Anis Rahman

Denis Pellerin

Dominique Houzet

Influence of number, location and size of faces on gaze in video

Keywords: faces, attention, eccentricity, eye movements, video

Many studies have reported the preference for faces and influence of faces on gaze, most of them in static images and a few in videos. In this paper, we study the influence of faces in complex free-viewing videos, with respect to the effects of number, location and size of the faces. This knowledge could be used to enrich a face pathway in a visual saliency model. We used eye fixation data from an eye movement experiment, hand-labeled all the faces in the videos watched, and compared the labeled face regions against the eye fixations. We observed that fixations made are in proximity to, or inside the face regions. We found that 50% of the fixations landed directly on face regions that occupy less than 10% of the entire visual scene. Moreover, the fixation duration on videos with face is longer than without face, and longer than fixation duration on static images with faces. Finally, we analyzed the three influencing factors (Eccentricity, Area, Closeness) with linear regression models. For one face, the E + A combined model is slightly better than the E model and better than the A model. For two faces, the three variables (E,A,C) are tightly coupled and the E + A + C model had the highest score.

Introduction

Gaze is highly influenced by faces in visual scenes compared to other object stimuli. Over the years, a number of studies have been conducted regarding the influence of faces on gaze, mostly using static stimuli [START_REF] Hasson | Eccentricity bias as an organizing principle for human high-order object areas[END_REF][START_REF] Rousselet | Is it an animal? is it a human face? fast processing in upright and inverted natural scenes[END_REF][START_REF] Jebara | Face or building superiority in peripheral vision reversed by task requirements[END_REF][START_REF] Jacques | The time course of visual competition to the presentation of centrally fixated faces[END_REF][START_REF] Bindemann | Viewpoint and center of gravity affect eye movements to human faces[END_REF][START_REF] Heisz | More efficient scanning for familiar faces[END_REF] and some on videos [START_REF] Rice | Parsing heterogeneity in autism spectrum disorders: Visual scanning of dynamic social scenes in school-aged children[END_REF][START_REF] Riby | Looking at movies and cartoons: eye-tracking evidence from williams syndrome and autism[END_REF][START_REF] Buchan | Spatial statistics of gaze fixations during dynamic face processing[END_REF][START_REF] Smith | Do the eyes really have it? dynamic allocation of attention when viewing moving faces[END_REF]. There is enough evidence that faces can be processed at the earliest after stimulus presentation [START_REF] Ro | Changing faces: A detection advantage in the flicker paradigm[END_REF][START_REF] Vuilleumier | Faces call for attention: evidence from patients with visual extinction[END_REF], and they are preferentially processed by the visual system compared to other object categories [START_REF] Rossion | The n170 occipitotemporal component is delayed and enhanced to inverted faces but not to inverted objects: an electrophysiological account of face-specific processes in the human brain[END_REF]. The preference is thought to be influenced by several This research was supported by Rhone-Alpes region (France) under the CIBLE project No. 2136. Thanks to Silvain Gerber, Lionel Granjon, Sophie Marat and Nathalie Guyader for helpful suggestions and eye movement experiment.

different factors like number of faces, eccentricity of face from fixation, face surface area, and closeness to other faces. However, these influences have rarely been reported for dynamic stimuli.

The number of faces limits the preference of faces, as they compete for limited attentional resources. Studies regarding event-related response to face stimulus (the N170), decrease considerably when more stimuli are presented in the visual field [START_REF] Miller | Suppression of visual responses of neurons in inferior temporal cortex of the awake macaque by addition of a second stimulus[END_REF][START_REF] Rolls | The responses of single neurons in the temporal visual cortical areas of the macaque when more than one stimulus is present in the receptive field[END_REF]. This suppression of neural representation for stimuli is referred to as competition [START_REF] Rahman | Influence of number, location and size of faces on gaze in video[END_REF][START_REF] Jacques | Concurrent processing reveals competition between visual representations of faces[END_REF], 2006). A recent study [START_REF] Jacques | Concurrent processing reveals competition between visual representations of faces[END_REF] showed that the response to foveal faces is reduced when another face is presented parafoveally. The suppression remained even when a scrambled face was presented as competing stimuli. This suggests that there is certainly some sensory competition rather than simply an effect of reduced spatial attention.

In videos, object stimulus patterns degrade due to the loss of information as it moves away from the foveal region. This degradation of information seems likely to influence the preference for faces in videos [START_REF] Sato | Interactions between two different visual stimuli in the receptive fields of inferior temporal neurons in macaques during matching behaviors[END_REF]. In still images, the influence of peripheral vision on face has been thoroughly studied, with controlled presentation of visual stimuli at predefined locations on rings of different eccentricities [START_REF] Paras | Face perception and configural uncertainty in peripheral vision[END_REF][START_REF] Reddy | Face identification in the near-absence of focal attention[END_REF][START_REF] Jebara | Face or building superiority in peripheral vision reversed by task requirements[END_REF][START_REF] Hershler | The wide window of face detection[END_REF][START_REF] Rigoulot | Fearful faces impact in peripheral vision: behavioral and neural evidence[END_REF]. Most found a drop in performance of object stimuli; in the case of faces in periphery, a steep drop in face-selective responses was observed. The main question in this study is to evaluate how face eccentricity-dependent sensitivity loss which occurs when viewing dynamic stimuli, could be used in an improved saliency model.

Both these limiting factors (number of faces and eccentricity) can be alleviated by the size of faces, which causes stimulus magnification to maintain foveal performance of faces and to diminish the effects of competition. Evidence shows that the accuracy and quality of visual performance in the periphery is identical to that in foveal vision [START_REF] Still | Peripheral image quality is almost as good as central image quality[END_REF][START_REF] Banks | Peripheral spatial vision: limits imposed by optics, photoreceptors, and receptor pooling[END_REF], but the drop is due to progressive undersampling of information presented away from fixation. Consequently, it limits the capacity of visual systems to extract information, which is important to attend to objects in a natural scene. This direct effect of eccentricity on stimuli can be compensated by using some linear eccentricity-dependent magnification, or size scaling [START_REF] Virsu | Visual resolution, contrast sensitivity, and the cortical magnification factor[END_REF][START_REF] Dow | Magnification factor and receptive field size in foveal striate cortex of the monkey[END_REF][START_REF] Levi | Vernier acuity, crowding and cortical magnification[END_REF][START_REF] Johnson | Size scaling compensates for sensitivity loss produced by a simulated central scotoma in a shape-from-texture task[END_REF].

The purpose of the current research is to study the influence of faces on gaze during free-viewing of videos, and analyze the effects of number, location and size of faces. We know that faces attract gaze in videos. We put forward the hypothesis that preference for faces depends not only on eccentricity but also on their area and number. The study reported here evaluates different eye fixation attributes: distance, proportion and duration. We also analyze the different influencing factors-number, location and size of faces-with statistical models, and test the hypothesis by analyzing different combinations of these influencing factors using a comparison criterion. The results obtained from this work could support the improvment of a separate face pathway to a visual saliency model-to more accurately predict eye movements.

Eye movement experiment

We used the eye position data from a previous experiment described in [START_REF] Marat | Modelling spatiotemporal saliency to predict gaze direction for short videos[END_REF]. The experiment aimed to record eye movements of participants when looking freely at videos with various contents. We used this data to understand the features that best explain eye movements and fixated locations. Here, we recall some of the main aspects of this experiment.

• Stimuli: Fifty-three videos (25 fps, 720×576 pixels per frame) were selected from different video sources, for example: indoor scenes, outdoor, scenes of day and night (Figure 1). The videos are converted to grayscale before presenting them to the participants. The videos were cut into 305 clip snippets each of 1-3s. This was done in order to obtain snippets with minimum change in plane. The aim here is to minimize potential top-down influence on eye movement. Finally, these clip snippets were strung together to obtain 20 clips of 30s. The duration of the clip was random, to eliminate any anticipation of transition made by the participants during viewing.

• Participants: Fifteen young adults (3 women and 12 men, aged 23-40 years) participated in the experiment. All participants had normal, or corrected to normal vision. Each participant, sitting with his/her head stabilized on a chin rest, in front of a monitor at 57cm viewing distance (40 • × 30 • field of view), was instructed to look at the videos without any task.

• Apparatus: An eye tracker (SR Research EyeLink II) was used to record eye movements. It is composed of three miniature cameras mounted on a helmet, two in front of each eye to provide binocular tracking, with the third on a head-band for head tracking. The recordings from the first two cameras, when compensated for head movements, give the gaze direction of the participant.

Method

In this study, we test a video database comprising faces to analyze their interest during free-viewing. We also evaluate the influence of different factors on the 7(2):5, 1-11 [START_REF] Rahman | Influence of number, location and size of faces on gaze in video[END_REF] Influence of number, location and size of faces on gaze in video interest of faces, such as number of faces, face eccentricity, face surface area, and closeness between two faces. In this section, we first present the data used for the evaluation that includes the hand-labeled faces of the entire video dataset, and the eye fixations recorded during the eye movement experiment. We use at most five fixations overtime that roughly equal 250ms after the scene onset. Second, we define these influencing factors. Third, we detail several evaluation metrics to analyze the influencing factors. Last, we summarize the methods used for statistical analysis of the data.

Database

The video database comprises a variety of face content, such as scenes with cases of one or more faces at different locations. Moreover, the faces are of different sizes. We labeled 14, 600 frontal and upright faces in total for the entire video database (14, 155 frames), to create a face ground-truth for this study. We also labeled turned faces when the facial features such as eyes and mouth regions were distinguishable. Moreover, background faces with blurred features were ignored in favor of foreground faces.

