Nicolas Gourier

Daniela Hall

James L Crowley

Facial Features Detection Robust to Pose, Illumination and Identity *

This paper addresses the problem of automatic detection of salient facial features. Face images are described using local normalized gaussian receptive fields. Face features are learned using a clustering of the Gaussian derivative responses. We have found that a single cluster provides a robust detector for salient facial features robust to pose, illumination and identity. In this paper we describe how this cluster is learned and which facial features have found to be salient.

Introduction

We are interested in automatically determining which facial features can be most reliably detected under variations in illumination, position, orientation and human identity. Our objective is to obtain a set of facial features that can serve as landmarks for tracking and recognition of facial expressions. We employ a fast, pixel level, detection algorithm to isolate and normalize the face region. Normalized face images are described by calculating a vector of scalenormalized Gaussian derivatives at each pixel. Salient facial features are detected using linear combinations of these descriptors. Such functions are learned using K-means clustering of the Gaussian derivative responses obtained from a set of training images. The resulting clusters specify linear combinations of Gaussian derivatives that act as detection functions for facial features that remain salient under variations in pose, illumination and identity.

Approaches to Facial Feature Detection

Facial feature detection may be performed using global or local features. A popular method for global analysis of face images is to project a normalized image into a linear subspace determined using a technique such as principal components analysis [START_REF] Turk | Eigenfaces for Recognition[END_REF]. However, PCA is sensitive to head orientation. Alternatively, global analysis may be performed by measuring the relative position of anatomical facial structures such as the eyes and lips [START_REF] Zobel | Robust Facial Feature Localization by Coupled Features[END_REF]. Nevertheless, global techniques tend to be sensitive to partial occlusions * 0-7803-8566-7/04/$20.00 c 2004 IEEE of facial features, and to the identity. Local approaches are less sensitive. We define salient features as features that draw attention. Features isolated in a dense feature space are salient features [START_REF] Walker | Automatically Building Appearance Models from Image Sequences using Salient Features[END_REF]. Determining such local feature points can be performed by partitionning the face image into several regions, by using textons as in [START_REF] Malik | Contour and Texture Analysis for Image Segmentation[END_REF] or finding generic features [START_REF] Mckenna | Tracking Facial Feature Points with Gabor Wavelets and Shape Models" 1st Int. Conf. on Audio-and Videobased Biometric Person Authentication[END_REF][START_REF] Schmid | Constructing Models for Content-Based Image Retrieval[END_REF][START_REF] Lisin | Extracting Salient Image Features for Reliable matching using Outlier Detection Techniques[END_REF].Facial features detection can be done using eigenfeatures [START_REF] Varchmin | Image based Recognition of gaze Direction using Adaptative methods[END_REF], blobs [START_REF] Howell | Active Vision Techniques for Visually Mediated Interaction[END_REF] or saddle points and maxima of the luminance distribution [START_REF] Otsuka | Real-time Estimation of Head Motion Using weak Perspective Epipolar Geometry[END_REF]. But such descriptors are sensitive to illumination and provide too many points, which can lead to accumulation errors. Interest points are not robust to pose, and are not well adapted to deformable objects such as the human face.

Our objective is to design descriptors that are robust to illumination, scale and orientation. A way to obtain more generic points is to use local feature vectors. Gabor wavelets can be used to detect scale-invariant feature points, as presented in [START_REF] Hall | Computation of Generic Features for Object Classification[END_REF], [START_REF] Wu | Wide-Range, Person-and Illumination-Insensitive Head Orientation Estimation[END_REF], [START_REF] Wiskott | Face Recognition by Elastic Bunch Graph Matching[END_REF] and [START_REF] Kruger | Gabor Wavelets Networks for Object Representation and Face Recognition[END_REF]. However, Gabor wavelets tend to be computionally expensive and have parameters that are difficult to adjust. Gaussian derivatives describe the appearance of neighborhoods and are an efficient means to compute scale and illumination robust local features. We adapt this method to detect facial salient features with interesting invariance properties.

Our approach is divided into several modules. First we employ a robust face tracker to detect and normalize the image of the face. As described in section 3, this step provides an important reduction in computation time. Further operations are performed on the normalized face image. We compute features for salient face regions with a learning process described in Section 4. Then we show and discuss our results in Section 5.

Face Image Normalization

We employ a robust video rate face tracker to focus processing on face regions. Our tracker uses pixel level detection of skin colored regions based on probability density function of chrominance [START_REF] Schwerdt | Robust face Tracking using Color" Automatic face and Gesture Recognition[END_REF].

Pixel Level Detection and Tracking using Skin Chrominance

To detect the face, we first detect skin regions in the image using intensity normalized color. The human face is a highly deformable surface and can be illuminated under several conditions. The color of image skin regions is determined by the product of the spectrum of skin pigments and illumination [START_REF] Klinker | A Physical Approach to Color Image Understanding[END_REF]. While such regions may have strong variations in intensity, chrominance will remain constant.

We use an intensity invariant feature vector for each pixel by normalizing the Red and Green component and dividing by the intensity to obtain a vector (r,g). The ratio of the histogram of known skin regions divided by the histogram of the entire image provides a lookup table that converts a pixel of chrominance (r,g) into the probability p(P ixel ∈ Skin|r, g) that the pixel is part of a skin region, as shown by Bayes rule. This lookup table gives us a direct relation (1) between intensity normalized color and probability.

p(P ix ∈ Skin|r, g) = p(r, g|P ix ∈ Skin)p(P ix ∈ Skin) p(r, g)

The skin probability map is obtained by computing p(P ixel ∈ Skin|r, g) for each pixel in a determined region. Face position and extent are tracked using a Kalman Filter. The first and second moments of the face are used to normalize the face position and orientation, as well as the size and resolution of the imagette that represents the face. The region of interest (ROI) for a face is maintained by a tracking process. In each image, the skin probability map is calculated within the region of interest predicted by using a zeroth order Kalman filter weighted by a Gaussian [START_REF] Schwerdt | Robust face Tracking using Color" Automatic face and Gesture Recognition[END_REF]. We estimate first and second moments with the following formulas (2):

µ x = 1 S p skin (x, y) • x • G(x, y, µ, C), µ y = 1 S p skin (x, y) • y • G(x, y, µ, C), σ 2 x = 1 S p skin (x, y)(x -µ x ) 2 G(x, y, µ, C), σ 2 y = 1 S p skin (x, y)(y -µ y ) 2 G(x, y, µ, C), σ xy = 1 S p skin (x, y)(y -µ y )(x -µ x )G(x, y, µ, C), (2)
where S = p skin (x, y) • G(x, y, µ, C)

Performance of the Face Tracker

To initialize our face tracker, we employ either the user's selection on the frame, or a generic ratio histogram. The choice of the number of histogram cells is crucial to provide a confident skin probability map. Histograms with too few cells will not be discriminative enough to compute the skin probability, whereas histograms with too many cells can contain empty cells.

We achieve real time processing by avoiding image copy. The ROI is scanned only once. The face tracker runs at video-rate on Pentium 800 MHz with images of 384x288 pixels. Eye detection rate on representative video sequences can be seen in table 1 and Figure 1. In this case, an error occurs when the computed ellipse does not contain an eye visible in the image.

An important property for a face tracker is stability. Stability is measured as the variation of the postion and size of the detected pixels of the face when the subject is at average distance from the camera and is not moving. We have calculated variances of the moments in pixels with regard to the size of the image on sequences of 20 seconds when the person's head has a certain pose and is not moving. Results are shown in Table 2. The face tracker can be perturbed when the subject is in profile because of the detction of his neck.

Normalized Face Imagette

Once the face is delimited with an ellipse in the image, the face is converted into a normalized luminance imagette (see Figure 2). The normalized face image offers several ad-

Generic Face Features Selection

In this section, we search for facial features robust to changes in illumination, pose and identity. We show how to describe an image with receptive fields, then how to automatically learn facial features with clustering and finally determine salient regions of a face.

Normalized Receptive Fields

Gaussian derivatives provide a feature vector for local appearance that can be made invariant. We use a five dimensional feature vector computed at each pixel by computing the convolution with the first derivative of a Gaussian in x and y direction (G x , G y ) and the second derivatives (G xx , G xy and G yy ). We use grey-level image of the face to be robust to chrominance variations of lights (sun, neon lights,...). We do not use the zeroth order Gaussian derivative in order to remain robust to changes in illumination intensity. Derivatives of higher order have been found to contribute little information for detection [START_REF] Hall | Viewpoint Independant Object Recognition from Local Appearence[END_REF].

The feature vector (G x ,G y ,G xx ,G xy ,G yy ) describes the local appearance of a neighboorhood and is determined using Gaussian derivatives that are normalized to the characteristic scale at each pixel. An example of feature vector of a pixel can be seen in Figure 3. The characteristic scale at each pixel is determined with the local maximum

Clustering operation

K-means clustering is used to determine a combination of Gaussian derivatives that provide a detection of salient facial features that is robust to variations. Gaussian derivatives vectors forms clouds of points in the feature space. The clustering operation finds these clouds. A distance metric for these feature vectors is defined by normalizing feature vectors by their variance. Gathering similar appearances captures the specificity of facial features. Each cluster can be used as a robust detector, because variances are learned. We have found that a single cluster provides a robust detector for salient facial features.

Robust Facial Features

Applying clustering to the feature vectors for multiple images from several faces provide appearance clusters for background, hair and different skin regions as well as salient facial features. For each pixel, we determine the most probable cluster. A pixel belongs to a certain cluster if the variance normalized distance between the appearance of the pixel and the cluster centroid is minimal in the feature space. Pixels of a same cluster are represented by a point cloud in the feature space and several connected regions in the image. In many experiments, one cluster corresponds to salient facial features and responds to: eyes, nose, mouth and chin.

Detection of facial features can also give rise to a number of small spurious detected regions. These can be eliminated by using a connected components analysis algorithm and compute the bounding box around. Regions with a small bounding box are eliminated. The remaining regions correspond to salient facial features. The connected components analysis also gives geometrical informations about robust face features in the face image. Computing the first and second moments of the connected components provides more informations. This information can be reprojected to the original image. The process of robust facial features detection is shown in Figure 4.

Experimental Results and Discussion

Training data

The choice of a good database is crucial for the learning step. To detect salient facial features that are robust under changing conditions, we have used 2 front images of 15 subjects to learn features vectors. Subjects are 20 to 40 years old. Five people have facial hair and 7 people wear glasses. Non-frontal images can introduce noise in the data, because some facial features have different appearances in different poses. Experiments on 2 people are shown in Table 3. Front pose provides more generic appearance for salient facial features, which remain robust on multiple poses after learning, whereas profile images provides appearance for salient features in profile, but not for front.

