Amir Hazem
email: amir.hazem@univ-nantes.fr

Emmanuel Morin
email: emmanuel.morin@univ-nantes.fr

Sebastian Pe Ña Saldarriaga

Bilingual Lexicon Extraction from Comparable Corpora as Metasearch

In this article we present a novel way of looking at the problem of automatic acquisition of pairs of translationally equivalent words from comparable corpora. We first present the standard and extended approaches traditionally dedicated to this task. We then reinterpret the extended method, and motivate a novel model to reformulate this approach inspired by the metasearch engines in information retrieval. The empirical results show that performances of our model are always better than the baseline obtained with the extended approach and also competitive with the standard approach.

Introduction

Bilingual lexicon extraction from comparable corpora has received considerable attention since the 1990s [START_REF] Rapp | Identify Word Translations in Non-Parallel Texts[END_REF][START_REF] Fung | A Statistical View on Bilingual Lexicon Extraction: From ParallelCorpora to Non-parallel Corpora[END_REF][START_REF] Fung | An ir approach for translating new words from nonparallel, comparable texts[END_REF][START_REF] Peters | Cross-language information retrieval: A system for comparable corpus querying[END_REF][START_REF] Rapp | Automatic Identification of Word Translations from Unrelated English and German Corpora[END_REF]Chiao and Zweigenbaum, 2002a;[START_REF] Déjean | Une nouvelle approche à l'extraction de lexiques bilingues à partir de corpus comparables[END_REF][START_REF] Gaussier | A Geometric View on Bilingual Lexicon Extraction from Comparable Corpora[END_REF][START_REF] Morin | Bilingual Terminology Mining -Using Brain, not brawn comparable corpora[END_REF]Laroche and Langlais, 2010, among others). This attention has been motivated by the scarcity of parallel corpora, especially for countries with only one official language and for language pairs not involving English. Furthermore, as a parallel corpus is comprised of a pair of texts (a source text and a translated text), the vocabulary appearing in the translated text is highly influenced by the source text, especially in technical domains. Consequently, comparable corpora are considered by human translators to be more trustworthy than parallel corpora [START_REF] Bowker | Working with Specialized Language: A Practical Guide to Using Corpora[END_REF]. Comparable corpora are clearly of use in the enrichment of bilingual dictionaries and thesauri (Chiao and Zweigenbaum, 2002b;Déjean et al., 2002), and in the improvement of cross-language information retrieval [START_REF] Peters | Cross-language information retrieval: A system for comparable corpus querying[END_REF].

According to [START_REF] Fung | A Statistical View on Bilingual Lexicon Extraction: From ParallelCorpora to Non-parallel Corpora[END_REF], bilingual lexicon extraction from comparable corpora can be approached as a problem of information retrieval (IR). In this representation, the query would be the word to be translated, and the documents to be found would be the candidate translations of this word. In the same way that as documents found, the candidate translations are ranked according to their relevance (i.e. a document that best matches the query). More precisely, in the standard approach dedicated to bilingual lexicon extraction from comparable corpora, a word to be translated is represented by a vector context composed of the words that appear in its lexical context. The candidate translations for a word are obtained by comparing the translated source context vector with the target context vectors through a general bilingual dictionary. Using this approach, good results on single word terms (SWTs) can be obtained from large corpora of several million words, with an accuracy of about 80% for the top 10-20 proposed candidates [START_REF] Fung | Finding Terminology Translations from Non-parallel Corpora[END_REF][START_REF] Rapp | Automatic Identification of Word Translations from Unrelated English and German Corpora[END_REF]. [START_REF] Cao | Base Noun Phrase Translation Using Web Data and the EM Algorithm[END_REF] have achieved 91% accuracy for the top three candidates using the Web as a comparable corpus. Results drop to 60% for SWTs using specialized small size language corpora (Chiao and Zweigenbaum, 2002a;[START_REF] Déjean | Une nouvelle approche à l'extraction de lexiques bilingues à partir de corpus comparables[END_REF][START_REF] Morin | Bilingual Terminology Mining -Using Brain, not brawn comparable corpora[END_REF].

In order to avoid the insufficient coverage of the bilingual dictionary required for the translation of source context vectors, an extended approach has been proposed [START_REF] Déjean | Une nouvelle approche à l'extraction de lexiques bilingues à partir de corpus comparables[END_REF][START_REF] Daille | French-English Terminology Extraction from Comparable Corpora[END_REF]. This approach can be seen as a query reformulation process in IR for which similar words are substituted for the word to be translated. These similar words share the same lexical environments as the word to be translated without appearing with it. With the extended approach, [START_REF] Déjean | Une nouvelle approche à l'extraction de lexiques bilingues à partir de corpus comparables[END_REF] obtained for single French-English words 43% and 51% precision out of the ten and twenty first candidates applied to a medical corpus of 100 000 words (respectively 44% and 57% with the standard approach) and 79% and 84% precision on the ten and twenty first candidates applied to a social science corpus of 8 million words (respectively 35% and 42% with the standard approach). Within this context, we want to show how metasearch engines can be used for bilingual lexicon extraction from specialized comparable corpora. In particular, we will focus on the use of different strategies to take full advantage of similar words.

The remainder of this paper is organized as follows. Section 2 presents the standard and extended approaches based on lexical context vectors dedicated to word alignment from comparable corpora. Section 3 describes our metasearch approach that can be viewed as the combination of different search engines. Section 4 describes the different linguistic resources used in our experiments and evaluates the contribution of the metasearch approach on the quality of bilingual terminology extraction through different experiments. Finally, Section 5 presents our conclusions.

Related Work

In this section, we first describe the standard approach dedicated to word alignment from comparable corpora. We then present an extension of this approach.

Standard Approach

The main work in bilingual lexicon extraction from comparable corpora is based on lexical context analysis and relies on the simple observation that a word and its translation tend to appear in the same lexical contexts. The basis of this observation consists in the identification of first-order affinities for each source and target language: First-order affinities de-scribe what other words are likely to be found in the immediate vicinity of a given word (Grefenstette, 1994a, p. 279). These affinities can be represented by context vectors, and each vector element represents a word which occurs within the window of the word to be translated (for instance a seven-word window approximates syntactical dependencies).

The implementation of this approach can be carried out by applying the following four steps [START_REF] Rapp | Identify Word Translations in Non-Parallel Texts[END_REF][START_REF] Fung | Finding Terminology Translations from Non-parallel Corpora[END_REF]:

Context characterization

All the lexical units in the context of each lexical unit i are collected, and their frequency in a window of n words around i extracted. For each lexical unit i of the source and the target languages, we obtain a context vector i where each entry, i j , of the vector is given by a function of the co-occurrences of units j and i. Usually, association measures such as the mutual information [START_REF] Fano | Transmission of Information: A Statistical Theory of Communications[END_REF] or the log-likelihood [START_REF] Dunning | Accurate Methods for the Statistics of Surprise and Coincidence[END_REF] are used to define vector entries.

Vector transfer

The lexical units of the context vector i are translated using a bilingual dictionary. Whenever the bilingual dictionary provides several translations for a lexical unit, all the entries are considered but weighted according to their frequency in the target language. Lexical units with no entry in the dictionary are discarded.

Target language vector matching

A similarity measure, sim(i, t), is used to score each lexical unit, t, in the target language with respect to the translated context vector, i. Usual measures of vector similarity include the cosine similarity [START_REF] Salton | Computer evaluation of indexing and text processing[END_REF] or the weighted jaccard index (WJ) (Grefenstette, 1994b) for instance.

Candidate translation

The candidate translations of a lexical unit are the target lexical units ranked following the similarity score.

Extended Approach

The main shortcoming of the standard approach is that its performance greatly relies on the coverage of the bilingual dictionary. When the context vectors are well translated, the translation retrieval rate in the target language improves. Although, the coverage of the bilingual dictionary can be extended by using specialized dictionaries or multilingual thesauri [START_REF] Chiao | The Effect of a General Lexicon in Corpus-Based Identification of French-English Medical Word Translations[END_REF][START_REF] Déjean | Une nouvelle approche à l'extraction de lexiques bilingues à partir de corpus comparables[END_REF], translation of context vectors remains the core of the approach.

In order to be less dependent on the coverage of the bilingual dictionary, [START_REF] Déjean | Une nouvelle approche à l'extraction de lexiques bilingues à partir de corpus comparables[END_REF] have proposed an extension to the standard approach. The basic intuition of this approach is that words sharing the same meaning will share the same environments. The approach is based on the identification of second-order affinities in the source language: Second-order affinities show which words share the same environments. Words sharing second-order affinities need never appear together themselves, but their environments are similar (Grefenstette, 1994a, p. 280).

Generally speaking, a bilingual dictionary is a bridge between two languages established by its entries. The extended approach is based on this observation and avoids explicit translation of vectors as shown in Figure 1. The implementation of this extended approach can be carried out in four steps where the first and last steps are identical to the standard approach [START_REF] Déjean | Une nouvelle approche à l'extraction de lexiques bilingues à partir de corpus comparables[END_REF][START_REF] Daille | French-English Terminology Extraction from Comparable Corpora[END_REF]:

Reformulation in the target language

For a lexical unit i to be translated, we identify the k-nearest lexical units (k nlu), among the dictionary entries corresponding to words in the source language, according to sim(i, s). Each nlu is translated via the bilingual dictionary, and the vector in the target language, s, corresponding to the translation is selected. If the bilingual dictionary provides several translations for a given unit, s is given by the union of the vectors corresponding to the translations. It is worth noting that the context vectors are not translated directly, thus reducing the influence of the dictionary.

