John-Eric Dufour

Emilie Gonnet

Sébastien Grau

François Hild
email: hild@lmt.ens-cachan.fr

On the analysis of canvas wrinkling via isogeometric stereocorrelation

Keywords: Climatic chamber, Optical technique, Punch test, Restoration, Wrinkling

Wrinkling is to be avoided in the restoration of works on canvas. In the worst cases, it may lead to paint cracking. Visual rendering of the paint may also change because of such mechanism. It is proposed to measure wrinkling under various conditions of contrast and loading via isogeometric stereocorrelation. This method allows low contrasted paints to be analyzed in addition to very large deformations that occur in punch tests on canvas.

Introduction

The restoration and transport of works on canvas requires their mechanical behavior to be understood as best as possible to avoid any damage. Further, in many cases, canvasses are not stress-free and thus very sensitive to the environment (e.g., relative humidity [START_REF] Berger | Interaction between canvas and paint lm in response to environmental changes[END_REF]Malowany et al., 5 2014)) in which they are stored and exhibited (e.g., museums, places of gathering). Being part of the cultural heritage, these works are handled with special care and the understanding of their deformations during these stages is very important. Over the last decades, non-destructive and possibly non-contacting techniques have been proposed to help the restoration of various works [START_REF] James | Radiographic techniques to evaluate paintings[END_REF][START_REF] Dulieu-Barton | Deformation and strain measurement techniques for the inspection of damage in works of art[END_REF][START_REF] Ambrosini | Lasers in the Conservation of Artworks[END_REF][START_REF] Mudge | Principles and practices of robust, photographybased digital imaging techniques for museums[END_REF][START_REF] Remondino | Review of geometric and radiometric analyses of paintings[END_REF][START_REF] Gavrilov | A review of imaging methods in analysis of works of art: Thermographic imaging method in art analysis[END_REF][START_REF] Sfarra | Non-destructive testing techniques to help the restoration of frescoes[END_REF]. Among various optical techniques that are utilized to monitor the 3D shape and deformation of canvasses, stereocorrelation is easy-to-implement and will be used herein.

One critical issue associated with stereocorrelation is to have sucient image contrast to enable the registration to succeed [START_REF] Khennouf | Assessing the feasibility of monitoring strain in historical tapestries using digital image correlation[END_REF][START_REF] Dureisseix | Followup of a panel restoration procedure through image correlation and nite element modeling[END_REF]. If natural contrast is not sucient, ad hoc procedures may be considered, for instance, by projecting a random pattern in addition to the negative image of the original surface [START_REF] Dureisseix | Followup of a panel restoration procedure through image correlation and nite element modeling[END_REF]. This method allows 3D shapes to be measured. However, it does not enable 3D surface motions to be evaluated. An alternative route consists of working on model canvas paintings [START_REF] Malowany | Application of 3D digital image correlation to track displacements and strains of canvas paintings exposed to relative humidity changes[END_REF] for which random speckle patterns (i.e., à la Roy Lichtenstein) are deposited onto the observed surfaces as classically performed in mechanical engineering [START_REF] Sutton | Computer vision-based, noncontacting deformation measurements in mechanics: A generational transformation[END_REF]. 3D shapes and surface displacements can then be measured via stereocorrelation (also referred to as 3D digital image correlation or StereoDIC [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF]).

The aim of the present work is to use global approaches to stereocorrelation [START_REF] Beaubier | CAD-based calibration of a 3D-DIC system: Principle and application on test and industrial parts[END_REF]Dufour et al., 2015a;[START_REF] Dubreuil | Mesh-based shape measurements with stereocorrelation[END_REF], which consist of measuring 3D shapes and surface displacements when registrations are performed over the whole surface of interest and not locally as in the aforementioned approaches [START_REF] Khennouf | Assessing the feasibility of monitoring strain in historical tapestries using digital image correlation[END_REF][START_REF] Dureisseix | Followup of a panel restoration procedure through image correlation and nite element modeling[END_REF][START_REF] Malowany | Application of 3D digital image correlation to track displacements and strains of canvas paintings exposed to relative humidity changes[END_REF]. Consequently, if some contrasted zones exist, the shape and displacement interpolations will allow the sought information to be assessed everywhere. Further, since wrinkling is monitored, the use of higher order displacement interpolations (e.g., provided by non uniform rational B-splines or NURBS [START_REF] Piegl | The NURBS Book -2nd Edition[END_REF]) is desirable. To prove the feasibility of isogeometric stereocorrelation, two test cases are analyzed. First, the eect of relative humidity variations will be studied for model canvas paintings and the Descent of the Cross, which was recently restored and attributed to the Swiss painter Sebastian Düring (1671-1723) [START_REF] Grau | Conservation et restauration de la Descente de Croix conservée dans le ch÷ur de la basilique de Saint-Maurice en Valais[END_REF]. For the actual work, the raw painted face will be analyzed (i.e., no speckle pattern was applied). Second, a punch test on modern canvas will be studied. This test is very challenging since very large out-of-plane motions occur. A new enrichment strategy will be implemented in the NURBS framework used herein to measure 3D shapes and their motions.

The outline of the paper is as follows. The main principles of isogeometric stereocorrelation are introduced in Section 2. Section 3 deals with the analysis of canvasses under dierent relative humidity levels. The rst part is devoted to the study of two model canvas paintings with dierent tensioning systems to minimize wrinkling. The second part then applies the best tensioning system to the restoration of the Descent of the Cross. Last, Section 4 is devoted to a punch test on canvas nailed on its frame. It required the NURBS-based kinematic basis to be gradually enriched to properly measure the complex deformation with numerous wrinkles.

Isogeometric stereocorrelation

Isogeometric stereocorrelation is a global approach based upon the gray level conservation between multiple images of the same object (Dufour et al., 2015b).

The stereo-rig, which is usually composed of 2 cameras, rst needs to be calibrated [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF]. This calibration procedure can be performed using dierent methods, namely, with a planar calibration target [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF][START_REF] Faugeras | Camera calibration for 3D computer Vision[END_REF][START_REF] Weng | Camera calibration with distortion models and accuracy evaluation[END_REF][START_REF] Zhang | A exible new technique for camera calibration[END_REF][START_REF] Salvi | A comparative review of camera calibrating methods with accuracy evaluation[END_REF] or with the object of interest itself [START_REF] Faugeras | Camera self-calibration: Theory and experiments[END_REF][START_REF] Fusiello | Uncalibrated euclidean reconstruction: a review[END_REF][START_REF] Beaubier | CAD-based calibration of a 3D-DIC system: Principle and application on test and industrial parts[END_REF]. In the present case two dierent strategies will be followed. First, the easel onto which painted works will be put will serve as calibration target. Second, an open book target will also be considered [START_REF] Besnard | Characterization of necking phenomena in high speed experiments by using a single camera[END_REF] in the punch test. The reason for choosing such calibration targets is related to the fact that only one pair of pictures will be used during this step.

Surface model

In global stereocorrelation, there are two ways of dening the surface of interest. First low order shape functions may be considered (e.g., meshes made 70 of triangular [START_REF] Dubreuil | Mesh-based shape measurements with stereocorrelation[END_REF] or quadrilateral [START_REF] Dufour | Displacement measurement using CAD-based stereo-correlation with meshes[END_REF] elements). Second higher order interpolations may be considered (e.g., Bézier or NURBS patches [START_REF] Beaubier | CAD-based calibration of a 3D-DIC system: Principle and application on test and industrial parts[END_REF]Dufour et al., 2015a)). Since wrinkling requires higher order interpolations of the measured displacement elds, the technique used herein is isogeometric stereocorrelation, namely, the surface of 75 interest is described by NURBS patches.

The three-dimensional shape X = (X, Y, Z) of an external surface is written as X(ξ), where ξ = (u, v) dene the parametric space, conventionally spanning the elementary square [0, 1] 2 . What is assumed to be known is the virtual (or nominal) shape of the object of interest. A NURBS patch is dened by its order, a network of control points with associated weights, and its knot vector [START_REF] Piegl | The NURBS Book -2nd Edition[END_REF]. The surface X(u, v) = (X, Y, Z) is expressed in the parametric space (u, v) as

X(u, v) = m i=0 n j=0 B ij (u, v)P ij (1)
where the blending functions are dened as

B ij (u, v) = N i,p (u)N j,q (v)ω ij m i=0 n j=0 N i,p (u)N j,q (v)ω ij (2) with ∀u ∈ [0, 1], N i,0 (u) =      1 when u i ≤ u ≤ u i+1 0 otherwise (3)
and

N i,p (u) = u -u i u i+p -u i N i,p-1 (u) + u i+p+1 -u u i+p+1 -u i+1 N i+1,p-1 (u) , (4)
where N i,p are mixing functions, P ij the coordinates of control points of the surface, ω ij the corresponding weights, (m + 1) × (n + 1) the number of control points and (p, q) the degrees of the surface.