To remain robust to identity, we have used images of our database, which is composed of series of images of 15 different people. These subjects can be gathered in two classes:

• Class A, in which people's face is "ordinary" or "common" with regard to people in the database. In our database, 73% of the subjects has white skin, european facial type and no beard.

• Class B, in which people's face differs from "common" faces in the database. These people can wear glasses, have a beard or different skin colors. In our database, 27% of the subjects has darker skin or oriental facial type or a beard.

We have observed performances of the learning process with people of different classes and obtained the following results:

• The clustering C(A) is done only with people belonging to class A.

-Regions obtained for facial features of a subject a ∈ A are significant and robust.

-Regions obtained for facial features of a subject b ∈ B are less significant and more noisy. These observations can be explained in the following way. The clustering C(A) done over "common" faces provides better results on subjects of class A, with "common" faces than subjects of class B, whose face's appearance differs from the subjects'faces of class A. Therefore the clustering C(A) is not well adapted for subjects of class B. We must then use other people in our learning process to remain robust to identity.

Adding a new subject a ∈ A in the clustering does not bring much more information, even on the subject (a). Furthermore, it can lead to a degradation of robustness and more noise, because it will specialize the learning for people of the class A only. Then the class A become more specific, and, as a consequence, regions do not remain generic and robust to identity. This can lead to overfitting.

Alternatively, adding a new subject b ∈ B provides better detection of facial features on (b), whose appearance differs from those of class A. The clustering C(A+b) adapts to the image of the face of (b) without becoming specialized. Furthermore, salient facial features are more often detected with C(A+b) than with the clustering C(A).

Influence of the number of clusters

The clustering step gathers feature vectors into K clusters. This step is an important part in the learning process and must be carried out. Therefore, the choice of the number of clusters K is crucial. If K is too small, appearance clusters won't be discriminative enough to detect salient features of the face. If K is too big, regions will be too small and too unstable in the image. During our experiments, we tested several K and obtained good results with K = 10. Resulting images with different number of clusters can be seen in Figure 5.

To measure the recall and the precision for each different K, we have employed a 10x15 grid on the normalized imagette of the face (see Table 4). Cases in the grid are manually labelled as follows : 1 if the case contain a facial salient features, 0 otherwise. During the tests, a case of the grid gets the value 1 if the ratio of the number of salient cluster pixels in the case over the total number of pixels in the case exceeds a fixed threshold. This threshold is called the detection threshold (see Table 5).

Facial feature detection performance

Tests have been made with representative people under changing lighting and pose conditions. The pose is determined by 2 angles (h,v), which vary from -90 degrees to +90 degrees. Each set contains 93 images of the same person at different poses. People are wearing glasses or not and having various skin color. We have calculated the detection 6).

With an average detection rate of 97 %, eyes are the most often detected feature. Eyes appearance does not vary as much as the other facial features because of their spherical shape. Furthermore, eyes can be detected as blobs on the face under several points of view. Glasses do not affect eyes detection. Mouth detection seems slightly higher for bearded subjects. A mouth is more salient when it is surrounded by a beard.

For 63% of the observed errors, the head pitch is inferior to -30 degrees, so the subject is looking down. This situation represents only 29% of all poses. Indeed, in this situation, eyes are no more visible in the image, but only eyebrows. Therefore, we have trained our algorithm on images on which subjects' head pitch is inferior to -30 degrees. In this case, the resulting clusters are less dicriminating and provides lower detection rate on face images. As a consequence, some facial features, such as chin and eyes, are less salient. Eyes detection is 59% less efficient with the algorithm trained with images of people looking down. The Figure 6: Examples of facial features detection [START_REF] Martinez | The AR Face Database[END_REF] Table 7: Eyes detection rate with regard to the size of the imagette in pixels Size 120x200 120x120 60x100 50x50 Detection 98,2 % 97,8 % 94,2 % 1,8 % nose has the worst detection average rate with 74%. It does not have as many symmetry properties as eyes and its appearance can suffer many variations. That is why the nose is less often detected than other facial features.

Influence of the size of the face imagette

To show the importance of the face image normalisation step, we have measured eyes detection rates with different sizes of the face imagette. Results of these experiments can be seen in table 7. Tests have been made on a sequence of 500 images in which the subject moves but has both eyes remaining visible on the screen. The head size changes from 50x50 to 20x20 pixels in the sequences. The last size, 50x50 pixels, corresponds to face image analysis without normalization, as the face in the sequnce has a maximal size of 50x50 pixels. We can see how the normalisation process enhances the detection rate. This provides the ability to deal with 20x20 pixels images of the head, such as panoramic or wide-angle public cameras im-ages. If this operation is not done, regions will be more imprecise and may not be found. Increasing the size of the normalized face image increases the accuracy of feature detection in the original image of the face. For our experiments, the face imagette has a size of 60x100 pixels.

Conclusions

Figure 1 :

 1 Figure 1: Example of face tracking. First and second moments provides an ellipse which delimit the face in the image

Figure 3 :

 3 Figure 3: Appearance based feature vector

Figure 4 :

 4 Figure 4: Robust facial features detection process Face image normalization, Mapping : Regions in red and green are considered as salient robust facial features and reprojected into a binary map, Connected components analysis

••

  The clustering C(A+a) is done with people belonging to class A and a new subject (a) belonging to class A too. Regions obtained for facial features of the subject (a) are less robust than those obtained with the previous clustering C(A). The clustering C(A+b) is done with people belonging to class A and a new subject (b) belonging to class B. Regions obtained for facial features of the subject (b) are less noisy and more salient than those obtained with the previous clustering C(A).

Figure 5 :

 5 Figure 5: Influence of the number of clusters Regions in red and green are considered as salient robust facial features Top left image is the original image Top right image is obtained with 5 clusters, which are not discriminative enough Bottom left image is obtained with 10 clusters Bottom right image is obtained with 15 clusters. Regions are too small to be relevant

Table 1 :

 1 Eye detection rate

	Sequence Number of images Eye Detection rate
	A	500	99,9 %
	B	700	99,8 %
	C	580	94,2 %
	D	300	93,1 %

A : Head slow translation B : Head fast translation C : Head zoom and inclination in the plane D : Head pitch and yaw

Table 2 :

 2 Stability of the position and the size of the detected face

	Pose	Front	Half-profile Profile
	X Center 0,31 %	1,13 %	3,23 %
	Y Center 0,64 %	1,05 %	1,58 %
	Width	0,55 %	1,08 %	1,38 %
	Height	0,64 %	1,14 %	1,38 %
	Figure 2: Face Image Normalization

vantages : position invariance of face features and scale invariance. This means that processing can be specialized to the normalized scale and processing time is independent of original face size. All further operations take place within this imagette.

Table 3 :

 3 Recall

		/ Precision in % when changing the head
	orientation in the training process
	Results obtained with a detection threshold of 0.25
	Images	Frontal	Near-Frontal	All
	Person 1 36,7 / 30,1	40,9 / 4,8	31,1 / 24,6
	Person 2	34 / 35,4	35,6 / 4,1	35,2 / 6,7

Table 4 :

 4 Results obtained with 30 front images and a detection threshold of 0.4

	Number K	2	3	5	7
	Recall	11,7 % 22,7 % 70,7 % 30,7 %
	Precision	2,3 % 13,1 % 18,2 % 21,5 %
	Number K	10	15	20
	Recall	40,2 % 12,2 % 6,1 %
	Precision	47,7 % 57,3 % 11,7 %

Table 5 :

 5 Recall/Precision with regard to detection threshold

	Detection Threshold	0,1	0,25	0,4
	Recall	46,3 %	34 %	23,2 %
	Precision	22,2 % 25,4 % 26,9 %
	Detection Threshold	0,5	0,66	0,75
	Recall	17,9 % 10,3 % 7,5 %
	Precision	27 %	27 %	27,3 %

Table 6 :

 6 Facial feature positive detection rate

	Feature	Person 1 Person 2 Person 3 Person 4
	Eyes	99 %	97 %	98 %	95 %
	Nose	70 %	82 %	61 %	82 %
	Mouth	85 %	90 %	95 %	85 %
	Chin	84 %	88 %	91 %	84 %
	Specificity	-	Glasses	Beard	Matt skin
	rate for each feature for 4 representative people (see Table

We have proposed a new approach to detect salient local face features which are robust to pose, illumination and identity. We do not need to constrain the image to allow our algorithm to work. The image is normalized in scale and orientation by a face tracker. Each pixel in the face image is associated to an appearance cluster. One particular cluster stands for salient robust face features which are: eyes, nose, mouth, chin. We have tried to extract and exploit the maximum of informations contained on a single image of a face and to limit the loss of generality.

These regions can be delimited with rectangles in the image. Identifying facial features using positions relative to the face image is difficult because of multiple variations of features possible. These variations are due to changing orientation, emotion and especially identity. Alternatively, a Bayesian classifier should be used to identify the regions. The rectangles can provide a grid on the image of the face too. Robust features can also be used for expression analysis under changing conditions.


Gwenn Le Fur
email: gwenn.le-fur@satimo.fr

Francisco Cano-Facila

Francesco Saccardi

Daniel Belot
email: daniel.belot@cnes.fr

Ludovic Durand

Luc Duchesne

Lars Jacob Foged

Jean-Marc Lopez

Anthony Bellion

Improvement of Antenna Measurement Results at Low Frequencies by Using Post-processing Techniques

Keywords: antenna measurement, low frequencies, VHF, post-processing techniques. I

This paper presents a part of a whole study dedicated to the design of an indoor near-field antenna measurement system in VHF range. During this study a new wide band probe covering the 50 to 400 MHz bandwidth was developed, manufactured and characterized. The paper will focus on the use of post-processing techniques to improve measurement results. Measurements have been done in two different systems. Results are given to show the capability of Time Domain filtering and Equivalent Current Approach to reduce ripples due to reflections and truncation effect.

INTRODUCTION

Recent enhancements in military telecommunication systems for monitoring and tracking in low VHF range imply the use of specific antenna measurement facilities to characterize either the antenna alone or the antenna mounted on a supporting structure which can be heavy and bulky. Conventional outdoor far-field antenna measurement ranges used at UHF frequency bands and above become poorly effective down in VHF frequency band due to ground effects and electromagnetic pollution. Other implementations such as indoor far-field or compact range are not suitable at these very large wavelengths. A brief state of the art of existing VHF measurement facilities has been done in [START_REF] Le Fur | Feasibility of indoor spherical near field antenna measurement facility in VHF range[END_REF]. The indoor Near-Field approach shows benefits in terms of compactness. However this approach involves issues due to high levels of reflectivity of the anechoic chamber, the Antenna Under Test (AUT) positioner and the measurement probe structure at these frequencies. Studies and characterizations of each component effect have been performed in a previous paper [START_REF] Le Fur | Design of a Spherical Near Field Antenna Measurement Facility in VHF Range[END_REF].