During the experiment, the eye tracker recorded participants' eye movements at 500 Hz-20 recordings for two eyes per frame and per participant. The recordings are then used to calculate corresponding fixations and saccades. In this study, we used these eye fixations to study different factors influencing the interest of faces in a dynamic scene. The distribution of eye fixations is shown in Figure 2. for all participants in a scene. These surface maps were created by placing a Gaussian with a diameter of 2 • of visual angle (equivalent to the fovea) centered at each fixation point, adding the Gaussians, and normalizing the height of the resulting sums.

In total, 23, 797 fixations were recorded for 15 participants, with 11, 155 fixations on scenes with at least one face. The number of fixations for one, two or more faces in a scene are summarized in Table 1.

Influencing factors

The study was designed to provide an insight into the extent to which different factors of faces affect their perceived interest. We annotate each frame in a scene with the number of faces present, and each face in the frame with its eccentricity and surface area. We also computed closeness of a face to another face in the frame. All these mentioned factors are measured as follows:

Number: is a simple count of faces present. It determines the complexity of the scene. For clarity, we only consider cases of frames with one face and two faces.

Eccentricity: is the relative distance from a participant's fixation on screen to the edge of nearest face ellipse in degrees. In Figure 3,(dr Area: is the two-dimensional surface of face ellipse in squared degrees. It is calculated as πr a r b , where r 1 and r 2 are the face ellipse's major and minor radii respectively.

Closeness: between the faces f 1 and f 2 in the case of two faces is the euclidean distance between the two face regions. In Figure 4, (d -(r(

Metrics

We used several evaluation metrics (minimum fixation distance, fixation proportion, fixation duration) to investigate the effects of faces on eye fixations during free-viewing of a visual scene. We analyzed the inflencing factors number, eccentricity, area, and closeness of 7(2):5, 1-11 [START_REF] Rahman | Influence of number, location and size of faces on gaze in video[END_REF] Influence of number, location and size of faces on gaze in video faces with linear regression models based on comparisons with face maps.

(C x , C y ) Fixation position (O x , O y ) face f a b α d r ( α ) E = d -r ( α )
O f 1 O f 2 face f 1 face f 2 a b d r ( α ) r ( β ) C = d - (r (α ) + r (β )) α β

Minimum fixation distance:

or Shortest Euclidean distance [START_REF] Wang | The attraction of visual attention to texts in real-world scenes[END_REF] from faces in a scene to fixation. The distance is computed from the fixation position to the face region of interest-the edge of face ellipse. Essentially, it is equal to the eccentricity E of the face closest to the fixation.

d min = arg min E

Fixation proportion

We categorized the fixations on scenes with faces into two types: fixations landing inside a face, called 'on-face' fixations (oF), and fixations landing outside a face, called 'not-on-face' fixations (nF). This was done by comparing fixation coordinates to a face, represented by an elliptical mask equal to the face dimensions plus 1 • of margin. In the study, we used the proportion of the two types of fixations, normalized by the total surface area of the faces. Here, we did not consider fixations for scenes with no faces to fixate upon.

Fixation duration.

Cognitive systems interact with the scene to determine where, and how long to fixate. The position of fixation points toward the region of interest, while its duration amounts to the attentional processing directed to that location [START_REF] Just | Eye fixations and cognitive processes[END_REF][START_REF] Rayner | Eye movements in reading and information processing: 20 years of research[END_REF][START_REF] Henderson | Regarding scenes[END_REF]. [START_REF] Le Meur | A coherent computational approach to model bottom-up visual attention[END_REF].

In this study, we are interested in analyzing the influence of faces in a scene. We used a comparison criterion to measure the correspondence between regions predicted to be fixated and regions fixated by participants, represented as face maps and eye fixation maps respectively.

• Face maps: We computed face map M f (Figure 7b) for each frame by hand-labeling the position of the face using a bounding box, and then applying a 2D Gaussian to it. The dimensions of the bounding box determine the variance of the 2D Gaussian from its origin in horizontal and vertical axis, whereas the amplitude of the function was kept constant for all faces. All values outside the Gaussianed elliptical face were set to zero.

• Eye fixation maps: The eye fixation maps were defined for each fixation made by a participant. It is simply the fixation position Gaussianed for one participant equivalent to 0.5 • of the visual field-the 7(2):5, 1-11 [START_REF] Rahman | Influence of number, location and size of faces on gaze in video[END_REF] Influence of number, location and size of faces on gaze in video size of the fovea with highest resolution. These maps, denoted as M h , were used to evaluate faces using the comparison criterion. A sample M h map is illustrated in Figure 5c. • Comparison criterion: To compute the comparison criterion, for instance for the first fixation, we compare M h for each participant to all M f maps for the entire duration of the fixation. The values are then averaged to get a score for the participant. Likewise, this process is repeated for all participants. Finally, all individual scores from all participants are averaged to get the score for this first fixation. We do the same for the five fixations and process a mean score of the five scores. We looked at each fixation separately but obtained no difference compared to the mean of five fixations. Note that in the case of face maps M f , the dimensions of the face define the standard deviations of the applied Gaussian where all values lying outside the resulting face ellipse are set to zero. In this study, we perform ROC (Receiver Operating Characteristics) analysis between two maps: a face map M f and a eye fixation map M h . The maps are processed as a binary classifier applied to every pixel; classified as fixated (or salient) or as not fixated (or not salient). A simple threshold is systematically moved between minimum and maximum values of the map. For each pair of thresholds, we get four numbers: the true positives (T P), the false positives (FP), the false negatives (FN) and the true negatives (T N). A ROC curve plots the false positive rate as a function of the true positive rate. The ROC area or the AUC (Area Under Curve) obtained by a trapezoid approximation, measures the classification performance. The trapezoidal rule used:

A = 1 2 N ∑ i=2 (x i -x i-1 ) • (y i + y i-1 )
It is usually taken as a scalar value, such that a value A = 0.5 reflects random forecasts, whereas A = 1.0 implies perfect forecasts.

Statistical analysis

To measure the influence of 'number of faces' in a scene, we compute a linear regression with levels (one face, two faces and no faces) for all video snippets-166 with one face, 120 with two faces and 191 with no faces. Likewise, this is done for all the evaluation metrics. To determine the influence of 'eccentricity of faces', 'area of faces' and 'closeness between two faces', the AUC comparison criterion was averaged across subjects for 166 video snippets in the three cases of video snippets. The correlation of the interest of faces with these influencing factors (independent variables) may be checked by multiple linear regression analysis. The statistical significance of the correlation (interaction effects) and of each independent variable (main effects) is given as an F statistic from which a p value can be determined. Note that regression was used because the dependent or predictor variables were continuous.

Results

In this study, we evaluate the interest in faces using different metrics: fixation duration, minimum fixation distance from face, 'on-face' and 'not-on-face' fixation proportion and comparison with face maps.

Minimum fixation distance

Distance of fixation from the closest face averaged across subjects for video snippets (166 with one face and 120 with two faces) with faces is shown as a function of fixation number in Figure 6. We observe that distance from the closest face decreased on scene onset, reached its minimum at the second fixation of about 2.5 • , and afterward remained steady in the following fixations. In the case of two faces, the distance to the closest face was smaller compared to one face, due to the presence of multiple regions of interest. This observation for fixation distance from face indicates one face attended yo in the first couple of fixations followed by exploratory fixations on the rest of the scene. This is not necessarily true in the case of two faces.

To get a baseline distance, we consider the fixation distribution of the entire eye fixation data. We take the median of this distribution as a baseline position, and compute the baseline using the distance of this position to the closest face, represented by a red line in Figure 6. The result shows that there is a significant difference between distance from a face and distance to the baseline point; t (570) = -13.534, p < 0.001 1 . In the Journal of Eye Movement Research 7(2):5, 1-11 [START_REF] Rahman | Influence of number, location and size of faces on gaze in video[END_REF] Influence of number, location and size of faces on gaze in video case of two faces, participants remain in proximity of faces over time. However, they comprise a significant proportion of the total fixations for scenes with faces. We conclude that faces are considered potential regions of interest when present in a scene, and participants remained in proximity to them over time.

Fixation proportion

Initially, the proportion of fixations made on face regions in a scene is smaller, which is attributed to a center-looking strategy on scene onset. As the scene progresses, at the second fixation F 2 , the proportion of fixations landing within the face regions becomes more than 50% of the total fixations. The preference of fixating faces becomes more prominent with the observation that the surface area occupied by face regions is much smaller compared to the rest of the visual stimuli.

Here we normalized the proportion with area and analyzed it over time and found this large proportion remains true in the following fixations for both one and two faces (Figure 7). Moreover, statistical test on proportions of first five fixations shows significant difference between 'on-face' (oF) and 'not-on-face' (nF) fixations; t (8) = 53.080, p < 0.001 in case of one face and t (8) = 30.283, p < 0.001 in case of two faces.

We conclude that faces, when present in a complex scene, are considered potential regions of interest. We found that participants remained in proximity to face over time. In addition to the inherent interest of faces, their eccentricity and area are also important, since these two factors could lead to less distorted information to form a recognizable object, resulting in fixations made much closer to a face in a scene.

Fixation duration

Fixation durations averaged across subjects for video snippets with faces (286 with one or two faces, and 192 with no faces) show that snippets with no faces have a shorter average duration of fixation (7.45 frames, 298ms) than snippets with faces (10.42 frames, 417ms), and the difference is significant t (476) = 6.288, p < 0.001. Since we use dynamic stimuli without sound, there is no impact of auditory cues on visual information extracted. We can imply that the presentation of visual stimuli alone leads to the centralization of gaze, resulting in longer fixations to extract maximum information from faces.

