic position of the target and then perform a translation of the image by the vector t = -δ. Indeed, noting U0 : R 2 → C the Shannon interpolate of u0 which is computed as U0 = U r 0 + i • U i 0 , where U r 0 and U i 0 denote the (real-valued) Shannon interpolates of the real and imaginary parts of u0 (see for instance [START_REF] Abergel | The Shannon Total Variation[END_REF] for the explicit definition), the translated signal is v0 :

k → U0(k + δ) and satisfies ∀k ∈ ω, v0(k) = U * 0 (k + δ) + A if k = k0 0 otherwise, (8)
where U * 0 denotes the Shannon interpolate of u * 0 . We see that, contrary to u0, the resampled signal v0 is not polluted anymore by the oscillations of the cardinal sine.

Since in practice, there may and will be numerous targets in a single image, a global translation will not be sufficient to accommodate all the targets of the image. For that reason, we propose to devise a local scheme to compute, at each position in the image, the translation that reduces the interference of the target-induced cardinal sine. Indeed, contrary to [START_REF] Li | Implementation of the RELAX algorithm[END_REF][START_REF] Tsao | Reduction of Sidelobe and Speckle Artifacts in Microwave Imaging: the CLEAN technique[END_REF], we made the choice to not explicitly detect targets to keep the process as robust as possible.

Local displacement vector field

Our approach consists in computing from u0 a dense field of displacements T = (Tx, Ty)

: ω → [-1 2 , 1 2 ] × [-1 2 , 1 2 ]
, and resample Fig. 3. Resampling targets at the subpixellic scale. We display in (a) the modulus of the pseudo-raw image u0 in the vinicity of a strong target. We display in (b) and (c) the modulus of the image u0 resampled by means of a translation of vector t = (0.3, 0) and t = (0.3, 0.1) respectively, yielding more localized signals. We display in the second row a plot of the real part of an horizontal section of u0, yielding the red dashed curve, and we do the same for the resampled signal displayed in (b), yielding the blue dotted curve. The green plain curve represents a pure cardinal sine, that is a function of the type x → A sinc(x) where A ∈ R. We see that the samples of both red and blue signals match very well with a sampling of the cardinal sine, the same observation holds for the imaginary parts of the considered signals.

the image u0 over the locally translated grid, which means to compute the image v0 : ω → C defined by

∀(k, l) ∈ ω, v0(k, l) = U0(k -Tx(k, l), l -Ty(k, l)) . (9)
In practice, the two components Tx and Ty of T will be estimated independently, we describe here the computation of Tx, that of Ty being totally similar. Given a locality parameter K ∈ N (we took K = 25 in all our experiments) and a position (k0, l0) ∈ ω, we associate to each translation tx ∈ [- 1 2 , 1 2 ] the mono-dimensional signal vt x ∈ C 2K+1 corresponding to the restriction of the horizontally translated signal (x, y) → U0(x -tx, y) to the horizontal 2K + 1 neighborhood of the pixel (k0, l0), that is,

∀p ∈ [-K, K] ∩ Z, vt x (p) = U0(k0 -p -tx, l0) .
We propose to select among all candidate translations the one that minimizes a particular cost function tx → J(vt x ) where J :

C 2K+1 → R, yielding Tx(k0, l0) = argmin - 1 2 ≤tx< 1 2 J(vt x ) . ( 10
)
We considered three different choices for J, denoted below as J1, J2 and J3.

(i) J1(vt x ) = -max p∈[-K,K] |vt x (p)|.
The idea underlying this choice is that when a target exists in the vicinity of (k0, l0), the appropriate translation is found by looking for the signal vt x having the highest maximal modulus (as it is for instance the case for the blue dotted curve in Fig. 3).

(ii) J2(vt x ) = TV d (v r tx ) + TV d (v i tx ), noting v r tx and v i tx the real and imaginary parts of vt x , and TV d the discrete total variation operator defined by

∀w ∈ R 2K+1 , TV d (w) = -K≤p<K |w(p + 1) -w(p)| .
The idea is that, when sampled on the appropriate grid, the discrete total variation of a target-induced cardinal sine is minimal, whereas it is always higher for all non integer displacements of the grid (the red dashed curve in Fig. 3 is more oscillatory than the blue dotted curve and exhibits a higher discrete total variation).

(iii) J3(vt x ) = TV d mask (v r tx ) + TV d mask (v i tx ), TV d mask being de- fined for any w ∈ R 2K+1 by TV d mask (w) = -K≤p<K p ∈{p 0 -1,p 0 } |w(p + 1) -w(p)| ,
where p0 denotes the position where |w| is maximal. Compared to the previous criterion, we choose to mask the contribution of the brightest pixel to the total variation so that the TV d mask of a pure cardinal sine function sampled on the appropriate grid is zero.