The proposed paper focuses on measurement results improvements of a new VHF probe using post processing techniques. First is briefly presented the new 50-400 MHz dual polarized probe. Then the two different measurement systems used are described. Post-processing techniques are then discussed and chosen to answer to inherents measurement system issues. Finally results are given to show the capability of Time Domain filtering and Equivalent Current Approach to reduce ripples due to reflections and truncation effect.

II. SATIMO VHF WIDE BAND PROBE

A new wide band and dual polarized VHF probe has been designed and manufactured. Mass and mechanical stability were critical points for this voluminous antenna as shown in Figure 1 with a weight around 70 kg and dimensions of 2 x 2 x 1.7 m. A mechanical analysis has been performed and the chosen main constituting material is the Metawell [START_REF] Gmbh | metal sandwich technology[END_REF]. Return losses for each polarization are below -10 dB as shown in Figure 1. Isolations between polarizations are better than 20 dB over the 50 to 400 MHz bandwidth.

III. MEASUREMENT SYSTEMS

A. CNES Toulouse France

The first measurement facility used is the single probe spherical near field measurement system located in the chamber of the CNES in Toulouse France. This facility is dedicated to perform antenna measurement from 80 MHz to 200 GHz [START_REF] Gdr Ondes | Rapport sur le recensement des chambres anéchoïques française et leurs perspectives[END_REF]. The chamber is shared by a compact range measurement system and a single probe near-field system. Above 400 MHz the compact range configuration is used. Below 400 MHz the near-field configuration is used. Nevertheless classical foam pyramidal absorbers are poorly efficient below 200 MHz. Therefore ripples due to reflections coming from the compact range reflector and from the chamber walls were expected.

B. Renault Aubevoye France

The second system used is the multi-probe hemi-spherical near-field range located at the Renault Center in Aubevoye France [START_REF] Gdr Ondes | Rapport sur le recensement des chambres anéchoïques française et leurs perspectives[END_REF]. The bandwidth of operation is from 70 MHz to 6 GHz. In this second case the very large absorbers present a better absorption than the CNES chamber ones being acceptable from 125 MHz. Moreover no reflecting plane is present therefore less ripples due to reflections were expected. However the main error in this system comes from the measurement surface which is truncated at 102 degrees in elevation. Then ripples due to this truncated surface were expected.

IV. POST-PROCESSING TECHNIQUES

As commented before, because the antenna is measured at very low frequencies, one important error contribution are the reflections coming from the surrounding walls which are covered with absorbers but which are not able to effectively remove them due to their poor reflectivity behavior at these frequencies. The number of approaches to analyze and cancel the effects of these unwanted contributions has increased in recent years. There are methods that are employed not to remove the reflection waves, but rather to characterize the chambers by a figure of merit, usually called the reflectivity level [START_REF] Appel-Hansen | Reflectivity level of radio anechoic chambers[END_REF]- [START_REF] Clouston | Measurement of anechoic chamber reflections by time-domain techniques[END_REF]. Other solutions within the category of so-called compensation methods can be applied to avoid measuring the antenna under test several times at different distances with respect to the probe. Test zone field compensation [START_REF] Nagatoshi | A method of pattern measurement to cancel reflection waves in anechoic chamber[END_REF] is one of these methods, and it is used to reduce the effects of undesired fields created by reflections, employing only the data from one measurement performed over a spherical surface. The main drawback of this last technique is the computational complexity, which can be reduced as shown in [START_REF] Leatherwood | Plane wave, pattern subtraction, range compensation[END_REF], in which the unwanted reflected components are associated with plane waves with which the error patterns are estimated. Other possible approaches include frequency decomposition techniques, where the signal measured in the frequencydomain is divided into a set of propagating components, including the direct and all the reflected contributions. Next, taking only the desired component, a result quite similar to the one obtained in a fully anechoic chamber may be achieved. The most popular techniques to carry out such decomposition use Prony's algorithm [START_REF] Van Blaricum | A technique for extracting the poles and residues of a system directly from its transient response[END_REF], pencil-of-function method [START_REF] Sarkar | Suboptimal approximation/identification of transient waveforms from electromagnetic systems by pencil-of-function method[END_REF] or matrix pencil method [START_REF] Adve | Extrapolation of time.domain responses from three-dimensional conducting objects utilizing the matrix pencil technique[END_REF]. Another solution is to perform the measurement with the antenna shifted from the origin and then to apply a modal filtering [START_REF] Hindman | Reflection suppression in large spherical near-field range[END_REF]. In this last technique, the antenna must be located several wavelengths far away from the origin, therefore, in our particular case where low frequencies are use, it is not possible to apply this technique. Other last group of techniques is based on diagnostic techniques with which the field is reconstructed on a surface close to the antenna [START_REF] Cano-Facila | New reflection suppression method in antenna measurement systems based on diagnostic techniques[END_REF]- [START_REF] Foged | Comparison of echo suppression techniques in near field antenna measurement applications[END_REF]. In this domain, if the desired contributions coming from the antenna and the undesired contributions coming from reflections are not spatially coincident a spatial filtering can be applied to suppress the second ones. In the case of a single probe system where the reflection contribution is variable relative to the antenna under test, this approach is less suitable.

The second error contribution is linked to the fact that in the Renault system the measurement is not performed over a surface which fully enclosed the antenna. As a consequence, the true far-field pattern cannot be determined in the whole sphere and is only valid within the so-called reliable region. A second effect, which is caused by the discontinuity of the measured field at the edge of the scan surface, is the presence of a ripple even within the reliable region. There are different approaches which attempt to reduce the mentioned ripple by the application of proper window functions to the near-field data [START_REF] Rousseau | The planar near-field measurement of a broad beam antenna using a synthetic subarray approach[END_REF]. Another ripple reduction approach is proposed in [START_REF] Joy | Windows '96 for planar near-field measurements[END_REF]- [START_REF] Joy | Test-zone field quality in planar near-field measurements[END_REF] in which raised cosine amplitude and quadratic phase windows are shown to provide high accuracy. Although windowing techniques can greatly reduce erroneous ripples, an extra scan area is needed to obtain the same reliable region. If the scan area is not extended with the use of one of these techniques, the extent of the reliable region is reduced. There is a second group of approaches which do not attempt to reduce the ripple within the region in which the far-field pattern is reliable, but they attempt to obtain a good estimation of the true pattern outside of that region. Another strategy to reduce truncation errors is to rotate the AUT about one or more axes [START_REF] Gregson | Development of wideangle pattern measurements using a probe-corrected polyplanar nearfield measurement technique[END_REF], measuring in different planes and combining them to increase the maximum validity angle. Logically, this technique requires a particular near-field to far-field transformation for each combination of plane acquisitions. In [START_REF] Bucci | A new strategy to reduce the truncation error in near-field far-field transformation[END_REF], the problem of truncation is addressed using a priori information about the AUT. The main idea of this approach is to estimate the nearfield data outside of the scanning area by extrapolating the measured data before calculating the far-field pattern. The a priori information is employed to obtain a non-redundant and non-uniform representation [START_REF] Bucci | Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples[END_REF] of the samples that are taken over the measurement surface. A new method that is based on the same principle, i.e., the use of a non-redundant sampling in the acquisition, was proposed in [START_REF] Bucci | A new method for avoiding the truncation error in near-field antennas measurements[END_REF]. In this method, the truncation error is practically eliminated by also addressing points in surfaces external to the actual scanning area. Therefore, this method can be applied whenever the set-up allows the variation of the distance between the AUT and the probe. Recent publications also use a priori information about the AUT in an iterative algorithm to extrapolate the reliable portion of the calculated far-field pattern. This method was proposed for the planar near-field case in [START_REF] Martini | Reduction of truncation errors in planar near-field aperture antenna measurements using the Gerchberg-Papoulis algorithm[END_REF] and generalized for the most common measurement systems in [START_REF] Cano-Facila | Reduction of truncation error in planar, cylindrical and partial spherical near-field antenna measurements[END_REF].

In our particular case, a time domain filtering technique and an equivalent current approach are employed to suppress reflections and truncation errors, respectively. More information about these two techniques is given in the two following sub-sections.

A. EQC approach

Equivalent Current Approach (EQC) also called Integral Equation Approach [START_REF] Petre | Planar near-field to far-field transformation using an equivalent magnetic current approach[END_REF] is performed by using the Insight software developed by Satimo [START_REF]Satimo Insight software[END_REF].

Based on the acquired NF data, Insight computes the equivalent electric and magnetic currents on an arbitrary shaped reconstruction surface conformal to the test object. From the equivalent currents the field on the full measured surface can be evaluated. This process mitigates the pattern error due to scan truncation that otherwise would show as a ripple due to the zero padding of the NF data in the traditional NF-FF transformation [START_REF] Foged | Comparative Investigation of Methods to Reduce Truncation Errors in Partial Spherical Near-Field Antenna Measurements[END_REF].

The equivalent electric and magnetic currents computed by Insight on an equivalent surface conformal to the shape of the VHF probe are shown in Figure 4.

It should be noted that the equivalent reconstruction surface also acts as spatial filter [START_REF] Foged | Comparison of echo suppression techniques in near field antenna measurement applications[END_REF], mitigating the unwanted effect of echoes or stray signals. B. Time domain filtering Time domain filtering techniques or time-gating techniques are well-known approaches which can be employed as an option to remove reflected contributions. The main idea is to separate the direct signal from the reflections using a time-domain representation. In the initial approaches [START_REF] Young | A comparison of time-and frequency-domain measurement techniques in antenna theory[END_REF], [START_REF] Burrell | Antenna radiation pattern measurement using time-to-frequency transformation (TFT) techniques[END_REF], the measurement is directly performed in the timedomain by utilizing pulse generators and sampling oscilloscopes. After that, a time filtering and a time-tofrequency transformation are required to infer a reliable frequency response. However, the complexity of the pieces of equipment is lower when using frequency-domain measurements as proposed in [START_REF] Loredo | Echo identification and cancellation techniques for antenna measurement in non-anechoic test sites[END_REF], which is the technique used in our particular case to remove the effect of unwanted reflections as much as possible. Firstly, the measurement is performed over a wide enough frequency range. Using these data, the time response is immediately calculated by applying an inverse Fourier transform. If there are contributions coming from different places in the measurement set-up (different distances and therefore different times of arrival), it is very simple to identify them in this new domain. After that, direct contribution is detected and gated, eliminating the delayed response due to the reflected and diffracted components. Finally, a Fourier transform is applied to return to the frequency-domain. Figure 5 shows the time domain response for one given direction. Direct and reflected contribution can easily be distinguished. The reflection coming from the CATR reflector is clearly identified regarding the time / distance occurrences. Figure 6 and Figure 7 present the frequency response before and after time filtering respectively for one given azimuth. Strong reduction of ripples can be observed.

