C Boué

S Holé

Depth evaluation of curvilinear cracks in metal using multi-speed laser lock-in thermography method

Keywords: Infrared thermography, lock-in thermography, non-destructive testing, crack depth, crack sizing, curvilinear crack

An original method using lock-in thermography with a laser excitation is proposed for the contactless estimation of open crack depths in metal with curvilinear shape. A continuous laser source regularly scans the structure under test leading to a periodical heating. The heat diffusion disturbances induced by a crack located in the thermal diffusion area are measured synchronously with the laser scans. The thermal signature of the crack is extracted from the amplitude of surface temperature images for various scanning speeds of the thermal source. The thermal signatures are analysed according to a length representative of the thermal diffusion length and to the radius curvature of the crack to give a local evaluation of the crack depth. The method, explained with 3D simulations, is experimentally implemented and tested with calibrated curvilinear cracks. The results demonstrate the potentiality of multi-speed laser lock-in thermography method as a contactless measurement tool for the evaluation of complex crack shapes up to 3 mm depth.

Introduction

Mechanical stresses in metals can develop damaging cracks with complex profiles on surface and in depth. The active infrared thermography which analyses the disturbances of heat diffusion from a heat source is a non-contact and non-destructive technique widely used for the inspection of component [START_REF] Shifani | A Study of Methods using Image Processing Technique in Crack Detection[END_REF]- [START_REF] Barakat | A one-dimensional approach towards edge crack detection and mapping using eddy current thermography[END_REF]. However, the discontinuity of the heat diffusion which reveals the cracks depends also on parameters like the sample surface state, the crack width or the crack shape and the spatial configuration of the heating excitation [START_REF] Mohan | Crack detection using image processing: A critical review and analysis[END_REF]- [START_REF] Shi | Quantitative Detection of Cracks in Steel Using Eddy Current Pulsed Thermography[END_REF]. Consequently, it is difficult to evaluate the crack depth, one of the most important parameters to assess the dangerousness of the crack, even for simple shape like linear cracks.

In [START_REF] Boué | Open crack depth sizing by multi-speed continuous laser stimulated lock-in thermography[END_REF] and [START_REF] Boué | Comparison between multi-frequency and multi-speed laser lock-in thermography methods for the evaluation of crack depths in metal[END_REF] an original method using lock-in thermography is presented to directly evaluate linear open crack depths without requiring any calibration procedure. A moving punctual continuous heating spot is used as a 3D probe of the crack depth, adjusting its thermal diffusion length by its moving speed. At the same time, an infrared camera acquires surface temperature images in synchronism with the heating spot displacements. The evolution of the spatial derivatives of the amplitude images obtained from the surface temperature for different speeds is representative of the crack depth. It has been chosen in [START_REF] Boué | Comparison between multi-frequency and multi-speed laser lock-in thermography methods for the evaluation of crack depths in metal[END_REF] to analyse the thermal response of an area back to the crack, in its "shadow". This robust method is validated for simple linear crack shape but more complex crack shapes which represent an industrial issue must be analysed.

In this work, the method described in [START_REF] Boué | Comparison between multi-frequency and multi-speed laser lock-in thermography methods for the evaluation of crack depths in metal[END_REF] is adapted to crack depth assessment for curvilinear geometry. Results from 3D Finite Element Method (FEM) simulation with COMSOL Multiphysics are used to introduce the adjustments required on crack indicators to take into account the crack curvature. These new developments applied for linear and curvilinear crack shapes are presented and analysed using the multi-speed laser lock-in thermography. They are then experimentally validated using thermo-erosion cracks of different curvature in steel samples.

Multi-speed laser lock-in thermography method

Fig. 1: experimental multi-speed lock-in thermography set-up

A continuous heating laser spot regularly moves along the crack from one end to the other on a distance L with speed v (Fig. 1). At the end of the path, the heating source is switched off to return back to its initial position as quickly as possible and restarts (switched on) its movement along the crack. Each heated point sees the thermal source periodically (a few periods) during a short time compared to the duration L/v. The thermal source moves in this way for several speeds (usually 5 speeds are sufficient). The heating source has to be located near the crack so that the crack acts as a thermal barrier which modifies locally the heat diffusion. In the case of non-linear crack shapes, the surface temperature distribution, which depends on the thermal diffusivity D of the studied material, is also sensitive to the crack shape curvature. Indeed, the heating spot can be on the concave or convex side of the crack. Consequently, the different speeds of the heating source along the crack must be correctly chosen. The thermal diffusion length µ used as a 3D scan parameter is:

π ∆ = π = µ t D v DL , ( 1
)
where ∆t is the source displacement duration for speed v for one scan.

The acquisition process of the infra-red camera is synchronized by an external trigger signal (Fig. 1) that is activated when the laser spot returns to its initial position. The camera lock-in module generates the amplitude and phase images of the surface temperature at the trigger frequency. In the absence of a lock-in module, for instance in the case of simulations, the amplitude and phase images are calculated by Fast Fourier Transform (FFT) of a series of surface images acquired during one scan of the heat source. One amplitude and one phase images are then extracted from the first harmonic (f0=1/∆t) of the FFT calculation for each heat source displacement speed.

The crack signature is extracted from the amplitude image Ash of an area located in the crack shadow area where heat diffusion and thus the surface temperature on the sample have been impaired after bypassing the crack (Fig. 2). The maximum value µmax of µ has to be chosen larger than the distance d between the heat spot and the crack.

Fig. 2: sample model in cross section (y,z directions).

The heat source is located on the left side of the crack and the crack shadow is located on the right side of the crack. d is the distance from the heat source to the crack and h is the crack depth.

In the procedure described in [START_REF] Boué | Comparison between multi-frequency and multi-speed laser lock-in thermography methods for the evaluation of crack depths in metal[END_REF], a depth indicator is obtained from the ratio between the temperature in the crack shadow area and a reference temperature measured symmetrically to the heat path which corresponds indeed to the temperature expected in the crack shadow area in case of the absence of the crack. The depth indicator has a poor dependency on the crack width if it is sufficiently small compared to crack depth [START_REF] Boué | Open crack depth sizing by multi-speed continuous laser stimulated lock-in thermography[END_REF][START_REF] Boué | Comparison between multi-frequency and multi-speed laser lock-in thermography methods for the evaluation of crack depths in metal[END_REF]. In the case of curved cracks, the reference and the shadow points must be determined according to geometrical considerations. Fig. 3 illustrates the two geometrical possibilities depending on whether the heating spot is in the concave or convex side of the crack path. Tetrahedral meshing is adapted to the different domains (bulk, crack) of the studied samples for the 3D Finite Element Method (FEM) simulations made with COMSOL Multiphysics. The number of mesh elements depends on the size of the crack (depth and width). It is between 250 000 and 800 000 for depths between 0.5 mm to 3 mm and for width of 200 µm or 50 µm. 100 surface temperature images are calculated during a scan and 5 scans are performed with 5 different speeds. Fig. 4a shows a typical mesh in (x,y,z) directions in case of a curvilinear crack of 20-mm curvature radius and 2mm crack depth. Fig. 4b represents the calculated infrared amplitude image A using a continuous moving heat spot (∆t = 16 s and L = 55 mm). The curved crack footprint is clearly visible at the right of the moving heat source. Each crack pixel P extracted from the spatial derivative of the amplitude image is associated with 3 aligned pixels: its local heat source which is its nearest hottest pixel, its reference and its shadow pixels. The crack shadow pixel is chosen at a distance ∆dsh of a few pixels ∆psh from P in the shadow area. The reference pixel is at the distance d + ∆dsh from the local heat source. Fig. 4c shows the localizations of the maximum heating positions (in green), the crack (in blue), the reference area (in black) and the shadow area for ∆psh = 4 pixels (in red). ∆psh value can be chosen as convenient to get rid of the pixel noise at the crack positions and to obtain a sufficient signal. The shadow amplitude Ash(P) and the reference amplitude A0(P) of the selected pixels P are defined for each speed values. Ash(P) depends on the crack geometry, the heating area and the speed of the heating spot whreas A0(P) does not depend on the crack depth.

Crack indicators for depth evaluation

Four samples with a curvilinear crack of 20-mm radius and with a constant depth (0.5 mm, 1 mm, 2 mm, 3 mm) are simulated. The shadow amplitude Ash(P) and the reference amplitude A0(P) of selected pixels are extracted from the calculated amplitude images obtained for different speeds. Fig. 5 shows Ash/A0 curves (5 speeds) and their polynomial fits h f as a function of normalized thermal diffusion length µ/µmax. µmax is the maximum value of the chosen thermal diffusion lengths µ. Each depth is identified by a well differentiated curve (color). The line bundle obtained for each depth in the simulations is mainly a consequence of the uncertainty of the calculated temperatures resulting from the meshing and the time sampling.

It was chosen in [START_REF] Boué | Comparison between multi-frequency and multi-speed laser lock-in thermography methods for the evaluation of crack depths in metal[END_REF] to extract from the fits two specific points called IA and IB indicators. The indicator IA, which can be defined as IA = 1-(Ash/A0)max for µ=µmax represents the thermal energy measured behind the crack at long time.