Comparison with face maps

Faces in a scene are certainly the regions of interest in dynamic stimuli, and they influence eye movements of the participants. In this section, we used a comparison criterion, the AUC score, to evaluate the face maps M f against the eye fixation maps M h . The aim was to evaluate the spatial importance of face regions.

Case of one face

AUC scores averaged across subjects for video snippets with one face (166 in total) are shown as a function of eccentricity and area in Figure 8. We observe that performance of faces decreased with increasing eccentricity; that is, faces presented in the foveal region of the current fixation tend to attract gaze compared to faces presented in the peripheral region in a quite linear way. In contrast, the performance of faces increased with increasing surface area. Faces with larger surface areas attract more gaze compared to ones with smaller surface areas.

Different models with eccentricity and area have been tested in the case of one face (Table 2). The results of linear regression for these models are presented in Table 3. For all cases, p-value < 0.05, then all proposed models (E, A, and E + A) are significantly better than the constant model.

Table 2 : List of statistical models in case of one face.

Short name Models Constant

Y = b 0 + ε E Y = b 0 + b 1 E + ε A Y = b 0 + b 2 A + ε E + A Y = b 0 + b 1 E + b 2 A + ε
In order to compare the different models, we consider f 2 criterion of Cohen [START_REF] Cohen | Statistical power analysis for the behavioral sciences[END_REF] whose defini-7(2):5, 1-11 [START_REF] Rahman | Influence of number, location and size of faces on gaze in video[END_REF] Influence of number, location and size of faces on gaze in video tion is

f 2 = R 2 /(1 -R 2 )
where R 2 is the square of the coefficient of multiple correlation. Cohen suggested that f 2 values of 0.02, 0.15 and 0.35 represent respectively small, medium and large effect size. We can use it to compare models with the same number of variables.

We also consider the Akaike information criterion (AIC) which is a measure of the relative quality of a statistical model for a given set of data. It takes into account both how well the model fits and its It provides mean for model selection (The best model corresponds to the largest negative value of AIC).

For one face (Table 3), f 2 criterion is very large with E model and medium with A model, thus E model is better than A model to explain the data. f 2 is large for E + A model too. AIC criterion shows that the best model is the E + A model, which confirms the interest of associating area variable to the classical eccentricity variable to explain the attraction of faces during eye movement. Let us note that we verified the applied condition of the models (independence, normality and homoscedasticity of residuals) by an analysis of studentized residuals.

Case of two faces

AUC scores averaged across subjects for video snippets with two faces (120 in total) are shown as a function of eccentricity of face, area of face and closeness between faces in Figure 9. Similar to one face, we observe that performance of faces decreases with increasing eccentricity to the nearest face and decreasing area of face. In addition, closer face regions result in higher scores compared to faces which are farther apart.

Different models with eccentricity, area and closeness were tested in case of two faces (Table 4). The results of linear regression for these models are presented in Table 5. For all cases (E, A, C, E + A, E + C, A + C, and E + A +C) p-value is < 0.05, and hence significantly better than the constant model. Table 4 : List of statistical models in case of two faces.

Short name Models

Constant

Y = b 0 + ε E Y = b 0 + b 1 E + ε A Y = b 0 + b 2 A + ε C Y = b 0 + b 3 C + ε E+A Y = b 0 + b 1 E + b 2 A + ε E+C Y = b 0 + b 1 E + b 3 C + ε A+C Y = b 0 + b 2 A + b 3 C + ε E+A+C Y = b 0 + b 1 E + b 2 A + b 3 C + ε
For two faces (Table 5), f 2 criterion is medium for all models. We can use it to compare models with the same number of variables. So, E model is better than A model and C model. Also, A +C model is better than E + A model and E +C model. AIC criterion shows that the best model is the E + A + C model, which confirms the interest of associating area and closeness variables to the eccentricity variable to explain the attraction of faces during eye movement. Let us note that we verified the applied condition of the models (independence, normality and homoscedasticity of residuals) by an analysis of studentized residuals.

Discussion

In summary, we measured and found significant effects of influencing factors on the interest of faces with dynamic stimuli. A number of studies have been conducted evaluating the influence of faces on gaze with static stimuli, very few with dynamic stimuli. They showed that participants tend to fixate faces based on saliency [START_REF] Birmingham | Saliency does not account for fixations to eyes within social scenes[END_REF], or due to their social importance [START_REF] Birmingham | Gaze selection in complex social scenes[END_REF]. Dynamic stimuli can offer more information compared to static stimuli, as it contains information about social status, identity, and emotions [START_REF] Buchan | Spatial statistics of gaze fixations during dynamic face processing[END_REF][START_REF] Foulsham | Gaze allocation in a dynamic situation: Speaking[END_REF].

Several works have been published using dynamic social scene-viewing.

Some compared visual fixation patterns of two groups, autistic and normal children [START_REF] Rice | Parsing heterogeneity in autism spectrum disorders: Visual scanning of dynamic social scenes in school-aged children[END_REF][START_REF] Riby | Looking at movies and cartoons: eye-tracking evidence from williams syndrome and autism[END_REF]. They measured fixation time proportion on scene parts (face, body, background) and response times showing differences between the two groups. Other works used calibrated stimuli from interviews. They evaluated how gaze is dynamically directed to the eyes, nose, or mouth [START_REF] Buchan | Spatial statistics of gaze fixations during dynamic face processing[END_REF][START_REF] Smith | Do the eyes really have it? dynamic allocation of attention when viewing moving faces[END_REF] on single large face with or without sound. They measured fixation proportions on face parts showing a higher proportion on mouth when sound is present. Recent work [START_REF] Song | Different types of sounds influence gaze differently in videos[END_REF] using complex scenes with free-viewing, showed that only sounds produced by humans on-screen influence participants gaze. In this case, human gaze was attracted by a talking or singing face.

Our contribution here is to evaluate dynamic stimuli on the impact of faces on gaze with several fixation attributes and influencing factors, for one face and for two faces. The database is based on dynamic complex scenes from films without calibration and without sound. The varying face content makes the video database useful to study the interest of face in real videos. Since little of the face content used comes from famous movies, less than 5%, the impact of familiar faces was negligible. Our study involved an experiment with free-viewing participants. [START_REF] Dorr | Variability of eye movements when viewing dynamic natural scenes[END_REF] using complex videos to evaluate the variability of gaze positions from free-viewing participants showing the interest of such an experiment. First, we quantified eye fixation attributes: distance of fixations to faces, proportion of fixations on faces, and duration of fixations. Then we evaluated three influencing factors (eccentricity, Influence of number, location and size of faces on gaze in video With the distance of fixations to face criterion, we conclude that participants remained in proximity to faces over time (mean of 2.5 • , similar to the literature [START_REF] Buchan | Spatial statistics of gaze fixations during dynamic face processing[END_REF]). The short distance is significantly lower than the baseline which is constant over time. We quantified the proportion of fixations on faces. We obtained a ratio of 50% of fixations on faces corresponding to only about 10% of screen. This is coherent with previous studies obtaining from 50% to 80% of fixations on face [START_REF] Rice | Parsing heterogeneity in autism spectrum disorders: Visual scanning of dynamic social scenes in school-aged children[END_REF][START_REF] Riby | Looking at movies and cartoons: eye-tracking evidence from williams syndrome and autism[END_REF][START_REF] Smith | Attentional synchrony and the influence of viewing task on gaze behavior in static and dynamic scenes[END_REF]. Fixation durations in a dynamic scene with faces are longer than with no face (around 400ms for one and two faces versus around 300ms with no face) because they offer more visual information for perception. We found that they were significantly longer in presence of faces, hinting that faces in a scene trigger fixations preceded by small saccades, essentially motivated to perform detailed analysis of the facial features. The durations were longer than usually reported in the literature for static images, reporting 200-250ms with no face and around 300ms with faces [START_REF] Pannasch | Visual fixation durations and saccade amplitudes: Shifting relationship in a variety of conditions[END_REF][START_REF] Smith | Attentional synchrony and the influence of viewing task on gaze behavior in static and dynamic scenes[END_REF][START_REF] Guo | Longer fixation duration while viewing face images[END_REF], which supports the idea of long explorations of few regions of interest. These results on fixation attributes are coherent with the literature and add new results on fixation duration on dynamic scenes with faces.

We also evaluated three influencing factors (eccentricity, area, closeness) with AUC criteria applied to comparison of fixation maps with ground truth face maps. First we consider the case of one face evaluating two variables (eccentricity and area). We observed that AUC drops as faces are presented farther away in the periphery or when their area is reduced. These observations are coherent with findings of different studies on static stimuli (with calibrated stimuli): that the performance of faces drops with increasing eccentricities with limited spatial information [START_REF] Thorpe | Detection of animals in natural images using far peripheral vision[END_REF][START_REF] Tomalski | Temporalnasal asymmetry of rapid orienting to face-like stimuli[END_REF][START_REF] Guo | I know you are beautiful even without looking at you: discrimination of facial beauty in peripheral vision[END_REF]). Area of face can limit and compensate for the effects of eccentricity already reported for static stimuli [START_REF] Johnson | Size scaling compensates for sensitivity loss produced by a simulated central scotoma in a shape-from-texture task[END_REF], and it can also reduce the crowding of facial information, increasing the influence of faces: that is, face becomes more recognizable when larger. We evaluated the effect of Eccentricity and Area with linear regression models (E, A and E + A models). f 2 criterion is high for E model showing that E model fits the data well. f 2 criterion is only medium for A model. AIC criterion shows that the E + A combined model is slightly better than the E model and better than the A model.

