Sylvie Castagnet
email: sylvie.castagnet@ensma.fr

David Mellier

Azdine Nait-Ali

Guillaume Benoit

In-situ X-ray computed tomography of decompression failure in a rubber exposed to high-pressure gas

Keywords: EPDM, Hydrogen, cavitation, Representative Volume Element (RVE), cavity field

Cavitation is a damage process often observed during or after pressure release in polymers exposed to high-pressure diffusive gases. Only a few characterizations of the phenomenon have been reported in the literature, all of them being based on 2D pictures of the sample taken after removal from the pressure chamber or, more recently, inside the chamber during pressure release. This study displays the first time-resolved 3D imaging of decompression failure in high-pressure gas exposed polymers, obtained from in-situ X-ray computed tomography. New data were provided about the out-of-plane shape and volume distribution of cavities. It allowed rigorous estimation of a Morphological Representative Volume Element for the cavity field. The bias resulting from the former 2D-projection methods could be discussed.

INTRODUCTION

Rubbers exposed to a high-pressure diffusive gas sometimes undergo cavitation or cracking during pressure release. This decompression-enhanced damage process occurs when the gas cannot desorb out of the polymer fast enough and locally expands. It has often been referred to as explosive decompression failure as most pioneer works were conducted with very fast decompression rates [START_REF] Denecour | Bubble formation in vulcanized rubbers[END_REF]. However, it may also occur under moderate or even slow pressure release rates, and exhibit rather slow growth kinetics. Decompression failure was reported in the literature a few decades ago in various gaspolymer systems [START_REF] Briscoe | Internal crack symmetry phenomena during gas-induced rupture of elastomers[END_REF][START_REF] Briscoe | Explosive decompression failure of rubber: a review of the origins of pneumatic stress induced rupture in elastomer[END_REF][START_REF] Embury | High-pressure gas testing of elastomer seals and a practical approach to designing for explosive decompression service[END_REF][START_REF] Gent | Cavitation in rubber: a cautionary tale[END_REF][START_REF] Gent | Nucleation and growth of gas bubbles in elastomers[END_REF][START_REF] Stewart | Nucleation and growth of bubbles in elastomers[END_REF][START_REF] Stevenson | Fracture of elastomers by gas decompression[END_REF]. It has not been that much investigated so far, especially in hydrogen. Interest for damage resistance of high-pressure hydrogen-exposed polymers was renewed about ten years ago, when hydrogen started to be considered as a new potential energy carrier. Since decompression failure may affect permeation properties and mechanical resistance, it is a major concern for industrial components designed for fuel cell vehicles and hydrogen charge stations. A better understanding of damage mechanisms is crucial to improve the design of components and the formulation of hydrogen-exposed rubbers. It faces strong limitations, among which is hydrogen manipulation in in-situ experiments. Only a few laboratories are able today to conduct physical and mechanical experiments under high-pressure hydrogen in polymers [START_REF] Yamabe | Influence of fillers on hydrogen penetration properties and blister fracture of rubber composites for O-ring exposed to high-pressure hydrogen gas[END_REF][START_REF] Yamabe | Application of acoustic emission method to detection of internal fracture of sealing rubber material by high-pressure hydrogen decompression[END_REF][START_REF] Castagnet | Mechanical testing of polymers into pressurized hydrogen: tension, creep and ductile fracture[END_REF][START_REF] Jaravel | On key parameters influencing cavitation damage upon fast decompression in a hydrogen-saturated elastomer[END_REF][START_REF] Koga | Evaluation on high-pressure hydrogen decompression failure of rubber O-ring using design of experiments[END_REF][START_REF] Jaravel | Real-time tracking and numerical simulation of cavity growth upon explosive decompression in hydrogen-saturated rubbers[END_REF][START_REF] Menon | Behaviour of Polymers in High Pressure Environments as Applicable to the Hydrogen Infrastructure[END_REF][START_REF] Duranty | An in situ tribometer for measuring friction and wear of polymers in a high pressure hydrogen environment[END_REF].

The most developed methods to track cavitation processes were based on the optical transmission of visible light in transparent samples exposed to several hydrogen pressure ranges [START_REF] Yamabe | Influence of fillers on hydrogen penetration properties and blister fracture of rubber composites for O-ring exposed to high-pressure hydrogen gas[END_REF][START_REF] Jaravel | On key parameters influencing cavitation damage upon fast decompression in a hydrogen-saturated elastomer[END_REF][START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions -Support to a relevant modeling framework[END_REF]. By time-resolved tracking using visible light transmission through a small window machined in the front door of the chamber, Jaravel et al. [START_REF] Yamabe | Influence of fillers on hydrogen penetration properties and blister fracture of rubber composites for O-ring exposed to high-pressure hydrogen gas[END_REF][START_REF] Jaravel | Real-time tracking and numerical simulation of cavity growth upon explosive decompression in hydrogen-saturated rubbers[END_REF] and Kane Diallo et al. [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions -Support to a relevant modeling framework[END_REF] studied the effect of decompression rate, saturation pressure and additional stretching on decompression failure in unfilled silicon and EPDM rubbers up to 40 MPa. Kane Diallo et al. led a statistical analysis of cavity distribution at the full-field scale (number of cavities, size distribution and spatial distribution), based on processed images of cavity fields during the decompression test. This evaluation of the spatial distribution of cavities was enriched by a covariogram method applied to each picture of the image stack. Covariogram corresponds to the probability C(h) of encountering a cavity at a distance h from another cavity along one given direction [START_REF] Nait-Ali | Catching the time evolution of microstructure morphology from dynamic covariograms[END_REF]. Among other data, this method allowed discussion of the features of a Morphological Representative Volume Element (MRVE) (isotropy, shape, size). MRVE data was useful for physically-motivated Finite Element modeling of the growth of pre-existing cavities on decompression in diffusive-mechanical conditions.

Visible light transmission techniques faced several strong limitations. First, these techniques were limited to transparent materials. Another limitation was the decreasing size of the window with increasing hydrogen pressure level. More importantly, these 2D techniques provided projected views of cavities through the full thickness of the sample, with possible artefacts due to overlapping, and strong assumptions about out-of-plane size and shape of cavities. In Kane Diallo et al.'s work for instance, cavities were supposed to be spherical. X-ray computed tomography appeared an alternate method to overcome both limitations and get a 3D indirect imaging of the cavity field. The main challenge was to get good enough space resolution with a time acquisition compatible with cavity growth kinetics. Over the last decade, X-Ray computed-tomography (CT) has become a reference nondestructive method to get indirect views of microstructure and defects in materials. Regarding polymer-based materials, it has been widely used in systems with good electronic density contrast, like between fiber and matrix in composites or between voids and polymer matrix (in foams for instance). Many studies in polymeric foams addressed the deformation and damage mechanisms under compression, first based on interrupted tests [START_REF] Daphalapurkar | Tomography and simulation of microstructure evolution of a closed-cell polymer foam in compression[END_REF][START_REF] Di Prima | Cyclic compression behavior of epoxy shape memory polymer foam[END_REF][START_REF] Patterson | Measure of morphological and performance properties in polymeric silicone foams by X-ray tomography[END_REF][START_REF] Youssef | Finite element modelling of the actual structure of cellular materials determined by X-ray tomography[END_REF] and, more recently, during in-situ testing [START_REF] Lachambre | In situ observation of syntactic foams under hydrostatic pressure using Xray tomography[END_REF][START_REF] Patterson | In-situ X-ray synchrotron tomographic imaging during the compression of hyper-elastic polymeric materials[END_REF] "Pure" bulk polymers are more complex systems. A first difficulty deals with the low electronic density contrast between phases: for instance, between the amorphous and crystalline layers in a semi-crystalline polymer, but also between carbon black and matrix in a filled rubber. A second difficulty is the scale of interest, usually rather fine regarding the microstructure and sometimes the defects. The size of nano-voids in a cavitated semicrystalline polymer or the fibrils diameter in a craze are close to the best spatial resolution now reached from X-ray CT using synchrotron beam radiation. The ex-situ X-Ray CT studies in polymers mainly addressed residual damage by mapping, either the attenuation coefficient as an indicator of the porosity level (e.g. [START_REF] Rosenberg | A mesoscale quantification method of cavitation in semicrystalline polymers using X-ray microtomography[END_REF] in polyvinylidene fluoride), or individualized defects (e.g. [START_REF] Saux | Evaluation of the fatigue defect population in an elastomer using X-ray computed micro-tomography[END_REF] about fatigue mechanisms in rubbers, [START_REF] Tanaka | Analysis of inner structure in highstrength biodegradable fibers by X-ray microtomography using synchrotron radiation[END_REF] for the voided structure of polymer fibers).