Calibration and 3D shape measurements

This rst part of a stereocorrelation analysis consists of a two-step procedure.

First the camera parameters are estimated (i.e., the projection matrices that link any 3D point to its 2D coordinates in the camera plane). Second the observed shape is measured according to its mathematical description [START_REF] Beaubier | CAD-based calibration of a 3D-DIC system: Principle and application on test and industrial parts[END_REF].

The calibration procedure consists in determining the elements of the projection matrices. Stereomatching is written as the minimization of the sum of squared dierences with respect to the elements of the projection matrices [START_REF] Beaubier | CAD-based calibration of a 3D-DIC system: Principle and application on test and industrial parts[END_REF])

η 2 tot ([M c1 ], [M c2 ]) = ROI (f c1 (ξ, [M c1 ]) -f c2 (ξ, [M c2 ])) 2 (5)
where [M ci ] denotes the projection matrix of the i-th camera, and f ci the image of the reference conguration shot by the i-th camera. A Gauss-Newton algorithm is used to perform the nonlinear least squares minimization.

Once the projection matrices are known, the mathematical model of the observed surface is updated by moving the control points. Stereomatching is performed again by minimizing the sum of squared dierences

η 2 tot ({P }) = ROI (f c1 (ξ, {P }) -f c2 (ξ, {P })) 2 (6)
with respect to the control point positions, which are all gathered in the column vector {P }. A Gauss-Newton algorithm is used again to perform the nonlinear least squares minimization.

3D displacement measurement

When the observed 3D shape has been determined, displacement measurements are performed to evaluate the surface deformation during various experiments on model canvas paintings (see Sections 3 and 4), or for monitoring purposes of a work restoration (see Section 3.5). In the following, the kinematic basis is also NURBS-based, namely, the displacement of any point of the surface is obtained by moving the control points, which will deform the surface according to the chosen parameterization.

The stereocorrelation functional becomes (Dufour et al., 2015b)

η 2 ({dP }) = 2 i=1 ROI (g ci (ξ, {P + dP }) -f ci (ξ, {P + dP })) 2 (7)
and is minimized via Gauss-Newton scheme with respect to the parameterization of the displacement eld {dP } of the control points for each considered time. g ci denotes the image of the deformed conguration shot by the i-th camera. The interested reader will nd additional details on such isogeometric approaches in [START_REF] Beaubier | CAD-based calibration of a 3D-DIC system: Principle and application on test and industrial parts[END_REF]Dufour et al., 2015a,b).

Restoration of the Descent of the Cross

The rst case study is associated with the restoration of the Descent of the Cross (Saint-Maurice basilica, Switzerland, see Figure 1) for which the tensioning system had to be studied [START_REF] Grau | Conservation et restauration de la Descente de Croix conservée dans le ch÷ur de la basilique de Saint-Maurice en Valais[END_REF]. Its size is 56.3 × 75.3 cm. Due to signicant humidity variations in the basilica, the canvas tension varied and may lead to wrinkles that are detrimental to the visual quality of the work and its aging. Two tensioning systems have been investigated on mockups (i.e., model canvas paintings) that were submitted to humidity variations. The 3D shapes and motions were measured via isogeometric stereocorrelation. Once the system was implemented on the Descent of the Cross painting, its 3D shape and motions were also monitored via isogeometric stereocorrelation to validate the chosen technical solution. The recorded data spanned from September 7, 2013 to October 13, 2014. For the analyzed 401 days the relative humidity varied between 77 % and 30 %

(with a mean level of 58 %). Meanwhile the mean temperature was equal to 18.7 °C (with a standard deviation of 1.6 °C). The spikes in the reported data correspond to instants of time when the heating system was turned o and on.

From these data it is concluded that the main parameter to be investigated is the relative humidity. The latter is critical for works on canvas [START_REF] Malowany | Application of 3D digital image correlation to track displacements and strains of canvas paintings exposed to relative humidity changes[END_REF] and wood [START_REF] Vici | Mechanical response of wooden boards subjected to humidity step variations: climatic chamber measurements and tted mathematical models[END_REF]. 1994). The latter no longer provides good support for the painted layer that deforms signicantly. Being only loosely maintained the canvas and pictorial layer may start wrinkling, which eventually may initiate paint cracking [START_REF] Berger | Interaction between canvas and paint lm in response to environmental changes[END_REF]. The severe conservation conditions reported in Figure 2 have consequences on the visual rendering of the painted work but more importantly they induce damage that may jeopardize its conservation. These degradation phenomena were observed on the surface of the Descent of the Cross (Figure 3). In order to avoid such phenomena to happen, two restoration routes consist of either adding interactive strainers or backing boards [START_REF] Roche | Inuence du type de châssis sur le vieillissement mécanique d'une peinture sur toile[END_REF][START_REF] Roche | Comportement mécanique des peintures sur toiles[END_REF].

The former uses springs that compensate for canvas motions. The latter consists of adding an element made of synthetic materials on the back of the strainer to create stable climatic conditions between the back of the work and the protection. In the present case, the choice of the tensioning system is driven by the strainer conservation and the location of the work in the basilica. In particular, the strainer being mounted on a wood trim so that its back is not in direct contact with the stone wall. Consequently, adding a backing wall was not considered. To limit the dimensional variations of canvas, one of the questions to be answered is whether the tensioning system can be mounted directly on the original strainer. Consequently, two dierent systems were evaluated.

Identical pinewood strainers were built with xed-angles and with all edges chamfered. The strainers were 28.2 × 37.6 cm in size, with 3.7 cm wide and 2.8 cm thick beams. Each canvas support was cut from linen (320 g/m 2 ), which had been wetted and stretched three times on a large loom. The canvasses were then sized with rabbit skin glue (10 %), primed with one layer of red ochre in linseed oil. Both size and ground were applied with a large brush.

The samples were left to dry for three months before being stretched onto the strainers using two dierent attachment congurations. One series of paintings had a strip lining made of monolament polyester fabric (18 g/m 2 ). The strainer proles were lined with Teon and paintings were stapled on the back face with 4-cm separation between two attachments (Figure 4 The calibration of the stereo-rigs was performed using the structure of the easel as the calibration target in order to have a 3D information, which was not available from the canvas alone (i.e., quasi planar surface). Since out of 180 plane displacements were expected to remain small, the dierent heights probed thanks to the easel were sucient. The initial shape of each canvas was measured starting from a simple planar model. Figure 6 shows an example of the initial surface model and the measured shape for one of the studied canvasses. In order to measure the real shape, cubic splines (i.e., 4 control points along each To monitor the relative humidity in the climatic chamber, the same thermohygrometer as for the basilica measurements was used, and was located in the center of the climatic chamber.

Analysis of two tensioning systems

By analyzing two sets of picture pairs acquired before any humidity variation the measurement resolution is estimated. In order to measure 3D displacements induced by wrinkles (i.e., leading to complex shapes), 13 control points along each direction are used, thereby leading to 13×13×3 degrees of freedom. In the present case, the standard displacement uncertainty is equal to 20 µm in the outof-plane direction, and 5 µm in in-plane directions. During the stereocorrelation analyses the levels of gray level residuals remain very low (i.e., their root mean square remains of the order of 1 % of the dynamic range of the gray level pictures in the reference conguration). Consequently, the following results are deemed trustworthy.

Figure 8 shows the surface shape of two studied canvasses. In both cases small deviations to atness are observed (i.e., corresponding to the three components of displacements shown in the gure). They are due to the fact that the strainers were not perfectly at and that the tension was applied manually.

Their amplitudes remain very small, namely, less than ±0.5 mm for system no. 2, and even lower for system no. 1. To characterize the deviation from surface atness, the root mean square (RMS) level is also reported. The RMS atness gap is equal to 13 µm for system no. 1 and 23 µm for system no. 2.

These very small values cannot be distinguished by bare eyes but quantied via global stereocorrelation. Figure 9 shows pronounced canvas wrinkling due to the fact that the ten-sioning systems cannot totally compensate for canvas extensions. In both cases wrinkles form immediately when the relative humidity is increased to 70-80 %.

Moreover, when it is decreased back to 40 % levels, the canvasses instantaneously recover their initial shape. It is worth noting that more control points are used to measure the 3D deformations (Figure 8) in comparison with the initial shape (Figure 6) because of the expected deformation complexity induced by wrinkling (i.e., 13 × 13 instead of 4 × 4 points).

