OPTICAL TECHNIQUES FOR RELIEF STUDY OF MONA LISA'S
WOODEN SUPPORT
Fabrice Brémand, Pascal Doumalin, Jean-Christophe Dupré, Franck Hesser, and Valéry Valle
Laboratoire de Mécanique des Solides, CNRS, UMR 6610, Université de Poitiers
SP2MI Bd M. et P. Curie, Téléport 2, BP 30179, 86962 Futuroscope Chasseneuil cedex, France
[email protected]
ABSTRACT
The aim of our study is to obtain a whole field 3D profile of the wood support of the famous painting: Mona Lisa, on
front and back faces several times in a day. These data allow understanding the mechanical impact of the frame in
which the panel is maintained attached and can be used to describe the hygromechanical behaviour of the wooden
painting. Out of plane displacements and thickness have been measured and have been used to validate a numerical
simulation of the hygromechanical behaviour of the panel. Two techniques have been tested: shadow moiré method
and fringe pattern profilometry. After a previous test realized at the C2RMF (Centre de Recherche et de Restauration
des Musées de France) laboratory on a painting of the same period, shadow moiré method has been chosen. It is
more accurate but especially less sensitive to the dark tints, to the changes of contrast and colour of the picture, or its
luster. During the testing day, we have observed the out of plane deformation of the panel. A contraction of the order
of 3/10 mm was detected early in the middle right part of the panel. This deformation is due to the wood’s behaviour
variations in response to atmospheric conditions of humidity and temperature , which are not the same in the room as
in the case. The greatest distortion was observed when removing the frame (±1 mm)., the out of plane displacement
and curvatures are maximal near the crack (upper middle left part) and also in the lower part of the panel.
Introduction
We present in this paper an original application of the use of optical methods for the study of the famous painting:
Mona Lisa. Several laboratories [1] have participated to this study in order to evaluate the degradation risk (specially
in relation with the existing crack) and to optimise the conservation conditions (regarding both the humidity regulation
and the design of the frame). Mona Lisa is painted on a poplar support, so our own contribution was to obtain a whole
field 3D profile of the panel on front and back face of Mona Lisa. Furthermore, these values allow understanding the
mechanical impact by the frame in which the panel is maintained attached and are used to describe the
hygromechanical behaviour of the wooden painting by many measurements performed during several hours. These
experimental data have been used to realize a numerical model of the panel, and also to validate the numerical
simulations of the mechanical behaviour of the panel [1][2].
The difficulties of this work are different than the classical ones in laboratory. For safety problem, we were able to
study the painting only a single day. No test can be realized before and after. So large number factors: like brightness,
contrast of the painting and room's lighting in the Louvre museum, are unknown. The experimental conditions impose
that the measurement technique uses a low lightening of the painting and a minimum apparatus. The known
2
parameters are the dimensions of the studied zone, relatively large 800x600 mm , the attended uncertainty allowing a
accurate comparison with the numerical simulation (about 0.01 mm) and the speed of measurement (several minutes)
which allows studying the hygromechanics behaviour. So, two techniques have been chosen: shadow moiré and
projection moiré. The first technique is the more precise but more complicated to put into practice (specially the use of
a reference grating put on the front of the specimen). It is generally used to study the behaviour of materials under
mechanical stresses, but also to determine the form of small objects. The second technique, is less sensitive, but
easier to employ, it can be used for large objects. In the present protocol dedicated to Mona Lisa, which is entirely
without the help of invasive techniques, these two methods were clearly indicated, since they would not involve
physical contact with the paint layer.
Principle of shadow moiré technique (SMT)
The shadow moiré technique (SMT) [3][4][5][6] consists in projecting a line grating in front of the object with the help of
a punctual light source, as shown on Figure 1. Superposing the grating and its shadow on the object leads to a moiré
phenomenon. Moiré fringes correspond to the contour lines of the relief of the object. The relief Z at the point of x coordinate is determined by the following relation [6]:
Z SMT = k ⋅
p
x
d−x
+
hS + Z hO + Z
(1)
where k is the fringe order or the number of contour lines, d the distance between the light source S and the observer
O, hS the distance between S and the plane of the grating pitch p, ho the distance between O and the grating.
It makes sense to consider, that projected and observed beams are collimated. Indeed, this assumption can be made
if the relief Z is small compared to ho and hs. Furthermore if the distance hS is equal to the observation distance ho
(h = ho = hS) then the relief is proportional and linear to contour lines. This relief can be determined as a function of the
phase ϕ of the grating obtained by a phase shifting process, as shown in the following relations from equation (1):
Z SMT =
ϕ p.h
ϕ
⋅
=
∆Z
2π d
2π
(2)
where ∆Z , the sensitivity factor, corresponds to the relief between two contour lines.