Two major improvements resulted from the increasing access to synchrotron beam facilities over the last decade. The first was the increasing space resolution, even for ex-situ observations. This allowed imaging of smaller defects, especially in semi-crystalline polymers: micronic cavities in stretched polypropylene [START_REF] Farge | A study of the deformation-induced whitening phenomenon for cavitating and non-cavitating semicrystalline polymers[END_REF], permanent voids in necked HDPE [START_REF] Morgeneyer | Nanovoid morphology and distribution in deformed HDPE studied by magnified synchrotron radiation holotomography[END_REF], voids/fibrils microstructures in polyamide 6 [START_REF] Laiarinandrasana | Damage of semicrystalline polyamide 6 assessed by 3D X-ray tomography: from microstructural evolution to constitutive modeling[END_REF][START_REF] Selles | Comparison of voiding mechanisms in semi-crystalline polyamide 6 during tensile and creep tests[END_REF], polyoxymethylene [START_REF] Ricard | Damage and fracture mechanisms of polyoxymethylene: Multiscale experimental study and finite element modeling[END_REF] or isotactic polypropylene [START_REF] Sokkar | Experimental investigation of craze morphology of isotactic polypropylene using computed tomography[END_REF]. The second interest was the ability to perform in-situ measurements during mechanical loading. Rosenberg et al. investigated the enhanced voiding in the center part of a notched specimen of polyvinylidene fluoride [START_REF] Rosenberg | A mesoscale quantification method of cavitation in semicrystalline polymers using X-ray microtomography[END_REF]. Poulet et al. did the same in notched samples of polyamide 11 under tension [START_REF] Poulet | Observations by in-situ X-ray synchrotron computed tomography of the microstructural evolution of semi-crystalline Polyamide 11 during deformation[END_REF]. Almost all mechanical loadings applied to polymers in in-situ tomographic work were tensile tests, although the shape of samples could locally promote triaxiality. The work from Lachambre et al. in syntactic polypropylene or polyurethane foams was an exception regarding hydrostatic pressure loading [START_REF] Lachambre | In situ observation of syntactic foams under hydrostatic pressure using Xray tomography[END_REF]. Ethanol was used as the pressure fluid. Using a high-resolution X-ray tomograph in a synchrotron facility, the spatial resolution was 3.5 x 3,5 x 3.5 µm 3 with an acquisition time of 5 min. In a totally different field but with analogies regarding the damage process, Babin et al. used in-situ X-ray CT to track the expansion and coalescence of bubbles during breadmaking [START_REF] Babin | Fast X-ray tomography analysis of bubble growth and foam setting during breadmaking[END_REF]. The average radius of bubbles was from several tens to several hundred microns. The kinetics was much slower than in the present case, with a radius rate of about 100 µm per hour. To the authors' knowledge, the in-situ tracking of decompression failure in hydrogenexposed polymers is reported here for the first time. This original experiment is depicted first, with special interest paid to the necessary compromise between acquisition parameters and cavity growth kinetics. Results are mainly focused on the contribution of 3D data: (i) the evaluation of the bias induced by the 2D projection of cavity field on the estimation of voided volume ratio or Morphological Representative Volume Element in visible light methods, (ii) the distribution of cavities through the thickness, (iii) the 3D anisotropy of voids.

EXPERIMENTAL

Material and decompression test 2. X-ray Computed-Tomography

The evolution of cavitation during decompression was followed by X-ray tomography, using an Ultratom tomograph from RX Solutions® fitted with the cylindrical pressure chamber visible in Figure 1. In the present work, the chamber was filled with hydrogen (nitrogen or carbon dioxide were also possible) up to a pressure of 20 MPa. Tests were conducted at room temperature, but the maximum possible temperature was 150°C. The device was remotely controlled; compression and decompression rates were regulated. Inside the chamber, the sample was centered on the rotation axis, 36-mm from the X-ray source. It was scanned using a micro X-Ray source (voltage 80 kV, current 375 µA). The detector was a flat panel with 1920x1536 sensitive elements (binning mode for this study). This X-ray detector consisted of an X-ray CsI scintillator screen settled on an amorphous silicon layer. Image slices were reconstructed based on a conventional filtered backprojection algorithm. Computed tomography reconstructs a 3D image of the internal structure (cavity field here) from the recording of a set of radiographs, obtained by tilting the sample at different angles. Each radiograph (i.e. for a given angle) displays a map of the local absorption of X-ray radiation. As for any transient phenomenon tracked by in-situ tomography, the present challenge was to come to a compromise between the quality of the acquired data (i.e. the stack of images used for one reconstruction) and the kinetics of the physical phenomenon. On one hand, the number of images of the stack must be large enough, and the spatial resolution small enough to ensure good quality of 3D reconstructions. Usually, the better the acquisition, the longer the time needed. On the other hand, the time elapsed between the first and the last image of the stack must be short enough to ensure slight evolution of the cavities to be reconstructed. This possible distortion effect limited the range of accessible growth kinetics. Several situations were tested to optimize the acquisition parameters. For the present decompression conditions (saturation pressure and pressure release rate), the retained configuration was a set of 200 projections, taken within 360°, acquired over a total time of 100 seconds. After reconstruction, the voxel edge size was 16µm.

Since the surfaces of the sample (initially flat) were blistered during decompression, images were cropped before segmentation. Indeed, the rubber matrix, the cavities and the surrounding medium had close grey levels. Without preliminary crop, some areas of the external medium around the blistered surface could be locally assimilated to cavities during segmentation. To avoid this, an identical number of slices was cropped from each side of the full imaged volume. The cropped volume was called "Observed Volume" in the following. This number of slices (five) was calibrated in order to remove the blistered surface but keep the maximum volume enclosing all cavities at any time. It was defined a posteriori and kept identical for the full test, so that the resulting Observed Volume was constant all along the test (9.68 mm x 9.64 mm x 1.84 mm). However, due to swelling of the sample during decompression, the Observed Volume corresponded to a variable fraction of the real sample volume. A rigorous calculation of the voided volume ratio (which was not the aim of the present study) should be based on an updated measurement of the sample size at any time of decompression, without any crop. Again, this was not trivial because of the above mentioned difficulty of surface detection. So, in order to compare the voided volume ratio obtained from the 2D and 3D methods in section 3.1, the voided volume was divided by the initial volume of the sample in both case, making the comparison possible. Segmentation was processed from slices along the Z direction, i.e. along the sample thickness.

Error estimation

A preliminary study was conducted using Matlab software® to estimate the error associated with the evolution of the cavity size during the acquisition of the stack. A numerical 3D field of perfectly spherical cavities was generated and evolved with a controlled diameter growth rate. This rate of 1µm/s was that experimentally measured in 2D by Kane Diallo et al. in the same material submitted to similar decompression conditions [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions -Support to a relevant modeling framework[END_REF]. This numeric sample was processed in the same way as the tomographic experiment would do: virtual radiographs were computed from successive 2D projections of the numerical field, following a step-increasing tilt angle. Due to the inflation of cavities by this time, the radiographs of the stack (obtained at different angles) corresponded to slightly different values of the cavity diameter. Two configurations were simulated. A first virtual stack of radiographs (A) was generated in "ideal" acquisition conditions, i.e. a large number of radiographs (1440 radiographs over 360°) and good spatial resolution (8x8x8 µm 3 voxel). The good reconstruction obtained from this stack A was considered as a reference. A second virtual stack of radiographs (B) was built-up in the same degraded mode as the real experiment, i.e. with a voxel size twice as large as the reference (A) and fourteen times less radiographs. The growth rate of cavities was kept unchanged for both configurations (A) and (B), as well as the time between the first and the last image of a given stack. The reconstruction algorithm was unchanged too. The error was calculated as the number of voxels which do not belong to the same phase (cavity or matrix) between configurations A and B. For the present experimental conditions, the error was 3.2%. Compared with the repeatability of the phenomenon shown later, it could be considered as acceptable.

