Henrique B Da Silva

Zenilton K G ‡ Do Patrocínio
email: zenilton@pucminas.br

Guillaume Gravier

Laurent Amsaleg
email: laurent.amsaleg@irisa.fr

Arnaldo De A Araújo
email: arnaldo@dcc.ufmg.br

Silvio Jamil
email: sjamil@pucminas.br

F Guimarães

Near-Duplicate Video Detection Based on an Approximate Similarity Self-Join Strategy

The huge amount of redundant multimedia data, like video, has become a problem in terms of both space and copyright. Usually, the methods for identifying near-duplicate videos are neither adequate nor scalable to find pairs of similar videos. Similarity self-join operation could be an alternative to solve this problem in which all similar pairs of elements from a video dataset are retrieved. Nonetheless, methods for similarity self-join have poor performance when applied to highdimensional data. In this work, we propose a new approximate method to compute similarity self-join in sub-quadratic time in order to solve the near-duplicate video detection problem. Our strategy is based on clustering techniques to find out groups of videos which are similar to each other.

I. INTRODUCTION

In recent years, research involving video retrieval has been increased, mainly due to the popularity of sites for video sharing and viewing over the Web, e.g., YouTube. However, according to [START_REF] Wang | GPU-based MapReduce for large-scale near-duplicate video retrieval[END_REF], several videos are either identical or almost identical when they are compared to each other. Usually, they differ in the format, encoding parameters, and the use of edition operations for the addition and/or removal of frames. This kind of videos was defined by [START_REF] Wu | Real-Time Near-Duplicate Elimination for Web Video Search With Content and Context[END_REF] as near-duplicate as their visual content is similar. As reported by [START_REF] Wu | Practical elimination of nearduplicates from web video search[END_REF], in a query for videos done in the Web, the search results presented 27% of redundancy and 20% of exact duplicate videos among all near-duplicate videos.

There are many works addressing the problem of Near Duplicate Video Retrieval (NDVR) and Near Duplicate Video Localization (NDVL). According to [START_REF] Chou | Pattern-Based Near-Duplicate Video Retrieval and Localization on Web-Scale Videos[END_REF], NDVR and NDVL can be classified into: (i) frame-level approaches, that figure out video similarity by computing similarity frame-by-frame from the videos; (ii) spatiotemporal methods using representative segments from videos based on spatiotemporal features [START_REF] Chou | Pattern-Based Near-Duplicate Video Retrieval and Localization on Web-Scale Videos[END_REF], [START_REF] Shang | Real-time large scale near-duplicate web video retrieval[END_REF]; (iii) video-level approaches, in which videos are represented by a global signature; and (iv) hybrid hierarchical methods, that perform a filter-and-refine approach, in which, non-near duplicate videos are filtered out (filtering step) and a ranking of the remain videos is performed (refine step) [START_REF] Wu | Practical elimination of nearduplicates from web video search[END_REF]. Some methods are also based on hash approaches to match similar videos and solve NDVR problem using global or local visual features as video representation [START_REF] Song | Multiple feature hashing for real-time large scale near-duplicate video retrieval[END_REF]. As the amount of data grows, some works address the near-duplicate video detection problem by using parallel solutions such as MapReduce framework or GPU for large-scale NDVR problem [START_REF] Wang | GPU-based MapReduce for large-scale near-duplicate video retrieval[END_REF]. When using the video-level strategy, the common approach to identify near-duplicate videos is based on the following: (i) videos are segmented into shots that are represented by key-frames; (ii) each key-frame is described by a signature (or descriptor), usually belonging to a high-dimensional space; (iii) then a global signature is computed over the key-frame descriptors to represent the whole video; and (iv) the similarity among videos can be computed by using either the set of signatures of key-frames or the global signature.

Unfortunately, the common strategy is not efficient for finding near-duplicate objects in a huge amount of highdimensional multimedia data since it is necessary to compute the similarity between each pair of elements. The retrieval of all similar pairs of videos from two collections is called similarity join, hereafter. More formally, let R and S be two video collections. The similarity join between R and S is the set of all pairs of elements r ∈ R and s ∈ S such that d(r, s) ≤ λ, in which d(r, s) and λ are a distance function and a given threshold, respectively. If both video collections are the same, this operation is called similarity self-join.

From an application point of view, similarity join has been adopted to find pairwise elements in two datasets, for instance: (i) similar pairs of strings from two collections of strings which is an important operation in data integration and cleansing [START_REF] Jiang | String Similarity Joins: An Experimental Evaluation[END_REF]; (ii) near-duplicate objects (regarding Web pages documents) [START_REF] Xiao | Efficient similarity joins for near-duplicate detection[END_REF]; (iii) data groups using density-based clustering [START_REF] Kim | DBCURE-MR: An efficient density-based clustering algorithm for large data using MapReduce[END_REF]; and (iv) similar time-sequence data [START_REF] Chen | Efficient Similarity Join for Time Sequences Using Locality Sensitive Hash and Mapreduce[END_REF]. To find near-duplicate objects by similarity join operation, a naive (or standard) method computes a distance between each pair of objects in O(n 2 ) time, where n is the size of the dataset [START_REF] Bryan | Compact Similarity Joins[END_REF]. Unfortunately, this computation is too expensive for many applications. In order to reduce the computational effort of similarity join, many approaches have been proposed [START_REF] Chen | Efficient Similarity Join for Time Sequences Using Locality Sensitive Hash and Mapreduce[END_REF], [START_REF] Bryan | Compact Similarity Joins[END_REF], [START_REF] Arasu | Efficient exact set-similarity joins[END_REF], [START_REF] Rong | Efficient and Scalable Processing of String Similarity Join[END_REF], [START_REF] Wang | Intelligent Similarity Joins for Big Data Integration[END_REF]. Moreover, it is also desirable to avoid the output explosion problem by obtaining all groups of elements that are similar to each other in the output instead explicitly enumerating all qualifying pairs, as [START_REF] Bryan | Compact Similarity Joins[END_REF] does.

Thereby, due to limitations imposed by quadratic compu- tational time to perform similarity join operation, the use of an approximate algorithm is a promising direction and it has attracted considerable attention. Even though many approaches have been proposed to reduce the computational effort to solve similarity join, high-dimensional data are still hard to manipulate due to the curse of dimensionality. Thus, in this work, we address near-duplicate video detection problem by using approximate similarity self-join operation to find groups of videos which are similar to each other in a database. In Figure 1, we outline our proposed method. In a general way, as the retrieved elements in each cluster are similar to each other, we can consider them as near-duplicate elements. It is important to note that we applied our method to a known dataset used to identify near-duplicate videos. We also applied our strategy to a dataset containing SIFT points in order to evaluate the performance of our strategy in a larger database.

As video descriptor, we adopted a high-dimensional global signature (with thousands of dimensions) based on BossaNova Video Descriptor (BNVD), introduced by [START_REF] Caetano | Pornography detection using BossaNova video descriptor[END_REF]. To describe video content, first we have used SIFT descriptors that were combined into BossaNova, a mid-level image representation [START_REF] Avila | Pooling in image representation: The visual codeword point of view[END_REF]. Finally, BNVD is obtained by applying max function on BossaNova vectors [START_REF] Caetano | Pornography detection using BossaNova video descriptor[END_REF].

The rest of this paper is organized as follows. In Section II, we review some related work. In Section III, we present the proposed approach to solve the near-duplicate video detection problem. In Section IV, we present the results of the experiments performed. Finally, in Section V, we draw some conclusions and discuss future work.

II. SIMILARITY SELF-JOIN

The main goal of the similarity self-join operation is to retrieve all pairs of similar elements belonging to a dataset. Regarding similarity self-join, the first kind of strategy for avoiding the enumeration of all pairs is called filter-andverification approach. This strategy is based on the identification of candidate pairs followed by the verification of those pairs. The identification of candidate pairs is used to filter out pairs which are not in the solution space in order to decrease the number of pairs that need to be verified. This strategy was successfully used by [START_REF] Jiang | String Similarity Joins: An Experimental Evaluation[END_REF], [START_REF] Wang | Intelligent Similarity Joins for Big Data Integration[END_REF] to cope with similarity self-join in the string domain. Several recent approaches, such as qChunk, PPJoin, Ed-Join, adaptJoin, developed strategies for improving the filter step in order to achieve efficiency on the search process. These algorithms were exhaustively compared in [START_REF] Jiang | String Similarity Joins: An Experimental Evaluation[END_REF]. According to [START_REF] Wang | Intelligent Similarity Joins for Big Data Integration[END_REF], parallel approaches by using MapReduce framework have widely been used in large datasets in order to reduce the computation time needed to perform a similarity (self-)join. For more information, the reader should refer to [START_REF] Kim | DBCURE-MR: An efficient density-based clustering algorithm for large data using MapReduce[END_REF], [START_REF] Fries | PHiDJ: Parallel similarity self-join for high-dimensional vector data with MapReduce[END_REF], [START_REF] Deng | MassJoin: A mapreducebased method for scalable string similarity joins[END_REF].

Despite the simplicity of the filter-and-verification approach, its main drawback is the difficulty to produce high performance when applied to large datasets and to high-dimensional spaces. To cope with these issues, some authors have proposed the computation of similarity join directly by methods based on tries [START_REF] Feng | Trie-join: a trie-based method for efficient string similarity joins[END_REF]. Even if the filter-and-verification approach and trie-based methods work well for small datasets, they still have poor performance on large datasets in high-dimensional spaces since the number of distance computation is very high. In this case, there are many approaches that try to cope with high-dimensional data using hash-based techniques. For instance, the method proposed in [START_REF] Zhao | Sim-min-hash[END_REF], called Sim-min-Hash, a generalization of min-Hash, solves image search and link problem handling billions of high-dimensional local descriptors in a fast way. In most cases, high-dimensional data has been handled by application of Locality Sensitive Hashing (LSH), due to its suitable performance on high-dimensional space, as was presented in [START_REF] Satuluri | Bayesian Locality Sensitive Hashing for Fast Similarity Search[END_REF], a Bayesian algorithm, called BayesLSH, for performing candidate pruning and similarity estimation by LSH. Additionally, another method based on hash, proposed in [START_REF] Gao | DSH: Data Sensitive Hashing for High-dimensional K-nnsearch[END_REF], called Data Sensitive Hashing (DSH), solves k-NN problem by improving hashing family through the combination of adaptive boosting and spectral techniques to hash, with large probability, k-NN pairs together.

Accordingly, there are many applications to similarity join using high-dimensional data, like near-duplicate video detection, for instance. However, there are several challenges related to the similarity join problem. Many different types of data are represented in high dimensional space, generally, using in many cases hundred or thousand dimensions. Furthermore, many applications have to deal with a huge amount of data. Similarity join in large datasets has become a difficulty task. Thus, another promising direction to solve similarity join problem is the approximate approach. Approximate algorithm can solve similarity join problem missing some results. In this paper, we aim to solve the near-duplicate video detection problem using an approximate similarity self-join approach based on a clustering algorithm.