There is a clear dierence between the two solutions since system no. 1 leads to a maximum amplitude of the order of 1 mm, as opposed to the second one for which the maximum amplitude is three times the former. More importantly, the wavelength of the former is equal to ≈ 2/5 the width of the frame. For the latter the wavelength is ≈ 2/3 of the frame width. The RMS atness gap is equal to 74 µm for system no. 1 and 158 µm for system no. 2.

(a) (b)

Figure 9: Comparison of the deformed shape of canvas using two dierent tension devices for the largest relative humidity variation. The dimensions are expressed in cm. Online version: corresponding movie for 80 analyzed steps These comparisons were repeated ve times under similar experimental conditions. The same type of results was observed (i.e., number of wrinkles, deformation levels). From this analysis, system no. 1 was selected (Figure 1). The rst reason was to avoid as much as possible large amplitude wrinkles, which would be detectable by bare eyes. The second reason was associated with the Teon tape that allows canvas tension to be as uniform as possible [START_REF] Young | The role of canvas attachments in the strain distribution and degradation of easel paintings[END_REF], which is more dicult to ensure with system no. 2. In both cases, the wrinkles are due to the fact that the sizing absorbs water vapor from the air, which causes dimensional variations. The latter ones induce extensional strains of canvasses and all the other layers, which cannot be compensated because they are stapled on xed stretchers.

In-situ monitoring

With the previous results, it was decided to apply the tension under given hygrometric condition (i.e., the mean relative humidity monitored in Saint Maurice basilica). Similarly, the temperature was close to that of the basilica (i.e., 21 °C). In order to check how the tension was applied, the Descent of the Cross was monitored for 24 h after being restored. Pairs of pictures are shot in 15minute intervals for dierent relative humidity levels. It is worth noting that the studied work has numerous areas with very low contrast (Figure 1). Even under such demanding conditions, isogeometric stereocorrelation was run and did not induce any convergence issue. Even though the raw painted side has numerous low contrast zones (Figure 1), it was analyzed via (global) isogeometric stereocorrelation with no additional patterning. Thanks to the global span of NURBS interpolations these zones are part of the global analysis. The calibration of the stereo-rig was performed using the easel as calibration object. The surface model used in this case was a cubic spline (i.e., single patch composed of 4 × 4 × 3 degrees of freedom) with all the corresponding weights equal to 1.

This choice is again dictated by the fact that the initial shape is very close to a plane. In order to better capture wrinkles, the displacement measurement is performed using a single patch made of 4 × 7 × 3 degrees of freedom.

By following the same procedure as before, the standard displacement uncertainty is equal to 25 µm in the out-of-plane direction, and 5 µm in in-plane directions. The correlation residual levels observed in the present analyses were of the same order of magnitude as for model canvas paintings. This result validates the registration procedure under such challenging conditions. Figure 10 shows the displacements of the canvas and the corresponding relative humidity level. The displacement amplitudes remain less than 1 mm along the normal direction to the pictorial layer. Such small amplitudes are expected to be undetectable by bare eyes. This in-situ analysis has allowed displacement elds to measured during canvas tensioning. Most of the motions occur during the rst ve hours during which the tensions are equilibrated [START_REF] Hedley | Relative humidity and the stress/strain response of canvas paintings: uniaxial measurements of naturally aged samples[END_REF]. This redistribution of tensions is inuenced by the relative humidity level. The restored Descent of the Cross has been brought back to the basilica of Saint-Maurice on December 20, 2014. The canvas tension is man-controlled each season. Having shown that stereocorrelation is a viable alternative, it could also be used to have a more quantitative assessment of in-situ canvas deformations.

Punch test on canvas

The second study was concerned with the restoration of the Madonna with Child (Home of Charity, Poissy, France, see Figure 11). It consisted of many steps due to the severe state of aging of the pictorial layer (i.e., dusting, cleaning, unwinding, lling, unwrapping, tear consolidation, lling, varnish application, retouching [START_REF] Gonnet | Restauration d'une Vierge à l'Enfant, la redécouverte d'une ÷uvre grâce à sa restauration[END_REF]). One additional issue was related to the reliability of nailed bonds of canvas on strainers. Since such type of study is destructive, it could not be performed on the actual work. Mockups were again considered for which a speckle pattern was sprayed on the back face. There are many experimental procedures to test textiles [START_REF] Lebrun | Evaluation of bias-extension and picture-frame test methods for the measurement of intraply shear properties of pp/glass commingled fabrics[END_REF][START_REF] Cao | Characterization of mechanical behavior of woven fabrics: Experimental methods and benchmark results[END_REF]Chen, 2010). To characterize the mechanical behavior of canvas and its bond with the strainer a punch test was selected. It enabled the canvas deformation to be studied under standard bond with the strainer, namely, with nails. The external strainer size is 28 × 36 cm. In that case, the 3D motions are very important and the studied canvas wrinkles as more displacements are prescribed. These challenging conditions require the NURBS-based kinematics to be enriched during the dierent phases of displacement measurement. Consequently, an open book target was used to calibrate the stereo system (Figure 13). One of its advantages also lies in the fact that being non planar, only one set of images is sucient for calibration purposes [START_REF] Besnard | Characterization of necking phenomena in high speed experiments by using a single camera[END_REF]. changing from a simple rectangular plate (shown in Figure 14), to a much more complex geometry. To measure the initial 3D shape, 4 × 4 control points are chosen (for the same reasons as before). However, the deformed shape asked for additional degrees of freedom, which are gradually added. The renement procedure used herein is very simple. At specic intervals, 2 × 2 control points are added to the knot vector (one on each side of the central value 0.5 to preserve symmetry), thus creating new knot spans in knot vector, and new `elements' inside the NURBS patch. In the present case, 2 new knot spans (thus control points) are added every 5 image. Consequently, the surface model is rened starting from the center and moving to the edges. This kind of strategy (i.e., knot renement) is a standard way to increase the number of degrees of freedom without increasing the degree of the solution [START_REF] Cottrell | Isogeometric Analysis: Toward Integration of CAD and FEA[END_REF] (i.e., in a similar way as the so-called h-renement technique in nite element methods [START_REF] Zienkievicz | The Finite Element Method[END_REF]). It also preserves maximum continuity inside the patch (i.e., through `elements'), thus allowing smooth solutions to be accurately captured. it is observed that as the sphere displacement level increases, the registration quality degrades in the vicinity of the sphere pole. Yarn motions occur, which are not compatible with the chosen kinematics. Moreover, at the end of the experiment, some yarns presumably fail. This (discontinuous) mechanism cannot be captured by the (continuous) kinematic basis used herein (NURBS).

However, it is detected by the correlation residuals.

The deformed shape of the tested canvas is shown in Figure 17 in addition to the out-of-plane displacement. As more displacements are applied to the sphere, more wrinkles appear. These wrinkles come from the method used to attach the canvas to the strainer (i.e., with nails). As there is a nite number of nails on the strainer, the tension in the canvas is (spatially) oscillating thereby inducing wrinkles of the canvas. At the end of the experiment, the nailed bond was still working properly while canvas degradation occurred (see Figures 15(b Isogeometric stereocorrelation was successfully applied to study canvas wrinkling. This deformation mechanism is to be reduced as much as possible for painted works on canvas. Conversely, in a punch test wrinkling will occur and needs to be assessed. Because higher order kinematics was sought, NURBSbased stereocorrelation was considered for the initial 3D shape and its deformation induced either by relative humidity variations or during a punch test.

The rst case was associated with the restoration of the Descent of the Cross for which the tensioning system had to be studied. Due to signicant humidity variations, canvas tension varied, which may lead to wrinkles that are detrimental to the visual quality and aging of the work. Two tensioning systems have been investigated on model canvas paintings that are submitted to humidity variations. The 3D shapes and motions were measured via isogeometric stereocorrelation. In the present case, the studied canvasses were patterned with random speckles. The two tensioning system did not lead to the same curvature elds. Once the system was selected for the Descent of the Cross, its 3D shape and motions were also monitored via isogeometric stereocorrelation to validate the chosen technical solution even though the painted layer contained areas with very low contrast (the whole pictorial layer could be analyzed thanks to the global support of NURBS).

The second study was concerned with the restoration of the Madonna with Child. The mechanical characterization of canvas and its bond was performed via a punch test on a model canvas nailed on its strainer. In that case, the 3D motions were very important and the studied canvas wrinkled as more displacements were prescribed. These challenging conditions required the kinematic description based on NURBS to be enriched during the dierent phases of measurement. Further, the analysis of the registration residuals showed that early stages of yarn motions and breakage are detected.