RESULTS

Figure 2 qualitatively illustrates some 3D views of the Observed Volume, obtained at different times of cavity expansion. These views showed that a non-negligible number of cavities were nucleated in the sample. With the current spatial resolution, the first ones were detected 4 minutes after the beginning of pressure release, i.e. at about 3 MPa of hydrostatic pressure. Figure 2-(c) shows the delayed nucleation of a second population of cavities, mainly clustered around the first ones, as shown in the last views of Figure 2. This phenomenon has been reported previously in the literature in other gas-rubber systems [START_REF] Gent | Cavitation in rubber: a cautionary tale[END_REF][START_REF] Gent | Nucleation and growth of gas bubbles in elastomers[END_REF]. The delayed nucleation process made the distribution of cavities rather heterogeneous at the sample scale. Due to the inability to really measure the distance between primary and secondary cavities from 2D imaging (out-of-plane distances were not accessible), clustering has never been quantitatively investigated so far, and its origin has never been elucidated. Interaction effects between close cavities were not the purpose of the present article, but the new tomographic device presented here should open new routes to address them in the future. Figure 3 plots the evolution of the number of cavities and total voided volume during and after pressure release. Each graph displays three curves, corresponding to three different tests performed with the same decompression conditions. Both graphs demonstrated the good repeatability of the method, with a maximum relative departure from the average of about 15 %. This was similar to the scatter previously obtained by Kane Diallo et al. from a 2D optical method [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions -Support to a relevant modeling framework[END_REF]. Compared to the former 2D methods, the present 3D tomographic one provided additional data about the thickness and the volume. The following sections focus on the bias induced by the 2D projection (inherent to the 2D optical methods) on four characteristics of single cavities or cavity fields. The contribution of in-situ tomography to each of them is detailed.

Estimation of the voided volume ratio

Tomographic reconstructions were processed to get the time evolution of the voided volume ratio, plotted in Figure 4 (red circles). The kinetics was expected to be the same as the total volume of cavities, plotted in Figure 3.

From the previous 2D visible light transmission methods, only an estimation of the voided surface ratio (i.e. a 2D projection of cavities through the thickness) could be accessed. The full cavity field was then integrated through the thickness and divided by the surface. Estimation of the volume ratio required a strong assumption about the shape of cavities (e.g. sphericity) in order to compensate the lack of out-of-plane data. Moreover, a non-negligible bias could arise from some overlapping of cavities. In order to discuss the bias of the 2D projection estimation, the present 3D data were also processed in the same way as done from the 2D method. The cavity field was projected through the thickness of the sample. The projected voided area was calculated and the 2D surface ratio of cavities was deduced (purple triangles in Figure 4). The huge error associated with 2D projection (680% at the maximum voided state) was clearly evidenced. As a matter of comparison, Figure 4 also displayed the 2D surface ratio measured by Kane Diallo et al. [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions -Support to a relevant modeling framework[END_REF] in 20x20 mm samples of the same EPDM, during similar pressure release (2.5 MPa/min) but from a higher exposure pressure of 15 MPa (grey squares in Figure 4). The experimental surface ratio significantly exceeded the estimation from tomographic data. However, the exposure pressure is known to promote cavitation. The higher exposure pressure could explain the enhanced void ratio in their case. The assumption of spherical cavities (made by Kane Diallo et al. for image processing of cavity fields pictures) could also be invoked, as discussed later in section 3.3.

Figure 4: Time evolution of the voided volume ratio obtained by in-situ X-ray tomography and projection through the thickness, for comparison purpose with the voided surface ratio obtained from

2D optical methods (* from [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions -Support to a relevant modeling framework[END_REF]).

Distribution of cavities through the thickness

At any decompression time, the volume of each cavity could be determined from in-situ Xray tomography reconstructions and correlated with its location through the thickness. In each graph of Figure 5, each dot corresponds to the volume of one cavity (plotted in micron 3 ) as a function of the abscissa of the gravity center in the thickness direction of the Observed Volume. The indicated time (in seconds) corresponds to the beginning of the stack acquisition. The first nucleated cavities were spread over the thickness, as observed at 200 s. The following graphs clearly evidenced that the largest cavities were located in the core of the sample. The largest cavities reached a maximum volume almost equivalent to a 700µmdiameter sphere. Many small cavities were simultaneously observed all through the thickness at any time, but with a higher concentration in the core.

Anisotropy of cavities

The visible light transmission methods available so far only displayed 2D views of cavities. In the same material as used here, Kane Diallo et al. showed that cavities were almost circular [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions -Support to a relevant modeling framework[END_REF]. The present tomography data allowed discussion of the anisotropy of cavities in a complete 3D context. This section first focuses on the 3D morphology and anisotropy of single cavities. Anisotropy at the scale of the field will be discussed in the next section. As a representative example, Figure 6 displays views of one single cavity extracted from the field (red square) in all planes. The isotropy of cavities in the XY plane, previously observed from the 2D method, was confirmed here, but a non-negligible flattening was clearly evidenced in the sample thickness direction. This was visible in both XZ and YZ planes, respectively, in Figure 6-(c) and (d). This result partly explained the overestimation of voided volume when calculated from the 2D surface ratio, assuming spherical cavities.

Cavity anisotropy was quantified based on the smallest convex hull, which was computed first and found very close to an ellipsoid. Three orthogonal axes were attached to each convex hull. The first axis corresponded to the maximum distance between two points of the convex hull. The second axis was orthogonal to the first one, and contained the most distant point to it on the convex hull. The third axis was obtained as the vector product of the other two directions. In case of non-uniqueness, the first occurrence was kept at each step. The anisotropy ratio of the cavity was finally defined as the ratio between the shortest and the largest axis. This ratio equalled 1 when the cavity was perfectly spherical. The above defined anisotropy ratio was very close to the Feret ratio in the present case of almost ellipsoidal cavities.

This long computational procedure has been carried out for each time and each of the three experiments. The anisotropy ratio was computed for each cavity at any acquisition time of the three experiments. Calculations were performed on the supercomputer facilities at the "Mésocentre de calcul SPIN" hosted by University of Poitiers. The time evolution of the average anisotropy ratio of the cavity field was plotted in Figure 7, for one of the three tests. On average, cavities appeared more spherical at the time of maximum cavitation (maximum number and volume ratio of cavities), without reaching perfect sphericity. The flattening effect was stronger at the beginning of inflation and at the end of deflation. However, the average anisotropy ratio only made sense depending on the shape of the distribution for the full cavity field. Then, the corresponding histograms were plotted in Figure 8 at different times after the beginning of decompression. At 200 seconds, the first population of cavities was at variable stages of the nucleation and growth process, with a highly distributed anisotropy ratio. Between 450 and 1500 seconds, histograms exhibited a major population of cavities almost spherical, with an anisotropy ratio close to 0.9. The delayed nucleation of secondary cavities was responsible for a shoulder of the main peak, visible between 300 and 800 seconds. After 200 seconds, cavities heterogeneously deflated, with again a distributed anisotropy ratio.

Figure 8: Evolution of the distribution of anisotropy ratio at different times after the beginning of decompression (same data set as used for the average calculation in Figure 7).

Conclusions deduced from the full cavity field were consistent with previous observations at the single cavity scale in Figure 6. Several phenomena could contribute to anisotropy. Due to the sample shape (smaller thickness than width), the diffusion boundary conditions could be anisotropic. Material anisotropy also could be incriminated, due to compression-molding of the initial rubber sheets.