In the case of two faces we evaluated three variables (eccentricity, area and closeness) with AUC criteria. This case is particularly difficult because it involves complex scenes with various faces of different size and position, and evolving in front of complex backgrounds. We observed for one face that AUC drops for increasing eccentricity and decreasing area. Moreover, the competing faces influence each other based on their relative location (closeness). The closer the two faces are, the greater their spatial strength to attract gaze is, hence limiting the effects of competition between the two faces. Thus, the AUC score increases when faces are closer. The results obtained on competition of faces corroborate previous findings on static images with calibrated stimuli showing that competition reduces performance of faces [START_REF] Jacques | Concurrent processing reveals competition between visual representations of faces[END_REF][START_REF] Nagy | Sensory competition in the face processing areas of the human[END_REF]. The performance with increasing eccentricity drops further compared to one 7(2):5, 1-11 [START_REF] Rahman | Influence of number, location and size of faces on gaze in video[END_REF] Influence of number, location and size of faces on gaze in video face. In a related study on static stimuli [START_REF] Guo | I know you are beautiful even without looking at you: discrimination of facial beauty in peripheral vision[END_REF] when a face is presented alongside similar faces (with same characteristics), the performance to fixate a face drops, possibly a consequence of the limited information-processing capacity of peripheral vision. We evaluated the effect of the three variables: Eccentricity, Area and Closeness, with seven linear regression models (using E, A and C). In fact the three variables are not independent, they are tightly coupled. AIC scores are comparable for E and A models due to face competition (C model is a bit lower). The three variables provide information, then the combined models (E + A, A + C, E + C) obtain higher scores than single variable models. For the two faces case, the adding of a closeness variable to the traditional eccentricity and area variables improves the quality of the model (highest score for E +A+C model).

In conclusion, we have evaluated with statistical models the influence of one or two faces that decreases with increasing eccentricity, with decreasing area and with increasing closeness. This study is important to understand eye movements for free viewing of complex object categories like faces in videos, in particular to build computational models for visual attention. The results could be helpful to support the adding of a modulation to a face pathway as for recently proposed visual saliency models [START_REF] Cerf | Predicting human gaze using low-level saliency combined with face detection[END_REF][START_REF] Marat | Improving visual saliency by adding 'face feature map' and 'center bias[END_REF], to better predict eye movements. The models already take into account area of faces by modifying the representative Gaussians in face saliency maps. These could be improved by modulating the amplitude of Gaussians based on closeness. Furthermore, a coefficient based on area and closeness could be introduced to weight the face saliency map during the fusion phase with other saliency maps of the model. However, the use of eccentricity for modulation is not straightforward as it depends on human eye fixation unlike area and closeness. Nevertheless, factors like moving faces, emotion/expression could be investigated to improve the face pathway, and then the modulation.

Figure 1 .

 1 Figure 1. : Some sample frames from different video source.

Figure 2 .

 2 Figure 2. : Representation of the positions of all fixations

  (α)) is eccentricity E of face ellipse with origin (O x , O y ) from fixation position (C x ,C y ).

  α) + r(β)) is the closeness C between the face ellipses with origins O f 1 and O f 2 .

Figure 3 .

 3 Figure 3. : Eccentricity of face E presented on screen from fixation. Consider a face ellipse f with major and minor radii, r a and r b , corresponding to face dimensions. r(α) is the radius to the position on the ellipse at angle α of a right angle triangle with sides of length a and b. The angle is measured from the axis of the face ellipse f to the fixation position (C x ,C y ). Finally, the radius r(α) is subtracted from the euclidean distance d between the origin of the face ellipse (O x , O y ) and fixation position to obtain the eccentricity of the face.

Figure 4 .

 4 Figure 4. : Closeness C between two faces presented on screen. Consider two face ellipses f 1 and f 2 with major and minor axis equal to the respective dimensions of the faces. r(α) and r(β) are the radii to positions on the ellipses at angles α and β of a right angle triangle with sides of length a and b. The angles are measured from the major axis of one face ellipse to the origin of the counterpart face ellipse. Finally, the radii r(α) and r(β) are subtracted from the euclidean distance d between the origins O f 1 and O f 2 to obtain closeness between faces.

Figure 5 .

 5 Figure 5. : From left to right (a) the input frame with faces, (b) the face map M f for the input frame with 2D-Gaussianed faces from the ground truth, and (c) the frame's corresponding fixation Gaussianed for one participant (σ = 0.5 • ), referred to as M h .

Figure 6 .

 6 Figure 6. : Minimum fixation distance from face for number of faces-one and two faces. We took the first five fixations {F 1 , F 2 , F 3 , F 4 , F 5 } after current scene onset and fixation F -1 from the previous clip snippet (fixation just before onset of current scene).

Figure 7 .

 7 Figure 7. : Area-normalized proportion of fixations made on or off face regions, denoted as 'on-face' (oF) and 'not-onface' (nF) fixations respectively. The pie chart represents the surface area of the face ellipses f i computed as ∑ i πr a r b and (40 • × 30 • ) -∑ i πr a r b for oF and nF regions respectively. The latter region uses the 40 • × 30 • field of view. Here, We took the first five fixations {F 1 , F 2 , F 3 , F 4 , F 5 } after current scene onset.

  Figure 9. : Scatterplots with regression lines for AUC comparison criterion for two faces.

Table 1 :

 1 Total number of face frames and fixations for one, two, or more faces in a scene. In the study, we consider frames, with corresponding fixations, for scenes with one or two faces.

	One Two More than
	face faces two faces

Table 3 :

 3 Results of linear regression for models tested in case of one face.

	Model	F	p	f 2	AIC
	E	F(1, 164) = 204.8 < 0.001	1.248	-239.3
	A	F(1, 164) = 38.67 < 0.001	0.235	-139.9
	E + A	F(2, 163) = 104.8 < 0.001	1.280	-240.0

Table 5 :

 5 Results of linear regression for models tested in case of two faces.

	Model	F	p	f 2	AIC
	E	F(1, 118) = 20.11 < 0.001	0.170	-141.7
	A	F(1, 118) = 16.23 < 0.001	0.137	-138.3
	C	F(1, 118) = 6.634	0.011	0.056	-129.4
	E + A	F(2, 117) = 12.68 < 0.001	0.216	-144.4
	E +C	F(2, 117) = 11.1	< 0.001	0.189	-141.7
	A +C	F(2, 117) = 13.29 < 0.001	0.227	-145.4
	E + A +C F(3, 116) = 10.15 < 0.001	0.262	-146.8

572 samples in total =

distance values per {166 (one face) +120 (two faces) video snippets}

[START_REF] Rahman | Influence of number, location and size of faces on gaze in video[END_REF]


S Wu

B Guillet

L Méchin

J M Routoure
email: routoure@greyc.ensicaen.fr

A systematic study of the impact of geometry on the low frequency noise in patterned La 0.7 Sr 0.3 MnO 3 thin films at 300 K

Keywords: 1/f noise, manganite, thin films PACS: 73.50 T, 75.47 Lx

We report systematic measurements of low frequency noise performed at room temperature in La 0.7 Sr 0.3 MnO 3 (LSMO) thin films (thickness =150 nm) patterned with different lengths (50 µm to 300 µm) and widths (20 µm to 400 µm). Noise measurements were performed using two probe configuration, four probe configuration and even six probe configuration. Different 1/f noise contributions were observed for the film, for the current contacts and also for the voltage contacts. For the smallest devices, the noise spectral density of the film contribution does not follow the classical quadratic dependence with the DC voltage. The current contact contribution is due to current crowding at the metal/LSMO interface as already reported. The voltage contact contribution could be attributed to DC current circulation into the voltage contacts.

INTRODUCTION

Manganite thin films are promising for next generation devices and sensors [1]. In the La 0.7 Sr 0.3 MnO 3 (LSMO) case, it has been demonstrated that high sensitivity bolometers can be realized [2,3]. High performance room temperature spintronic devices, magnetoresistances as well as strain gauge could also be obtained due to the low value of the low frequency (LF) noise [4]. A systematic study of the impact of length, width and thickness and appropriated geometries on the low frequency (LF) noise level is therefore necessary for future sensor developments [5].

Systematic measurements of LF noise at room temperature in patterned La 0.7 Sr 0.3 MnO 3 (LSMO) thin films of thickness 150 nm, various lengths L (50 µm, 100 µm, 150 µm, 200 µm and 300 µm) and two widths W (20 µm and 100 µm) have been performed. The spatial homogeneity of the film was checked by resistivity measurements; magnetic characterizations versus temperature have revealed a Curie temperature close to 350 K and the temperature of maximal resistance was found to be about 390 K. These values are very close to the bulk ones proving the good crystallinity of the thin films. The noise measurements have been performed directly on prober using a Karlsuss PM5 with two, four or six probes.

EXPERIMENTAL CONDITIONS

Devices geometry

Deposition process as wall as the patterning technique of the devices can be found in [5]. An optical photography of one device is given in the figure (1). It consists of bridges with different lengths and widths with two current pads IP and IM and various voltage pads (V1 to V4 on one side of the bridge and V1' to V4' on the other side) which allows various device lengths. The metallic pads are realized using gold. The bridge lengths between V1 -V2, V2 -V3 and V3 -V4 are 100µm, 50µm and 150µm respectively.

Two probe, four probe and six probe configurations

The experimental set-up mainly consists in one low noise high output impedance DC current source previously described in [START_REF] Routoure | A low-noise high output impedance DC current source[END_REF] and up to two identical voltage amplifiers. It is assumed here that the DC current source is ideal: its output impedance is infinite and its noise contribution is negligible. We also suppose that the noise contribution of the voltage amplifier is negligible and that the input impedance of the amplifier is very high so that no DC current flows in its inputs. The device is connected at the output of the DC current source and the DC voltage and the fluctuations are measured using the voltage amplifiers. A spectrum analyzer (HP3562A or HP89410A) is connected at the output of the amplifiers. It calculates the noise spectral density or the cross spectral density in the case of the six probe configuration.