Figure 1 : Principle of projection fringe method
Figure 2 : Set-up of fringe pattern profilometry
As a master grating is required in front of the object, this measuring method cannot be applied on extra large object.
To eliminate geometrical distortions, the relief of a reference plane can be subtracted to the one of the object.
Principle of fringe pattern profilometry (FPP)
In order to allow the study of larger objects, Durelli [8], Pirroda [9] and Theocaris [10, 11] have developed the
projection moiré method or fringe pattern profilometry (FPP). They have used the set-up shown on Fig. 2.
The principle of this method consists in projecting a fringe pattern first on a reference plane and then on the studied
object [7][8][9][10]. The first image corresponds to the master grating of shadow moiré. The second image comes from
the grating projected onto the object and corresponds to the shadow of the grating used in shadow moiré. Fringes
similar to those of shadow moiré can appear by adding both images. Nowadays, thanks to the numeric tool which is
easier to use, both images are recorded independently and analysed separately by means of phase shifting described
in the next paragraph. In shadow moiré, the introduction of phase shifting is quite difficult to carry out (without
modifying contour fringes). However, in projection moiré, we need only to shift the grating in its plane.
If the assumption of collimated beams is kept, the relief is then proportional to the phase difference (φ) between the
values obtain on the reference plane (ϕR) and the second one calculated on the object (ϕO). Relief is then determined
by the following simplified expression, which is similar to the shadow moiré relationship:
Z FPP =
(ϕ O − ϕ R ) Pproj
φ Pproj.h
⋅
=
⋅
2π
tan α 2π
d
with tan α =
d ,
p
Pproj =
h
cos α
and h = ho = hp
(3)
Another solution, which is not described here, consists in applying a particular calibration in order to avoid the
reference plane [11].
Principle of fringes analysis
For both techniques, the light intensity (I) recorded by the CCD camera can be expressed, at each pixel (i,j), by
I (i, j ) = I 0 (i, j ) + I1 (i, j ) sin (ϕ (i, j ))
(4)
with I0 as the background and I1 as the amplitude of the fringe pattern.
The accuracy of the relief measurement is function to the phase analysis process. The more accurate fringe analysis
procedures consist in recording several images and introducing a phase shift θk (eq. 4).
I k (i, j ) = I 0 (i, j ) + I1 (i, j ) sin (ϕ (i, j ) + θ k )
(5)
From at least three values of θk, and the corresponding images, one can solve equation 5 [11][12][13]. We have to
use a procedure given a high accuracy with a low sensitivity of background and amplitude variation. These two last
points are very important because the painting has dark or bright zones. Furthermore, the lighting can not be constant
during the test. So we have chosen a solution allowing to analyse the frequency of a signal and not the intensity,
which allows to minimize those effects [14][15]. This technique is reliable and has already been used for moiré and
photoelasticity studies [14][16]. It consists in shifting the signal in order to temporally obtain a series of images with an
integer number of period at each pixel. The temporal computation of the Fourier transform for each pixel in the series
of images is done in order to extract several harmonics. For example, for a shift of one period, the phase can be
calculated by:
1
 Im(C (i, j )) 
+1
ϕ (i, j ) = tan−1 

4
Re(
C
+1 (i, j )) 

(6)
With C+1 which represents the first harmonic of the complex Fourier spectrum.
A phase unwrapping process allows to obtain the continuity of the derivatives of the phase field [17][11]. This
procedure can be used for SMT or for FPP. For shadow moiré method, the introduction of phase shifting is obtained
by moving the master grating in the z direction. For the second technique, it is simply obtained by numerically shifting
the projected grating.
Previous tests have been performed to evaluate phase accuracy. These tests have been made on plate of
polycarbonate painted in white. The obtained accuracy is about 1/200 of wave lenght (≈ 3°). This value is twice as
greater as the one for phase shifting with 3 images.
From equations (2) or (3), we can evaluate uncertainties function to phase values. The both expressions are similar;
the accuracy is function to the geometric parameters, the period of the grating and the uncertainty on the measured
phase.
∆Z SMT =
p.h
∆ϕ
2πd
∆Z FPP =
p proj .h
2πd
∆φ
For a similar device, the differences are due to the period of the grating respectively p and pproj . Experimentally, the
pitch for the FPP is five to ten times as bigger as the one for SMT, then the accuracy decreases in the same way.