Morphological Representative Volume Element

The above depicted cavity field raised the question of the size and composition of a Representative Volume Element, especially for further modelling purpose. A covariogram method can be applied to access the size of a Morphological Representative Volume Element (MRVE) of a damage field [START_REF] Nait-Ali | Catching the time evolution of microstructure morphology from dynamic covariograms[END_REF]. Covariogram corresponds to the probability C(h) of encountering a cavity at a distance h from another cavity along one given direction. A correlation length can be extracted from the covariogram in any given direction. It corresponds to the distance beyond which the statistical influence of the cavity field becomes negligible in the considered direction. The correlation length can be viewed as the half-size of a MRVE in the considered direction. Kane Diallo previously applied this method to 2D-projection pictures of cavity fields [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions -Support to a relevant modeling framework[END_REF]. Only two orthogonal directions of the sample plane (X and Y) were processed. For moderate decompression conditions, the MREV was shown to be isotropic in the sample plane, with a slightly larger width than the average size of cavities. The MRVE could be represented as a unit cell containing one or two cavities. More severe decompression conditions (i.e. fast decompression rates or high pressures for saturation) strongly promoted cavity nucleation. The MREV size drastically increased and included a large number of cavities on average. However, no information about MRVE was provided through the thickness. The evidence of two regimes had major consequences on the modeling strategy. In the former case of moderate decompression conditions, modeling could be based on a unit cell containing one c D c D or two cavities. In the latter case of faster decompression, the MREV became very large compared to the sample size and the concept of MRVE itself was questioned, suggesting the need for alternate modeling approaches.

In the present work, covariograms were computed from the successive tomographic reconstructions (i.e. at different times) in the three main space directions: X and Y in the sample plane and Z in the thickness direction. The correlation distance extracted from each covariogram was plotted in Figure 9. The time evolution of along X and Y were compared to each other; this confirmed that the MRVE was almost isotropic in the sample plane all along decompression, as reported by Kane Diallo et al. from the 2D method [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions -Support to a relevant modeling framework[END_REF]. On the other hand, the correlation length in the thickness direction Z was always smaller than in the plane direction. This meant that the MRVE was flattened in the thickness direction. This could not be detected from the 2D method. Cavitation was maximum after 600 seconds, as shown from the evolution of the number and volume ratio of cavities in Figure 3. At that time, the corresponding MRVE size was (2x0,72) x (2x0,736) x (2x0,416) mm 3 = 1.76 mm 3 . This size appeared close to the average size of clusters visible in Figure 2, containing a large primary cavity surrounded by several smaller ones. An important result was that the MRVE estimated from the 3D analysis in the present decompression conditions was significantly smaller than that previously estimated in the same decompression conditions, but in a 2D context, by Kane Diallo et al. [START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions -Support to a relevant modeling framework[END_REF]. In this latter case, the MRVE size was very close to the size of the sample itself. This MRVE analysis was further proof of the bias induced by 2D projection methods. A 3D characterization of cavitation appeared essential to support modelling strategy.

CONCLUSIONS

A 3D in-situ X-ray tomography experiment was developed for better understanding of decompression failure mechanisms in gas-exposed polymers. The acquisition configuration and testing conditions were optimized to minimize artefacts and provide reliable quantitative results. A major concern of this article was to discuss and complete the alternate method used so far in the literature, i.e. the observation of 2D projections of the cavity field using visible light transmission. In the present unfilled EPDM, the time evolution of the number of cavities was found to be similar to that measured from a 2D optical method. Even if the previously estimated surface ratio could not be directly compared to the present volume ratio, both parameters consistently evolved with time. A major interest of this new experiment was the additional data provided in the out-of-plane direction. First, the 3D shape of cavities could be clarified. In these samples, cut from plates, cavities were not isotropic. They were flattened in the sample plane at the early stage of growth. They then became isotropic, but flattened again at the end of deflation. Regarding the out-of-plane distribution, coarser cavities were located in the core of the sample. They co-existed with a population of small cavities, spread throughout the sample thickness. This 3D experiment also allowed complete characterization of the size and shape of a Morphological Representative Volume Element (MRVE) for the cavity field. The MRVE size was estimated from a covariogram analysis. It was also found to be flattened in the sample plane, with a size corresponding to that of clusters of primary cavity surrounded by several secondary ones. This size was much smaller than estimated from the 2D method in the same decompression conditions. This is one case of bias induced by 2D projection in the visible light transmission method. Being able now to characterize the cavity field in 3D during decompression allows more precise investigation about key issues like the relationships between microstructure and damage resistance, or the transition from cavitation to cracking activated during pressure cycles. An EPDM, exposed to high-pressure hydrogen, was tracked by in-situ computed tomography.

FIGURE CAPTIONS

Novel and quantitative data were provided about the out-of-plane morphology and volume distribution of cavities. Some bias, induced by 2D projection in the former visible light transmission methods, were discussed.

The experiment opens new routes for a better understanding and accurate quantification of decompression failure.

Figure 1 :

 1 Figure 1: Pressure chamber for in-situ X-ray tomography of hydrogen exposure and decompression.

Figure 2 :

 2 Figure 2: 3D views of cavities expansion in unfilled EPDM saturated at 12 MPa and decompressed at 2.5 MPa/min (view of the Observed Volume).

Figure 3 :

 3 Figure 3: Repeatability of the (a) number of cavities and (b) total volume of cavities during three decompression tests in unfilled EPDM (exposure pressure 12 MPa; decompression rate 2.5 MPa/min) as detected from in-situ X-ray tomography.

Figure 5 :

 5 Figure 5: Mapping of the volume of all cavities as a function of their location through the thickness of the Observed Volume, at different times after the beginning of decompression.

Figure 6 :

 6 Figure 6: a) view of the cavity field 3000s after the beginning of decompression and focus on the shape of one single cavity (red square) in planes (b) XY, (c) XZ) and (d) YZ.

Figure 7 :

 7 Figure 7: Time evolution of the average anisotropy ratio of cavities.

Figure 9 :

 9 Figure 9: Time evolution of the correlation distances extracted from covariograms along the two principal axes in the sample plane (X and Y) and along the thickness direction (Z) during and after decompression.

Figure 1 :

 1 Figure 1: Pressure chamber for in-situ X-ray tomography of hydrogen exposure and decompression.

Figure 2 :

 2 Figure 2: 3D views of cavities expansion in unfilled EPDM saturated at 12 MPa and decompressed at 2.5 MPa/min (view of the Observed Volume).

Figure 3 :

 3 Figure 3: Repeatability of the (a) number of cavities and (b) total volume of cavities during three decompression tests in unfilled EPDM (exposure pressure 12 MPa; decompression rate 2.5 MPa/min) as detected from in-situ X-ray tomography.

Figure 4 :

 4 Figure 4: Time evolution of the voided volume ratio obtained by in-situ X-ray tomography and projection through the thickness, for comparison purpose with the voided surface ratio obtained from 2D optical methods (* from[START_REF] Kane-Diallo | Time-resolved statistics of cavity fields nucleated in a gas-exposed rubber under variable decompression conditions -Support to a relevant modeling framework[END_REF]).

Figure 5 :

 5 Figure 5: Mapping of the volume of all cavities as a function of their location through the thickness of the Observed Volume, at different times after the beginning of decompression.

Figure 6 :

 6 Figure 6: a) view of the cavity field 3000s after the beginning of decompression and focus on the shape of one single cavity (red square) in planes (b) XY, (c) XZ) and (d) YZ.

Figure 7 :

 7 Figure 7: Time evolution of the average anisotropy ratio of cavities.

Figure 8 :

 8 Figure8: Evolution of the distribution of anisotropy ratio at different times after the beginning of decompression (same data set as used for the average calculation in Figure7).

Figure 9 :

 9 Figure 9: Time evolution of the correlation distances extracted from covariograms along the two principal axes in the sample plane (X and Y) and along the thickness direction (Z) during and after decompression.

HIGHLIGHTS

  Time-resolved 3D views of decompression failure in rubbers are reported for the first time.