The indicator IB represents the reduced thermal diffusion length at which the heat begins to bypass the crack. IB corresponds to the value µ/µmax when fh = s, s being generally chosen as a decreasing linear function of µ/µmax,; here, s=1-µ/µmax. Fig. 5: Ash/A0 and their fits as a function of µ/µmax for simulated depth values h = 0.5 mm (in red), h = 1 mm (in blue), h = 2 mm (in yellow), h = 3 mm (in green) for 10 selected pixels P. The abscissa of the intersection of A sh /A 0 and the line s(µ/µmax ) in black gives the indicator IB for the crack depth. The value 1 -(Ash/A0)max at µ/µmax = 1 gives the indicator IA for the crack depth. Simulated parameters: maximum of the thermal diffusion length is µmax = 4.7 mm; number of pixels for the probing area definition is ∆psh = 4 pixels; Finally, to improve the robustness of the depth evaluation, the global crack indicator is calculated with:

I = IA*IB (2)
Fig. 6 presents the indicator I calculated with (4) as a function of the x pixel position of 300 selected pixels along the crack for 4 constant depths. (The crack extremity pixels are not considered). These curves show the good sensitivity of indicator I to the crack depth although the indicator seems to slightly depend on the crack position, more importantly for larger depth.

Geometry considerations for depth evaluation in case of curvilinear crack shapes

In order to assess the impact of the crack curvature, the multi-speed laser lock-in thermography method is applied to various curvilinear crack shapes. The presented simulated results are calculated using a moving continuous heat spot along a curved crack shape with a constant crack depth h. In these simulations, the crack width is chosen equal to the presented experimental artificial crack width (200 µm).

Localisation of the heating spot with respect to crack curvature

In case of a curvilinear curve, the heating spot can be positioned at a distance d on the convex or on the concave crack side. Fig. 7 shows the simulated amplitude images for a 20 mm radius curved crack. In Fig. 8, Ash (in blue) and A0 (in red) are extracted along the x direction for 5 heating spot speed (∆t between 2 s to 16 s) for the selected pixels along the crack for the concave (Fig. 8a) or the convex (Fig. 8b) heating configuration. In the concave configuration (Fig. 8a), A0 is higher and Ash is slightly lower than in the convex configuration (Fig. 8b). Consequently, Ash/A0 is lower for the concave configuration. Consequently, the resulting I indicator is higher if the heating spot is in concave configuration compared to that obtained in convex configuration. The model might not be suitable when the heating spot is in convex configuration, for example when Ash/A0 is near or greater than 1.

I indicators obtained for the concave and convex configurations are presented Fig. 9 for two d distances. Calculated I obtained for the heating spot in a concave configuration (in blue) leads to better differentiation of depths. In the convex configuration (in red), I indicator loses sensitivity for small crack depth and greatly depends on distance d compared to the concave configuration.

(a) (b) Fig. 9: indicator I (a.u.) as a function of x position (pixel) for curvilinear crack shapes for 4 crack depths (h = 0.5 mm, h = 1 mm, h = 2 mm, h = 3 mm) and for concave configuration (in blue) and convex configuration (in red).

u.) as a function of simulated crack depth (mm) for a 20 mm curvilinear crack shapes (heating spot in a concave configuration (in blue) and in a convex configuration (in brown)). The linear line is the fitting curve for the linear crack response (in green). The distance d = 2 mm.

For a distance of 2 mm (Fig. 10), I obtained in the convex heating position (in brown) are underestimated compared to those obtained for a linear configuration (in green). Conversely, I obtained for the concave heating position (in blue) are overestimated.

Radius of the curvilinear crack

The surface amplitude is simulated with different crack curvatures (R = 10 mm, R = 20 mm, infinity R). 9) are overestimated compared to those obtained for a linear crack. Furthermore, the indicator values are higher for the concave heating compared to those obtained with the convex heating. These observations show the I dependence to the crack geometry in x,y directions and that a geometrical correction has to be taking into account.

Fig. 11: indicator I (a.u.) for linear and curvilinear crack shapes as a function of crack depths (mm) for different heating configuration (convex or concave). The linear line is the fitting curve for the linear crack response. The crack radius is R = 20 mm or R = 10 mm, the distance is d = 1.25 mm.

The results obtained for the concave configuration better differentiate the different depths. Thus they are used afterwards to determine a basic geometrical correction. The chosen geometric factor takes into account the radius of curvature of the heated area (R-d) as well as the distance between the heating area and the shadow area (d+dsh). The most suitable geometric model is the square function. For a large radius of curvature, the model tends to the linear model.

The depth evaluation h ev is deduced from the indicator I using only the concave results by: [START_REF] Netzelmann | Induction thermography: principle, applications and first steps towards standardisation[END_REF] where G is the correction deduced from the slope of the linear response (line in Fig. 11). Since the indicator depends little on the crack opening [START_REF] Boué | Comparison between multi-frequency and multi-speed laser lock-in thermography methods for the evaluation of crack depths in metal[END_REF], G is virtually independent of the crack width. Fig. 12 shows the evaluated depth obtained using [START_REF] Ciampa | Recent Advances in Active Infrared Thermography for Non-Destructive Testing of Aerospace Components[END_REF]. The G value deduced from the linear (infinity R) response is equal to 5 for d = 1.25 mm and 6 for d = 2 mm for material whose thermal diffusivity is close to the one of steel. These simulated results obtained for curvilinear crack (R = 10 mm and R = 20 mm) validate the method for curvilinear cracks. The proposed geometrical correction for the depth indicators is efficient if the heating spot is localised in the crack concave area.

Experimental results

Experimental tests were carried out to access to the depth evaluation of artificial vertical open cracks made by electro-erosion in steel cubes. Four steel samples have a curvilinear crack of radius 20 mm with a constant depth (0.5 mm, 1 mm, 2 mm, 3 mm). One sample is a steel block with a linear crack with no constant 0-4 mm depth. The measured thermal diffusivity of the plate [START_REF] Oswald-Tranta | Time and frequency behaviour in TSR and PPT evaluation for flash thermography[END_REF] is equal to 4.5 ± 0.3 10 -6 m 2 s -1 . The shape of the cracks is necessary to correctly position the laser spot and to determine the local radius of curvature necessary for the geometric correction. In this study, the crack shapes are well identified. A prior localization step is therefore not necessary. In less favorable cases, the image of the crack location obtained without optimizing the position of the laser [START_REF] Fedala | Infrared lock-in thermography crack localization on metallic surfaces for industrial diagnosis[END_REF], provides the necessary information both to adjust the trajectory of the laser spot and to determine the local radius of curvature of the crack.

Using the multi-speed laser lock-in method, the experimental results are obtained from 5 amplitude infrared images of 5 laser displacement speeds (L scanned in 2 s to 16 s). The experimental parameters (speed v and distance d) depend on the sample thermal diffusivity and on the power of the thermal source. The thermal diffusion length µ is obtained by [START_REF] Shifani | A Study of Methods using Image Processing Technique in Crack Detection[END_REF]. The thermal sensitivity of the method is better for low speed which induces maximum thermal length µmax, but the synchronous acquisition process may shift near DC measurements. Fig. 13 to 15 show the amplitude images obtained for a sample with a curvilinear crack and for the sample with a linear crack. The laser is positioned in the concave area of the crack curve (Fig. 13), in the convex area (Fig. 14) or near the linear crack (Fig. 15). Fig. 13b, Fig. 14b and Fig. 15b show the localization of the crack pixels, the hottest, normalized and shadow pixels extracted from measured amplitude image (Fig. 13a, Fig. 14b and Fig. 15b). The presented process is first validated using the linear crack with a non-constant depth. The I indicator responses calculated using (2) are presented Fig. 16. The G value deduced from the slope of the line is equal to 6 which corresponds more or less to the one obtained by simulation. 17 shows the Ash/A0 curves (5 measurements and their fits) for the four steel samples having a curvilinear crack from which are extracted the indicators IA and IB. Fig. 17a, the heating laser spot in in concave area: the depth information is relevant. Fig. 17b, the heating laser spot in the convex area: the depth information is lost for shallow depths. With a 2W continuous laser power and a high sensitivity infrared camera, the multi-speed laser lock-in method allows to evaluate 0-3 mm crack depth for linear crack as well as for curvilinear crack. The total camera acquisition duration is around 2 min for 5 multi-speeds lock-in method.

Conclusion

The multi-speed lock-in infrared thermography method is an experimental method which allows to evaluate linear and curvilinear crack depth, independently of their width.

The evolution of the depth indicator is analysed by simulations for curvilinear crack. The simulation results allow to propose a simple geometrical correction to take into account the crack curvature in the calculations of depth evaluations. Measurements with an infrared camera obtained with linear and controlled curvilinear "cracks" in steel blocks validate the proposed indicator for curvilinear shapes. The geometrical correction is made on simple cases of curvilinear cracks with constant radius of curvature. For more complex geometries, it is then necessary to take into account local radii of curvature. Taking into account the shape of the crack allows to study cracks other than linear.

The multi-speed lock-in infrared thermography method evaluates cracks of a few centimetre long. This method is indicated for the investigation of curvilinear geometries as long as the displacement of a laser beam onto the surface is possible. No surface preparation and no calibration procedure are required. The depth evaluation method is nonpolluting, non-destructive and is implemented with simple optical adjustments.

Declarations: to be used for all articles, including articles with biological applications Funding: not applicable Conflicts of interest/Competing interests: not applicable

Fig. 3 :

 3 Fig. 3: top view of the sample model (x,y directions). The heating spot is located (a) on concave side or (b) on convex side of the crack path. The crack shadow is located behind the crack on a line perpendicular to the crack path and passing through the heating spot. The reference point is located on the same line, symmetrically to the crack shadow point with respect to the heating spot.