III. SIMILARITY SELF-JOIN FOR NEAR DUPLICATE VIDEO DETECTION

A. BossaNova Video Descriptor (BNVD)

Let V be a video sequence. V = f i , i ∈ [1, N ]
, where f i is the keyframe1 of the shot i and N is the number of keyframes. Let Z be a set of BossaNova vectors computed for the video

V. Z = z i , i ∈ [1, N ],
where z i is a BossaNova vector extracted for the keyframe f i . The BNVD of Z can be modeled by a function h, according to [START_REF] Caetano | Pornography detection using BossaNova video descriptor[END_REF], as follows:

h : R Z -→ R Z , Z -→ h({z i }) = [[o m,b ] , p m ] T , o m,b = median(z i m,b ), p m = median(t i m ), (1)
where Z ⊂ {1, ..., M } × {1, ..., B + 1}, and

z i = z i m,b , t i m
T , M is the number of visual codewords, and B denotes the number of bins of each BossaNova histogram.

B. Near Duplicate Video Detection

Let us remember the definition of the similarity join operation. Let R and S be two datasets containing n and m elements, respectively. The result of the similarity join between R and S is the set of all pairs of elements r ∈ R and s ∈ S such that d(r, s) ≤ λ, in which d(r, s) and λ are a distance function and a given threshold, respectively. If both datasets are the same, this operation is called similarity selfjoin. The computational cost of the naive algorithm to compute this operation is O(n 2 ). Even if this cost can be considered reasonable, it is unfeasible for huge datasets. Thus, in this work, we are interested in computing a similarity self-join by using a sub-quadratic method trying to maintain the same quality measures, in terms of precision and recall, of the exact solution which runs in a quadratic time.

In order to decrease the computational time, it is mandatory to avoid distance computations between all pairs. For that, our method for approximate similarity self-join can be divided in if isViable(B,score(s k ),type) then

// Function index returns index of pair in s 14:

 ← index(s k ) 15:
// Add point to bucket 16:

B  ← B  ∪ {pi} 17:
end if

18:
end for 19: end for 20: return B two main tasks, as illustrated in Figure 1: (i) the clustering; and (ii) the bucket assignment. The clustering step aims at identifying salient elements to well represent all dataset, and then, we assign similar elements to the same bucket in which all elements could be considered as similar to each other. However, before applying the proposed similarity self-join strategy to near-duplicate video detection, the videos must be transformed into only one feature vector, and in this work, we have used BossaNova mid-level representation (see [START_REF] Avila | Pooling in image representation: The visual codeword point of view[END_REF] for more information) for describing the video.

Thereby, we propose a general strategy to solve similarity self-join problem. It is an approximate approach based on clustering algorithm to find groups of similar elements in which the dataset is composed by high-dimensional points. There are some advantages to use clustering algorithms. The first one, a set of similar elements are represented by centroids; thus, clustering algorithm aids putting together elements that are similar to each other in the same cluster. Another one is to handle centroids instead to handle all elements from the datasets, avoiding the quadratic time in processing. We should only process centroids to assign all points from the dataset, which is less expensive, specially when using a small list of centroids. Considering that one element could be similar to several other elements, even if they are in different clusters, the bucket assignment should adopt a soft assignment strategy, in which one element is assigned to several buckets.

Basically, our solution of soft assignment, after the K-means clustering, can be summarized in two main steps: (i) the p points of the dataset are assigned to, at most, m nearest clusters; and (ii) the clusters contain all similar points, that can be enumerated for identifying the near-duplicate videos. Moreover, a verification step may be applied in order to Algorithm 2 Similar pairs enumeration algorithm

Require: B = {B1, . . . , BK } Buckets, Bj ⊆ R N ×d Ensure: A ⊆ R d × R d
List of similar pairs 1: // Generates all pairs of points within the same bucket 2: A ← ∅ 3: for l = 1 to K do end for 11: end for 12: return A remove dissimilar videos. Algorithm 1 formalizes the assignment task of our method to approximate similarity self-join with quantization of a set of high-dimensional points p, given a set of clusters identified by their centroids c. Analogously to [START_REF] Bryan | Compact Similarity Joins[END_REF], Algorithm 1 generates a compact representation of the similarity self-join result which is a set of buckets of points that satisfies the threshold of similarity-self join. Thus, for a given number of clusters, the output is linear rather than quadratic in the number of points. Moreover, Algorithm 2 can be used to enumerate all pairs of similar points if it is needed.

As can be seen in Algorithm 1, the function isViable controls the insertion of elements in each bucket, that could represent a similar element (for instance, a video). We have used, in this work, three different strategies to control the insertion of an element p i in the bucket j, which is represent by the centroid c j : (i) without any threshold; (ii) with a threshold; and (iii) with an adaptive threshold. In the first, the element p i is included in the bucket j without any verification. For the specified threshold strategy, the element p i is inserted in the bucket j, if d(p i , c j ) ≤ λ. Regarding the adaptive strategy, an average value of the similarities between all elements of the bucket j and the centroid c j is used to control the insertion of the element p i . This average value can be obtained by Equation 2,

d(p i , c j ) ≤ | Bj | l=1 d(p l , c j ) | Bj | (2)
in which Bj represents the set of points that are already assigned to the cluster related to centroid c j .

In terms of computational cost, the proposed Algorithm 1, runs in O(n × (K + K × log m + m)) to solve the assignment step of the similarity self-join problem, which is the most important one. Considering that m ≤ K, this order can be re-written as O(n × K × log m). It is worth to mention that this strategy identifies the groups of similar elements.

IV. EXPERIMENTS

In order to evaluate the performance of our algorithm for similarity self-join, we performed two different experiments. The first one used a subset of BIGANN dataset [START_REF] Jégou | Product Quantization for Nearest Neighbor Search[END_REF], in which we studied the behavior of the proposed method by using different amount of elements, from 10,000 to 100,000 SIFT vectors. The other experiment is related to the near-duplicate video detection by using the CC WEB VIDEO dataset [START_REF] Wu | Real-Time Near-Duplicate Elimination for Web Video Search With Content and Context[END_REF], [START_REF] Wu | Practical elimination of nearduplicates from web video search[END_REF]. This dataset consists of 13,137 videos downloaded from the Web divided into 24 categories. Differently from [START_REF] Wu | Real-Time Near-Duplicate Elimination for Web Video Search With Content and Context[END_REF], in which the authors used the fusion of content and context information, we used only video content information to identify the near-duplicate videos. In Figure 2, we illustrate some keyframes for each category.

In order to evaluate the results on the BIGANN dataset, the ground-truth was generated by using the naive method for computing the distance between each pair of points. On the other hand, in the CC WEB VIDEO dataset, the ground-truth was modified: instead of representing all similar videos for a given query, we identified all pairs of similar videos through human analysis.

A. Experimental Setup

For the representation of the video, we used BossaNova mid-level representation, which is a global signature. BossaNova is a representation for content-based concept detection in images and videos, which enriches the Bag-of-Words model [START_REF] Avila | Pooling in image representation: The visual codeword point of view[END_REF]. Here, the video is represented by a collection of SIFT descriptors [START_REF] Lowe | Distinctive Image Features from Scale-Invariant Keypoints[END_REF], which are extracted from the key-frames. These SIFT descriptors are encapsulated by a BossaNova descriptor, thus each video is represented by only one descriptor. For SIFT descriptor, we obtained the code from OpenCV repository [START_REF] Bradski | The OpenCV Library[END_REF], and the code for BossaNova is available at http://www.npdi.dcc.ufmg.br/bossanova. BossaNova Video Descriptor was adopted to represent the video frames. We kept the BossaNova parameter values the same as in [START_REF] Caetano | Pornography detection using BossaNova video descriptor[END_REF] and we used max as aggregating function.

Furthermore, in our experiments, we applied K-means algorithm [START_REF] Douze | The Yael library[END_REF] to cluster the dataset by using 30% of the elements of the dataset. In CC WEB VIDEO dataset, we adopted three different values for K: 16, 32, and 64. We performed experiments with each video class separately, and we did not use large values of K, since some classes are very small and a compact video representation (one high-dimensional feature per video) was adopted.

In BIGANN dataset, we adopted four different values for K: 64, 128, 256, and 512. Our soft-assignment algorithm assigns all elements to clusters. BossaNova descriptors were 3009-dimensional features for each key-frame. We performed our algorithm on subsets from BIGANN dataset with 128dimensional SIFT points. We defined a subdataset x-BIGANN as the set of elements of BIGANN with x elements. We also applied Nested Loop (NL) algorithm in order to compare its performance to our approach. Nested Loop algorithm for similarity join computes the distance between all pairs and check if their distance is no larger than a given threshold.

In order to present a comparative analysis, we performed experiments using fixed threshold and without using threshold value to filter out pairs. As mentioned before, our method computes all centroids by K-means clustering and assigns each point p of dataset to the m nearest clusters. We used five different values for m: 10%, 20%, 30%, 40%, and 50% of the number of clusters (K). The threshold value for the Nested Loop algorithm was empirically defined as 75% of the λ.

Experiments were conducted on a computer under Ubuntu 12.04 LTS OS with Intel Xeon Quad-Core 2.4 Ghz and 72 GB RAM. We evaluated the performance of our algorithm by measuring the recall and precision rates of several assessments of near-duplicate detection problem. The presented results are the average values of the five runs of our method.

B. Experimental Quantitative Results

In this subsection, we analyze the experimental results of our approximate similarity join approach. One of the main problems in dealing with Web videos is that there must exist a lot of near-duplicate videos which differ in size, format and long and short versions. The benchmark CC WEB VIDEO dataset brings these variations. In the clustering step of the algorithm, K-means is performed by a sample from all database.

Figure 3 shows the best average results of our method for similarity self-join in the BIGANN dataset experiments with K equals to 256 and m equals to 10% of the number of clusters (K). These results are expressed in terms of F-Measure average. Our approximate algorithm reached more than 90% of F-Measure in average for all datasets using fixed and adaptive threshold. This strategy avoids the similarity selfjoin algorithm to assign points far away to each other in the same cluster. It improves the result of the algorithm. In Table I, we present the comparative results in terms of time to process all experiments for all datasets from BIGANN. Our method outperformed, in terms of time, the Nested Loop approach missing only a couple of results. Beside that, time processing of our algorithm did not grow as Nested Loop, instead. We also present the speedup for computing the similarity self-join as a relation between the time for computing Nested Loop and the time for computing the clustering and assignment tasks.