For example for h≈d and p=1mm, we obtain ∆zFPP≈0.1 mm taking into account the accuracy of the phase:
∆zSMT≈0.01mm and for pproj=10mm
Figure 3 : Details of recorded image from the FPP and steps for phase calculation (Marco d'Oggiono "Mary
breastfeeding Jesus ".
First Experimental test
If SMT gives better accuracy, the specific conditions of the present work does not allow us to choose between FPP
and SMT. A first experimental case has been made to definitively adopt the best solution. So a quick experimental test
th
has been realized the day before. This study has been performed on a same period painting (beginning of XVI
century) ,which is supposed having the same contrast and brightness (Figure 3). A profile of a detail is plot on Figure
4. We can see that SMT gives better results in the light zone (factor 2) than in the dark zone (factor 4).The RMS error
is about 0.03 mm for SMT and 0.11 mm to 0.05 mm for FPP. This calculation involves small variations due to the
painting and the master grating (reference or projected). This test confirms that SMT is the more efficient method for
this study.
0.25
Dark zone
Light zone
0.2
0.15
FPP
SMT
relief (mm)
0.1
0.05
coordinates
0
0
25
50
75
100
125
150
175
200
-0.05
-0.1
-0.15
-0.2
Figure 4 : Comparison of the relief profile obtain by the two techniques
Experimental test procedure
The set-up is shown on Figure 5. The locations of the grid, the CCD camera and the light source have been choosing
related to the conditions of relationship 2. The panel has been analysed with a resolution equal to 600x900 pixels and
the pitch of the grid is 1 line/mm.
Several measurements have been realized during the day. This made it possible, on one hand, to optimize the
measuring method by improving the acquisition conditions and, on the other, to observe the picture’s behaviour when
it was removed from its glass case. The first (TEST.I) attempt was realized just after the panel was taken from its
case, the second (TEST.II) one after a period of two and a half hours and a series of tests for which the crossbars
which maintain the panel into the frame were unscrewed, and then screwed back on. The third one was carried out
after the panel had been removed from its frame, and concerned both the front (TEST.III.f) and back (TEST.III.b) of
the panel. The last attempt (TEST.IV) was performed after the panel had been refitted into its frame, six hours after
the picture was taken from its case.
To take account of the manipulations and movements of the painting in assessing the data, a specific
procedure was added to our program. This consisted of calculating an average plane for the panel, which was
subsequently subtracted from the measurements recorded. Without positing any hypothesis, a relief relative to that
average plane was obtained at a fixed point on the picture. The panel’s average plane was calculated from its upper
and lower edges, where there is less distortion.
Figure 5: Experimental device: h0=4,30 m ; hS= 4,30 m; d=2,65 m.
Relief measurement
A first qualitative examination was quickly performed. By placing the picture behind the reference grid, it was possible
to observe the relief directly in the form of contour lines (Figure 6-a). Taking account of the experimental device, we
had about two millimetres between two fringes. We see concentric fringes, which allowed us to conclude that the
picture is not flat, but bulged on the right hand side.
The quantitative measurement is obtained by the presented procedure compiled in our software which calculates the
relief for every point. One result is presented on Figure 6-b. The precision obtained is a function of the contrast of the
picture and the state of its surface, with levels of uncertainty changing from place to place overall, these being globally
5/100 of a millimetre and ranging between 2/100 mm and 1/10 mm. In order to increase accuracy, a low-pass filter
(3x3 pixels) is used.
a :Recorded image of the
shadow moiré fringes
b :Relief (mm)
Figure 6 : TEST.I (on the face of the painting)
Out of plane displacement
By comparing the measured relief during the day, we have calculated the out of plane displacements of the panel. A
contraction of the order of 3/10 of a millimetre was detected early on in the centre right part of the panel between
TEST.III.f and Test.I (Figure 7-a). Two causes for this may be advanced, the first related to the unscrewing and
screwing back of the crossbars, and the second linked to variations in thermohygrometric conditions of its
environment, inasmuch as the panel had been out of its case for two and a half hours.
The highest displacement was observed on removal from the frame (±0.75 mm) between TEST.III.f and Test.I (Figure
7-b), the deformation and curvatures are largest close to the crack and also in the lower part of the panel. This is an
indication that these both zones are more strongly affected by bending effect.