These two case studies prove the feasibility of isogeometric stereocorrelation to be applied in the context of restoration of canvas paintings. Thanks to its versatility, it could be adapted to the two studied situations that induced either very small deformations of canvas, very large levels, or low contrasted works. If more local analyses are required, namely, the displacements at the yarn levels, the same formalism can be used by considering the NURBS or FE model of canvas [START_REF] Naouar | Mesoscale FE analyses of textile composite reinforcement deformation based on X-ray computed tomography[END_REF].

Figure 1 :

 1 Figure 1: Front (top) and back (bottom) faces of the Descent of the Cross prior to (left) and after (right) restoration. The size of the work is 56.3 × 75.3 cm

Figure 2 :

 2 Figure 2: Change of relative humidity and temperature inside the Saint-Maurice basilica from September 7, 2013 to October 13, 2014. The dashed vertical lines correspond to transitions from even to odd months

Figure 3 :

 3 Figure 3: Damage induced by canvas wrinkling of the Descent of the Cross prior to its restoration

Figure 4 :

 4 Figure 4: Studied tensioning systems

Figure 6 :

 6 Figure 6: Initial (a) and measured (b) 3D shape of one of the studied canvasses. The blue dots depict the control points of the NURBS patch for the initial shape measurement. Dimensions are expressed in cm

Figure 7 :

 7 Figure 7: Relative humidity cycles for the two model canvas paintings

Figure 8 :

 8 Figure 8: Comparison of the initial shape of canvas using two dierent tension devices. Dimensions are expressed in cm. The blue dots depict the control points of the NURBS patch used for the subsequent displacement measurements

Figure 10 :

 10 Figure 10: Displacement of the restored Descent of the Cross at the end of the relative humidity variation. All dimensions are expressed in cm. Online version: corresponding movie for 278 analyzed steps

Figure 11 :

 11 Figure 11: Front and back faces of the Madonna with Child prior to restoration. The size of the work is 33 × 39 cm

Figure 12

 12 Figure 12(a) shows the experimental conguration of the punch test. A sphere 50-mm in diameter made of steel was selected. It was moved at a speed of 50 µm/s by the actuator of the servohydraulic testing machine. As in the previous study, a model canvas painting was considered. It could be speckled with black and white paint on its back face (Figure 12(b)). The loaded strainer was simply supported with respect to the frame of the testing machine. The stereo-system consisted of 2 Canon EOS 60D cameras with 50-mm objective lenses. It was located above the tested canvas. The interframe time was 10 s.

Figure 13 :

 13 Figure 13: Image pair that was used for the calibration of the stereo-rig with an open book target

Figure 14 :

 14 Figure 14: Initial canvas shape. All dimensions are expressed in mm. The blue dots depict the control points of the NURBS patch

  Figure 15: (a) Change of number of kinematic degrees of freedom with the image number during the punch test. (b) Load displacement curve of the punch test (the open circles correspond to the analyzed picture pairs)

  ) and 16(b)). This observation validates the nail spacing chosen in the present study.

Figure 17 :

 17 Figure 17: Deformed canvas and out-of-place displacements via isogeometric stereocorrelation for the peak load. All dimensions are expressed in mm. The blue lines depict the knot spans (similar to elements in a nite element formulation) of the NURBS formulation. Online version: corresponding movie for 117 analyzed steps

Acknowledgements

This work was partially supported under the PRC Composites, French research project funded by DGAC, involving SAFRAN Group, ONERA and CNRS.


Ante Buljac

Thibault Taillandier-Thomas

Lukas Helfen

Thilo F Morgeneyer

François Hild
email: hild@lmt.ens-cachan.fr

Evaluation of Measurement Uncertainties of DVC applied to Laminography Data

Keywords: Digital Volume Correlation, Kinematic measurements, Resolution, Synchrotron laminography, Uncertainty quantication

The eect of dierent data acquisition parameters on the displacement and strain uncertainties of kinematic elds is assessed via Digital Volume Correlation (DVC) applied to three-dimensional (3D) data sets imaged by synchrotron laminography. The measurement uncertainty is estimated for i) dierent materials and associated varying contrast, ii) repeated scans, i.e., uncertainty due to the imaging system and 3D reconstruction iii) rigid body motions between two subsequent scans, iv) rescaling of the gray level histogram during 32-bit oating point to 8-bit integer data conversion, v) changes in the beam properties, i.e., use of a monochromator or an undulator, and vi) changes in camera/detector characteristics. It is found that the amount of image contrast is not the only parameter that controls uncertainties for different materials. Further, the rigid body motion procedure should be preferred over repeated scans as it provides more conservative measurement uncertainty values. The applied 32 to 8-bit conversion procedure, beam tuning and detector characteristics hardly aect the measurement uncertainty.

Introduction

Laminography [START_REF] Gondrom | X-ray computed laminography: an approach of computed tomography for applications with limited access[END_REF][START_REF] Helfen | On the implementation of computed laminography using synchrotron radiation[END_REF] can be considered as a 3D imaging technique complementary to computed tomography since it is particularly adapted to twodimensionally extended specimens. Employing synchrotron radiation [START_REF] Helfen | High-resolution three-dimensional imaging of at objects by synchrotron-radiation computed laminography[END_REF][START_REF] Helfen | Investigation of hybrid pixel detector arrays by synchrotron-radiation imaging[END_REF] the main advantages oered by laminography are that regions of interest in large plate-like samples can be imaged with high spatial resolutions [START_REF] Helfen | Investigation of hybrid pixel detector arrays by synchrotron-radiation imaging[END_REF], high energies [START_REF] Houssaye | Three-dimensional pelvis and limb anatomy of the Cenomanian hind-limbed snake Eupodophis descouensi (Squamata, Ophidia) revealed by synchrotron-radiation computed laminography[END_REF][START_REF] Maurel | Three-dimensional investigation of thermal barrier coatings by synchrotron-radiation computed laminography[END_REF] and using dierent phase-contrast imaging modalities [START_REF] Helfen | Phase-contrast and holographic computed laminography[END_REF][START_REF] Altapova | Phase contrast laminography based on Talbot interferometry[END_REF].

In materials science, applications with panel-and sheet-like samples can be investigated in-situ contrary to tomography, where samples close to match stick or rod shapes [START_REF] Baruchel | X-Ray Tomography in Material Sciences[END_REF][START_REF] Maire | On the Application of X-ray Microtomography in the Field of Materials Science[END_REF][START_REF] Buère | In Situ Experiments with X ray Tomography: an Attractive Tool for Experimental Mechanics[END_REF] are required. The liberty to employ plate-and sheet-like specimens via synchrotron laminography allows more engineeringrelevant loading conditions to be applied, say, during the study of damage initiation and growth inside materials. For instance, the development of damage has been observed for low stress triaxialities ahead of notched samples during ductile tearing [START_REF] Morgeneyer | Ductile crack initiation and propagation assessed via in situ synchrotron radiation computed laminography[END_REF][START_REF] Morgeneyer | In situ 3D observation of early strain localisation during failure of thin al alloy (2198) sheet[END_REF][START_REF] Morgeneyer | On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3)[END_REF].

It is however worth noting that the sampling of the 3D Fourier domain of the region of interest is incomplete in laminography [START_REF] Helfen | On the implementation of computed laminography using synchrotron radiation[END_REF], which leads to imaging artifacts [START_REF] Xu | Comparison of image quality in computed laminography and tomography[END_REF]. These artifacts are assumed to degrade the displacement and strain resolutions when digital volume correlation (DVC) techniques are applied [START_REF] Morgeneyer | 3D digital volume correlation of synchrotron radiation laminography images of ductile crack initiation: An initial feasibility study[END_REF]. Laminography can still be considered as a relatively new technique, especially when considering its use for measuring kinematic elds and strain changes inside material microstructures. It should also be noted that the attainable spatial resolution in synchrotron laminography is higher in directions perpendicular to the rotation axis than in the direction parallel to it. One should therefore also expect higher strain measurement uncertainties along the rotation axis direction.

The aim of the present paper is to assess in a systematic manner the displacement and strain uncertainties measured via DVC when considering synchrotron laminography data. There are various ways of estimating the uncertainties of correlation techniques. When only one acquisition is available, articial motions i.e., generally uniform translations are applied to create a new volume that is subsequently registered with the reference volume [START_REF] Liu | Accuracy and precision of digital volume correlation in quantifying displacements and strains in trabecular bone[END_REF][START_REF] Roux | Three dimensional image correlation from X-ray computed tomography of solid foam[END_REF][START_REF] Benoit | 3D analysis from micro-MRI during in situ compression on cancellous bone[END_REF][START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using three-dimensional correlation of laboratory X-ray microtomography images[END_REF][START_REF] Hild | Three dimensional analysis of a tensile test on a propellant with digital volume correlation[END_REF][START_REF] Madi | Computation of full-eld displacements in a scaold implant using digital volume correlation and nite element analysis[END_REF]. This rst type of analysis mainly probes the interpolation scheme of the gray levels (GL) to achieve sub-voxel resolutions.