In Figure 4 we show the best results of F-Measure average obtained by our method when applied to CC WEB VIDEO dataset with K equals to 16 and m equals to 40% of the number of clusters (K). The method without threshold

V. CONCLUSION

In this work, we addressed the problem of near-duplicate video detection by using similarity self-join operation in order to identify groups in which the elements are similar to each other. Experimental results were used to analyze the behavior of video representation using BNVD and BIGANN dataset. The results showed that our solution can solve similarity selfjoin problem in approximated way with more the 90% of F-Measure in 100, 000-BIGANN dataset reaching almost 93% of speedup in relation to nested loop strategy. We also solved near-duplicate video detection problem using approximate solution to similarity self-join, and our method outperformed the nested loop approach, in terms of time, without a significant loss of performance, in terms of F-Measure.

As further work, we consider to explore this kind of strategy to cope with very large datasets, mainly by studying the behavior of other clustering methods and other way to define the insertion in the buckets. Furthermore, we also intend to study the amount of redundant pairs that may be generated by our enumeration algorithm.

Fig. 2 .

 2 Fig. 2. CC WEB VIDEO Dataset.

Algorithm 1 3 :

 13 Approximate similarity self-join algorithm: the assigment task Require: p = {p1, . . . , pN } Data points, pi ∈ R d Require: c = {c1, . . . , cK } Centroids, cj ∈ R d Ensure: B = {B1, . . . , BK } Buckets, Bj ⊆ R N ×d 1: // Initialization of a bucket B i for each cluster 2: for j = 1 to K do Bj ← ∅ 4: end for 5: // Assign each point to at most m buckets and fill in buckets 6: for i = 1 to n do 7: for j = 1 to K do Fill in similarity array s 8: sj ← (j, d(pi, cj))

4 : 6 :

 46 for i = 1 to size(B l ) do 5: pi ← i-th element of B l for j = i + 1 to size(B l ) do 7: pj ← j-th element of B l 8:A ← A ∪ (pi, pj)

1Fig. 3 .

 3 Fig. 3. F-Measure average results for similarity self-join in the BIGANN dataset in which K = 256 and m = 10% of the number of clusters (K).

  0

  the others in most of the video classes without losing time performance. As the experiment was performed in each video class separately, most of the videos in the same class are very similar to each other. Then, specifying a threshold (even an adaptive threshold) can deteriorate the performance. In TableII, we present the comparative results in terms of time to process all experiments for each video class from CC WEB VIDEO, and as expected, our method outperformed, in terms of time, the Nested Loop approach without losing performance. As we can see, our solution based on clustering can perform similarity join much faster than Nested Loop algorithm. The presented time value represents the sum of K-means algorithm plus similarity self-join algorithm.

			Performance of similarity self-join applied to near-duplicate video detection.
		1
		0.8
	F-measure	0.6
		0.4				Adaptative thresholding
						Specified thresholding
		0.2				Without thresholding
		2	4	6	8	10	12	14	16	18	20	22	24	26
							Video Classes
	Fig. 4. F-Measure average results for near-duplicated video detection by using similarity in which K = 16 and m = 40% of the number of clusters (K).

outperformed

TABLE I TIME

 I (SECONDS) TO PERFORM SIMILARITY SELF-JOIN FOR BIGANN DATASET USING SPECIFIED THRESHOLD STRATEGY (CLUSTERING TIME PLUS SELF-JOIN TIME), IN WHICH K = 256 AND m = 10% OF THE NUMBER OF CLUSTERS (K).

	Size ×10 3	From clustering to enumeration Clustering (C) Join (J) Pairs	NL	Speedup (NL/(C+J))
	10	1.56	3.02	0.45	109.80	23.95
	20	6.90	6.13	1.77	475.64	36.48
	30	13.59	9.06	3.77	1060.25	46.79
	40	21.66	12.13	6.83	1908.83	56.47
	50	27.65	15.12 10.29	2891.34	67.60
	60	33.80	18.10 14.76	4170.56	80.34
	70	42.56	21.10 19.18	5691.56	89.39
	80	68.07	24.17 26.61	7424.54	80.49
	90	60.51	27.04 32.19	9194.06	105.01
	100	93.08	30.22 40.91	11404.03	92.49
					average	41.197

TABLE II TIME

 II (SECONDS) TO PERFORM SIMILARITY SELF-JOIN FOR CC WEB VIDEO DATASET WITHOUT THRESHOLD STRATEGY (CLUSTERING TIME PLUS SIMILARITY SELF-JOIN TIME), IN WHICH K = 16 AND m = 40% OF THE NUMBER OF CLUSTERS (K).

	Video class	From clustering to enumeration Clustering (C) Join (J) Pairs	NL	Speedup (NL/(C+J))
	1	0.121	0.326 0.004	13.240	29.62
	2	0.127	0.239 0.002	7.360	20.11
	3	0.049	0.180 0.001	4.130	18.03
	4	0.040	0.142 0.001	2.570	14.12
	5	0.054	0.164 0.001	3.410	15.64
	6	0.251	0.317 0.003	13.038	22.95
	7	0.049	0.151 0.001	2.894	14.47
	8	0.128	0.221 0.002	6.196	17.75
	9	0.042	0.098 0.001	1.206	8.61
	10	0.068	0.123 0.001	1.917	10.04
	11	0.034	0.156 0.001	3.087	16.25
	12	0.192	0.360 0.004	16.993	30.78
	13	0.078	0.172 0.001	3.732	14.93
	14	0.026	0.045 0.001	0.246	3.46
	15	0.422	0.725 0.017	68.796	59.98
	16	0.019	0.086 0.001	0.917	8.73
	17	0.115	0.267 0.002	9.155	23.97
	18	0.077	0.202 0.002	5.190	18.60
	19	0.075	0.234 0.002	6.957	22.51
	20	0.023	0.081 0.001	0.821	7.89
	21	0.079	0.184 0.001	4.383	16.67
	22	0.067	0.174 0.001	3.882	16.11
	23	0.315	0.540 0.011	38.344	44.85
	24	0.047	0.119 0.001	1.794	10.81
	average	0.104	0.221 0.003	9.177	28.22

A keyframe is a frame that represents the content of a logical video unit, like a shot or scene, for example.

ACKNOWLEDGMENTS

The authors are grateful to FAPEMIG/INRIA/MOTIF, CNPq, CAPES and STIC-AmSud for the partial financial support of this work.


Jerôme Bascou
email: jerome.bascou@univ-st-etienne.fr

Bernard Henry

René-Pierre Ménot

Minoru Funaki

Guilhem Barruol

Jérôme Bascou

Contribution of AMS measurements in understanding the migmatitic terrains of Pointe Géologie, Terre Adélie (East-Antarctica)

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Migmatites and anatexites result from progressive melting of rocks under high-grade metamorphism conditions (Wimmenauer and Bryhni, 2007 and references therein). These rocks are often the only key to determine the structural patterns in the deep crust. However, the structural analysis of migmatites is often not very easy because of their complex and multiphase evolution. Moreover, leucosomes and melanosomes produced by partial melting of an initially highly deformed rock acquire a fabric that mostly cannot be determined by classical structural approaches. In partially melted gneisses, i.e. migmatites, this fabric can differ from that of the initial parent rock. In dominantly melted rocks, the only visible fabric is mostly related to the orientation of rafts of melanosomes or mesosomes, i.e. respectively restites or protoliths, and therefore this fabric only offers an incomplete signature of the tectonic events suffered by the rocks during melting. Melanosomes and mesosomes usually exhibit features of metamorphic rocks whereas the leucosomes display an igneous-like appearance. Therefore, one of the characteristics of the migmatite structures is their local variability and then their analysis implies a large collection of data together with a statistical approach. To this aim, the field observations are limited to the few visible strain markers and generally not well-adapted.

The anisotropy of magnetic susceptibility (AMS) technique (see [START_REF] Borradaile | Tectonic applications of magnetic susceptibility and its anisotropy[END_REF][START_REF] Borradaile | Anisotropy of magnetic susceptibility (AMS), magnetic petrofabrics of deformed rocks, magnetic fabrics[END_REF][START_REF] Graham | Magnetic susceptibility anisotropy: an unexploited element of petrofabric[END_REF][START_REF] Tarling | The Magnetic Anisotropy of RocksFirst edition[END_REF] and references therein) has been widely used for structural applications. It offers several advantages relatively to field observations revealing not only foliation and lineation but also intensity and shape of the fabric and it can be applied to a set of samples representative of a significant area. [START_REF] Ferré | Magnetic susceptibility anisotropy: a new petrofabric tool in migmatites[END_REF], [START_REF] Kruckenberg | Viscoplastic flow in migmatites deduced from fabric anisotropy: an example from the Naxos dome, Greece[END_REF] and more recently [START_REF] Viegas | Fabrics of migmatites and the relationships between partial melting and deformation in high-grade transpressional shear zones: the Espinho Branco anatexite (Borborema Province, NE Brazil)[END_REF] used the magnetic anisotropy for migmatite study. They showed that despite rheological variations during deformation (solid-state deformation followed by viscous flow), AMS axes were consistent with observed regional field structures and thus highlighting the interest of the AMS data for the strain analysis in hightemperature metamorphic rocks. In paramagnetic migmatites, for which biotite is the main carrier of the magnetic susceptibility, the degree and the shape of the AMS ellipsoid could be different in granites and diatexites, and in metatexites [START_REF] Hasalovà | Origin of migmatites by deformation-enhanced melt infiltration of orthogneiss: a new model based on quantitative microstructural analysis[END_REF]. Others authors have shown that anatexites could have specific AMS signature, associated with rheological properties and that AMS characterizations could therefore help to better model the different stages of migmatitic terrains [START_REF] Charles | The Montagne Noire migmatitic dome emplacement (French Massif Central): new insights from petrofabric and AMS studies[END_REF][START_REF] Kratinová | AMS record of brittle dilation, viscous-stretching and gravity-driven magma ascent in area of magma-rich crustal extension (Vosges Mts., NE France)[END_REF][START_REF] Schulmann | Influence of melt induced mechanical anisotropy on the magnetic fabrics and rheology of deforming migmatites, Central Vosges, France[END_REF].

The Pointe Géologie archipelago (Terre Adélie, East Antarctica) represents a mid-crustal section that experienced a long-lived anatectic event during Paleoproterozoic times, 1.69 Ga ago (Ménot et al., 2007 and references therein). This archipelago is partially covered by the ice cap and limited outcrops are key zones to constrain the Terre Adélie Craton tectonics. [START_REF] Pelletier | Tectonique transpressive en Terre Adélie au Paléoprotérozoïque (East Antarctica)[END_REF] and [START_REF] Gapais | Paleoproterozoic tectonics in the Terre Adélie Craton (East Antarctica)[END_REF] proposed a detailed structural and kinematic analyses of the area based on foliation and lineation measurements of migmatites. But we believe these structural data can be supplemented by taking into account the magnetic fabric of the leucosomes and anatexites, which correspond to more than 50% of the outcrops and which are devoid of clear strain markers. This would greatly improve the knowledge of the structural pattern of this region.