Vector matching against reformulations

The similarity measure, sim(s, t), is used to score each lexical unit, t, in the target language with respect to the k nlu. The final score assigned to each unit, t, in the target language is given by:

sim(i, t) = s∈kNLU sim(i, s) × sim(s, t) (1)
An alternate scoring function has been proposed by [START_REF] Daille | French-English Terminology Extraction from Comparable Corpora[END_REF]. The authors computed the centroid vector of the k nlu, then scored target units with respect to the centroid.

The Metasearch Approach

Motivations

The approach proposed by [START_REF] Déjean | Une nouvelle approche à l'extraction de lexiques bilingues à partir de corpus comparables[END_REF] implicitly introduces the problem of selecting a good k. Generally, the best choice of k depends on the data. Although several heuristic techniques, like cross-validation, can be used to select a good value of k, it is usually defined empirically.

The application of the extended approach (EA) to our data showed that the method is unstable with respect to k. In fact, for values of k over 20, the precision drops significantly. Furthermore, we cannot ensure result stability within particular ranges of values. Therefore, the value of k should be carefully tuned.

Starting from the intuition that each nearest lexical unit (nlu) contributes to the characterization of a lexical unit to be translated, our proposition aims at providing an algorithm that gives a better precision while ensuring higher stability with respect to the number of nlu. Pushing the analogy of IR style approaches [START_REF] Fung | An ir approach for translating new words from nonparallel, comparable texts[END_REF] a step further, we propose a novel way of looking at the problem of word translation from comparable corpora that is conceptually simple: a metasearch problem.

In information retrieval, metasearch is the problem of combining different ranked lists, returned by multiple search engines in response to a given query, in such a way as to optimize the performance of the combined ranking [START_REF] Javed | Models for Metasearch[END_REF]. Since the k nlu result in k distinct rankings, metasearch provides an appropriate framework for exploiting information conveyed by the rankings.

In our model, we consider each list of a given nlu as a response of a search engine independently from the others. After collecting all the lists of the selected nlu's, we combine them to obtain the final similarity score. It is worth noting that all the lists are normalized to maximize in such a way the contribution of each nlu. A good candidate is the one that obtains the highest similarity score which is calculated with respect to the selected k. If a given candidate has a high frequency in the corpus, it may be similar not only to the selected nearest lexical units (k), but also to other lexical units of the dictionary. If the candidate is close to the selected nlu's and also close to other lexical units, we consider it as a potential noise (the more neighbours a candidate has, the more it's likely to be considered as noise). We thus weight the similarity score of a candidate by taking into account this information. We compare the distribution of the candidate with the k nlu and also with all its neighbours. This leads us to suppose that a good candidate should be closer to the selected nlu's than the rest of its neighbours, if it's not the case there is more chances for this candidate to be a wrong translation.

Proposed Approach

In the following we will describe our extension to the method proposed by [START_REF] Déjean | Une nouvelle approche à l'extraction de lexiques bilingues à partir de corpus comparables[END_REF]. The notational conventions adopted are reviewed in Table 1. Elaborations of definitions will be given when the notation is introduced. In all our experiments both terms and lexical units are single words.

Symbol

Definition l a list of a given lexical unit. k the number of selected nearest lexical units (lists). f req(w, k) the number of lists (k) in which a term appears. n all the neighbours of a given term. u all the lexical units of the dictionary. w l a term of a given list l. s(w l )

the score of the term w in the list l. max l the maximum score of a given list l. max All the maximum score of all the lists. s norm (w l )

the normalized score of term w in the list l. s(w)

the final score of a term w. θ w the regulation parameter of the term w. The first step of our method is to collect each list of each nlu. The size of the list has its importance because it determines how many candidates are close to a given nlu. We noticed from our experiments that, if we choose lists with small sizes, we should lose information and if we choose lists with large sizes, we could keep more information than necessary and this should be a potential noise, so we consider that a good size of each list should be between 100 and 200 terms according to our experiments.

After collecting the lists, the second step is to normalize the scores. Let us consider the equation 2 :

s norm (w l ) = s(w l ) × max l max All (2)
We justify this by a rationale derived from two observations. First, scores in different rankings are compatible since they are based on the same similarity measure (i.e., on the same scale). The second observations follows from the first: if max (l) ≫ max (m), then the system is more confident about the scores of the list l than m.

Using scores as fusion criteria, we compute the similarity score of a candidate by summing its scores from each list of the selected nlu's :

s(w) = θ w × k l=1 s norm (w l ) n l=1 s norm (w l ) (3)
the weight θ is given by :

θ w = f req(w, k) × (u -(k -f req(w, k))) (u -f req(w, n)) (4)
The aim of this parameter is to give more confidence to a term that occurs more often with the selected nearest neighbours (k) than the rest of its neighbours. We can not affirm that the best candidate is the one that follows this idea, but we can nevertheless suppose that candidates that appear with a high number of lexical units are less confident and have higher chances to be wrong candidates (we can consider those candidates as noise). So, θ allows us to regulate the similarity score, it is used as a confident weight or a regulation parameter. We will refer to this model as the multiple source (MS) model. We also use our model without using θ and refer to it by (LC), this allows us to show the impact of θ in our results.

Experiments and Results

Linguistic Resources

We have selected the documents from the Elsevier website 1 in order to obtain a French-English specialized comparable corpus. The documents were taken from the medical domain within the subdomain of 'breast cancer'. We have automatically selected the documents published between 2001 and 2008 where the title or the keywords contain the term 'cancer du sein' in French and 'breast cancer' in English. We thus collected 130 documents in French and 118 in English and about 530,000 words for each language. The documents comprising the French/English specialized comparable corpus have been normalized through the following linguistic pre-processing steps: tokenisation, part-of-1 www.elsevier.com speech tagging, and lemmatisation. Next, the function words were removed and the words occurring less than twice (i.e. hapax) in the French and the English parts were discarded. Finally, the comparable corpus comprised about 7,400 distinct words in French and 8,200 in English.

The French-English bilingual dictionary required for the translation phase was composed of dictionaries that are freely available on the Web. It contains, after linguistic pre-processing steps, 22,300 French single words belonging to the general language with an average of 1.6 translations per entry.

In bilingual terminology extraction from specialized comparable corpora, the terminology reference list required to evaluate the performance of the alignment programs are often composed of 100 single-word terms (SWTs) (180 SWTs in [START_REF] Déjean | Une nouvelle approche à l'extraction de lexiques bilingues à partir de corpus comparables[END_REF], 95 SWTs in (Chiao and Zweigenbaum, 2002a), and 100 SWTs in [START_REF] Daille | French-English Terminology Extraction from Comparable Corpora[END_REF]). To build our reference list, we selected 400 French/English SWTs from the UMLS2 meta-thesaurus and the Grand dictionnaire terminologique3 . We kept only the French/English pair of SWTs which occur more than five times in each part of the comparable corpus. As a result of filtering, 122 French/English SWTs were extracted.

Experimental Setup

Three major parameters need to be set to the extended approach, namely the similarity measure, the association measure defining the entry vectors and the size of the window used to build the context vectors. [START_REF] Laroche | Revisiting Context-based Projection Methods for Term-Translation Spotting in Comparable Corpora[END_REF] carried out a complete study about the influence of these parameters on the quality of bilingual alignment.

As similarity measure, we chose to use the weighted jaccard index:

sim(i, j) = t min (i t , j t ) t max (i t , j t ) (5)
The entries of the context vectors were determined by the log-likelihood [START_REF] Dunning | Accurate Methods for the Statistics of Surprise and Coincidence[END_REF], and we used a seven-word window since it approximates syntactic dependencies. Other combinations of parameters were assessed but the previous parameters turned out to give the best performance.

Results

To evaluate the performance of our method, we use as a baseline, the extended approach (EA) proposed by [START_REF] Déjean | Une nouvelle approche à l'extraction de lexiques bilingues à partir de corpus comparables[END_REF]. We compare this baseline to the two metasearch strategies defined in Section 3: the metasearch model without the regulation parameter θ (LC); and the one which is weighted by theta (MS). We also provide results obtained with the standard approach (SA).

We first investigate the stability of the metasearch strategies with respect to the number of nlu considered. Figure 2 show the precision at Top 20 as a function of k. In order to evaluate the contribution of the parameter θ, we chose to evaluate the metasearch method starting from k = 4, this explains why the precision is extremely low for low values of k. We further considered that less than four occurrences of a term in the whole lexical units lists can be considered as noise. On the other side, we started from k = 1 for the extended approach since it makes no use of the parameter θ. Figure 2 shows that extended approach reaches its best performance at k = 7 with a precision of 40.98%. Then, after k = 15 the precision starts steadily decreasing as the value of k increases.

