FULL 3D STRAIN MEASUREMENT BY DIGITAL VOLUME CORRELATION
FROM X-RAY COMPUTED AND OPTICAL SCANNING TOMOGRAPHIC
IMAGES
A. Germaneau, P. Doumalin and J.C. Dupré
Laboratoire de Mécanique des Solides, CNRS, UMR 6610, Université de Poitiers
SP2MI Bd Marie et Pierre Curie, Téléport 2, 86960 Futuroscope Chasseneuil, France
[email protected]
ABSTRACT
Digital Volume Correlation (DVC) which is the 3D extension of DIC technique allows us to determine the full 3D strain tensor
and to analyse 3D mechanical phenomena in the core of materials or structures without hypothesis on the strain kinematic
formulation nor using a numerical simulation. Nevertheless, DVC needs volume images which contain a 3D random
distribution of grey levels following displacements of material. In this study, we present 3D measurements with images from Xray Micro-Computed Tomography (XµCT) and Optical Scanning Tomography (OST). In order to define the fields of application
of both techniques, we show their performances and we evaluate their measurement uncertainty in the cases of rigid body
translations and a homogeneous strain test.
Introduction
Currently, the determination of mechanical data fields during a loading is generally performed by non-contact optical
techniques like Digital Image Correlation techniques (DIC). However, these methods are only surface measurement
techniques and even if we add the measurement of out-plan displacements (for example obtained by stereoscopy), we cannot
calculate all the three-dimensional strain components. To obtain the full 3D strain tensor, DIC techniques [1-4] have been
extended to Digital Volume Correlation (DVC) [5-11]. So we are able to investigate 3D mechanical phenomena in the core of
materials or structures without hypothesis on the strain kinematics formulation nor using a numerical simulation. Up to the
present, DVC technique has generally been applied on volume images generated by X-ray Micro-Computed Tomography
(XµCT). This technique enables to have volume images with a good quality in particular on materials having a native
heterogeneous microstructure like trabecular bone. This kind of microstructure provides a 3D random distribution of grey levels
and DVC can be applied [5-7]. XµCT has mainly applications in medicine and biology but is also currently employed to
observe heterogeneous materials like ceramics [12] or to detect defects in industrial structures [13-15]. All materials do not
present a natural contrast sufficient to use DVC and so, it can be necessary to include small dense markers involving density
gradients during the elaboration of the specimen. But that is possible only in the case of an elaboration process which insures
to visualize these particles inside the specimen after the material manufacture, for example the ones obtained by powder
metallurgy route [8]. So, XµCT study by using DVC requires a costly and complex device. We have developed recently a
technique to obtain volume images in transparent materials [9-11]. The Optical Scanning Tomography (OST) is based on the
scattered light phenomenon and the optical slicing of the volume. The specimen is cast with a transparent resin containing
added particles. In this paper, we present DVC coupled with OST and with X-ray µCT. Then we study performances of both
methods from several mechanical tests in order to determine uncertainty of displacement and strain measurement.
Digital Volume Correlation
DIC technique, usually used in mechanics to measure plane or 3D displacements of loading surfaces [1-4,17,18], has been
extended to DVC for full 3D displacement and strain measurement [5-11]. The displacement field between a reference state
and a deformed state of a studied sample is measured on a 3D virtual grid. The displacement of each point of this grid is
calculated by intercorrelation of the grey levels of the neighbourhood D surrounding the considered point in both states. D is
composed of several voxels and corresponds to a subset of the volume. By noting X and x the coordinates (in voxels) of a
same point in the reference state and the deformed state, both configurations are linked by the 3D material transformation
φ: x = φ(X). For a subset D centred at the point X0 in the reference state, φ is approximated by its expansion at the first order
corresponding to a rigid body motion combined with a homogeneous deformation:
φ( X ) = X + U ( X ) ≈ X + U ( X 0 ) +
∂U
( X 0 ).( X − X 0 )
∂X
(1)
The displacement U(X0) of the subset centre gives the intensity (u,v,w) of the rigid body translation. The local displacement
∂U
( X 0 ) , sometimes neglected in the case of small local strain, includes the rigid body rotation and the
∂X
∂u ∂u ∂u ∂v ∂v ∂v ∂w ∂w ∂w
local stretch of the subset volume and is characterized by 9 parameters:
. The best
, , , , , ,
,
,
∂x ∂y ∂z ∂x ∂y ∂z ∂x ∂y ∂z
gradient expression
parameters characterizing the approximation (3 or 12 scalars) are those which minimize a correlation coefficient C which
measures the degree of similarity of grey level distributions in D and its transformed one by φ. The chosen formulation of C is
insensitive to small contrast and brightness fluctuations which can appear in images: a normalized cross correlation
formulation based on grey level gaps in respect to the average on the subset.