The first aim of this paper is thus to specify the deformation framework associated to the Paleoproterozoic event in the Terre Adélie Craton from a large sampling in the Pointe Géologie rocks. For this, a detailed AMS study was performed in the migmatites, the larger granitic dykes and the anatexites observed in the field. Finally, we focus on various migmatitic structures of centimetric to decametric size; they correspond to various phases of segregation, extraction and transfer of melts. We discuss the relationships between these different migmatitic habitus and their related AMS signature in order to constrain the rheological evolution of the molten crust.

Geological setting

The Terre Adélie Craton (TAC) is located in the easternmost part of the East Antarctic Shield. It crops out as islands and capes scattered along the coast between 135°and 145°E but most of the outcrops are located in three main archipelagos, from West to East: around Pointe Géologie (66°40′S, 140°E), East of Port Martin (66°49′S, 141°24′E) and East of Cape Denison (67°00′S, 142°40′E) up to the Mertz glacier. This crystalline basement consists of Late Archean and Paleoproterozoic formations devoid of any significant reworking since 1.5 Ga and then it is considered as part of the Mawson block or Mawson continent [START_REF] Fanning | The "Mawson Block": once contiguous Archean to Proterozoic crust in the East Antarctic Shield and Gawler Craton, Australia[END_REF][START_REF] Fanning | A closer examination of the direct links between Southern Australia and Terre Adélie and Georges V land[END_REF]. At the scale of the TAC (Fig. 1a), the Late Archean and Paleoproterozoic terrains are distributed westwards and eastwards from Port Martin, respectively [START_REF] Ménot | Geology of the Terre Adélie Craton[END_REF][START_REF] Monnier | Actualisation des données géologiques sur Terre Adélie (Antarctique Est): mise en évidence d'un collage tectonique au Protérozoïque[END_REF].

This paper deals only with the Paleoproterozoic formations of the Pointe Géologie area, i.e. around the permanent French Polar Station Dumont d'Urville (DDU). As described by Bellair (1961a,b), the main rock types are migmatitic to anatectic gneisses that are cross cut by dykes of apparently isotropic coarse-grained pink granites. The detailed mapping, carried on during the GEOLETA project, improves this previous description and points out some new and significant features [START_REF] Gapais | Paleoproterozoic tectonics in the Terre Adélie Craton (East Antarctica)[END_REF][START_REF] Monnier | Actualisation des données géologiques sur Terre Adélie (Antarctique Est): mise en évidence d'un collage tectonique au Protérozoïque[END_REF][START_REF] Peucat | The Terre Adélie basement in the East-Antarctica Shield: geological and isotopic evidence for a major 1.7 Ga thermal event; comparison with the Gawler Craton in south Australia[END_REF]. Migmatitic gneisses and anatexites derived from a sedimentary sequence dominated by aluminous pelites together with subordinate greywackes, quartzites and calc-silicates. Metagreywakes appear as intercalations ranging from tens of centimetres to hundreds of metres in thickness. Quartzites and calc-silicates form dismembered layers and enclaves of mesosomes within the migmatites and they mark out a poorly preserved sedimentary layering (Fig. 2a). Amphibolitized dolerites and gabbros occur as dykes and sills and they are coeval with the anatectic event as shown by magma mingling features between the mafic intrusions and the granitic magmas. Isotopic data indicate Paleoproterozoic sources for both sediments and magmas (Peucat et al., 1999 and references therein). According to [START_REF] Monnier | Actualisation des données géologiques sur Terre Adélie (Antarctique Est): mise en évidence d'un collage tectonique au Protérozoïque[END_REF], mesosomes, neosomes and anatexites represent respectively 10%, 60% and 30% of the rock types in the Pointe Géologie archipelago.

These migmatites and anatectites formations underwent a polyphased tectonic and metamorphic evolution with three phases of deformation leading to a composite sub-vertical regional S 0 -S 1 -S 2 foliation [START_REF] Monnier | Actualisation des données géologiques sur Terre Adélie (Antarctique Est): mise en évidence d'un collage tectonique au Protérozoïque[END_REF][START_REF] Pelletier | Tectonique transpressive en Terre Adélie au Paléoprotérozoïque (East Antarctica)[END_REF][START_REF] Pelletier | From partial melting to retrogression in the Pointe Geologie migmatitic complex: a history of heterogeneous distribution of fluids[END_REF]. Neosomes display two successive parageneses related to both the S 1 and S 2 metamorphic foliations and bearing more or less comparable Sil ± Kfs ± Pl ± Bt ± Cd ± Grt ± Ilm ± Qtz mineral assemblages. The leucosomes are enriched in Qtz, Kfs and Pl and also contains Crd and Sil and subordinate Bt and Grt. Moreover, mineral chemistry of the S 1 and S 2 parageneses and of the successive generations of leucosomes is similar and argues for a single continuous Low Pressure-High Temperature metamorphic event, the peak conditions reaching 750-700 °C at 0.6-0.4 GPa. Large amounts of anatectic melts have been produced all along the tectonic and metamorphic evolution as suggested by the various habitus of the leucosomes and their chronology relative to the composite foliation S 1 -S 2 (Fig. 2). The oldest leucosomes appear as mylonitic lenses isoclinally folded by the S 1 -S 2 transposition, but most of them are undeformed and concordant within the S 2 foliation and the latest melts cross cut the S 2 planes. But all the successive generations display the same mineral assemblages and chemistry. A retrograde imprint with Bt ± Ms ± And assemblages marks the later evolution to about 550-450 °C and 0.5-0.4 GPa [START_REF] Gapais | Paleoproterozoic tectonics in the Terre Adélie Craton (East Antarctica)[END_REF][START_REF] Monnier | Actualisation des données géologiques sur Terre Adélie (Antarctique Est): mise en évidence d'un collage tectonique au Protérozoïque[END_REF][START_REF] Pelletier | Tectonique transpressive en Terre Adélie au Paléoprotérozoïque (East Antarctica)[END_REF].

According to [START_REF] Pelletier | Tectonique transpressive en Terre Adélie au Paléoprotérozoïque (East Antarctica)[END_REF] and [START_REF] Gapais | Paleoproterozoic tectonics in the Terre Adélie Craton (East Antarctica)[END_REF], the Pointe Géologie archipelago shows two types of structural domains: N-S to N340°striking vertical shear zones and domes, both of them developed during the metamorphic and anatectic event (Fig. 1b). Between the vertical zones, the foliation attitudes point out dome-shaped structures marked by flat lying to gently plunging fabrics. Domes display a mineral lineation outlined by a preferred orientation of sillimanite crystals while such a lineation is almost invisible in the vertical zones. The transition from flat lying to vertical foliation zones can be sharp, with development of vertical foliation planes cutting across the horizontal foliation of domes. It can also correspond to gradual folds with horizontal axes. Such a transition from dome to vertical shear zone never exceeds 500 m in thickness. Frequently, the vertical structures seem to be younger than the flat-lying ones but flat-lying structures reworking vertical ones have also been observed. Moreover, all these structures were developed under similar Pressure-Temperature and syn-anatectic conditions and then can be considered as nearly coeval [START_REF] Monnier | Le socle protérozoïque de Terre Adélie (Antarctique Est): son évolution tectonométamorphique et sa place dans les reconstitutions du protogondwana[END_REF][START_REF] Pelletier | Tectonique transpressive en Terre Adélie au Paléoprotérozoïque (East Antarctica)[END_REF]. Such patterns could be explained in the frame of a regional convergence, toward a cool rigid Archean craton, of a hot and weak juvenile crust leading to horizontal crustal flow and strain partitioning with localized deformation along transpressional shear zones.

Bellair and Delbos (1962) assumed a 1.5 Ga age (Rb/Sr on micas) for the partial melting event. More recently, new radiometric data [START_REF] Peucat | The Terre Adélie basement in the East-Antarctica Shield: geological and isotopic evidence for a major 1.7 Ga thermal event; comparison with the Gawler Craton in south Australia[END_REF], U-Pb on zircons and monazites, Sm-Nd and Rb-Sr on whole rocks and minerals) gave precise time constraints on the geological evolution of the meta-volcano-sedimentary formations of Pointe Géologie. Time of sediments deposition is bracketed between the inherited zircon ages (1.76-1.72 Ga) and the peak of metamorphism and partial melting occurred at 1.69 Ga. The later cooling path was constrained by Nd garnet ages (650-600 °C at 1.63-1.60 Ga) and Rb-Sr muscovite and biotite ages (450-350 °C at 1.5 Ga).

High temperature rock types and regional structures

While the lithology is very homogeneous in the scale of the archipelago, field relationships between different rock types are much more complex in the outcrops scale. Four main rock types have been sampled for AMS measurements: migmatites including leucosomes and melanosomes, coarse-grained pink granites, anatexites and strongly mylonitic gneisses.

Migmatitic gneisses

They have been described previously as the K-feldspar gneisses [START_REF] Monnier | Actualisation des données géologiques sur Terre Adélie (Antarctique Est): mise en évidence d'un collage tectonique au Protérozoïque[END_REF][START_REF] Pelletier | From partial melting to retrogression in the Pointe Geologie migmatitic complex: a history of heterogeneous distribution of fluids[END_REF] and they are the dominant facies, especially on the Pétrels Island (Dumont d'Urville station, Fig. 1). We will discuss more specifically the space and time relationships between leucosomes and melanosomes in the migmatitic gneisses. As partial melting developed all along a polyphased tectonic and metamorphic event, leucosomes display various habitus relative to the composite foliation S 1 -S 2 , which is marked on the field by more or less continuous layers for melanosomes and septas of mesosomes (Fig. 2a):

(i) The earliest leucosomes occur as elongated and mylonitized lenses, sometimes affected by synfolial folds leading to the S 2 foliation. They bear evidence of successive segregation and deformation processes, syn S 1 -ante S 2 and syn S 2 , respectively (Fig. 2a). (ii) The most common leucosomes are undeformed and concordant within the migmatites foliation, possibly affected later by post-foliation open folds. They represent syn-to late-S 1 -S 2 foliation melts (Fig. 2b). (iii) The latest leucosomes form dykelets crosscutting, and postdating, the S 1 -S 2 foliation. They can be connected to the concordant leucosomes, tapping off and transferring melts upwards (Fig. 2a,c).

Coarse-grained pink granites

They appear as dykes of decimetric to decametric thickness that cross cut the regional foliation (Fig. 2d). They are devoid of any visible markers of orientation. Such dykes, collecting the dykelets, drain the more evolved melts during the late-to post-tectonic stages of the partial melting and metamorphic event.