The metasearch strategy based only on similarity scores shows better results than the baseline. For every value of k ≥ 10, the LC model outperform the extended approach. The best precision (48.36%) is obtained at k = 14, and the curve corresponding to the LC model remains above the baseline regardless of the increasing value of the parameter k. The curve corresponding to the MS model is always above the (EA) for every value of k ≥ 10. The MS model consistently improves the precision, and achieves its best performance (60.65%) at k = 21.

We can notice from Figure 2 that the LC and MS models outperform the baseline (EA). More importantly, these models exhibit a better stability of the precision with respect to the k-nearest lexical units. Although the performance decrease as the value of k increases, it does not decrease as fast as in the baseline approach.

For the sake of comparability, we also provide results obtained with the standard approach (SA) (56.55%) represented by a straight line as it is not dependent on k. As we can see, the metasearch approach (MS) outperforms the standard approach for values of k bertween 20 and 30 and for greater values of k the precision remains more or less almost the same as the standard approach (SA). Thus, the metasearch model (MS) can be considered as a competitive approach regarding to its results as it is shown in the figure 2.

Finally, Figure 3 shows the contribution of each nlu taken independently from the others. This confirms our intuition that each nlu contribute to the characterization of a lexical unit to be translated, and supports our idea that their combination can improve the performances. Figure 3 shows the top 20 of each nlu. Notice that the nlu are ordered from the most similar to the lexical unit to be translated to the less similar, and that each one of the nearest lexical units contains information that it is worth taking into account.

Although each nlu can only translate few terms, by using the metasearch idea we are able to improve the retrieval of translation equivalents. The main idea of the metasearch paradigm is to take into account the information conveyed by all the k nlu, using either similarity scores, their behaviour with all the neighbours, in order to improve the performance of the alignment process.

Although significant improvements can be obtained with the metasearch models (comparatively to the EA and SA approach), especially concerning precision stability with respect to the k nlu, we believe that we need to address the estimation of k beforehand. Rather than fixing the same k for all the units to be translated, there is the possibility to adapt an optimal value of k to each lexical unit, according to some criteria which have to be determined. Finally, we present in table 2 a comparison between SA, EA and MS for the top 5, 10, 15 and 20. By choosing the best configuration of each method, we can note that our method outperforms the others in each top. In addition, for the top 10 our precision is very close to the precision of the standard approach (SA) at the top 20. we consider these results as encouraging for future work.

Discussion

Our experiments show that the parameter k remains the core of both EA and MS approaches. A good selection of the nearest lexical units of a term guarantee to find the good translation. It is important to say that EA and MS which are based on the k nlu's depends on the coverage of the terms to be translated. Indeed, these approaches face three cases : firstly, if the frequency of the word to be translated is high and the frequency of the good translation in the target language is low, this means that the nearest lexical units of the candidate word and its translation are unbalanced. This leads us to face a lot of noise because of the high frequency of the source word that is over-represented by its nlu's comparing to the target word which is under-represented. Secondly, we consider the inverse situation, which is: low frequency of the source word and high frequency of the target translation, here as well, we have both the source and the target words that are unbalanced regarding to the selected nearest lexical units. The third case, represents more or less the same distribution of the frequencies of source candidate and target good translation. This can be considered as the most appropriate case to find the good translation by applying the approaches based on the nlu's (EA or MS). Our experiments show that our method works well in all the cases by using the parameter θ which regulate the similarity score by taken into account the distribution of the candidate according to both : selected nlu's and all its neighbours. In resume, words to be translated as represented in case one and two give more difficulties to be translated because of their unbalanced distribution which leads to an unbalanced nlu's. Future works should confirm the possibility to adapt an optimal value of k to each candidate to be translated, according to its distribution with respect to its neighbours.

Conclusion

We have presented a novel way of looking at the problem of bilingual lexical extraction from comparable corpora based on the idea of metasearch engines. We believe that our model is simple and sound. Regarding the empirical results of our proposition, performances of the multiple source model on our dataset was better than the baseline proposed by [START_REF] Déjean | Une nouvelle approche à l'extraction de lexiques bilingues à partir de corpus comparables[END_REF], and also outperforms the standard approach for a certain range of k. We believe that the most significant result is that a new approach to finding single word translations has been shown to be competitive. We hope that this new paradigm can lead to insights that would be unclear in other models. Preliminary tests in this perspective show that using an appropriate value of k for each word can improve the performance of the lexical extraction process. Dealing with this prob-

Figure 1 :

 1 Figure 1: Illustration of the extended approach.

Figure 2 :

 2 Figure 2: Precision at top 20 as a function of the number of nlu.

Figure 3 :

 3 Figure 3: Precision at top 20 for each of the 20 nlu. The precision is computed by taking the each nlu independently from the others.

Table 1 :

 1 Notational conventions.

Table 2 :

 2 Precision(%) at top 5, 10, 15, 20 for SA, EA and MS.

Proceedings of the 4th Workshop on Building and Using Comparable Corpora, pages 35-43, 49th Annual Meeting of the Association for Computational Linguistics, Portland, Oregon, 24 June 2011. c 2011 Association for Computational Linguistics

http://www.nlm.nih.gov/research/umls

http://www.granddictionnaire.com/

lem is an interesting line for future research.

Acknowledgments

The research leading to these results has received funding from the French National Research Agency under grant ANR-08-CORD-013.


Olivier Morel

Ralph Seulin

David Fofi

Measurement of three-dimensional mirror parameters by polarization imaging applied to catadioptric camera calibration

Keywords:

A new efficient method of calibration for catadioptric sensors is presented in this paper. The method is based on an accurate measurement the three-dimensional parameters of the mirror by means of polarization imaging. While inserting a rotating polarizer between the camera and the mirror, the system is automatically calibrated without any calibration patterns. Moreover, this method permits most of the constraints related to the calibration of catadioptric systems to be relaxed. We show that, contrary to our system, the traditional methods of calibration are very sensitive to misalignment of the camera axis and the symmetry axis of the mirror. From the measurement of three-dimensional parameters, we apply the generic calibration concept to calibrate the catadioptric sensor. The influence of the disturbed measurement of the parameters on the reconstruction of a synthetic scene is also presented. Finally, experiments prove the validity of the method with some preliminary results on three-dimensional reconstruction.

I. INTRODUCTION

Conventional perspective cameras have limited fields of view that make them restrictive in some applications such as robotics, video-surveillance, and so on. One way to enhance the field of view is to place a mirror with a surface of revolution in front of the camera so that the scene reflects on the mirror omnidirectionally. Such a system, comprised of both lenses (dioptric) and mirrors (catoptric) for image formation, is called catadioptric. Several configurations exist and those satisfying the single viewpoint constraint are described in [START_REF] Baker | A theory of catadioptric image formation[END_REF].

Catadioptric vision systems available on the market have been extensively studied. Commercial devices are not adapted to our requirements, because optical components need to be placed between the camera and the mirror. Mirrors have therefore been produced in our own facilities thanks to the Plateform3D department [START_REF]Plateform3D Department -3 Dimensional Solid Modelling and Prototyping[END_REF]. A high speed machining centre has been used to produce a very high quality surface, which is polished after production.

We developed a new approach of calibrating catadioptric sensors by polarization imaging. This method enables the calibration of any mirror shape, since it is based on the measurement of three-dimensional parameters such as height and normal orientations of the surface.

The only constraint is that an orthographic camera has to be used. To calibrate the system, we apply the generic calibration concept developed by Sturm and Ramalingam [START_REF] Ramalingam | A generic structure-from-motion framework[END_REF][START_REF] Sturm | Multi-view geometry for general camera models[END_REF].

The article is structured as follows. The next section recalls previous work on paracatadioptric calibration, since the measurement of the surface normals by polarization imaging induces orthographic projection, and most of the calibration methods developed for catadioptric systems rely on the single viewpoint constraint. We show the misalignment sensitivity of these methods for the reconstruction of a synthetic scene. Then, after presenting some basic knowledge about polarization imaging, we detail how to calibrate the sensor with the generic calibration concept. In section 4, simulations are presented to illustrate the influence of the parameter measurement on the quality of the reconstruction. Preliminary results on a calibrated spherical mirror are also described. The paper ends with a conclusion and a few words about future work to be undertaken.

The most obvious calibration method that can be used is an approach based on the image of the mirror's bounding circle [START_REF] Kang | Catadioptric self-calibration[END_REF][START_REF] Yagi | Real-time omnidirectional image sensor (copis) for visionguided navigation[END_REF]. It has the main advantage of being easily automated, but the drawbacks are that the surface mirror has to be very accurate and the mirror boundary has to accurately encode the intrinsic parameters. Other self-calibration method can be found in [START_REF] Fabrizio | Calibration of panoramic catadioptric sensors made easier[END_REF][START_REF] Geyer | Structure and motion from uncalibrated catadioptric views[END_REF]. Another approach of calibrating catadioptric sensor is to use geometric invariants on the image such as lines or circles [START_REF] Geyer | Paracatadioptric camera calibration[END_REF][START_REF] Geyer | Catadioptric camera calibration[END_REF][START_REF] Barreto | Geometric properties of central catadioptric line images[END_REF][START_REF] Ying | Catadioptric camera calibration using geometric invariants[END_REF]. Finally the sensor can also been calibrated by using some calibration pattern with control points whose 3D world coordinates are known [START_REF] Grossberg | A general imaging model and a method for finding its parameters[END_REF][START_REF] Aliaga | Accurate catadioptric calibration for real-time pose estimation of room-size environments[END_REF][START_REF] Swaminathan | Caustics of catadioptric cameras[END_REF][START_REF] Mei | Single view point omnidirectional camera calibration from planar grids[END_REF][START_REF] Ramalingam | Theory and calibration algorithms for axial cameras[END_REF].