ACCEPTED MANUSCRIPT

ACKNOWLEDGEMENTS

This work was partially funded by the French Government programs "Investissements d'Avenir" LABEX INTERACTIFS (reference ANR-11-LABX-0017-01) and EQUIPEX GAP (reference ANR-11-EQPX-0018). Computations have been performed on the supercomputer facilities of the Mesocentre de calcul SPIN Poitou-Charentes. Authors are indebted to Pr. Nishimura from Hydrogenius Laboratory at Kyushu University (Japan), for kindly providing the material, and for fruitful discussions for many years now with him and members of his Research Group.


J Helbert

D Dyar

I Walter

D Wendler

T Widemann

E Marcq

G Guignan

A Maturilli

S Ferrari

N Mueller

D Kappel

M. D 'amore

A Boerner

C Tsang

G E Arnold

S Smrekar

R Ghail

The Venus Emissivity Mapper (VEM) -Obtaining Global Mineralogy of Venus from Orbit

Introduction:

The Venus Emissivity Mapper is the first flight instrument designed to focus on mapping the surface of Venus using atmospheric windows around 1 µm. After several years of development, VEM has a mature design with an existing laboratory prototype verifying an achievable instrument SNR of well above 1000, as well as predicted error in retrieval of relative emissivity of better than 1%, assuming the availability of an improved Venus topography.

It will provide a global map of rock type, iron contents and redox state of the surface by observing the surface with six narrow band filters, ranging from 0.86 to 1.18 µm. Three additional windows allow corrections for cloud composition and variability, two measure water abundance, and three compensate for stray light. Continuous observation of Venus' thermal emission will also place tighter constraints on current day volcanic activity. Eight of the channels provide measurements of atmospheric water vapor abundance as well as cloud microphysics and dynamics and permit accurate correction of atmospheric interference on the surface data. Combining VEM with a high-resolution radar mapper, such as on the ESA EnVision or NASA VERITAS mission proposals, will provide key insights into the divergent evolution of Venus and Earth.

VEM Design: The VEM system design, discussed in detailed in [START_REF] Helbert | Proceedings of SPIE. 9973[END_REF][START_REF] Helbert | [END_REF], is a pushbroom multispectral imaging system. It leverages a proven measurement technique pioneered by VIRTIS on Venus Express (VEX) [START_REF] Incecco | [END_REF][START_REF] Helbert | [END_REF][5][6][7][8][9][10], but it incorporates lessons learned from VIRTIS to achieve greatly improved sensitivity and spectral and spatial coverage: • a filter array (rather than a grating) provides wavelength stability (band-center and width-scatter) ~5× more stable and maximizes signal to the focal-plane array (FPA), and • first coverage of the spectral windows below 1µm, • a two-stage baffle decreases scattered light and improves sensitivity, • use of an InGaAs detector with an integrated thermal electric cooler (TEC) eliminates the need for cryogenic cooling. VEM's design draws strongly on DLR's BepiColombo MERTIS instrument (launching in 2018). This design maturity, combined with a standard camera optical design, leads to low development risk.

A first performance evaluation of the VEM prototype used two Venus analog samples heated to Venus surface temperatures [START_REF] Helbert | [END_REF]. The retrieved emissivities match the laboratory values, and the uncertainty for a single unbinned exposure is <0.35%. VEM uses onboard software developed for MERTIS to bin, coadd, and losslessly compress data upon uplink command. During the science orbits, VEM oversamples at 10 km spatial resolution (33×33 pixel binning). To further enhance SNR, VEM uses digitial TDI to provide 189× gain over single-pixel SNR. Using current performance of the laboratory prototype for a single unbinned exposure and SNR enhancement due to onboard processing, we expect a system SNR of well beyond 1000.

VEM atmospheric correction: Methodology for retrieving surface emissivity is complex but well understood and demonstrated. To distinguish between surface and atmospheric contributions, VEM uses an updated version of the extensively tested pipeline developed to process VIRTIS data [5], combined with a radiative transfer model (RTM) [11][12][13][14]. Surface emissivity retrieval techniques were developed based on Galileo NIMS observations at 1700, 1800 and 2300 nm [15]. VEM cloud bands occur at 1195, 1310, and 1510 nm [16], the first on the flank of the 1180-nm surface windows [17]. VEM's cloud bands are close to surface bands, providing near-optimal correction.

Conclusion: VEM builds on recent advances in the laboratory analog spectroscopy at PSL at DLR [START_REF] Helbert | Proceedings of SPIE. 9973[END_REF][START_REF] Gilmore | pdf 16th VEXAG Meeting 2018[END_REF]. It is the first flight instrument specially designed to focus on mapping the surface of Venus using the atmospheric windows around 1 µm. VEM has a mature design with an existing laboratory prototype verifying an achievable instrument SNR of well above 1000 as well as a predicted error in the retrieval of relative emissivity of better than 1%.


Bilal Faye
email: faye@lipn.univ-paris13.fr

Hanane Azzag
email: azzag@univ-paris13.fr

Mustapha Lebbah
email: mustapha.lebbah@uvsq.fr

Fangchen Fang
email: fangchen.feng@univ-paris13.fr

Normalisation Contextuelle : Une Nouvelle Approche pour la Stabilité et l'Amélioration des Performances des Réseaux de Neurones

L'apprentissage des réseaux de neurones est confronté à des défis majeurs liés au changement de distribution en couches, perturbant ainsi la convergence et les performances des modèles. La Normalisation par lot (BN) a révolutionné ce domaine, mais repose sur l'hypothèse simplifiée d'une seule composante gaussienne par lot. Pour remédier à cela, la Normalisation par Mélange (MN) a adopté une approche basée sur le modèle de mélange gaussien (GMM), mais avec des coûts computationnels importants liés à l'algorithme Espérance-Maximisation (EM) pour déterminer des composantes. Notre solution, la Normalisation Contextuelle (CN), regroupe des observations similaires en "contextes" pour une représentation locale, sans nécessiter d'algorithme de construction de ces contextes. Les paramètres de normalisation sont appris de manière similaire aux poids du modèle, assurant rapidité, convergence et performances supérieures par rapport à BN et MN.

Introduction

La normalisation est une opération courante en traitement des données, impliquant des transformations pour obtenir une représentation avec des propriétés statistiques spécifiques [START_REF] Kessy | Optimal whitening and decorrelation[END_REF]. Elle est utilisée pour égaliser les amplitudes des variables, accélérant ainsi la convergence dans les réseaux de neurones monocouches [START_REF] Lecun | Efficient backprop[END_REF]. Cependant, dans les réseaux multicouches, la normalisation des entrées peut ne pas avoir d'impact significatif sur les autres couches, car les distributions de données évoluent avec les mises à jour des poids du réseau. Pour garantir la stabilité de l'apprentissage à travers toutes les couches, diverses techniques de normalisation sont nécessaires, notamment la normalisation des activations, des poids et des gradients. La normalisation par lot (batch) (BN) [START_REF] Ioffe | Batch normalization : Accelerating deep network training by reducing internal covariate shift[END_REF] est une méthode courante pour stabiliser l'apprentissage dans les réseaux de neurones multicouches.

Elle standardise les activations de chaque couche en utilisant la moyenne et l'écart-type du batch auquel elles appartiennent. Cette méthode permet l'utilisation de taux d'apprentissage plus élevés, accélérant ainsi l'apprentissage. Cependant, BN présente deux inconvénients majeurs : elle dépend de la taille du batch d'entraînement, ce qui peut nuire aux performances avec de petits batchs, et elle suppose que les observations d'un même batch proviennent d'une même distribution, ce qui peut être trop restrictif pour des données complexes et variées. Des variantes spécialisées de la BN ont été proposées pour remédier aux limitations liées à la taille du batch [START_REF] Huang | Normalization techniques in training dnns : Methodology, analysis and application[END_REF]. Une autre approche, la normalisation par mélange (MN) [START_REF] Kalayeh | Training faster by separating modes of variation in batchnormalized models[END_REF], a été introduite pour contourner les limitations découlant de l'hypothèse d'une composante unique. Dans le contexte de MN, il est considéré que les observations d'un même lot ne sont pas nécessairement issues de la même distribution. Elle utilise l'algorithme Espérance-Maximisation (EM) Dempster et al. (1977) pour regrouper les observations en fonction de leurs composantes et normalise ensuite chaque observation en utilisant les statistiques de sa composante respective. MN améliore la performance et la convergence par rapport à BN, notamment pour les réseaux de neurones convolutifs. Cependant, l'utilisation d'EM dans MN entraîne une augmentation significative du temps de calcul, limitant ainsi son efficacité. Pour résoudre ce problème, nous proposons une nouvelle approche de normalisation nommée Normalisation Contextuelle (CN). CN partage l'hypothèse de MN concernant la distribution des observations au sein d'un batch, mais elle introduit une structure appelée "contexte" pour regrouper les observations en ensembles cohérents sans recourir à l'algorithme EM. Chaque contexte suit une distribution gaussienne dans un espace latent de représentation, et les observations d'un même contexte sont normalisées en utilisant des paramètres appris lors de la rétropropagation du gradient. Cette approche facilite la tâche du réseau de neurones, accélère l'apprentissage et surmonte les limitations de MN liées au temps de calcul élevé.