V. RESULTS

The time filtering approach is applied in the results obtained in both measurement systems (CNES and Renault) to reduce reflections. However the EQC approach is only applied in the Renault measurement results for truncation matter.

Results examples at 100 MHz of the post-processing are shown in Figure 8 (c) for the time filtering (TF) at CNES and in Figure 8 (e) for both time and EQC post-processing at Renault. The reference simulated result in shown Figure 8 (a). We observe good improvements of the results as shown in Figure 9 and Figure 10 for E plane at 100 MHz and 200 MHz respectively. More results at lower frequencies will be presented at the EuCAP Conference. Uncertainties estimation regarding these two measurement systems have now to be dressed when using these post-processing techniques. This paper presents the use of post-processing techniques to improve antenna measurement results at low frequency in two different measurement systems. Due to the large wavelengths considered measurement results are strongly impacted by reflections coming from chamber walls and system components. Time filtering is better suited compared to modal or spatial filtering at these low frequencies. To reduce truncation effect in the Renault system, equivalent current approach is added. Obtained results are conclusive and underline the capability of indoor measurement systems to measure antenna at frequencies below their specifications.

Figure 1

 1 Figure 1 Photograph of the 50-400 MHz dual polarized probe and measured return losses versus frequency

Figure 2

 2 Figure 2 Photograph of the Near Field measurement configuration inside the CNES Compact Range

Figure 3

 3 Figure 3 Photograph of the Renault Range

Figure 4

 4 Figure 4 Examples of equivalent current sources for Renault (left: J currents, right: M currents)

Figure 5 Figure 6 Figure 7

 567 Figure 5 Example of time response in CNES chamber

Figure 8 Figure 9 Figure 10

 8910 Figure 8 Examples of results at 100 MHz Theta component -Elevation vs. azimuth plots (a) Simulation (b) CNES not filtered (c) CNES after time filtering (d) Renault not filtered (e) Renault after time and Insight processing


Gwenn Le Fur
email: gwenn.le-fur@satimo.fr

Francisco Cano-Facila

Luc Duchesne

Daniel Belot
email: daniel.belot@cnes.fr

Lise Feat

Anthony Bellion

Kevin Elis

Romain Contreres

Uncertainty Analysis of Spherical Near Field Antenna Measurement System at VHF

This paper presents parts of investigations done in a VHF spherical Near-Field system in order to estimate measurement uncertainties. First is briefly presented the monoprobe Near-Field system and uncertainties estimations challenges due to low frequencies. Then are described the specific and appropriated approaches for the considered errors terms. Classical pattern comparison is used. Finally is provided a global budget error with and without time filtering to appreciate the benefit of such an approach. I.

INTRODUCTION

Recent needs for monitoring and tracking in low VHF range below 100 MHz imply the use of specific antenna measurement facilities to characterize either the antenna alone or the antenna mounted on a supporting structure which can be heavy and bulky. Conventional outdoor Far-Field antenna measurement ranges used at UHF frequency bands and above become poorly effective down in VHF frequency band due to ground effects and electromagnetic pollution. Other implementations such as indoor Far-Field or compact range are not suitable at these very large wavelengths. The indoor Near-Field approach shows benefits in terms of compactness. However this approach involves issues due to high levels of reflectivity of the anechoic chamber and system components at these frequencies. Studies and characterizations of each component effect have been performed [START_REF] Le Fur | Feasibility of indoor spherical near field antenna measurement facility in VHF range[END_REF] and post processing techniques have been used and compared previously [START_REF] Le Fur | Improvement of Antenna Measurement Results at Low Frequencies by Using Post-processing Techniques[END_REF] to assess the system performances.

The present paper focuses on measurement uncertainties estimation to qualify the CNES VHF Near-Field system. Based on the well-known 18-NIST terms [START_REF] Newell | Error analysis techniques for planar near-field measurements[END_REF] specific and appropriated approaches of estimation had to be found regarding the present perturbations in the chamber. These approaches are detailed for the main impacting considered terms. To build the budget error two main cases are taken; the unfiltered measurement results and the time domain filtered ones. Not all the error terms are estimated in this paper due to current lack of measurement. Full budget error could be provided for the conference. Thus this paper aims present the approaches of error terms estimation to finally appreciate performances of the low frequencies Near-Field system dealing with the present reflections. The paper is organized as follow; first are described the system, the approaches and the tools used for error estimations. Then are detailed the estimation for five main terms. Finally are provided two budgets error with and without time-filtering

II. SYSTEM DETAILS AND APPROACHES USED FOR ERROR

ESTIMATION

Uncertainties estimation can be performed either for a system (independent of the used antenna under test) or for a specific antenna within a system. In this paper we are in the second case and focus on uncertainty budget of the CNES Near-Field system for a particular Antenna Under Test (AUT).

A. CNES VHF Near-Field Facility

The measurement used is the single probe spherical Near-Field measurement system located in the chamber of the CNES in Toulouse France. This facility is dedicated to perform antenna measurement from 80 MHz to 200 GHz [START_REF] Gdr Ondes | Rapport sur le recensement des chambres anéchoïques française et leurs perspectives[END_REF]. The chamber is shared by a compact range measurement system and a single probe Near-Field system. Above 400 MHz the compact range configuration is used. Below 400 MHz the Near-Field configuration is used. Nevertheless classical foam pyramidal absorbers are poorly efficient below 200 MHz. Therefore ripples due to reflections coming from the compact range reflector and from the chamber walls are present. In order to extend the operational measurement bandwidth down 100 MHz a new wide band and dual polarized VHF probe has been designed and manufactured [START_REF] Le Fur | Improvement of Antenna Measurement Results at Low Frequencies by Using Post-processing Techniques[END_REF] (cf. Figure 1). All the measurement results presented in this paper have been done by using this probe.

B. Antenna Pattern Comparison Approach

The chosen alternative to estimate some of the error terms later presented in section III is the classical approach based on comparing two different patterns, one of them considered as reference. This strategy was already employed in [START_REF] Pivnenko | Comparison of Antenna Measurement Facilities With the DTU-ESA 12 GHz Validation Standard Antenna Within the EU Antenna Centre of Excellence[END_REF] where four different comparison approaches were presented in order to determine the accuracy of different measurement facilities. In our particular case, it was seen that the most appropriate comparison was the weighted logarithmic difference which is able to de-emphasize the noise and the large spikes. The weighted function is composed by the noise function, , and the pattern weighting function, with (see [START_REF] Pivnenko | Comparison of Antenna Measurement Facilities With the DTU-ESA 12 GHz Validation Standard Antenna Within the EU Antenna Centre of Excellence[END_REF] for more details). Thus the weighted logarithmic difference between two patterns ( ) and ( ) can be expressed as:

( ) ( ) (1)
with

( ) ( ) ( ) (2)
Once the difference has been computed, the mean and standard deviation can be calculated in a usual way. These two factors will be used as figures of merit to quantify the difference between the two patterns.

C. CNES Near-Field Coordinate system

The Near-Field coordinate system is as shown in the Figure 3 below where the acquisition on the full sphere is achieved by rotating the AUT around two axes corresponding to the azimuth (φ) and elevation (θ) of the measurement sphere. The "180 Phi" and "360 Phi" areas "seen" by the AUT are illustrated in the Figure 4-a and Figure 4-b, respectively. Due to the theta excursion, the "360 Phi" scan is expected to show more symmetric results than the "180 Phi" scan. Indeed in this last case perturbations coming from the CATR reflector are stronger than the ones coming from the opposite wall. Due to this fact, the "360 Phi" scan with time domain filtering (described in the next point) is taken as reference in the comparisons (this fact has been also validated by comparing with simulations, providing the "360 Phi" scan better results).

E. Time Domain Filtering Approach

Because the measurements are performed at VHF, as commented later, the main source of error are reflections coming from the walls of the chambers. Although the chamber is covered with absorbent material, the reflectivity of such a material at these frequencies is not so big. Due to this reason, a filtering in the time domain is applied in order to reduce the unwanted contributions. This filtering is a classical approach [START_REF] Loredo | Echo identification and cancellation techniques for antenna measurement in non-anechoic test sites[END_REF] and it is based on measuring over a wide enough frequency range. Using these data, the time response is immediately calculated by applying an inverse Fourier transform. If there are contributions coming from different places in the measurement set-up (different distances and therefore different times of arrival), it is very simple to identify them in this new domain. After that, direct contribution is detected and gated, eliminating the delayed response due to the reflected and diffracted components. Finally, a Fourier transform is applied to return to the frequency-domain.

III. ERRORS TERMS ESTIMATION

A. Probe alignment

This error appears when the probe is not well aligned with the measurement axis. To estimate the error due to this misalignment, we consider polar patterns obtained through the calibration measurement (360° phi scan on-axis). Once applied calibration coefficients on this calibration measurement result, we note the differences between 90° spaced values which relies the probe capability to discriminate cross-polarization level. As the angular sampling for this measurement is 1° we record the differences at +/-0.5° (the maximum error of alignment which could be committed for this given angular sampling step). Thus we obtain a error value which will be involved in the crosspolarization error budget. The variation of this error value versus frequency is shown in Figure 5.

B. Probe y position errors

To estimate the probe y-position error an additional measurement with a 2 cm offset in the y-direction was performed. The results of this measurement were compared in Far-Field with the ones obtained in the assumed good ylocation of the probe. The pattern comparison approach described in section II.B is applied and the variation of this error value versus frequency is shown in Figure 6 for both Co and Cross-polarization.