For all the configurations, the DC current source is connected between IP and IM probes. The mathematical derivation is not described here but it can be shown that :

in the two probe configuration, one voltage amplifier is connected to the IP and IM probes. The noise contribution of the film and the current contact can be estimated.

In the four probe configuration, one voltage amplifier is connected to the voltage probes. The noise contribution of the film and of the voltage contact can be estimated. In the six probe configuration, two voltage amplifiers are connected on each side of the bridge. With a spectrum analyzer that calculates the cross spectral density, all the uncorrelated noise sources disappear and only the contribution of the film can be estimated.

The LF noise spectral densities or cross spectral densities of the devices were measured for different DC current across the film in the two probe (S V 2p ), four probe (S V 4p ) and six probe configurations(S V 6p ).

RESULTS AND ANALYSIS

Only the results obtained for the device with L = 100 µm and W = 100 µm will be shown and discussed. The same behavior has been measured for the other lengths and widths. In figure (2a) are plotted S V 2p , S V 4p and S V 6p for f = 1 Hz versus the current in the device. It clearly shows that the three contributions for the film, the current contact and the voltage contact have to be taken into account. For the contact contribution the current contact noise level has a very high value as already reported [START_REF] Barone | [END_REF]8]. It is found that the noise spectral density follows a quadratic current dependency. It has been attributed to current crowding at the edge of the gold/LSMO interface.

The film contribution does not follow a quadratic DC current dependency. This may be attributed to lorentzian contributions in the spectra. Whatever the slope is, the bridge contribution is 2 times smaller than the voltage contact one. It shows another time that published normalized values of α/n (with α the Hooge constant and n the concentration ) may have been overestimated.

The last point concerns the fact that a LF noise contribution is found for the voltage contacts. In a first analysis, it is assumed that no DC current flows into the voltage contact since the input impedance of the voltage amplifier is very high: no 1/f noise should be found in the voltage contact. As shown in the figure (2b ), the current line penetrates more or less deeper in the voltage contact depending on the shape of the LSMO part of the voltage contact. This point has been verified using numerical simulation of the current flow in the device. This current circulation in the voltage contacts creates voltage contact noise contributions. These noise contributions disappear when the devices are characterized in the six probe configuration because the noise contributions that appear on each side of the bridge are not correlated.

CONCLUSION

The impact of the geometry of patterned LSMO thin film with different lengths and widths on the LF noise has been reported. It has been shown that three contributions exist for the film, the current contact and the voltage contact. The last one has been attributed to current flowing more or less deeper in the LSMO part of the voltage contact. This study shows that in order to obtain the correct value of the film noise, six probe configuration has to be used. Moreover, in the frame of sensor development, the shape of the voltage contact has to be designed carefully in order to limit as much as possible the LF noise contribution of the voltage contact.

FIGURE 1 .

 1 FIGURE 1. Optical photography of one 100 µm width bridge with the two current probes IP and IM and 2 x 4 voltage probes (V1...V4, V1'...V4') on each side of the bridge. The bridge lengths between V1 -V2, V2 -V3 and V3 -V4 are 100µm, 50µm and 150µm respectively.

FIGURE 2 .

 2 FIGURE 2. a. Spectral density measured in the bridge with L = 100 µm and W = 100 µm in the two probe (S V 2p ), the four probe (S V 2p ) and the six probe configuration (S V 6p ). b. Sketch showing the left part of the device with the IP pad and the two first voltage pads and the path of the current lines. The current lines close to the edge of the bridge penetrate more or less deeper in the LSMO part of the voltage contact depending on the shape of the voltage contact and the width of the bridge.


Mathieu Roche
email: mathieu.roche@cirad.fr

How to exploit paralinguistic features to identify acronyms in texts?

Keywords:

This paper addresses the issue of acronym dictionary building. The first step of the process identifies acronym/definition candidates, the second one selects candidates based on a letter alignment method. This approach has two advantages because it enables (1) to annotate documents, (2) to build specific dictionaries. More precisely, this paper discusses the use of a specific linguistic concept, the gloss, in order to identify candidates. The proposed method based on paralinguistic markers is independent of languages.

Introduction

Acronyms are numerous in specialized domain, e.g. biomedical and agronomy documents [START_REF] Chang | Creating an online dictionary of abbreviations from medline[END_REF]. An acronym is a set of characters corresponding to the first letters of a group of words, for instance, the acronym FAO is associated with the definition Food and Agriculture Organization. This paper summarizes a method to identify acronyms and expansions in documents. This automatic recognition enables to annotate these elements in texts. This work deals with the use of paralinguistic features in order to identify acronym/definition couples.

After the description of related work in the following section, Section 3. describes our approach based on 2 steps: Extraction of acronym/expansion candidates (Section 3.1.), Filtering of candidates (Section 3.2.). Finally, before Discussion and Conclusion sections, experiments of our approach are detailed in Section 4.

Related work

Among the several existing methods for acronym extraction in the literature, some significant work need to be mentioned. The acronym detection involves recognizing a character chain as an acronym and not as an unknown or misspelled word. Most acronym detecting methods rely on using specific linguistic markers.

Yates' method [START_REF] Yeates | Automatic extraction of acronyms from text[END_REF] involves the following steps: First, separating sentences by segments using specific markers (brackets, points) as frontiers. Then the acronym/expansion couples are tested. The acronym/definition candidates are accepted if the acronym characters correspond to the first letters of the potential definitions words. The last step uses specific heuristics to select the relevant candidates. These heuristics rely on the fact that acronyms length is smaller than their expansion length, that they appear in upper case, and that long expansions of acronyms tend to use stop-words such as determiners, prepositions, suffixes and so forth. Therefore, the pair "FAO/Food and Agriculture Organization" is valid according to these heuristics.

Other studies [START_REF] Chang | Creating an online dictionary of abbreviations from medline[END_REF][START_REF] Larkey | Acrophile: An automated acronym extractor and server[END_REF] use similar methods based on the presence of markers associated with linguistic and/or statistical heuristics. In this context [START_REF] Okazaki | Building an abbreviation dictionary using a term recognition approach[END_REF] propose statistical measurements from the terminology extraction area. Okazaki and Ananiadou apply the C-value measure [START_REF] Frantzi | Automatic recognition of multi-word terms: the C-value/NCvalue method[END_REF][START_REF] Nenadic | Terminology-Driven Mining of Biomedical Literature[END_REF] initially used to extract terminology. It favors a candidate term that doesn't appear often in a longer term. For instance, in a specialized corpus (i.e. Ophthalmology), the authors discovered that the term "soft contact" was irrelevant, while the frequent and longer term "soft contact lens" is relevant. An advantage of C-value measure is its independence from characters alignment (actually, a lot of acronyms/definitions are relevant while the letters are in a different order, e.g. "AW / water activity").

Other approaches based on supervised learning methods consist of selecting relevant expansions. In [START_REF] Xu | Using SVM to extract acronyms from text[END_REF], the authors use SVM approaches (Support Vector Machines) with features based on acronym/expansion information (e.g. length, presence of special characters, context, etc). [START_REF] Torii | A comparison study on algorithms of detecting long forms for short forms in biomedical text[END_REF] present a comparative study of the main approaches (supervised learning methods, rules-based approaches) by combining domainknowledge.

Larkey et al.'s method [START_REF] Larkey | Acrophile: An automated acronym extractor and server[END_REF] uses a search engine to enhance an initial corpus of Web pages useful for acronym detection. To do so, starting from a list of given acronyms, queries are built and submitted to the AltaVista 1 search engine. Query results are Web pages which URLs are explored, and possibly added to the corpus.

Acronym/expansion recognition

Our method of construction of acronym dictionaries is based on two steps detailed in the following subsections.

Step 1: Extraction of candidates

First, specific punctuation and character markers are taken into account in order to identify acronym/definition pairs (see Figure 1). In this paper, we investigate the extraction of candidates by exploiting the "glosses" of words and paralinguistic markers (i.e. brackets, punctuations, etc.) to detect acronym/definition candidates.

Glosses are spontaneous descriptions identifiable with specific markers (for example, called, i.e., and so forth). These ones highlight lexical semantic relationships, e.g. equivalence, specification of the meaning, nomination, hyponomy, hyperonomy.

The abstract pattern of glosses is given by the structure X marker Y 1 , Y 2 ...Y n where X and Y i can be acronyms and/or definitions. The identification and selection of glosses are based on the use of patterns and Web-mining approaches [START_REF] Mela | Lexical knowledge acquisition using spontaneous descriptions in texts[END_REF].

In this paper, we extract candidates based on the gloss markers "(" and ")":

• Local Pattern 1 [X=acronym, Y 1 =definition]:
The first pattern detects Y 1 (definition), between "(" and ")" following the acronym (X). For example, the sentence "relation empirique entre l'indice de végétation NDVI (Normalized Difference Vegetation Index), mesuré au maximum ..." allows to extract X = NDVI and Y 1 = Normalized Difference Vegetation Index.

• Local Pattern 2 [X=definition, Y 1 =acronym]:
The second pattern detects Y 1 (acronym), between "(" and ")" following the definition (X). The beginning of the definition is recognized with the first word of the phrase in upper case. For example, the sentence " ... la mesure Normalized Difference Vegetation Index (NDVI)" allows to extract X = Normalized Difference Vegetation Index and Y 1 = NDVI.