Fig. 4 :

 4 (a) surface mesh in (x,y,z) directions used for FEM simulations in case of curvilinear crack, (b) simulated amplitude images A (arbitrary units a.u.), (c) Localisation image of crack pixels (in blue), hottest (in green), reference (in black) and shadow pixels (in red) extracted from simulated amplitude image. 1 pixel represents 100 µm x 100 µm. Simulated crack h = 2 mm, width = 200µm, L= 55 mm. Distance of the heat source to the crack d = 1.25 mm, radius of the heat source r = 0.25 mm, thermal diffusivity D = 4.5e -6 m 2 /s, displacement duration of moving continuous heat source ∆t = 16 s, number of pixels for the probing area definition ∆psh = 4 pixels.

Fig. 6 :

 6 Fig. 6: Indicator I (arbitrary unit) along the crack (pixels in x direction) of a curvilinear crack (radius 20 mm) for 4 depth h values (0.5 mm in red, 1 mm in blue, 2 mm in yellow, 3 mm in green). The heating displacement is in the concave area. 1 pixel represents 100 µm.

Fig. 7 :

 7 Infrared simulated amplitude image A. (a): concave configuration, (b): convex configuration. The simulated depth is h = 2mm. 1 pixel represents 100 µm * 100 µm. The crack curvature is 20 mm. The distance of the heat source to the crack is d = 1.25 mm. The number of pixels for the probing area definition is ∆psh = 4 pixels. The thermal diffusivity is 4.5 e -6 m 2 /s.

Fig. 8 :

 8 Ash (in blue) and A0 (in red) (arbitrary units) as a function of position x, extracted from simulated amplitude images obtained for 5 spot speeds. (a): heating spot in the concave area. (b): heating spot in the convex area. Simulated depth h = 2mm. 1 pixel in x direction represents 100µm. The distance of the heat source to the crack is d = 1.25 mm. The crack curvature is 20 mm. The number of pixels for the probing area definition is ∆psh = 4 pixels. The thermal diffusivity is 4.5 e -

  (a) distance d = 1.25 mm. (b) distance d = 2 mm.

Fig. 10

 10 Fig. 10 presents more specifically the concave and convex mean responses with d = 2 mm which is the distance of the experimental results presented thereafter.

Fig. 10 :

 10 Fig. 10: indicator I (a.u.) as a function of simulated crack depth (mm) for a 20 mm curvilinear crack shapes (heating spot in a concave configuration (in blue) and in a convex configuration (in brown)). The linear line is the fitting curve for the linear crack response (in green). The distance d = 2 mm.

  Fig. 11 compares the I indicator obtained from linear (infinity R) and curvilinear heating displacements (which amounts to saying linear or curvilinear crack shapes) obtained for d = 1.25 mm. Fig 11 shows that the I values obtained for the curved crack (extracted from Fig.

Fig. 12 :

 12 Fig. 12: depth evaluation (mm) as a function of simulated h (mm) for the non-constant linear crack and for curved constant cracks (R=10 mm and R=20 mm) with G = 5 for d = 1.25 mm and G = 6 for d = 2 mm. The black line corresponds to the expected response.

Fig. 13 :Fig. 14 :Fig. 15 :

 131415 Fig. 13: heating displacement along the concave side of the curvilinear crack, (a) amplitude images A (arbitrary units a.u.), (b) localisation of crack pixels (in blue), hottest (in green), reference pixels (in black) and shadow pixels (in red) extracted from measured amplitude image. 1 pixel represents 100 µm x 100 µm. The crack width is 200 µm and crack depth is h = 2 mm. Thermal diffusivity is D = 4.5 e -6 m 2 /s. Displacement duration of moving continuous heat source is ∆t = 16 s. Number of pixels for the probing area definition is ∆psh = 4 pixels.

Fig. 16 :

 16 Fig. 16: indicator I (a.u.) as a function of x displacement (mm) for a non-constant 0.5-3.5 mm crack depth. The black line corresponds to the linear response. Laser power P = 2 W. Studied length along the crack ∆x = 30 mm.

Fig.

  Fig.17shows the Ash/A0 curves (5 measurements and their fits) for the four steel samples having a curvilinear crack from which are extracted the indicators IA and IB. Fig.17a, the heating laser spot in in concave area: the depth information is relevant. Fig.17b, the heating laser spot in the convex area: the depth information is lost for shallow depths.

Fig. 17 :

 17 Fig. 17: Ash/A0 measurements and their fits (a.u.) as a function of µ/µmax for crack depth values h = 0.5 mm (in red), h = 1 mm (in blue), h = 2 mm (in yellow), h = 3 mm (in green) for 40 selected P pixels. The intersection of an Ash/A0 curve and the line s(µ/µmax) in black gives the IB indicator of the crack depth. The value 1 -(Ash/A0)max at µ/µmax = 1 gives the IA indicator of the crack depth. (a): concave configuration, (b): convex configuration. Maximum of the thermal diffusion length µmax = 4.8 mm. Number of pixels for the probing area definition ∆psh = 4 pixels. Laser power 2W. The post processing procedure extracts IA and IB indicators from Fig. 17a to calculate I indicator following (2) (Fig. 18).

Fig. 18 :

 18 Fig. 18: indicator I (a.u.) along the crack (x direction) of curvilinear crack for four h values (0.5 mm in red, 1 mm in blue, 2 mm in yellow, 3 mm in green) in case of a concave configuration of the heating spot.

Fig. 19 :

 19 Fig. 19: crack depth evaluation (mm) as a function of h (mm) for the non-constant linear crack (4 measurements) and for the curvilinear constant depth electro erosion "cracks" with a heating spot displacement on the concave side of the crack. Width of the cracks 200 µm, G = 6. The black line corresponds to the expected response.

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Alfred Galichon

THE VAR AT RISK

Keywords: value-at-risk MSC 2000 subject classification: 91B06, 91B30, 90C08

firm is not neutral in respect to regulatory capital budgeted under rules which are based on the Value-at-Risk.

Introduction. The single most used measure of financial risk is undoubtedly the Valueat-Risk (VaR). The VaR at level 95% is defined as the minimal amount of capital which is required to cover the losses in 95% of cases. In statistical terms, the value-at-risk is the quantile of level α of the losses, namely V aR α (X) = inf {x : Pr (X ≤ x) > α} .

(note that unlike the most widely adopted convention in the literature, we chose to count positively an effective loss).

The widespread popularity of the Value-at-Risk is due to its adoption as a the "1st pillar" in the Basle II agreements. Despite its widespread use and simplicity, the Value-at-Risk is highly criticized among academics. The literature on risk measures classically defines a set of axioms which satisfactory risk measures should satisfy (see [START_REF] Artzner | Coherent measures of risk[END_REF]). Among these, subadditivity: a risk measure ρ should satisfy ρ (X + Y ) ≤ ρ (X) + ρ (Y ). While this axiom Rochet for a stimulating discussion and encouragements. Support from Chaire EDF-Calyon "Finance and Développement Durable," Chaire Société Générale "Risques Financiers" and Chaire Axa "Assurance et Risques Majeurs" is greatly acknowledged.

1 is widely accepted in the academic community, it is perhaps ironical that VaR fails to satisfy this subadditivity axiom as it has been widely documented (we come back to that below).

The subadditivity axiom is generally motivated by a loose invocation of risk aversion, or preference for diversification. In this note I propose a quite different argument to motivate the importance of the subadditivity axiom. Assuming the value-at-risk is used to budget regulatory capital requirements, and assuming that the managers have an incentive to minimize these capital requirements, I show that the lack of the subadditivity property induces the possibility for the management to optimally divide their risk in order to minimize their budgeted capital: the structure of the firm is not neutral to the aggregated capital requirement. More precisely, I show that for any level of the value-at-risk there is a division of the risk which sets the aggregated capital requirement to zero.

The problem. Consider a trading floor which is organized into N various desks. For each trading desk i = 1, ..., N , call X i the random variable of the contingent loss of trading floor i. The total random loss of the trading floor is X = N i=1 X i , we suppose that this amount is bounded: X ∈ [0, M ] almost surely.

We suppose that each desk budgets a regulatory capital equal to its VaR at level α ∈ (0, 1), V aR α (X i ). Consequently the total amount of regulatory capital which the management needs to budget is N i=1 V aR α (X i ). It is therefore easy to formulate the manager's problem:

inf N i=1 V aR α (X i ) s.t. N i=1
X i = X a.s., and V aR α (X i ) ≥ 0.

As the management has full control over the structure of the firm, it results that it has the choice over N and over the random variables (X 1 , ..., X N ) which satisfies the constraints.

Note that the economical risk which the trading floor bears is X = N i=1 X i , and thus the regulatory capital to be budgeted should be V aR α (X) > 0. However, I shall show that, under mild assumptions, there is a structure of the firm such that the capital budgeted under the rule above is zero. The assumption needed is the following:

Assumption. The distribution of X is absolutely continuous with respect to the Lebesgue measure.

The optimal structure. In that case, there exists a sequence of real numbers x 0 = 0 < x 1 < ... < x N = M such that Pr (X ∈ (x i , x i+1 )) < 1 -α for all i = 1, ..., N -1.

We are therefore going to consider the digital options 1

X i = X1 {X ∈ (x i , x i+1 )}
(note that these options can be approximated by linear combinations of standard calls and puts, or butterfly options). We have

N i=1 X i = X.

As we have Pr (X

i > 0) < 1 -α, it follows that V aR α (X i ) = 0.
Therefore the capital to be budgeted under this structure of the firm is the sum of the capital to be budgeted for the different trading desks, which is zero.

Remarks.

1. This example shows that it is possible to tear down the risk into small pieces which are indetectable by the Value-at-Risk.