C. Probe z position errors

This error appears when the distance from the center of rotation of the AUT to the phase center of the probe is different to that one considered in the Near-Field to Far-Field transformation process. If the phase center of the probe is wellknown, this error is negligible. However, the phase center could change in a great way in frequency, especially in low frequency. Therefore, when measuring in a very large frequency range because the measurement distance is considered constant for all the frequency points, there will not be a negligible z-position error. In fact, this is the situation in our measurement. The Near-Field distance is considered to be equal to 4.4 m. However, the phase center position of our probe is different depending on the frequency. One method to estimate the impact of this error is to compute and compare the Far-Field patterns assuming that the acquisition has been performed over a sphere of 4.4 m radius and 4.4 + ∆z m radius. ∆z is considered to be equal to the maximum variation of the phase center (around 0.8 m, value which has been calculated both from simulations and measurements). The impact of an error of 0.8 m (worst case) in the z-position of the probe introduces the Far-Field errors depicted in the following figure. Logically, this error can be avoided by using for each frequency the appropriate Near-Field distance and for that, the phase center positions of the probe as a function of the frequency must be known.

D. Room scattering

Usual approach to estimate the room scattering impact consists in moving the whole measurement system in the chamber by λ/4 steps. For each system location is measured the whole antenna pattern. Such an approach is not implementable in our case considering the wavelengths and the system setup. Consequently we propose here to use the available "180 Phi" and "360 Phi" measurement results of the same antenna to estimate the room scattering effect. "360 Phi" scan results with time filtering are taken as reference. To this reference are compared the "180 Phi" scan results with and without time filtering. Results versus frequency for Co and Cross polarization are presented in Figure 8. We can clearly appreciate through this plot the benefit of the time filtering for this particular AUT.

In this contribution is assumed to be included the "Multiple Reflections" term between probe and antenna under test.

E. Random errors

A two-step method is employed to determine the impact of random errors in Far-Field results. The first step is based on estimating the noise power in the current Near-Field data. For that purpose, the spherical wave coefficients obtained from simulated Near-Field noise with different power levels are compared with the spherical wave coefficients calculated from the acquired Near-Field data. As observed in Figure 9, the lower order modes contain the information about the AUT and after a given value modes only have information about the noise. Therefore, by comparing the higher order modes of the spherical expansions commented before it is possible to determine the noise power present in the Near-Field data. In our system, for example, the noise power is 43 dB below the maximum, that is, the signal-to-noise ratio in Near-Field is 43 dB.

Figure 9 Modal contains of the measurement and several noise powers

Once the noise power is known in Near-Field (determined spherical wave spectrum analysis), it is possible to assess the behavior of such a noise in Far-Field by simply using Montecarlo method. This method typically runs simulations many times over in order to obtain the distribution of an unknown probabilistic entity. Logically, the larger the number of simulations is, the better the estimation of the random variable is. In our particular case, we want to determine the properties of the Far-Field noise due to a known Near-Field noise. This is achieved by following the next steps:  A reference Far-Field pattern is computed by using Near-Field data.

 Near-field data is corrupted by adding a noise with the power level determined in the first stage and the corresponding corrupted Far-Field pattern is computed.

 Then, the reference and corrupted Far-Field patterns are compared, obtaining the error pattern (noise in Far-Field).

 The two previous steps are repeated several times in order to obtain a better estimation of the properties of the noise in Far-Field.

 Finally, all the error patterns are used to determine the mean and the standard deviation of the Far-Field noise.

Figure 1

 1 Figure 1 Photograph of the dual polarized probe in the CNES VHF Near-Field systemCopyright CNES

Figure 2

 2 Figure 2 Example of patterns comparison (a) and corresponding weigthed logarithmic difference (b)

Figure 3 CNES

 3 Figure 3 CNES Near-Field coordinate system

Figure 4

 4 Figure 4 Illustration of the "180 phi" scan (a) and "360 phi" scan (b) configuration

Figure 5

 5 Figure 5 STD values of probe alignment error versus frequency

Figure 6

 6 Figure 6 STD values of probe y-position error versus frequency

Figure 7

 7 Figure 7 STD values of probe z-position error versus frequency

Figure 8

 8 Figure 8 STD values of probe room scattering error versus frequency

Figure

  Figure 10 STD values of random error versus frequency

IV. RESULTS

Tables I and II present the budgets error for the unfiltered and the time-filtered cases respectively. For each frequency point are provided the mean and the standard deviation for the Co and the Cross polarization results. These power budget rely measurement uncertainties by considering the contribution of the five error terms previously analyzed. We globally observe the expected improvement by applying time-filtering. This observation comes from two main reasons; firstly the room scattering error is one of the bigger impacts at low frequency and time domain filtering aims to reduce this impact. Secondly the reference measurement result used in pattern comparison for the estimation of several terms is time filtered to be closer to simulation results.

Another observation at high frequency is the fact that we retrieve the behavior of the noise contribution in both error budgets. This is partly due to the absence of time-filtering in the random error term estimation.

V. CONCLUSION AND PROSPECTS

Through investigations done in the CNES VHF spherical Near-Field system this paper provides a partial estimation of measurement uncertainties. First was presented the mono-probe near field system and uncertainties estimations challenges due to low frequencies. Specific and appropriated approaches for the estimation of five considered errors terms are detailed. Finally was provided a global budget error with and without time filtering to appreciate the benefit of such an approach.

This study provides a first estimation of the system measurement uncertainties for one particular antenna under test. Ongoing works aim to build a full system error budget independent of the measured antenna.


Gwenn Le Fur
email: gwenn.le-fur@satimo.fr

Francisco Cano-Facila

Luc Duchesne

Daniel Belot
email: daniel.belot@cnes.fr

Lise Feat

Anthony Bellion

Romain Contreres

Kevin Elis

Comparison of antenna measurement results in disturbed environment using a VHF spherical near field system

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

INTRODUCTION

Recent needs for monitoring and tracking in low VHF range below 100 MHz imply the use of specific antenna measurement facilities to characterize either the antenna alone or the antenna mounted on a supporting structure which can be heavy and bulky. Conventional outdoor far-field antenna measurement ranges used at UHF frequency bands and above become poorly effective down in VHF frequency band due to ground effects and electromagnetic pollution. Other implementations such as indoor far-field or compact range are not suitable at these very large wavelengths. The indoor Near-Field approach shows benefits in terms of compactness. However this approach involves issues due to high levels of reflectivity of the anechoic chamber and system components at these frequencies. Studies and characterizations of each component effect have been performed [START_REF] Le Fur | Feasibility of indoor spherical near field antenna measurement facility in VHF range[END_REF] and post processing techniques have been used and compared previously [START_REF] Le Fur | Improvement of Antenna Measurement Results at Low Frequencies by Using Post-processing Techniques[END_REF] to assess the system performances.

The proposed paper focuses on measurement configurations to properly characterize a very narrowband antenna taken as a reference. Considering the very narrow operational bandwidth the time filtering cannot be applied without removing critical antenna information. Thus this paper aims to propose an appropriate way to perform measurements dealing with the present reflections. By comparing the measurement results coming from two scan configurations to the simulation ones, is found a proper way to measure the considered reference antenna in the considered CNES Near-Field system. Statistical approach is used to compare pattern results difference and a way to qualify the pattern results symmetry is proposed.

II. CNES VHF NEAR-FIELD FACILITY

The measurement used is the single probe spherical near field measurement system located in the chamber of the CNES in Toulouse France. This facility is dedicated to perform antenna measurement from 80 MHz to 200 GHz [START_REF] Gdr Ondes | Rapport sur le recensement des chambres anéchoïques française et leurs perspectives[END_REF]. The chamber is shared by a compact range measurement system and a single probe near-field system. Above 400 MHz the compact range configuration is used. Below 400 MHz the nearfield configuration is used. Nevertheless classical foam pyramidal absorbers are poorly efficient below 200 MHz. Therefore ripples due to reflections coming from the compact range reflector and from the chamber walls are present. In order to extend the operational measurement bandwidth down 100 MHz a new wide band and dual polarized VHF probe has been designed and manufactured [START_REF] Le Fur | Improvement of Antenna Measurement Results at Low Frequencies by Using Post-processing Techniques[END_REF] (cf. Figure 1). All the measurement results presented in this paper have been done by using this new probe.

III. DETAILS OF INVESTIGATIONS

A. Antenna Under Test

The Antenna Under Test (AUT) used for these investigations is a crossed double dipole mounted on a ground plane and working around 140 MHz in a narrow bandwidth. The AUT is considered as a reference antenna i.e with a wellknown far-field pattern and a confident simulation model. The both dipoles are fed with equal amplitude signals with 90 degree phase difference. The expected symmetric far-field pattern covers a half sphere: φ [0; 360°] and θ [0; 90°].

B. Antenna Pattern Comparison Approach

The strategy to compare measured patterns in the far-field region follows is the one used in [START_REF] Pivnenko | Comparison of Antenna Measurement Facilities With the DTU-ESA 12 GHz Validation Standard Antenna Within the EU Antenna Centre of Excellence[END_REF] within the EU Antenna Centre of Excellence. Two far-field antenna patterns are compared using the weighted logarithmic difference to deemphasize the noise and the large spikes. The weighted function is composed by the noise function and the pattern weighting function ( ) [START_REF] Pivnenko | Comparison of Antenna Measurement Facilities With the DTU-ESA 12 GHz Validation Standard Antenna Within the EU Antenna Centre of Excellence[END_REF]. Thus the weighted logarithmic difference between two patterns ( ) and( ) can be expressed by:

( ) ( ) (1)
With

( ) ( ) ( ) (2)
Each measurement result are compared to the simulated pattern (obtained by using FEKO EM simulation software) taken as reference. Two classical figures of merits are used to quantify the difference between both antenna patterns namely the mean value and the standard deviation (STD). Results are reported in section IV.B.

C. Measurement Configurations

Considering the fixed near field probe (cf. Figure 2) and its location inside the anechoic chamber, the AUT is moving around two axes (phi and theta) to perform the full sphere scan. As shown in Figure 2 the near field system is not located on a chamber axis of symmetry. Consequently the AUT could see different perturbations along the full sphere scan. Thus we considered and compared two ways of scanning the full sphere The "180 Phi" and "360 Phi" areas "seen" by the AUT are illustrated in the Figure 2 and Figure 3 respectively. Due to the theta excursion the "360 Phi" scan is expected to show more symmetric results than the "180 Phi" scan. Indeed in this last case perturbations coming from the CATR reflector are strongly different to the ones coming from the opposite wall. The amount of the perturbation should not be necessary lower but their effects are expected to be more homogeneous on the whole measurement sphere by using the "360 Phi" scan. Figure 4 shows the 2D patterns results obtained through simulation and both measurement configurations. Both components of the field are plotted (in dB with 0 dB for max value). We observe at this stage a better agreement between the "360 Phi" scan and the simulation results. The "180 Phi" shows a globally good behavior but seems to be more impacted by perturbations. Moreover an asymmetry can be observed on the "180 Phi" results. These first observations are confirmed by plotting several pattern cuts shown in Figure 5. At high patterns levels a good agreement is obtained between the three results. Differences can be observed for lower levels on some cuts.