The second type of approach also uses only one reference volume but consists of adding noise to create a new volume, which is correlated with its noisefree reference [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF][START_REF] Leclerc | Digital volume correlation: What are the limits to the spatial resolution[END_REF]. The sensitivity to noise is addressed in this type of procedure. To be fully representative the model should incorporate the full covariances associated with radiograph acquisition and 3D reconstruction.

Last, the previous two procedures can be applied experimentally [START_REF] Liu | Accuracy and precision of digital volume correlation in quantifying displacements and strains in trabecular bone[END_REF][START_REF] Limodin | Analysis and artifact correction for volume correlation measurements using tomographic images from a laboratory x-ray source[END_REF]. This method consists of analyzing two consecutive scans of the sample in a given loading state (it may also be unloaded) without rigid body motion (called bis in the following) between the two acquisitions or with a deliberately introduced one (called rbm). The main advantage is that both sources of error are investigated at the same time for the material of interest. This last method will be adopted in this work.

The standard uncertainties are reported to have levels higher than those observed in standard 2D-DIC techniques when using local or global DVC approaches [START_REF] Liu | Accuracy and precision of digital volume correlation in quantifying displacements and strains in trabecular bone[END_REF][START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using three-dimensional correlation of laboratory X-ray microtomography images[END_REF][START_REF] Limodin | Analysis and artifact correction for volume correlation measurements using tomographic images from a laboratory x-ray source[END_REF][START_REF] Madi | Computation of full-eld displacements in a scaold implant using digital volume correlation and nite element analysis[END_REF]. For laminography, an additional degradation has been observed [START_REF] Morgeneyer | 3D digital volume correlation of synchrotron radiation laminography images of ductile crack initiation: An initial feasibility study[END_REF] for an aluminum alloy with a very low volume fraction of markers. Since laminography is a very recent technique, there are very few results dealing with DVC calculations [START_REF] Roux | Digital volume correlation from tomographic images: Results and challenges[END_REF][START_REF] Morgeneyer | 3D digital volume correlation of synchrotron radiation laminography images of ductile crack initiation: An initial feasibility study[END_REF][START_REF] Taillandier-Thomas | Localized strain eld measurement on laminography data with mechanical regularization[END_REF][START_REF] Buljac | Slant strained band development during at to slant crack transition in AA 2198 T8 sheet: in situ 3D measurements[END_REF][START_REF] Morgeneyer | On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3)[END_REF]. In particular, measurement uncertainties have been evaluated [START_REF] Morgeneyer | 3D digital volume correlation of synchrotron radiation laminography images of ductile crack initiation: An initial feasibility study[END_REF] for one given microstructure (AA 2139) that was deemed very dicult for DVC calculations since the vol-ume fraction of secondary particles was less than 1 %. This microstructure is also considered hereafter in addition to two additional ones. A second class of materials presented in this work is nodular graphite cast iron. The microstructure scales and topology signicantly dier from aluminum alloys.

Hence, X-ray projections, reconstructed volume contrast and the nal conditions for DVC registrations are changed.

In the reported analyses a global approach to DVC is chosen in which the displacement eld is based on discretizations made of 8-noded elements (i.e., C8-DVC [START_REF] Roux | Three dimensional image correlation from X-ray computed tomography of solid foam[END_REF], see Appendix A). The studied materials and the way the laminography data are obtained are rst presented. Next, the measurement uncertainties are evaluated for the chosen microstructures for two dierent cases:

• measurement resolution applied to repeated scans (bis ) of the object;

• measurement uncertainty applied to scans of the object moved (rigid body motion -rbm) prior the second acquisition.

Possible relationships between the underlying microstructure, sample geometry, experimental setup and uncertainty levels are discussed. Last, additional eects are analyzed on a more limited amount of scans:

• inuence of 32-bit oating point to 8-bit integer data conversion (i.e., rescaling of the gray level histogram);

• inuence of the beam properties (i.e., use of monochromator or undulator);

• inuence of the employed optical camera/detector.

The listed items will be detailed in Section 2.3.

Analyzed Materials and Congurations

Four dierent materials will be studied herein:

• AA2139 grade (i.e., Al-Cu-Mg alloy), whose intermetallic volume fraction has been determined to be 0.45 % with an initial void volume fraction of 0.34 % (see Figure 1(a)). A feasibility study and analysis was performed in terms of measurement uncertainties [START_REF] Morgeneyer | 3D digital volume correlation of synchrotron radiation laminography images of ductile crack initiation: An initial feasibility study[END_REF]. Rather high levels of uncertainties were reported and one question is related to the representativeness of these results with other materials such as those studied hereafter. Samples with two dierent heat treatments are examined, namely, T3 denotes solution heat-treated, stretched between 2-4 % and naturally aged to obtain the T351 (T3) condition.

Conversely, T8 stands for an articial aging treatment. These heat treatments should not change the image contrast at micrometer scale.

The at to slant transition was studied for this alloy in its T3 state [START_REF] Morgeneyer | On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3)[END_REF].

• AA2198 grade (i.e., Al-Cu-Li alloy), which has no initial porosity and 0.3 % volume fraction of metallic inclusions (see Figure 1(b)). For this grade, the at to slant transition associated with ductile fracture was also analyzed thanks to DVC measurements [START_REF] Morgeneyer | In situ 3D observation of early strain localisation during failure of thin al alloy (2198) sheet[END_REF][START_REF] Buljac | Slant strained band development during at to slant crack transition in AA 2198 T8 sheet: in situ 3D measurements[END_REF]. This material has the lowest volume fraction of secondary phases (i.e., particles and voids). Again, two dierent heat treatments (T3 and T8) will be studied.

• AA6061 grade (i.e., Al-Mg-Si alloy) is a precipitation hardened aluminum alloy, with 0.25 % volume fraction of coarse Mg 2 Si particles, 0.57 % of iron-rich intermetallics and 0.05 % of porosity (see Figure 1(c)). Compared to Figure 1(a-b), less ring artifacts (i.e., concentric with the specimen rotation center) are apparent. Crack initiation and propagation were investigated for such grade [START_REF] Shen | Three-dimensional quantitative in situ study of crack initiation and propagation in aa6061 aluminum alloy sheets via synchrotron laminography and nite-element simulations[END_REF] but no DVC analysis was carried out so far. In that work the T6 heat treatment was inspected.

• Commercial nodular graphite cast iron (EN-GJS-400) represents a second class of materials with dierent microstructural scales (Figure 1(d)). Holes were drilled in the examined plates. In these analyses, an experimental/numerical framework was developed to validate simulations at the microscopic scale [START_REF] Buljac | Numerical Validation Framework for Micromechanical Simulations based on Synchrotron 3D Imaging[END_REF][START_REF] Shakoor | On the choice of boundary conditions 51 for micromechanical simulations based on 3D imaging[END_REF]. The physical size of one voxel is 0.7 µm for aluminum alloys and 1.1 µm for cast iron. Especially for sub-gures (a) and (b), ring artifacts from static features on the projection radiographs are clearly visible

Aluminum alloys

The alloy forming directions will dene the measurement frame used herein (even for cast iron). The rolling direction is denoted as L, the transverse direction as T, and the short-transverse direction in the through-thickness as S. The 3D images of aluminum alloys were obtained at the imaging beamline ID19 [START_REF] Weitkamp | Parallelbeam imaging at the ESRF beamline ID19: current status and plans for the future[END_REF] of the European Synchrotron Radiation Facility (Grenoble, France).

An inclination angle of the specimen rotation axis of approximately 25

• with respect to the beam normal (θ ≈ 65 • ) was chosen as well as a monochromatic beam of around 25 keV X-ray energy obtained from a single-bounce multilayer monochromator or a short-period single-harmonic undulator source providing approximately the same monochromaticity. A series of 1500 radiographs is then used to reconstruct 3D volumes by using a ltered-back-projection algorithm [START_REF] Myagotin | Ecient Volume Reconstruction for Parallel-Beam Computed Laminography by Filtered Backprojection on Multi-Core Clusters[END_REF]. For aluminum alloys all examined reconstructed volumes belong to compact tension (CT)-like specimen regions ahead of the notch root (AA2139 and AA2198) or crack tip (AA6061). The experimental setup is illustrated in Figure 3. The CT-like specimen has dimensions 60×70×1 mm. It contains either a machined notch of radius 0.17 mm produced by electrical discharge machining or fatigue pre-cracking has been utilized for AA6061. The rst case studied in detail is a CT-like sample made of AA2139 T3.