Since our approach requires a telecentric lens, we recall some methods devoted to paracatadioptric camera calibration (the Single View Point constraint involve a telecentric lens and a paraboloidal mirror). In this case, the more robust methods are based on the fitting of lines projected onto the mirror [START_REF] Geyer | Paracatadioptric camera calibration[END_REF][START_REF] Barreto | Paracatadioptric camera calibration using lines[END_REF][START_REF] Vanderportaele | A new linear calibration method for paracatadioptric cameras[END_REF]. This approach also has some shortcomings: lines have to be precisely detected and the optical axis of the camera is assumed to be aligned with the symmetry axis of the paraboloid. To illustrate the misalignment effect, the three-dimensional reconstructions of a synthetic scene based on calibrations using the three preceding methods are simulated (Figure 1).

The scene represents a room of size 500 × 500 × 250 cm with elements such as windows, doors, and a table. Three images of the catadioptric sensor are used to triangulate the points of the scene. For the calibration process, 20 lines are computed and perfectly detected on the mirror. Then the calibration parameters are used to reconstruct the scene, according to the linear-eigen method [START_REF] Hartley | Triangulation[END_REF]. As presented in Figure 2, the misalignment of the paracatadioptric system leads to the introduction of an important error on the reconstruction even if the calibration is performed with perfect line fittings.

To deal with the alignment errors between the mirror and the lens, more sophisticated algorithms were introduced [START_REF] Gonzalez-Barbosa | Vision panoramique pour la robotique mobile : stéréovision et localisation par indexation d'images[END_REF][START_REF] Gonzalez-Barbosa | Fast dense panoramic stereovision[END_REF][START_REF] Ieng | Geometric construction of the caustic curves for catadioptric sensors[END_REF]. In [START_REF] Gonzalez-Barbosa | Vision panoramique pour la robotique mobile : stéréovision et localisation par indexation d'images[END_REF][START_REF] Gonzalez-Barbosa | Fast dense panoramic stereovision[END_REF], the calibration process, which requires a large number of parameters to estimate, involves a three step algorithm in order to compute all the extrinsic and intrinsic parameters. In [START_REF] Ieng | Geometric construction of the caustic curves for catadioptric sensors[END_REF], a more flexible solution based on caustic curves is proposed: the main advantage of this method is that it is applicable for every surface of revolution. In order to relax constraints on the mirror shape, the method presented in this paper is based on the generic calibration concept: it leads to a flexible, easy-to-perform, and shape-independent method of calibration.

B. The generic calibration concept

The previous calibration methods for omnidirectional catadioptric sensors assume that:

(i) the mirror shape is perfectly known; (ii) the alignment of the sensor is perfect so that the single viewpoint constraint is satisfied; and (iii) the projection model can be easily parameterized. Some methods relax the second constraint and a few relax the first one, but before recent work [START_REF] Sturm | Multi-view geometry for general camera models[END_REF][START_REF] Grossberg | A general imaging model and a method for finding its parameters[END_REF][START_REF] Pless | Using many cameras as one[END_REF] calibrating methods always underlie an explicit parametric model of projection. This new model has the advantage of working for any type of camera (catadioptric systems, central cameras with or without distortion, axial cameras, etc.) and of handling heterogeneous systems [START_REF] Ramalingam | A generic structure-from-motion framework[END_REF] (for instance, a sensor composed of an omnidirectional camera and a perspective camera). However, developing an efficient and easy-to-use calibration method based on this model is not trivial. In this paper, the proposed new method enables catadioptric sensor calibration by polarization imaging. It relaxes the three constraints listed above ((i), (ii), and (iii)). Moreover, the calibration can even be performed by a non-specialist as it only requires an optical apparatus and no image processing.

III. POLARIZATION IMAGING

Polarization imaging enables the study of the polarization state of a light wave. The most common applications in artificial vision involve segmenting dielectric and metallic objects [START_REF] Wolff | Polarization-based material classification from specular reflection[END_REF] and to detect transparent surfaces. Polarization imaging likewise enables three-dimensional information of specular objects to be detected ("Shape from polarization" method [START_REF] Miyazaki | Transparent surface modeling from a pair of polarization images[END_REF][START_REF] Rahmann | Reconstruction of quadrics from two polarization views[END_REF]). The physical principle behind "Shape from polarization" is the following: after being reflected, an unpolarized light wave becomes partially linearly polarized, depending on the surface normal and the refractive index of the media it impinges on.

Partially linearly polarized light has three parameters: the light magnitude I, the degree of polarization ρ and the angle of polarization ϕ.

To calibrate the mirror used in our catadioptric sensor, the polarization state of the reflected light has to be measured. A rotating polarizer placed between the camera and the mirror is used. The complete sensor (mirror and camera) and the polarizer are placed into a diffuse light environment composed of a backlit cylindrical diffuser (Figure 3). The light intensity of each pixel is linked to the angle of the polarizer and to the polarization parameters by the following equation:

I p (α) = I 2 (ρ cos (2α -2ϕ) + 1) , ( 1
)
where α is the polarizer angle. The purpose of polarization imaging is to compute the three parameters, I, ϕ, and ρ, by interpolating this formula. Because three parameters need to be determined, at least three images are required. Images are taken with different orientations of the polarizer. To get an automatic calibration of the catadioptric system, a liquid-crystal polarization rotator is used instead of the polarizer. It acts as a rotating polarizer, which has the ability to be electrically controlled. Figure 4 shows an image of the degree and the angle of polarization of a spherical mirror.

A. The relationship between the polarization parameters and the normals

Wolff and Boult have demonstrated how to determine constraints on surface normals by using the Fresnel reflectance model [START_REF] Wolff | Constraining object features using a polarization reflectance[END_REF]. The surface of the mirror is assumed to be continuous and described by a Cartesian expression: z = f (x, y). Therefore, each surface normal is given by the following non-normalized expression:

n =      -∂f (x,y) ∂x -∂f (x,y) ∂y 1      =      p = tan θ cos φ q = tan θ sin φ 1      . ( 2
)
The aim of the "Shape from polarization" technique is to compute the normals from the angles θ and φ. By combining the Fresnel formulae and the Snell-Descartes law one can find a relationship between the degree of polarization ρ and the zenith angle θ [START_REF] Miyazaki | Transparent surface modeling from a pair of polarization images[END_REF]. For specular metallic surfaces, the following formula can be applied [28]:

ρ(θ) = 2n tan θ sin θ tan 2 θ sin 2 θ + |n| 2 , (3)
where n = n(1 + iκ) is the complex refractive index of the mirror. In the case of a perfect specular mirror, the formula (3) can be directly used if the complex refractive index of the material is known. However, if the refractive index is unknown or if the mirror is not perfectly specular a pseudo-refractive index [START_REF] Morel | Active lighting applied to 3d reconstruction of specular metallic surfaces by polarization imaging[END_REF] can be applied. This consists of estimating a pseudo-refractive index that best fits the relation between the angle θ and the degree of polarization ρ measured on a known shape of the same material. Finally, this pseudorefractive index, which has no physical meaning, enables the degree of polarization image of the new object to be computed using the same relation [START_REF] Sturm | Multi-view geometry for general camera models[END_REF].

The azimuth angle φ is linked to the angle of polarization ϕ since the reflected light becomes partially linearly polarized according to the normal of the plane of incidence. Because our imaging system uses a telecentric lens, orthographic projection onto the sensor is assumed and the azimuth angle φ can be inferred from the angle of polarization ϕ according to the following equation:

φ = ϕ ± π 2 . ( 4
)

B. Disambiguation of the normals

From equations ( 3) and ( 4), the surface normals are determined with an ambiguity.

Since mirrors used in catadioptric vision are of convex and revolution shape, a segmented image I quad can be directly computed from the near center of the mirror (Figure 5(a)). This segmented image is an image with four gray levels that represent the four quadrants oriented with an angle in ]0, π/2[. The algorithm of the disambiguation process described in [START_REF] Morel | Active lighting applied to 3d reconstruction of specular metallic surfaces by polarization imaging[END_REF] is applied with the segmented image I quad and with the angle of polarization image ϕ. The azimut angle φ is computed as follows:

1. φ = ϕ -π 2 , 2. φ = φ + π if [(I quad = 0) ∧ (φ ≤ 0)] ∨ [I quad = 1] ∨ [(I quad = 3) ∧ (φ ≥ 0)] ,
where ∧ and ∨ represent, respectively, the logical operators AND and OR. The result of the disambiguation is presented in Figure 5(b).