B. Statistical data to quantify differences

Pattern comparison approach briefly described before is applied on the two scan configurations. Differences are computed on the whole patterns cuts (θ ≤ 180°) between each scan configuration and simulation. Table 1 and Table 2 synthetize the statistical values obtained for the "360 Phi" and "180 Phi" scan respectively. Both field components and four cuts are considered. Mean and STD values are computed and reported for each case. Looking on STD results the "180 Phi" scan shows differences slightly larger than the "360 Phi" scan. Mean values are even better for the "180 Phi" scan. Thus the preliminary qualitative observations about pattern asymmetry are not confirmed by this statistical approach. This is due to the use of STD and means values on the whole patterns. Consequently is proposed hereafter a small observation approach of the pattern symmetry disparity.

C. Pattern symmetry disparity

In order to add an information about the expected pattern symmetry, we proposed here a simple way to appreciate it. On Figure 6 is plotted an example of weighted logarithmic difference for the both scan configurations. We can clearly see that this difference is more symmetrical along the pattern for the "360 Phi" scan. To appreciate the symmetry disparity of the weighted difference, a second logarithmic difference is computed between the first one and itself "theta flipped". This simple approach is illustrated on Figure 7 for a cut of the "360 Phi" scan results where the red curve shows the symmetry disparity in dB on the whole cut i.e. the difference between the weighted difference on the considered cut and this same difference flipped in theta.

Figure 7 Weigthed logarithmic difference for the "360 Phi" and symmetry disparity -Cut Phi 0° -Phi Component On Figure 8 are reported the symmetry disparity for the two scan configurations. Four Phi cuts (0° / 45° / 90° / 135°) for the Phi component are considered. An important gap between the two scan configurations is observed for all the considered cuts. The "360 Phi" scan results appear much more symmetric than the ones obtained by the "180 Phi" scan. This qualitative observation in addition with the statistical values obtained shows the benefit to perform the full sphere scan using the "360 Phi" scan configuration in the Near-Field CNES system. This paper proposes an appropriate way to perform measurements of a narrowband antenna dealing with the reflections present in the Near-Field system used. By comparing the measurement results coming from two scan configurations to the simulation ones, a proper way to measure the considered reference antenna in the CNES Near-Field system is found. Statistical approach is used to compare pattern results difference and an additional way to qualify the pattern results symmetry is proposed to choose the best measurement configuration. Additional global error budget of the system could be provided in the presentation.

Figure 1

 1 Figure 1 Photograph of the dual polarized probe in the CNES VHF Near-Field systemCopyright CNES

  [5]: -The "180 Phi" scan : θ [-180; 180]° and φ [0; 180]° -The "360 Phi" scan : θ [0; 180]° and φ [0; 360]°

Figure 2 Figure 3 Figure 4

 234 Figure 2 Illustration of the area seen by the AUT in the "180 phi" scan configurationtop view

Figure 5

 5 Figure 5 Comparison of radiation pattern cuts @ 140 MHz Cut Phi 0° -(a) Theta component -(b) Phi component Cut Phi 90° -(c) Theta component -(d) Phi component

Figure 6

 6 Figure 6 Weigthed logarithmic difference for the "360 Phi" and "180 Phi" scans -Cut Phi 0° -Phi Component

Figure 8

 8 Figure 8 Symmetry disparity the "360 Phi" and "180 Phi" scans -Cut Phi 0°/45°/90°/135° -Phi Component

TABLE 1 STATISTICAL

 1 FIGURES OF MERITS FOR THE WEIGHTED DIFFERENCE BETWEEN "360 PHI" SCAN AND SIMULATION

		ETheta component	EPhi component
	Cuts	Mean (dB) STD (dB) Mean (dB) STD (dB)
	Phi 0	0.38	0.82	-0.30	0.48
	Phi 45	0.01	0.78	-0.01	1.47
	Phi 90	0.03	0.70	0.93	0.88
	Phi 135	0.31	1.27	-1.10	1.63
	TABLE 2 STATISTICAL FIGURES OF MERITS FOR THE WEIGHTED
	DIFFERENCE BETWEEN "180 PHI" SCAN AND SIMULATION
		ETheta component	EPhi component
	Cuts	Mean (dB) STD (dB) Mean (dB) STD (dB)
	Phi 0	0.03	1.07	-0.10	1.14
	Phi 45	-0.01	0.59	0.68	1.18
	Phi 90	-0.01	0.88	0.36	1.07
	Phi 135	0.21	1.41	-0.02	1.67


G Le Fur

F Cano-Facila

L Duchesne

D Belot

K Elis

L Feat

A Bellion

R Contreres

Investigations on Probe Phase Center Impact in Antenna Measurement Results Uncertainty for Spherical Near-Field Systems

Keywords: antenna measurement, measurement results uncertainties, Near Field system, probe phase center. I

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

INTRODUCTION

Near Field techniques show good benefits for antenna measurement for many years. Uncertainties analysis for Near-Field antenna measurement system is well established by the well-known NIST 18 terms [START_REF] Newell | Error analysis techniques for planar near-field measurements[END_REF]. In this paper we focus investigations on the "probe z-position error". This error appears when the distance from the center of rotation of the Antenna Under Test (AUT) to the phase center of the probe is different to that one considered in the Near-Field to Far-Field transformation process. If the phase center of the probe is wellknown and consequently the measurement radius, this error is negligible. However, the phase center could change in a great way in frequency, especially by using wide band probes. Therefore, when measuring one a very large frequency range because the measurement distance is often considered constant for all the frequency points, there will not be a negligible zposition error. In fact, this is the situation in many measurement systems. Then it is interesting to investigate the error made as a function of the ratio between the maximum displacement of the probe phase center in the z direction (Δz) and the measurement distance considered. This error was estimated for a specific measurement system in [START_REF] Le Fur | Uncertainty Analysis of Spherical Near Field Antenna Measurement System at VHF[END_REF]. This impact must also be shown by considering the wavelength of the measurement and the potential mutual coupling between the measurement probe and the AUT. The paper is organized as follow. First is briefly described the considered measurement system. Then is detailed the approach for the error estimation with the antenna pattern comparison chosen and the different cases studied. Finally are given the results uncertainties and some prospects.

II. CNES VHF NEAR-FIELD FACILITY

The measurement system considered is the single probe spherical near field measurement system located in the chamber of the CNES in Toulouse France. This facility is dedicated to perform antenna measurement from 50 MHz to 200 GHz [START_REF] Gdr Ondes | Rapport sur le recensement des chambres anéchoïques française et leurs perspectives[END_REF]. The chamber is shared by a compact range measurement system and a single probe Near-Field system. Above 400 MHz the compact range configuration is used. Below 400 MHz the Near-Field configuration is used. Nevertheless classical foam pyramidal absorbers are poorly efficient below 200 MHz. Therefore ripples due to reflections coming from the compact range reflector and from the chamber walls are present. In order to extend the operational measurement bandwidth down to 50 MHz a new wide band and dual polarized VHF probe has been designed and manufactured [START_REF] Le Fur | Uncertainty Analysis of Spherical Near Field Antenna Measurement System at VHF[END_REF] (cf. Fig. 1).

III. INVESTIGATION DETAILS

A. Proposed approach for error estimation

In order to appreciate the error due to the probe phase center displacement, two main cases are considered. The first one consists in using the Mean Radius of Measurement (MRM) obtained by considering the mean location of the probe phase center on the measured frequency band. The second one consists in using the corrected Radius of Measurement (RM) obtained by considering the probe phase

Copyright CNES AUT Probe center location for each frequency point. These two distances -MRM and RMare involved in the Near-Field to Far-Field transform. Far-Field patterns are then compared to the reference pattern obtained by simulation. The approach used to compare the patterns is briefly described here after.

B. Antenna pattern comparison approach

The strategy to compare measured patterns in the Far-Field region is the one used in [START_REF] Pivnenko | Comparison of Antenna Measurement Facilities With the DTU-ESA 12 GHz Validation Standard Antenna Within the EU Antenna Centre of Excellence[END_REF] within the EU Antenna Centre of Excellence where four different comparison approaches were presented in order to determine the accuracy of different measurement facilities. In our particular case, it was seen that the most appropriate comparison was the weighted logarithmic difference which is able to de-emphasize the noise and the large spikes. Two Far-Field antenna patterns are then compared using the weighted logarithmic difference. The weighted function 𝑊 𝑙𝑜𝑔 is composed by the 𝑊 𝑛,3 noise function and the pattern weighting function 𝑊 𝑝 (𝛽 = 0.3) [START_REF] Pivnenko | Comparison of Antenna Measurement Facilities With the DTU-ESA 12 GHz Validation Standard Antenna Within the EU Antenna Centre of Excellence[END_REF]. Thus the weighted logarithmic difference between two patterns 𝑓 1 (𝜃, 𝜑) and 𝑓 2 (𝜃, 𝜑) can be expressed by:

∆ w,log (θ, φ) = W log ∆ log (θ, φ) (1) With ∆ 𝑙𝑜𝑔 (𝜃, 𝜑) = 20 𝑙𝑜𝑔 10 𝑓 1 (𝜃, 𝜑) -20 𝑙𝑜𝑔 10 𝑓 2 (𝜃, 𝜑) (2)
Each measurement result are compared to the simulated pattern (obtained by using FEKO EM simulation software) taken as reference. Once the difference has been computed, the mean (μ) and standard deviation (STD) values can be calculated in a usual way. These two factors are used as figures of merit to quantify the difference between the two patterns. Results are reported in section IV.

C. Using simulations of the measurement setup

First investigations are performed by using simulation of the CNES spherical Near-Field system including the model of the VHF wide band probe and an antenna under test. None other component of the system is present. AUT and probe are perfectly aligned and the AUT is rotated around ϕ and θ axis to obtain a full sphere Near-Field dataset. In such a way the two main errors considered are the probe z-position error and the mutual coupling between probe and AUT. Two physical distances are considered. The second one (10 m) is twice than the first one (5 m) (see Fig. 2). From those, MRM and RM electrical length are used to compute Far-Field pattern. The frequency range considered is from 50 to 200 MHz and the minimum sphere diameter including the AUT is 2.60 m. As an example, for this given AUT size and considering a measurement radius of 5 m, Far-Field region is reached for frequencies below 125 MHz. Above this frequency the effect of the probe phase center is thus expected bigger than the mutual coupling one. On the other hand below this frequency the mutual coupling effect is expected to be predominant. By increasing the measurement distance both effects should be mitigated. Results of the pattern comparisons in terms of μ and STD values are presented in section IV.