Nodular graphite cast iron

Commercial nodular graphite cast iron (serial code EN-GJS-400) is also analyzed. This material has a broader gray level distribution due to the presence of two highly contrasted materials (i.e., ferrite and graphite, see Figures 1 and2). Additional (i.e., morphological) characterizations of cast iron are reported in Appendix D. It is believed that such microstructure is a priori more favorable to DVC analyses. For the cast iron grade considered herein two tensile samples with dierent geometries will be examined. First, a plate with a central hole that localizes stresses in the surrounding ligaments is analyzed (Figure 7). One of the ligaments has been scanned during the experiment as shown in Figure 7. The inspected region has a size of 512 × 512 × 512 voxels.

The results from this ROI have been used to drive micromechanical simula-tions [START_REF] Buljac | Numerical Validation Framework for Micromechanical Simulations based on Synchrotron 3D Imaging[END_REF]. The position of the ROI in the global coordinate system is shown in Figure 7 while in the through-thickness direction (S) the ROI and sample mid-thickness planes coincide. No rbm scans were performed with this sample geometry. The second geometry, which is inspired by the work of Weck et al. [START_REF] Weck | Experimental investigation of void coalescence in metallic sheets containing laser drilled holes[END_REF],

is shown in Figure 8. The scanned region incorporates two holes while the ROI employed in DVC calculations is mainly concentrated in the ligament between the holes. Figure 8 shows the location of the Region of Interest (ROI)

in the reference conguration. The rbm conguration is moved by u T = 68 voxels and u L = 267 voxels. The results from this ROI have been used

to outline all the advantages of using measured DVC boundary conditions when running micromechanical numerical simulations [START_REF] Shakoor | On the choice of boundary conditions 51 for micromechanical simulations based on 3D imaging[END_REF]. To be able to keep large ROI sizes, the original reconstructed volumes are a priori coarsened, each 8 neighboring voxels are averaged to form one supervoxel (i.e., binning of 2 × 2 × 2). In the following, no distinction will be made between this coarsened conguration and the other ones and the term voxel will still be used irrespective of such operation. Five dierent procedures will be applied to analyze the displacement and strain uncertainties of DVC applied to laminography data:

• repeated scan -bis: two scans are acquired without any motion between the two acquisitions. This procedure is standard when aiming for the resolution of the measurement technique [START_REF]International Vocabulary of Metrology -Basic and General Concepts and Associated Terms, VIM. International Organization for Standardization[END_REF].

• rigid body motion -rbm: two scans are acquired when a rigid body motion is applied prior to the second acquisition. This case is deemed dicult [START_REF] Morgeneyer | 3D digital volume correlation of synchrotron radiation laminography images of ductile crack initiation: An initial feasibility study[END_REF] because the reconstruction artifacts (e.g., rings) do not follow the motion and may severely bias the displacement measurements with low contrasted images as studied herein (Figure 2). To assess the contribution induced by rigid body motions bis and rbm cases are compared.

• 32-bit oating point to 8-bit integer conversion: the reconstruction of 3D volumes from radiographs involves at one stage to map the 32bit oating point values onto 8-bit integer digitizations (i.e., rescaling)

given the high level of noise. The transformation is performed by choosing the gray level range for the data to be rescaled. The same gray level range is generally applied for conversion of a series of in situ scans. This may depart from the gray level conservation, which is the underlying principle of DVC (see Equations ( 3) and ( 4)). The inuence of dierent rescaling parameters is studied in the case of the AA2139 T3 sample.

• changes in the beam properties: since the data originate from dierent experimental sessions, the inuence of the experimental setup, particulary the monochromator, is also evaluated. Experimental data available for this analysis are from AA2139 T3 and AA2198 T8 alloys observed with a monochromator opposed to AA2139 T8 and AA2198 T3 cases scanned without monochromator but using X-rays from a singleharmonic undulator.

• detector characteristics: the inuence of optical camera/detector is also probed. The scans available for AA2198 T8 alloy are sampled with two dierent cameras.

The overview of listed procedures and cases in which they will be studied are summarized in Table 1. a linear interpolation and measured elds is an estimation of the measurement errors [START_REF] Limodin | Analysis and artifact correction for volume correlation measurements using tomographic images from a laboratory x-ray source[END_REF]. The root mean square error associated with each displacement component is obtained. From that quantity, the displacement uncertainty σ u corresponds to the mean of these three values. Eight-noded cubes are chosen in the nite-element based DVC code (i.e., C8-DVC [START_REF] Roux | Three dimensional image correlation from X-ray computed tomography of solid foam[END_REF], see Appendix A).

The size of each C8 element is the length expressed in voxels of any edge.

Dierent sizes are considered, namely, 16, 24, 32, 48 and 64-voxel elements.

However for one case (AA2139 T8) calculations could not be performed with 16-voxel elements because of divergence.

The logarithmic (or Hencky) strains are evaluated by using the mean value of the displacement gradient tensor in each C8 element [START_REF] Hild | Three dimensional analysis of a compression test on stone wool[END_REF]. The reason for that choice is to account for possible large rotations, even though very small stretches are expected in the following. In the present case, von

Mises' equivalent strain eq (to assess plastic strains [29, 9, 32]) is reported.

The reason for this choice will become clearer in the sequel since the level of uncertainties will generally be higher than elastic strains in the studied aluminum alloys. Such small uncertainties can only be achieved by considering regularized DVC [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF][START_REF] Leclerc | Digital volume correlation: What are the limits to the spatial resolution[END_REF][START_REF] Taillandier-Thomas | Localized strain eld measurement on laminography data with mechanical regularization[END_REF], which is not used herein. Further, it has been checked that the mean strain error was at least one order of magnitude less than the corresponding standard deviation σ . Only the latter will be reported hereafter.

3 Uncertainty quantications

Repeated scanbis case

For bis cases no rigid body motions are carried out between the two acquired scans so that the eect of noise associated with laminography, reconstruction and DVC will be probed. For this analysis the experimental data are from AA2139 T8, AA2198 T3, AA6061, and cast iron samples (Table 1). For cast iron, two repeated scans (i.e., bis cases) are available for the central hole experiment (Figure 7), namely, 0 mm in the undeformed state and one more in a deformed state (i.e., 0.744 mm) for the applied displacement.

For the two-hole experiment as for all aluminum alloys, only the reference conguration is used for uncertainty quantications.

Figure 9(a-c) shows the standard displacement uncertainties along the three directions as functions of the element size for all studied congurations.

There is no clear eect of the three directions on the level of displacement uncertainty. This observation is conrmed when each component uncertainty is compared with the mean value reported in Figure 9(d). Compared with earlier results [START_REF] Morgeneyer | 3D digital volume correlation of synchrotron radiation laminography images of ductile crack initiation: An initial feasibility study[END_REF], the present uncertainties are lower and there is no clear dierence between the direction along the rotation axis and those in the perpendicular plane. One of the reasons is that in present case, no motions are applied between the two acquisitions. This eect will be investigated in Section 3.2.

σ u = A α+1 α and σ = B σ u (1)
where A is a constant expressed in voxels, and B a dimensionless quantity that depends on the way the strains are computed from the measured displacements. However, in the case of AA6061 T6, it can be noted that B is not constant, but starts to decrease after reaching certain element sizes resulting in lower strain uncertainty levels compared with AA2198 T3 and AA2139 T8 grades for the element size = 64 voxels. This can be partly explained by fact that DVC benets from reaching certain element sizes that overcome characteristic voids and intermetallic particles spacings (see Figure 6).

Eect of rigid body motionrbm case

For rbm cases rigid body motions are applied between the two scans acquired.

The experimental data available for this analysis are from AA2139 T3, AA2139 T8, AA2198 T3, AA2198 T8, AA6061 and cast iron samples (Table 1). Dierent levels and directions of rigid body motions have been applied

in each case (see Section 2), the minimum level being for AA2198 T8 where only small motions in the S-direction occurred.

Figure 12 shows the displacement uncertainties for the three components and the mean level. For the samples that have experienced very large displacements along the L-direction (i.e., AA2139 T8, AA2198 T3, AA6061 T6), the corresponding displacement uncertainty degrades in comparison with the other displacement components. Displacements along the S-direction have also a marked eect (for sample AA6061 T6). The fact that the reconstruction axis has moved with respect to microstructure features has a clear eect on the measurement uncertainties (when compared with bis cases), especially for the cases investigated herein, which are deemed very dicult since the volume fraction of markers is very low (Table 2). However, the overall trend for rbm cases remains the same as in bis cases in terms of dependence with the element size. Along with previous statement, the AA2198 T8 microstructure showed the lowest uncertainties since the position of the reconstruction axis and consequently artifacts remained almost the same. Hence, this rbm case is close to bis cases reported before.