C. Calibration

To calibrate our imaging system, we use the generic calibration concept introduced by Sturm and Ramalingam [START_REF] Sturm | A generic concept for camera calibration[END_REF]. The concept considers an image as a collection of pixels, and each pixel measures the light along a particular 3D ray. Thus, calibration can be seen as the determination of all projection rays and their correspondence with pixels. A 3D ray is represented here by a couple of points (A, B) that belong to the ray. The points' coordinates are defined as described in equation ( 5):

A =      x a y a z a      , B =      x b y b z b      . ( 5
)
To calibrate the sensor, let us take point A, that belongs to both the mirror surface and the 3D-ray, to be the first point of the ray (Figure 6). Since orthographic projection is assumed, the x A and y A coordinates can be directly deduced from the (u, v) coordinates of each pixel (up to a scale factor given by the lens). Finally, the computation of the z a coordinate requires the determination of the 3D mirror surface. The 3D surface of the mirror can be computed from the normals (given by polarization imaging) thanks to the Frankot-Chellappa algorithm [START_REF] Frankot | A method for enforcing integrability in shape from shading algorithms[END_REF]. Denoting the Fourier transforms of, respectively, the surface height and the x, y gradients as f , p and q, we have:

∀ (u, v) = (0, 0) , f (u, v) = -jup -jv q u 2 + v 2 . ( 6
)
The three-dimensional surface is obtained by taking the inverse Fourier transform of the former equation. This integration process gives us the surface height of the mirror with a constant of integration. Nevertheless, this constant is not required because of the orthographic projection assumption.

As shown in Figure 6, the second point B of the ray can be written as:

B = A + k      tan 2θ cos φ tan 2θ sin φ 1      , ( 7
)
where k is a non-null constant.

IV. EXPERIMENTS

In the previous section, we showed that the three-dimensional parameters of the mirror, θ, φ, and z, are required to calibrate the catadioptric system, according to the generic calibration concept introduced by Sturm. The azimuth and zenith angles, φ and θ, are directly given by the measurement of the polarization parameters (ρ, ϕ) of the light reflected by the mirror. The surface height z, that represents the 3D shape of the mirror, is obtained by an integration process.

In this section, simulations are first presented to illustrate the influence of the parameters on reconstruction quality. Then, preliminary results on a calibrated spherical mirror show the good accuracy of the three-dimensional parameters measurement by polarization imaging. Experiments on the reconstruction of a real scene are also presented.

A. Simulations

To simulate the three-dimensional reconstruction error, the synthetic scene introduced in Section 2, Figure 1 is reconstructed thanks to the generic calibration concept. The normals angles θ and φ were computed from a perfect parabolic mirror with a 7°misalignment between the optical axis and the symmetry axis of the mirror. The parameters are then disturbed by adding various levels of Gaussian noise. At each level of noise, the experiment is regenerated 50 times, and the average error reconstruction is computed. Figure 7(a)

and Figure 7(b) show, respectively, the reconstruction error of the scene induced by noisy measurement of θ and φ. The synthetic scene is reconstructed with or without mirror reconstruction, meaning that the integration process from equation ( 6) is carried out or not (Figure 8).

Figure 7 shows that scene reconstruction is quite sensitive to the measurement of the parameters θ and φ; on the other hand, the integration process is not required, and we can assume that the z parameter is negligible. In addition, Figure 9 shows the reconstruction error of the scene by only adding Gaussian noise to the mirror height z. In this case, the normals are assumed to be perfectly measured and noise is only added on the surface height of the mirror. The reconstruction quality remains good even if the mirror height is very noisy (the mirror height is 1 cm and the radius is 2 cm) or badly reconstructed.

B. Preliminary results

Simulated scene

Preliminary results were carried out with a catadioptric sensor made of a camera with a telecentric lens and a calibrated spherical mirror (radius = 1 cm). Let us notice that our system does not satisfy the single view point constraint. Nevertheless, this property is not required here for the three-dimensional reconstruction of the scene. As described in section 3, our catadioptric sensor is calibrated by measuring the three-dimensional parameters of the mirror with a liquid crystal polarization rotator placed between the camera and the mirror.

To evaluate the accuracy of our system, we compare the parameters (θ, φ, and z) obtained with our system to the theoretical parameters of the mirror (Figure 10).

The mean quadratic errors of the angles θ and φ are, respectively, 0.49°and 1.02°. Fig-

ure 11 shows the reconstruction of the synthetic scene by using the calibration made by polarization imaging. Since the mirror is spherical, three-dimensional reconstruction errors increase highly. Nevertheless, the synthetic scene is well reconstructed with an average error of 9.68 cm.

Real scene

As presented in Figure 12, the three-dimensional reconstruction of a real scene was also carried out. The imaging system is made of a CCD camera with a telecentric lens and a spherical mirror. In order to precisely control its displacement, it has been placed on a precision three axes stage. The x × y × z operating area is 1.5m × 1.6m × 0.4m, and the x×y×z room size to reconstruct is about 6m×5m×2.5m. After calibrating our catadioptric sensor by polarization imaging, 3D points of interest have been triangulated by moving the system.

Once the imaging system is automatically calibrated by polarization imaging, three images with three different known positions have been acquired. Several points of interest were picked on each image: four rectangular targets, two edges of the ceiling, three windows and one fluorescent light (Figure . 12(b,c,d)). The result of the triangulation process presented in Figure 13 shows that the reconstruction is qualitatively satisfactory. The global topology of the scene and the relative distances are respected; right angles appear as (near-) right angles. In fact, the overall reconstruction is as expected, considering: the small displacement of the sensor in comparison with the size of the scene;

the fact that we only performed linear triangulation without refinement; and the reconstruction also depends on the accuracy of point picking (the correspondence has been established manually and a slight error in the location of 2D points lead to erroneous 3D reconstruction).

V. CONCLUSION

In this paper, a new efficient calibration method for catadioptric sensors has been presented. This method is based on the measurement of three-dimensional parameters of the mirror thanks to polarization imaging. The calibration can be performed "in one click" even by a non-specialist, because it only requires an optical apparatus, no image processing and no calibration pattern. Contrary to traditional methods, it deals with misalignment of the sensor and works for any shape mirror (regular or not). Experimental results prove that the sensor is properly calibrated and a satisfactory three-dimensional reconstruction of the scene can be obtained. We have also shown that the 3D-shape of the mirror can be neglected in comparison with the normals orientations. The plateform3D department is currently manufacturing a parabolic mirror and future work will consist of creating a paracatadioptric

Figure 1 :

 1 Figure 1: Simulation of the three-dimensional reconstruction: the theoretical scene, in blue, represents a room with elements such as windows, door and table ; black dots depict the three locations of the sensor. The reconstructed scene with the Vanderportaele calibration method (misalignment of 2°) is drawn in red.

Figure 2 :

 2 Figure 2: Reconstruction error induced by the misalignment between the optical axis of the camera and the symmetry axis of the paraboloid. Calibration method used are Geyer, Barreto and Vanderportaele. (room of size 500 × 500 × 250 cm)

Figure 3 :Figure 4 :

 34 Figure 3: Polarization imaging: after being reflected by the mirror, the light becomes partially linearly polarized.

Figure 5 :

 5 Figure 5: Disambiguation of the azimuth angle: (a) segmented image (I quad ∈ {0, 1, 2, 3}), (b) image of the resulting azimuth angle φ (φ ∈ [-π/2, 3π/2]).

Figure 6 :

 6 Figure 6: Description of the 3D-ray given by the points A and B.

Figure 7 :Figure 8 :

 78 Figure 7: Reconstruction error induced by noisy measurement of the normals parameters: (a) θ angle, (b) φ angle.

  error (cm)Noise standard deviation added on z (cm)

Figure 9 :

 9 Figure 9: Reconstruction error induced by noisy calculation of the mirror height z.

Figure 10 :

 10 Figure 10: Measurement errors of the three-dimensional parameters: (a) angle θ, (b) angle φ and (c) deviation map of the mirror z.

Figure 11 :Figure 12 :

 1112 Figure 11: Simulation of the three-dimensional reconstruction by using the calibration done from the polarization imaging.

Figure 13 :

 13 Figure 13: Results of the three-dimensional reconstruction: reconstruction according to the lineareigen and mid-point methods are respectively drawn in red and blue, the gray points represent the 3 positions of the sensor.

sensor in order to compare our method to other methods known in the literature for real scenes.


Mark S Ashbaugh

Antoine Henrot

Richard S Laugesen

Rebuttal of Donnelly's paper on the spectral gap

The spectral gap conjecture of M. van den Berg [2, formula (65)] asserts that λ 2λ 1 ≥ 3π 2 for all convex euclidean domains of diameter 1, where λ 1 and λ 2 denote the first two eigenvalues of the Dirichlet Laplacian. Notice that equality holds for the 1-dimensional unit interval, which can be regarded also as a degenerate n-dimensional rectangular box. The gap estimate is conjectured to hold more generally for Schrödinger operators with convex potentials, under Dirichlet boundary conditions; see the work of S.-T. Yau and collaborators [START_REF] Singer | An estimate of the gap of the first two eigenvalues in the Schrödinger operator[END_REF][START_REF] Yau | Nonlinear analysis in geometry[END_REF]. This Schrödinger gap conjecture was proved some time ago in 1 dimension by R. Lavine [START_REF] Lavine | The eigenvalue gap for one-dimensional convex potentials[END_REF], and more recently in all dimensions by B. Andrews and J. Clutterbuck [START_REF] Andrews | Proof of the fundamental gap conjecture[END_REF].