After the panel had been refitted into the frame, the out of plane displacements between TEST.IV and Test.I are about
–3/10 and +12/10 of a millimetre. Its shape was practically identical to that obtained after unscrewing and screwing
back the crossbars (Figure 7-a). In other words, the convexity observed at the beginning was maintained.
These values were used by J. Grill et al to validate their hygromechanical simulation using a 1D or 2D version of
Transpore software [18][2]. Their algorithm gives predictions for the reaction of a panel portion in the both extreme
situations of free or blocked curvature [1].
Taking as state of reference the second relief (TEST.II) obtained two and a half hours after the extraction and refitting
into the frame, we were able to eliminate effects due to handling and the first variations due to thermohygrometric
behaviour of the wood. The convex area is better defined (Figure 8-a). Its maximum follows an inclined axis of 10°
compared to vertical and goes through the split in the panel. From a mechanical point of view, the panel is subject to a
combined effect of bending and torsion. In order to visualize the zones of high curvature, where the stresses are
concentrated, the second derivative of the out of plane displacement according to horizontal was calculated given
values proportional to the bending moment. The conclusion is that stresses are concentrated at the level of the split
but also at the bottom of the picture (Figure 8-b).
Thickness determination
A mapping of thickness was calculated from the relief obtained for the front and back surfaces (TEST.III). The
positioning of the two surfaces was effected in the light of depths measured by the French Centre for Museum
Research and Restoration at the four corners of the picture.
From this map, a computer model representing the picture’s volume was used for virtual visualization purposes
(Figure 9) and/or computer calculations. This model made it possible, with the help of a rapid prototyping machine, to
produce a small-scale manipulable replica (Figure 9).
(a) TEST.II
(b) TEST.III.f
Figure 7: Out of plane displacement (mm) of the panel versus TEST.I
-1
(a) Out of plane displacement (mm),
(b) Corresponding curvature (mm )
Figure 8: (TEST.III.f) versus ( TEST.II)
Figure 9: Mapping of the thickness (mm): reverse side view (horizontal and vertical scale reduced to=14% and true
relief scale)
Conclusion
SMT used for this study was chosen because it is more accurate but specially less sensitive to the dark tints, changes
of contrast and colour of the picture, or its lustre. Furthermore SMT made it possible to image the picture’s relief as
contour lines. Since it avoids physical contact, it is a technique that neither damages the picture nor disturbs the
quantities that are to be measured. It therefore allows measurements to be taken from the panel’s painted side. Tests
were undertaken in the actual room in the museum in which the picture is displayed without particular precautions
regarding lighting. Just a few seconds were all it took to obtain the images, which made it possible to perform multiple
measurements and to witness change, even fast change, over time. Relief is obtained with an average precision of
5/100 of a millimetre over a surface of 518 x 770 mm, and a resolution of the order of 600 x 900 points. The quality of
performance of SMT is directly linked to the line grating placed in front of the object. This grating, produced by
ourselves, had several defects, which detracted from the method’s quality of performance on this occasion. A more
uniform grating will improve accuracy and above all render it more homogeneous.
During the day of tests, the data collected on the relief of the panel made it possible to produce a digital 3-D model
and to observe the mechanical behaviour of the panel. Both are essential to effect realistic numerical simulations of
the interaction between the frame and the panel and its reaction to exterior conditions. A contraction of the order of
3/10 of a millimetre was detected early on in the centre right part of the panel. It appeared during the earliest hours of
handling (during the unscrewing and screwing back of the crossbars) and remained even after refitting. This distortion
may originate in changes in the wood’s behaviour in response to atmospheric conditions of humidity and temperature,
which are not necessarily the same in the room as in the glass case. The greatest distortion was observed on removal
from the frame (±1 mm), when the panel assumed a bulging shape. The distortion and curvatures are at their greatest
at the level of the split and also in the lower part of the panel. This is an indication that these two zones are more
strongly affected by forces of flexion and are critical areas calling for careful monitoring.
It is conceivable that, by adapting the system whereby the line grating is fixed, it might be possible to test the picture
through the glass of its case. If so, it would be possible to establish a procedure to check the panel frequently without
handling it or even opening the case. Such an arrangement would allow long-term continuous assessment to monitor
the panel’s changes and anticipate any deterioration in the paint layer.
Acknowledgments
We thank Jean-Pierre Mohen, Michel Menu and Bruno Mottin (C2RMF, UMR 171), Cécille Scallierez and Vincent
Pomarède, Curators in the Louvre Museum for supporting the research work.
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