Figure 13 shows the mean displacement uncertainties and the correspond-ing equivalent strain uncertainties. Even though the scatter in terms of strain uncertainties is higher than in the previous case (see Figure 11), the overall levels are very close and do not seem to be aected in a signicant way by the amount of motion.

Element size (voxels)

Comparison between bis and rbm cases

In Figure 14 displacement and strain resolutions are explicitly compared for bis and rbm cases for the studied aluminum alloys and cast iron. After rigid body motions the reconstruction artifacts have changed their relative position with respect to the microstructural features, which is reected in higher uncertainty levels for the rbm cases. This eect is clearly observed on the mean displacement uncertainties in Figure 14(a). It can be noted that the microstructure with no voids and lower contrast (i.e., unfavorable gray level gradient for DVC registration) is more sensitive to rigid body motions.

This eect is less pronounced for strain uncertainties (Figure 14(b)).

Element size (voxels) These last results show that to fully evaluate measurement uncertainties it is better to move the sample before the second acquisition so that the rotation axis no longer coincides with that of the rst reconstructed volume. Hence, in rbm cases all the detrimental eects are accounted for and the reported uncertainty levels represent the stable limit above which mechanical signal is distinguished from the noise even for large displacement amplitudes.

Let us note that in some bis/rbm cases presented above the trends between displacement and strain resolutions dier signicantly, e.g., for AA2198 T3 there is a signicant gap between bis and rbm curves for displacement uncertainties while in the strain resolution graph the values almost coincide. This is probably due to the fact that measured rotations in the displacement uncertainty procedure are extracted globally on the ROI scale, while in the strain resolution procedure the Hencky strains exclude rotations more locally, i.e., at the C8 element level.

Inuence of other parameters

Three additional eects are analyzed with a more limited set of scans, namely, digitization level, monochromator and acquisition device. These parameters can be varied at the synchrotron facility where the scans were acquired.

Beam properties

The monochromator's main function is to extract from an incoming polychromatic X-ray beam a monochromatic beam of adjustable wavelength.

The experimental data for AA2139 T3 and AA2198 T8 are obtained with a multilayer monochromator opposed to AA2139 T8 and AA2198 T3 scanned without monochromator. The multilayer monochromator was needed in order to limit the dose rate, beam hardening artifacts and high X-ray energies from the broad-band undulator source employed, which has rather low fundamental energies and multiple higher harmonics. The cross-section of the beam diracted by the multilayer shows a characteristic modulated intensity pattern [START_REF] Rack | Comparative study of multilayers used in monochromators for synchrotron-based coherent hard X-ray imaging[END_REF] stemming from long-range substrate and coating imperfections.

These intensity modulations are prone to provoke ring artifacts after laminographic reconstruction. The AA2139 T8 and AA2198 T3 data were scanned without monochromator with radiation from a more recent short-period undulator providing mainly a single harmonic of suciently high energy and the higher harmonics are greatly reduced in intensity in comparison with the rst harmonic. In that conguration, no monochromator is therefore needed anymore.

In Figure 16 both mean displacement and equivalent strain resolutions for the available data are shown. The data obtained with the monochromator are depicted with dashed lines and sux M in the legend. There is no clear evidence of an inuence of the monochromator for AA2198 alloy. As mentioned, AA2198 T8 rbm case is the result of small amounts of rigid body motion, hence these data can be understood as a bis case. This is illustrated in Figure 16(a) when compared to 2198 T3 bis data where a good agreement is found except for the larger element sizes. Hence, the monochromator presence may induce some additional uncertainty with longer characteristic wavelengths (i.e., rings), which results in a saturation of the uncertainty levels as shown in Figure 16(a) (see gap between AA2198 T3 bis and AA2198 T8 rbm M for large element sizes).

Element size (voxel) The fact that the AA2139 T3 rbm M case signicantly deviates from AA 2139 T8 rbm should not be directly assigned to the monochromator inuence. It is rather a consequence of dierent levels of applied rigid body motions and more importantly, the volume of the notch present in the reconstructed volume and ROI position with respect to it. Hence, the solution to probe this eect would be the existence of AA2139 T8 bis M data to compare it with its non-monochromator correspondent AA2139 T8 bis in conditions where there is no inuence of rigid body motions.

Detector

The experimental data for AA2198 T8 material are obtained with two dierent types of detectors, namely the ESRF inhouse-designed FReLoN camera with 20x (i.e., 10x objective plus 2x eye-piece) optical magnication and the PCO.edge camera (PCO GmbH, Kelheim Germany) with 10x optical magnication (but both with similar scintillators converting the X-rays to visible light).

The FReLoN camera uses a sensor based on the charge-coupled device (CCD) principle while the PCO.edge camera uses a scientic CMOS sensor.

Both converters have a dynamic range of about 14 bits but the full-well capacity of the PCO.edge is ≈ 5× lower so that less X-ray photons are needed to saturate the detector. Consequently, also less photon statistics may be expected, resulting in noisier reconstructed images. To compensate for this eect to some extent, the number of X-ray projection images is

Discussion

The eect of three additional parameters (i.e., rescaling, monochromator and detector) was studied in the present section. In all reported cases, it was found that their inuence was not as important in comparison with the results discussed in Section 3.

Conclusion

Laminography is still a recent imaging technique [START_REF] Helfen | High-resolution three-dimensional imaging of at objects by synchrotron-radiation computed laminography[END_REF][START_REF] Helfen | Synchrotron-radiation computed laminography for high-resolution three-dimensional imaging of at devices[END_REF]. The use of reconstructed volumes for kinematic measurements has only started a few years ago [START_REF] Roux | Digital volume correlation from tomographic images: Results and challenges[END_REF][START_REF] Morgeneyer | 3D digital volume correlation of synchrotron radiation laminography images of ductile crack initiation: An initial feasibility study[END_REF][START_REF] Taillandier-Thomas | Localized strain eld measurement on laminography data with mechanical regularization[END_REF][START_REF] Buljac | Slant strained band development during at to slant crack transition in AA 2198 T8 sheet: in situ 3D measurements[END_REF][START_REF] Morgeneyer | On strain and damage interactions during tearing: 3D in situ measurements and simulations for a ductile alloy (AA2139-T3)[END_REF]. Consequently, there is a need for metrological assessment of the measured displacement elds and the corresponding strain elds. The present paper aims at discussing ways of evaluating measurement uncertainties in this context. The procedures introduced herein are applied to aluminum alloys leading to weakly contrasted volumes when imaged by X-ray sources and cast iron with low contrast ferritic zones when imaged at high resolutions.

In the Al alloys studied, for any location of the region of interest, the uncertainty level is uniform for repeated scanning (bis ) cases. However, when rigid body motions are applied between two acquisitions the ROI position close to the notch degrades the uncertainty level for AA2198 and AA6061 aluminum grades. The amount of initial voids or secondary particles is not necessarily sucient to reach the lowest uncertainty level. In the present case, the AA2198 grade has the lowest volume fraction of the three aluminum alloys but also the lowest uncertainty. This result calls for caution and it is advisable to acquire at least two scans in the reference conguration, if possible by slightly moving the setup between the two acquisitions, to evaluate the practical displacement and strain resolutions of the studied material. Further, the strain resolution for cast iron virtually coincides with bis and rbm resolution levels for aluminum alloys, which supports the aforementioned limited inuence of microstructure properties on the nal measurement uncertainty levels in laminography.

By comparing the uncertainty levels measured with and without rigid body motions, for aluminum alloys and cast iron, it appears that the uncertainty mainly comes from the laminography technique and its associated artifacts [START_REF] Helfen | On the implementation of computed laminography using synchrotron radiation[END_REF][START_REF] Xu | Comparison of image quality in computed laminography and tomography[END_REF]. The uncertainty levels are equal or higher with rigid body motions since some of the artifacts are not attached to the microstructure (like, e.g., phase-contrast edge enhancement) but intrinsic of the acquisition set-up (e.g., ring artifacts).