( f ( X ) − f D ).( g (φ( X )) − g D )
X ∈D
C =1−
( f ( X ) − f D )2 .
X ∈D
(2)
( g (φ( X )) − g D ) 2
X ∈D
where X refers to voxels in D, f and g are respectively the grey levels in the initial and deformed images, f D and g D are their
averages over D and φ(D). The parameters of the approximation φ are resumed in the vector P = (u , v, w)
P = u , v, w,
or
∂u ∂u ∂u ∂v ∂v ∂v ∂w ∂w ∂w
, , , , , ,
,
,
if the local gradient is taking into account.
∂x ∂y ∂z ∂x ∂y ∂z ∂x ∂y ∂z
The success of minimization process depends on the start point P0. In order to avoid the convergence on a local minimum, P0
is chosen near the final solution and according to the retained formulation, is equal to (u 0 , v0 , w0 ) or
(u
0
, v0 , w0 ,0,0,0,0,0,0,0,0,0) . This estimation in entire voxels is obtained by a direct systematic calculus which explores all the
possible entire translations in the neighbourhood of a position which has been deduced from the known translation of the
precedent point. This process is the first step of the measurement procedure. In a second time, we research a finer solution of
P with an automatic first-gradient minimization procedure whose mathematical expression is given by the following formula:
P n +1
where 0 < a ≤ 1 and
∂C
(P n )
∂P
= Pn − a
∂C
(P n )
∂P
(3)
. is the Euclidian norm.
The parameter a defines the distance between two successive solutions initially equal to 0.5 and then is divided by 2 when C
increases again. A trilinear interpolation of the grey levels in the deformed image is used in order to calculate the grey level
variations between two adjacent voxels. In this way, it is possible to achieve the position of the subset D in fractions of voxels
(subvoxel precision).
DVC gives a discrete displacement field, the displacements of the centres of the subsets. In order to determine, at each point
X0 of coordinates (X0,Y0,Z0), the gradient of the transformation F defined by:
F=
∂x
∂U
=I+
∂X
∂X
(4)
we choose 3 vectors (dX1,dX2,dX3) from the 6 neighbouring points of X0 in the initial state. These 3 vectors describe a 3D cross
centred at X0 and define a parallelepipedic volume where we can consider F homogeneous. With these vectors and their
homologous ones in the deformed state noted (dx1,dx2,dx3), we can write the following system of equation:
dxi = Fxx dX i + Fxy dYi + Fxz dZ i
dX i
dy i = Fyx dX i + Fyy dYi + Fyz dZ i with dX i = dYi
dz i = Fzx dX i + Fzy dYi + Fzz dZ i
dZ i
and
dxi
dx i = dy i
dz i
i = 1,2,3
(5)
The numerical resolution of this linear system gives all the components of F. In our case, we choose an initial regular grid of
points where the vectors (dX1,dX2,dX3) are respectively in the directions x,y,z, of the orthonormed basis. The expressions of the
system (5) are simplified and we can express the components of the gradient ∂U by finite differences (6):
∂X
∂α α( X 0 + l 0 , Y0 , Z 0 ) − α( X 0 − l 0 , Y0 , Z 0 )
=
2.l 0
∂x
∂α α( X 0 , Y0 + l 0 , Z 0 ) − α( X 0 , Y0 − l 0 , Z 0 )
=
2.l 0
∂y
∂α α( X 0 , Y0 , Z 0 + l 0 ) − α( X 0 , Y0 , Z 0 − l 0 )
=
2.l 0
∂z
(6)
where α = u,v,w and l0 is the step of the 3D uniform grid. In this case, the gauge length is equal to 2l0.
Then the Green-Lagrange strain tensor E can be calculated by:
E=
1 T
( F .F − I )
2
(7)
DVC can be applied to any type of 3D images if the grey level of each voxel represents a data which follows the material
movement. OST and XµCT are two techniques allowing us to generate these images.