2. Note that the case we have here, which is the worse case (at least for the regulator's point of view), is the case where all the X i 's are comonotonic. Comonotonicity indeeds turns out to be the regulator's worse case, which justifies the comonotonic additivity axiom put forward a large literature: see [START_REF] Kusuoka | On law invariant coherent risk measures[END_REF].

3. The above considerations have lead [START_REF] Galichon | Comonotonic measures of multivariate risks[END_REF] to define the axiom of strong coherence, which is a natural requirement so that the structure of the firm be neutral to risk measurement.

In that paper, strong coherence is shown to be equivalent to the classical risk measures axioms.

4.

The assumption that the distribution of the risk X be absolutely continuous is crucial.

There are connections to be explored with the theory of portfolio diversification under thick-tailedness, see [START_REF] Ibragimov | Portfolio diversification and value at risk under thick-tailedness[END_REF].

Possible extension. It would be interesting to consider the above problem with overhead costs, namely when dividing the firm into several desks is costly. The problem would then become inf N i=1 V aR α (X i ) + α (N ) s.t.

N i=1

X i = X a.s., and V aR α (X i ) ≥ 0.

where α (N ) is increasing with N and can be interepreted as an a priori penalization of the firm's complexity by the regulator. The problem of determining an optimal α (N ) from the point of view of the regulator in a properly defined setting is of interest. † École polytechnique, Department of Economics. E-mail: alfred.galichon@polytechnique.edu

Date:

  First version is June 3, 2008. The present version is of May 18, 2009. Correspondence address: Département d'Economie, Ecole polytechnique, 91128 Palaiseau, France. The author thanks Jean-Charles

1

  this is a slight abuse of terminology as a digital option would more correctly characterize xi1 {X ∈ (xi, xi+1)}, but both expressions are close when xi and xi+1 are close.


Martin Dornier
email: martin.dornier@interdigital.com

Philippe-Henri Gosselin

Christian Raymond

Yann Ricquebourg

Bertrand B Coüasnon

Yann Ricque- Bourg

StyleGAN-based heatmap generator

Keywords: face alignment, semi-supervised training, active learning, transfer learning

may come

Introduction

The success of Deep Learning comes mainly from its ability to automatically learn optimal features for the target task instead of relying on hand-crafted features such as SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] or HOG [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]. A subdomain of Machine Learning is Representation Learning; the goal, in this case, is to learn useful features, e.g., interpretable or that can be used for transfer learning.

One application of representation learning is reducing the number of annotated samples needed to train a model on a downstream discriminative task, such as classification or regression. Popular representation learning methods make use of auto-encoders [START_REF] Browatzki | 3fabrec: Fast few-shot face alignment by reconstruction[END_REF][START_REF] Higgins | beta-vae: Learning basic visual concepts with a constrained variational framework[END_REF][START_REF] Kim | Disentangling by factorising[END_REF], contrastive learning [START_REF] Caron | Unsupervised learning of visual features by contrasting cluster assignments[END_REF][START_REF] Chen | A simple framework for contrastive learning of visual representations[END_REF][START_REF] Zheng | General facial representation learning in a visual-linguistic manner[END_REF], or masked inputs [START_REF] Devlin | BERT: Pre-training of deep bidirectional transformers for language understanding[END_REF][START_REF] He | Masked autoencoders are scalable vision learners[END_REF]. While some methods also use Generative Adversarial Networks (GANs) [START_REF] Chen | Infogan: Interpretable representation learning by information maximizing generative adversarial nets[END_REF][START_REF] Donahue | Adversarial feature learning[END_REF][START_REF] Dumoulin | Adversarially learned inference[END_REF][START_REF] Xu | Generative hierarchical features from synthesizing images[END_REF][START_REF] Nitzan | Large: Latent-based regression through gan semantics[END_REF], we think that this idea is very promising and has not been fully explored, especially with the newest GANs architectures.

Among GANs, StyleGAN is a popular approach thanks to its disentangled latent space that can be distilled into multiple styles [START_REF] Karras | A style-based generator architecture for generative adversarial networks[END_REF][START_REF] Karras | Analyzing and improving the image quality of stylegan[END_REF][START_REF] Karras | Alias-free generative adversarial networks[END_REF]. Each style controls a semantic characteristic of the image such as color scheme, object shape, or background. Because of its lack of an inference module, many approaches propose to invert StyleGAN in order to project real images into its latent space and perform semantic editing of these images [START_REF] Alaluf | Restyle: A residual-based stylegan encoder via iterative refinement[END_REF][START_REF] Gu | Image processing using multi-code gan prior[END_REF][START_REF] Kang | Gan inversion for out-of-range images with geometric transformations[END_REF][START_REF] Nitzan | Large: Latent-based regression through gan semantics[END_REF][START_REF] Richardson | Encoding in style: a stylegan encoder for image-to-image translation[END_REF][START_REF] Tov | Designing an encoder for stylegan image manipulation[END_REF][START_REF] Wang | High-fidelity gan inversion for image attribute editing[END_REF][START_REF] Xu | Generative hierarchical features from synthesizing images[END_REF][START_REF] Yao | Feature-style encoder for style-based gan inversion[END_REF]. However, most of these works focus on image-to-image translation tasks, such as attribute editability, superresolution, or inpainting. While some methods have studied the use of Style-GAN for regression tasks [START_REF] Xu | Generative hierarchical features from synthesizing images[END_REF][START_REF] Nitzan | Large: Latent-based regression through gan semantics[END_REF], they usually focus on semantic attribute regression such as face pose or age estimation [START_REF] Nitzan | Large: Latent-based regression through gan semantics[END_REF] or don't evaluate in depth how well their method performs compared to modern supervised methods or semi-supervised methods in the setting of limited training data.

In this work, we study the use of StyleGAN for a challenging regression task: facial landmark prediction (also called face alignment) with limited training data. Face alignment tries to localize some pre-defined facial anatomical keypoints (such as the corners of the mouth, the eyes, the boundaries of the face, ...) in a face image. Many downstream tasks rely on the predicted landmarks such as face swapping or facial expression recognition.

Through the years, the performance of face alignment models has increased, especially since the rise of Deep Learning, although most of the recent improvements come more from specifically designed training schemes like complex training losses [START_REF] Wang | Adaptive wing loss for robust face alignment via heatmap regression[END_REF][START_REF] Huang | Adnet: Leveraging error-bias towards normal direction in face alignment[END_REF] rather than better network architectures. Also, because labeling facial keypoints is time-consuming and can be challenging on images with large poses or occlusions, non-synthetic facial landmark datasets are usually relatively small (a few thousand samples) compared to other computer vision tasks such as image classification, making the trained models prone to overfitting. In this paper, we study whether it is possible to train a facial landmark detector with only a few samples and still get good accuracy and generalization. To tackle this challenge, we propose a new architecture. We transform a pre-trained StyleGAN • We study the use of active learning to improve our results even further.

The rest of our paper is organized as follows: first, we present in Section 2 the StyleGAN architecture and existing works on StyleGAN inversion. We also introduce the face alignment task and sum up methods dealing with limited training data for this task. For the last part of this section, we explain the active learning principle and list several existing methods and applications. In Section 3, we present our proposed framework for face alignment with limited training data. Section 4 reports the results of our different experiments. Finally, Section 5 concludes this paper.

Related Work

This section relates works related to our method, based on StyleGAN, and its purpose: face alignment with limited training data. The active learning principle is also explained and different existing methods are presented.

StyleGAN

Generative Adversarial Networks (GANs) [START_REF] Goodfellow | Generative adversarial nets[END_REF] have greatly improved the quality of image generation over the last few years. StyleGAN [START_REF] Karras | A style-based generator architecture for generative adversarial networks[END_REF] differs from previous GAN architectures by its generative process. Instead of starting from Gaussian noise z ∈ Z (the latent representation) and progressively increasing the spatial dimensions through the network layers, z is first projected to an intermediate latent space W via a non-linear mapping network f : Z → W which produces an intermediate latent code w ∈ W. The input of the generator is a constant learned vector c1 and at each layer, w is transformed by an affine transformation (different for each layer i) into a style vector y i and injected into the current feature map via an AdaIn [START_REF] Huang | Arbitrary style transfer in real-time with adaptive instance normalization[END_REF] operation. Each style vector controls a specific aspect of the generated image, style vectors corresponding to low-resolutions control high-level attributes such as, for face images, pose, face shape, or hairstyle while high-resolution style vectors control fine-grained aspects such as the color scheme or microstructure.

StyleGAN Inversion

To semantically edit a synthetic image generated by StyleGAN, we can modify some of its style vectors. To modify in the same way a real image, we first need to approximate its StyleGAN latent vector, this is called StyleGAN inversion. StyleGAN inversion methods are divided into three families. The optimization-based methods iteratively refine a latent code by minimizing the reconstruction error [START_REF] Gu | Image processing using multi-code gan prior[END_REF][START_REF] Kang | Gan inversion for out-of-range images with geometric transformations[END_REF]. The encoder-based methods train an encoder to predict the latent code [START_REF] Nitzan | Large: Latent-based regression through gan semantics[END_REF][START_REF] Richardson | Encoding in style: a stylegan encoder for image-to-image translation[END_REF][START_REF] Tov | Designing an encoder for stylegan image manipulation[END_REF][START_REF] Wang | High-fidelity gan inversion for image attribute editing[END_REF][START_REF] Xu | Generative hierarchical features from synthesizing images[END_REF][START_REF] Yao | Feature-style encoder for style-based gan inversion[END_REF]. Finally, hybrid methods train an encoder to predict an initial latent code which is refined through optimization [START_REF] Alaluf | Restyle: A residual-based stylegan encoder via iterative refinement[END_REF]. Optimization-based and hybrid-based methods usually have better reconstruction errors but are much slower than encoder-based.