Anatexites

They outcrop widely on the western area of Pointe Géologie (e.g., Gouverneur, Balance, Taureau and La Vierge Islands, Fig. 1b). They derived from a higher degree of partial melting than migmatitic gneisses and show a mineral assemblage comparable to that of leucosomes but more enriched in Qtz and depleted in Crd. Anatexites display more isotropic textures but a flow fabric or a palimpsest structure of the parent gneisses is often marked by some oriented rafts of migmatitic gneisses or schlierens of biotite and oxides (Fig. 2e).

Mylonitic gneisses

They crop-out in a restricted and narrow zone on the western flank of the Pétrels Island (Rocher Jakobsen) and some islands of Pointe Géologie, i.e. along a major N-S to N340°striking vertical shear zone (sites II and V, Fig. 1b). They consist of Bt + Grt gneisses with only tiny Grt-bearing leucosomes and are characterized by a vertical mylonitic foliation. They derive from the K-feldspar migmatitic gneisses and then may be considered as late-to post-migmatisation tectonites (Fig. 2f).

Sampling and methodological procedures

Because of outcrops restriction (areas non accessible because of ice cover or of birds protection) and high variability of the visible deformation, sampling was often made in sites relatively close one to another in the main island (Pétrels Island). Other sites were scattered in different smaller islands (Fig. 1b). Most sampled sites correspond to a surface up to 50 m 2 .

For the magnetic fabric study, cores were sampled with a gasolinepowered portable drill. All the cores were oriented only with a sun compass (Fig. 2f) because of the immediate proximity of the south magnetic pole. This introduced a strong limitation for sampling because of climatic conditions. The sampling was made during three different summer fieldworks, in 1992, 2009 and 2011. As a whole, 819 rock cores were obtained in 33 sites, including 20 on the Pétrels Island (Fig. 1b). When possible, different facies (from gneiss to anatexites) were sampled on the same site. The cores were cut in the laboratory in specimen of standard paleomagnetic size. Magnetic susceptibility and its anisotropy have been measured using Kappabridges KLY3 and MKF1 (AGICO, Brno). The determined AMS ellipsoid yields the three principal susceptibility axes defined as K 1 ≥ K 2 ≥ K 3 . K 1 is the magnetic lineation and K 3 is perpendicular to the magnetic foliation and the bulk magnetic susceptibility is Km = (K 1 + K 2 + K 3 ) / 3. The [START_REF] Jelínek | Characterization of the magnetic fabrics of rocks[END_REF] parameters P′ and T were used to describe the intensity and the shape of the magnetic fabric, respectively. Data for a group of samples were analysed using normalized tensor variability [START_REF] Hext | The estimation of second-order tensors, with related tests and designs[END_REF][START_REF] Jelínek | Statistical processing of anisotropy of magnetic susceptibility measured on groups of specimens[END_REF] statistics.

Thermomagnetic experiments were performed on representative samples by heating in argon or in air using the CSL equipment and the CS2-3 oven (AGICO, Brno) associated with the Kappabridges. Hysteresis loops were obtained for small cores (cylindrical samples of about 3 cm 3 ) using a laboratory-made translation inductometer within an electromagnet capable of reaching 1.6 T.

Magnetic mineralogy

Thin sections study

As the magnetic characteristics of the rocks are tightly controlled by their mineralogical content, the various assemblages recorded in the different rock types need to be specified. This is particularly important for migmatitic gneisses where partial melting induces a redistribution of refractory minerals from the mesosomes to both the melanosomes and leucosomes.

Magnetite and ilmenite are the most abundant oxide minerals. They form discrete grains or clusters with direct contact between magnetite and ilmenite. The larger grains (millimetric in size) are generally found in the melanosomes and therefore in the foliation and they appear as elongated parallel to the lineation defined by the sillimanite when the latter is present (Fig. 3). In addition to the observed lengthening of magnetite and ilmenite grains parallel to this lineation, measurement of lattice preferred orientations (LPO) of these minerals by indexation of electron back-scattered diffraction (EBSD) patterns (e.g., [START_REF] Bascou | Titanohematite lattice preferred orientation and magnetic anisotropy in high-temperature mylonites[END_REF][START_REF] Prior | The application of electron backscatter diffraction and orientation contrast imaging in the SEM to textural problems in rocks[END_REF] shows that for ilmenite, deformation also induces LPO-related structural axes (measurements performed using a SEM-JSM 5600 at the University of Montpellier -France). Pole figure projections (Mainprice's software, Geosciences Montpellier) of ilmenite crystallographic planes from gneisses of the vertical shear zones show relatively strong LPO which is characterized by poles of (0001) planes strongly concentrated close to the normal to the foliation, and by the poles of (2-1-10) and (10-10) prism planes concentrated close to the lineation (Fig. 4). Oxides are also present in the leucosomes but the fabric (grains alignment) appears to be more scattered than in the melanosomes. At the melanosome-leucosome boundary, elongated grains, maybe rotated, have been observed with high angle to the foliation. In the leucosomes, the oxide grains become more discrete and more rounded perpendicular to the lineation (and parallel to the foliation for samples from vertical shear zones; Fig. 3).

Cordierite is also present in migmatitic gneisses and granitic dykes of Pointe Géologie. This mineral that occurs mainly as grains of globular shape (up to a few millimetres in size) is carefully considered because cordierite presents specific magnetic properties. It carries an inverse magnetic fabric (i.e. characterized by a switch between the principal maximum K 1 and minimum K 3 axes of the AMS ellipsoid with respect to the grain shape; [START_REF] Potter | Single domain particles in rocks and magnetic fabric analysis[END_REF][START_REF] Rochette | Rock magnetism and the interpretation of anisotropy of magnetic susceptibility[END_REF] that could lead to misinterpretation of the measured magnetic fabrics. However, the mean magnetic susceptibility (Km) for magnetite, which is ubiquitous in studied thin sections, is significantly higher than for cordierite (Km ratio > 1000). Therefore magnetite should be the main carrier of measured AMS. For relatively large grains of magnetite, the K 1 and K 3 axes correspond to the magnetic lineation and magnetic foliation poles, respectively.

Rock-magnetism

The mean magnetic susceptibility Km (average 5.2 • 10 -2 SI) of the studied samples is different according to the studied fraction of the migmatites (Fig. 5). The gneisses with subordinate melted component and dominant melanosome fraction have very high susceptibility (average 7.4 • 10 -2 SI). Km values for dominant granitic fraction are lower (average 1.5 • 10 -2 SI). Migmatite samples including both leucosome and melanosome fractions present intermediate Km values. This variability in leucosome versus melanosome content of gneissic specimens largely explains the relatively large distribution of susceptibilities (Fig. 5). The anatexite samples show the highest variation in Km values.

All thermomagnetic KT curves (susceptibility K as a function of temperature T) show a sharp susceptibility decrease around 580 °C during heating (Fig. 6), clearly indicating the presence of magnetite. In addition low temperature experiments confirm the presence of this mineral by a large susceptibility increase around -160 °C (Fig. 6). This variation corresponds to the Verwey transition that is characteristic of pure magnetite. Few thermomagnetic curves are perfectly reversible. Susceptibility values from the cooling curve are slightly higher than the initial values, in air as in argon atmosphere (but sometimes much higher as for the sample AP35). Partial cooling loops during heating evidenced a weak mineral alteration due to heating around 300 to 400 °C that probably reflects transformation of a weak amount of maghemite in hematite. New magnetite is also often formed at the highest temperatures.

Hysteresis loops show generally a full saturation of the magnetization for moderate field (Fig. 7). This saturation was partial only for the sample AP35, evidencing the presence of another component of high coercivity. This component is probably hematite. In most cases, ferrimagnetic minerals have a largely dominant effect for the susceptibility, but two of the studied granite samples highlight also a strong contribution of the paramagnetic or even diamagnetic minerals. Day plot [START_REF] Day | Hysteresis properties of titanomagnetites: grain size and compositional dependence[END_REF] for mainly magnetite-bearing samples clearly evidences a large size of the magnetite particles (Fig. 8).

Main magnetic mineral is therefore magnetite with a large grain size. Obtained AMS data are then directly usable for a structural analysis (no inverse magnetic fabric due to small Single Domain particles -Potter and Stephenson, 1988).

Magnetic fabric

Structures from AMS

Data for migmatitic gneisses and for granites are presented separately (for migmatitic gneissic including minor melted parts and "granitic" samples small amounts of melanosome assemblage this separation has been made then considering the dominant presence of one of these components). Moreover, from the regional structural field pattern, two distinct zones have been distinguished for the AMS analyse: the sub-vertical zones and the dome-shaped zones (Figs. 9 and 10). Maps of the AMS foliations and lineation are presented in Fig. 11.

Structures in sub-vertical domains

In migmatitic gneisses from sub-vertical shear zones (sites A, D, O, R, T, and XIII of Pétrels Island -Fig. 10), melanosome layers show generally both well clustered magnetic foliation pole (K 3 ) and magnetic lineation (K 1 ). The magnetic foliation is dominantly sub-vertical N-S consistent with the observed field foliation, and the magnetic lineation presents a weak plunge and N-S trend. The granitic rich layers (site XIIIγ) display more diffuse fabric, in particular the K 1 axes that tend to form a girdle perpendicular to K 3 . Well-individualized leucosomes (e.g. Fig. 2C) show a different magnetic fabric signature than melanosome layers characterized by strongly inclined to sub-vertical magnetic lineation (site XIIIγ -Fig. 10). The K 3 axes are more scattered than the K 1 axes. They are in the horizontal plane with two dominant directions: N-S and E-W.

In large coarse-grained pink granite dykes (sites Aγ, Dγ, Eγ -Fig. 10) the K 3 axes are relatively well clustered defining a subvertical magnetic foliation of strike close to that of neighbouring gneisses (sites Dγ, Eγ) or showing a divergent strike (sites Aγ). The magnetic lineation pattern is characterized by the verticalization of K 1 axes in comparison to magnetic lineation measured in the melanosome layers of neighbouring gneisses. They form girdle perpendicular to K 3 (site Eγ) or strongly inclined to sub-vertical local maximum (sites Dγ and Aγ, respectively).

In anatexites of Gouverneur, Taureau, La Vierge, Balance Islands (sites VIIγ, XVγ, XIVγ, XVIIIγ, XIXγ -Fig. 9) and South of Pétrels Island (site XXγ -Fig. 10) K 1 tends to have high inclination or to be sub-vertical. For Gouverneur anatexites (sites VIIγ, XVγ) the clustering of K 1 is strong. K 3 lies in the horizontal plane, relatively scatted or forms clusters at 90°from each other with the N-S and E-W directions that prevails for south Gouverneur Islands (site XVγ). Sampling sites XIVγ and XVIIIγ (Fig. 9) show heterogeneities in the partial melting degree that is expressed in the field by the frequent occurrence of migmatitic gneiss rafts in the anatexites. For site XVIIIγ (La Vierge Island) anatexites and migmatitic gneisses show distinct magnetic axes orientations characterized by K 1 preferentially sub-vertical and K 3 horizontal N-S (i.e., magnetic foliation vertical E-W) in anatexites while K 1 is horizontal to strongly plunging N-S and K 3 is horizontal E-W (i.e., magnetic foliation vertical N-S) in neighbouring gneisses. The N-S orientation of magnetic foliation in gneisses fit quite well with the regional vertical shear zone structures. In opposite, the E-W vertical magnetic foliation that appears in the larger dykes and anatexites is less well-constrained by large-scale field structures (Fig. 9). For site XIVγ (Taureau Island) the distinction in the field between the anatexites and the migmatitic gneisses was difficult to make, explaining the relatively wide confidence zones at 95% for K 1 and K 3 . However the magnetic fabric is mainly oriented vertical E-W for the foliation and strongly-inclined (westward) for the lineation.