Optical Scanning Tomography
In order to obtain 3D images in transparent materials, we have developed recently [9-11] a method based on the phenomenon
of scattered light created by randomly distributed particles, added in the specimen during its elaboration. This technique using
optical slicing and embedded particles is akin to 3D photoelasticity by scattered light technique [19-21] for solid materials and
Particle Image Velocimetry (PIV) [22,23] for fluids. With our procedure, at each step of loading, a 3D image is obtained by
scanning the specimen with a plane laser beam in the z direction and with a motorized translation stage (with a minimum
increment equal to 0.625 µm) controlled by an integrated linear-scale encoder. At each position of the beam, we record a x-y
2D image of the illuminated section where a random pattern due to scatterers appears. The volume is constituted by the
succession of these 2D images. The plane laser beam is obtained by using a laser source, a convergent lens and a cylindrical
lens (Figure 1). The random distribution of the grey levels within the volume image is given by the scattered light phenomenon.
To create this phenomenon, several kinds of particles with different sizes or shapes can be employed. However, the particle
size has to be larger than the laser wavelength in order to avoid the speckle laser phenomenon which can disturb the
measurement. In order to have cubic voxels, it is necessary to have the same spatial resolution along z direction and along x
and y directions of a slice. For that, the resolution of the CCD (Charged Coupling Device) camera, corresponding to the x-y
spatial resolution, is equal to the step between to successive slices imposed with the motorized translation stage. We also
choose the same value for the thickness of the plane laser beam giving thereby a fill factor of voxels equal to 100%. For this
study, according to our experimental setup, we have chosen a spatial resolution of voxels equal to 60 µm/voxel.
Plane beam
Mirror
Specimen
Cylindrical lens
Convergent lens
Translation
stage
CCD
camera
Laser source
Mirror
PC
y
x
z
Figure 1. Experimental device for Optical Scanning Tomography
X-Ray Micro-Computed Tomography
Generally, volume images are generated by XµCT [5-8] and several kinds of devices are employed like a medical or laboratory
tomograph or a synchrotron radiation device [16]. This latter device is employed in order to achieve a resolution near to one
micrometer and a better image quality (smaller level noise…). In this study, volume images are generated by a laboratory
tomograph allowing us to have a minimal resolution of about 5 µm. The principle consists in recording by a CCD camera
transmitted X-ray intensity field through a specimen under different angular positions (Figure 2). For one position, we obtain a
digital image named radiography.. From all radiographies, a 3D image of the variations of the linear attenuation coefficient in
the specimen is reconstructed by an filtered back-projection algorithm [24]. The distribution of grey levels in 3D images is due
to local differences of density and so gives a 3D representation of the microstructure of the studied specimen. X-rays are well
adapted to study materials with a native heterogeneous microstructure like tissues and organs as applications in medicine or
biology but also enable the observation of other heterogeneous materials as ceramics or geomaterials… However, all
materials do not present, by XµCT, a natural local contrast sufficient to use DVC. Then it is necessary to include some small
markers during the making of the material. The density of particles must be different to the one of the material. In order to be in
accordance with the study made by OST, we choose a voxel size equal to 60µm.
X-ray
detector
Specimen
y
X-ray source
z
x
Figure 2. X-ray Micro-Computed Tomography
Specimens manufacture
For our study, specimens have been realised by casting from liquid resins. This process makes easier to add and blend some
particles in order to obtain a homogeneous distribution. For OST, specimens are made with polyurethane (PU) which is a
transparent material and we add some particles in order to involve scattered light phenomenon. Several kinds of marks can be
employed with different shapes but their size has to be larger than the laser wavelength to avoid the speckle laser
phenomenon. We have shown [11] that a good quality of images can be obtained with specimens made in polyurethane by
including particles of polyamide with a size of approximately 150 µm. For XµCT, we use dense particles of copper. In order to
have a homogeneous distribution of these particles, we use silicone which is a more viscous resin than PU. Silicone has not
been used with OST because it is not a transparent material. As both materials have a low Young’s modulus, we can study
different mechanical situations from small to large strains. To have the same resolution than the one of OST, we use particles
with a size going from 50 µm to 150 µm.
Figure 3 shows volume images obtained by OST and by XµCT. Figure 3-a presents grey level variations for a volume of
3
300x850x140 voxels on the surface and at the core. The particles appear with a sufficient contrast and are almost spherical.