Cette étude présente trois contributions significatives : -CN : Méthode novatrice pour une normalisation adaptative grâce à des contextes prédéfinis, simulant des métaclasses et des clusters dans des espaces latents gaussiens. Les paramètres, acquis pendant l'entraînement, supplantent l'algorithme EM. -Une couche adaptable à diverses architectures de réseaux de neurones, y compris les Transformeurs et les réseaux de neurones convolutifs. -Inspirés par des résultats prometteurs en classification, CN est déployée en adaptation de domaine, boostant performance et convergence de l'algorithme AdaMatch. Berthelot et al. (2021).

2 État de l'art

Normalisation par lot (BN)

Soit x ∈ R N ×C×H×W , un tenseur d'activation dans un réseau de neurones convolutif, où les axes N , C, H, et W représentent respectivement le lot (batch), le nombre de canaux, la hauteur et la largeur. On pose B, le sous-ensemble résultant de l'aplatissement de x sur tous les axes sauf celui des canaux. Pour toute activation

x i dans B = {x 1:m : m ∈ [1, N ] × [1, H] × [1, W ]}, la standardisation avec BN s'écrit sous la forme : xi = x i -µ B σ 2 B + ϵ , (1)
où µ B = 1 m m i=1 x i et σ 2 B = 1 m m i=1 (x i -µ B )
2 représentent respectivement la moyenne et la variance de B, tandis que ϵ > 0 est une petite valeur qui gère les instabilités numériques. La transformation décrite dans l'Équation (1) génère une distribution ayant une moyenne nulle et une variance unitaire si les observations à l'intérieur du batch proviennent de la même distribution (hypothèse utilisée par BN). Cette contrainte de moyenne nulle et de variance unitaire permet de stabiliser la distribution des activations et, par conséquent, d'accélérer l'apprentissage du réseau de neurones. Bien que la BN présente de bonnes performances, elle est sensible à la taille du batch et repose sur l'hypothèse limitée d'une composante unique pour l'ensemble des observations d'un même batch. Afin de relever ces défis, de nombreuses variantes et méthodes alternatives ont été développées [START_REF] Kalayeh | Training faster by separating modes of variation in batchnormalized models[END_REF]; [START_REF] Huang | Normalization techniques in training dnns : Methodology, analysis and application[END_REF], et nous en présenterons quelques-unes dans la section suivante.

Variantes de la BN

Quelques extensions de BN ont été proposées, notamment la normalisation par couche (LN) la normalisation par instance (IN), la normalisation par groupe (GN), et la normalisation par mélange (MN) [START_REF] Kalayeh | Training faster by separating modes of variation in batchnormalized models[END_REF]; [START_REF] Huang | Normalization techniques in training dnns : Methodology, analysis and application[END_REF]. La transformation générale x → x, commune à toutes ces variantes, est appliquée sur x avec un aplatissement sur la dimension spatiale L = H × W , et peut être formulée de la manière suivante :

v i = x i -E Bi (x), xi = v i E Bi (v 2 ) + ϵ , (2)
où MN : Utilise un modèle de mélange gaussien (GMM) pour normaliser les activations en fonction de plusieurs modes de variation dans leur distribution sous-jacente, adapté lorsque la distribution des activations présente plusieurs modes. MN normalise chaque activation x i dans B i en utilisant la moyenne et l'écart-type de la composante du mélange à laquelle x i appartient. La fonction de densité de probabilité p θ peut être représentée sous la forme d'un GMM.

B i = {j : j N ∈ [1, N ], j C ∈ [i C ], j L ∈ [1, L]}, et i = (i N , i C , i L )
Soit x ∈ R D , et θ = {λ k , µ k , Σ k : k = 1, ..., K}, alors on a : p(x) = K k=1 λ k p(x|k), s.t. ∀ k : λ k ≥ 0, K k=1 λ k = 1, où p(x|k) = 1 (2π) D/2 |Σ k | 1/2 exp - (x-µ k ) T Σ -1 k (x-µ k ) 2
, est la kème composante, µ k la moyenne et Σ k la matrice de covariance. La probabilité que x ait été généré par la k-ème composante gaussienne peut être définie comme suit :

τ k (x) = p(k|x) = λ k p(x|k) K j=1 λj p(x|j)
. En se basant sur ces hypothèses et la transformation générale dans l'Équation (2), la normalisation de x i , est définie comme suit :

xi = K k=1 τ k (x i ) √ λ k xk i , (3)
avec xj ) , est la contribution normalisée de x i sur B i dans l'estimation des statistiques de la k-ème composante gaussienne. Avec cette approche, MN peut être alors appliquée en deux étapes :

v k i = x i -E Bi [τ k (x).x], xk i = v k i E Bi [τ k (x).(v k ) 2 ] + ϵ , ( 4
) où τk (x i ) = τ k (xi) j∈B i τ k (
1. Estimation des paramètres de chaque composante Gaussienne θ = {λ k , µ k , Σ k : k = 1, ..., K} à l'aide de l'algorithme Espérance-Maximisation (EM) Dempster et al. (1977).

2. Normalisation de chaque x i avec les paramètres de la composante à laquelle elle appartient (Équation (3)).

En comparaison avec la BN, la MN favorise une meilleure convergence et des performances accrues dans les réseaux de neurones convolutifs sur des tâches d'apprentissage supervisé.

3 Méthode Proposée : Normalisation Contextuelle (CN)

Description de la Méthode

CN, à l'instar de la normalisation par mélange, modélise efficacement les données sous l'hypothèse d'un mélange de composantes gaussiennes dans un espace latent. Dans cette approche, un "contexte" est défini comme un groupe d'observations partageant des caractéristiques similaires. Les observations au sein d'un même contexte sont regroupées et normalisées avec des paramètres appris par le réseau, le tout sans nécessiter le recours à l'algorithme EM coûteux. Chaque contexte est identifié par un identifiant unique r, simplifiant ainsi le processus de normalisation. CN peut être formulée comme l'équation 2 lorsque

B i = {j : j N ∈ [i N ], j C ∈ [i C ], j L ∈ [1, L]}

Apprentissage des paramètres

Considérons un tenseur d'activation x dans l'espace R N ×C×H×W , où B i , un groupe d'activations résultant de l'aplatissement de x le long de l'axe L = H × W . Chaque x i dans B i est normalisée avec les paramètres (µ r , σ r ) du contexte r auquel il est associé :

xi ← x i -µ r σ 2 r + ϵ (5)
La moyenne (µ r ) et l'écart-type (σ r ) sont appris comme des poids du réseau de neurones par normalisation des activations résultantes de la transformation des observations du contexte r, Algorithme 1 : CN-Transformation : Normalisation d'une activation Entrée :

x i ∈ B i ; contexte r associé à x i Sortie : { xi = CN(x i , r)} 1 α r ← onehot(r) // encodage de la variable catégorielle r 2 µ r ← Encodeur µ (α r ) // estimation de la moyenne 3 σ 2 r ← Encodeur σ (α r ) //
     α r ← onehot(r) µ r ← Encodeur µ (α r ) : W µ α r + b µ σ 2 r ← Encodeur σ (α r ) : W σ α r + b σ (6)
Dans l'Équation 6, W et b sont les paramètres de Encodeur, et r est le contexte associé à x i . La fonction onehot(.) effectue l'encodage one-hot de la variable catégorielle r, pour donner α r . Durant la phase d'apprentissage du réseau de neurones avec une normalisation contextuelle, conformément à l'Algorithme 2, il est nécessaire de rétropropager le gradient de la fonction de perte, ℓ, à travers la transformation. De plus, les gradients par rapport aux paramètres de la transformation CN doivent être calculés. Cela est réalisé en utilisant la règle de la chaîne, comme le montre l'expression suivante (avant simplification) :