In mylonitic gneisses of Rocher Jakobsen (sites II, V -Fig. 10) the magnetic foliation (K 1 -K 2 ) is dominantly sub-vertical N-S consistent with the observed field structures (Fig. 2f). In contrast, the AMS measurement reveals uncommon strongly plunging to sub-vertical lineations.

Structures in dome-shaped zones

In the migmatitic gneisses of the dome-shaped zones (sites G, H, J, K, S, VI, XVI, XVII -Fig. 9; sites B, C, F, i, Q, I, III, IV, VIII, XII -Fig. 10) magnetic fabric displays a stronger variability of K 1 and K 3 axes orientations from one site to another one in comparison to structured vertical shear zones. A relatively strong dip (about 45°) for the magnetic foliation is frequently observed, in particular for the sites located in the flanks of domes (sites H, K -Fig. 9; sites Q, III, IV -Fig. 10) for the eastern flanks of the Pétrels-Curie Islands dome. The magnetic foliations are generally in agreement with the field measured strikes and dip of melanosome layers. The magnetic lineations show gently plunge and dominant NNW strike. The innermost zones of domes (sites XVI, XVII -Fig. 9; sites B, I, VIII -Fig. 10) show also a strong variability of AMS structure patterns correlated to the complexity of migmatitic structures observed at the domes apex. In these areas are observed the less well-defined magnetic directions (with the widest confidence zones at 95%: site XVI -Fig. 9; site I -Fig. 10). Sampling in a granitic dykelet of centimetric width crosscutting flat-lying gneissic foliation was performed in site XVIIγ (Fig. 9). The related AMS pattern is characterized by vertical N-S magnetic foliation and sub-vertical magnetic lineation and then differs from that of gneisses. Large dykes of coarse-grained pink granites sampled in sites Fγ and XIIγ (Fig. 10) are characterized by vertical magnetic lineation, which contrasts with the magnetic lineation in the melanosomes of gneisses. This feature is similar to that observed in the granitic bodies from the sub-vertical shear zones. The magnetic foliation in the granitic dykes is either concordant with that of host gneisses (site Fγ) or discordant (site XIIγ).

AMS parameters (P′ and T)

Fig. 12 presents the characteristics of the susceptibility ellipsoid. The degree of anisotropy P′ is very high in migmatitic gneisses (mean 1.48; minimum 1.04; maximum 1.90). It is lower in granites (mean 1.22; minimum 1.04; maximum 1.41). The shape of the susceptibility ellipsoid is dominantly oblate, in gneisses (for T value: mean 0.30; minimum -0.08; maximum 0.66) as in granites (mean 0.25; minimum -0.10; maximum 0.62). In granites, a clear relationship (Fig. 5, R 2 = 0.70) between P′ (intensity of the fabric) and Km (related to the concentration in ferrimagnetic minerals) highlights the dominant effect of the magnetite in the AMS [START_REF] Henry | Chemical remagnetizations in the Illizi basin (Saharan craton, Algeria) and their acquisition process[END_REF]. In addition, for granite, the anatexites show the highest variability in P′ and T values, including the extreme values (Fig. 12).

Discussion

Mean susceptibility (Km) and anisotropy (P′) signature

The amount of magnetite is clearly different in migmatitic gneisses and in granites (leucosomes and dykes), as shown by the susceptibility values (Fig. 5). This variation of magnetic susceptibility between melanocratic and leucocratic layers is mainly due to the variations in magnetite content. Magnetic studies suggest also mineralogical changes affecting magnetite with formation of ferromagnetic oxides of lower susceptibility like maghemite or hematite. Concerning this mineralogical change, gneiss sample AP35 (close to site R, Fig. 1) is the only one with hematite; it also presents the strongest formation of a lot of new magnetite at high temperature (susceptibility becomes 14 times higher on the cooling KT curve). It is therefore probably due to a strong local mineralogical alteration (interaction with deep fluids, weathering?) and remains an exception. In addition, in site test R, magnetic and field structures (foliations and lineations) show a good agreement (Figs. 10 and 11), indicating that the primary AMS signature is relatively unaffected by such secondary mineralogical alteration.

The intensity of magnetic anisotropy, illustrated by the P′ values, is lower in granites than in gneisses (Fig. 12). In granites, it presents a relationship with the Km values, suggesting that this difference could be explained by the lower amount of magnetite in granites, AMS then resulting mainly from strong magnetite anisotropy and weak anisotropy of a low susceptibility component. Owing to its Km values, this last component is the paramagnetic minerals and not the maghemite. Inversely to granitic magmas constituting leucosomes and dykes, a wide range in Km values, according to samples within a single site, is observed in anatexites. It is due to the presence of more refractory gneissic rafts with high concentration in oxides grains.

AMS in migmatitic gneisses and granites

In addition to the Km and P′ values, the main differences in magnetic fabrics between melanosome layers of gneisses and granitic magmas concern the orientation of the K 1 axis (Figs. 9,10 and 11b). K 1 axis is dominantly close to the horizontal in melanosomes and close to vertical in granites. In some granitic sites (e.g., sites Eγ, XIIγ, Fig. 10), its orientation is relatively scattered within a vertical plane, probably because of the presence of melanosome fragments within the granite. Another difference in orientation of susceptibility axes concerns K 3 . Whereas in melanosome the K 3 orientation fits with the field structures, it appears that granites show a more complex magnetic orientation pattern with two dominant orientations. The first one matches that of the gneissic structures (for example, the magnetic foliation is vertical N-S in the shear zone area) and the second is at 90°of the first one. In some sites the two clustering are present (e.g., site XVγ -Fig. 9 and site XIIIγ -Fig. 10). The first orientation is associated to the strike-slip deformation, which is at the origin of the large-scale sub-vertical corridors. The cause of the second orientation is less straightforward. In Pétrels Island, the migmatitic layering is locally cut by vertical large granitic dykes of NE-SW to E-W direction [START_REF] Monnier | Le socle protérozoïque de Terre Adélie (Antarctique Est): son évolution tectonométamorphique et sa place dans les reconstitutions du protogondwana[END_REF]. Then emplacement of such E-W trend dykes is tectonically controlled and consistent with a component of NNW-SSE to N-S dextral strike-slip [START_REF] Gapais | Paleoproterozoic tectonics in the Terre Adélie Craton (East Antarctica)[END_REF]. In the western area (Gouverneur Island area, Fig. 9) where rafts of melanocratic restites are still packed in anatexites, E-W magnetite could correspond to inheritance of older tectonic foliations. Gneisses in sites II and V (Rocher Jakobsen), on the south-western border of the Pétrels Island also present mainly sub-vertical K 1 axes (Figs. 10 and11b). However, they are intensely deformed, unlike the granitic leucosomes, dykes and anatexites that have not recorded pervasive solid-state deformation. Thus, AMS fabric is interpreted as resulting of high-strained, late-to post-migmatisation solid-state deformation. The strong K 3 clustering indicating N-S vertical magnetic foliation and the highly tilted to sub-vertical magnetic lineation could be related to deformation in transpression regime.

Thin section analyses show a change of orientation of magnetite grains from melanosome to leucosome. In melanosome, the oxide grains are within the foliation and elongated parallel to the lineation also marked by the sillimanite alignment mainly developed in the dome-like domains. They are interpreted as resulting from solid-state deformation that occurred during the Paleoproterozoic metamorphic event (1.69 Ga ago). In addition, LPO of ilmenite, which strongly correlated with the tectonic fabrics, suggests dislocation glide on (0001) as a main deformation mechanism. The basal slip for hematite-ilmenite mineral could reflect high-temperature solid-state deformation (e.g., [START_REF] Bascou | Titanohematite lattice preferred orientation and magnetic anisotropy in high-temperature mylonites[END_REF][START_REF] Siemes | Texture, microstructure, and strength of hematite ores experimentally deformed in the temperature range 600-1100 °C and at strain rates between 10 -4 and 10 -6 s -1[END_REF]. Migmatitisation processes are also associated with segregation and transfer of melts that lead to leucosomes, pink granitic dykes and then anatexites when partial melting degree is higher. As discussed in [START_REF] Vigneresse | Rheological transitions during partial melting and crystallisation with application to felsic magma segregation and transfer[END_REF] and recently in [START_REF] Holtzman | Effects of stress-driven melt segregation on the viscosity of rocks[END_REF], the increase of melt fraction produce changes in rheological behaviour from plastic deformation evidenced by elongation of oxide minerals that are abundant in the melanosome, to magma flow with transport of isolated solid particles in leucosomes, dykes and anatexites.

Such different and successive deformation regimes affecting magnetite grains first deformed under solid-state conditions and then involved as passive rigid particles during melting, have been already described in migmatitic terrains (e.g., [START_REF] Ferré | Magnetic susceptibility anisotropy: a new petrofabric tool in migmatites[END_REF]. In the Pointe Géologie migmatites, these successive regimes generate distinctive AMS signatures. The clearest difference concerns the K 1 orientation, which tends to a vertical plunge during segregation and circulation of felsic melt pointing out a dominantly upward transport. The K 3 orientation in granitic layers appears more sensitive to the degree of partial melting and the structuration of country rock. At lower degree of partial melting, liquid segregation remained strongly constrained by the framework of the unmelted gneissic material. Then the felsic melt, forming dykelets and connected lenses is concordant with the foliation of gneisses. For higher degree of melting, the melt was injected as concordant or crosscutting large veins (Fig. 2c,d) resulting in the various orientations of the magnetic foliation.