Rather than predicting the true latent code z ∈ Z or the intermediate latent code w ∈ W, most methods predict a code for each style: w + = (w 1 , w 2 , ..., w n ) ∈ W + , n being the number of styles [START_REF] Nitzan | Large: Latent-based regression through gan semantics[END_REF][START_REF] Richardson | Encoding in style: a stylegan encoder for image-to-image translation[END_REF][START_REF] Tov | Designing an encoder for stylegan image manipulation[END_REF][START_REF] Wang | High-fidelity gan inversion for image attribute editing[END_REF]. This gives more flexibility and improves the reconstruction error. Some methods predict along w + a feature map f ∈ F which replaces the first layers of the generator [START_REF] Kang | Gan inversion for out-of-range images with geometric transformations[END_REF][START_REF] Yao | Feature-style encoder for style-based gan inversion[END_REF]. This feature map improves the reconstruction error but also makes it possible to encode images that do not follow the training dataset alignment (e.g., FFHQ [START_REF] Karras | A style-based generator architecture for generative adversarial networks[END_REF]) and Celeba-HQ [START_REF] Karras | Progressive growing of gans for improved quality, stability, and variation[END_REF] always have the face centered in the image and eyes at the same level). For example, translated or rotated images can still be faithfully encoded and reconstructed.

StyleGAN for regression

While the most common use cases for StyleGAN inversion are image-toimage translation tasks, such as attribute edition or super-resolution [START_REF] Alaluf | Restyle: A residual-based stylegan encoder via iterative refinement[END_REF][START_REF] Gu | Image processing using multi-code gan prior[END_REF][START_REF] Kang | Gan inversion for out-of-range images with geometric transformations[END_REF][START_REF] Richardson | Encoding in style: a stylegan encoder for image-to-image translation[END_REF][START_REF] Tov | Designing an encoder for stylegan image manipulation[END_REF][START_REF] Wang | High-fidelity gan inversion for image attribute editing[END_REF][START_REF] Yao | Feature-style encoder for style-based gan inversion[END_REF], it can also be used for discriminative tasks like classification or regression. GHFeat [START_REF] Xu | Generative hierarchical features from synthesizing images[END_REF] proposes a StyleGAN encoder which predicts the style vectors {y i }. Then for each discriminative task, such as face verification and face alignment, a fully connected layer with the predicted style vectors as input is trained for the task. However, compared to modern supervised methods, its results are quite poor, and it does not study how well its method would perform when training with limited data. LARGE [START_REF] Nitzan | Large: Latent-based regression through gan semantics[END_REF] does not propose a new StyleGAN encoder architecture but notices that the linear directions in the latent space which affect image attributes are also linear in terms of the magnitude of change, so it is possible to create a linear regression model using a few calibration samples. Unfortunately, its method is restricted to image attribute regression, such as age or head pose estimation, and can't be applied for more complex tasks like facial landmark prediction.

Face Alignment

Rather than directly predicting the position of the landmarks in the image, most of the recent 2D face alignment models predict facial landmark heatmaps [START_REF] Newell | Stacked hourglass networks for human pose estimation[END_REF], one for each landmark. The landmark position is then inferred from the best local maximum of the heatmap.

In the case of 3D face alignment, we try to detect the true anatomical position of the keypoints, for example, the landmarks on the outline of the face remains at the same anatomical position even if there are occluded (the reader can compare the Figure 2 and 4 for a visual explanation). For this task, methods can be divided into two families, those that attempt to detect the landmarks directly [START_REF] Bulat | How far are we from solving the 2d & 3d face alignment problem?(and a dataset of 230,000 3d facial landmarks)[END_REF] or those that aim to fit a 3D face model and then obtain the landmarks from this model [START_REF] Zhu | Face alignment across large poses: A 3d solution[END_REF][START_REF] Guo | Towards fast, accurate and stable 3d dense face alignment[END_REF][START_REF] Ruan | Sadrnet: Self-aligned dual face regression networks for robust 3d dense face alignment and reconstruction[END_REF]. Wu et al. [START_REF] Wu | Synergy between 3dmm and 3d landmarks for accurate 3d facial geometry[END_REF] learn both tasks at the same time.

Semi-supervised methods try to alleviate the problem of facial landmark annotations we explained in the introduction of this paper. To do so they use annotated and non-annotated data during the training. Honori et al. [START_REF] Honari | Improving landmark localization with semi-supervised learning[END_REF] impose equivariance of predicted landmarks to geometric transformations. TS 3 [START_REF] Dong | Teacher supervises students how to learn from partially labeled images for facial landmark detection[END_REF] uses pseudo-labeling with a teacher-student method. Some methods are based on transfer learning: 3FabRec [START_REF] Browatzki | 3fabrec: Fast few-shot face alignment by reconstruction[END_REF] trains an auto-encoder to reconstruct face images and then modifies its decoder to generate facial landmark heatmaps, SCAF [START_REF] Dornier | Scaf: Skip-connections in auto-encoder for face alignment with few annotated data[END_REF] improves 3FabRec by adding skip-connections to the auto-encoder and using active learning. FaRL [START_REF] Zheng | General facial representation learning in a visual-linguistic manner[END_REF] uses masked image modeling and image-text contrastive learning on a large text/image pair dataset to pre-train a network and then use it for several facial downstream tasks including face alignment. Transfer learning is an interesting principle to train with limited annotated data but the whole pipeline can be computationally expensive and we think the carbon footprint for training a model should be taken into account. In the case of 3FabRec and SCAF, the self-supervised training of the encoder is done using millions of face images and lasts for days. Also, its sole purpose is the face alignment task. FaRL, in its principle, is more efficient because the pre-trained network can be used for different facial downstream tasks such as face parsing or facial attribute recognition and not only facial alignment. But its architecture is based on a visual Transformer [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF], a heavy architecture, and trains on a dataset of 20 million samples using 32 Nvidia V-100 GPUs so the pre-training is very expensive.

Our approach is also based on transfer learning but we don't need to do Another solution to tackle the challenge of facial landmark annotations is to use synthetic samples. Qian et al. [START_REF] Qian | Aggregation via separation: Boosting facial landmark detector with semi-supervised style translation[END_REF] train a network to produce multiple images with different styles but the same face pose from an input image to increase the training dataset size. Wood et al. [START_REF] Wood | 3d face reconstruction with dense landmarks[END_REF] use a fully synthetic dataset created with computer graphics [START_REF] Wood | Fake it till you make it: Face analysis in the wild using synthetic data alone[END_REF] to train a facial landmark detector. With computer graphics, it is possible to generate a lot of samples with perfect annotations, but generating such a dataset requires much computation power and there is always a domain gap between the generated images and real face images which may deteriorate the performance of the model on real test images. Also, if you need a new kind of annotations (additional landmarks for example), you need to generate again the whole dataset.

While Wood et al. [START_REF] Wood | 3d face reconstruction with dense landmarks[END_REF] obtain fair results on the face alignment dataset 300-W [START_REF] Sagonas | 300 faces in-the-wild challenge: The first facial landmark localization challenge[END_REF], it is hard to determine how well their model would perform on more challenging face alignment datasets such as WFLW [START_REF] Wu | Look at boundary: A boundary-aware face alignment algorithm[END_REF].

Active learning

In academic research, most of the time, scientists randomly choose the labeled samples from the whole labeled training set to demonstrate the effectiveness of their model in the few-shot setting. Yet, at first, this labeled training set does not exist for real-world applications, and one must choose the samples to annotate among an unlabeled training dataset.

The goal of active learning is to select the most valuable samples to annotate rather than sampling them randomly from the initial unlabeled dataset. Indeed, in the case of random sampling, very similar or easy samples might be sampled even though they won't help the model to improve during the training. Active learning tries to prevent this kind of situation. When annotating samples is very time-consuming, such as annotating facial landmarks, active learning can be very valuable.

Active learning is an iterative procedure: from a non-annotated dataset U N , an initial set L 0 is annotated and then the model is trained on L 0 . Once the training is over, all the remaining unlabeled samples from U N are ranked using an acquisition function which depends on the model predictions, the K best samples are annotated and added to L 0 giving a new annotated set L 1 .

Then, the model is trained again from scratch on this new annotated set and U N samples are ranked again, etc. This scheme repeats until the exhaustion of the annotation budget.

The crucial part of active learning is the choice of the acquisition function. The acquisition functions can be divided into two families even though works combine both approaches [START_REF] Kirsch | Batchbald: Efficient and diverse batch acquisition for deep bayesian active learning[END_REF]. The first family follows the "uncertainty sampling" principle [START_REF] Gal | Deep bayesian active learning with image data[END_REF][START_REF] Yoo | Learning loss for active learning[END_REF][START_REF] Dornier | Scaf: Skip-connections in auto-encoder for face alignment with few annotated data[END_REF], where the acquisition function tries to choose the samples where the model is the least confident in its predictions; the acquisition function mimics the training loss. The second family of acquisition functions is based on "diversity sampling" [START_REF] Sener | Active learning for convolutional neural networks: A core-set approach[END_REF], this time, the acquisition function tries to select samples that represent the diversity of the unlabeled dataset and tries to avoid selecting samples too similar.