∂ℓ ∂µ r = ∂ℓ ∂ xi . ∂ xi ∂µ r = - ∂ℓ ∂ xi .(σ 2 r + ϵ) -1/2 ∂ℓ ∂σ 2 r = ∂ℓ ∂ xi . ∂ xi ∂σ 2 r = µ r + x i 2(σ 2
r + ϵ) 3/2 CN est une opération différentiable qui normalise les activations dans les réseaux de neurones, favorisant l'adaptation aux distributions d'entrée pour de meilleures performances après convergence. Après cette convergence, CN crée des composantes gaussiennes dans un espace latent, fournissant une représentation adaptée à la tâche.

Lors de l'inférence, nous avons le choix entre normaliser les activations avec CN ou considérer tous les contextes collectivement. Cette amélioration est appelée CN+ et est décrite dans l'Algorithme 3. Soit T , le nombre de contextes, pour un x i quelconque dans B i , en reformulant l'Équation 7, CN+ s'écrit comme suit :

xi = √ T T r=1 τ r (x i )x r i , (7)
avec v r i = x i -E Bi [τ r (x).x], xr i = v r i E Bi [τ r (x).(v r ) 2 ] + ϵ , où τr (x i ) = τ r (x i ) j∈Bi τ r (x j )
, en supposant que les probabilités a priori (λ r = 1 T , r = 1, ..., T ) sont constantes.

Ensembles de données

Les expériences de cette étude s'appuient sur plusieurs jeux de données de référence couramment utilisés en classification. CIFAR-10 et CIFAR-100 comprennent respectivement 50,000 images d'entraînement et 10,000 images de test, avec une taille de 32 × 32 pixels Krizhevsky et al. (2009a,b). CIFAR-100, une extension de CIFAR-10, se divise en 100 classes regroupées en 20 superclasses. Tiny ImageNet est une version réduite d'ImageNet avec 200 classes [START_REF] Le | Tiny imagenet visual recognition challenge[END_REF]. MNIST digits contient 70,000 images représentant les 10 chiffres Le-Cun et Cortes (2010). SVHN focalise sur la reconnaissance des chiffres dans des scènes naturelles, totalisant plus de 600,000 images [START_REF] Sermanet | Convolutional neural networks applied to house numbers digit classification[END_REF]. Cette consolidation permet d'optimiser l'espace tout en préservant les informations essentielles.

Etude Comparative : Normalisation Contextuelle (CN) vs. Normalisation par Mélange (MN) et Normalisation par Lot (BN) sur CIFAR et Tiny ImageNet

Dans cette expérience, nous utilisons un réseau de neurones convolutifs (ConvNet+BN) peu profonde. Ce réseau est composé de quatre couches de convolution avec activation ReLU, chacune suivie par une couche de BN. Une des limites de BN est la baisse de performance lors de l'application de fonctions non linéaires (par exemple, ReLU) après la normalisation des activations. À travers ConvNet+BN, nous illustrons comment la CN résout ce problème et améliore la convergence et les performances globales en remplaçant une couche de BN par une couche de CN sur le réseau. Pour ce faire, nous commençons par appliquer l'algorithme d'Estimation du Maximum de Vraisemblance (MLE) Dempster et al. (1977) sur chaque ensemble de données (CIFAR-10, CIFAR-100 et Tiny ImageNet) afin de déterminer les paramètres de chaque composante gaussienne, en utilisant k = 3. Ensuite, nous créons trois modèles distincts sur chaque ensemble de données : le premier modèle, ConvNet+BN, consiste en des couches de normalisation BN uniquement. Le deuxième modèle, nommé ConvNet+MN, est obtenu en remplaçant la troisième couche BN de ConvNet+BN par une couche de MN. Enfin, le troisième modèle, appelé ConvNet+CN, est obtenu en remplaçant la troisième couche BN de ConvNet+BN par une couche de CN. Les paramètres des composantes gaussiennes ainsi obtenus sont utilisés par la couche MN dans ConvNet+MN pour normaliser les activations, tandis que la couche CN dans ConvNet+CN utilise ces composantes gaussiennes comme contextes, pour normaliser les activations du même contexte, avec des paramètres (µ r et σ 2 r avec r = 1, ..., 3) qui sont appris pendant l'entraînement du réseau ConvNet+CN.

Les modèles sont entraînés dans des conditions uniformes sur 100 époques. Pendant la phase d'apprentissage, l'accuracy de chaque modèle est enregistré tous les 25 époques sur les données de validation, ce qui permet de suivre de près l'évolution des performances, comme le montre la Table 1. Le modèle ConvNet+CN, qui intègre la normalisation contextuelle, présente des performances supérieures sur l'ensemble des époques enregistrés, et cet avantage s'intensifie quel que soit le nombre de classes dans l'ensemble de données à prédire (10, 100 et 200). Ce même effet est également observé lors de l'utilisation de l'approche CN+ en inférence sur le modèle ConvNet+CN, comme décrit dans la Table 2. En plus d'améliorer les performances, la normalisation contextuelle CN dans ConvNet+CN accélère la convergence du réseau de neurones, comme illustré sur la Figure 1. Ce phénomène est cohérent avec les améliorations de performances observées. En résumé, à travers les résultats présentés dans la Table 1 et la Figure 1, notre méthode de normalisation contextuelle démontre sa capacité à améliorer les performances et à accélérer la convergence des réseaux de neurones convolutifs complexes. Elle surmonte les limitations de la BN liées à l'utilisation de fonctions non linéaires (comme ReLu dans cet exemple) et dépasse même l'approche utilisant MN.

L'approche présentée dans cette expérimentation utilise des composantes gaussiennes trouvées via MLE comme contextes, mais c'est fastidieux. Dans les sections suivantes, nous simplifierons la définition des contextes sans recourir à des algorithmes complexes. La MN ne sera plus utilisée, car elle exige la préconstruction de composantes gaussiennes (une limite). Étant donné que CN se comporte de manière similaire à CN+, elle sera privilégiée pour l'inférence, étant plus économique en ressources.

Normalisation Contextuelle avec les Superclasses de CIFAR-100 comme Contextes

Cette expérience vise à créer des contextes dans un ensemble de données, sans nécessiter d'algorithme, contrairement à celle décrite dans la Section 4.2. Nous utilisons l'exemple de CIFAR-100, qui comprend 100 classes à prédire et 20 superclasses regroupant ces classes. Les superclasses sont utilisées comme contextes pour notre approche de normalisation CN. Dans cette approche, les observations appartenant au même contexte (superclasse) r sont normalisées avec les mêmes paramètres (µ r et σ r ) appris pendant l'entraînement du réseau de neurones. Le réseau de base utilisé est un modèle ViT issu de Keras [START_REF] Salama | Implementing the vision transformer (vit) model for image classification[END_REF]. Deux variantes sont construites à partir de ce modèle. La première incorpore une couche de normalisation par lots (BN) en tant que première couche pour normaliser les images (ViT+BN), tandis que la seconde intègre directement notre couche de normalisation contextuelle (CN) en tant que première couche du modèle ViT pour normaliser les entrées (ViT+CN). Les deux modèles sont entraînés avec des paramètres identiques sur 200 époques.