Nevertheless, the plane laser beam is enlarged during its propagation in the specimen due to particles and so these ones
3
appear slightly stretched according to the width z. Figure 3-b is a reconstruction (300x500x140 voxels ) of the volume
corresponding to the specimen studied by XµCT. We obtain a significant contrast because only copper particles appear due to
their large density compared to the one of the material. The volume image process is performed by a rotation according to yaxis and there are some reconstruction artefacts according to x-z plane due to prompt spatial variations of X-ray attenuation
inside the specimen: it appears in particular some well-known rings and shadow areas close to dense particles of copper
(Figure 3-c). Scanning was conducting at 120 kV and 0.1 mA. An angular step of one degree has been chosen after several
tests with our specimen, which is a good ratio between acquisition time and reconstruction quality.
(a)
(b)
(c)
x
z
y
z
y
x
z
x
Figure 3. Volume image generated (a) by OST, (b) by XµCT and (c) an x-z slice inside the volume generated by XµCT.
Experiments
First, we evaluate displacement measurement uncertainty obtained by both techniques. For that, we impose successive
subvoxel displacements on a range of one voxel on specimens with a micro translation stage. In assuming that displacements
are homogeneous in the whole specimen, we determine gaps between imposed and measured displacements at each point of
a 3D uniform grid. For each displacement component (α = u,v,w), a statistical process on the studied volume provides the
σα for
average gap which corresponds to the systematic error and the standard deviation σα which gives the uncertainty ∆α = 2σ
a level of confidence of 95%. This procedure is repeated for each imposed translation in order to take into account the errors
relative to all subvoxel displacement.. Furthermore, we calculate the global uncertainty of displacement from all gaps which
gives a suitable estimation of the displacement uncertainty.
In a second way, we estimate the strain measurement uncertainty. For that in both cases, we impose homogeneous strain
rates on specimens by making a tensile test. A classical loading system is easily employed on the OST device. To make the
test inside the X-ray tomography device, we have developed an experimental setup made in an X-ray transparent material
(PMMA). So, we can perform similar tests by using XµCT device and OST device. By assuming that the displacement gradient
is homogeneous in the zone of interest, the accuracy of each gradient component can be evaluated by a statistical process
(average and standard deviation) similar to the one made for rigid body displacement test. It has been shown that to include
the local gradient in the 3D material transformation (1) improves the measurement uncertainty [9,11] and so the results
presented below take into account this formulation.
Results and discussion
Displacement tests
Figure 4 presents the differences between measured displacements and imposed subvoxel translations. On Figure 4-a, for
OST technique, we present error displacement values (average and standard deviation) only according to z direction (w
component) and to x direction (u component) knowing that we observe the same values of error in x and y directions because
of the optical slicing is performed along the plane x-y. As well, by XµCT (Figure 4-b), we show only results according to x (u)
and y (v) because error values are similar along x or z knowing that volume images are generated with a rotation according to
3
the y-axis. For both techniques, correlation subset size used is equal to 31x31x31 voxels and the step between two
successive subsets is of 20 voxels. In both cases, we obtain the well-known sinusoidal evolution of the error similar to the one
obtained by DIC [25,26]. For OST technique (Figure 4-a), measurement error along z-direction is about twice or three times as
large as the one along x-direction. Along z-direction, the geometry of the representation of particles is lightly stretched and
involve some larger measurement uncertainties. By XµCT, displacement error is slightly larger according to y than the one
along x. This difference is due to reconstruction process of the tomography technique: the volume is the juxtaposition of
reconstructed slices along y. For each slice, reconstruction process gives values more smoothed in opposition to y values.
From Figure 3, we calculate the global uncertainty of displacement from all the translations tests. On one hand, we obtain
σu(OST) = 0.015 voxel and σw(OST) = 0.058 voxel with OST technique. On the other hand the results with XµCT are
σu(µCT) = 0.043 voxel and σv(µCT) = 0.056 voxel. So we can remark that OST presents a better accuracy than the one of XµCT,
except the z direction where particles appear lightly stretched. We find the same value of uncertainty than the one obtained by
DIC. However, XµCT is less accurate. The difference can be explained by the presence of the reconstruction process which
adds some own errors and can involve smaller accuracy by DVC.