For works related to our task, Yoo et al. [START_REF] Yoo | Learning loss for active learning[END_REF] compute the spatial entropy of body part heatmaps to assess their quality. SCAF [START_REF] Dornier | Scaf: Skip-connections in auto-encoder for face alignment with few annotated data[END_REF] proposes a new acquisition function: the Negative Neighborhood Magnitude which computes the sum of heatmap pixel values in a window centered on the predicted landmark position, and uses it for the face alignment task.

Method

Overview

Noticing the quality of StyleGAN [START_REF] Karras | A style-based generator architecture for generative adversarial networks[END_REF] for image generation, we propose to modify its generator so that it generates facial landmarks heatmaps instead.

To apply it to real images, we also need to project them into StyleGAN latent space. This leads to our proposed framework: from an input face image, we first approximate its StyleGAN latent vector using a pre-trained Feature-Style encoder [START_REF] Yao | Feature-style encoder for style-based gan inversion[END_REF], then we use the modified StyleGAN generator to predict facial landmark heatmaps from this latent vector. The generator is modified in a way similar to 3FabRec [START_REF] Browatzki | 3fabrec: Fast few-shot face alignment by reconstruction[END_REF]: additional convolution layers are interleaved with the generator layers and trained to generate facial landmark heatmaps.

Our full architecture can be seen in Fig. 1.

StyleGAN inversion encoder

We need to choose a StyleGAN inversion model for our architecture. We don't consider optimization-based and hybrid models because their computational time would lead to intractable training time. FFHQ [START_REF] Karras | A style-based generator architecture for generative adversarial networks[END_REF], the training dataset of most StyleGAN generators for face images contains only aligned images, the face is always perfectly centered, the eyes are always at the same horizontal position and the zoom is also the same for all images. Many StyleGAN encoders can only faithfully reconstruct images that follow this alignment. But images from face alignment datasets do not follow this alignment, it depends on the face detector used or the provided ground truth bounding boxes, so we need a StyleGAN encoder that can still reconstruct non-aligned images. That's why we choose Feature-Style encoder [START_REF] Yao | Feature-style encoder for style-based gan inversion[END_REF] which outputs the extended latent vector w + but also a feature map f which makes it possible to handle non-aligned images.

Modified StyleGAN generator

3FabRec [START_REF] Browatzki | 3fabrec: Fast few-shot face alignment by reconstruction[END_REF] and then SCAF [START_REF] Dornier | Scaf: Skip-connections in auto-encoder for face alignment with few annotated data[END_REF] showed that the decoder of an autoencoder trained to reconstruct face images can be modified to predict facial landmark heatmaps. Indeed, during the generation of a face image from a latent representation, generator layers must extract (among other face attributes) face shape information from the latent vector to generate the image.

Facial landmark heatmaps can be seen as a face image where only information about face shape has been kept. So, we can try to perform some kind of style transfer by modifying the already trained generator to generate not RGB face images but facial landmark heatmaps instead. While 3FabRec and SCAF use a small inverted ResNet-18 as a decoder, we propose to use a more advanced network: a StyleGAN generator. We detail this architecture in the following paragraphs.

As explained in Section 2.1, the StyleGAN generator starts from a constant vector c 1 followed by StyleGAN blocks which progressively increase the resolution. Each block is composed of an upsampling operation followed by two convolutions equipped with an AdaIn [START_REF] Huang | Arbitrary style transfer in real-time with adaptive instance normalization[END_REF] operation to inject the layerspecific style vector. From an image, the Feature-Style encoder predicts 2 codes: the feature code f and the extended latent vector w + = (w 1 , w 2 , ...).

During image generation, the first layers of the generator (five in this paper) are replaced by the feature code f . Then, each latent vector w i (except for the first ones) from w + is transformed into a style vector by an affine transformation and injected into the corresponding StyleGAN layer through

AdaIN. The last block outputs the reconstructed image.

We modify this architecture to predict facial landmark heatmaps. From a chosen generator block until the landmark heatmap resolution is reached, in a similar way to 3FabRec and SCAF, we interleave the generator blocks with Interleaved Transfer Layers (ITLs) [START_REF] Browatzki | 3fabrec: Fast few-shot face alignment by reconstruction[END_REF]. An ITL is a 3 × 3 convolutional layer located directly after a StyleGAN block. It takes as input the output of the StyleGAN block and outputs a feature map with the same spatial dimensions and number of channels of the input, except for the last ITL. This last ITL generates the landmark heatmaps so this time the number of channels is equal to the number of landmarks. Fig. 1 shows the full architecture. During the supervised training, the StyleGAN blocks parameters are frozen, but not the encoder ones which are fine-tuned while the ITLs are trained.

Active learning

Optionally, we use active learning to select our samples when training with limited data. We use the Negative Neighborhood Magnitude (NNM) [START_REF] Dornier | Scaf: Skip-connections in auto-encoder for face alignment with few annotated data[END_REF] as the acquisition function. For each predicted heatmap H, the NNM sums the heatmap pixels values in a square window W i centered on the predicted landmark position li (the argmax of the heatmap in our case). To finish the NNM computation, all the heatmaps sums are then summed again and we take negative so that the less confident the model is, the greater NNM is.

N N M ( H) = - L i=1 u,v∈W i Hi (u, v) (1)
This acquisition function selects heatmaps with low magnitude around the predicted landmark positions, it assumes that when the model is not confident about its prediction, the heatmap magnitude will be low so adding this sample to the training set would improve the model performance. As in SCAF [START_REF] Dornier | Scaf: Skip-connections in auto-encoder for face alignment with few annotated data[END_REF], we discard the top-10% NNM samples from the potential candidates to avoid selecting outliers.

Experiments

2D Face alignment datasets

We evaluated our method on three 2D face alignment datasets commonly used in the literature.

AFLW: This dataset [START_REF] Koestinger | Annotated facial landmarks in the wild: A largescale, real-world database for facial landmark localization[END_REF]

3D Face alignment datasets

We also evaluated our architecture on the 3D face alignment task to see if it can also generate accurate 3D facial landmark heatmaps. For this task, we use two different datasets, one for training and another for evaluation. By 3D landmarks we actually mean the 2D projections of the 3D landmarks because to predict the full 3D landmarks we would need to modify our architecture.

300-W-LP:

This is a synthetic dataset [START_REF] Zhu | Face alignment across large poses: A 3d solution[END_REF] created from 300-W images using the profiling method of [START_REF] Zhu | Face alignment across large poses: A 3d solution[END_REF] to render its faces into larger poses. This dataset contains 122,450 face images with a face pose yaw angle ranging from -90°to 90°. 68 2D and 3D landmarks annotations are provided for each face.

We train our model on this dataset to predict the 3D landmarks.

AFLW2000-3D: This dataset [START_REF] Zhu | Face alignment across large poses: A 3d solution[END_REF] was constructed by re-annotating the first 2,000 images of AFLW annotated with 68 3D landmarks consistent with the ones of 300-W-LP. The face pose also ranges from -90°to 90°. We use this dataset to evaluate our models trained on 300-W-LP. The dataset can be divided into 3 subsets according to the absolute face pose: a subset with almost frontal faces ([0°, 30°]), another one with medium poses ([30°, 60°])

and the last one with profile views ([60°, 90°]).

Evaluation

As metrics, we use the usual Normalized Mean Error (NME) and Area Under Curve (AUC) of the Cumulative Error Distribution (CED) to compare our model to other methods. The NME is defined as:

N M E(%) = 1 N N i=1 ||s i -si || d * 100 (2)
where s i and si are the ground truth and predicted location of landmark i, N the number of landmarks and d a normalization distance.

For 300-W and WFLW, we use the distance between the outer eye corners as the normalization distance for the NME (NME inter-ocular ). For AFLW, because of the large number of profile faces, we report both the NME normalized with the diagonal of the ground truth bounding box (NME diag ) or the square root of the ground truth bounding box area (

√ w bbox * h bbox ) (NME box ).
We also report the AUC at 7% NME box (AUC 7 box ) for this dataset. For AFLW2000-3D, as the normalization distance, we also use the square root of the bounding box area. Because no bounding box is provided for this dataset, it is computed from the ground truth 3D landmarks.

Architecture and training parameters

The StyleGAN inversion encoder is a Feature-style encoder [START_REF] Yao | Feature-style encoder for style-based gan inversion[END_REF] pretrained on FFHQ [START_REF] Karras | A style-based generator architecture for generative adversarial networks[END_REF]. The generator is a StyleGAN2 generator [START_REF] Karras | Analyzing and improving the image quality of stylegan[END_REF] pretrained on the same dataset. Input images are resized to 256 × 256 pixels.

We use 5 ITLs (see Section 4.7.2 for details) and output heatmaps of size 128 × 128 pixels.

Our models are implemented with PyTorch [START_REF] Paszke | Pytorch: An imperative style, high-performance deep learning library[END_REF]. We train all our models for 200,000 training steps using a batch size of 8 on a Nvidia V100 GPU with 16 GB of memory. For the training loss, we use a standard MSE loss between the predicted and ground truth heatmaps. As optimizer, we use Adam [START_REF] Kingma | A method for stochastic optimization[END_REF] (β 1 = 0.9, β 2 = 0.99) with an initial learning rate of 0.0001 for the ITLs and 0.00002 for the pre-trained encoder. Both learning rates are decayed by a factor of 0.995 every 10 epochs. We use random vertical flip (p=50%), rotation (±30°), translation (±4%), scaling (±5%), occlusions (p=50%, PyTorch default settings for the bounding box size), Gaussian blur (p=20%), brightness (p=45%, ±80%) and contrast changes (p=45%, 30%-200%) as data augmentations.