Since our numerical expriments revealed that the third choice led to the most satisfying results, it was systematically used in all the experimental results displayed below. The computation of the resampled image v0 defined by (9) from the pseudo-raw image u0 is summarized in Algorithm 1, and some experimental results are displayed and commented in Fig. 4 and Fig. 5.

Algorithm 1: pseudo-raw image resampling

Input: a pseudo-raw image u0 : ω → C, a locality parameter K, a discrete set of NT candidate translations T = -1 2 + 1 N T • {0, . . . , NT -1}, and a cost function J (in all our experiments, we took K = 25, NT = 20 and J = J3 defined in Section 3.2).

Output: a resampled pseudo-raw image v0 : ω → C, and a translation map T : ω → T × T.

Initialization: precompute the horizontally and vertically translated signals

v x t = U0(ω -(t, 0)), v y t = U0(ω -(0, t)) for all t ∈ T. for (k, l) ∈ ω do ∆ωx ← (k + [-K, K] ∩ Z) × {l} ∆ωy ← {k} × (l + [-K, K] ∩ Z) t * x ← argmin tx∈T J(v x tx (∆ωx)) t * y ← argmin ty ∈T J(v y ty (∆ωy)) T (k, l) ← (t * x , t * y ) v0(k, l) ← U0(k -t * x , l -t * y ) (

Statistical properties of the resampled image

We investigate the statistical properties of the resulting image and we show that, under a reasonable assumption, our sampling scheme produces a signal that is completely faithful to the underlying signal. We make the assumption that the scene is the superposition of a signal stemming from a bright target of amplitude A and a fully-developed speckle. Under this model, the term u * 0 (k) in (7) corresponds to the sampling of a band-limited Gaussian white second-order stationary process [START_REF] Oliver | Understanding Synthetic Aperture Radar Images[END_REF]. This means that the correlation between samples distant by an integer value is zero. The process being Gaussian, the zero correlation implies independence. A consequence of the bandlimited property of the SAR-signal is that the original discrete image allows one to recover the true values of U0(x) for real x by means of Shannon interpolation of the available discrete signal. When examining the criteria J1,2,3 proposed in Section 3.2, one can see that for the entire line the optimal value of tx is the same for all pixels, provided the computation window of the criterion is large enough. Thus, provided that our estimated tx equals to δ the final discrete result of our resampling will be, according to (8), U * 0 (k + δ) except at pixel k = k0 (the target appears here) which are integer distant samples from the underlying fully-developed speckle and hence i.i.d Gaussian variables. We reduced the statistical accuracy of the resulting signal to the one of our estimator of tx. One can show that this estimator is unbiased and has a variance depending on the ratio between the power of the u * 0 process and A. In Fig. 6, we show how the output image v0 both respects the theoretical Gaussian distribution of the samples and dramatically decreases the correlation between neighboring pixels, supporting our claim of statistical accuracy of the resampled image.

Kx-2 k=0

Ky -1 l=0

w(k +1, l)•w(k, l), where w : IK x × IK y → C denotes the considered patch).

  This work is supported by the ANR through the MIRIAM project.

  (a) amplitude image (b) Fourier spectrum Fig. 1. Example of TerraSAR-X image data. We display in (a) the modulus of a SAR image u : Ω → C, and in (b) the modulus of u over a single period Ω (high values are displayed in dark, low values in bright, white meaning zero). The Fourier spectrum (b) reveals the presence of zero-padding ( u is zero everywhere outside of a rectangular sub-frequency domain ω Ω delimited by the red dashed line), showing that the image u was oversampled. 2. FROM THE SAR IMAGE DATA TO THE PSEUDO-RAW SPECTRUM 2.1. Pseudo-raw image and pseudo-raw spectrum Let u : Ω → C denote a complex-valued SAR image with domain Ω = IM × IN , where IK = {0, . . . , K -1} . We denote by u the discrete Fourier transform (DFT) of u, which is the two-dimensional and (M, N )-periodic signal defined by

Fig. 4 .

 4 Fig. 4. Resampling the pseudo-raw image. We display in (a) and (b) the modulus of the pseudo-raw image u0 and the modulus of the resampled image v0 computed using Algorithm 1. In the second row, we display some close-up views of the images (a) and (b). The blue arrows indicate the values of the computed translation fields T at each pixel location. One can see that the resampled image v0 exhibits a similar level of details than u0 but is free of target induced cardinal sine phenomenon.

Fig. 5 .

 5 Fig. 5. Multi-Look comparison between apodized data and resampled pseudo-raw data. We display in (a) the multi-look realvalued image u ML ω = ( 1 26

Fig. 6 .

 6 Fig.6. First order statistics of a region with constant radiometry. We display here the empirical distribution of a 50 × 60 patch of the real part of the apodized image uω and the resampled image v0 where the signal corresponds to pure speckle. In both cases, the empirical probability density function is very well approached by a Gaussian function. Pixels of uω are highly correlated (0.49) while those of v0 exhibit a one hundred times smaller correlation (we computed corr(w) = 1 w 2