Methodological implication

All these observations evidence that the AMS study of migmatites cannot be considered as a routine method. To be as reliable as possible, the AMS approach requires a preliminary detailed observation of the various leucosomes habitus and then a large number of magnetic measurements of rocks characterized by various degrees of melting. For low degrees of partial melting, the leucocratic layers remain strongly constrained by the earlier structure defined by the unmelted material. In that case, the AMS principal axes are representative of the local and regional finite strain axes. This fits with the conclusions of [START_REF] Ferré | Magnetic susceptibility anisotropy: a new petrofabric tool in migmatites[END_REF] for migmatites of the Superior Province in Minnesota. For more pronounced partial melting degree, well-individualized leucocratic layers can developed specific AMS patterns associated to flow regime. In the Pointe Géologie migmatites, the difference with the local and regional finite strain axes is highlighted by the direction of K 1 -axes. Large variations of the partial melting rates increase the probability to measure composite AMS and the difficulty to isolate pre-and syn-melting magnetic fabrics. The strongest variations of AMS principal directions and parameters recorded in anatexites are related to the heterogeneity of these rocks, which enclose number of schlierens and refractory material.

Regional-scale implication

The structural pattern of Pointe Géologie consists of two domains with respectively flat-lying and sub-vertical foliations related to long-lived tectonics, metamorphic and anatectic event, both associated with partial melting [START_REF] Monnier | Le socle protérozoïque de Terre Adélie (Antarctique Est): son évolution tectonométamorphique et sa place dans les reconstitutions du protogondwana[END_REF][START_REF] Pelletier | Tectonique transpressive en Terre Adélie au Paléoprotérozoïque (East Antarctica)[END_REF][START_REF] Pelletier | From partial melting to retrogression in the Pointe Geologie migmatitic complex: a history of heterogeneous distribution of fluids[END_REF]. Commonly, stretching and mineral lineations are not easy to observe within foliations, especially within the sub-vertical shear zones. Combining structural field evidences, determination of Pressure-Temperaturetime metamorphic evolution and analogue modelling of the mantlecrust ductile deformation [START_REF] Gapais | Paleoproterozoic tectonics in the Terre Adélie Craton (East Antarctica)[END_REF] proposed a model in which the Paleoproterozoic hot and weak lithosphere converges toward a cooler and more rigid craton (Neoarchean in age). In this model, deformation of the Paleoproterozoic crust combines both transpression and horizontal flow and/or lateral escape parallel to the Archean basement and perpendicular to the shortening direction.

AMS structural pattern determination provides complementary structural information such as magnetic lineations. They are mainly sub-horizontal in sub-vertical foliated gneisses. It underlines the importance of the strike-slip component during the deformation. In mylonitic gneisses that are considered as late-to post-migmatisation tectonites, nearly vertical lineations are observed. This is considered as the effect of transpressional regime, which combines strike-slip and acrossstrike shortening. In migmatites with flat-lying foliation, the magnetic lineation displays a moderate plunge and is commonly parallel to the mineral stretching lineation. Such features are in good agreement with horizontal crustal flow mechanism.

Increasing amounts of partial melting lead to a loss of the rock cohesion related to the migration of the melt trough a solid-state made of previously deformed minerals. Thus the former tectonic structures will be progressively erased. Melt segregation and transfer occur under a viscous flow regime, mainly driven by gravity forces. In such a regime, the well-defined leucosomes, dykes and anatexites will develop a magnetic fabric corresponding to the upward transport of magmas (Fig. 13). There, AMS measurements can better map the mass transfers associated with intensive melting at peak temperatures and to characterize the general framework of a warm and buoyant lithosphere. Hence, rheological changes related to partial melting generate strain partitioning between gneisses and strongly melted rocks in a regional transpressional tectonics context.

Conclusions

Pointe Géologie archipelago in Terre Adélie represents a mid-crustal section affected by intensive anatectic processes during Paleoproterozoic

  Fig. 1. a. Simplified geological map of the Terre Adélie Craton (after Ménot et al., 2007). Studied zone (indicated by a red star) is located in the 1.7 Ga domain, close to the French Polar Dumont d'Urville station. Inset: location and limits of the Terre Adélie Craton (TAC) in East Antarctica and of the Gawler Craton (GC) in South Australia. b. Synthetic structural map of the Pointe Géologie archipelago and location of AMS sites.

Fig. 2 .

 2 Fig. 2. Examples of high-temperature rock types outcropping in Pointe Géologie archipelago. (a) Migmatitic gneiss structures illustrating the HT polyphased tectonic in the Paleoproterozoic formation of the TAC: (1) septas of mesosomes-quartzite layer and (2) more continuous layers of melanosomes (Bt + Sil + Crd ± Grt) bearing foliation; successive generations of melt with (3) earliest elongate lenses locally affected by intrafolial folds within the foliation, (4) undeformed and concordant leucosomes within foliation and (5) later thin dykes crosscutting foliation. (b) Folded leucosomes with axial plane parallel to the foliation. (c) Melt segregation and upwards transfer processes generating concordant leucosomes feeding crosscutting dykelets (Pétrel Island). (d) Coarse-grained pink granites forming metric to decametric thickness dykes crosscutting the regional foliation (Pétrels Island). (e) Anatexite showing melanocratic schlierens and rafts bearing the foliation (Balance Island). (f) Mylonitic gneisses characterized by vertical foliation (Rocher Jackobsen, Pétrels Island). The solar compass used to determine the geographical orientations of tectonic structures gives the photograph scale.

Fig. 3 .

 3 Fig.3. Photomicrograph of migmatitic gneiss in the vertical shear zone (Lamarck Island), perpendicular to the foliation (S) and parallel to the lineation. Sillimanite (sil) and elongated oxide grains (mainly magnetite (mag) in this sample) are parallel to the lineation in the melanocratic layers. The shape of oxide grains tends to be more rounded perpendicular to the lineation and parallel to the foliation in more leucocratic areas (see white arrows).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Ilmenite lattice-preferred orientations (LPO) of migmatitic gneiss in the vertical shear zone, Pétrels Island (sample AP85). LPO are represented on equal area, lower hemisphere projection. A black line marks the foliation and the lineation corresponds to the structural axis X. The density contours are in Multiple Uniform Distribution (MUD), shadinginverse log; N is the number of measured grains.

Fig. 6 .Fig. 7 .

 67 Fig.6. Normalized susceptibility K/Ko as a function of the temperature T in gneisses (samples AP35 and AP85) and in granites (09cjb75, 11cjb41 and 11cjb71).

Fig. 8 .

 8 Fig.8. Ratios of hysteresis parameters[START_REF] Day | Hysteresis properties of titanomagnetites: grain size and compositional dependence[END_REF]: Jrs/Js (Jrs: remanence at saturation; Js: magnetization at saturation) as a function of Hcr/Hc (Hcr: remanent coercive force; Hc: coercive force). Areas for Multi-Domain (MD) and Pseudo Single Domain (PSD) grain size for pure magnetite.

Fig. 9 .

 9 Fig.10

Fig. 10 .

 10 Fig.10. Projection of AMS data of each site of Pétrels Island. Distribution of maximum axes (K 1 ; blue squares) and minimum axes (K 3 ; violet circles) with confidence ellipses is in lower hemisphere equal area projection. White, salmon and green colours of stereonets background represent the AMS projections of gneisses, granitic dykes and anatexites, respectively.

Fig. 12 .Fig. 11 .

 1211 Fig.12. T parameter as a function of P′ values[START_REF] Jelínek | Characterization of the magnetic fabrics of rocks[END_REF] for high temperature main rock types of the Pointe Géologie archipelago.

Fig. 13 .

 13 Fig. 13. 3D diagram modified after Gapais et al., 2008. Deformation of a hot and mid-crustal section (Pointe Géologie) in response to a W-E transpressional tectonics. White arrow indicates the horizontal flow marked by mineral stretching lineation in the gneisses. Green arrows indicate the vertical flow point out by AMS analyses only in granitic dykes and anatexites.

Acknowledgements

We are very grateful to the French polar institute "Institut Paul Emile Victor" for support of the programme ArLiTA (Architecture de la Lithosphere de Terre Adélie) and to the Japanese "National Institute of Polar Research" for participation to the international exchange programme under the Antarctica Treaty and for invitation of BH in Tokyo. This paper has benefited of the reviews of Federico Rossetti, Fabrizio Storti and Denis Gapais.

times, 1.69 Ga ago. This generated different High Temperature rock types such as: migmatites including leucosomes and melanosomes, coarse-grained pink granites dyke and anatexites. Partial melting affects indifferently the regional structures characterized by the juxtaposition of flat-lying and sub-vertical deformation zones.

Magnetic mineralogy study shows that oxide minerals are mainly large and solid state deformed grains of magnetite. In the leucosomes, oxides appear more scattered than in the melanosomes and their rounded isometric shape suggests rotations of the former elongated grain during melt segregation. This contrasted magnetite grains distribution will generate specific AMS signatures for melanocratic layers on the one hand and felsic leucosomes, dykes and anatexites on the other hand. The mean magnetic susceptibility (Km) and the anisotropy degree (P′) are higher in the gneisses and much lower in granitic samples. Orientation of principal susceptibility axes also differs. In migmatitic gneisses, magnetic foliations are coherent with the local and regional structural pattern. Nevertheless AMS technique provides complementary information on the orientation of lineations when they are difficult to observe in the field. The magnetic lineations are dominantly sub-horizontal in the shear zone while they tend to be vertical in granitic dykes, anatexites and well-defined leucosomes. Such a changing of orientation is related to a rheological contrast between the solid-state deformation suffered by the oxide grains and their reorientation in a viscous flow during the aggregation of felsic melt.

Magnetic structure pattern is in good agreement with the regional tectonic framework assumed by [START_REF] Gapais | Paleoproterozoic tectonics in the Terre Adélie Craton (East Antarctica)[END_REF]. Based on metamorphic and field structural data together with analogic simulation, these authors proposed a model of deformation that implies the convergence of a hot and weak Paleoproterozoic lithosphere toward a cooler rigid Archean craton. They explained the structural pattern of the Pointe Géologie and surrounding areas by horizontal, orogen-parallel, sub-constrictional flow that could result from oblique convergence and/or from particular boundary conditions facilitating lateral escape. A comparable model was also assumed from numerical modelling to image the deformation of the hot Neoarchean crust of the Eastern part of the Terre Adélie Craton [START_REF] Duclaux | Orogen-parallel flow during continental convergence: numerical experiments and Archean field examples[END_REF]. The magnetic structures map highlights the role of the gravity-induced upwelling of a crust undergoing partial melting.

From methodological point of view, this work emphasizes on the importance of an acute petrological characterization of sites to be selected for AMS measurements in order to avoid pre-and syn-melting composite magnetic fabrics. Finally, it underlines the difficulty of AMS analyses in anatectic rocks, which lead to a very large variability for AMS signature; this mainly due by the preservation of refractory rafts and by the heterogeneity of such rocks.