Last, let us note that the procedures discussed herein and applied to reconstructed volumes obtained by synchrotron laminography can also be applied to synchrotron or lab tomography. All these techniques require reconstruction steps from the raw data (i.e., radiographs) to the 3D volumes that induce additional artifacts depending on the equipment itself but also on the reconstruction algorithms.

by considering its L2-norm with respect to kinematic unknowns. Since a global approach is used in this work [START_REF] Roux | Three dimensional image correlation from X-ray computed tomography of solid foam[END_REF], C 0 continuity over the Region of Interest (ROI) is prescribed to the solution and the global residual to be minimized reads

Φ 2 c = ROI ρ 2 (x) (4)
The displacement eld is parameterized by using a kinematic basis consisting of shape functions Ψ p (x) and nodal displacements u p u

(x) = p u p Ψ p (x) (5)
Among a whole range of available elds, nite element shape functions are particularly attractive because of the link they provide between the measurement of the displacement eld and numerical models [START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF][START_REF] Hild | Toward 4D mechanical correlation[END_REF].

Thus, a weak formulation based on C8 nite elements with trilinear shape functions is chosen [START_REF] Roux | Three dimensional image correlation from X-ray computed tomography of solid foam[END_REF]. After successive linearizations and corrections the current system to be solved reads

[M]{δu} = {b} (6)
where

M ij = ROI (∇f • Ψ i )(x)(∇f • Ψ j )(x) (7)
represents the DVC matrix, while vector {b}

b i = ROI (f (x) -g(x))(∇f • Ψ i )(x) (8)
contains the current dierence between the reference volume f and the corrected deformed volume g = g(x + ũ(x)), ũ being the current estimate of the displacement eld. The global residual needs to decrease so that convergence 2. Taking into account the low precision of the proposed method the results show there are no signicant deviations (i.e., same order of magnitude) for any of the studied alloys. AA6061 T6 alloy has the lowest number of features available per element volume, but this needs to be taken with caution since AA6061 contains larger voids and particles (Figure 18(e)) where multiple elements can benet from a single object, i.e., its boundary where gray level gradient is higher. This is outlined in the second row of Table 2 where the number of surface voxels (from all objects) per element volume is displayed. 2).

Figure 1 :

 1 Figure 1: Mid-thickness L -T slice in 3D reconstructed volumes of three

Figure 2 :

 2 Figure 2: Gray level histograms of the investigated 3D volume images for

Figure 3 :

 3 Figure 3: Experimental setup schematically shown for the aluminum CT-like

Figure 4 (Figure 4 :

 44 Figure 4(a) shows the location of the Region of Interest (ROI) in the reference and rigid body motion (rbm ) conguration moved by u T = 22 voxels, u L = 1113 voxels so that the rotation axis and reconstruction artifacts (e.g., rings) are positioned dierently in the inspected region. Its size is 1184 × 512 × 992 voxels. All examined ROIs for aluminum alloys have the

Figure 5 :Figure 6 :

 56 Figure 5: Region of interest considered in the uncertainty analysis of the

Figure 7 :

 7 Figure 7: Schematic view of the sample with the scanned region close to the

Figure 8 :

 8 Figure 8: The sample geometry with the scanned region between the two

Figure 9 :

 9 Figure 9: Standard displacement resolutions as functions of the element size for bis cases

Figure 10 :Figure 10 )Figure 11 :

 101011 Figure 10: The largest 100 eigenvalues of the inverse DVC matrix [M] -1 for

Figure 12 :

 12 Figure 12: Standard displacement uncertainties as functions of the element size for rbm cases

Figure 13 :

 13 Figure 13: Standard displacement (a) and strain (b) uncertainties as functions of the element size for rbm cases

Figure 14 :

 14 Figure 14: Standard displacement (a) and strain (b) resolutions as functions of the element size for bis and bis cases

4. 1 Figure 15 :

 115 Figure 15: Standard displacement (a) and strain (b) resolutions as functions of the element size for rescaling cases. Two dierent rescaling parameters resulted in 2 dierent set of scans, namely: reference scan A and rbm scan A; reference scan B and rbm scan B. Three combinations are analyzed to measure displacement elds: reference scan A vs. rbm scan A, reference scan B vs. rbm scan B, and reference scan A vs. rbm scan B (crossed)

Figure 16 :

 16 Figure 16: Standard displacement (a) and strain (b) resolutions as functions

2 × 2 Figure 17 :

 2217 Figure 17: Standard displacement (a) and strain (b) resolutions as functions

Figure 18 :

 18 Figure 18: Isometric isolated view of voids (blue) and particles (yellow) (+

Figure 19 :

 19 Figure 19: Procedure followed when isolating voids and particles for AA2139

Figure 21 )

 21 . Beside the isolated nodules one can notice a secondary void population in the ferritic matrix. The results, in terms of number of contrasted objects and surface voxels per element volume (isotropic element size = 32 voxels) are 1.5 and 458 respectively. These values are in the range of levels for aluminum alloys (Table

Figure 21 :

 21 Figure 21: Isometric isolated view of nodules and secondary void population in 3D reconstructed volume (256 × 256 × 256 voxels) of the studied cast iron

Table 1 :

 1 Uncertainty assessment: procedures and available data 32 to 8 bit conversionrescaling All the previous cases are analyzed as follows. DVC analyses are run to measure displacement elds. The latter ones are interpolated to account for rigid body motions (i.e., translations and rotations). The dierence between

		√
	beam properties	√	√	√	√
	detector inuence				√

Table 2 :

 2 Number of features per element when the element size is = 32 vox-volume with size 256 × 256 × 256 voxels is extracted from the ROI used in DVC analyses (

	els
	Al alloy	AA2139 T3 AA2139 T8 AA2198 T3 AA2198 T8 AA6061 T6
	features per el.	3.5	2.6	4.7	3.3	1.0
	surface voxels per el.	396	248	223	178	642

representative

Acknowledgments

The nancial support of the Fédération Francilienne de Mécanique and Agence Nationale de la Recherche (ANR-14-CE07-0034-02 grant for COMIN-

SIDE project) is gratefully acknowledged. Constellium C-Tech is acknowledged for aluminum alloys supply. M. Kuna, L. Zybell and M. Horn from IMFD, TU Freiberg, are thanked for cast iron supply and machining. We would like to acknowledge the European Synchrotron Radiation Facility for provision of beamtime at beamline ID15A and ID19 (experiments MA1006, MI1149, ME1366, MA2183).

Appendices Appendix A: Volumetric Correlation

The DVC technique used herein is an extension of 2D global Digital Image Correlation [START_REF] Besnard | Finite-Element Displacement Fields Analysis from Digital Images: Application to Portevin-Le Châtelier Bands[END_REF][START_REF] Hild | Comparison of local and global approaches to digital image correlation[END_REF]. The reconstructed volume is represented by a discrete scalar matrix of spatial coordinate x of (in this case 8-bit deep) gray levels determined by the microstructure absorption of X-rays. The principle of DVC consists of registering the gray levels f in the reference conguration x and those of the deformed conguration g, which satisfy gray level conservation

where u is the sought displacement eld. However, due to acquisition noise, reconstruction artifacts [START_REF] Xu | Comparison of image quality in computed laminography and tomography[END_REF] and the correlation procedure itself [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF], ideal matching is not achieved in real examples. Consequently, the solution consists of minimizing the gray level residual

is achieved, i.e., the displacement corrections {δu} become vanishingly small. In this form, the displacement eld is regularized in the sense of a continuity requirement that is a priori assumed for the kinematic solution. Moreover, additional mechanical knowledge may be added to help convergence. This type of procedure is referred to as mechanical regularization [START_REF] Taillandier-Thomas | Localized strain eld measurement on laminography data with mechanical regularization[END_REF]. It will not be used hereafter.

Appendix B: Morphological characterization of Al alloys

A priori analysis of contrast features (i.e., particles + voids) serves as quantication of material natural contrast, i.e., to predict the quality of volume gray level gradient that will serve as sensitivity for DVC registrations (see Equation ( 7)). Representative volumes with size 256 × 256 × 256 voxels are extracted from ROIs used in DVC analyses (Figure 18). Since it is case sensitive, slightly dierent threshold gray levels have been used to isolate voids and particles for the studied Al alloys. Before creating binary images all the original volumes are ltered in the open source software Fiji [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF][START_REF] Schindelin | The imagej ecosystem: An open platform for biomedical image analysis[END_REF][START_REF] Schneider | NIH Image to ImageJ: 25 years of image analysis[END_REF] by applying Gaussian blur with length of 1 voxel.

Appendix C: Reconstruction artifact

The sampling of the 3D Fourier domain of the region of interest is incomplete in laminography. Consequently the most outward voxels have only information from a limited range of viewing angles. This is illustrated in Figure 20 for

Appendix D: Morphological characterization of cast iron

The analysis of contrasted features is conducted for cast iron by following the same procedure as for aluminum alloys described in Appendix B. The