0.12
0.12
(a)
0.04
0
-0.04
(b)
0.08
0
0.2
0.4
0.6
0.8
1
1.2
Error (voxel)
Error r (voxel)
0.08
u,v (OST)
-0.08
-0.12
0.04
0
-0.04
0
0.2
0.4
0.6
0.8
-0.12
Imposed Displacement (voxel)
1.2
u,w (XµCT)
-0.08
w (OST)
1
v (XµCT)
Imposed Displacement (voxel)
Figure 4. Displacement error (average and uncertainty) on rigid body translation tests performed by (a) OST and by (b) XµCT
Imposed strain
To determine strain measurement uncertainty obtained by OST or by XµCT, we have performed in both cases a tensile test
with an imposed strain according to y-axis going from 0.1% to 30%. The dimensions of the studied area are
3
170x500x100 voxel . Figure 5 presents evolution of average values of Green-Lagrange tensor components according to
imposed strain. Figure 5-a shows that in both cases the measured strain corresponds to imposed strain. Evolution of average
values of shear components of Green-Lagrange tensor is shown on Figure 5-b. These values are approximately equal to zero
for small strain and remain low compared to diagonal components even for large strain. The larger values of Eyz can be
explained by a non-pure tensile test. Figure 6 presents standard deviation values for each component of Green-Lagrange
tensor according to the imposed strain. Until 10% of imposed strain, standard deviations remain low and constant. Only
standard deviation of the component Ezz obtained by OST is larger than the other values because of the large uncertainty of w.
For larger strain, standard deviation values increase proportionally to the imposed strain and are particularly large for the
components obtained by XµCT. This error can be due to the evolution of grey levels of images which can be disturbed by the
strain process. Furthermore, we remark that the error evolution is faster in the z direction for OST and the y direction for XµCT.
In both cases, these directions are the directions of juxtaposition of slices. Once again, optical slicing and reconctruction
process involve additional errors.
(a)
0.35
0.002
Exx (OST)
Eyy (OST)
0.2
Ezz (OST)
0.15
Exx (XµCT)
0.1
Eyy (XµCT)
0.05
Ezz (XµCT)
0.01
0.1
1
Average Measured Strain
Average Measured Strain
0.3
0.25
0
-0.050.001
(b)
0.003
0.001
0
-0.0010.001
0.1
1
Exy (OST)
-0.003
Exz (OST)
-0.004
Eyz (OST)
-0.005
-0.1
-0.006
-0.15
-0.007
Imposed Strain
0.01
-0.002
Exy (XµCT)
Exz (XµCT)
Imposed Strain
Eyz (XµCT)
Figure 5. (a) Average values of diagonal components of Green-Lagrange tensor; (b) Average values of shear components
(a)
0.025
0.01
Exx (OST)
0.02
Eyy (OST)
STD
Ezz (OST)
Exx (XµCT)
0.01
Eyy (XµCT)
Exy (OST)
Exz (OST)
0.008
STD
0.015
Eyz (OST)
0.006
Exy (XµCT)
Exz (XµCT)
0.004
Ezz (XµCT)
0.005
0
0.001
(b)
0.012
Eyz (XµCT)
0.002
0.01
0.1
0
0.001
1
Imposed Strain
0.01
0.1
1
Imposed Strain
Figure 6. (a) Standard deviation values of diagonal components of Green-Lagrange tensor; (b) Standard deviation values of
shear components
Conclusion
This paper presents DVC coupled with OST or XµCT allowing us to analyse 3D mechanical effects in materials and structures.
We have observed on experimental tests that uncertainties for displacement and strain measurement are of the same order of
magnitude, with a lightly smaller uncertainty for OST. So according to the kind of study, OST or XµCT can be employed but it
seems natural that the latter is more destined to materials study in particular if this one can present a native heterogeneous
microstructure. OST technique which works on transparent models seems better adapted for structure studies. In this case, a
model structure can easily be elaborated with an adequate transparent material coupled with a loading system.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Chu, T.C., Ranson, W.F., Sutton, M.A. and Peters, W.H., “Applications of Digital Image Correlation techniques to
experimental mechanics,” Experimental Mechanics, 25(3), 232-244 (1985).
Bruck, H.A., McNeill, S.R., Sutton, M.A. and Peters, W.H., “Digital Image Correlation using Newton-Raphson method of
partial differential correction,” Experimental Mechanics, 29(2), 261-267 (1989).
Vendroux, G. and Knauss, W.G., “Submicron deformation filed measurements: Part 2. Improved Digital Image
Correlation,” Experimental Mechanics, 38(2), 86-92 (1998).
Bornert, M. and Doumalin, P., “Micromechanical applications of digital image correlation techniques,” Jacquot P., Fournier
J. Eds., Interferometry in Speckle Light, Theory and Applications, Springer, 67-74 (2000).
Bay, B.K., Smith, T.S., Fyrhie, D.P. and Saad, M., “Digital Volume Correlation: three-dimensional strain mapping using Xray tomography,” Experimental Mechanics, 39(3), 217-226 (1999).