Note that these patterns are independent of languages because the method is based on paralinguistic markers (i.e., brackets in this work). This is very important when languages are mixed, for instance in specialized domains. The example of Figure 1 shows a definition in English (expansion of "NDVI") in an abstract written in French.

In this situation, we are 4 different cases of results:

1 www.altavista.com/

• Case 1: the relevant definition is returned (like previous examples),

• Case 2: the extracted phrase contains the relevant definition (i.e. partially relevant, but too large),

• Case 3: the extracted phrase is a part of the relevant definition (i.e. partially relevant, but too specific),

• Case 4: the extracted phrase is irrelevant.

Both proposed patterns will be evaluated in Section 4. of this paper.

Step 2: Filtering of candidates

The second step aims at removing irrelevant acronym/definition pairs and deleting irrelevant word(s) from candidate definitions. For this process, we propose to align the acronym letters with the potential definition words, by mapping each acronym letter with the first character of each definition word, respecting the order of words. If the first letter of the candidate definition word can not be aligned with the acronym corresponding character, the following characters (of the word) are taken into account. For instance, this method allows to find that "Extraction Itérative de la Terminologie" is a possible definition of the French acronym EXIT.

Evaluation

This paper focuses on the study of a corpus of 2000 paper abstracts provided by Cirad2 : French research centre working with developing countries to tackle international agricultural and development issues. Table 1 shows that better results are given with the second local pattern. But a lot of cases are partially relevant (i.e. ∼ 40%), so we have to improve and enrich this pattern approach. [START_REF] Roche | Managing the acronym/expansion identification process for textmining applications[END_REF]. We plan to apply the second step of the process (see Section 3.2.) with the pattern approach described in Section 3.1. on the Cirad corpus. Note that our previous work [START_REF] Roche | Managing the acronym/expansion identification process for textmining applications[END_REF] uses more global patterns ; then a lot of noise is returned. The pattern approach described in this paper is more specific with better results in term of precision (∼ 40% in this current work vs. 15% in our previous work).

Patterns

Discussion: Towards a Web-mining approach

In this section, we propose to integrate Web-mining measures in order to automatically validate results returned by our approach [START_REF] Turney | Mining the Web for synonyms: PMI-IR versus LSA on TOEFL[END_REF][START_REF] Mela | Lexical knowledge acquisition using spontaneous descriptions in texts[END_REF].

For instance, we can query a search engine with the acronym "BSV" and its possible definition to check on the Web if this association exists. This query should be a disjunction (i.e. OR operator) of the acronym and its possible definition returned with our process (i.e. Banana streak virus). This one returns a larger amount of documents. The conjunction of the acronym and the expansion (i.e. AND operator) enables to return a lower number of documents.

But the returned documents are more relevant (i.e. the precision is improved).

In our case, we choose to consider the "hits" given by Google3 on the examples of Table 2 (i.e. number of pages returned by the search engine based on conjunction). For instance, we have tested the query "BSV" AND "Banana streak virus" that returns 7580 pages4 . All the results (i.e. hits) are given in Table 3. This table shows that hits have generally very high values, this allows us to automatically validate acronym/definition couples. Note that hits of irrelevant couples return lower values (for instance, with the couples "ETM"/"environ 5.000 m3.ha-1", "SIPSA"/"indicateurs, documents, cartes", and so on).

Moreover, we can integrate this kind of information in classical similarity measures, e.g. Dice measure [START_REF] Smadja | Translating collocations for bilingual lexicons: A statistical approach[END_REF]. Dice measure can be used to compute a sort of relationship between an acronym (i.e. acro) and a definition (i.e. def ). In our context, Dice measure (formula (1)) is based on the number of Web pages given by search engines (i.e. hits).

W eb Dice (acro, def ) = 2 × hits(acro, def ) hits(acro) + hits(def )

(1) This measure returns the following result with the previous example:

W eb Dice (BSV, Banana streak virus) = 2×hits("BSV " AN D "Banana streak virus") hits("BSV ")+hits("Banana streak virus") = 2×7580 2840000+15400 = 0.0053 W eb Dice can be applied in order to rank couples (see Table 4). This enables to detect relevant acronym/definition pairs (i.e. couples with high W eb Dice values).

Conclusion

The process described in this paper is based on the use of specific linguistic markers to detect acronyms. In future work we plan to integrate statistical information and Webmining approaches in order to improve our methods based on linguistic rules. Our text-mining system allows us to enrich specialized thesaurus (e.g. MeSH5 , Agrovoc6 ). These thesaurus are useful to automatically annotate texts.

Moreover we plan to investigate a contrastive analysis of English/French corpora in order to give a new point of view of the phenomenon of spontaneous descriptions. A first study on aligned English/French texts reveals frequent regularities of glosses in a multilingual context. The alignment enables to improve the multilingual lexical acquisition of new words and their translations.

Table 1 :

 1 Evaluation of extracted definition with patterns.

		Local pattern 1 Local pattern 2
	Number of
	extracted	78	64
	definitions
	Case 1	31	28
	(relevant)	(39.7%)	(43.7%)
	Case 2	3	6
	(partially relevant)	(3.8%)	(9.3%)
	Case 3	1	18
	(partially relevant)	(1.3%)	(28.1%)
	Case 4	43	12
	(irrelevant)	(55.1%)	(18.7%)

The evaluation of the acronym/expansion extraction method is conducted on a corpus (general domain) having a reasonable size (7465 words). The experiments based on standard evaluation measures of data-mining domain highlight acceptable results (i.e. Precision: 66.7%, Recall: Figure 1: Recognition of the couple NDVI / Normalized Difference Vegetation Index in AGRITROP database.

Table 2 :

 2 Examples of acronyms/definitions.

80%, F-measure: 72.7%)

Table 3 :

 3 Examples of acronym/definition and hits scores.

	Acronym	Possible definition	Hits (Google)
	NRPS	NonRibosomal Peptide Synthetase	230000
	VLE	Virtual Laboratory Environment	36900
	BMR	Bois Massif Reconstitué	9270
	ATPSM	Agricultural Trade Policy Simulation Model	27700
	ASA	Articulation du Semi-aride	663
	CLF	Corynespora Leaf Fall	22800
	BASIC	Brésil, Afrique du Sud, Inde, Chine	21100
	CIAT	Centro international de agricultura tropical	75000
	BSV	Banana streak virus	7580
	ER	Ehrlichia ruminantium	121000
	CSSV	Cacao swollen shoot virus	2040
	MAE	Mesures agrienvironnementales	951
	ACMV	African cassava mosaic virus	90200
	TYLCV	Tomato yellow leaf curl virus	354000

Table 4 :

 4 Acronym/definition couples ranked with W eb Dice .

	Acronym	Possible definition	W eb Dice
	ATPSM	Agricultural Trade Policy Simulation Model	1.3014
	TYLCV	Tomato yellow leaf curl virus	0.7167
	NRPS	NonRibosomal Peptide Synthetase	0.4423
	CIAT	Centro international de agricultura tropical	0.1408
	ACMV	African cassava mosaic virus	0.0970
	CSSV	Cacao swollen shoot virus	0.0245
	VLE	Virtual Laboratory Environment	0.0222
	CLF	Corynespora Leaf Fall	0.0208
	BSV	Banana streak virus	0.0053
	BMR	Bois Massif Reconstitu	0.0046
	ER	Ehrlichia ruminantium	0.0004
	BASIC	Brsil, Afrique du Sud, Inde, Chine	0.0001
	ASA	Articulation du Semi-aride	0
	MAE	Mesures agrienvironnementales	0

http://www.cirad.fr/en/home-page

http://www.google.fr/

Queries performed on the 20th of March 2014.

http://www.nlm.nih.gov/mesh/

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Acknowledgements

This work was supported in part by the French National Research Agency under JCJC program, grant ANR-12-JS02-01001, as well as by University Montpellier 2 and CNRS. The author thanks DIST (Scientific and Technical Information Service) for the acquisition of Cirad corpus.


Daniel Barlet
email: daniel.barlet@univ-lorraine.fr

Barlet Daniel

Two semi-continuity results for the algebraic dimension of compact complex manifolds

Keywords: AMS Classification 2010 : 32G05-32A20-32J10 Key words : Algebraic dimension-Relative codimension 1 cycle-space-Moishezon spaces

    Copyright Two semi-continuity results for the algebraic dimension of compact complex manifolds.

1 Statement of results.

This Note has been inspired by Dan Popovici article [P.13], and we show here that using relative codimension 1 cycle-space, we may give another proof of its theorem 1.2 and 1.4 and also obtain from this proof much more general results. In fact, very few is new in this approach, because we simply uses the ideas in [P.13] and combined them with the classical tools introduced in [B.75], [B.78] and [C.80]; see also [B-M.14]. This gives a more geometric view on the use of the existence of a strongly Gauduchon form (see definition above) and of the existence of a relative Gauduchon metric on a family of compact complex manifolds using a geometric approach to (relative) algebraic reduction in term of codimension 1 (relative) cycles.

We shall prove the following generalizations of the theorem 1.2 and 1.4 of [P.13].

Theorem 1.0.1 Let π : X → D be a proper holomorphic submersion on an open disc D ∈ C with center 0, where X is a connected complex manifold. Assume that for each t ∈ D X t := π -1 (t) is a compact connected complex manifold of dimension n. Assume also that we have i) For each t ∈ D \ {0} the manifold X t is a Kähler manifold.

ii) For each t ∈ D \ {0} the algebraic dimension of X t is at least equal to a ≥ 1.

iii) The dimension h 0,1 (t

) := dim H 1 (X t , O) is independant of t ∈ D.
Then the algebraic dimension of the manifold X 0 is at least equal to a.