Alexander Fedotov

F R Éd

Éric Klopp

STARK-WANNIER LADDERS AND CUBIC EXPONENTIAL SUMS

On L 2 (R), we consider the Schrödinger operator (1.1) H ǫ = -∂ 2 ∂x 2 + v(x) -ǫx, where v is a real analytic 1-periodic function and ǫ is a positive constant. This operator is a model to study a Bloch electron in a constant electric field ( [START_REF] Avron | The lifetime of Wannier ladder states[END_REF]). The parameter ǫ is proportional to the electric field. The operator (1.1) was studied both by physicists (see, e.g., the review [START_REF] Gluck | Wannier-Stark resonances in optical and semiconductor superlattices[END_REF]) and by mathematicians (see, e.g., [START_REF] Sacchetti | Existence of the Stark-Wannier quantum resonances[END_REF]). Its spectrum is absolutely continuous and fills the real axis. One of main features of H ǫ is the existence of Stark-Wannier ladders. These are ǫ-periodic sequences of resonances, which are poles of the analytic continuation of the resolvent kernel in the lower half plane through the spectrum (see, e.g., [START_REF] Buslaev | A Bloch electron in an external field[END_REF]). Most of the mathematical work studied the case of small ǫ (see, e.g., [START_REF] Sacchetti | Existence of the Stark-Wannier quantum resonances[END_REF][START_REF] Buslaev | Imaginary parts of Stark-Wannier resonances[END_REF] and references therein). When ǫ is small, there are ladders exponentially close to the real axis. Actually, only the case of finite gap potentials v was relatively well understood. For these potentials, there is only a finite number of ladders exponentially close to the real axis. It was further noticed that the ladders non-trivially "interact" as ǫ changes, and conjectured that the behavior of the resonances strongly depends on number theoretical properties of ǫ (see, e.g., [START_REF] Avron | The lifetime of Wannier ladder states[END_REF]).

In the present note, we only consider the periodic potential v(x) = 2 cos(2πx) and study the reflection coefficient r(E) of the Stark-Wannier operator (1.1) in the lower half of the complex plane of the spectral parameter E. The resonances are the poles of the reflection coefficient. We show that, as Im E → -∞, the function E → 1 r(E) can be asymptotically described in terms of a regularized cubic exponential sum that is a close relative of the cubic exponential sums often encountered in analytic number theory. This explains the dependence of the reflection coefficient on the arithmetic

The present work was supported by the Russian foundation of basic research under grant 14-01-00760-a. A.F. acknowledges support by the Fondation Sciences Mathématiques de Paris. F.K. acknowledges support by the Chebyshev Laboratory and the French Embassy in Russia through the Chaire Lamé. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme "Periodic and Ergodic Spectral Problems" supported by EPSRC Grant Number EP/K032208/1. F.K. also acknowledges support from the Simons Foundation during his stay at the INI.. nature of ǫ. For π 2 3ǫ ∈ Q, we describe the asymptotics of the Stark-Wannier ladders situated far from the real axis.

Let us recall the definition of the reflection coefficient for (1.1) following [START_REF] Buslaev | A Bloch electron in an external field[END_REF]. Consider the equation

(1.2) -ψ ′′ (x) + (v(x) -ǫx)ψ(x) = Eψ(x), x ∈ C,
For the sake of simplicity, assume that the potential v is entire. Assume also

1 0 v(x) dx = 0.
For any E ∈ C, there are unique solutions ψ ± to (1.2) that admit the asymptotic representations

(1.3) ψ -(x, E) = 1 4 √ -ǫx-E e --E/ǫ x √ -ǫt-E dt+o(1) , x → -∞, ψ + (x, E) = 1 4 √ ǫx+E e i x -E/ǫ √ ǫt+Edt+o(1) , x → +∞,
where the determinations of √ • and 4 √ • are analytic in C \ R -and positive along R + .

Consider also the solution ψ * + (x, E) = ψ + (x, Ē). The solutions ψ + and ψ * + being linearly independent, one has This estimate is locally uniform in ǫ > 0.

(1.4) ψ -(x, E) = w(E)ψ * + (x, E) + w * (E)ψ + (x, E), x ∈ R,
Clearly, the asymptotic behavior of 1/r(E) as Im E → -∞ is determined by the Fourier series terms with large positive m, and so, roughly,

(1.6) 1 r(E) ≈ a(ǫ) P(E/ǫ), P(s) = m≥1 √ m e -2πiωm 3 -2m log (2πm/e)+2πims .
It is worth to compare the function P with the cubic exponential sums Such sums were extensively studied in analytic number theory, see, e.g., [START_REF] Davenport | Analytic methods for Diophantine equations and Diophantine inequalities[END_REF]. They were proved to depend strongly on the arithmetic nature of ω. This appears to be true in our case too. We have Theorem 2. Let v(x) = 2 cos(2πx). Assume that ω ∈ Q and represent it in the form ω = p q , where 0 ≤ p < q are co-prime integers. If p = 0, we take q = 1. For ξ ∈ R, we set I q (ξ) :

= {m ∈ Z : |ξ -m q | ≤ 1/2}. As Im E → -∞, one has (1.7) r -1 (E) = b(ǫ) ρ q m∈Iq(ξ)
S q (p, m)e ρe iπ(ξ-m/q) +iπ(ξ-m/q)+O(log 2 ρ/ρ) + e O( ρ ln ρ ) , where b(ǫ) = π 3 2 e iπ/4 / √ 2ǫ, ξ = Re E/ǫ, ρ = e -π Im E/ǫ , and

S q (p, m) = q-1 l=0 e -2πi pl 3 -ml q .
The error estimates are locally uniform in ǫ > 0.

Let us discuss this result. First, assume that ω = 0. By Theorem 2, The first statement immediately follows from Theorem 2; to prove the second one has to use Jensen formula and Levin lower bounds for the absolute values of entire functions, see, e.g., [START_REF] Levin | of Translations of Mathematical Monographs[END_REF].

(1.8) (b(ǫ)r(E)) -1 = √ z e √ z+O( ln 2 z √ z ) + e O( √ z ln z ) , z = e 2iπE/ǫ , where the determination of √ • is analytic in C \ R -and positive along R + . Recall that 1/r is ǫ-periodic. Let B ǫ = {E ∈ C : Im E ≤ 0, 0 ≤ Re E ≤ ǫ}. Representa- tion (1.8) implies
When ω = 0, it is difficult to obtain the asymptotics of the resonances as, in a neighborhood of the line Re E/ǫ = 1/2 mod 1, they are determined by the first Fourier coefficients of 1/r, i.e., by p(m) with m = 1, 2, 3 . . . . Hence, the problem is not asymptotic in nature.

If ω = 0, then the description of the resonances is determined by the values of S q (p, m) for m = 1, 2, . . . q -1 (the map m → S q (p, m) is q-periodic). The S q (p, m) are cubic complete rational exponential sums, see, e.g., [START_REF] Korobov | Exponential sums and their applications[END_REF]. One easily checks Lemma 1. For any q ∈ N,

q-1 m=0 |S q (p, m)| 2 = q 2 .
This implies that, for q ≥ 1, there is at least one integer 0 ≤ m 0 < q -1 such that S q (p, m 0 ) = 0.

If S q (p, m) is non zero for only one 0 ≤ m 0 < q (this happens, for example, for q = 2, 3, 6), then one can characterize the resonances as when ω = 0. Now, they live near the lines {Re E/ǫ = m 0 /q + 1/2 + n}, n ∈ Z.

For large q, there are actually many non-zero values S q (p, m):

Lemma 2. There exists a constant C > 0 such that, for any co-prime q > p > 0, one has #{0 ≤ m < q : S q (p, m) = 0} ≥ Cq 2 3 . This statement follows from Lemma 1 and the well-known upper bound for general complete rational exponential sums of Hua ( [START_REF] Korobov | Exponential sums and their applications[END_REF]).

In general, the behavior of m → S q (p, m) is nontrivial; it is known to depend strongly on the prime factorization of q. Computer calculations lead to the following conjecture: if q is prime, 0 < p < q, and 0 < m < q, then S q (p, m) = 0.

If S q (p, m) is non zero for at least two values of m such that 0 ≤ m < q, then, using (1.5), one can describe asymptotically all the resonances with sufficiently negative imaginary part. One has Corollary 2. Assume that, for some integers m 1 < m 2 such that m 2 -m 1 < q, one has S q (p, m 1 ) = 0, S q (p, m 2 ) = 0, and S q (p, m) = 0 for all m 1 < m < m 2 . Then, for sufficiently large y > 0, in the vertical half-strip

E ∈ C : -Im E ≥ ǫy, m 1 q ≤ Re E ǫ ≤ m 2 q ,
there are resonances, and they are described by the asymptotic formulas:

(1.9)

E ǫ = -i ln(πk) π -ln sin π(m 2 -m 1 ) q + m 2 + m 1 q + o(1), k ∈ N,
where o(1) → k→+∞ 0.

This statement easily follows from Theorem 2. Finally, let us describe very briefly the ideas leading to Theorems 1 and 2. Buslaev's solutions ψ ± used to define the reflection coefficient (see (1.3)) are entire functions of x and E; they satisfy the relations ψ ± (x + 1, E) = ψ ± (x, E + ǫ). It appears that the analytic properties of such solutions can naturally be described in terms of a system of two first order difference equations on the complex plane (see, for example, [START_REF] Fedotov | Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case[END_REF]). To get the asymptotics of the Fourier coefficients of the reflection coefficient, we study the solutions of this system far from the origin. The idea leading from Theorem 1 to Theorem 2 is analogous to one used to study the behavior of the exponential sums N n=1 e -2πiωn 3 with ω ∈ Q for large N, see [START_REF] Davenport | Analytic methods for Diophantine equations and Diophantine inequalities[END_REF]. However, to use it successfully, one has to carry out a non trivial analysis of properties of the error term in (1.5).

  where the coefficient w(E) is independent of x and the function E → w(E) is entire. The ratio r(E) = w * (E)/w(E) is the reflection coefficient. It is an ǫ-periodic meromorphic function of E. The reflection coefficient is analytic in C + , and, for E ∈ R, one has |r(E)| = 1. The poles of r are the resonances of H ǫ . Let us now state the first of our results. Represent 1/r by its Fourier series 1/r(E) = m∈Z e 2πniE/ǫ p(m) for Im E ≤ 0. Let a(ǫ) = 2 ǫ πe iπ/4 . One has Theorem 1. Let v(x) = 2 cos(2πx). Then, as m → ∞, (1.5) p(m) = a(ǫ) √ m e -2πiωm 3 -2m log (2πm/e)+δ(m) , ω = π 2 3ǫ , where, for x real, {x} denotes the fractional part of x, and δ(m) = O(log 2 m/m).

e

  -2πiωn 3 .

Corollary 1 .

 1 Assume ω = 0. The resonances located in B ǫ have the following properties : • for sufficiently large y > 0, the resonances with Im E < -ǫy are located in the domain | Re E -ǫ/2| ≤ Cǫ 2 /| Im E|, where C > 0 is a constant; • let n(y) be the number of resonances in the rectangle [0, ε] -i [0, εy]; then, one has n(y) = 1 π e πy+o(1) as y → ∞.