La Figure 2 illustre une convergence plus rapide du modèle ViT+CN par rapport au modèle ViT+BN. Cette observation est cohérente sur les ensembles d'entraînement et de validation, renforçant l'idée que la normalisation contextuelle CN accélère l'apprentissage des réseaux de neurones. En plus de cette accélération, CN améliore la performance, comme démontré dans la Table 3, où des différences significatives sont notées sur l'ensemble des métriques. L'incorporation de CN, comme illustré par ViT+CN, permet d'obtenir une augmentation de l'accuracy de 10%, un niveau d'accuracy jamais atteint par un modèle ViT entraîné à partir de zéro sur l'ensemble de données CIFAR-100.

Cette expérience démontre une méthode simple pour créer des contextes sans algorithme, accélérant la convergence et améliorant la performance des réseaux de neurones. Cependant, la structure de superclasse n'est pas toujours facile à établir. La prochaine section présente une application typique de CN, où les contextes se révèlent clairement.

La Normalisation Contextuelle en Adaptation de Domaine

L'adaptation de domaine [START_REF] Farahani | A brief review of domain adaptation[END_REF] transfère des connaissances du domaine source au domaine cible malgré les différences de distribution. Les modèles d'apprentissage automatique rencontrent des défis en raison de ces variations de distribution. En utilisant la normalisation contextuelle, on crée des représentations spécifiques à chaque domaine en fonction de la tâche cible, résolvant les problèmes liés aux différences de distribution, améliorant les performances et accélérant la convergence. Afin d'illustrer cette approche, nous prenons l'exemple de l'algorithme AdaMatch Berthelot et al. (2021), en considérant les domaines source et cible comme des contextes distincts (r ∈ {domaine source, domaine cible}). Nous utilisons un modèle de base, AdaMatch Paul (2019), créé à partir de réseaux résiduels de grande taille [START_REF] Zagoruyko | Wide residual networks[END_REF], pour servir de modèle de référence. À partir de ce modèle de référence, nous en construisons un autre (AdaMatch+CN) en intégrant une couche CN en tant que première couche du réseau. Cette couche permet de normaliser les données directement en fonction de leur contexte ou de leur domaine d'origine. Dans cette expérience, l'ensemble de données MNIST est utilisé en tant que domaine source, avec des étiquettes connues, tandis que l'ensemble de données SVHN sert de domaine cible, avec des étiquettes inconnues.

Afin de surveiller la stabilité des deux modèles, AdaMatch et AdaMatch+CN, lors de 4, où l'on observe une augmentation de l'accuracy de 18% sur le domaine cible. En somme, l'adaptation de domaine est un exemple concret de l'efficacité de la normalisation contextuelle. En associant chaque contexte à un domaine, cette approche réduit la variabilité du gradient, favorisant la stabilité de l'apprentissage. Cela se traduit par une convergence plus rapide, comme le montre l'exemple d'AdaMatch, et des performances de généralisation améliorées sur l'ensemble des domaines (source et cible).

FIG. 3 -Variation de la variance du gradient sur le domaine cible (SVHN) pendant l'entraînement des deux modèles : AdaMatch et AdaMatch+CN. Graphique de gauche : Variation du gradient maximal. Graphique de droite : Variation du gradient moyen.

Conclusion

La normalisation contextuelle (CN) représente une avancée significative dans l'entraînement des réseaux de neurones. En partageant une philosophie similaire avec la normalisation par mélange (MN), CN élimine le besoin de l'estimation coûteuse de l'algorithme Espérance-Maximisation (EM) utilisée par MN, en apprenant directement les paramètres des distributions gaussiennes à partir de groupes de données nommés "contextes". Nos expériences ont démontré que CN surpassait à la fois la normalisation par lots (BN) et la normalisation par mélange (MN), offrant une meilleure convergence et des performances de généralisation améliorées. Trois méthodes ont été explorées pour définir les contextes : l'utilisation des composantes d'un modèle de mélange gaussien (GMM), l'identification des superclasses dans un ensemble de données, et l'application en adaptation de domaine en considérant chaque domaine comme un contexte distinct.

À court terme, nous visons à créer une variante non supervisée de CN en éliminant le besoin d'entrée du contexte, optant plutôt pour son estimation pendant l'apprentissage du réseau de neurones.

Références

Berthelot, D., R. Roelofs, K. Sohn, N. Carlini, et A. Kurakin (2021). Adamatch : A unified approach to semi-supervised learning and domain adaptation. arXiv preprint arXiv :2106.04732.

Dempster, A. P., N. M. Laird, et D. B. Rubin (1977). Maximum likelihood from incomplete data via the EM algorithm. JOURNAL OF THE ROYAL STATISTICAL SOCIETY, SERIES B 39(1), 1-38.

Algorithme 3 :

 3 CN-Inférence Entrée : Réseau de neurones N et tr CN avec les paramètres Θ ∪ {µ r , σ r } T r=1 (réf. Algorithme 2) ; B i = {x i } m i=1 ; choix ∈ {CN, CN+} ; Sortie : Réseau de neurones avec des paramètres gelés N et inf CN ; B i normalisé 1 N et inf CN ← N et tr CN // Initialisation du réseau de neurones pour l'inférence avec des paramètres gelés 2 si choix = CN alors 3 pour i ← 1 à m faire -Pour un contexte r donné -Prendre les paramètres µ r et σ r correspondants -Normaliser x i : xi ← CN(x i , r) 4 si choix = CN+ alors 5 pour i ← 1 à m faire -Calculer xi avec l'Équation 7 // Normalisation + agrégation 4 Expériences Dans cette section, nous comparons la normalisation contextuelle (CN) à d'autres méthodes, notamment la normalisation par lot (BN) et la normalisation par mélange (MN), sur différentes architectures, dont les Réseaux de Neurones Convolutifs (CNN) (voir Section 4.2 et Section 4.4) et les Transformeurs de Vision (ViT) Dosovitskiy et al. (2020) (voir Section 4.3). Nous évaluons ces approches sur diverses tâches, notamment la classification et l'adaptation de domaine.

  FIG. 1 -Evolution de l'erreur durant l'entraînement des modèles ConvNet+BN, ConvNet+MN et ConvNet+CN, sur les ensembles de données CIFAR-10, CIFAR-100 et Tiny ImageNet.

  FIG. 2 -Évolution de l'erreur sur les ensembles d'entraînement et de validation de CIFAR-100 pendant l'entraînement des modèles ViT+BN et ViT+CN.

  Dosovitskiy, A., L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, et al. (2020). An image is worth 16x16 words : Transformers for image recognition at scale. arXiv preprint arXiv :2010.11929.

  estimation de la variance Réseau de neurones N et tr CN normalisé 1 N et tr CN = N et // Initialisation des paramètres du réseau de neurones normalisé avec CN

4 xi = CN(x i , r) : Normaliser x i avec l'Équation 5 Algorithme 2 : CN-Entraînement : Normalisation d'une activation Entrée : Réseau de neurones N et avec des paramètres Θ ; B i = {x i } m i=1 ; Sortie : 2 pour i ← 1 à m faire -Identifier le contexte r associé à x i -Appliquer l'Algorithme 1 : xi = CN(x i , r) -Modifier chaque couche de N et tr CN en remplaçant x i par xi 3 Propager le gradient dans N et tr CN pour optimiser : Θ = Θ ∪ {µ r , σ r } T r=1 comme indiqué dans l'Algorithme 1. Dans l'algorithme 1, nous utilisons une transformation affine de la forme :

  TAB. 4 -Évaluation de la performance des modèles AdaMatch et AdaMatch+CN sur les données de validation des ensembles de données source (MNIST) et cible (SVHN). l'entraînement sur le domaine cible, nous enregistrons la variabilité du gradient. Une faible variance du gradient est un indicateur de la stabilité du réseau, ce qui peut accélérer la convergence et améliorer les performances globales. Sur la Figure3, on peut clairement observer que le modèle AdaMatch+CN affiche une variance du gradient considérablement réduite par rapport au modèle AdaMatch. Cela met en évidence le rôle bénéfique de la normalisation contextuelle (CN) dans la stabilisation du réseau en influençant le gradient, ce qui a un impact positif sur l'accélération de la convergence du modèle. Cet effet se traduit par une amélioration significative des performances, tant sur le domaine source (MNIST) que sur le domaine cible (SVHN), comme en témoigne la Table