Theorem 1.0.2 Let π : X → D be a proper holomorphic submersion on an open disc D ∈ C with center 0, where X is a complex manifold. Assume that, for each t ∈ D * , the compact connected complex manifold of dimension n X t := π -1 (t) is Kähler. Assume also that X 0 admits a smooth 2n -2 real d-closed form such that its (n -1, n -1) part is positive definite. Then, if for each t ∈ D \ {0} the algebraic dimension of X t is at least a ≥ 1, the algebraic dimension of X 0 is also at least a. some remarks.

1. When the map π is kählerian in the sense of [C.81], which implies that X t is in the class C of A. Fujiki 1 for all t ∈ D, then the semi-continuity result of the algebraic dimension of the fibers of π is proved in [C.81].

But the kählerian assumption for π is rather strong as its implies that X 0 is in the class C and also properness of the irreducible components of the relative cycles spaces for any dimension of the cycles. Here we only use properness for relative codimension 1 cycles and no Kähler assumption on X 0 . Note that in the absolute case, this compactness for irreducible components of the codimension 1 cycle space is always true (for any compact complex space, see [C.82]) but this is far to be true for smaller cycles of positive dimension.

2. In both cases the key point is to produce a (n -1, n -1) smooth positive D-relative form in X such the integral of it, on the generic member of an analytic family of relative (n -1)-cycles giving for generic t ∈ D \ {0} an algebraic reduction for X t , is locally bounded near 0. In the second case this is an easy consequence of the d-closedness of this form ; in the first case, it uses only the ∂ ∂-closedness of such a form combined with a nice argument of D. Popovici [P.13] using the constancy of the numbers h 0,1 (t).

2 Algebraic reduction.

2.1 Absolute case.

For a compact irreducible complex space X of dimension n the irreducible components of the complex space C n-1 (X) are compact Moishezon spaces. As it is difficult to find an explicit proof of this resul, despite the fact that it is well known for the specialists since 30 years (see [C.82]), I shall give a proof in the appendix. Note that in codimension > 1 this result is not true in general : see, for instance [C.91].

For any compact irreducible analytic subset Γ of dimension d ≥ 1 in C n-1 (X) there exists a "natural" meromorphic map

K Γ : X C d-1 (Γ)
called the Kodaira map of Γ. This means that there exists a (natural) proper (thanks to [B.78]) modification τ Γ : X Γ → X and a holomorphic map (also denote

K Γ ) K Γ : X Γ → C d-1 (Γ)
associating to the generic point x of X the (d -1)-cycle in Γ which is the subset of Γ parametrizing the (n -1)-cycles containing x.

We shall denote its image Q Γ . Note that Q Γ is always a compact irreducible Moishezon space. We shall recall some more details on this construction in the relative case in the next paragraph.

The algebraic dimension a(X) of X is the maximal of the dimension of the Q Γ when Γ is an irreducible component of C n-1 (X). In fact, for a given X there exists a Γ such its Kodaira map is an algebraic reduction (see [C.80] or [C.81]).

Claim. In any case, the algebraic dimension of X is bigger or equal to the dimension of Q Γ .

proof. If the generic member of Γ is not irreducible (but reduced), there exists a proper finite map g : Γ → Γ and an analytic family of (n -1)-cycles parametrized by Γ, with generic irreducible member, obtained by Stein factorization of the projection G Γ → Γ. If Γ ′ is the image of Γ by the corresponding classifying map, we have a direct image g * :

C d-1 ( Γ) → C d-1 (Γ)
which is proper an finite and induces a genrically finite surjective meromorphic map

Q Γ Q Γ .
So the dimension of Q Γ is again bounded by the algebraic dimension of X.

In the case of non reduced generic member in the family parametrized by Γ we may consider the pull back of the set theoretic graph |G Γ | on the normalisation Γ 1 of Γ which is the graph of an analytic family of (n -1)-cycles (see [B.75] or [B-M.14]) and replace Γ by the image Γ ′ of Γ 1 in C n-1 (X) by the corresponding classifying map. Then it is easy to see that Q Γ and Q Γ ′ are bimeromorphic, so again dim Q Γ is at most the algebraic dimension of X.

The relative case.

We consider now a proper surjective holomorphic map π : X → D of an irreducible complex space X to an open disc D ∈ C with center 0. Let dim X = n + 1 ; note that for each t ∈ D the fiber X t := π -1 (t) is a compact complex space with pure dimension n. It will be convenient to assume that for generic t in D the compact space X t is irreducible2 . Consider now an irreducible D-proper analytic subset of dimension d, Γ ⊂ C n-1 (π), of the space of D-relative (n -1)-cycles in X. We say that Γ is a good filling for π : X → D if the following conditions are satisfied i) The generic member of the universal family parametrized by Γ is irreducible.

ii) The graph G Γ ⊂ Γ × D X of the universal family parametrized by Γ projects surjectively on X by the second projection.

Then it is clear that G Γ is irreducible, proper on D and has dimension d + n -1. Note that the second condition implies that d + n -1 ≥ n + 1, so that d ≥ 2.

Claim. The closed analytic subset Y Γ ⊂ X which is the locus of the fibers of dimension ≥ d -1 of the projection pr :

G Γ → X has empty interior in each X t , t ∈ D. proof. If X t has an irreducible component Z in Y Γ , then pr -1 (Z) has dimen- sion ≥ d + n -1 = dim G Γ .
So, as G Γ is irreducible, we see that this contradicts the fact that pr : G Γ → X is surjective.

So we have a proper D-modification3 τ Γ : X Γ → X and a holomorphic D-map

K Γ/D : X Γ → C d-2 (Γ/D)
obtained by composition of the "fiber-map" of pr : G Γ → X and the D-relative direct-image map of relative (d -2)-cycles via the D-map p :

G Γ → Γ (see [B.75] or [B-M.14].
Claim. The D-map K Γ/D is proper, and so Q Γ is proper over D.

proof. Let K be a compact set in C d-2 (Γ/D) and let L its projection on D. Then any relative cycle in K is contained in p -1 (L) where p : Γ → D is the projection which is assumed to be proper. Then the pull-back of the compact set p -1 (L) on G Γ is a compact set M and also its image pr(M). Now K -1 Γ/D (K) ⊂ pr(M) and is compact. The propernes of Q Γ on D is then easy.

Denote Q Γ the image of K Γ/D . As for each t ∈ D the family Γ(t) is a finite union of irreducible compact analytic subsets in C n-1 (X t ), with at least one good filling set, the compact analytic set Q Γ (t) is a positive dimensional compact Moishezon space.

Assume now that for t ∈ T , where T is an uncountable subset of D, the family Γ(t) gives an algebraic reduction of X t . Then we have, for such

t ∈ T dim Q Γ (t) = a(X t ).
This implies, by semi-continuity of the dimension of the fibers of the projection Q Γ → D, that for each t in D, using the result at the end of paragraph 2.1, we have dim a(X t )

≥ dim Q Γ (t) ≥ inf t∈T dim Q Γ (t) = inf t∈T a(X t ).
Conclusion. Assume that we know the following :

i) The irreducible component Γ of C n-1 (π) is proper on D.
ii) For t in T where T is uncountable, we have dim Q Γ (t) = a.

Then we conclude that for each t ∈ D we have a(X t ) ≥ a.

3 A sufficient condition for properness of the irreducible components of the space of relative divisors.

We begin by a definition which is equivalent to the notion of a "strongly Gauduchon metric" introduced in [P.13].

Definition 3.0.1 Let X be a reduced complex space of pure dimension n. A 2n -2 smooth form ω on X will be called a sG-form on X when it satisfies:

i) The form ω is d-closed.

ii) The (n -1, n -1) part of ω is positive definite on X.

Recall that the strict positivity above means that in a local embedding in an open set U of some C N , the (n -1, n -1) part of ω may be induced on X by a smooth, strictly positive in the sense of Lelong (n -1, n -1) form on U.

We also need the relative version of this notion :

Definition 3.0.2 Let π : X → S be a surjective proper n-equidimensional morphism of reduced complex spaces. A smooth S-relative 2n -2 form ω /S will be called a S-relative sG-form if it induces a sG-form on each X s := π -1 (s).

Note that in the absolute case the irreducible components of the space C n-1 (X) of compact (n -1)-cycles of a compact pure n-dimensional complex space X are always compacts, thanks to the existence of a meromorphic algebraic reduction X Y , where Y is a projective variety (see [C.81] and [C.82]). The existence of a S-relative sG-form for a proper n-equidimensional surjective map π : X → S implies a relative version of this result.

Proposition 3.0.3 Let π : X → S be a surjective proper n-equidimensional morphism of reduced complex spaces admitting a S-relative sG-form ω. Then each connected component of the S-relative cycle space C n-1 (π) is proper over S.

Proof. For a (n -1)-relative cycle C of π define F (C) := C ω. This is a continuous function of C (see [B.75] or [B-M.14]) and on any given compact set in S, this function is bigger than ǫ times the volume of C for a continuous hermitian metric defined on X , because in the integration, only the (n -1, n -1) part of ω /S is relevant. As the function F is locally constant on the fibers of the projection C n-1 (π) → S thanks to the d /S -closeness of ω /S , this is enough to prove the properness of the projection C n-1 (π) → S thanks to E. Bishop's theorem (for instance, see [B-M.14]).