D. Using Measurements

Measurement campaign is just finishing in the CNES-VHF facility. Different AUTs with different sizes have been measured. Frequency ranges used are down to 50 MHz and up to 400 MHz. In contrast with the simulation of the measurement setup described above several error terms are involved in the measurement results uncertainties especially the one coming from the chamber reflectivity. Measurement results are currently processing in the same manner in order to appreciate the probe phase center contribution. Then only some preliminary patterns are provided as examples in the next section.

IV. RESULTS

A. Using simulations of the measurement setup

Results presented below are obtained by comparing four main cuts of the Far-Field patterns. Worst errors values are then retained. Plots report errors in terms of mean and standard deviations values of the weighted logarithmic differences on the whole frequency band. Obtained errors for the copolarization, the directivity and the -10 dB and -20 dB below peak are reported. Solid lines refer to the results using the MRM and dashed lines to the ones using the good RM. A large benefit is observed by correcting the probe phase center location for the first measurement distance (see Fig. 3.a and Fig. 3.b). The z-probe phase center location was obtained through simulation. Its location relatively to the mean value over the frequency band is plotted in Fig. 5. We observe bigger relative displacement for the low and the high frequency points. This behavior can be observed in the results obtained by using the small MRM where the Near-Field to Far-Field transform process is strongly involved. ). This can be first explained by Far-Field distance already reached for most part of the frequency band. A second explanation is the decreased ratio Δz/MRM and then the reduced impact of the probe phase center displacement. However error levels are drastically reduced especially at low frequencies for the present case. By increasing the measurement distance is also reduced the mutual coupling between the AUT and the probe which involves in measurement uncertainties.

B. Using measurements

As mentioned before measurement results processing are still under progress. Global results will be given for the conference. As examples are plotted two results of Far-Field patterns at 50 MHz. The minimum sphere size including the AUT is 1.4 m diameter and the measurement distance is 5.8 m. Fig. 6.a presents the patterns obtained by using the MRM and Fig. 6.b the ones by using the corrected RM. Co-polarization and cross-polarization for the two main cuts are reported to be compared with the simulation results. Very small differences can be observed between the two cases. Slight improvements are present on low levels e.g. on the cross-polarization patterns. Considering the AUT size, the frequency and the measurement distance these first results seem to show agreements with results for the larger distance taken in the simulation of the measurement setup. This paper proposes partial investigation on the probe phase center impact in antenna measurement results for a spherical Near-Field system. Pattern comparison approach is used to evaluate the probe z-position error. By using simulation of the actual considered system setup in VHF band reduced measurement results uncertainties are shown by correcting properly the measurement radius. The large frequency band and different considered cases give insights on behaviors of the perturbations involved. For low frequency, increasing the measurement distance shows benefit on results uncertainties. This could be explained by the decrease of the mutual coupling and lower relative displacement of the probe phase center. Measurement data have to be provided to ensure these first observations. Different frequency ranges, measurement radiuses and AUT sizes must be considered in order to obtain a global observation of the probe phase center impact in spherical Near-Field measurement.

Fig. 1 .

 1 Fig. 1. Photograph of the dual polarized probe in the CNES VHF Near-Field system

Fig. 2 .

 2 Fig. 2. Coordinate system and measurement radiuses

Fig. 3 .

 3 Fig. 3. Using MRM (a) Mean values of the weigthed logarithmic differences (b) STD values of the weigthed logarithmic differences -Peak to peak errors

Fig. 4 .

 4 Fig. 4. Using 2xMRM (a) Mean values of the weigthed logarithmic differences (b) STD values of the weigthed logarithmic differences -Peak to peak errors

Fig. 6 .

 6 Fig. 6. Measured patterns results at 50 MHz (a) using MRM -(b) using RM


Léon Matar-Tine

Inverse problem for cell division rate in population dynamics

. The estimation strategy is based on the observation of the long-term solution of the model for a size structured population when the growth and division rates are fully general. The issue of the theoretical and numerical results are discussed here.

Introduction

The observation of a population of cells (proteins, bacteria, etc.) allows to understand the dynamics of the considered population such as its distribution, the behavior of its parameters (growth rate, death rate, division rate). This understanding depends on technologies that biologists have for experiences. For instance, one knows that from cytometric flow technology it is possible to measure at each time the distribution of the population what is not the case for instance for the measurement of the division rate because of the lack of technology able to follow a single cell from its birth to its division. So, to estimate this division rate which is a crucial parameter (in the case of pathogenic cells as E. Coli bacteria), mathematical and numerical approaches are needed. Then without lost of generality let consider the following growth-fragmentation type model:

       * t n(t, x) + * x (g(x)n(t, x)) + B(x)n(t, x) = 2 ∞ 0 B(y) (x, y)n(t, y) dy, n(0, x) = n 0 (x), x ≥ 0, g(0)n(t, 0) = 0, t ≥ 0, (1)
where n(t, x) is the density repartition function of the cells structured by their size variable x at time t. We assume the cell population to evolve only by growth and division process without death and birth events. The growth rate g is a given or measured function which depends on the size x. The total rate a e-mail: leon-matar.tine@univ-lyon1.fr This is an Open Access article distributed under the terms of the Creative Commons Attribution License 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article available at http://www.itm-conferences.org or http://dx.doi.org/10.1051/itmconf/20150401003 of division for cell of size x is denoted by B(x), so the product B(y) (x, y) is the rate at which a single cell of size y divides in to cells of size respectively x and yx. The kernel division (x, y) is required to satisfy:

(x, y) = 0 ∀ x > y, ∞ 0 (x, y) dx = 1, (x, y) = (y -x, y), ∞ 0 x (x, y) dx = y 2 • (2)
With this model [START_REF] Doumic | Estimating the division rate for the growth-fragmentation equation[END_REF] we can formally deduce by simple integration that, in one hand the time variation of the total number of cells depends on the division rate B what means

d dt ∞ 0 n(t, x) dx = ∞ 0 B(x)n(t, x) dx.
In the other hand, one deduce that the time variation of the total mass of the population depends on the growth rate g what means

d dt ∞ 0 xn(t, x) dx = ∞ 0 g(x)n(t, x) dx.
The aim of this paper is to show how to estimate the division rate B from the knowledge of measurements on the density repartition n(t, x) and on the growth rate g(x). Let us recall that these assumed known quantities can be experimentally measured by cytometric flows technology, while the division rate need theoretical and numerical approaches for its estimation. Estimating this division rate is crucial if one is interested of pathologic cells (as bacteria or misfolding proteins) which cause disease. Work in this theoretical estimation exist for the simple case g(x) ≡ 1 and where the individual cell divides with equal mitosis (see [START_REF] Perthame | On the inverse problem for size structured population model[END_REF][START_REF] Doumic | Numerical solution of an inverse problem in sizestructured population dynamics[END_REF]), that means, (x, y) = x=y/2 . This equal mitosis change the integral term in (1) into 4B(2x)n(t, 2x). Here we investigate the case where the kernel division is quite general and the growth rate g not necessarily constant. Of course, for the model [START_REF] Doumic | Estimating the division rate for the growth-fragmentation equation[END_REF], there are many results on the well-posedness and the essential properties. For more detail one can refer to [START_REF] Heijmans | On the stable size distribution of populations reproducing by fission into two unequal parts[END_REF][START_REF] Metz | The dynamics of physiologically structured populations[END_REF][7][START_REF] Michel | General entropy equations for structured population models and scattering[END_REF][START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF]. The fact that, at any time we can have observation of the size distribution of the cell population allows us to focus the estimation of the division rate on the time asymptotic behavior of the model [START_REF] Doumic | Estimating the division rate for the growth-fragmentation equation[END_REF]. So the paper is organized as follow: in Sect. 1, we present the asymptotic behavior of the model and related results proved by generalized relative entropy principle (GRE). In Sect. 2 we present the inverse problem of estimation the division rate B thanks to two regulation methods. The Sect. 3, is devoted to the numerical methods and results. The last section is for the conclusion.

Asymptotic behavior of the model

The asymptotic behavior of the model ( 1) is based on the GRE principle (see [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF]) which need basic hypothesis on the parameters.

With the notation

P := f ∈ L 1 loc (R * + ), f ≥ 0 : ∃ , ν ≥ 0, lim x→∞ sup x -f (x) < ∞, lim x→∞ inf x ν f (x) > 0 ,
we assume: the kernel division fulfill in addition to (2), the relations

∃ c > 0, ∀p ≥ 2, y 0 x p y p (x, y)d ≤ c < 1 2 • (3)
The growth rate and the division rate satisfy

B ∈ L 1 loc (R * + ) ∩ P, ∃ 0 ≥ 0, ∀t ≥ 0, g ∈ L ∞ loc (R + , x 0 dx) ∩ P (4) 01003-p.2
WMLS 2014 and for all compact set

K ⊂ R + , ∃ m K > 0 such that g(x) ≥ m K ∀x ∈ K ∃ b ≥ 0, suppB = [b, ∞[. (5)
The cells of size zero forming is avoided thanks to the assumption

∃ C > 0, ≥ 0, such that x 0 (z, y) dz ≤ min 1, C x y and x g(x) ∈ L 1 0 . ( 6
)
For the cells of small size, the growth is assumed to be more important than de division, that means

B(x) g(x) ∈ L 1 0 with L 1 0 = L 1 (0, a), a > 0. ( 7
)
For the cells of big size, in order to avoid the formation of infinite size cells at finite time (gelation phenomenon), the growth and division rates satisfy

lim x→+∞ x B(x) g(x) = +∞, ∀ t ≥ 0. ( 8
)
Thanks to the general assumptions ( 3)-( 8), the existence, uniqueness and properties of eigenelements associated to (1) is proved in [START_REF] Doumic | Eigenelements of a General Aggregation-Fragmentation Model[END_REF]. In particular it is shown by the GRE principle that

∞ 0 |n(t, x)e -t -0 N (x)|(x) dx -→ t→∞ 0; 0 = ∞ 0 n 0 (x)(x)dx,
where ( , N, ) is the unique eigentriplet solution of the following stationary equation and its adjoint equation:

       c* x (g(x)N(x)) + (B(x) + )N (x) = 2 +∞ 0 B(y) (x, y)N (y) dy, gN(x = 0) = 0; N (x) ≥ 0; ∞ 0 N (x) dx = 1, > 0. ( 9
)      cg(x)* x (x) -(B(x) + )(x) = -2B(x) x 0 (y, x)(y) dy, (x) ≥ 0; ∞ 0 (x)N (x) dx = 1. (10)
The estimation approaches of the division rate we develop here is based on the problem ( 9) and the assumption on the knowledge of g, N or their experimental measurements. Of course there is two fundamental estimates for the eigenvalue from simple integration of (9) which are written as follow

= ∞ 0 B(x)N (x) dx and/or ∞ 0 x N(x) dx = 1 ∞ 0 g(x)N (x) dx.