The proof in this journal by H. Donnelly [START_REF] Donnelly | Spectral gap for convex planar domains[END_REF] of the original gap conjecture in 2 dimensions (for the Dirichlet Laplacian with zero potential) is not correct. The Editors of Mathematische Zeitschrift have asked us to describe the flaws in the proof, in order to clarify the state of the literature.

Donnelly's approach to the problem is a natural one: first perform a shape optimization to rule out a non-degenerate minimizing domain, and then analyze the spectral gap for a sequence of domains degenerating to an interval, with the help of results by D. Jerison [START_REF] Jerison | The diameter of the first nodal line of a convex domain[END_REF]. (For some history on this approach, and on the gap conjecture more generally, see the report on AIM meeting "Low Eigenvalues of Laplace and Schrödinger Operators" [START_REF]Workshop on Low Eigenvalues of Laplace and Schrödinger operators; background material and open problems[END_REF], especially page 12 of the open problems list.)

The error lies in the proof of the shape optimization step, as we now explain. Donnelly wishes to prove that no minimizing domain can exist for He supposes that a minimizing domain Ω does exist, and aims for a contradiction by deforming the domain under the flow of a vector field V . He correctly observes that by the Hadamard formula for the shape derivative of an eigenvalue [3, Section 2], "for any deformation preserving convexity and diameter normalization, we have the equality"

∂Ω ∂u 1 ∂n 2 V • n dσ = ∂Ω ∂u 2 ∂n 2 V • n dσ. (1)
(Here u 1 and u 2 denote the first two eigenfunctions of the Dirichlet Laplacian on Ω.) The derivation of this first-variation condition (1) requires the deformed domain to remain convex and have diameter 1 under both the forwards and backwards deformation of Ω.

The next sentence of the paper incorrectly deduces that ( ∂u 1 ∂n ) 2 = ( ∂u 2 ∂n ) 2 on the boundary. (In fact slightly more is claimed, that ∂u 1 ∂n = ∂u 2 ∂n , but that discrepancy is unimportant here.) The deduction is incorrect because the class of deformations that preserve convexity and the diameter normalization might not be sufficiently numerous to enable the values of V • n to cover a dense subset of L 2 (∂Ω). For example, the boundary of Ω might contain a curve that is not strictly convex, such as a straight line segment, and this curve would support no local perturbations that preserve convexity of the domain under both the forwards and backwards flow. Indeed, in shape optimization problems with convexity constraints, optimality conditions generally hold only on strictly convex parts of the boundary. See, for example, some of the methods developed in the literature for dealing with convexity constraints [4, Theorem 4.2.2], [START_REF] Jerison | The direct method in the calculus of variations for convex bodies[END_REF]Section 7], and [START_REF] Lamboley | Polygons as optimal shapes with convexity constraint[END_REF].

The diameter constraint is also problematic for Donnelly's argument. Nowhere does his paper explain how one should ensure that the perturbing flow fixes the diameter. The problem is serious. For example, if Ω is a domain of constant width such as a disk or a Reuleaux triangle, then most perturbing flows will change the diameter.

A final reason why we believe Donnelly's approach to be beyond repair is that if his argument were correct, then it would apply equally well to the modified gap functional 2λ 2 -λ 1 , showing that this functional cannot have a minimizing domain (among convex domains of diameter 1) and hence must be minimized by some degenerating sequence of domains. That is impossible, because 2λ 2λ 1 ≥ λ 1 → ∞ as the sequence of domains degenerates to an interval.


E Wandersman

R Candelier

G Debrégeas

A Prevost

Texture-induced modulations of friction force: the fingerprint effect

Keywords: numbers: 46.55.+d, 68.35.Ct, 81.40.Pq

Dry solid friction is often accompanied by force modulations originating from stick-slip instabilities. Here a distinct, quasi-static mechanism is evidenced leading to quasi-periodic force oscillations during sliding contact between an elastomer block, whose surface is patterned with parallel grooves, and finely abraded glass slides. The dominant oscillation frequency is set by the ratio between the sliding velocity and the period of the grooves. A mechanical model is proposed that provides a quantitative prediction for the amplitude of the force modulations as a function of the normal load, the period of the grooves and the roughness characteristics of the substrate. The model's main ingredient is the non-linearity of the friction law. Since such non-linearity is ubiquitous for soft solids, this "fingerprint effect" should be relevant to a large class of frictional configurations and might in particular have important consequences in human (or humanoid) active digital touch.

Surface texture engineering by nano-or micropatterning techniques has proven to be an efficient tool to tune frictional or adhesive properties at solid/solid [1][2][3] or solid/liquid interfaces [4]. Many of the existing designs are directly inspired by natural surfaces. Superhydrophobic surfaces for example, reproduce the microscopic pattern observed on lotus flower leaves [5]. Surfaces with a shark skin-like texture provide significant drag reduction in turbulent flows [6]. Geckos feet fibrillar structure provides them with remarkable adhesion capabilities, and has allowed the design of functional adhesives [7].

In comparison, the effect of micro-patterning in solid/solid friction has so far attracted much less attention. In a recent paper however, we suggested that epidermal ridges -the regular pattern characteristics of humans and primates glabrous skin surface -strongly modify the skin internal stress elicited upon actively rubbing the fingertip over a finely abraded surface, thus enhancing tactile sensitivity [8]. More precisely, such patterns were shown to produce a quasi-periodic modulation of the texture-induced subcutaneous stress field at a frequency set by the ratio between the scanning velocity and the pattern period. This finding led to the integration of fingerprint-like structure in several robotic hand designs [9,[START_REF] Damian | IEEE/RSJ 2010 Int. Conf. Int. Robots & Syst[END_REF]. Interestingly, this effect not only shows up in local stress measurements but is also observed in the global friction force signal measured while scanning a human finger on a rough surface [START_REF] Prevost | [END_REF]. Since the friction force signal conveys a large part of the tactile information for texture discrimination, as recently evidenced in psychophysical assays [12], it is crucial from both biological and robotic point of views to understand the physical origin of such texture-induced friction force modulations and to identify the parameters which control their spectral characteristics.

To address this question, we investigate in this Letter the friction dynamics between a patterned elastomer block and a finely abraded glass surface in a sphere-on-plane geometry (Fig. 1). The elastomer blocks (thickness ≈18 mm) were obtained by molding liquid PolyDiMethylSiloxane-crosslinker mixture (PDMS, Sylgard 184, Dow Corning) in concave spherical lenses of radius of curvature R = 128.8 mm. Prior to molding, the lenses surface was patterned with a periodic square grating of depth 40 µm and spatial period λ = 125, 218 or 760 µm (width λ/2) using soft photolithography techniques [8]. The elastomer Young's modulus was measured to be E = 2.2 ± 0.1 MPa. The abraded planes consisted in microscope glass slides sandblasted or abraded using a Silicon Carbide powder-water mixture. The surface topography was measured with an optical profilometer (M3D, Fogale Nanotech) from which the root meansquared height h rms and surface profile autocorrelation spectrum C(q) were computed [13]. Two substrates were considered, referred to as "rough-" (h rms ≈ 2.3 µm) and "rough+" (h rms ≈ 5 µm), respectively. For both, C(q) displays a self-affine behavior as usually observed for real surfaces [14] of the form C(q) = C 0 1 + (q/q c ) 2 -α with (C 0 , q c , α) = (1.3 10 -22 m 4 , 6 10 4 m -1 , 1.29) and (1.3 10 -20 m 4 , 8.5 10 3 m -1 , 1.32) for substrate rough-and rough+, respectively. Substrates were rubbed against the elastomer surface along the x direction at constant velocity v (range 0.01 -0.5 mm.s -1 ) with a motorized translation stage (LTA-HL, Newport), under constant normal force F N (range 0.02 -2N). F N and tangential force F S were measured with a 1 mN accuracy at 1 kHz by monitoring the deflections of two orthogonal cantilevers (Fig. 1) using capacitive position sensors (MCC-10 and MCC-20, Fogale Nanotech). The apparent contact zone, imaged in a light-transmitted geometry (Fig. 1) had a diameter ranging from 1 to 6 mm, much larger than the substrate roughness characteristic scale. Typical F S signals in steady sliding are shown on Fig. 2a-d as a function of the scanned distance u = v.t with v = 0.1 mm.s -1 perpendicular to the ridges direction and F N = 0.5 N. Minute but clearly measurable quasi-periodic oscillations are observed, with no equivalent in the normal force signals. Their amplitude weakly depends on λ and increases with F N . They are significantly more pronounced with the rough+ than with the rough-substrate (Fig. 2b and2d). The effect entirely vanishes when the ridges direction is aligned with the direction of motion. In the range of velocities explored, these modulations are independent of v. All experiments presented further are thus done at v = 0.1 mm.s -1 and u is taken as the time-varying variable.