When training with limited data (without active learning), the training samples are chosen randomly from among the full training set before each run. We report the means and standard deviations over 5 runs for all training sizes except 50, for which we use 10 runs. During active learning, the initial training set contains 10 random samples. K, the number of samples added after training depends on the final training set size. We make it large enough so that there are no more than 5 trainings in total.

Results on 2D face alignment

Comparison with fully supervised methods

Table 1 shows comparisons of our method with state-of-the-art (SOTA) on 300-W and WFLW datasets when training with the full training dataset (except for Wood et al. [START_REF] Wood | 3d face reconstruction with dense landmarks[END_REF] which is trained on a synthetic dataset). Our method achieves results comparable to the SOTA of 2019 but is surpassed by current SOTA methods. For AFLW (see Table 2), we are second on all metrics, just behind another semi-supervised method [START_REF] Zheng | General facial representation learning in a visual-linguistic manner[END_REF]. We hypothesize that these better results, compared to 300-W and WFLW are caused by the fact that AFLW is relatively easier (only 19 landmarks) and many images are closer to the ones found in FFHQ, the training dataset for the StyleGAN encoder and generator. While these results are interesting, our main goal with our method is to train the model with limited training. These results are discussed in the next subsection. Figure 2 shows predictions of our model on some images of the 300-W Full test.

Comparison with semi-supervised methods

We compare our method to other semi-supervised methods training with limited data. On 300-W (Table 4), we surpass other methods for all limited training sizes ranging from 20% (630 samples) of the full training set [START_REF] Browatzki | 3fabrec: Fast few-shot face alignment by reconstruction[END_REF]148 samples in total) to only 50 samples. When training with this last training set size, we get comparable or better results than other methods training with 20% of the dataset. Also, the performance of our algorithm does not degrade significantly when the training set size goes down, even for 50 samples. For the more challenging dataset WFLW (see Table 3), we also surpass other methods when training on limited data. Methods reported their results on AFLW with the NME diag or NME box so we computed both metrics (see Table 5). For the NME diag , we surpass other semi-supervised methods on all training set sizes on both the Full and Frontal test sets. For the NME box , we compared our method with FaRL [START_REF] Zheng | General facial representation learning in a visual-linguistic manner[END_REF],

they only report results for 100%, 10%, and 1% sizes but we can see that even though they achieve better results when training with the full training set, their results are even with us on the 10% training set size and we surpass them on the 1% training set size. For WFLW, using active learning always improves the performance of our model, proving again its efficiency. One should also notice that for both datasets, the use of active learning also reduces the NME standard deviations;

Results with active learning

that would tend to demonstrate that the samples selected by active learning are similar across the runs.

On AFLW, this time the active learning does not improve much the performance on the Full test set. It decreases it a bit on the Frontal test set, meaning active learning improves the performance on challenging images such as profile faces but with the cost of a slightly reduced accuracy on easy faces.

Results on 3D face alignment

We compare our model to other which train on 300-W-LP and evaluate on the AFLW2000-3D dataset. Although our method only predicts landmark heatmaps and does rely on estimating a 3D face model like most of the recent 3D face alignment models [START_REF] Guo | Towards fast, accurate and stable 3d dense face alignment[END_REF][START_REF] Ruan | Sadrnet: Self-aligned dual face regression networks for robust 3d dense face alignment and reconstruction[END_REF][START_REF] Wu | Synergy between 3dmm and 3d landmarks for accurate 3d facial geometry[END_REF], our method still obtains decent results. For the low pose test subset ([0°, 30°]), we obtain a similar performance to the state-of-the-art. However, for larger poses, our model falls behind recent This may be explained by the fact there is no profile face in the pre-training dataset (FFHQ [START_REF] Karras | A style-based generator architecture for generative adversarial networks[END_REF]) and because our method relies on heatmaps, it may be difficult for some images to detect precisely occluded landmarks. Our model still obtains decent results on many profile face images (see Figure 4).

Results when training with limited training data

We evaluate our model on AFLW2000-3D after training on data sampled from 300-W-LP. We use three different sampling methods, "Random": fully random sampling, "Balanced": random sampling but the low (0-30°), medium (30-60°) and large pose (60-90°) subsets must have equal size and "Active": sampling using active learning (same procedure as used for 2D face alignment). We use different training set sizes, 300, 150, and 48, all divisible by 3 so we can have perfectly balanced training sets for the "Balanced" sampling method. 2 to 6 ITLs, the NMEs are very close and most of the time the Wilcoxon's test null hypothesis can't be rejected when comparing two models. For small training set sizes, such as 50 samples, we need at least 3 layers to have the best performance. We suppose that when training with limited data, the encoder can't be totally fine-tuned because of its large number of parameters compared to ITLs so more ITLs make the training easier. However, from 3 to 6 layers the standard deviations are overlapping, and again the Wilcoxon's tests can't tell apart the models, so there is no clear winner. For other experiments in this paper, we used 5 ITLs.

Conclusion

In this paper, we have demonstrated that StyleGAN [START_REF] Karras | A style-based generator architecture for generative adversarial networks[END_REF] in proving the efficiency of our method rather than beating the state-of-theart, we didn't use any specific trick used in recent face alignment models, such as facial boundaries heatmaps [START_REF] Wu | Look at boundary: A boundary-aware face alignment algorithm[END_REF], complex training loss [START_REF] Wang | Adaptive wing loss for robust face alignment via heatmap regression[END_REF][START_REF] Huang | Adnet: Leveraging error-bias towards normal direction in face alignment[END_REF], or attention maps [START_REF] Ruan | Sadrnet: Self-aligned dual face regression networks for robust 3d dense face alignment and reconstruction[END_REF] to boost our model. They could be easily applied to our method to improve its performance.

  generator, through the use of additional layers, into a facial landmark heatmap generator which can be trained with limited training data. Paired with a StyleGAN encoder, we can perform face alignment on real images. With this new framework, we achieve competitive results on multiple facial landmark datasets, even on images very different from the StyleGAN generative distribution, and beat state-of-the-art in the low-data regime. Our contributions are as follows: • We propose a new architecture derived from StyleGAN to perform face alignment with limited training data. On multiple 2D and 3D face alignment datasets, our method achieves competitive results in the fully supervised setting and systematically beats state-of-the-art in the low training data setting.

  any dedicated pre-training prior to our face alignment training. Rather, we promote StyleGAN pre-trained generators and encoders, which have been trained for a completely different purpose: face image generation for generators and face attribute modification for encoders. It makes the carbon footprint for the redaction of this paper very low compared to the previously discussed transfer learning methods.

Figure 1 :

 1 Figure 1: Our network architecture. We use a pre-trained Feature-style encoder and StyleGAN2 generator. Additional convolution layers are interleaved with some generator convolution blocks and are trained to predict landmark heatmaps.

Figure 2 :

 2 Figure 2: Our model predictions on some images of 300W Full test set. Top row shows landmark predictions and bottom row shows heatmap predictions.

Figure 3

 3 Figure 3 shows examples of images selected by the Negative Neighborhood Magnitude (NNM), the active learning acquisition function used. Heatmaps with low magnitude have various causes: network confusion between the outline of the face and hair or beard (first two pictures), unusual face pose (third picture), or almost closed eyes (fourth picture). Also, landmarks on the outline of the face being the most ambiguous, they usually are the ones with the lowest heatmap magnitude.

Figure 3 :

 3 Figure 3: Some images of 300-W selected by the NNM, the active learning acquisition function, after a first training on 10 random images. First row: ground truth landmarks. Second row: predicted landmarks. Third row: ground truth heatmaps. Fourth row: predicted heatmaps. The NNM successfully selects images where some landmark predictions are not accurate. These samples will be added to the training labeled set for the next training iteration and should improve the model performance.

Figure 4 :

 4 Figure 4: Our model predictions on some images of AFLW2000-3D. The top row shows landmark predictions and the bottom row shows heatmap predictions. Even occluded landmarks are correctly predicted.

5. 1 .

 1 Future workThanks to the Feature-Style encoder[START_REF] Yao | Feature-style encoder for style-based gan inversion[END_REF], we were able to perform our training and testing on unaligned images which do not belong to the original StyleGAN generative distribution. An interesting work would be to align face alignment dataset images to make them follow FFHQ alignment. Would this improve performance because images would lie closer to the original generative distribution, or make it worse because of less face pose diversity during training, is an open question. Also, because we were more interested

  This dataset[START_REF] Sagonas | 300 faces in-the-wild challenge: The first facial landmark localization challenge[END_REF] contains face images annotated using 68 landmarks. Following the common splits[START_REF] Dong | Teacher supervises students how to learn from partially labeled images for facial landmark detection[END_REF][START_REF] Browatzki | 3fabrec: Fast few-shot face alignment by reconstruction[END_REF], the training set contains 3,148 images while the Full test set contains 689 images and is divided into a Common test set of 554 images and a Challenging test set of 135 images.

	into several (partially overlapping) subsets, each focused on a specific char-
	acteristic: pose, expression, illumination, make-up, occlusion, or blur.
	WFLW: This dataset introduced in [45] contains 7,500 training images
	and 2,500 testing images annotated with 98 landmarks. The test set is split

contains 24,386 face images annotated with 21 landmarks. Following usual practice

[START_REF] Dong | Teacher supervises students how to learn from partially labeled images for facial landmark detection[END_REF][START_REF] Browatzki | 3fabrec: Fast few-shot face alignment by reconstruction[END_REF]

, we ignore the landmarks of the ears and use 20,000 training images and 4,386 testing images. We evaluate our model on the Full test set and the Frontal test set, a subset that contains only images with a frontal view.