Lemma 3.0.4 [see [P.13]] Let π : X → D be a proper holomorphic family of compact connected complex manifolds of dimension n parametrized by an open disc with center 0 in C. Assume that X 0 := π -1 (0) has a G-form ω 0 . Then we may find a small open disc D ′ ⊂ D with center 0 and a relative G-form ω on π -1 (D ′ ) inducing ω 0 on X 0 .

Proof. Thanks to Ehresmann theorem, there exists an open disc D 1 ⊂ D with center 0 and a C ∞ trivialisation θ : π -1 (D 1 ) → D 1 × X 0 of the fibration π inducing the identity on X 0 . Define ω := θ * (ω 0 ). As ω 0 is d-closed, so is ω. We shall consider ω as a relative d-closed form. It induces ω 0 on X 0 . As the complex structure of X t varies continuously with t ∈ D 1 , the (n -1, n -1) part of the relative form ω varies continuously. As it is definite positive at t = 0 there exists an open disc D ′ ⊂ D 1 with center 0 where it stays definite positive on each fiber ; so ω induces a relative G-form on π -1 (D ′ ). .

In the situation of this corollary define X ′ := π -1 (D ′ ). Now the proposition above gives that the connected components of C n-1 (X ′ ) are proper over D ′ .

Remark. In the situation of theorems 1.0.1 and 1.0.2 we may apply this lemma to see that any irreducible component of C n-1 (π * ) is proper over D * , where D * = D \ {0} and π * is the restriction to π -1 (D * ), because we assume that each X t , t ∈ D * , is Kähler, so admits a G-form.

4 Adaptation of two key results of [P.13].

We shall adapt in this section two key results in [P.13] in order to use them in our proofs.

Adaptation of the remark 2.1

Let π : X → D be a proper holomorphic submersion of a (n + 1)-dimensional connected complex manifold X to a disc. To follow the point of view in [P.13], we trivialize π as a C ∞ -submersion4 and we shall assume that X ≡ X × D, π given by the second projection, the complex manifold X t := π -1 (t) is then defined by a complex structure J t on X which depends smoothly on t ∈ D.

We shall denote by X * := π -1 (D * ) where D * := D \ {0}, and π * : X * → D * the restriction of π. Note that this morphism is (locally on D * ) a Kähler morphism, so that the irreducible components of C n-1 (π * ) are proper on D * (see the final remark of the previous paragraph).

Proposition 4.1.1 In the situation above, assume that the algebraic dimension a(X t ) of X t is at least equal to a ≥ 1 for all t ∈ D \ {0}. Then there exists an irreducible component Γ * of the space C n-1 (π * ) with the following properties : i) Γ * is proper and surjective over D * .

ii) Γ * fills up X * (i.e. the graph G Γ * ȯf Γ * surjects onto X * ).

iii) The generic cycle in Γ * is irreducible.

iv) The dimension of Q Γ * is at least a + 1 and the projection Q Γ * → D * is proper and surjective.

proof. Note first that, as any irreducible component of C n-1 (π * ) is proper over D * , the only condition in i) is the surjectivity. For each t ∈ D * there exists an irreducible (n -1)-cycle C t in X t which is the generic member of a compact irreducible analytic subset Γ(t) in C n-1 (X t ) such that the image Q Γ (t) of the corresponding Kodaira map as dimension at least a. Then Γ(t) is a compact analytic subset of some irreducible component Γ * of C n-1 (π * ). As there are only countably many Γ * and uncountably many t ∈ D * , there exists at least one Γ * which contains uncountably many such Γ(t). Then this Γ * must satisfies i). The corresponding Q Γ * has uncountably many fibers over D * which have dimension at least a. So the condition iv) is also fullfilled, as properness of Q Γ * on D * is automatic form the properness of Γ * on D * . This implies condition ii) because the fact that Q Γ * surjects to D * implies that the generic fiber of Γ * over D * has dimension ≥ 1. The condition iii) is also clear.

In this situation we shall denote by α(Γ * ) ∈ H 2 (X, Z), the (topological) fundamental class of any (relative) cycle in the family parametrized by Γ * .

4.2 Adaptation of the proposition 3.1.

Recall first that in our situation there exists, thanks to [G.77], a smooth relative Gauduchon metric on X . For our purpose we shall simply use the fact that there exists a smooth family γ t of positive definite (1, 1)-forms on X t , t ∈ D with the condition that ∂ t ∂t γ ∧(n-1) t = 0 for each t ∈ D. Then the variant of the proposition 3.1. of [P.13] we shall use is the following. Proposition 4.2.1 Let α ∈ H 2 (X, Z) be the fundamental class of the relative (n -1)-cycles in the family parametrized by Γ * chosen as in the proposition 4.1.1. Note [C t ] the integration current on X t of a member of the family Γ * contained in X t for some t ∈ D * . If h 0,1 (t) is independent of t ∈ D, there exists a constant C > 0 such that for t ∈ D * near enough 0 we have

< [C t ], γ ∧n-1 t > ≤ C < +∞ where γ ∧(n-1) t
is the (n -1)-th exterior power of a relative Gauduchon metric on X .

proof. Let ω be a smooth d-closed real 2-form on X which is a de Rham representative of the class α ∈ H 2 (X, Z). So for each t ∈ D * there exists a real 1-current β t on X t such that [C t ] = ω + dβ t . As β t is real, we have

β t = β 1,0 t + β 1,0 t . Note that type consideration shows that β 0,1 t := β 1,0 is solution of the equation ∂t β 0,1 t = -ω 0,2 t .
So following [P.13], we shall define β0,1 as the (unique) solution of this equation with minimal L 2 norm defined by the Gauduchon metric γ t on X t . Defining βt = β0,1 + β0,1 we have now that the real current

[C t ] -ωt -d( βt )
is d-exact and of type (1, 1) on X t . Then, as the complex compact manifold X t is Kähler, it satisfies the ∂ ∂-lemma (see for instance [V.86]), and there exists a 0-current ϕ t on X t such that [C t ] = ωt + d( βt ) + i∂ t ∂t ϕ t .

An easy consequence is now, as ∂ t ∂t γ ∧(n-1) t = 0, that

< [C t ], γ ∧n-1 t >=< ωt , γ ∧n-1 t > + < d( βt ), γ ∧n-1 t > .
As the function t →< ωt , γ ∧n-1 t > is continuous on D, to bound the left handside, it is enough to bound near t = 0 the term < d( βt ), γ ∧n-1 t >= ± < βt , dγ ∧n-1 t > .

Then it is enough to follow the argument concluding the proof of the proposition 3.1 in [P.13], showing that under the assumption that h 0,1 (t) is constant, t → βt depends continuously of t ∈ D to conclude.

An obvious consequence of this result is that the volume (for the relative Gauduchon metric we have fixed, but this property is independant of the choice of a continuous relative hermitian metric) of members in Γ * is uniformely bounded, up to shrink the disc D around 0. This implies that the closure Γ of Γ * in C n-1 (π) which is an irreducible analytic subset, is proper over D thanks to Bishop's theorem (see [B-M.14]).

Proofs of the two theorems. In both cases we have an irreducible analytic subset Γ of C n-1 (π) which is proper surjective over D, with an irreducible generic member, such that its graph surjects to X and such that the corresponding Q Γ has generic fibers of dimension ≥ a over D. Then its fiber at t = 0 has dimension at least a which implies that the algebraic dimension of X 0 is at least a, thanks to the conclusion of the section 2.

Note that it is proved in [V.89] that a compact complex manifold is in the class C if and only if it is bimeromorphic to a compact Kähler manifold.

Using a Stein reduction of π, this assumption is not restrictive.

This means that it induces a proper modification on each X t , t ∈ D.

this is not restrictive, up to localize the situation on D; see the begining of the proof of the lemma 3.0.4.

It is enough to treat this case for our result.

Appendix

We give here a proof of the following (classical) statement : Theorem 5.0.2 Let X be a compact irreducible complex space of dimension n. Then the irreducible components of the space C n-1 (X) of (n -1)-cycles in X are compact and Moishezon.

proof. Using Hironaka's desingularization theorem it is not restrictive to assume that X is a compact connected manifold of dimension n and that we have a holomorphic surjective map r : X → P where P is a projective manifold of dimension a where a is the algebraic dimension of X. This comes from the fact that by a proper modification of X we add only finitely many effective irreducible (n -1)-cycles, and then, for any irreducible component Γ of the space of (n -1)-cycle of our initial X, there exists an irreducible component Γ of space of (n -1)-cycles of the smooth modification of X such that the direct image of cycles gives a modification Γ → Γ. Recall now that the number of non polar effective irreducible divisors in X is bounded thanks to [C.82]. Let now Γ be an irreducible component of C n-1 (X) of positive dimension with generic member irreducible 5 . Then each member of the corresponding family is polar and so has a (a -1)-dimensional image in P . This means that the image of the graph G Γ ⊂ Γ × X by the map id Γ ×r is proper and (a -1)-equidimensional on Γ. Up to replace Γ by its normalization, we have the graph of an analytic family of (a -1)-cycles in P . As Γ is irreducible and P projective, the volume of these cycles in P is uniformely bounded. Now using any continuous hermitian metric on X, we have an uniform bound for the volume of the generic fibers of r thanks to [B.78]. Then a Fubini type argument implies that the volume of the generic member of our initial family, which is bounded by the volume of the pull-back by r of its image by r is uniformely bounded. Now Bishop's theorem implies that Γ is relatively compact in C n-1 (X), as it is closed, it is compact. To conclude we have to remember that the normalization of Γ dominates a compact analytic subspace of an irreducible component of C a-1 (P ) and that this map is generically finite, because the pull-back by r of an irreducible effective divisor in P contains only finitely many irreducible effective divisors in X. So Γ is a compact irreducible Moishezon space.