The power spectrum S [δFS] (q) of δF S (u) = F S (u) -FS (the bar stands for time averaging) was computed by averaging over a 20 mm travel distance. All spectra exhibit well defined peaks at q λ = 2π/λ and at corresponding harmonics (Fig. 2e). S [δFS ] at q = q λ increases with F N as AF ν N (Fig. 3) where values of A and ν are collected in Table 1. As mentioned above, A depends weakly on λ and is significantly higher for the rough+ than for the rough-substrate. The exponent ν is close to 1 for all experimental configurations.

In parallel to the force measurements, rapid imaging of the contact zone was performed using a fast camera (Fastcam APX-RS, Photron, Japan) operating at 60 Hz. At all loads, contact occurs only at the ridges summits, thus yielding a large contrast between the top and bottom of the pattern and allowing for a precise tracking of the ridges edges with ≈ 10 µm accuracy. No stick-slip motion of the ridges was observed. Thus, the measured force oscillations cannot be accounted for by periodic stick-slip events whose frequency is set by the period of the pattern as reported in [2]. This is further supported by the observation that the fluctuations are quasi-static (v independent).

In the rest of this Letter, we aim at understanding the physical origin of the observed friction force modulation and the dependence of its amplitude with the normal load, pattern period and substrate roughness characteristics. We first consider a perfectly smooth elastomer surface (no patterning) rubbed against an abraded glass slide. We denote p(x, y, u) and τ (x, y, u) the normal and tangential stress field at a given position u of the substrate, such that F N (u) = p(x, y, u) dx dy and F S (u) = τ (x, y, u) dx dy. Under a local Amontons-Coulomb friction law assumption i.e. τ (x, y, u) = µ 0 p(x, y, u) with µ 0 a uniform (material dependent) friction coefficient, the friction force reads F S (u) = µ 0 F N and is thus expected to be time-independent. We therefore hypothesize a weakly non-linear relationship between p(x, y, u) and τ (x, y, u) which is equivalent to postulating a pressure dependence of the local friction coefficient, i.e. µ = τ /p = µ(p). The friction force now reads

F S (u) = µ(p)p(x, y, u) dx dy (1)
The spatial variations of p(x, y, u) can be decomposed into a time-averaged component p(x, y) set by the macroscale plane-on-sphere contact geometry and a timefluctuating component δp(x, y, u) associated with the microscopic roughness of the glass slide. For simplicity, we further assume that the elastomer is in intimate contact with the glass surface. This assumption is expected to fail at the micro-asperities scale but should be valid at intermediate scale such as the inter-ridge distance λ. In this limit, the texture-induced pressure modulations are set by the topography of the glass substrate and are independent of the local mean pressure imposed by the Hertzian geometry p(x, y). The pressure modulation field δp(x, y, u) is thus a sole function of the position of the substrate with respect to the contact zone (u -x, y), so that the pressure field reads

p(x, y, u) = K(u) (p(x, y) + δp(u -x, y)) (2)
The normalization factor K(u) ensures that F N remains constant during sliding. This condition imposes

K(u) = 1 + 1 F N δp(u -x, y) dx dy -1 ≈ 1 - 1 F N δp(u -x, y) dx dy (3)
The latter expansion is valid when the contact diameter is much larger than the texture scale. Indeed, in this 0.64 λ =218 µm, rough+ 3.2 10 -9 0.81 1.9 10 -9 0.88 λ =760 µm, rough-6.7 10 -10 0.75 2.0 10 -10 0.4 limit, the integral of the roughness-induced pressure field becomes vanishingly small with respect to the confining force F N . Rewriting Eq. 1, one obtains

δF s (u) = F s (u) -F s = (µ(p) -µ )δp(u -x, y) dx dy
where µ is defined as the ratio FS /F N .

The friction force fluctuations thus appear as the convolution product of a function characterizing the friction coefficient spatial heterogeneities and the textureinduced pressure modulations field. The presence of regular ridges at the surface of the elastomer imposes that the stress between ridges vanishes. This can be accounted for, in first approximation, by introducing a λ-periodic Heavyside function H(x) = θ(sin(2πx/λ)) under the integral, which directly results in a spectral selection of the associated spatial mode.

In Fourier space [13], the fluctuating component of the friction force thus reads

S [δFS ] (q u ) = (2π) 4 dq v |F (H(x)(µ(x, y) -µ ))| 2 S [δp]
Wave vector q u (resp. q v ) refers to the parallel (resp. perpendicular) direction to the sliding direction. Under the assumption of intimate contact, S [δp] can be simply expressed as a function of the roughness spectrum C(q) [13] as S [δp] (q) = (E * /2) 2 q 2 C(q), where q is the norm of (q u , q v ) and E * the reduced Young's modulus. The friction coefficient spatial field, on the other hand, is estimated from global force measurements. For all experimental configurations, a power-law dependence is observed between both force components FS = B.F γ N with γ = 0.87 ± 0.04 (Fig. 4). We postulate a power-law relationship between the local shear stress and the local pressure, τ (x, y) = β pm (x, y) [START_REF] Scholz | The Mechanics of Earthquakes and Faulting[END_REF]. Assuming that p(x, y) follows a Hertz profile, the friction force F S can be derived by analytically integrating the latter equation over the contact area. This procedure yields approximated values for β and m as function of B and γ. Figure 4 shows a comparison between the FS versus F N obtained experimentally and through integration of τ (x, y) = β pm (x, y) over the contact area.

Both predicted and measured amplitudes of S [δFS ] (q λ ) with F N are presented on Fig. 3, for all experimental configurations, using the set of parameters (β, m, C 0 , q c , α) obtained as described earlier. The predicted amplitude of F S modulations is in reasonable agreement with the data, without any adjustable parameter. The model captures in particular the analytic increase of S [δFs] (q λ ) with F N , with an exponent ν comparable to its experimental counterpart (see Table 1 and Fig. 3). By tuning systematically the input of the model (m, λ), ν is found to increase linearly with m and to decrease linearly with λ in the investigated range (not shown). The large uncertainty of ν (∼ 0.1) in the model is thus mostly due to the experimental inaccuracy in evaluating the parameter m. The predicted amplitude A is trivially proportional to C 0 , slightly decreases with λ and falls rapidly to zero when m approaches unity, i.e. when Amontons-Coulomb friction's law is verified locally. This study shows that any non-linearity in the friction law leads to the development of texture-induced friction force fluctuations. Since purely linear friction laws are scarcely observed, the present mechanism should be relevant to most practical situations. However, such fluctuations are expected to be in practice hardly detectable when the fixed substrate (here the PDMS block) is smooth and the contact zone diameter is much larger than the typical roughness scale. In contrast, the presence of a regular pattern at the surface of the block operates a spectral filtering of the texture induced pressure modulation which induces a mode selection of the force fluctuations at a particular frequency. The amplitude of these force fluctuations can then become an order of magnitude larger than in the smooth case as shown by the present study. This mechanism was here examined for the simplest possible pattern, i.e. regular parallel stripes, but richer force signal spectra should be similarly obtained with more complex patterns.

In essence, the macroscopic friction force signal carries information about the spectral content of the microscopic substrate topography at frequencies defined by the elastomer micro-pattern. A practical consequence is that F S fluctuations amplitudes could be used to discriminate surfaces having small differences in their roughness at scales much smaller than the contact extension. Hence, under the same conditions, force fluctuations are roughly 10 times larger for the substrate having a h rms ≈ 5µm than for the one with h rms ≈ 2.3µm. This mechanism could also facilitate the detection of motion. The transition between static and sliding contact should manifest as the sudden appearance of large single mode oscillations of the friction force. One may finally foresee the possibility to evaluate the scanning velocity, based on the sole friction force signal, by continuously extracting the peak frequency of these oscillations. We believe that this "spectrotribometry" approach may be relevant to human digital touch and prehension, owing to the presence of fingerprints, and that it could be easily implemented in tactile robotic sensing devices.

FIG. 1 .

 1 FIG. 1. (color on line) Sketch of the experimental setup. Inset: image of a typical contact at FN = 0.5 N with λ = 125 µm. The red circle defines the apparent contact perimeter.

FIG. 2 .

 2 FIG. 2. (color on line) (a)-(d) Blow up of FS/ FS fluctuations (resp. FN / FN in thin black lines) versus u/λ at FN =0.5 N for different λ and both substrates in steady sliding. (e) Averaged power spectra of FS fluctuations (λ=125 µm, FN =0.5 N, "rough-" substrate), both with v perpendicular (•) and parallel (•) to the ridges.

2 FIG. 3 .

 23 FIG. 3. (color on line) (a)-(d) S [δF S ] (q λ ) versus FN . Dashed lines are best experimental fits, solid lines are model predictions. Shaded areas bound the values of A by tuning the parameters of the model.

FIG. 4 .

 4 FIG. 4. (color on line) Time-averaged friction force Fs versus FN . Different symbols correspond to different configurations. Inset: Average dynamic friction coefficient FS/FN versus FN . On both graphs, lines are power law fits with γ = 0.87 ± 0.04. The dashed line has a slope of 1.

TABLE I .