Estimation of the division rate

In this part, we investigate the question of recovering the division rate B when the growth rate g is given and a sequence of measurements N such that, with an adequate norm they fulfill, N -N < and | -| < for and N satisfying Eq. ( 9).

01003-p.3

ITM Web of Conferences

The possible noise on the measures can make the problem ill posed what is inherent to almost inverse problem. What's happen in the problem [START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF] when N is replaced by N and one wants to quantify the term * x (g(x)N (x)). To be more clear, lets consider the following sample.

Example 3.1: Let consider f ∈ C ∞ (R) and for arbitrary n ∈ N * . We define for all > 0, the approximation f (x) = f (x) + sin nx . It is obvious to see that ff ∞ ≤ means that f as an good approximation of f . But when one looks to the derivative f (x) = f (x) + n cos nx , it fulfill ff ∞ = n and we imagine when n is chosen as big as possible.

In order to come up this difficulty of controlling the term * x (gN ), we propose two methods of regularization named Filtering and Quasi-reversible methods that we present here below by introducing the new unknown H = BN, solution of the approximated problem

H (x) -2 ∞ 0 (x, y)H (y) dy = -c * x gN (x) -N (x).

Filtering method

Here, we introduce the mollifier

= 1 x ; ∈ C ∞ c (R + ); ∞ 0 (x) dx = 1; > 0,
and define the following regularized problem

H , (x) -2 ∞ 0 (x, y)H , (y) dy = * -c , * x gN (x) -N (x) . (11)
For this regularized problem we prove the existence and uniqueness (see [START_REF] Doumic | Estimating the division rate for the growth-fragmentation equation[END_REF]) by using the Lax-Miligram theorem which is based on the bilinear form

A(u, v) = ∞ 0 u(x)v(x)x p dx -2 ∞ 0 u(x) ∞ x (x, y)v(y) dy x p dx =< u, v > L 2 (x p dx) -B(u, v), with B(u, u) = 2 ∞ 0 ∞
x x p u(x)u(y) (x, y) dy dx, that we show the continuity and coercivity. In addition we have by priori estimates and Minkowski inequality, the following approximations

|c , -c| ≤ C(p, , N ) , ||B , -B|| L 2 (x p N 2 dx) ≤ C(p, , N ) + ,
where C(p, , N) is a constant depending on p, and N .

Quasi-reversible method

The key point of this approach is to add another derivative term in the problem in order to balance the term source of ill posed term. So the regularized problem is casted as follow for k ∈ R + , k ≥ 2 and > 0.

x -k * x x k+1 H , + H , -2 ∞ 0 (x, y)H , (y) dy = -c , * x gN -N . ( 12
)
Here, we prove the existence and uniqueness of the solution (see [START_REF] Doumic | Estimating the division rate for the growth-fragmentation equation[END_REF]) by the Banach fixed point theorem. We begin by writing the regularized problem with a general unknown u as:

x * x u(x) + (k + 1) + 1 u = 2 ∞ 0 (x, y)u(y) dy + F (x), 01003-p.4
WMLS 2014

where F (x) = -c , * x gN -N . So we define the application in L 1 (R + , x dx) such that

T : v → u with u = T (v) is the explicit solution of x * x u(x) + (k + 1) + 1 u = 2 ∞ 0 (x, y)v(y) dy + F (x).
We prove that T is a contracting Lipschitz function.

In addition we have by priori estimates and Minkowski inequality, the following approximations

|c , -c| ≤ C(p, N ) , ||B , -B|| L 2 (x p N 2 dx) ≤ C(p, N ) + ,
where C(p, N) is a constant depending on p and N .

Numerical simulation for the inverse problem of estimating B

This part is devoted to the numerical test of the estimated division rate B from the two proposed regularization methods. Experimental data on the long time behavior of the cell repartition function and on the growth rate are needed. In [START_REF] Doumic | Estimating the division rate for the growth-fragmentation equation[END_REF], we have no data available, so we numerical generate synthetic data on N by solving first the direct problem (1) with given g, andB; what allows to obtain the asymptotic profile of the size distribution function of cells. Once this asymptotic profile is obtained, we keep the same parameters g, and we forget the division rate B that we want to recover. By adding noise on N , we numerically recover the division rate B thanks to our two regularized methods and compare B and B which was taken into account in the resolution of the direct problem (1).

Results on the direct problem

The simulation of the direct problem follows a finite volume method applied to (1). For the length domain L and grid number points n max we have

x i = i x, 0 ≤ i ≤ n max , x = L n max
. So, for a time step t we define t k = k t, k ∈ N and take the following approximations

n k i = 1 x x i+ 1 2 x i-1 2 n(k t, y) dy, n k+1 i = 1 t t 0 n(k t + s, x i ) ds.
We use time splitting and solve first the growth part of (1), * t n + * x (g n) = 0, as follows

n k+1/2 i = n k i - t x (gn) k i+1/2 -(gn) k i-1/2 ,
where (gn) k i±1/2 are incoming and outgoing flows. In second, we solve the gain and lost part by cell division as follows

n k+1 i = (1 -t B i )n k+1/2 i + 2 t F k i with F k i ≈ x nmax x i B(y)n k+1/2 (y) (x i , y) dy.
In order to get the asymptotic profile, we renormalize the discrete solution: For different initial conditions and parameter values on g, and B satisfying the general assumption of existence and uniqueness of solution our scheme converge to the same asymptotic behavior as one can see in figure (Fig. 1) below. Let recall that the time step is computed with respect the CFL (Courant-Friedrichs-Lewi) stability condition, t < 1/ max 0≤i≤n max B i + g i x , with the notation B i = B(i x) and g i = g(i x). Once results on the direct problem are obtained, means the asymptotic behavior, we numerically simulate the regularized problems in order to estimate the division rate.

ñk+1 = n k+1 n max j =1

Results on the regularized problems

For the quasi-reversibility method, the associated regularized problem ( 12) is discretized as follows

-x -k i x k+1 i+1 H i+1 -x k+1 i H i x + H i -2 ka j =i H j (i, j ) x = L i , i = 1, • • • , n max . 01003-p.6
WMLS 2014 ). Down center: g(x) = x 1/3 , B is here a tray-function (see 13).

We then rewrite this discrete equation under matrix shape A × H = L where A appears as a triangular matrix, so we deduce the solution H by solving this simple iteration

           H n max = L nmax A nmax ,nmax ; H i = 1 A i,i   L i - n max j =i+1 A i,j H j   , ∀i = n max -1, ..., 1.
The non-oscillatory of this scheme is ensured by the constraint A i,i = 1 + i -2 (i, i) x > 0. For the filtering method, we apply the same discretization as in quasi-reversibility case to (11). The convolution term is computed by FFT (Fast Fourier Transform) followed by an IFFT (Inverse Fast Fourier Transform) what leads to the approximations We then write the discrete equation of (11) under matrix shape A × H = L with A an upper triangular matrix and deduce the approximation of H = BN as follows

           H n max = L nmax A nmax ,nmax ; H i = 1 A i,i   L i - n max j =i+1 A i,j H j   , ∀i = n max -1, ..., 1.
The numerical results show a good accuracy of our regularization methods as depicted in the pictures below in the case where we assume that we have exact measures ( = 0) that means N = N . These pictures compare the exact value on B taken in the direct problem and its estimated values obtained by 01003-p.8

WMLS 2014 solving the quasi-reversibility or filtering methods. We define a Tray-function for B as follows

B(x) =      0 f o r x < 2, ( x-2 13 ) 2 for x ∈ [2, 15], 1 f o rx > 15. (13)
In order to take into account the uncertainty in measurement, we add some multiplicative noise which is uniformly distributed in N = max(N(1 + l ), 0) with l ∈ -1 2 , 1 2 and ∈ [0, 1].

So, with N considered as the measured data, we obtain the following results with various choice of and the fixed parameters g(x) = x, (x, y) = 1 y I {x<y} , B(x) = exp(-0.08(x -13) 2 . In the simulations we investigate for any , the optimal regularization parameter which gives the best estimation B , of the division rate B by minimizing the L 2 -norm of the following relative error

err = B , -B L 2 BN L 2 •

Conclusion

This study presents how to recover the division rate for a general growth-fragmentation model for a size structured population where measurements on the time asymptotic profile of the density. Two method of regularization are highlighting here in order to overcome the difficulty of the ill-posed inverse problem. Theoretical estimates are built with strong analysis on the well-posedness of the regularized problem and numerical methods are implemented. As show on the above numerical illustrations, the estimation of the division rate is quite good. However, some issues remain to be addressed, specially for small size x, our algorithm need to be performed such as the appearance of oscillations.

5 Figure 1 .

 51 Figure 1. Direct problem g = x 1/3 , (x, y) = 1 y I {x<y} : top left step initial function. Top right gaussian initial function. Down left steady solution with B(x) ≡ 1. Down right steady solution with B(x) = exp(-0.08(x -13) 2 ). Mesh size is n max = 500 and L = 30.

Figure 2 . 1 y

 21 Figure 2. Comparison of BN in the direct problem and BN in the regularized problem for = 0 and (x, y) = 1 y I {x<y} . Top left: g(x) = x, B(x) = exp(-0.08(x -12) 2 ). Top right: g(x) = x 1/2 , B(x) = min(1, x 2 10). Down center: g(x) = x 1/3 , B is here a tray-function (see 13).

Figure 3 .

 3 Figure 3. Comparison between a given B on the direct problem and its estimations thanks to the regularized problems with noisy data. Top left: = 0.01. Top right: = 0.03. Down left: = 0.05. Down right: = 0.05.

This proceeding is a part of a work paper [1] done in collaboration with Marie Doumic, head of the MAMBA team of INRIA. The work was partially financed by the Agence Nationale de la Recherche, Grant No. ANR-09-BLAN-0218 TOPPAZ. I thank Abdelkader Lakmeche of Sidi Bel-Abbes university (Algeria) who gave me the opportunity to present this work during the Workshop WMLS'14 SBA.