300-W:

Table 1 :

 1

	300-W	WFLW

NME inter-ocular (%) (↓) on the 300-W Common, Challenging and Full test sets, and on the WFLW Full test set. *Trained on a synthetic dataset.

Table 2 :

 2 Comparison with state-of-the-art methods on the AFLW Full and Frontal test sets.

		WFLW dataset
	Method		Training set size
		7500	1500	750	375	50
		(100%) (20%) (10%) (5%) (0.67%)
	AVS [41]	4.39	6.00	7.20	-	-
	3FabRec [3]	5.62	6.51	6.73	7.68	8.39
	SCAF [39]	5.50	6.07	6.28	6.72	8.06
	SCAF+AL [39]	-	-	6.24	6.59	7.60
	Ours	4.62	5.09	5.44	5.80	7.78
	Ours std	0.03	0.08	0.07	0.08	0.20
	Ours+AL	-	4.94	5.18 5.45	7.30
	Ours+AL std	-	0.04	0.03	0.04	0.12

Table 3 :

 3

NME inter-ocular (%) (↓) when training with limited training set size on the WFLW Full test set. AL stands for Active Learning.

Table 4 :

 4 NME inter-ocular (%) (↓) when training with limited training set size on 300-W on the Common, Challenging and Full test sets (first, second and third columns respectively for each training set size). AL stands for Active Learning.

						300-W dataset
	Method							Training set size
		3148			630			315			168		50
		(100%)			(20%)		(10%)			(5%)		(1.59%)
		Com. Ch. Full Com. Ch. Full Com. Ch. Full Com. Ch. Full Com. Ch. Full
	RCN+ [37]	3.00 4.98 3.46	-	6.12 4.15	-	6.63 4.47	-	9.95 5.11	-	-	-
	AVS [41]	3.21 6.49 3.86 3.85	-	-	4.27	-	-	6.32	-	-	-	-	-
	TS 3 [38]	2.91 5.90 3.49 4.31 7.97 5.03 4.67 9.26 5.64	-	-	-	-	-	-
	3FabRec [3]	3.36 5.74 3.82 3.76 6.53 4.31 3.88 6.88 4.47 4.22 6.95 4.75 4.55 7.39 5.10
	SCAF [39]	3.48 5.89 3.95 3.66 6.23 4.17 3.87 6.60 4.40 3.93 6.84 4.50 4.33 7.60 4.97
	SCAF+AL [39]	-	-	-	-	-	-	3.99 6.49 4.48 4.19 6.78 4.70 4.29 6.93 4.81
	Ours	2.97 5.30 3.42 3.14 5.66 3.64 3.22 5.87 3.74 3.33 6.05 3.86 3.57 6.62 4.16
	Ours std	0.01 0.09 0.02 0.02 0.04 0.02 0.03 0.06 0.03 0.03 0.13 0.03 0.05 0.21 0.07
	Ours+AL	-	-	-	3.12 5.53 3.59 3.20 5.67 3.68 3.32 5.83 3.81 3.54 6.24 4.06
	Ours+AL std	-	-	-	0.03 0.05 0.02 0.02 0.05 0.02 0.03 0.04 0.03 0.01 0.13 0.02
						AFLW dataset
	Method							Training set size
		20000		4000		2000		1000	200	50
		(100%)		(20%)	(10%)		(5%)	(1%)	(0.25%)
		Full Fr. Full Fr. Full Fr. Full Fr. Full Fr. Full Fr.
								NME box (%) ↓
	RCN+ [37]	1.61	-		-	-	-	-	2.17	-	2.88	-	-	-
	TS 3 [38]	-	-		1.99 1.86 2.14 1.94 2.19 2.03	-	-	-	-
	3FabRec [3]	1.87 1.59 1.96 1.74 2.03 1.74 2.13 1.86 2.38 2.03 2.74 2.23
	Ours	1.45 1.28 1.60 1.39 1.63 1.41 1.66 1.43 1.79 1.53 2.05 1.71
	Ours std	<.01 <.01 <.01 0.02 <.01 0.01 <.01 0.03 0.01 0.01 0.03 0.02
	Ours+AL	-	-		-	-	-	-1.66 1.49 1.77 1.56 2.03 1.75
	Ours+AL std -	-		-	-	-	-<.01 0.02 <.01 <.01 0.03 0.04
								NME diag (%) ↓
	FaRL [8]	0.94 0.82 -	-1.15 -	-		-	1.35	-	-	-
	Ours	1.02 0.90 1.13 0.98 1.15 1.00 1.17 1.01 1.27 1.08 1.45 1.21
	Ours std	<.01 <.01 <.01 0.01 <.01 0.01 <.01 0.02 0.01 <.01 0.02 0.02
	Ours+AL	-	-		-	-	-	-1.17 1.05 1.25 1.11 1.44 1.24
	Ours+AL std -	-		-	-	-	-<.01 0.02 <.01 <.01 0.02 0.03

Table 5 :

 5 Comparison with other semi-supervised methods when training with limited training set size on AFLW on the Full and Frontal test sets (first and second columns respectively for each training set size). AL stands for Active Learning On 300-W, while the use of active learning greatly reduces the NME on the Challenging test, especially when the training size is very small, there is no real improvement on the Common test set. It proves the efficiency of active learning to select hard samples from the unlabeled training set, similar to the ones found in the Challenging test set.

Table 6 :

 6 Table 7 reports the NME box of the models on the

	AFLW2000-3D pose subsets. It also reports the face pose distribution of the
	training datasets sampled from 300-W-LP. The whole 300-W-LP contains
	25% low pose images, 37% medium pose images, and 38% large pose im-
	ages so our "Random" samplings are close to this distribution. "Balanced"
	samplings have by definition a (33%, 33%, 33%) pose distribution. In the

NME box (%) (↓) on different subsets of AFLW2000-3D divided by face pose (yaw angle). "Balanced" column is the average of the first 3 columns. "Mean" column reports the mean NME over the whole AFLW2000-3D dataset.

Table 7 :

 7 Comparison of different sampling methods for several training set sizes according to the face pose yaw angle. This table reports the NME of models trained on some 300W-LP samples and evaluated on the AFLW2000-3D subsets. It also reports the face pose yaw distribution of the training datasets. "Random": training samples are chosen randomly.

	4.7. Ablation studies
	4.7.1. Encoder fine-tuning
	Table 8 reports results on the 300-W and WFLW Full test sets for different
	training set sizes whether we fine-tune or not the encoder while training the
	ITLs. Fine-tuning improves the performance for all training set sizes on both
	datasets, especially on WFLW.
	4.7.2. Number of Interleaved Transfer Layers
	We made experiments to know if there is an optimal number of Interleaved
	Transfer Layers (ITLs), results are reported in Table 9. When training with
	the full training set, using only one layer makes the model perform a bit worse
	(also confirmed by Wilcoxon's signed tests not reported in this table). From

"Balanced": training samples are random but the low, medium and large face pose subsets must have equal size. "Active": training samples are selected using active learning.

Table 8 :

 8 NME inter-ocular (%) (↓) on 300-W and WFLW Full test sets for different training set sizes with and without encoder fine-tuning.

	Num. ITLs		300-W			WFLW
		Training set size	Training set size
		3148	315	50	7500	750	50
		(100%) (10%) (1.59%) (100%) (10%) (0.67%)
	1	3.54	3.76	4.26	4.75	5.68	8.72
	std	0.03	0.02	0.09	0.04	0.11	0.35
	2	3.44	3.72	4.22	4.65	5.50	8.20
	std	0.04	0.03	0.03	0.02	0.04	0.32
	3	3.43	3.74	4.13	4.66	5.46	7.65
	std	0.04	0.01	0.02	0.08	0.05	0.23
	4	3.46	3.73	4.16	4.63	5.43	7.94
	std	0.04	0.02	0.03	0.05	0.04	0.27
	5	3.42	3.74	4.16	4.62	5.44	7.78
	std	0.02	0.03	0.07	0.03	0.07	0.20
	6	3.45	3.70	4.15	4.61	5.42	7.82
	std	0.03	0.01	0.06	0.4	0.03	0.29

Table 9 :

 9 NME inter-ocular (%) (↓) on 300-W and WFLW Full test sets, depending on the number of Interleaved Transfer Layers (ITLs) and the training set size.

  can be used, not only for generative tasks such as attribute edition or single attribute regression like age estimation but also for complex discriminative tasks like face alignment, even when the training data is limited. Our method can generate accurate facial landmark heatmaps and obtains competitive results on multiple face alignment datasets for both 2D and 3D facial landmarks. It also systematically surpasses other semi-supervised methods when training with limited data. Another advantage of our method is that we don't need to perform any dedicated computationally expensive unsupervised training on large databases[START_REF] Browatzki | 3fabrec: Fast few-shot face alignment by reconstruction[END_REF][START_REF] Dornier | Scaf: Skip-connections in auto-encoder for face alignment with few annotated data[END_REF][START_REF] Zheng | General facial representation learning in a visual-linguistic manner[END_REF], prior to the supervised training because our model is based on StyleGAN and pre-trained weights are already available.

Acknowledgments

This work was granted access to the HPC resources of IDRIS under the allocation 2022-AD011012376 made by GENCI. Bertrand Coüasnon received his Ph.D. in computer science from the University of Rennes 1, France, in 1996. Since 1996 he is an Associate Professor at INSA Rennes, France. His research interests are knowledge formalization, generic methods for document structure recognition, combining deep learning and syntactical methods, and handwriting recognition.