 I : Experimental and model parameters. =125 µm, rough-1.6 10 -10 1.1 1.7. 10 -10 0.87 λ =218 µm, rough-2.3 10 -10 0.92 4.8 10 -10

	Configuration	A exp	ν exp	A model	ν model
	λ

The authors deeply thank F. Petrellis for illuminating mathematical surgery, F. Zalamea for his valuable help, A. Chateauminois and C. Frétigny for fruitful discussions, and acknowledge financial support from ANR-DYNALO NT09-499845.


Christoph Grunau
email: christoph.grunau@univ-perp.fr

Jérôme Boissier

No evidence for lateral gene transfer from salmonids to Schistosomes (or vice versa)

No evidence for lateral gene transfer between salmonids and schistosomes.

are parasitic helminths that cause a severe disorder called bilharziosis or schistosomiasis in humans and several animals. The deduction of gene transfer was based on the analysis of an EST library for Schistosoma japonicum (Adult SjC 7/94, GenBank accession number BU712912) and on PCR on genomic and cDNA of S. mansoni and S. japonicum. The study identified salmon transposon-like sequences in schistosomes, and these sequences were deposited in GenBank as schistosomesspecific, salmon-like repeats (GenBank accession numbers AY834394 to AY834403).

Naturally, the report stirred up a considerable interest in the scientific community since the evolutionary conclusions were far-reaching [2][3][4] . Nowadays, schistosomatidae are parasitic helminths of mammals and birds, and a relation to fish that would be close enough to allow for gene transfer made it necessary to reconsider the evolutionary history of these blood-flukes. We show here that we find no evidence for the proposed transfer of repetitive sequences from salmonids to S. mansoni and S.

japonicum.

We re-sequenced the genomes of female and male S. mansoni using the Genome Analyzer II (Illumina) by single end sequencing according to the manufacturers protocol. We obtained a total of 17,955,578 reads (~645 Mb). Sequences are available at the NCBI sequence read archive (study accession number SRA012151.6). These sequences were aligned to the 1278 repeats (total number of bp: The sample history of the EST library "Adult SjC 7/94" is well documented in the "note" section of GenBank accession number BU712912. The authors of the database entry mention there that 2-3% of the clones contain inserts with homology to salmon DNA and apparently they considered this as contamination. Salmon sperm DNA has traditionally been used as a carrier material in many laboratories. We hypothesize that such a contamination had misguided the authors of the original article. Cross-species and vector contamination would not be surprising and are found in many databases 5 .

We hope to initiate with the present work a re-examination of evolutionary theories concerning Schistosomatidae and Salmonoidae that were based on the initial -and as we think erroneous -report of horizontal gene transfer between these clades.

Supplementary note

Lateral gene transfer could only be imagined if in one point of the evolutionary history, salmonid fishes were in close physical contact to schistosomes (e.g. parasited by the latter). Consequently, both groups of organisms must have lived at the same time in the same place and schistosomes must have got inside (and out) the salmonids somehow. We refer to this as encounter and compatibility filters 1 . From what is known about the biology of both taxonomic groups, both filters are probably closed in a hypothetical Salmonid-Schistosome host-parasite system. First, salmonids are typically cold-water fishes living higher the northern tropic, while schistosomes and their intermediate hosts occur in the intertropical zones. Moreover, both species, salmon and snail, are highly sensible to temperature variations: schistosomes do not infest under 20°C, while upper lethal limits of salmonids is around 25°C. For instance, in the Biomphalaria glabrata / S. mansoni host parasite system, the mortality rate of infected molluscs is 96% at 16°C, and after 2 weeks at 15°C the cercarial production cease completely 2 . In the Oncomelania hupensis / S. japonicum host parasite system, Zhou et al. 3 and Yang et al. 4 described that at 21°C the duration of development of the parasite inside its mollusc host is two times longer compared to 30°C. For salmonids, the negative effects of high temperatures (>15°) are also extensively documented in the literature (see 5 for review). Warm temperature reduces fecundity, decrease egg survival, retard growth of fry and smolts, and increase mortality in all developmental stages. For instance, the maximum temperature for spawning and for egg or alevin to survive is only 12°C. In other words, when schistosomes start to infect, their hypothetical salmon hosts start to die (and vice versa).

Second, schistosomes parasitize only endothermic animals (birds and mammals) [START_REF] Brant | [END_REF] . Even if schistosome cercariae could be found in the same area as a salmon species (i.e. the encounter filter would be open) it remains highly improbable for the parasite cercariae to infect the fish because host temperature is a key component of the host finding and host penetration for both S. mansoni and S. japonicum [7][8][9] . S. mansoni cercariae are attracted by a thermal gradient 7 and do not attach to a cold support (< 25°C) 9 . Also S. japonicum cercariae migrate in the direction of high temperature and penetration starts only at 30°C surface temperature 8 . The infection of an ectothermic host is not impossible but highly improbable. One might argue that even if today a Salmonid-Schistosome host-parasite system is impossible, gene transfer could have occurred in an earlier system. Indeed, a sister group of the schistosomatidae, the sanguinicolidae, are parasites of fishes and one could imagine a common ancestor that was a parasite of ancient salmonid fishes. The genetic distance of schistosomatidae to sanguinicolidae was estimated using the 18S and 28S RNA genes, and the mitochondrial cytochrome c oxidase CO1 gene. Sequence divergence is 8%, 11.5% and 30%, respectively [START_REF] Brant | [END_REF] . Sequence divergence of the putative S.

mansoni salmon-style repeats to the closest salmonid sequences is 3-5% with the exception of GenBank acc. number AY834399, a 123 bp long SINE fragment (divergence 9.6%) (supplementary table 2). These sequence divergence values correspond to the genetic distance to the closest salmon species (supplementary table 2).

That is to say the genetic distance of schistosomatidae to a sister group is larger than the genetic distance to the putative donor species for the gene transfer. Consequently, either there is stronger selective pressure on the repeats than on essential genes such as rRNA,

or the transfer has occurred after the separation of the sister groups. Neither the first nor the latter (as outlined above) is likely in our opinion.

table 1 .

 1 Alignment of the sequence of BU711870.1 spanning the primer binding sites with the sequence of the PCR product delivered 47.6% sequence identity (Needle, http://www.ebi.ac.uk/Tools/emboss/align/). Comparison with the genomic sequence of S.mansoni identified the PCR product as part of a unique sequence on scaffold 000213 (position 409215-409650, 99% identity). For the Hpa primer pairs 1 , the PCR product has a size of 175 bp instead of 159 bp predicted size, and alignment with AY834401.1 gives only 45% similarity. Blast finds a unique 100% match on S. mansoni genomic scaffold 000001 (position 1882577-1882721). For the Igf primer pair, the predicted size is 266 bp, the observed size is 1041 bp and the similarity to AY834397.1 is very low (18.7%). Primer sequences are given in supplementary In short, we did not observe PCR amplification with the previously used primers

	1,305,387) in the

whole-genome shotgun reads and the nucleotide collection at the NCBI Blast server (http://blast.ncbi.nlm.nih.gov/Blast.cgi, blastn default parameters) was searched but did not deliver significant hits ("50% coverage, "50% identity) other than the abovementioned GenBank entries. Finally, we attempted to PCR amplify the Sm_salmonid sequences. First we used primers that were designed based on the GenBank entry AY834402 and genomic DNA extracts of S. mansoni larvae (cercariae) or adult worms of African and Middle American origin (Oman, Egypt, Guadeloupe and Brazil) as template. Rainbow trout (Oncorhynchus mykiss) DNA served as control for a touch-up PCR with 30 cycles starting at 42°C (18 degrees below the calculated Tm, i.e. low stringency). In no case, amplification occurred for the schistosomes (supplementary figure

1 A

). We then tested the primer pairs that were originally used by Melamed et al.

1

(personal communication) with genomic DNA of S. mansoni and primer pairs Igf, PRL and LH (supplementary figure

1 C

). The PCR products were cloned and sequenced. For LH, a PCR product of 461 bp was amplified. This is larger than the expected size of 424 bp based on GenBank entry BU711870.1 cited in

1

.

for the putative salmonid-like repeat sequences in S. mansoni or S. japonicum. Taken together, our experimental results, our in-silico analysis, and the schistosome and salmon eco-ethology and the literature (see supplementary note) do not support the view that gene transfer occurred from salmonids to schistosomes or between their ancestors.

One might wonder how such a relation could have been established in the first place.

Acknowledgments

The authors wish to thank Michael Blouin (Oregon State University, Corvallis, USA) for the gift of salmon DNA, Egyptian and Oman strains of S. mansoni were a gift of Hélène Moné and S. japonicum adults a gift of André Theron. Jean-François Allienne, Nathalie Arancibia, Bernard Dejean, and Anne Rognon provided technical support. Rémi Emans set up the local Blast databases. Illumina sequencing was done at the Plateforme MGX, Institut de Génomique Fonctionnelle, Montpellier. France. Discussion with Claude Combes helped to improve the manuscript tremendously. The authors are grateful to Philippa Melamed for providing information about the PCR primers and conditions that were used by her.

Author contributions

C.G. and J.B. designed the experiment and wrote the manuscript. C.G. performed the experimental work and data analysis

Competing financial interests

The authors declare no competing financial interests.

Abbreviation used in the text

Forward primer

Reverse primer