Smith, T.S., Bay, B.K. and Rashid, M.M., “Digital volume correlation including rotational degrees of freedom during
minimization,” Experimental Mechanics, 42(3), 272-278 (2002).
Verhulp, E., Rietbergen, B. and Huiskes, R., “A three-dimensional Digital Image Correlation technique for strain
measurements in microstructures,” Journal of Biomechanics, 37, 1313-1320 (2004).
Bornert, M., Chaix, J.M., Doumalin, P., Dupré, J.C., Fournel, T., Jeulin, D., Maire, E., Moreaud, M. and Moulinec, H.,
“Mesure tridimensionnelle de champs cinématiques par imagerie volumique pour l’analyse des matériaux et des
structures,” I2M, 3-4, 43-88 (2005).
Germaneau, A., Doumalin, P. and Dupré J.C., “Full 3D measurement of strain field by scattered light for analysis of
structures,” Experimental Mechanics, To appear.
Germaneau, A., Doumalin, P. and Dupré J.C., “Improvement of accuracy of strain measurement by Digital Volume
Correlation for transparent materials,” Proc. of Photomechanics 2006, edited by M. Grédiac and J. Huntley, Clermont
Ferrand, France.
Germaneau, A., Doumalin, P. and Dupré J.C., “3D Strain field Measurement by correlation of volume images using
scattered light: recording of images and choice of marks,” Strain, To appear.
Phillips, D.H. and Lanutti, J.J., “Measuring physical density with X-ray computed tomography,” NDT&E Int., 30, 339-350
(1997).
Lambrineas, P., Davis, J.R., Suendermann, B., Wells, P., Thomson, K.R, Woodward, R.L., Egglestone, G.T. and Challis,
K., “X-ray computed tomography examination of inshore mine-hunter hull composite material,” NDT&E Int., 24, 207-213
(1991).
Maire, E., Buffière, J.Y., Salvo, L., Blandin, J.J., Ludwig, W. and Letang, J.M., “On the application of X-ray
microtomography in the field of materials science,” Advanced Engineering Materials, 3(8), 539-546 (2001).
15. Benouali, A.H., Froyen, L., Delerue, J.F. and Wevers, M., “Mechanical analysis and microstructural characterization of
metal foams,” Materials Science and Technology, 18(5), 489-494 (2002).
16. Babout, L., Maire, E., Buffière, J.Y. and Fougères, R., “Characterization by X-ray computed tomography of decohesion,
porosity growth and coalescence in model matrix composites,” Acta Materiala, 49, 2055-2063 (2001).
17. Garcia, D., Orteu, J.J. and Penazzi, L., “A combined temporal tracking and stereocorrelation technique for accurate
measurement of 3D displacements,” Journal of Materials Processing Technology, 125-126, 736-742 (2002).
18. Schreier, H.W., Garcia, D. and Sutton, M.A., “Advances in light microscope stereo vision,” Experimental Mechanics, 44(3),
278-288 (2004).
19. Desailly, R. and Lagarde, A., “Sur une méthode de photoélasticimétrie tridimensionelle non destructive à champ complet,”
Journal de Mécanique Appliquée, 4(1), 3-30 (1980).
20. Dupré, J.C. and Lagarde, A., “Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image
processing,” Experimental Mechanics, 37(4), 393-397 (1997).
21. Germaneau, A., Doumalin, P. and Dupré, J.C., “3D Photoelasticity and Digital Volume Correlation applied to 3D
mechanical stuides,” Proc. of ICEM 13 Alexandroupolis, Greece, (2007).
22. Maas, H.G., Gruen, A. and Papantoniou, D., “Particle tracking velocimetry in three dimensional flows, part 1.
Photogrammetric determination of particles coordinates,” Experiments in Fluids, 15, 133-146 (1993).
23. Brücker, Ch., Digital-Particle-Image-Velocimetry (DPIV) in a scanning light-sheet: 3D starting flow around a short
cylinder,” Experiments in Fluids, 19, 255-263 (1995).
24. Baruchel, J., Buffière, J.Y., Maire, E., Merle, E. and Peix, G., “X-ray Tomography in Material Science,” Hermès Science
Publications (2000).
25. Sutton, M.A., Mc Neill, S.R., Jang, J. and Babai, M., “Effects of subpixel image restoration on digital image correlation
estimates,” Optical Engineering, 27, 870-877 (1988).
26. Choi, S. and Shah, S.P., “Measurement of deformations on concrete subjected to compression using image correlation,”
Experimental Mechanics, 37(3), 307-313 (1997).